Thermomechanics of Solids With Lower-Dimensional Energetics: On the Importance of Surface, Interface, and Curve Structures at the Nanoscale. A Unifying Review

Javili, A.; McBride, A.; Steinmann, P.

doi:10.1115/1.4023012

Abstract

Surfaces and interfaces can significantly influence the overall response of a solid body. Their behavior is well described by continuum theories that endow the surface and interface with their own energetic structures. Such theories are becoming increasingly important when modeling the response of structures at the nanoscale. The objectives of this review are as follows. The first is to summarize the key contributions in the literature. The second is to unify a select subset of these contributions using a systematic and thermodynamically consistent procedure to derive the governing equations. Contributions from the bulk and the lower-dimensional surface, interface, and curve are accounted for. The governing equations describe the fully nonlinear response (geometric and material). Expressions for the energy and entropy flux vectors, and the admissible constraints on the temperature field, all subject to the restriction of non-negative dissipation, are explored. A particular emphasis is placed on the structure of these relations at the interface. A weak formulation of the governing equations is then presented that serves as the basis for their approximation using the finite element method. Various forms for a Helmholtz energy that describes the fully coupled thermomechanical response of the system are given. They include the contribution from surface tension. The vast majority of the literature on surface elasticity is framed in the infinitesimal deformation setting. The finite deformation stress measures are, thus, linearized and the structure of the resulting stresses discussed. The final objective is to elucidate the theory using a series of numerical example problems.

Introduction

The surface of a solid body typically exhibits properties that differ from those of the encased bulk. These differences, caused by processes such as surface oxidation, ageing, coating, atomic rearrangement, and the termination of atomic bonds, are present in comparatively thin boundary layers. Similarly, interfaces within the bulk can be viewed as two-sided internal surfaces. The mechanical and thermal properties of the interface can also differ significantly from the surrounding bulk. Surface and interface effects are especially significant for nanomaterials due to their large surface-to-volume ratio. These effects were appreciated already in the early 1900s by Pawlow [1], for example, who predicted the decrease in the melting temperature of a particle as its size reduces.

There exist two principal approaches to study the thermodynamics of surfaces and interfaces:

the zero-thickness layer or Gibbs (geometrical) method wherein a mathematical surface with zero thickness is introduced to capture excess quantities on the surface [2]
the finite-thickness layer method that dates back to van der Waals wherein a layer of finite thickness can be imagined instead of the interface

The reader is referred to Guggenheim [3] for further details and a comparison of the two approaches.

Following Gibbsean thermodynamics, various models have been proposed to endow the surface and interface with their own distinct properties, e.g., Adam [4], Shuttleworth [5], Herring [6], Bikerman [7], Orowan [8], and Cahn [9,10]. The physical chemistry of surfaces and interfaces for liquids and solids is studied in Refs. [4,11–15], among others.

The thermodynamic fundamentals of surface science were reviewed by Rusanov in Refs. [16,17]. Müller and Saul [18] presented a review on the importance of stress and strain effects on surface physics. The role of stress at solid surfaces was critically examined by Ibach [19]. Leo and Sekerka [20] investigated the equilibrium conditions for interfaces using a variational approach wherein the excess energy associated with the interface is allowed to depend on both the deformation of the interface and the crystallographic normal to the interface. Cammarata et al. [21,22,23,24] highlighted the surface and interface stress effects on thin films and nanoscaled structures. Fischer et al. [25] studied the role of surface energy and surface stress in phase-transforming nanoparticles and reported on the thermodynamics of a moving surface.

Gutman [26] recognized an inconsistency between the notion of the surface stress defined by, e.g., Shuttleworth, and that of the continuum elasticity theory. Bottomley and Ogino [27] examined an alternative to the Shuttleworth formulation of surface stress in solids (see also Ref. [28]). Kramer and Weissmüller [29] criticized Refs. [26–28] and clarified various misleading statements in the literature. Fischer et al. [25] circumvented such issues by a careful handling of the actual and reference configurations. Furthermore, Fischer et al. utilized the concept of configurational forces [30,31] as a convenient framework for distinguishing between surface energy, surface tension, and surface stress.

A widely adopted model, proposed by Gurtin and Murdoch [32,33], gives the surface its own tensorial stress measures (see, e.g., Refs. [22,34–36], for applications in nanomaterials). Murdoch [37], Gurtin and Struthers [38], and Gurtin et al. [39] extended this approach to consider interfaces within a solid. An important extension of the Gurtin and Murdoch model to account for the flexural resistance of the surface was developed by Steigmann and Ogden [40] and further studied in Refs. [41,42]. Moeckel [43] followed a different approach to Gurtin and Murdoch [32] for a moving interface within a thermomechanical solid. A master-balance equation is derived and applied to various key fields on the interface (see also Ref. [44]). The resulting equations are then restricted using the entropy principles [45] and constitutive relations for material interfaces derived. An alternative approach to develop general governing equations for the interface is to integrate the known equations for the bulk over the thickness of the interfacial layers (see, e.g., Ref. [46]). Daher and Maugin [47] used the method of virtual power [48,49] to derive the governing equations for an interface within a thermomechanical solid. This flexible approach is well suited to describing complex phenomena. For example, Daher and Maugin [50] applied the method to a model of deformable semiconductors with interfaces (see also Ref. [51]). Murdoch [52] addressed various aspects of surface modeling. Šilhavý [53] proved the existence of equilibrium of a two-phase state with an elastic solid bulk and deformation dependent interfacial energy. Park et al. [54], Park and Klein [55] developed an alternative continuum framework based on the surface Cauchy–Born model, an extension of the classical Cauchy–Born model, to include surface stresses (see also Refs. [56,57]).

The effect of surface energetics for ellipsoidal inclusions and the size-dependent elastic state of embedded inhomogeneities was investigated by Sharma et al. [58], Sharma and Ganti [59], and Sharma and Wheeler [60]. They utilized the classical formulation of Eshelby [61,62] for embedded inclusions and modified it by incorporating surface energies. Duan et al. [63] extended the Eshelby formalism for inclusion/inhomogeneity problems to the nanoscale. Effective mechanical and thermal properties of heterogeneous materials containing nanoinhomogeneities based on the generalized Eshelby formalism is investigated in Refs. [63–65]; see also related works Refs. [66–72].

Zöllner et al. [73] recently reported that to explore the biomechanical interaction during tissue expansion, one could model skin growth using a boundary energy. The thermomechanical behavior of low-dimensional systems was reviewed by Sun [74] together with a theoretical analysis elaborating on existing approximations in continuum mechanics and quantum computations. Johnson [75] studied the thermodynamics of a coherent interface separating two nonhydrostatically stressed crystals.

The objective of this presentation is to derive the equations governing the fully nonlinear coupled transient thermomechanical response of a body composed of a bulk, intersected by an interface, and encased by a surface, all of which are assumed to be energetic. Furthermore, an energetic curve that accounts for the interaction between the interface and the surface is also present. The importance of capturing such an interaction when modeling ionic nanowires was recently observed in Ref. [76].

A particular emphasis is placed here on the thermomechanical response of the interface. The governing equations for the various parts of the body are obtained from the balances of several key properties, namely linear and angular momentum, energy, and entropy. The corresponding constitutive relations then arise as thermodynamic restrictions on the dissipation. The diffusion of mass or chemical species is not accounted for in this contribution (see Ref. [77] for the case of species diffusion in thermomechanical solids with energetic surfaces).

Interfaces can be classified as follows. A material interface is one that does not move independently of the surrounding bulk material. An energetic interface is understood here to imply that the interface possesses its own thermomechanical structure in the form of an internal energy, entropy, constitutive relations, and dissipation. Such an interface is termed a thermodynamical singular surface by Daher and Maugin [47]. A thermal interface is defined as one that allows for heat conduction along the interface but possesses no energetic structure. Finally, a standard interface does not allow heat conduction along the interface and possesses no energetic structure. Daher and Maugin [47] term such an interface a free singular surface.

An interface can be classified further according to its thermal properties as follows:

Thermally perfect interface: The jump in the temperature and in the normal heat flux across the interface is zero.
Generic imperfect interface: In this general case neither the jump in the temperature nor the jump in the normal heat flux across the interface need be zero. A thermal interface is an example of a generic imperfect interface. Özdemir et al. [78], for example, developed a thermomechanical cohesive zone model for generic imperfect interfaces. The following two models are specializations of the generic imperfect interface model (see Refs. [79–81] and references therein).
- (i)
  Weakly conducting (Kapitza) interface: This type of imperfect interface is modeled using Kapitza's concept of thermal resistance.¹ The model allows for a temperature jump across the interface. The normal heat flux is, however, continuous across the interface. The Kapitza model for thermal interfaces has been widely investigated (see, e.g., Ref. [83] and references therein).
- (ii)
  Highly conducting (HC) interface: This type of imperfect interface models the temperature as continuous across the interface, while allowing a jump in the normal component of the heat flux. Yvonnet et al. [84] have used the HC model to numerically investigate the effective conductivity of nanocomposites containing interfaces. For nonenergetic interfaces, a jump in the heat flux across the interface is only possible if there is an associated heat flux along the interface (see, e.g., Ref. [85]). This restriction does not hold for energetic interfaces [86].

Mosler and Scheider [87] recently described the admissible assumptions for the motion of the interface for a class of damage-type cohesive models. In a similar spirit, the relation of the interface temperature to that in the surrounding bulk is discussed here in detail. In particular, it is shown that the interface temperature is not, in general, equal to the average of the temperature across the interface.

Steinmann and Häsner [88] considered an energetic interface (with zero dissipation across the interface) within a thermomechanical solid subject to the assumption of infinitesimal deformation. Steinmann [89] presented extensive details on energetic surfaces using both the deformational and configurational frameworks of continuum mechanics. That investigation did not consider thermal contributions, interfaces, or numerical aspects. Javili and Steinmann [90,91,92,93] extended the work to account for the thermomechanics of energetic surfaces and the implementation of the model using the finite element method. The current review includes an extension of these works to account for the role of the interface and curve. Related works by Yvonnet et al. [83,84] focus on stationary heat conduction across thermal interfaces.

The focus of this work is on energetic elasticity as compared to entropic elasticity. For entropic elastic materials, e.g., polymers, the temperature increases due to stretching. On the contrary, energetic elastic materials, e.g., metals, cool down when stretched. The thermomechanical coupling, whether for energetic or entropic elasticity, is termed, henceforth, the Gough–Joule effect. The Gough–Joule effect is often used for entropic elastic materials as it was originally realized for rubbers (see Gough [94] and Joule [95]). It is noteworthy, that the Gough–Joule effect is not solely due to the deformation, but it is rather the rate of a deformation that induces a change in the temperature field.

In summary, the key objectives and contributions of this work are as follow:

To systematically derive the equations governing the response of a thermomechanical body, intersected by a material interface, and enclosed by a surface, undergoing finite deformations coupled to transient heat conduction. The procedure used to derive the equations is general and could be applied to a range of complex phenomena.
To account for the interaction between the interface and the surface.
To derive the thermodynamically consistent constitutive relations, dissipation inequalities, and temperature evolution equations in the various parts of the body (bulk, surface, interface, and curve).
To briefly present the weak formulation of the governing equations for the general case of a thermomechanical solid possessing an energetic surface, interface, and curve. Various other models are shown to arise as restrictions of this general case.
To elaborate various aspects of material (constitutive) modeling and discuss the admissible range of material parameters.
To elucidate the theory with numerical examples based on a finite element approximation of the governing equations.

This manuscript is organized as follows. The notation and certain key concepts are introduced briefly in Sec. 2. The equations governing the response of the various parts of the body are derived from fundamental balance principles in Sec. 3. A thermodynamic framework is then utilized in Sec. 4 to determine the form of the thermodynamically consistent constitutive relations. The nature of the coupling between the primary fields (i.e., the displacement and the temperature) and the various parts of the body are made clear. The dissipation inequality for the interface is analyzed in detail to show the possible relations between the interface temperature and that of the surrounding bulk. The weak formulation of the governing equations is then given in Sec. 5. Various important problems, which result as restrictions of this general weak formulation, are highlighted. Aspects of material modeling are elaborated in Sec. 6. A surface Helmholtz energy is introduced and the relationship between surface material parameters is discussed. The admissibility of surface negative material parameters is investigated. A series of numerical example problems, based on a finite element approximation of the weak formulation, is presented in Sec. 7 to elucidate the theory. Section 8 concludes this work.

Preliminaries

The purpose of this section is to summarize certain key concepts in nonlinear continuum mechanics and to introduce the notation adopted here. Detailed expositions on nonlinear continuum mechanics can be found in Refs. [96–101], among others. For further details concerning the continuum mechanics of deformable surfaces and interfaces the reader is referred to Refs. [38,43,47] and the references therein.

Notation and Definitions

Direct notation is adopted throughout. Occasional use is made of index notation, the summation convention for repeated indices being implied. The three-dimensional Euclidean space is denoted $E$ ³. The scalar product of two vectors a and b is denoted $a \cdot b$ = [a]_i[b]_i. The scalar product of two second-order tensors A and B is denoted $A : B = {[A]}_{ij} {[B]}_{ij}$ ⁠. The composition of two second-order tensors A and B, denoted $A \cdot B$ ⁠, is a second-order tensor with components [ $A \cdot B$ ]_ij = [A]_im[B]_mj. The vector product of two vectors a and b is denoted a × b with [a × b]_k = [ε]_ijk[a]_i[b]_j, where ε denotes the third-order permutation (Levi–Civita) tensor. Analogously, the vector product of a vector a and a second-order tensor B is denoted a × B with [a × B]_kl = [ε]_ijk[a]_i[B]_jl. The nonstandard products of a fourth-order tensor $C$ and a vector b are defined by ${[b \bar{\cdot} C]}_{ikl} = {[C]}_{ijkl} {[b]}_{j}$ and ${[C \bar{\cdot} b]}_{ijl} = {[C]}_{ijkl} {[b]}_{k}$ ⁠. Other nonstandard products of a fourth-order tensor $C$ and a vector b are defined by ${[b \underline{\cdot} C]}_{ijl} = {[C]}_{ijkl} {[b]}_{k}$ and ${[C \underline{\cdot} b]}_{ikl} = {[C]}_{ijkl} {[b]}_{j}$ ⁠. The nonstandard product of a fourth-order tensor $C$ and a second-order tensor A is defined by ${[C \bar{:} A]}_{ik} = {[C]}_{ijkl} {[A]}_{jl}$ ⁠. The nonstandard minor transposes of a fourth-order tensor $C$ are defined by ${[^{t} C]}_{ijkl} = {[C]}_{jikl} and {[C^{t}]}_{ijkl} = {[C]}_{ijlk}$ ⁠. The action of a second-order tensor A on a vector a is given by [ $A \cdot a$ ]_i = [A]_ij[a]_j. The tensor product of two vectors a and b is a second-order tensor D = a $\otimes$ b with [D]_ij = [a]_i[b]_j. The tensor product of two second-order tensors A and B is a fourth-order tensor $D$ = A $\otimes$ B with [ $D$ ]_ijkl = [A]_ij[B]_kl. The two nonstandard tensor products of two second-order tensors A and B are the fourth-order tensors ${[A \bar{\otimes} B]}_{ijkl} = {[A]}_{ik} {[B]}_{jl}$ and ${[A \underline{\otimes} B]}_{ijkl} = {[A]}_{il} {[B]}_{jk}$ ⁠.

Quantities or operators corresponding to the bulk, surface, interface, and curve are denoted as {•}, ${\overset{\land}{•}}$ ⁠, ${\bar{•}}$ ⁠, and ${\tilde{•}}$ ⁠, respectively, unless specified otherwise. Note, ${\overset{\land}{•}}$ denotes a surface-quantity that is not necessarily tangent to the surface. Similarly, a bulk quantity {•} on the surface implies the evaluation of {•} on the surface and is not, in general, equivalent to the corresponding surface quantity ${\overset{\land}{•}}$ ⁠.

The expression the various parts of the body implies the bulk, the surface, the interface, and the curve.

Key definitions are given in Tables 1 and 2. For the derivations thereof and extensive technical details see classical manuscripts on differential geometry [102,103,104], among others. Basic concepts and terminologies of surfaces and curves can be found in the fundamentals of differential geometry on surfaces and curves briefly reviewed in Appendix A.

Table 1

Summary of the notation used to denote key quantities and operators in the bulk, on the surface and interface, and on the curve. Quantities and operators corresponding to the material and spatial configurations are distinguished by a semicolon.

	Bulk	Surface	Interface	Curve
Domain	$B_{0}$ ; $B_{t}$	$S_{0}$ ; $S_{t}$	$I_{0}$ ; $I_{t}$	$C_{0}$ ; $C_{t}$
Normal		N ; n	$\bar{N}$ ⁠, ${\bar{N}}^{\mp}$ ; $\bar{n}$ ⁠, ${\bar{n}}^{\mp}$	${\tilde{N}}_{c}$ ⁠, ${\tilde{N}}_{i}$ ⁠, ${\tilde{N}}_{s}$ ⁠, ${\tilde{N}}_{s}^{\pm}$ ; ${\tilde{n}}_{c}$ ⁠, ${\tilde{n}}_{i}$ ⁠, ${\tilde{n}}_{s}$ ⁠, ${\tilde{n}}_{s}^{\pm}$
Tangent		${\tilde{N}}_{s}$ ⁠, $\overset{\land}{N}$ ; ${\tilde{n}}_{s}$ ⁠, $\overset{\land}{n}$	${\tilde{N}}_{i}$ ; ${\tilde{n}}_{i}$	$\tilde{N}$ ; $\tilde{n}$
Unit tensor	I ; i	$\overset{\land}{I}$ ; $\overset{\land}{i}$	$\bar{I}$ ; $\bar{i}$	$\tilde{I}$ ; $\tilde{i}$
Differential element	dV ; dv	dA ; da	dA ; da	dL ; dl
Oriented element	—	dA ; da	dA ; da	dL ; dl
Control regions	$B$ ₀ ; $B$ _t	$S$ ₀ ; $S$ _t	$I$ ₀ ; $I$ _t	$C$ ₀ ; $C$ _t
Curvature	—	$\bar{C}$ ; $\bar{c}$	$\bar{C}$ ; $\bar{c}$	$\tilde{C}$ ; $\tilde{c}$
Covariant basis	G₁, G₂, G₃ ; g₁, g₂, g₃	${\overset{\land}{G}}_{1}$ ⁠, ${\overset{\land}{G}}_{2}$ ; ${\overset{\land}{g}}_{1}$ ⁠, ${\overset{\land}{g}}_{2}$	${\bar{G}}_{1}$ ⁠, ${\bar{G}}_{2}$ ; ${\bar{g}}_{1}$ ⁠, ${\bar{g}}_{2}$	${\tilde{G}}_{1}$ ; ${\tilde{g}}_{1}$
Contravariant basis	G¹, G², G³ ; g¹, g², g³	${\overset{\land}{G}}^{1}$ ⁠, ${\overset{\land}{G}}^{2}$ ; ${\overset{\land}{g}}^{1}$ ⁠, ${\overset{\land}{g}}^{2}$	${\bar{G}}^{1}$ ⁠, ${\bar{G}}^{2}$ ; ${\bar{g}}^{1}$ ⁠, ${\bar{g}}^{2}$	${\tilde{G}}^{1}$ ; ${\tilde{g}}^{1}$
Placement	X ; x	$\overset{\land}{X}$ ; $\overset{\land}{x}$	$\bar{X}$ ; $\bar{x}$	$\tilde{X}$ ; $\tilde{x}$
Tangent map	F ; f	$\overset{\land}{F}$ ; $\overset{\land}{f}$	$\bar{F}$ ; $\bar{f}$	$\tilde{F}$ ; $\tilde{f}$
Normal map	Cof F ; cof f	$\overset{\land}{Cof} \overset{\land}{F}; \overset{\land}{cof} \overset{\land}{f}$	$\bar{Cof} \bar{F}; \bar{cof} \bar{f}$	$\tilde{Cof} \tilde{F}; \tilde{cof} \tilde{f}$
Jacobian	J ; j	$\overset{\land}{J}$ ; $\overset{\land}{j}$	$\bar{J}$ ; $\bar{j}$	$\tilde{J}$ ; $\tilde{j}$
Velocity	V ; v	$\overset{\land}{V}$ ; $\overset{\land}{v}$	$\bar{V}$ ; $\bar{v}$	$\tilde{V}$ ; $\tilde{v}$
Divergence	Div ; div	$\overset{\land}{Div}$ ; $\overset{\land}{div}$	$\bar{Div}$ ; $\bar{div}$	$\tilde{Div}$ ; $\tilde{div}$
Gradient	Grad ; grad	$\overset{\land}{Grad}$ ; $\overset{\land}{grad}$	$\bar{Grad}$ ; $\bar{grad}$	$\tilde{Grad}$ ; $\tilde{grad}$
Determinant	Det ; det	$\overset{\land}{Det}$ ; $\overset{\land}{\det}$	$\bar{Det}$ ; $\bar{\det}$	$\tilde{Det}$ ; $\tilde{\det}$
Cofactor	Cof ; cof	$\overset{\land}{Cof}$ ; $\overset{\land}{cof}$	$\bar{Cof}$ ; $\bar{cof}$	$\tilde{Cof}$ ; $\tilde{cof}$

Table 2

Summary of the notation used to denote key quantities and operators in the bulk, on the surface and interface, and on the curve corresponding to the material and spatial configurations

Material configuration	Spatial configuration
$I : = G_{1} \otimes G^{1} + G_{2} \otimes G^{2} + G_{3} \otimes G^{3}$	$i : = g_{1} \otimes g^{1} + g_{2} \otimes g^{2} + g_{3} \otimes g^{3}$
$\begin{matrix} Grad {•} : = \partial {•} / \partial X & Div {•} : = Grad {•} : I \end{matrix}$	$\begin{matrix} grad {•} : = \partial {•} / \partial x & div {•} : = grad {•} \end{matrix} : i$
$Det {•} : = [{•} \cdot G_{1}] \cdot [[{•} \cdot G_{2}] \times [{•} \cdot G_{3}]] / [G_{1} \cdot [G_{2} \times G_{3}]]$	$det {•} : = [{•} \cdot g_{1}] \cdot [[{•} \cdot g_{2}] \times [{•} \cdot g_{3}]] / [g_{1} \cdot [g_{2} \times g_{3}]]$
$\begin{matrix} Cof {•} : = Det {•} {•}^{- t} & d v = J d V & d a = Cof F \cdot d A \end{matrix}$	$\begin{matrix} cof {•} : = \det {•} {•}^{- t} & d V = j d v & d A = cof f \cdot d a \end{matrix}$
$\overset{\land}{I} : = {\overset{\land}{G}}_{1} \otimes {\overset{\land}{G}}^{1} + {\overset{\land}{G}}_{2} \otimes {\overset{\land}{G}}^{2} = I - N \otimes N$	$i : = {\overset{\land}{g}}_{1} \otimes {\overset{\land}{g}}^{1} + {\overset{\land}{g}}_{2} \otimes {\overset{\land}{g}}^{2} = i - n \otimes n$
$\begin{matrix} \overset{\land}{Grad} {\overset{\land}{•}} : = Grad {\overset{\land}{•}} \cdot \overset{\land}{I} & \overset{\land}{Div} {\overset{\land}{•}} \end{matrix} : = \overset{\land}{Grad} {\overset{\land}{•}} : \overset{\land}{I}$	$\begin{matrix} \overset{\land}{grad} {\overset{\land}{•}} : = grad {\overset{\land}{•}} \cdot \overset{\land}{i} & \overset{\land}{div} {\overset{\land}{•}} : = \overset{\land}{grad} {\overset{\land}{•}} : \overset{\land}{i} \end{matrix}$
$\overset{\land}{Det} {\overset{\land}{•}} : = \| [{\overset{\land}{•}} \cdot {\overset{\land}{G}}_{1}] \times [{\overset{\land}{•}} \cdot {\overset{\land}{G}}_{2}] \| / \| {\overset{\land}{G}}_{1} \times {\overset{\land}{G}}_{2} \|$	$\overset{\land}{det} {\overset{\land}{•}} : = \| [{\overset{\land}{•}} \cdot {\overset{\land}{g}}_{1}] \times [{\overset{\land}{•}} \cdot {\overset{\land}{g}}_{2}] \| / \| \overset{\land}{g_{1}} \times \overset{\land}{g_{2}} \|$
$\begin{matrix} \overset{\land}{Cof} {\overset{\land}{•}} : = \overset{\land}{Det} {\overset{\land}{•}} {\overset{\land}{•}}^{- t} & d a = \overset{\land}{J} d A & d l = \overset{\land}{Cof} \overset{\land}{F} \cdot d L \end{matrix}$	$\begin{matrix} \overset{\land}{cof} {\overset{\land}{•}} : = \overset{\land}{det} {\overset{\land}{•}} {\overset{\land}{•}}^{- t} & d A = \overset{\land}{j} d a & d L = \overset{\land}{cof} \overset{\land}{f} \cdot d l \end{matrix}$
$\bar{I} : = {\bar{G}}_{1} \otimes {\bar{G}}^{1} + {\bar{G}}_{2} \otimes {\bar{G}}^{2} = I - \bar{N} \otimes \bar{N}$	$\bar{i} : = {\bar{g}}_{1} \otimes {\bar{g}}^{1} + {\bar{g}}_{2} \otimes {\bar{g}}^{2} = i - \bar{n} \otimes \bar{n}$
$\begin{matrix} \bar{Grad} {\bar{•}} : = Grad {\bar{•}} \cdot \bar{I} & \bar{Div} \end{matrix} {\bar{•}} : = \bar{Grad} {\bar{•}} : \bar{I}$	$\begin{matrix} \bar{grad} {\bar{•}} : = grad {\bar{•}} \cdot \bar{i} & \bar{div} {\bar{•}} \end{matrix} : = \bar{grad} {\bar{•}} : \bar{i}$
$\bar{Det} {\bar{•}} : = \| [{\bar{•}} \cdot {\bar{G}}_{1}] \times [{\bar{•}} \cdot {\bar{G}}_{2}] \| / \| {\bar{G}}_{1} \times {\bar{G}}_{2} \|$	$\bar{\det} {\bar{•}} : = \| [{\bar{•}} \cdot {\bar{g}}_{1}] \times [{\bar{•}} \cdot {\bar{g}}_{2}] \| / \| {\bar{g}}_{1} \times {\bar{g}}_{2} \|$
$\begin{matrix} \bar{Cof} {\bar{•}} : = \bar{Det} {\bar{•}} {\bar{•}}^{- t} & d a = \bar{J} d A & d l = \bar{Cof} \bar{F} \cdot d L \end{matrix}$	$\begin{matrix} \bar{cof} {\bar{•}} : = \bar{det} {\bar{•}} {\bar{•}}^{- t} & d A = \bar{j} d a & d L = \bar{cof} \bar{f} \cdot d l \end{matrix}$
$\tilde{I} : = \tilde{G_{1}} \otimes {\tilde{G}}^{1} = \tilde{N} \otimes \tilde{N}$	$\tilde{i} : = \tilde{g_{1}} \otimes {\tilde{g}}^{1} = \tilde{n} \otimes \tilde{n}$
$\begin{matrix} \tilde{Grad} {\tilde{•}} : = Grad {\tilde{•}} \cdot \tilde{I} & \tilde{Div} {\tilde{•}} : = \tilde{Grad} {\tilde{•}} : \tilde{I} \end{matrix}$	$\begin{matrix} \tilde{grad} {\tilde{•}} : = grad {\tilde{•}} \cdot \tilde{i} & \tilde{div} {\tilde{•}} \end{matrix} : = \tilde{grad} {\tilde{•}} : \tilde{i}$
$\tilde{Det} {\tilde{•}} : = \| {\tilde{•}} \cdot {\tilde{G}}_{1} \| / \| {\tilde{G}}_{1} \|$	$\tilde{det} {\tilde{•}} : = \| {\tilde{•}} \cdot {\tilde{g}}_{1} \| / \| {\tilde{g}}_{1} \|$
$\begin{matrix} \tilde{Cof} {\tilde{•}} : = \tilde{Det} {\tilde{•}} {\tilde{•}}^{- t} & d l = \tilde{J} d L & \tilde{n} = \tilde{Cof} \tilde{F} \cdot \tilde{N} \end{matrix}$	$\begin{matrix} \tilde{cof} {\tilde{•}} : = \tilde{det} {\tilde{•}} {\tilde{•}}^{- t} & d L = \tilde{j} d l & \tilde{N} = \tilde{cof} \end{matrix} \tilde{f} \cdot \tilde{n}$

Kinematics

Consider a continuum body that takes the open set $B_{0} \subset R^{3}$ at the time t = 0, as depicted in Fig. 1. The boundary of the body $B_{0}$ is denoted by $S_{0}$ := ∂ $B_{0}$ ⁠. The body $B_{0}$ is partitioned into two disjoint subdomains, denoted by the open sets $B_{0}^{+}$ and $B_{0}^{-}$ ⁠, by a two-sided interface $I_{0}$ ⁠. The interface $I_{0}$ is an open set; thus, $B_{0} = B_{0}^{+} \cup I_{0} \cup B_{0}^{-}$ ⁠. The boundary of the interface, a two-sided curve, is denoted as $C_{0}$ := $\partial I_{0}$ ⁠. In a similar fashion to the interface, the curve $C_{0}$ partitions the surface $S_{0}$ into two open sets $S_{0}^{+}$ and $S_{0}^{-}$ such that $S_{0}^{+} \cap S_{0}^{-} = ∅⁣$ and $S_{0}$ = $S_{0}^{+} \cup C_{0} \cup S_{0}^{-}$ ⁠.

Fig. 1

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The domains $B_{0}$ ⁠, $S_{0}$ ⁠, $I_{0}$ ⁠, and $C_{0}$ and the various unit normals

Remark. The boundary $S_{0}$ is closed and is assumed to be smooth. For the more general case where the boundary $S_{0}$ is nonsmooth, the curve $C_{0}$ can be understood as a geometrical entity. For the restricted case considered here where the boundary is smooth, the curve $C_{0}$ accounts for the interaction between the interface and the surface and should be understood as a physical entity. The extension of this work to nonsmooth boundaries is straightforward. Nevertheless, it involves introducing additional geometric concepts and further notation while providing little additional insight into the fundamental concepts.

The bulk is defined by $B_{0}$ := $B_{0}^{+} \cup B_{0}^{-}$ and is the reference placement of material particles labeled $X \in B_{0}$ ⁠. The two sides of the interface $I_{0}$ are denoted $I_{0}^{+} : = \partial B_{0}^{+} \cap I_{0}$ and $I_{0}^{-} : = \partial B_{0}^{-} \cap I_{0}$ ⁠, with $I_{0} = I_{0}^{+} \cup I_{0}^{-}$ ⁠. Material particles on the interface are labeled $\bar{X} \in I_{0}$ and, by definition, $\bar{X} : = X |_{I_{0}}$ ⁠. The outward unit normals to $I_{0}^{+}$ and $I_{0}^{-}$ are denoted ${\bar{N}}^{+}$ and ${\bar{N}}^{-}$ ⁠, respectively, with ${\bar{N}}^{+} (\bar{X})$ = $- {\bar{N}}^{-} (\bar{X})$ ⁠. The unit normal to $I_{0}$ is denoted $\bar{N} (\bar{X}) : = \bar{N^{-}} (\bar{X})$ ⁠.

The surface is defined by $S_{0} : = S_{0}^{+} \cup S_{0}^{-}$ with outward unit normal N. Material particles on the surface are labeled $\overset{\land}{X} \in S_{0}$ and, by definition, $\overset{\land}{X} = X |_{I_{0}}$ ⁠. The two sides of the curve $C_{0}$ are denoted $C_{0}^{+} : = \partial I_{0}^{+}$ and $C_{0}^{-} : = \partial I_{0}^{-}$ ⁠, with $C_{0} = C_{0}^{+} \cup C_{0}^{-}$ ⁠. Material particles on the curve $C_{0}$ are labeled $\tilde{X} \in C_{0}$ ⁠, where $\tilde{X} : = X |_{C_{0}}$ ⁠. The outward unit normals to $C_{0}^{+}$ and $C_{0}^{-}$ ⁠, tangential to $I_{0}$ and identical at a point $\tilde{X}$ ⁠, are denoted ${\tilde{N}}_{i}$ ⁠. The unit normals to $C_{0}^{+}$ and $C_{0}^{-}$ ⁠, tangential to $S_{0}$ ⁠, are denoted ${\tilde{N}}_{s}^{+}$ and ${\tilde{N}}_{s}^{-}$ ⁠, respectively, with ${\tilde{N}}_{s}^{+} (\tilde{X}) = - {\tilde{N}}_{s}^{-} (\tilde{X})$ ⁠. The unit normal to $C_{0}$ ⁠, tangential to $S_{0}$ ⁠, is defined by ${\tilde{N}}_{s} (\tilde{X}) : = {\tilde{N}}_{s}^{-} (\tilde{X})$ ⁠. The unit normals to $C_{0}$ in the sense of the Frénet–Serret formulae and identical at a point $\tilde{X}$ are denoted ${\tilde{N}}_{c}$ ⁠. Note that ${\tilde{N}}_{c}$ does not necessarily coincide with either ${\tilde{N}}_{i}$ or ${\tilde{N}}_{s}$ ⁠.

Let $T$ = [0, T] $\subset R_{+}$ denote the time domain. A motion of the reference placement for a time $t \in T$ is denoted by the orientation-preserving map φ: $B_{0} \times T \to R^{3}$ ⁠. The current placement of the bulk associated with the motion φ is denoted $B_{t} = ϕ (B_{0} (X), t)$ with particles designated as x = φ (X, t) $\in B_{t}$ (see Fig. 2).²

Fig. 2

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The material and spatial configurations of a continuum body and the associated motions, deformation gradients, and velocity measures in the various parts of the body

The restriction of the motion φ to the surface $S_{0}$ is denoted $\overset{\land}{ϕ}$ ⁠. The current placement of the surface is denoted $S_{t} = \overset{\land}{ϕ} (S_{0} (\overset{\land}{X}), t)$ with particles $\overset{\land}{x} = \overset{\land}{ϕ} (\overset{\land}{X}, t) \in S_{t}$ . It is assumed that particles on the surface of the body $S_{0}$ constitute the surface for all times $t \in T$ and $\overset{\land}{ϕ} = ϕ |_{S_{0}}$ ⁠, consequently $\overset{\land}{x} = x |_{S_{t}}$ ⁠. This assumption, referred to as kinematic slavery by Steinmann and Häsner [88], mimics coherence in interfaces.

The restriction of the motion φ to the positive and negative sides of the interface $I_{0}$ is denoted ${\bar{ϕ}}^{+}$ and ${\bar{ϕ}}^{-}$ ⁠, respectively. The placements of the positive and negative sides of the interface at time t are denoted $I_{t}^{+} = {\bar{ϕ}}^{+} (I_{0}^{+} (\bar{X}), t)$ and $I_{t}^{-} = {\bar{ϕ}}^{-} (I_{0}^{-} (\bar{X}), t)$ ⁠, respectively, with particles ${\bar{x}}^{+} = {\bar{ϕ}}^{+} (\bar{X}, t) \in I_{t}^{+}$ and ${\bar{x}}^{-} = {\bar{ϕ}}^{-} (\bar{X}, t) \in I_{t}^{-}$ ⁠. The material interface is assumed to be coherent: The motion is not necessarily smooth across the interface, however, it is continuous across the interface, smooth away from the interface, and smooth up to the interface from either side [105]. Thus, ${\bar{ϕ}}^{+} = {\bar{ϕ}}^{-} = ϕ |_{I_{0}}$ and, consequently, ${\bar{x}}^{+} = {\bar{x}}^{-} = x |_{I_{t}}$ ⁠. The restriction of the motion φ to the interface is defined by $\bar{ϕ} : = ϕ |_{I_{0}}$ with particles designated as $\bar{x} = \bar{ϕ} (\bar{X}, t) \in I_{t}$ ⁠.

Remark. Given that this work focuses on coherent material interfaces it may seem superfluous to distinguish between $I_{t}^{+}$ and $I_{t}^{-}$ ⁠. However, coherence concerns only the continuity of the motion across the interface and jumps in other fields, e.g., temperature, are permitted.□

Following from the aforementioned assumptions, the restriction of the motion φ to the curve $C_{0}$ at the junction of the interface and the surface is denoted $\tilde{ϕ}$ ⁠. The current placement of the curve is denoted $C_{t} = \tilde{ϕ} (C_{0} (\tilde{X}), t)$ with particles $\tilde{x} = \tilde{ϕ} (\tilde{X}, t) \in C_{t}$ . Finally, $\tilde{ϕ} = ϕ |_{C_{0}}$ and $\tilde{x} = x | C_{t} = \bar{x} | C_{t} = \hat{x} | C_{t}$ ⁠.

For the normal vectors to the surface, interface, and curve in the spatial configuration the same convention as in the material configuration is used, however, with lowercase n instead of uppercase N.

Deformation Gradients

The deformation gradient

F : T B_{0} \to T B_{t}

⁠, that is the (invertible) linear tangent map between spatial and material line elements

d x \in T B_{t}

(tangent space to

B_{t}

⁠) and

d X \in T B_{0}

⁠, and the associated (material) velocity V are defined by

F (X, t) : = Grad ϕ (X, t) and V : = D_{t} ϕ (X, t)

Similarly, the various deformation gradients for the remainder of the body are given by

\begin{matrix} \overset{\land}{F} : T S_{0} \to T S_{t}, & \overset{\land}{F} (\overset{\land}{X}, t) : = \overset{\land}{Grad} \overset{\land}{ϕ} (\overset{\land}{X}, t), \\ \bar{F} : T I_{0} \to T I_{t}, & \bar{F} (\bar{X}, t) : = \bar{Grad} \bar{ϕ} (\bar{X}, t), \\ \tilde{F} : T C_{0} \to T C_{t}, & \tilde{F} (\tilde{X}, t) : = \tilde{Grad} \tilde{ϕ} (\tilde{X}, t) \end{matrix}

The surface, interface, and curve deformation gradients (⁠

\overset{\land}{F}, \bar{F,} and \tilde{F}

⁠) are noninvertible linear tangent maps between the respective spatial and material elements (see Table 1 for further information). Although these deformation gradients are noninvertible, due to rank deficiency, they each possess an inverse in the following generalized sense:

\overset{\land}{F} \cdot {\overset{\land}{F}}^{- 1} = \overset{\land}{i} and {\overset{\land}{F}}^{- 1} \cdot \overset{\land}{F} = \overset{\land}{I}, \bar{F} \cdot {\bar{F}}^{- 1} = \bar{i} and {\bar{F}}^{- 1} \cdot \bar{F} = \bar{I}, \tilde{F} \cdot {\tilde{F}}^{- 1} = \tilde{i} and {\tilde{F}}^{- 1} \cdot \tilde{F} = \tilde{I}

The volume elements in the bulk, the area elements on the surface, and interface and the line elements on the curve in the material and spatial configurations can be mapped into each other by means of the Jacobian determinants of their associated deformation gradients and their inverses in the various parts of the body as

\begin{matrix} d v = J d V with J : = Det F, \\ d a = \overset{\land}{J} d A with \overset{\land}{J} : = \overset{\land}{Det} \overset{\land}{F}, \\ d a = \bar{J} d A with \bar{J} : = \bar{Det} \bar{F}, \\ d l = \tilde{J} d L with \tilde{J} : = \tilde{Det} \tilde{F}, \end{matrix} \begin{matrix} d V = j d v with j : = det F^{- 1}, \\ d A = \overset{\land}{j} d a with \overset{\land}{j} : = \overset{\land}{det} {\overset{\land}{F}}^{- 1}, \\ d A = \bar{j} d a with \bar{j} : = \bar{det} {\bar{F}}^{- 1}, \\ d L = \tilde{j} d l with \tilde{j} : = \tilde{det} {\tilde{F}}^{- 1} \end{matrix}

Control Regions

The equations governing the coupled response of the various parts of the body are obtained from the balances of linear and angular momentum, energy, and entropy performed over a control region. These balances are then localized at arbitrary points in the bulk $B_{0}$ ⁠, on the surface $S_{0}$ ⁠, on the interface $I_{0}$ ⁠, or on the curve $C_{0}$ ⁠.

The canonical control region is defined as one that possibly has as part of its boundary the surface $S_{0}$ and is denoted

B_{0} \subset B_{0}

with boundary

\partial B_{0}

(see Fig. 3). The orientable external surface of the control region is defined by

S_{0}^{ext} : = \partial B_{0} \cap S_{0}

while the interior surface is

S_{0}^{int} : = \partial B_{0} \ S_{0}^{ext}

⁠. The outward unit normals to

S_{0}^{int}

and

S_{0}^{ext}

are denoted M and N, respectively. The surface of the control region is defined by

S_{0} : = S_{0}^{ext} \cup S_{0}^{int}

⁠. The boundary of

S_{0}^{ext}

⁠, a curve, is defined by

L_{0} : = \partial S_{0}^{ext}

⁠. The unit normal to the curve

L_{0}

is denoted

\overset{\land}{N}

and is tangent to the surface

S_{0}^{ext}

⁠.

Fig. 3

View large Download slide

The bulk $B_{0}$ and the canonical control region $B_{0}$ and its representation in the bulk, and on the surface, interface, and curve

The control region

B_{0}

can include a segment of the interface

I_{0}

defined by

I_{0} : = B_{0} \cap I_{0}

⁠. Thus,

B_{0}

can be partitioned into two subdomains

B_{0}^{+}

and

B_{0}^{-}

⁠. The interface

I_{0}

has two sides

I_{0}^{+} : = I_{0} \cap \partial B_{0}^{+}

and

I_{0}^{-} : = I_{0} \cap \partial B_{0}^{-}

with

I_{0} = I_{0}^{+} \cup I_{0}^{-}

⁠.

The boundary of the interface $\partial I_{0}$ is a curve denoted as

C_{0}

⁠. The, possibly empty, intersection of the curves

C_{0}

and

C_{0}

is denoted as

C_{0}^{ext} : = C_{0} \cap C_{0}

⁠. The interior boundary of the interface

I_{0}

is

C_{0}^{int} : = C_{0} \ C_{0}^{ext}

⁠. The outward unit normal to the curve

C_{0}^{int}

is denoted

\bar{M}

and is tangent to the interface

I_{0}

⁠. The unit normal to the curve

C_{0}^{ext}

⁠, tangent to the interface

I_{0}

⁠, is denoted

{\tilde{N}}_{i}

⁠. The unit normal to the curve

C_{0}^{ext}

⁠, in the sense of Frénet–Serret formulae, is denoted

{\tilde{N}}_{c}

⁠. The unit normal to the curve

C_{0}^{ext}

⁠, tangent to the surface

S_{0}

⁠, is denoted

{\tilde{N}}_{s}

⁠. The unit normal to the boundary of the curve

\partial C_{0}^{ext}

⁠, tangent to the curve

C_{0}^{ext}

is denoted

\tilde{N}

⁠. The curve

C_{0}^{ext}

has two sides

^{+} C_{0}^{ext} : = C_{0}^{ext} \cap \partial I_{0}^{+}

and

^{-} C_{0}^{ext} : = C_{0}^{ext} \cap \partial I_{0}^{-}

with

C_{0}^{ext} =^{+} C_{0}^{ext} \cup^{-} C_{0}^{ext}

⁠. Analogously,

^{+} C_{0}^{int} : = C_{0}^{int} \cap \partial I_{0}^{+}

and

^{-} C_{0}^{int} : = C_{0}^{int} \cap \partial I_{0}^{-}

with

C_{0}^{int} =^{+} C_{0}^{int} \cup^{-} C_{0}^{int}

⁠.

Key Definitions and Identities

Various key definitions and identities that are required in the remainder of the presentation are now introduced. Several other useful relations are recorded in Appendix B.

Jump and Average Relations

The following identity relates the jump of the product of two quantities (scalars or tensors, arbitrarily denoted a and b) to the products of their jumps and averages:

〚 a ★ b 〛 = 〚 a 〛 ★ {{b}} + {{b}} ★ 〚 a 〛

(1)

where the operator

⋆

denotes either scalar multiplication or a contraction. The average and the jump of a quantity over the interface

I_{0}

is defined as

{{{\bar{•}}}} : = \frac{1}{2} [{\bar{•}} |_{I_{0}^{+}} + {\bar{•}} |_{I_{0}^{-}}] and 〚 {\bar{•}} 〛 : = {\bar{•}} |_{I_{0}^{+}} - {\bar{•}} |_{I_{0}^{-}}

Integral Relations

The extended form of the divergence theorem in the material configuration for the various parts of the body is now given. The (bulk) divergence theorem relates the material divergence of a quantity over the control volume

B_{0}

entirely within the bulk, possibly containing an interface

I_{0}

⁠, to the flux of the quantity over the boundary

\partial B_{0}

⁠, and the jump of the flux in the quantity over the interface

I_{0}

(see, e.g., Ref. [96] for further details). For a tensor field {

•

}

\int_{B_{0}} Div {•} d V = \int_{\partial B_{0}} {•} \cdot M d A - \int_{I_{0}} [[{•}]] \cdot \bar{N} d A

(2)

Similarly, the corresponding surface, interface, and curve divergence theorems for tensorial quantities on the surface {

\overset{\land}{•}

}, interface {

\bar{•}

}, and curve {

\tilde{•}

} are, respectively, given by

\int_{S_{0} ext} \overset{\land}{Div} {\overset{\land}{•}} d A = \int_{L_{0}} {\overset{\land}{•}} \cdot \overset{\land}{N} d L - \int_{S_{0}^{ext}} \overset{\land}{C} {\overset{\land}{•}} \cdot N d A - \int_{C_{0}^{ext}} [[{\overset{\land}{•}}]] \cdot \tilde{N_{s}} d L where \overset{\land}{C} : = - \overset{\land}{Div} N

(3)

\int_{I_{0}} \bar{Div} {\bar{•}} d A = \int_{C_{0}^{int}} {\bar{•}} \cdot \bar{M} d L + \int_{C_{0}^{ext}} {\bar{•}} \cdot {\tilde{N}}_{i} d L - \int_{I_{0}} \bar{C} {\bar{•}} \cdot \bar{N} d A where \bar{C} : = - \bar{Div} \bar{N}

(4)

\int_{C_{0}^{ext}} \tilde{Div} {\tilde{•}} d L = \sum_{\partial C_{0}^{ext}} {\tilde{•}} \cdot \tilde{N} - \int_{C_{0}^{ext}} \tilde{C} {\tilde{•}} \cdot \tilde{N_{c}} d L where \tilde{C} : = - \tilde{Div} {\tilde{N}}_{c}

(5)

Piola Identity and Piola Transform

The local balance equations expressed in spatial quantities can be obtained from the material versions using the standard (bulk) Piola identity

Div (Cof F) = 0

(6)

The surface, interface, and curve divergence theorems together with their corresponding Nanson's formulae result in the following surface, interface, and curve Piola identities:

\overset{\land}{Div} (\overset{\land}{Cof} \overset{\land}{F}) = \overset{\land}{J} \overset{\land}{c} n, \bar{Div} (\bar{Cof} \bar{F}) = \bar{J} \bar{c} \bar{n}, \tilde{Div} (\tilde{Cof} \tilde{F}) = \tilde{J} \tilde{c} {\tilde{n}}_{c}

(7)

A bulk Cauchy stress, heat flux, or entropy, collectively denoted as a Cauchy quantity γ, transforms into its corresponding Piola quantity

Γ

via the (bulk) Piola transform

Γ = γ \cdot Cof F

(8)

Analogously, a surface, interface, and curve Cauchy quantity denoted as

\overset{\land}{γ}

⁠,

\bar{γ}

⁠, and

\tilde{γ}

⁠, respectively, transform into their corresponding Piola quantities

\overset{\land}{Γ}

⁠,

\bar{Γ}

⁠, and

\tilde{Γ}

via the Piola transforms

\overset{\land}{Γ} = \overset{\land}{γ} \cdot \overset{\land}{Cof} \overset{\land}{F}, \bar{Γ} = \bar{γ} \cdot \bar{Cof} \bar{F}, \tilde{Γ} = \tilde{γ} \cdot \tilde{Cof} \tilde{F}

(9)

Superficial and Tangential Tensor Fields

Material second-order tensors and vectors on the surface, interface, and curve can be, respectively, classified as superficial (in their tangent spaces) or tangential. Superficial material second-order tensors on the surface, interface, and curve, respectively, possess the orthogonality properties

{\overset{\land}{•}} \cdot N = 0, {\bar{•}} \cdot \bar{N} = 0, {\tilde{•}} \cdot {\tilde{N}}_{c} = 0

If the arbitrary quantities in the preceding relation are vectors, they are termed tangential. Tangential material second-order tensors on the surface, interface, and curve, respectively, possess the orthogonality properties

\begin{matrix} {\overset{\land}{•}} \cdot N = 0 and N \cdot {\overset{\land}{•}} = 0, {\bar{•}} \cdot \bar{N} = 0 and \bar{N} \cdot {\bar{•}} = 0, {\tilde{•}} \cdot {\tilde{N}}_{c} = 0 and {\tilde{N}}_{c} \cdot {\tilde{•}} = 0 \end{matrix}

or, alternatively,

\begin{matrix} \overset{\land}{I} \cdot {\overset{\land}{•}} \cdot \overset{\land}{I} = {\overset{\land}{•}}, & \bar{I} \cdot {\bar{•}} \cdot \bar{I} = {\bar{•}}, & \tilde{I} \cdot {\tilde{•}} \cdot \tilde{I} = {\tilde{•}} \end{matrix}

Spatial second-order tensors and vectors can be defined in the same fashion by replacing the material quantities with the spatial ones. Note that the set of all tangential second-order tensors is contained within the set of all superficial second-order tensors.

Governing Equations

The coupled governing equations for the thermomechanical problem in the various parts of the body are now derived using an arbitrary (canonical) control region

B_{0} \subset B_{0}

⁠. Thereafter, the balances of momentum, energy, and entropy, i.e., the governing equations in integral form, are given. Finally, these integral balance expressions are localized at arbitrary points in the bulk

B_{0}

⁠, on the surface

S_{0}

⁠, or interface

I_{0}

⁠, or on the curve

C_{0}

⁠, thereby giving the local (strong) form of the governing equations.³

The integral form of the governing equations over the control region, in its most general form, consists of eight terms that account for the contributions from the various parts of the body; that is,

\begin{matrix} 0 (or 0) = \int_{B_{0}} {•} d V + \int_{S_{0}^{ext}} {\overset{\land}{•}} d A + \int_{I_{0}} {\bar{•}} d A + \int_{C_{0}^{ext}} {\tilde{•}} d L + \int_{S_{0}^{int}} {°} \cdot M d A + \int_{L_{0}} {\overset{\land}{°}} \cdot \overset{\land}{N} d L + \int_{C_{0}^{int}} {\bar{°}} \cdot \bar{M} d L + \sum_{\partial C_{0}^{ext}} {\tilde{°}} \cdot \tilde{N} \end{matrix}

where ${•}$ denotes a source-like term and ${°}$ is the corresponding flux. The corresponding integral expressions for the balance of linear and angular momentum are derived and, subsequently, localized in Sec. 3.1. The localization process is explained in detail. In Sec. 3.2 the integral and resulting local forms of the energy balance are given. Finally, in Sec. 3.3 the process is repeated for the entropy balance.

Mechanical Power

The global working, i.e., the mechanical power, in the material configuration is denoted

W_{0}

and given by

\begin{matrix} W_{0} = W_{0} (V, \overset{\land}{V}, \bar{V}, \tilde{V}) : = \int_{B_{0}} V \cdot B^{p} d V + \int_{S_{0}^{ext}} \overset{\land}{V} \cdot {\overset{\land}{B}}^{p} d A + \int_{I_{0}} \bar{V} \cdot {\bar{B}}^{p} d A + \int_{C_{0}^{ext}} \tilde{V} \cdot {\tilde{B}}^{p} d L + \int_{S_{0}^{int}} V \cdot [P \cdot M] d A + \int_{L_{0}} \overset{\land}{V} \cdot [\overset{\land}{P} \cdot \overset{\land}{N}] d L + \int_{C_{0}^{int}} \bar{V} \cdot [\bar{P} \cdot \bar{M}] d L + \sum_{\partial C_{0}^{ext}} \tilde{V} \cdot [\tilde{P} \cdot \tilde{N}] \end{matrix}

(10)

The prescribed bulk (body) force per unit reference volume of

B_{0}

is denoted B^p (N/m³) where the superscript p indicates a prescribed quantity. The prescribed tractions per unit reference area of the surface

S_{0}^{ext}

and the interface

I_{0}

are denoted

{\overset{\land}{B}}^{p}

(N/m²) and

{\bar{B}}^{p}

(N/m²), respectively. The prescribed traction per unit reference length of the curve

C_{0}^{ext}

is denoted

{\tilde{B}}^{p}

(N/m). As with the bulk force

B^{p}

⁠, the prescribed tractions on the surface, interface, and curve can be understood as prescribed surface, interface, and curve forces, respectively. The bulk, surface, interface, and curve Piola stresses are denoted

P

⁠,

\overset{\land}{P}

⁠,

\bar{P}

⁠, and

\tilde{P}

⁠, respectively. The terms

P \cdot M, \overset{\land}{P} \cdot \overset{\land}{N}, \bar{P} \cdot \bar{M},

and

\tilde{P} \cdot \tilde{N}

denote the tractions acting on

S_{0}^{int}, L_{0}, C_{0}^{int},

and

\partial C_{0}^{ext}

⁠, respectively.

Remark. It assumed that the Cauchy theorem holds for the bulk, surface, interface, and curve Piola stresses. Furthermore, $\overset{\land}{P}$ ⁠, $\bar{P}$ ⁠, and $\tilde{P}$ are superficial tensor fields in their tangent spaces.□

Invoking arguments concerning the invariance of the working under superposed rigid-body motions (see Ref. [92] for details concerning the case of thermomechanical solids with surface energy only) yields the global balances of momentum and angular momentum

\begin{matrix} \int_{B_{0}} B^{p} d V + \int_{S_{0}^{ext}} {\overset{\land}{B}}^{p} d A + \int_{I_{0}} {\bar{B}}^{p} d A + \int_{C_{0}^{ext}} {\tilde{B}}^{p} d L + \int_{S_{0}^{int}} P \cdot M d A + \int_{L_{0}} \overset{\land}{P} \cdot \overset{\land}{N} d L \int_{C_{0}^{int}} \bar{P} \cdot \bar{M} d L + \sum_{\partial C_{0}^{ext}} \tilde{P} \cdot \tilde{N} = 0 \end{matrix}

(11)

\begin{matrix} \int_{B_{0}} R \times B^{p} d V + \int_{S_{0}^{ext}} \overset{\land}{R} \times {\overset{\land}{B}}^{p} d A + \int_{I_{0}} \bar{R} \times {\bar{B}}^{p} d A + \int_{C_{0}^{ext}} \tilde{R} \times {\tilde{B}}^{p} d L + \int_{S_{0}^{int}} R \times [P \cdot M] d A + \int_{L_{0}} \overset{\land}{R} \times [\overset{\land}{P} \cdot \overset{\land}{N}] d L + \int_{C_{0}^{int}} \bar{R} \times [\bar{P} \cdot \bar{M}] d L + \sum_{\partial C_{0}^{ext}} \tilde{R} \times [\tilde{P} \cdot \tilde{N}] = 0 (12) \end{matrix}

(12)

where R, $\overset{\land}{R}, \bar{R}$ ⁠, and $\tilde{R}$ are vectors to the points in the bulk, surface, interface, and curve, respectively, relative to a fixed point in $R^{3}$ ⁠, e.g., the origin.

Bulk

The global statements and are now localized at arbitrary

X \in B_{0}

by considering a control region

B_{0}

such that

\partial B_{0}

does not contain any portion of the surface or interface, i.e.,

\partial B_{0} \cap S_{0} = \partial B_{0} \cap I_{0} = ∅⁣

⁠. Under such restrictions the limits of the integrals in Eqs. and are

\begin{matrix} B_{0} \neq ∅⁣, & S_{0}^{ext} = ∅⁣, & I_{0} = ∅⁣, & C_{0}^{ext} = ∅⁣, \\ S_{0}^{int} = \partial B_{0} \neq ∅⁣, & L_{0} = ∅⁣, & C_{0}^{int} = ∅⁣, & \partial C_{0}^{ext} = ∅⁣ \end{matrix}

From the arbitrariness of the control region

B_{0}

⁠, and taking into account the corresponding extended (bulk) divergence theorem together with Eq. ₁, one obtains the familiar local balances of linear and angular momentum in the bulk

B_{0}

Div P + B^{p} = 0

(13)

F \cdot P^{t} = P \cdot F^{t}

(14)

Surface

The global statements and are now localized at arbitrary

\overset{\land}{X} \in S_{0}

by considering a control region

B_{0}

where

\partial B_{0}

contains a portion of the surface, i.e.,

\partial B_{0} \cap S_{0} \neq ∅⁣

but does not contain any portion of the interface, i.e.,

\partial B_{0} \cap I_{0} = ∅⁣

⁠. Under such restrictions the limits of the integrals in Eqs. and are

\begin{matrix} B_{0} \neq ∅⁣, & S_{0}^{ext} \neq ∅⁣, & I_{0} = ∅⁣, & C_{0}^{ext} = ∅⁣, \\ S_{0}^{int} = \partial B_{0} / S_{0}^{ext} \neq ∅⁣, & L_{0} \neq ∅⁣, & C_{0}^{int} = ∅⁣, & {\partial C}_{0}^{ext} = ∅⁣ \end{matrix}

The volume of the control region is then decreased to zero, i.e., the limit

B_{0} \to ∅⁣

is taken and; consequently,

S_{0}^{int} = S_{0}^{ext}

with M =

- N

and R =

\overset{\land}{R .}

From the arbitrariness of the control region

B_{0}

and taking into account the corresponding extended surface divergence theorem together with Eq. ₂, one obtains the local balances of linear and angular momentum on the surface

S_{0}

as

\overset{\land}{Div} \overset{\land}{P} + {\overset{\land}{B}}^{P} - P \cdot N = 0

(15)

\overset{\land}{F} \cdot \overset{\land}{P^{t}} = \overset{\land}{P} \cdot {\overset{\land}{F}}^{t}

(16)

Interface

Following the general approach outlined for the bulk and the surface, the global statements and are now localized at arbitrary

\bar{X} \in I_{0}

⁠. Thus, a control region

B_{0}

is considered such that

\partial B_{0}

contains a portion of the interface, i.e.,

\partial B_{0} \cap I_{0} \neq ∅⁣

⁠, but

\partial B_{0}

does not contain any portion of the surface, i.e.,

\partial B_{0} \cap S_{0} = ∅⁣

⁠. Under such restrictions the limits of the integrals in Eqs. and are

\begin{matrix} B_{0} \neq ∅⁣, & S_{0}^{ext} = ∅⁣, & I_{0} \neq ∅⁣, & C_{0}^{ext} = ∅⁣, \\ S_{0}^{int} = \partial B_{0} / I_{0} \neq ∅⁣, & L_{0} = ∅⁣, & C_{0}^{int} = \partial I_{0} \neq ∅⁣, & \partial C_{0}^{ext} = ∅⁣ \end{matrix}

The volume of the control region is then decreased to zero, i.e., the limit

B_{0} \to ∅⁣

is taken and, consequently,

S_{0}^{int} = I_{0}

with M =

\bar{N}

on

I_{0}^{+}

⁠, M =

- \bar{N}

on

I_{0}^{-}

⁠, and R =

\bar{R}

⁠. Once again, from the arbitrariness of the control region

B_{0}

and taking into account the corresponding extended interface divergence theorem together with Eq. ₃, one obtains the local balance of linear and angular momentum on the interface

I_{0}

as

\bar{Div} \bar{P} + {\bar{B}}^{p} + [[P]] \cdot \bar{N} = 0

(17)

\bar{F} \cdot {\bar{P}}^{t} = \bar{P} \cdot {\bar{F}}^{t}

(18)

Curve

Finally, the global statements and are localized at arbitrary

\tilde{X} \in C_{0}

⁠. Thus, a control region

S_{0}^{ext}

is considered such that

S_{0}^{ext}

contains a portion of the curve, i.e.,

S_{0}^{ext} \cap C_{0} \neq ∅⁣

⁠. Under such restrictions the limits of the integrals in Eqs. and are

\begin{matrix} B_{0} = ∅⁣, & S_{0}^{ext} \neq ∅⁣, & I_{0} = ∅⁣, & C_{0}^{ext} \neq ∅⁣, \\ S_{0}^{int} = S_{0}^{ext} \neq ∅⁣, & L_{0} \neq ∅⁣, & C_{0}^{int} = C_{0}^{ext} \neq ∅⁣, & \partial C_{0}^{ext} \neq ∅⁣ \end{matrix}

The limit

S_{0}^{ext} \to ∅⁣

is then taken such that

\partial S_{0}^{ext} = C_{0}^{ext}

with

\overset{\land}{N} = {\tilde{N}}_{s}

on

^{+} C_{0}^{ext}, \overset{\land}{N} = - {\tilde{N}}_{s}

on

^{-} C_{0}^{ext}

and

\overset{\land}{R} = \tilde{R}

⁠. From the arbitrariness of the control region

S_{0}^{ext}

⁠, and taking into account the corresponding extended divergence theorem together with Eq. _4, one obtains the local balance of linear and angular momentum on the curve

C_{0}

as

\tilde{Div} \tilde{P} + {\tilde{B}}^{p} - \bar{P} \cdot {\tilde{N}}_{i} + [[\overset{\land}{P}]] \cdot \tilde{N_{s}} = 0

(19)

\tilde{F} \cdot {\tilde{P}}^{t} = \tilde{P} \cdot {\tilde{F}}^{t}

(20)

Remark. The local balance of momentum on the interface Eq. (17) relates the interface Piola stress to the jump of the bulk Piola stress across the interface. The curve Piola stress, however, is coupled not only to the jump of the surface Piola stress across the curve but also to a contribution from the interface Piola stress according to Eq. (19).□

Material and Spatial Formulations

In summary, the local balances of linear and angular momentum for the various parts of the body in material quantities read

\begin{matrix} \begin{matrix} Div P + B = 0 & and F \cdot P^{t} = P \cdot F^{t} in B_{0} \end{matrix} \\ \begin{matrix} where B = B^{p}, \end{matrix} \\ \begin{matrix} \overset{\land}{Div} \overset{\land}{P} + \overset{\land}{B} = 0 & and \overset{\land}{F} \cdot {\overset{\land}{P}}^{t} = \overset{\land}{P} \cdot {\overset{\land}{F}}^{t} in S_{0} \end{matrix} \\ \begin{matrix} where \overset{\land}{B} = {\overset{\land}{B}}^{p} - P \cdot N, \end{matrix} \\ \begin{matrix} \bar{Div} \bar{P} + \bar{B} = 0 & and \bar{F} \cdot {\bar{P}}^{t} = \bar{P} \cdot {\bar{F}}^{t} in I_{0} \end{matrix} \\ \begin{matrix} where \bar{B} = {\bar{B}}^{p} + [[P]] \cdot \bar{N} \end{matrix} \\ \begin{matrix} \tilde{Div} \tilde{P} + \tilde{B} = 0 & and \tilde{F} \cdot {\tilde{P}}^{t} = \tilde{P} \cdot {\tilde{F}}^{t} in C_{0} \end{matrix} \\ \begin{matrix} where \tilde{B} = {\tilde{B}}^{p} - \bar{P} \cdot {\tilde{N}}_{i} + [[\overset{\land}{P}]] \cdot {\tilde{N}}_{s} \end{matrix} \end{matrix}

(21)

Invoking the balances of momentum in the various parts of the body , the expression for the mechanical power can be reformulated in terms of the stresses (the proof is given in Appendix C) as

W_{0} = \int_{B_{0}} P : Grad V d V + \int_{I_{0}} \bar{P} : \bar{Grad} \bar{V} d A + \int_{S_{0}^{ext}} \overset{\land}{P} : \overset{\land}{Grad} \overset{\land}{V} d A + \int_{C_{0}^{ext}} \tilde{P} : \tilde{Grad} \tilde{V} d L

(22)

The (bulk) Cauchy stress σ and the surface, interface, and curve tangential Cauchy stresses, denoted

\overset{\land}{σ}

⁠,

\bar{σ}

⁠, and

\tilde{σ}

⁠, respectively, are related to their corresponding Piola stresses via the Piola transforms and . The working in Eq. can, thus, be reformulated in terms of spatial quantities as

\begin{matrix} W_{t} = \int_{B_{t}} σ : grad v d v + \int_{I_{t}} \bar{σ} : \bar{grad} \bar{v} d a + \int_{S_{t}^{ext}} \overset{\land}{σ} : \overset{\land}{grad} \overset{\land}{v} d a + \int_{C_{t}^{ext}} \tilde{σ} : \tilde{grad} \tilde{v} d l \end{matrix}

where the spatial velocity fields in the bulk on the surface, interface, and curve are denoted $v (x, t), \overset{\land}{v} (\overset{\land}{x}, t), \bar{v} (\bar{x}, t)$ ⁠, and $\tilde{v} (\tilde{x}, t)$ ⁠, respectively.

Using the Piola identities and , the local momentum balance Eq. can be expressed in terms of spatial

\begin{matrix} \begin{matrix} div σ + b = 0 \\ where b = b^{p}, \\ \overset{\land}{div} \overset{\land}{σ} + \overset{\land}{b} = 0 \\ where \overset{\land}{b} = {\overset{\land}{b}}^{p} - σ \cdot n, \\ \bar{div} \bar{σ} + \bar{b} = 0 \\ where \bar{b} = {\bar{b}}^{p} + [[σ]] \cdot \bar{n}, \\ \tilde{div} \tilde{σ} + \tilde{b} = 0 \\ where \tilde{b} = {\tilde{b}}^{p} - \bar{σ} \cdot {\tilde{n}}_{i} + [[\overset{\land}{σ}]] \cdot {\tilde{n}}_{s} \end{matrix} & \begin{matrix} and σ = σ^{t} in B_{t} \\ and \overset{\land}{σ} = {\overset{\land}{σ}}^{t} in S_{t} \\ and \overset{\land}{σ} = {\overset{\land}{σ}}^{t} in I_{t} \\ and \tilde{σ} = {\tilde{σ}}^{t} in C_{t} \end{matrix} \end{matrix}

(23)

where the prescribed bulk force per unit current volume of $B_{t}$ is denoted b^p := jB^p. In an identical fashion, the prescribed surface, interface, and curve forces are denoted ${\overset{\land}{b}}^{p} : = \overset{\land}{j} {\overset{\land}{B}}^{p}, {\bar{b}}^{p} : = \bar{j} {\bar{B}}^{p} and {\tilde{b}}^{p} : = \tilde{j} {\tilde{B}}^{p}$ ⁠, respectively.

Thermal Power

In order to derive the internal energy balance equation one needs a relation for the global heating. The global heating, i.e., the global external thermal power, expressed in the material configuration is denoted

Q_{0}

and takes the form

Q_{0} : = \int_{B_{0}} Q^{p} d V + \int_{S_{0}^{ext}} \overset{\land}{Q^{p}} d A + \int_{I_{0}} {\bar{Q}}^{p} d A + \int_{C_{0}^{ext}} {\tilde{Q}}^{p} d L + \int_{S_{0}^{int}} [- Q \cdot M] d A + \int_{L_{0}} [- \overset{\land}{Q} \cdot \overset{\land}{N}] d L + \int_{C_{0}^{int}} [- \bar{Q} \cdot \bar{M}] d L + \sum_{\partial C_{0}^{ext}} [- \tilde{Q} \cdot \tilde{N}]

(24)

The prescribed bulk (body) heat source per unit reference volume (and per unit time) in

B_{0}

is denoted Q^p (N/s m²). The prescribed heat sources per unit reference area (and per unit time) of the surface

S_{0}^{ext}

and the interface

I_{0}

are denoted

{\overset{\land}{Q}}^{p}

(N/s m) and

{\bar{Q}}^{p}

(N/s m), respectively. The prescribed heat source per unit length (and per unit time) of the curve

C_{0}^{ext}

is denoted

{\tilde{Q}}^{p}

(N/s). The bulk, surface, interface, and curve Piola heat flux vectors are denoted Q,

\overset{\land}{Q}

⁠,

\bar{Q}

⁠, and

\tilde{Q}

⁠, respectively. The terms Q

\cdot

M,

\overset{\land}{Q}

\cdot

\overset{\land}{N}

,

\bar{Q}

\cdot

\bar{M}

⁠, and

\tilde{Q}

\cdot

\tilde{N}

denote the heat fluxes on

S_{0}^{int}

⁠,

L_{0}

C_{0}^{int}

⁠, and

\partial C_{0}^{ext}

⁠, respectively. It is assumed that the Cauchy theorem holds for the heat flux vectors in all parts of the body. Note that

\overset{\land}{Q}

⁠,

\bar{Q}

⁠, and

\tilde{Q}

are tangential vectors fields.

The rate of change of total internal energy expressed in the material configuration

E

₀ is given by

E_{0} : = \int_{B_{0}} D_{t} E d V + \int_{S_{0}} D_{t} \overset{\land}{E} d A + \int_{I_{0}} D_{t} \bar{E} d A + \int_{C_{0}} D_{t} \tilde{E} d L

where the bulk internal energy per unit reference volume, the surface and interface internal energies per unit reference area, and the curve internal energy per unit reference length are denoted E (N/m²), $\overset{\land}{E}$ (N/m), $\bar{E}$ (N/m), and $\tilde{E}$ (N), respectively.

The internal energy balance states that working and heating cause a change in the internal energy, that is,

W_{0} + Q_{0} = E_{0}

(25)

The global statement is now localized at arbitrary X

\in

B_{0}

,

\overset{\land}{X}

\in

S_{0}

,

\bar{X}

\in

I_{0}

⁠, and

\tilde{X}

\in

C_{0}

⁠. The resulting local balances of energy for the various parts of the body expressed in material quantities are

\begin{matrix} P : Grad V - Div Q + Q = D_{t} E & in B_{0} \\ where {Q = Q}^{p}, \\ \overset{\land}{P} : \overset{\land}{Grad} \overset{\land}{V} - \overset{\land}{Div} \overset{\land}{Q} + \overset{\land}{Q} = D_{t} \overset{\land}{E} & in S_{0} \\ where \overset{\land}{Q} = {\overset{\land}{Q}}^{p} + Q \cdot N, \\ \bar{P} : \bar{Grad} \bar{V} - \bar{Div} \bar{Q} + \bar{Q} = D_{t} \bar{E} & in I_{0} \\ where \bar{Q} = {\bar{Q}}^{p} - [[Q]] \cdot \bar{N,} \\ \tilde{P} : \tilde{Grad} \tilde{V} - \tilde{Div} \tilde{Q} + \tilde{Q} = D_{t} \tilde{E} & in C_{0} \\ where \tilde{Q} = {\tilde{Q}}^{p} + \bar{Q} \cdot {\tilde{N}}_{i} - [[\overset{\land}{Q}]] \cdot {\tilde{N}}_{s} \end{matrix}

(26)

The Piola identities for the various parts of the body allow Eq. (26) to be expressed solely in terms of spatial quantities. The (bulk) Cauchy heat flux vector q and the surface, interface, and curve tangential Cauchy heat flux vectors, denoted $\overset{\land}{q}$ ⁠, $\bar{q}$ ⁠, and $\tilde{q}$ ⁠, respectively, are related to their corresponding Piola heat flux vectors via the Piola transforms (8) and (9).

Thus, the local balances of energy for the various parts of the body in terms of spatial quantities are

\begin{matrix} σ : grad v - div (q + e v) + q = d_{t} e & in B_{t} \\ where {q = q}^{p}, \\ \overset{\land}{σ} : \overset{\land}{grad} \overset{\land}{v} - \overset{\land}{div} (\overset{\land}{q} + \overset{\land}{e} \overset{\land}{v}) + \overset{\land}{q} = d_{t} \overset{\land}{e} & in S_{t} \\ where \overset{\land}{q} = {\overset{\land}{q}}^{p} + q \cdot n, \\ \bar{σ} : \bar{grad} \bar{v} - \bar{div} (\bar{q} + \bar{e} \bar{v}) + \bar{q} = d_{t} \bar{e} & in I_{t} \\ where \bar{q} = {\bar{q}}^{p} - [[q]] \cdot \bar{n}, \\ \tilde{σ} : \tilde{grad} \tilde{v} - \tilde{div} (\tilde{q} + \tilde{e} \tilde{v}) + \tilde{q} = d_{t} \tilde{e} & in C_{t} \\ where \tilde{q} = {\tilde{q}}^{p} + \bar{q} \cdot {\tilde{n}}_{i} - [[\overset{\land}{q}]] \cdot {\tilde{n}}_{s} \end{matrix}

(27)

where the prescribed heat source per unit current volume (and unit time) in $B_{t}$ is denoted q^p := jQ^p. The bulk internal energy per unit current volume is denoted e := jE. In an identical fashion to the bulk, the prescribed surface, interface, and curve heat sources are denoted ${\overset{\land}{q}}^{p} : = \overset{\land}{j} {\overset{\land}{Q}}^{p}, {\bar{q}}^{p} : = \bar{j} {\bar{Q}}^{p}$ ⁠, and ${\tilde{q}}^{p} : = \tilde{j} {\tilde{Q}}^{p}$ respectively. Similarly, $\overset{\land}{e} : = \overset{\land}{j} \overset{\land}{E}, \bar{e} : = \bar{j} \bar{E}, and \tilde{e} : = \tilde{j} \tilde{E}$ denote the surface, interface, and curve internal energies, respectively. Recall that d_t denotes the spatial time derivative. Thus, convective contributions of the type ev appear in the corresponding divergence theorems.

Entropy Power

The global entropy input rate in the material configuration, denoted

H

₀, is given by

H_{0} : = \int_{B_{0}} H^{p} d V + \int_{S_{0}^{ext}} {\overset{\land}{H}}^{p} d A + \int_{I_{0}} {\bar{H}}^{p} d A + \int_{C_{0}^{ext}} {\tilde{H}}^{p} d L + \int_{S_{0}^{int}} [- H \cdot M] d A + \int_{L_{0}} [- \overset{\land}{H} \cdot \overset{\land}{N}] d L + \int_{C_{0}^{int}} [- \bar{H} \cdot \bar{M}] d L + \sum_{\partial C_{0}^{ext}} [- \tilde{H} \cdot \tilde{N}]

(28)

The prescribed bulk (body) entropy source per unit reference volume (and unit time) in

B_{0}

is denoted

H^{p}

(N/sKm²). The prescribed entropy sources on the surface

S_{0}^{ext}

⁠, and the interface

I_{0}

are denoted

{\overset{\land}{H}}^{p}

(N/sKm) and

{\bar{H}}^{p}

(N/sKm), respectively. The prescribed entropy source on the curve

C_{0}^{ext}

is denoted

{\tilde{H}}^{p}

(N/sK). In the global entropy input the terms H

\cdot

M,

\overset{\land}{H}

\cdot

\overset{\land}{N}

⁠,

\bar{H}

\cdot

\bar{M}

⁠, and

\tilde{H}

\cdot

\tilde{N}

denote the entropy fluxes on

S_{0}^{int}

⁠,

L_{0}

⁠,

C_{0}^{int}

⁠, and

\partial C_{0}^{ext}

⁠, respectively. The vectors H,

\overset{\land}{H}

⁠,

\bar{H}

⁠, and

\tilde{H}

are the bulk, surface, interface, and curve Piola entropy flux vectors, respectively. The vectors

\overset{\land}{H}

⁠,

\bar{H}

⁠, and

\tilde{H}

are tangential tensor fields in their tangent spaces.

The total entropy production rate in the material configuration, denoted

P_{0}

⁠, is defined by

P_{0} : = \int_{B_{0}} Π d V + \int_{S_{0}^{ext}} \overset{\land}{Π} d A + \int_{I_{0}} \bar{Π} d A + \int_{C_{0}^{ext}} \tilde{Π} d L \geq 0

where II ≥ 0, $\overset{\land}{Π}$ ≥ 0, $\bar{Π}$ ≥ 0, and $\tilde{Π}$ ≥ 0 are the entropy production rates per unit reference volume of the bulk, per unit reference area of the surface and interface, and per unit reference length of the curve, respectively, all constrained to be positive.

The rate of the change of total entropy in the material configuration, denoted

N

_0, is defined by

N_{0} : = \int_{B_{0}} D_{t} Ξ d V + \int_{S_{0}^{ext}} D_{t} \overset{\land}{Ξ} d A + \int_{I_{0}} D_{t} \bar{Ξ} d A + \int_{C_{0}^{ext}} D_{t} \tilde{Ξ} d L

where the bulk entropy per unit reference volume is denoted $Ξ$ (N/Km²). The surface and interface entropies per unit reference area are denoted $\bar{Ξ}$ (N/Km) and $\overset{\land}{Ξ}$ (N/Km), respectively. The curve entropy is denoted $\tilde{Ξ}$ (N/K).

The second law of thermodynamics imposes the physical restriction that the rate of increase of entropy in a body is not less than the total entropy supplied to the body or, equivalently, that the entropy increase is due to positive entropy production and entropy input, that is,

N_{0} = P_{0} + H_{0} with N_{0} \geq H_{0} since P_{o} \geq 0

(29)

The global statement of balance of entropy is then localized at arbitrary X

\in

B_{0}

,

\overset{\land}{X}

\in

S_{0}

,

\bar{X}

\in

I_{0}

⁠, and

\tilde{X}

\in

C_{0}

⁠. The resulting local balances of entropy for the various parts of the body in material quantities are

\begin{matrix} - Div H + Π + H = D_{t} Ξ & and Π \geq 0 in B_{0} \\ where {H = H}^{p}, \\ - \overset{\land}{Div} \overset{\land}{H} + \overset{\land}{Π} + \overset{\land}{H} = D_{t} \overset{\land}{Ξ} & and \overset{\land}{Π} \geq 0 in S_{0} \\ where \overset{\land}{H} = {\overset{\land}{H}}^{p} + H \cdot N \\ - \bar{Div} \bar{H} + \bar{Π} + \bar{H} = D_{t} \bar{Ξ} & and \bar{Π} \geq 0 in I_{0} \\ where \bar{H} = {\bar{H}}^{p} - [[H]] \cdot \bar{N,} \\ - \tilde{Div} \tilde{H} + \tilde{Π} + \tilde{H} = D_{t} \tilde{Ξ} & and \tilde{Π} \geq 0 in C_{0} \\ where \tilde{H} = {\tilde{H}}^{p} + \bar{H} \cdot \tilde{N_{i}} - [[\overset{\land}{H}]] \cdot \tilde{N_{s}} \end{matrix}

(30)

The (bulk) Cauchy entropy flux vector h and the surface, interface, and curve tangential Cauchy entropy flux vectors, denoted $\overset{\land}{h}$ ⁠, $\bar{h}$ ⁠, and $\tilde{h}$ ⁠, respectively, are related to their corresponding Piola entropy flux vectors via the Piola transforms (8) and (9).

Following the same procedures employed in the previous sections, the local balances of entropy for the various parts of the body in terms of spatial quantities are

\begin{matrix} - div (h + ξ v) + π + h = d_{t} ξ & and π \geq 0 in B_{t} \\ where h = h^{p}, \\ - \overset{\land}{div} (\overset{\land}{h} + \overset{\land}{ξ} \overset{\land}{v}) + \overset{\land}{π} + \overset{\land}{h} = d_{t} \overset{\land}{ξ} & and \overset{\land}{π} \geq 0 in S_{t} \\ where \overset{\land}{h} = {\overset{\land}{h}}^{p} + h \cdot n, \\ - \bar{div} (\bar{h} + \bar{ξ} \bar{v}) + \bar{π} + \bar{h} = d_{t} \bar{ξ} & and \bar{π} \geq 0 in I_{t} \\ where \bar{h} = {\bar{h}}^{p} - [[h]] \cdot \bar{n,} \\ - \tilde{div} (\tilde{h} + \bar{ξ} \bar{v}) + \tilde{π} + \tilde{h} = d_{t} \tilde{ξ} & and \tilde{π} \geq 0 in C_{t} \\ where \tilde{h} = \tilde{h^{p}} + \bar{h} \cdot {\tilde{n}}_{i} - [[\overset{\land}{h}]] \cdot {\tilde{n}}_{s} \end{matrix}

(31)

where the bulk prescribed entropy source, entropy and entropy production per unit current volume in $B_{t}$ are denoted $h^{p} : = {jH}^{p}, ξ : = j Ξ, and π : = j Π$ ⁠, respectively. In an identical fashion to the bulk, the prescribed surface, interface, and curve entropy sources are denoted ${\overset{\land}{h}}^{p} : = \overset{\land}{j} {\overset{\land}{H}}^{p}, {\bar{h}}^{p} : = \bar{j} {\bar{H}}^{p}, and {\tilde{h}}^{p} : = \tilde{j} {\tilde{H}}^{p}$ ⁠, respectively. Similarly, $\overset{\land}{ξ} : = \overset{\land}{j} \overset{\land}{Ξ}, \bar{ξ} : = \bar{j} \bar{Ξ}$ ⁠, and $\tilde{ξ} : = \tilde{j} \tilde{Ξ}$ denote the surface, interface, and curve entropies, respectively. Finally, the prescribed surface, interface, and curve entropy productions are denoted $\overset{\land}{π} : = \overset{\land}{j} \overset{\land}{Π}, \bar{π} : = \bar{j} \bar{Π}, and \tilde{π} : = \tilde{j} \tilde{Π}$ ⁠, respectively.

To proceed, the commonly made assumption that the entropy fluxes are the corresponding heat fluxes divided by the corresponding absolute temperatures is used, that is:

\begin{matrix} H : = Q Θ^{- 1}, & \overset{\land}{H} : = \overset{\land}{Q} {\overset{\land}{Θ}}^{- 1}, & \bar{H} : = \bar{Q} {\bar{Θ}}^{- 1}, & \tilde{H} : = \tilde{Q} {\tilde{Θ}}^{- 1}, \\ H^{p} : = Q^{p} Θ^{- 1}, & {\overset{\land}{H}}^{p} : = {\overset{\land}{Q}}^{p} {\overset{\land}{Θ}}^{- 1}, & {\bar{H}}^{p} : = {\bar{Q}}^{p} {\bar{Θ}}^{- 1}, & {\tilde{H}}^{p} : = {\tilde{Q}}^{p} {\tilde{Θ}}^{- 1} \end{matrix}

(32)

The variables $Θ$ > 0, $\overset{\land}{Θ}$ > 0, $\bar{Θ}$ > 0, and $\tilde{Θ}$ > 0 denote the absolute temperatures in the bulk, on the surface, interface, and curve, respectively. Note that the assumption (32) is not true in general but holds for simple thermodynamic processes (see, e.g., Refs. [30,106,107]). The validity and necessity of this assumption should be explored using the procedure outlined in Müller [108] and later works.

The dissipation power for the various parts of the body are denoted

D, \overset{\land}{D}

,

\bar{D}

⁠, and

\tilde{D}

and are defined by

D : = Θ Π \geq 0, \overset{\land}{D} : = \overset{\land}{Θ} \overset{\land}{Π} \geq 0, \bar{D} : = \bar{Θ} \bar{Π} \geq 0, \tilde{D} : = \tilde{Θ} \tilde{Π} \geq 0

Substituting these relations into the balances of entropy for the various parts of the body yields the localized dissipation inequalities

\begin{matrix} D = Θ D_{t} Ξ + Div Q - Q \cdot Grad \ln Θ - Q^{p} \geq 0 in B_{0}, \overset{\land}{D} = \overset{\land}{Θ} D_{t} \overset{\land}{Ξ} + \overset{\land}{Div} \overset{\land}{Q} - \overset{\land}{Q} \cdot \overset{\land}{Grad} \ln \overset{\land}{Θ} - [{\overset{\land}{Q}}^{p} + \frac{\overset{\land}{Θ}}{Θ} Q \cdot N] \geq 0 in S_{0}, \bar{D} = \bar{Θ} D_{t} \bar{Ξ} + \bar{Div} \bar{Q} - \bar{Q} \cdot \bar{Grad} \ln \bar{Θ} - [{\bar{Q}}^{p} - \bar{Θ} [[\frac{Q}{Θ}]] \cdot \bar{N}] \geq 0 in I_{0}, \tilde{D} = \tilde{Θ} D_{t} \tilde{Ξ} + \tilde{Div} \tilde{Q} - \tilde{Q} \cdot \tilde{Grad} \ln \tilde{Θ} - [{\tilde{Q}}^{p} + \frac{\tilde{Θ}}{\bar{Θ}} \bar{Q} \cdot {\tilde{N}}_{i} - \tilde{Θ} [[\frac{\overset{\land}{Q}}{\overset{\land}{Θ}}]] \cdot {\tilde{N}}_{s}] \geq 0 in C_{0} \end{matrix}

(33)

The localized dissipation inequalities can be reformulated using the energy balance Eq. as

\begin{matrix} D = Θ D_{t} Ξ - D_{t} E + P : Grad V - Q \cdot Gradln Θ & \geq 0 in B_{0}, \\ \hat{D} = \hat{Θ} D_{t} \hat{Ξ} - D_{t} \hat{E} + \hat{P} : \hat{Grad} \hat{V} - \hat{Q} \cdot \hat{Grad} ln Θ + [1 - \frac{\hat{Θ}}{Θ}] Q \cdot N & \geq 0 in S_{0}, \\ \bar{D} = \bar{Θ} D_{t} \bar{Ξ} - D_{t} \bar{E} + \bar{P} : \bar{Grad} \bar{V} - \bar{Q} \cdot \bar{Grad} ln \bar{Θ} + [\bar{Θ} 〚 \frac{Q}{Θ} 〛 - 〚 Q 〛] \bar{\cdot N} & \geq 0 in I_{0}, \\ \tilde{D} = \tilde{Θ} D_{t} \tilde{Ξ} - D_{t} \tilde{E} + \tilde{P} : \tilde{Grad} \tilde{V} - \tilde{Q} \cdot \tilde{Grad} ln \tilde{Θ} + [\tilde{Θ} 〚 \frac{\hat{Q}}{\hat{Θ}} 〛 - 〚 \hat{Q} 〛] \cdot \tilde{N_{s}} + [1 - \frac{\tilde{Θ}}{\bar{Θ}}] \bar{Q} \cdot \tilde{N_{i}} & \geq 0 in C_{0} \end{matrix}

(34)

Next, the Helmholtz energy, obtained as a Legendre transformation of the internal energy in terms of entropy and temperature, is introduced. Separate Helmholtz energies in the bulk, on the surface, interface, and curve, denoted

Ψ

⁠,

\overset{\land}{Ψ}

⁠,

\bar{Ψ}

⁠, and

\tilde{Ψ}

⁠, respectively, are permitted and are defined by

Ψ : = E - Θ Ξ, \overset{\land}{Ψ} : = \overset{\land}{E} - \overset{\land}{Θ} \overset{\land}{Ξ}, \bar{Ψ} : = \bar{E} - \bar{Θ} \bar{Ξ}, \tilde{Ψ} : = \tilde{E} - \tilde{Θ} \tilde{Ξ}

(35)

From the definitions of the Helmholtz energies, the extended Clausius–Duhem dissipation inequalities can be obtained as

\begin{matrix} D = P : Grad V - D_{t} Ψ - Ξ D_{t} Θ - Q \cdot Gradln Θ & \geq 0 in B_{0,} \\ \hat{D} = \hat{P} : \hat{Grad} \hat{V} - D_{t} \hat{Ψ} - \hat{Ξ} D_{t} \hat{Θ} - \hat{Q} \cdot \hat{Grad} ln \hat{Θ} + [1 - \frac{\hat{Θ}}{Θ}] Q \cdot N & \geq 0 in S_{0,} \\ \bar{D} = \bar{P} : \bar{Grad} \bar{V} - D_{t} \bar{Ψ} - \bar{Ξ} D_{t} \bar{Θ} - \bar{Q} \cdot \bar{Grad} ln \bar{Θ} + Q [\bar{Θ} 〚 \frac{Q}{Θ} 〛 - 〚 Q 〛] \bar{\cdot N} & \geq 0 in I_{0,} \\ \tilde{D} = \tilde{P} : \tilde{Grad} \tilde{V} - D_{t} \tilde{Ψ} - \tilde{Ξ} D_{t} \tilde{Θ} - \tilde{Q} \cdot \tilde{Grad} ln \tilde{Θ} + [\tilde{Θ} 〚 \frac{\hat{Q}}{\hat{Θ}} 〛 - 〚 \hat{Q} 〛] \cdot \tilde{N_{s}} + [1 - \frac{\tilde{Θ}}{\bar{Θ}}] \bar{Q} \cdot \tilde{N_{i}} & \geq 0 in C_{0} \end{matrix}

(36)

Thermohyperelastic Helmholtz Energy

To exploit the dissipation inequalities further, a thermohyperelastic constitutive model is specified by selecting a particular list of arguments for the Helmholtz energies. For the case of thermohyperelastic behavior, the arguments of the Helmholtz energies are chosen as follows:

Ψ : = Ψ (F, Θ), \overset{\land}{Ψ} : = \overset{\land}{Ψ} (\overset{\land}{F}, \overset{\land}{Θ}, n), \bar{Ψ} : = \bar{Ψ} (\bar{F}, \bar{Θ}, \bar{n}), \tilde{Ψ} : = \tilde{Ψ} (\tilde{F}, \tilde{Θ}, \tilde{n})

(37)

Note that the surface and interface Helmholtz energies are allowed to depend on their corresponding normals in the spatial configuration to capture possible anisotropic behavior. The anisotropy of the curve, however, is modeled by dependence of the curve Helmholtz energy on its tangent in the spatial configuration. It should be emphasized that in deriving the balance equations in Sec. 3, no isotropy assumptions were made.

Constitutive Equations

Exploiting the dissipation inequalities in the various parts of the body using a Coleman–Noll-like procedure yields the constitutive laws

\begin{matrix} P = \frac{\partial Ψ}{\partial F}, & Ξ = - \frac{\partial Ψ}{\partial Θ} & in B_{0,} \\ \overset{\land}{P} = \frac{\partial \overset{\land}{Ψ}}{\partial \overset{\land}{F}} - n \otimes \overset{\land}{S} & with & \overset{\land}{S} : = \frac{\partial \overset{\land}{ψ}}{\partial n} \cdot \overset{\land}{i} \cdot \overset{\land}{cof} \overset{\land}{F}, & \overset{\land}{Ξ} = - \frac{\partial \overset{\land}{Ψ}}{\partial \overset{\land}{Θ}} & in S_{0,} \\ \bar{P} = \frac{\partial \bar{Ψ}}{\partial \bar{F}} - \bar{n} \otimes \bar{S} & with & \bar{S} : = \frac{\partial \bar{ψ}}{\partial \bar{n}} \cdot \bar{i} \cdot \bar{cof} \bar{F}, & \bar{Ξ} = - \frac{\partial \bar{Ψ}}{\partial \bar{Θ}} & in I_{0}, \\ \tilde{P} = \frac{\partial \tilde{Ψ}}{\partial \tilde{F}} - \tilde{s} \otimes \tilde{N} & with & \tilde{s} : = \frac{\partial \tilde{ψ}}{\partial \tilde{n}} \cdot [i - \tilde{i}], & \tilde{Ξ} = - \frac{\partial \tilde{Ψ}}{\partial \tilde{Θ}} & in C_{0} \end{matrix}

(38)

where the Helmholtz energies in the spatial configuration, i.e., per unit deformed area for the surface and interface and per unit deformed length for the curve, are defined by $\overset{\land}{ψ} : = \overset{\land}{j} \overset{\land}{Ψ}, \bar{ψ} : = \bar{j} \bar{Ψ},$ and $\tilde{ψ} : = \tilde{j} \tilde{Ψ}$ ⁠, respectively. The quantities $\overset{\land}{S}$ ⁠, $\bar{S}$ ⁠, and $\tilde{s}$ denote the surface shear, interface shear, and curve shear, respectively (see Refs. [105,109]). Note that the terms $n \otimes \overset{\land}{S}$ ⁠, $\bar{n} \otimes \bar{S}$ ⁠, and $\tilde{s} \otimes \tilde{N}$ map from the tangent spaces to the material surface, interface, and curve, to the normals to the spatial surface, interface, and curve, respectively.

Reduced Dissipation Inequalities

Inserting the constitutive relations derived in Sec. 4.1 in the dissipation inequalities , results in the following reduced dissipation inequalities:

\begin{matrix} D = - Q \cdot Grad ln Θ & \geq 0 in B_{0}, \\ \hat{D} = \underset{{\hat{D}}_{∥}}{\underset{︸}{- \hat{Q} \cdot \hat{Grad} ln \hat{Θ}}} + \underset{{\hat{D}}_{⊥}}{\underset{︸}{[1 - \frac{\hat{Θ}}{Θ}] Q \cdot N}} & \geq 0 in S_{0}, \\ \bar{D} = \underset{{\bar{D}}_{∥}}{\underset{︸}{- \bar{Q} \cdot \bar{Grad} ln \bar{Θ}}} + \underset{{\bar{D}}_{⊥}}{\underset{︸}{[\bar{Θ} 〚 \frac{Q}{Θ} 〛 - 〚 Q 〛] \cdot \bar{N}}} & \geq 0 in I_{0}, \\ \tilde{D} = \underset{{\tilde{D}}_{∥}}{\underset{︸}{- \tilde{Q} \cdot \tilde{Grad} ln \tilde{Θ}}} + \underset{{\tilde{D}}_{⊥}}{\underset{︸}{[\tilde{Θ} 〚 \frac{\hat{Q}}{\hat{Θ}} 〛 - 〚 \hat{Q} 〛] \cdot \tilde{N_{s}} + [1 - \frac{\tilde{Θ}}{\bar{Θ}}] \cdot \bar{Q} \cdot \tilde{N_{i}}}} & \geq 0 in C_{0} \end{matrix}

(39)

Note that the first terms in Eq. (39)_2–4 represent the dissipation along the surface, interface and curve and are denoted

{\overset{\land}{D}}_{∥}

⁠,

{\bar{D}}_{∥},

and

{\tilde{D}}_{∥}

⁠, respectively. In contrast, the remaining terms represent the dissipation across the surface, interface, and curve, and are denoted as

{\overset{\land}{D}}_{⊥}

⁠,

{\bar{D}}_{⊥}

⁠, and

{\tilde{D}}_{⊥}

⁠, respectively.

Dissipation Inequalities in the Bulk and Along the Surface, Interface, and Curve

In order to partially ensure thermodynamic consistency by satisfying the dissipation inequalities along the surface, interface, and curve, and in the spirit of Fourier's law, the heat flux vectors are assumed to be of the form

\begin{matrix} Q = - J F^{- 1} \cdot k \cdot F^{- T} \cdot Grad Θ in B_{0}, \overset{\land}{Q} = - \overset{\land}{J} {\overset{\land}{F}}^{- 1} \cdot \overset{\land}{k} \cdot {\overset{\land}{F}}^{- T} \cdot \overset{\land}{Grad} \overset{\land}{Θ} in S_{0}, \bar{Q} = - \bar{J} {\bar{F}}^{- 1} \cdot \bar{k} \cdot {\bar{F}}^{- T} \cdot \bar{Grad} \bar{Θ} in I_{0}, \tilde{Q} = - \tilde{J} {\tilde{F}}^{- 1} \cdot \tilde{k} \cdot {\tilde{F}}^{- T} \cdot \tilde{Grad} \tilde{Θ} in C_{0} \end{matrix}

(40)

where k, $\overset{\land}{k}$ ⁠, $\bar{k}$ ⁠, and $\tilde{k}$ denote the bulk, surface, interface, and curve positive semidefinite spatial thermal conductivity tensors, respectively.

Dissipation Inequalities Across the Surface, Interface, and Curve

Assuming that the dissipation inequalities in the bulk and along the surface, interface, and curve are satisfied, the further reduced dissipation inequalities across the surface, interface, and curve are given by

\begin{matrix} {\overset{\land}{D}}_{⊥} = \overset{\land}{Θ} [\frac{1}{\overset{\land}{Θ}} - \frac{1}{Θ}] Q \cdot N & \geq 0, \\ {\bar{D}}_{⊥} = \bar{Θ} [[[\frac{Q}{Θ}]] - \frac{[[Q]]}{\bar{Θ}}] \cdot \bar{N} & \geq 0, \\ {\tilde{D}}_{⊥} = \tilde{Θ} [[[\frac{\overset{\land}{Q}}{\overset{\land}{Θ}}]] - \frac{[[\overset{\land}{Q}]]}{\tilde{Θ}}] \cdot {\tilde{N}}_{s} + \tilde{Θ} [\frac{1}{\tilde{Θ}} - \frac{1}{\bar{Θ}}] \bar{Q} \cdot {\tilde{N}}_{i} & \geq 0 \end{matrix}

(41)

The requirements for these inequalities to be satisfied are now discussed.

Surface.

The most obvious choice to sufficiently satisfy the further reduced dissipation inequality on the surface ₁ is

\overset{\land}{Θ} = {Θ |}_{S_{0}}

(42)

that is, to impose so-called thermal slavery on the surface. It should be emphasized that the thermal slavery condition on the surface is not an a priori assumption but arises, rather, as a natural consequence of the thermodynamics. Alternatively McBride et al. [77] propose a Robin-type boundary condition between the bulk and surface.

Interface.

The conditions to satisfy the further reduced dissipation inequality on the interface Eq. ₂ are now presented. Equation ₂ can be reformulated⁴ using either of the two identities

\begin{matrix} [[\frac{Q}{Θ}]] = [[Q]] {{\frac{1}{Θ}}} + {{Q}} [[\frac{1}{Θ}]], \frac{[[Q]]}{\bar{Θ}} = [[\frac{Q}{Θ}]] \frac{{{Θ}}}{\bar{Θ}} + {{\frac{Q}{Θ}}} \frac{[[Θ]]}{\bar{Θ}} \end{matrix}

(43)

as

\begin{matrix} \Rightarrow {\bar{D}}_{⊥} = \bar{Θ} [{{\frac{1}{Θ}}} - \frac{1}{\bar{Θ}}] [[Q]] \cdot \bar{N} + \bar{Θ} [[\frac{1}{Θ}]] {{Q}} \cdot \bar{N} \geq 0 ({using Eq. (43)}_{1}), \Rightarrow {\bar{D}}_{⊥} = [\bar{Θ} - {{Θ}}] [[\frac{Q}{Θ}]] \cdot \bar{N} - [[Θ]] {{\frac{Q}{Θ}}} \cdot \bar{N} \geq 0 ({using Eq. (43)}_{2}) \end{matrix}

(44)

The further reduced dissipation inequality on the interface (41)₂ is satisfied if either of the two equivalent forms in Eq. (44) is satisfied.

By defining the coldness

ϑ

as the temperature inverse, i.e.,

ϑ : = 1 / Θ

⁠, Eq. can be expressed in an alternative form as

\begin{matrix} \Rightarrow - {\bar{D}}_{⊥} = [\bar{ϑ} - {{ϑ}}] - \frac{[[Q]]}{\bar{ϑ}} \cdot \bar{N} - [[ϑ]] \frac{{{Q}}}{\bar{ϑ}} \cdot \bar{N} \leq 0 ({corresponding to Eq. (44)}_{1}), \Rightarrow {\bar{D}}_{⊥} = [\bar{Θ} - {{Θ}}] [[\frac{Q}{Θ}]] \cdot \bar{N} - [[Θ]] {{\frac{Q}{Θ}}} \cdot \bar{N} \geq 0 ({corresponding to Eq. (44)}_{2}) \end{matrix}

(45)

Coldness-based discussion.

The sufficient conditions to satisfy Eq. ₁ are examined first. The inequality is satisfied if the following conditions hold:

[\bar{ϑ} - {{ϑ}}] [[Q]] \cdot \bar{N} \leq 0 and [[ϑ]] {{Q}} \cdot \bar{N} \geq 0

(46)

A natural choice to satisfy the first relation ₁ is a Fourier-type law for the jump of the heat flux across the interface, that is,

\bar{ϑ} - {{ϑ}} = - {\bar{α}}_{ϑ} [[Q]] \cdot \bar{N}

(47)

where

{\bar{α}}_{ϑ}

≥ 0 is a coldness sensitivity coefficient across the interface. To satisfy the second condition ₂ a Fourier-type law for the average heat flux across the interface can be chosen, that is,

[[ϑ]] = {\bar{r}}_{ϑ} {{Q}} \cdot \bar{N}

(48)

where ${\bar{r}}_{ϑ}$ ≥ 0 is a heat flux resistance coefficient across the interface.

Remark. For a thermal interface, the jump in the normal component of the heat flux is given by Eq. ₃ as

[[Q]] \cdot \bar{N} = - \bar{Div} \bar{Q}

(49)

which is the thermal Young–Laplace equation on the interface. Equation (49) states that the normal jump of heat flux is equal to the negative of the divergence of the heat flux along the interface. In particular, in the absence of heat flux along the thermal interface the normal jump of the heat flux, i.e., the left-hand side of the Eq. (49) vanishes, i.e., the standard interface, and, therefore, Eq. (46)₁ is trivially satisfied.□

Remark. A sufficient condition for a thermal interface to satisfy Eq. (46)₁ is to choose the interface coldness equal to the average of the coldness across the interface. This conclusion shall be compared with Daher and Maugin [47] who viewed it as a necessity.□

Remark. Using the relation $[[ϑ]] = - {{ϑ}} [[Θ]] / {{Θ}}$ ⁠, the condition (46)₂ can be reformulated as $[[Θ]] {{Q}} \cdot \bar{N} \leq 0$ that can be satisfied by a Fourier-type law for the average heat flux across the interface $[[Θ]] = - {\bar{r}}_{Q} {{Q}} \cdot \bar{N}$ where ${\bar{r}}_{Q} \geq 0$ denotes the Kapitza heat flux resistance coefficient.□

Temperature-based discussion

The sufficient conditions to satisfy Eq. ₂ are now examined. The inequality is satisfied if the following conditions hold:

[\bar{Θ} - {{Θ}}] [[\frac{Q}{Θ}]] \cdot \bar{N} \geq 0 and [[Θ]] {{\frac{Q}{Θ}}} \cdot \bar{N} \leq 0

(50)

A natural choice to satisfy condition ₁ is a Fourier-type law for the jump in the entropy flux across the interface, that is,

\bar{Θ} - {{Θ}} = {\bar{α}}_{Θ} [[H]] \cdot \bar{N}

(51)

where

{\bar{α}}_{Θ} \geq 0

is a temperature sensitivity coefficient across the interface. Likewise, to satisfy the condition ₂ a Fourier-type law for the average entropy flux across the interface can be chosen, that is,

[[Θ]] = - {\bar{r}}_{Θ} {{H}} \cdot \bar{N}

(52)

where ${\bar{r}}_{Θ} \geq 0$ is the entropy flux resistance coefficient across the interface.

Remark. For a standard interface, the jump of the entropy flux; in contrast to the jump of the heat flux, does not necessarily vanish according to Eq. (30)₃. That is, for a standard interface Eq. (50)₁ is not trivially satisfied. Thus, a sufficient condition to satisfy Eq. (50)₁ is to choose the interface temperature equal to the average of the temperature across the interface.□

Remark. For a highly conducting interface the jump of the temperature across the interface vanishes, which corresponds to zero normal dissipation across the interface, i.e.,

{\overset{\land}{D}}_{⊥} = 0

⁠. For this case

\bar{Θ} : = {{Θ}}

or

1 / \bar{Θ} : = {{1 / Θ}}

⁠. Therefore, for a highly conducting interface, the interface temperature (resp. coldness) is trivially the average of the temperature (resp. coldness) across the interface since

{Θ |}_{I_{0}^{+}} = {Θ |}_{I_{0}^{-}} = Θ

⁠.□

Remark. Using the relation $[[Θ]] = - {{Θ}} [[ϑ]] / {{ϑ}}$ ⁠, the condition (50)₂ can be reformulated as, $[[ϑ]] {{H}} \cdot \bar{N} \geq 0$ ⁠, which can be satisfied by a Fourier-type law for the average entropy flux across the interface $[[ϑ]] = {\bar{r}}_{H} {{H}} \cdot \bar{N}$ where ${\bar{r}}_{H} \geq 0$ denotes an entropy flux resistance coefficient.□

Summary.

Thus, in summary, the sufficient conditions to satisfy the further reduced dissipation inequality on the interface ₂ are

\bar{ϑ} - {{ϑ}} = - {\bar{α}}_{ϑ} [[Q]] \cdot \bar{N} and ([[ϑ]] = {\bar{r}}_{ϑ} {{Q}} \cdot \bar{N} or [[Θ]] = - {\bar{r}}_{Q} {{Q}} \cdot \bar{N})

or

\bar{Θ} - {{Θ}} = {\bar{α}}_{Θ} [[H]] \cdot \bar{N} and ([[Θ]] = - {\bar{r}}_{Θ} {{H}} \cdot \bar{N} or [[ϑ]] = {\bar{r}}_{H} {{H}} \cdot \bar{N})

Curve.

The further reduced dissipation inequality on the curve ₃ has two parts. The first term involves jumps and formally resembles the further reduced dissipation inequality on the interface ₂. The second term formally resembles the further reduced dissipation inequality on the surface ₁. The sufficient condition to satisfy the further reduced dissipation inequality on the curve is to ensure that each of these two terms are non-negative. Analogous to the interface, we have two sets of conditions and one of them has to be satisfied to ensure that the first term in Eq. ₃ is non-negative, that is,

\frac{1}{\tilde{Θ}} = {{\frac{1}{\overset{\land}{Θ}}}} and [[\overset{\land}{Θ}]] = - {\tilde{r}}_{Q} {{\overset{\land}{Q}}} \cdot {\tilde{N}}_{s}

(53)

or

\tilde{Θ} = {{\overset{\land}{Θ}}} and [[\frac{1}{\overset{\land}{Θ}}]] = {\tilde{r}}_{H} {{\overset{\land}{H}}} \cdot {\tilde{N}}_{s}

(54)

where ${\tilde{r}}_{Q} \geq 0$ and ${\tilde{r}}_{H} \geq 0$ are heat flux and entropy flux resistance coefficients, respectively. Analogous to the surface, we can impose the thermal slavery on the curve, i.e., $\tilde{Θ} = {\bar{Θ} |}_{C_{0}}$ ⁠, as the most obvious way to guarantee that the second term in Eq. (41)₃ is non-negative. A second novel option is to impose a constraint between the temperatures on the curve $\tilde{Θ}$ and interface $\bar{Θ}$ and the heat flux $\bar{Q} \cdot {\tilde{N}}_{i}$ that resembles a Robin-type boundary condition between the curve and the interface.

Temperature Evolution Equations

The thermohyperelastic constitutive equations together with the local energy balance equations give the temperature evolution equations in the bulk, on the surface, interface, and curve as

\begin{matrix} c_{F} D_{t} Θ = - Div Q + Q + Θ \frac{\partial P}{\partial Θ} : D_{t} F in B_{0}, \\ {\overset{\land}{c}}_{\overset{\land}{F}} D_{t} \overset{\land}{Θ} = - \overset{\land}{Div} \overset{\land}{Q} + \overset{\land}{Q} + \overset{\land}{Θ} \frac{\partial \overset{\land}{P}}{\partial \overset{\land}{Θ}} : D_{t} \overset{\land}{F} in S_{0}, \\ {\bar{c}}_{\bar{F}} D_{t} \bar{Θ} = - \bar{Div} \bar{Q} + \bar{Q} + \bar{Θ} \frac{\partial \bar{P}}{\partial \bar{Θ}} : D_{t} \bar{F} in I_{0}, \\ {\tilde{c}}_{\tilde{F}} D_{t} \tilde{Θ} = - \tilde{Div} \tilde{Q} + \tilde{Q} + \tilde{Θ} \frac{\partial \tilde{P}}{\partial \tilde{Θ}} : D_{t} \tilde{F} in C_{0} \end{matrix}

(55)

whereby

c_{F}

,

{\overset{\land}{c}}_{\overset{\land}{F}}

⁠,

{\bar{c}}_{\bar{F}}

⁠, and

{\tilde{c}}_{\tilde{F}}

denote the heat capacity coefficients in the bulk, on the surface, interface, and curve, respectively, defined as

\begin{matrix} c_{F} : = - Θ \frac{\partial^{2} Ψ}{\partial Θ \partial Θ}, {\overset{\land}{c}}_{\overset{\land}{F}} : = - \overset{\land}{Θ} \frac{\partial^{2} \overset{\land}{Ψ}}{\partial \overset{\land}{Θ} \partial \overset{\land}{Θ}}, {\bar{c}}_{\bar{F}} : = - \bar{Θ} \frac{\partial^{2} \bar{Ψ}}{\partial \bar{Θ} \partial \bar{Θ}}, {\tilde{c}}_{\tilde{F}} : = - \tilde{Θ} \frac{\partial^{2} \tilde{Ψ}}{\partial \tilde{Θ} \partial \tilde{Θ}} \end{matrix}

(56)

The derivation of the interface temperature evolution Eq. (55)₃ is given in Appendix D.

Weak Formulation

In order to establish a principle of virtual work like statement and as a prerequisite for finite element method approximations (to be presented in a subsequent contribution), the mechanical weak form, i.e., the weak form of the linear momentum balance Eq. (21) and the thermal weak form, i.e., the weak form of the temperature evolution Eq. (56), are required.

Weak Formulation—Mechanical

To derive the mechanical weak form, the local linear momentum balance equations for the various parts of the body (21)_1–4 are tested from left with vector-valued test functions

δ ϕ \in H_{0}^{1} (B_{0})

⁠,

δ \overset{\land}{ϕ} \in H_{0}^{1} (S_{0})

⁠,

δ \bar{ϕ} \in H_{0}^{1} (I_{0})

⁠, and

δ \tilde{ϕ} \in H_{0}^{1} (C_{0})

⁠, respectively. The result is integrated over the corresponding domains in the material configuration to give the global weak form of the balance of linear momentum as

Find

ϕ \in H^{1} (B_{0})

⁠,

\overset{\land}{ϕ} \in H^{1} (S_{0})

⁠,

\bar{ϕ} \in H^{1} (I_{0})

⁠, and

\tilde{ϕ} \in H^{1} (C_{0})

such that

\begin{matrix} \int_{B_{0}} δ ϕ \cdot [Div P + B^{p}] d V + \int_{S_{0}} δ \overset{\land}{ϕ} \cdot [\overset{\land}{Div} \overset{\land}{P} + [{\overset{\land}{B}}^{p} - P \cdot N]] d A \\ + \int_{I_{0}} δ \bar{ϕ} \cdot [\bar{Div} \bar{P} + [{\bar{B}}^{p} + [[P]] \cdot \bar{N}]] d A \\ + \int_{C_{0}} δ \tilde{ϕ} \cdot [\tilde{Div} \tilde{P} + [{\tilde{B}}^{p} - \bar{P} \cdot {\tilde{N}}_{i} + [[\overset{\land}{P}]] \cdot {\tilde{N}}_{s}]] d L = 0, \\ \begin{matrix} \forall δ ϕ \in H_{0}^{1} (B_{0}), & \forall δ \overset{\land}{ϕ} \in H_{0}^{1} (S_{0}), & \forall δ \bar{ϕ} \in H_{0}^{1} (I_{0}), & \forall δ \tilde{ϕ} \in H_{0}^{1} (C_{0}) \end{matrix} \end{matrix}

Using Eq. _1–4 and the extended divergence theorems , the orthogonality properties of the Piola stress measures in the various parts of the body (which cause the integrals containing the curvature terms to vanish) and from the kinematic slavery condition

\begin{matrix} {{δ ϕ}} |_{I_{0}} = δ \bar{ϕ}, & {{δ \overset{\land}{ϕ}}} |_{C_{0}} = δ \tilde{ϕ}, & {δ ϕ |}_{S_{0}} = δ \overset{\land}{ϕ}, & {δ \bar{ϕ} |}_{C_{0}} = δ \tilde{ϕ} \end{matrix}

(57)

the weak form of the balance of linear momentum becomes

\begin{matrix} \int_{B_{0}} P : Grad δ ϕ d V + \int_{S_{0}} \overset{\land}{P} : \overset{\land}{Grad} δ \overset{\land}{ϕ} d A + \int_{I_{0}} \bar{P} : \bar{Grad} δ \bar{ϕ} d A + \int_{C_{0}} \tilde{P} : \tilde{Grad} δ \tilde{ϕ} d L - \int_{B_{0}} δ ϕ \cdot B^{p} d V - \int_{S_{0}} δ \overset{\land}{ϕ} \cdot {\overset{\land}{B}}^{p} d A - \int_{I_{0}} δ \bar{ϕ} \cdot {\bar{B}}^{p} d A - \int_{C_{0}} δ \tilde{ϕ} \cdot {\tilde{B}}^{p} d L = 0 (58) \end{matrix}

(58)

where the test functions satisfy the conditions given in Eq. (57).

Weak Formulation—Thermal

In order to derive the thermal weak form we proceed formally. The local temperature evolution equations for the various parts of the body (55)_1–4 are tested from left with scalar test functions

δ Θ \in H_{0}^{1} (B_{0})

⁠,

δ \overset{\land}{Θ} \in H_{0}^{1} (S_{0})

⁠,

δ \bar{Θ} \in H_{0}^{1} (I_{0})

⁠, and

δ \tilde{Θ} \in H_{0}^{1} (C_{0})

⁠, respectively. The result is integrated over the corresponding domains in the material configuration to give the global weak form of the temperature evolution equation as:

Find

Θ \in H^{1} (B_{0})

⁠,

\overset{\land}{Θ} \in H^{1} (S_{0})

⁠,

\bar{Θ} \in H^{1} (I_{0})

⁠, and

\tilde{Θ} \in H^{1} (C_{0})

such that

\begin{matrix} \int_{B_{0}} δ Θ [c_{F} D_{t} Θ + Div Q - Q^{p} - Θ P,_{Θ} : D_{t} F] d V \\ + \int_{S_{0}} δ \overset{\land}{Θ} [{\overset{\land}{c}}_{\overset{\land}{F}} D_{t} \overset{\land}{Θ} + \overset{\land}{Div} \overset{\land}{Q} - [{\overset{\land}{Q}}^{p} + Q \cdot N] - \overset{\land}{Θ} {\overset{\land}{P}}_{, \overset{\land}{Θ}} : D_{t} \overset{\land}{F}] d A \\ + \int_{I_{0}} δ \bar{Θ} [{\bar{c}}_{\bar{F}} D_{t} \bar{Θ} + \bar{Div} \bar{Q} - [{\bar{Q}}^{p} - [[Q]] \cdot \bar{N}] - \bar{Θ} {\bar{P}}_{, \bar{Θ}} : D_{t} \bar{F}] d A \\ + \int_{C_{0}} δ \tilde{Θ} [{\tilde{c}}_{\tilde{F}} D_{t} \tilde{Θ} + \tilde{Div} \tilde{Q} - [{\tilde{Q}}^{p} + \bar{Q} \cdot {\tilde{N}}_{i} - [[\overset{\land}{Q}]] \cdot {\tilde{N}}_{s}] \\ - \tilde{Θ} {\tilde{P}}_{, \tilde{Θ}} : D_{t} \tilde{F}] d L = 0, \\ \begin{matrix} \forall δ Θ \in H_{0}^{1} (B_{0}), & \forall δ \overset{\land}{Θ} \in H_{0}^{1} (S_{0}), & \forall δ \bar{Θ} \in H_{0}^{1} (I_{0}), & \forall δ \tilde{Θ} \in H_{0}^{1} (C_{0}) \end{matrix} \end{matrix}

(59)

Using Eq. _1–4 and the extended divergence theorems given in Eqs. , the orthogonality properties of the heat flux vectors for the various parts of the body (which cause the integrals containing the curvatures to vanish), Eq. , and the assumption of thermal slavery for the surface and the curve, i.e.,

\overset{\land}{Θ} = {Θ |}_{S_{0}}

and

\tilde{Θ} = {\bar{Θ} |}_{C_{0}}

so that

{δ Θ |}_{S_{0}} = δ \overset{\land}{Θ}, {δ \bar{Θ} |}_{C_{0}} = δ \tilde{Θ}

(60)

the global weak form of the temperature evolution equation becomes

\begin{matrix} - \int_{I_{0}} [[δ Θ]] {{Q}} \cdot \bar{N} + [{{δ Θ}} - δ \bar{Θ}] [[Q]] \cdot \bar{N} d A - \int_{C_{0}} [[δ \overset{\land}{Θ}]] {{\overset{\land}{Q}}} \cdot {\tilde{N}}_{s} + [{{δ \overset{\land}{Θ}}} - δ \tilde{Θ}] [[\overset{\land}{Q}]] \cdot {\tilde{N}}_{s} d L = \int_{B_{0}} Q \cdot Grad δ Θ - δ Θ c_{F} D_{t} Θ + δ Θ Q^{p} + δ Θ Θ P_{, Θ} : D_{t} F d V + \int_{S_{0}} \overset{\land}{Q} \cdot \overset{\land}{Grad} δ \overset{\land}{Θ} - δ \overset{\land}{Θ} {\overset{\land}{c}}_{\overset{\land}{F}} D_{t} \overset{\land}{Θ} + δ \overset{\land}{Θ} {\overset{\land}{Q}}^{p} + δ \overset{\land}{Θ} \overset{\land}{Θ} {\overset{\land}{P}}_{, \overset{\land}{Θ}} : D_{t} \overset{\land}{F} d A + \int_{I_{0}} \bar{Q} \cdot \bar{Grad} δ \bar{Θ} - δ \bar{Θ} {\bar{c}}_{\bar{F}} D_{t} \bar{Θ} + δ {\bar{Θ Q}}^{p} + δ {\bar{Θ Θ P}}_{, \bar{Θ}} : D_{t} \bar{F} d A + \int_{C_{0}} \tilde{Q} \cdot \tilde{Grad} δ \tilde{Θ} - δ \tilde{Θ} {\tilde{c}}_{\tilde{F}} D_{t} \tilde{Θ} + δ \tilde{Θ} {\tilde{Q}}^{p} + δ \tilde{Θ} \tilde{Θ} {\tilde{P}}_{, \tilde{Θ}} : D_{t} \tilde{F} d L \end{matrix}

(61)

where the test functions satisfy the conditions given in Eq. (60).

Comparison With Several Other Models

The objective of this section is to briefly derive several models given in the literature by simplifying the weak forms (58) and (61). In particular, the focus is on the thermal weak form (61). The following cases are examined:

standard thermomechanical solids [110]
highly-conducting interfaces [84]
Kapitza interfaces [83]
thermomechanical surfaces [93]
thermomechanical HC interfaces [86]
generalized Kapitza interfaces [111]

Note that none of the aforementioned models account for the contributions from the curve and, therefore, all the corresponding integrals vanish identically.

Standard Thermomechanical Solids

For standard thermomechanical solids with neither energetic nor thermal interfaces, surfaces, or curves, the left-hand side of Eq. and the integrals over the interface, surface, and curve vanish. Therefore, the thermal weak form reads

\begin{matrix} \int_{B_{0}} Q \cdot Grad δ Θ - δ Θ c_{F} D_{t} Θ + δ Θ Q^{p} + δ Θ Θ P_{, Θ} : D_{t} F d V = - \int_{S_{0}} δ \overset{\land}{Θ} {\overset{\land}{Q}}^{p} d A \end{matrix}

The numerical implementation of finite strain thermoelasticity is well documented in the literature (see, e.g., Ref. [110] and references therein).

Highly Conducting Interfaces

Thermal solids with highly conducting interfaces do not allow for a jump in the temperature field across the interface, i.e.,

[[Θ]] = 0

and

[[δ Θ]] = 0

⁠. Thus,

{{Θ}} = \bar{Θ}

and

{{δ Θ}} = δ \bar{Θ}

⁠. Therefore, the stationary thermal weak form in the absence of (bulk) heat sources reads

\int_{B_{0}} Q \cdot Grad δ Θ d V + \int_{I_{0}} \bar{Q} \cdot \bar{Grad} δ \bar{Θ} d A = - \int_{S_{0}} δ \overset{\land}{Θ} {\overset{\land}{Q}}^{p} d A

(62)

For details of the numerical implementation see, e.g., Yvonnet et al. [84] and the references therein.

Kapitza Interfaces

For the case of thermal solids containing Kapitza interfaces, the normal heat flux is continuous across the interface, i.e.,

[[Q]] \cdot \bar{N} = 0

⁠, but a jump in the temperature field is admissible via the relation

[[Θ]] = - {\bar{r}}_{Q} {{Q}} \cdot \bar{N}

⁠. Therefore, the stationary thermal weak form reads

\int_{B_{0}} Q \cdot Grad δ Θ d V - \int_{I_{0}} \frac{1}{{\bar{r}}_{Q}} [[δ Θ]] [[Θ]] d A = - \int_{B_{0}} δ Θ Q^{p} d V

(63)

The last term on the left-hand side vanishes for a standard interface; this assumption has been made by Yvonnet et al. [83].

Thermomechanical Surfaces

For thermomechanical solids with energetic surfaces, the left-hand side of Eq. and the integrals over the interface and curve vanish. Thus, the thermal weak form reads

\begin{matrix} \int_{B_{0}} Q \cdot Grad δ Θ - δ Θ c_{F} D_{t} Θ + δ Θ Q^{p} + δ Θ Θ P_{, Θ} : D_{t} F d V + \int_{S_{0}} \overset{\land}{Q} \cdot \overset{\land}{Grad} δ \overset{\land}{Θ} - δ \overset{\land}{Θ} {\overset{\land}{c}}_{\overset{\land}{F}} D_{t} \overset{\land}{Θ} + δ \overset{\land}{Θ} {\overset{\land}{Q}}^{p} + δ \overset{\land}{Θ} \overset{\land}{Θ} {\overset{\land}{P}}_{, \overset{\land}{Θ}} : D_{t} \overset{\land}{F} d A = 0 \end{matrix}

which is identical to the one given in Javili and Steinmann [93].

Thermomechanical Highly Conducting Interfaces

For thermomechanical solids with HC energetic interfaces, the left-hand side of Eq. vanishes since

[[Θ]] = 0

⁠,

[[δ Θ]] = 0

⁠,

{{Θ}} = \bar{Θ}

⁠, and

{{δ Θ}} = δ \bar{Θ}

⁠. Thus, the thermal weak form reads

\begin{matrix} \int_{B_{0}} Q \cdot Grad δ Θ - δ Θ c_{F} D_{t} Θ + δ Θ Q^{p} + δ Θ Θ P_{, Θ} : D_{t} F d V + \int_{S_{0}} δ \overset{\land}{Θ} {\overset{\land}{Q}}^{p} d A + \int_{I_{0}} \bar{Q} \cdot \bar{Grad} δ \bar{Θ} - δ \bar{Θ} {\bar{c}}_{\bar{F}} D_{t} \bar{Θ} + δ \bar{Θ} {\bar{Q}}^{p} + δ \bar{Θ} \bar{Θ} {\bar{P}}_{, \bar{Θ}} : D_{t} \bar{F} d A = 0 \end{matrix}

This model, as compared to Eq. (62), accounts for thermomechanical coupling in the bulk and on the interface and is not limited to the stationary case; see Javili et al. [86] for further details.

Generalized Kapitza Interfaces

The Kapitza interface is termed generalized in the sense that the interface is mechanically energetic in contrast to classical Kapitza interfaces. For the case of thermomechanical solids containing generalized Kapitza interfaces, a jump in the temperature field is admissible via the relation

[[Θ]] = - {\bar{r}}_{Q} {{Q}} \cdot \bar{N}

while

[[Q]] \cdot \bar{N} = 0

⁠. Therefore, the thermal weak form reads

\begin{matrix} \int_{B_{0}} Q \cdot Grad δ Θ - δ Θ c_{F} D_{t} Θ + δ Θ Q^{p} + δ Θ Θ {P,}_{Θ} : D_{t} F d V + \int_{S_{0}} δ \overset{\land}{Θ} {\overset{\land}{Q}}^{p} d A - \int_{I_{0}} \frac{1}{{\bar{r}}_{Q}} [[δ Θ]] [[Θ]] d A = 0 \end{matrix}

This model, as compared to Eq. (63), not only endows the interface with its own mechanically energetic structure but also accounts for thermomechanical coupling in the bulk and is not limited to the stationary case; see Javili et al. [111] for further details.

Material Modeling

The objective of this section is to elaborate various aspects of material (constitutive) modeling. From the perspective of material modeling, the surface and the interface are essentially identical. Furthermore, the derivations on the surface and interface capture all the key features of the material model for the curve. Therefore, without any loss of generality, we shall focus the majority of the subsequent discussion on the material modeling of the bulk and the surface. However, in order to provide sufficient background material to explain the numerical examples in Sec. 7, we also briefly present details on the thermomechanical modeling of interfaces. The influence of the surface on the overall response of a structure at the nanoscale is one of the key applications for models of surface elasticity in the literature. We also restrict attention to thermoelastic materials. Inelastic behavior at the nanoscale is recently reported in Ref. [112].

The material modeling of bulk materials is a mature field with many standard references (see, e.g., Ref. [100]). This is not the case for the surface. Modeling the constitutive response of the surface surrounding the bulk introduces additional complexity for the following reasons:

The surface deformation gradient $\overset{\land}{F}$ is rank deficient. Thus, the derivations of the surface stress measures and their linearizations are more complicated.
The surface is a two-dimensional manifold embedded in three-dimensional space. Thus, the admissible range for the surface material parameters differs from those of the bulk (see Refs. [92,113] for further details).
Surface tension, which corresponds to a constant energy per unit area of the spatial configuration, introduces a nonstandard term into the surface Helmholtz energy. The surface Helmholtz energy for the strain-free configuration no longer vanishes. Hence, the classical (bulk) growth conditions [114,115] are no longer valid.
The notions of the surface energy, surface tension and surface stress are often used inappropriately in the literature (for a detailed discussion and clarification see Ref. [25] and references therein).
The definition of the surface elasticity theory in the seminal work of Gurtin and Murdoch [32] has a serious defect. This error was reported by the authors in Ref. [116] but has been overlooked by many others.
Gurtin and Murdoch used the letter S to denote the Piola stress and called it the Piola–Kirchhoff stress. The letter S is widely used in the literature to denote the (second) Piola–Kirchhoff stress.

For these reasons, there exists considerable confusion and inconsistency in the literature on surface elasticity (e.g., Refs. [25,26,29,117] aim to report and clarify such issues). In order to clarify the theory and to present a self-contained and consistent framework, a Helmholtz energy for the surface is introduced based on measures of the finite deformation, and the resulting stresses derived. The vast majority of the literature on surface elasticity theory considers the small-strain setting. Thus, the stress measures are linearized at the material configuration to obtain their small-strain counterparts. The result of the linearization of the finite deformation stress measures needs to be interpreted carefully. It is shown here that the linearization of the surface Piola stress $\overset{\land}{P}$ at the material configuration is different from that of the surface Piola–Kirchhoff stress $\overset{\land}{S}$ ⁠. Recall that for the bulk the linearizations of the Piola stress P and the Piola–Kirchhoff stress S at the material configuration are identical.

First a standard hyperelastic material model for the bulk is introduced and corresponding Piola and Piola–Kirchhoff stresses given. Thereafter, in Sec. 6.2, the derivations of the surface Piola and Piola–Kirchhoff stresses corresponding to a hyperelastic material model for the surface are performed. The surface material parameters and their admissible ranges for the linear elastic case are also presented. Furthermore, in order to elucidate the role of surface tension and the dependence of the surface operators upon the curvature within a simplified setting, an example involving the Young–Laplace equation is given. Finally, the hyperelastic material models are extended to the case of thermohyperelasticity in Sec. 6.3.

Remark. The form of the Helmholtz energy on the interface, and indeed in the bulk, can be obtained from fundamental reasoning or from atomistic modeling (see, e.g., Refs. [25,118]). Dingreville and Qu [119] developed a semianalytic method to compute the surface elastic properties of crystalline materials. Moreover, an interface energy can be constructed using the surface Cauchy–Born hypothesis [56]. In Ref. [76] the surface elastic parameters are extracted from ab initio calculations. Similar strategies can be employed to extract the interface parameters. Thus, in general, thermal and mechanical constants can be obtained using inverse parameter identification.

Hyperelastic Material in the Bulk

The bulk response is assumed to be hyperelastic. Specifically, a neo-Hookean-type Helmholtz energy Ψ of the following form is chosen:

\begin{matrix} Ψ (F) = \frac{1}{2} λ \ln^{2} J + \frac{1}{2} μ [F : F - 3 - 2 \ln J] & with & J = Det F > 0 \end{matrix}

(64)

where the Helmholtz energy is characterized by the two Lamé constants μ and λ. Alternatively, and in order to guarantee objectivity, the Helmholtz energy is expressed in terms of the right Cauchy–Green stretch tensor

C : = F^{t} \cdot F

as

\begin{matrix} Ψ (C) = \frac{1}{2} λ \ln^{2} J + \frac{1}{2} μ [Tr C - 3 - 2 \ln J] & with & J = Det F = {[Det C]}^{\frac{1}{2}} \end{matrix}

where Tr {•} = {•}: I denotes the trace operator. The Piola stress P and Piola–Kirchhoff stress S follow from Eq. as

P (F) : = \partial_{F} Ψ (F) = λ \ln J F^{- t} + μ [F - F^{- t}]

(65)

S (C) : = 2 \partial_{C} Ψ (C) = λ \ln J C^{- 1} + μ [I - C^{- 1}]

(66)

It is of particular interest to linearize these two stress measures at the material configuration in order to obtain their small strain counterparts. In doing so, recall that F − I = Grad u where u denotes the displacement vector and the infinitesimal strain is defined by ε := [Grad u]^sym. The symmetric part of a second-order tensor ${•}$ can be obtained via ${•}^{sym} = I^{sym} : {•}$ where $I^{sym} : = (1 / 2) [I \bar{\otimes} I + I \underline{\otimes} I]$ denotes the fourth-order symmetric identity tensor. The fourth-order volumetric identity tensor is defined by $I^{vol} : = (1 / 3) I \otimes I$ ⁠. Furthermore, ${•}^{vol} : = {•} : I^{vol}$ represents the volumetric part of a second-order tensor ${•}$ ⁠.

The linearization of the Piola stress P in Eq. at the material configuration is given by

Lin {P |}_{F = I} = {P |}_{F = I} + {A |}_{F = I} : [F - I] = 3 λ ɛ^{vol} + 2 μ ɛ

(67)

where the tangent

A

is

A : = \partial_{F} P = λ [F^{- t} \otimes F^{- t} + \ln J D] + μ [I - D]

with

\begin{matrix} D : = \partial_{F} F^{- t} = - F^{- t} \underline{\otimes} F^{- 1} & and & I : = \partial_{F} F = i \bar{\otimes} I \end{matrix}

Next, the Piola–Kirchhoff stress S given in Eq. is linearized at the material configuration

Lin {S |}_{C = I} = {S |}_{C = I} + \frac{1}{2} {C |}_{C = I} : [C - I] = 3 λ ɛ^{vol} + 2 μ ɛ

(68)

where

C : = 2 \partial_{C} S = λ [\frac{1}{2} C^{- 1} \otimes C^{- 1} + \ln J H] - μ H

with

H : = \partial_{C} C^{- 1} = - \frac{1}{2} [C^{- 1} \bar{\otimes} C^{- 1} + C^{- 1} \underline{\otimes} C^{- 1}]

Comparing Eqs. and , it is obvious that

Lin P {|_{F = I} = Lin S |}_{c = I} = : σ_{Lin} or σ_{Lin} = E : ɛ

(69)

with

E : = [3 λ I^{vol} + 2 μ I^{sym}]

where σ_Lin denotes the linearized stress tensor and $E$ is the fourth-order tensor of elastic moduli.

It is common to decompose the linearized stress σ_Lin and infinitesimal strain ε into their volumetric and deviatoric contributions as

σ_{Lin} = σ_{Lin}^{vol} + σ_{Lin}^{dev} and ɛ = ɛ^{vol} + ɛ^{dev}

It follows from Eq. that

σ_{L i n}^{v o l} = 3 κ ε^{v o l} . and σ_{L i n}^{d e v} = 2 μ ε^{d e v}

where κ := λ + 2/3μ is the bulk compression modulus. For the sake of completeness, the elastic modulus and Poisson ratio are denoted as E and v, respectively. The relation between the various material parameters in the bulk are summarized in Table 3 (see, e.g., the standard Ref. [120] for further details).

Table 3

Classical relations between material parameters in the bulk where $Δ : = E^{2} + 9 λ^{2} + 2 E λ$

	$λ$	$μ = G$	$E$	$ν$	$κ$
$\begin{matrix} λ & μ \end{matrix}$	$λ$	$μ$	$\frac{μ (3 λ + 2 μ)}{λ + μ}$	$\frac{λ}{2 (λ + μ)}$	$λ + \frac{2}{3} μ$
$\begin{matrix} λ & E \end{matrix}$	$λ$	$\frac{E - 3 λ + \sqrt{Δ}}{4}$	$E$	$\frac{2 λ}{E + λ + \sqrt{Δ}}$	$\frac{E + 3 λ + \sqrt{Δ}}{6}$
$\begin{matrix} λ & ν \end{matrix}$	$λ$	$\frac{λ (1 - 2 ν)}{2 ν}$	$\frac{λ (1 + ν) (1 - 2 ν)}{ν}$	$ν$	$\frac{λ (1 + ν)}{3 ν}$
$\begin{matrix} λ & κ \end{matrix}$	$λ$	$\frac{3}{2} (κ - λ)$	$\frac{9 κ (κ - λ)}{3 κ - λ}$	$\frac{λ}{3 κ - λ}$	$κ$
$\begin{matrix} μ & E \end{matrix}$	$\frac{μ (E - 2 μ)}{3 μ - E}$	$μ$	$E$	$\frac{E - 2 μ}{2 μ}$	$\frac{μ E}{3 (3 μ - E)}$
$\begin{matrix} μ & ν \end{matrix}$	$\frac{2 μ ν}{1 - 2 ν}$	$μ$	$2 μ (1 + ν)$	$ν$	$\frac{2 μ (1 + ν)}{3 (1 - 2 ν)}$
$\begin{matrix} μ & κ \end{matrix}$	$κ - \frac{2}{3} μ$	$μ$	$\frac{9 κ μ}{3 κ + μ}$	$\frac{3 κ - 2 μ}{6 κ + 2 μ}$	$κ$
$\begin{matrix} E & ν \end{matrix}$	$\frac{E ν}{(1 + ν) (1 - 2 ν)}$	$\frac{E}{2 (1 + ν)}$	$E$	$ν$	$\frac{E}{3 (1 - 2 ν)}$
$\begin{matrix} E & κ \end{matrix}$	$\frac{3 κ (3 κ - E)}{9 κ - E}$	$\frac{3 κ E}{9 κ - E}$	$E$	$\frac{3 κ - E}{6 κ}$	$κ$
$\begin{matrix} ν & κ \end{matrix}$	$\frac{3 κ ν}{1 + ν}$	$\frac{3 κ (1 - 2 ν)}{2 (1 + ν)}$	$3 κ (1 - 2 ν)$	$ν$	$κ$

Hyperelastic Material on the Surface

As in the bulk, we assume a hyperelastic response for the surface. The surface Helmholtz energy

\overset{\land}{Ψ}

is chosen as follows [121]:

\begin{matrix} \overset{\land}{Ψ} (\overset{\land}{F}) = \overset{\land}{γ} \overset{\land}{J} + \frac{1}{2} \overset{\land}{λ} \ln^{2} \overset{\land}{J} + \frac{1}{2} \overset{\land}{μ} [\overset{\land}{F} : \overset{\land}{F} - 2 - 2 \ln \overset{\land}{J}] with \overset{\land}{J} = \overset{\land}{Det} \overset{\land}{F} > 0 \end{matrix}

(70)

The surface Helmholtz energy consists of a surface tension part characterized by the surface tension

\overset{\land}{γ}

⁠, and a neo-Hookean part, formally similar to that of the bulk given, characterized by the two surface Lamé constants

\overset{\land}{μ}

and

\overset{\land}{λ}

⁠. The surface tension part represents a constant Helmholtz energy per unit area of the spatial configuration and accounts for possible fluid-like behavior of the surface. The neo-Hookean part represents the elastic, solid-like response of the surface. Alternatively, the surface Helmholtz energy can be expressed in terms of the surface right Cauchy–Green stretch tensor

\overset{\land}{C} : = {\overset{\land}{F}}^{t} \cdot \overset{\land}{F}

as follows:

\begin{matrix} \overset{\land}{Ψ} (\overset{\land}{C}) = \overset{\land}{γ} \overset{\land}{J} + \frac{1}{2} \overset{\land}{λ} \ln^{2} \overset{\land}{J} + \frac{1}{2} \overset{\land}{μ} [\overset{\land}{Tr} \overset{\land}{C} - 2 - 2 \ln \overset{\land}{J}] with \overset{\land}{J} = \overset{\land}{Det} \overset{\land}{F} = {[\overset{\land}{Det} \overset{\land}{C}]}^{\frac{1}{2}} > 0 \end{matrix}

where $\overset{\land}{Tr} {•} = {•} : \overset{\land}{I}$ denotes the surface trace operator.

The surface Piola stress

\overset{\land}{P}

and the surface Piola–Kirchho stress

\overset{\land}{S}

are, respectively, given by

\overset{\land}{P} (\overset{\land}{F}) = \partial_{\overset{\land}{F}} \overset{\land}{Ψ} (\overset{\land}{F}) = \overset{\land}{γ} \overset{\land}{J} {\overset{\land}{F}}^{- t} + \overset{\land}{λ} \ln \overset{\land}{J} {\overset{\land}{F}}^{- t} + \overset{\land}{μ} [\overset{\land}{F} - {\overset{\land}{F}}^{- t}]

(71)

\overset{\land}{S} (\overset{\land}{C}) : = 2 \partial_{\overset{\land}{C}} \overset{\land}{Ψ} (\overset{\land}{C}) = \overset{\land}{γ} \overset{\land}{J} {\overset{\land}{C}}^{- 1} + \overset{\land}{λ} \ln \overset{\land}{J} {\overset{\land}{C}}^{- 1} + \overset{\land}{μ} [\overset{\land}{I} - {\overset{\land}{C}}^{- 1}]

(72)

In an identical fashion to the bulk, these stress measures are linearized at the material configuration to obtain their small-strain counterparts. Recall that $\overset{\land}{F} - \overset{\land}{I} = \overset{\land}{Grad} \overset{\land}{u}$ where $\overset{\land}{u}$ denotes the surface displacement vector and the infinitesimal surface strain is defined by $\overset{\land}{ɛ} : = {[\overset{\land}{Grad} \overset{\land}{u}]}^{sym}$ The symmetric part of a surface second-order tensor ${\overset{\land}{•}}$ can be obtained as ${\overset{\land}{•}}^{sym} = {\overset{\land}{I}}^{sym} : {•}$ where ${\overset{\land}{I}}^{sym} : = (1 / 2) [\overset{\land}{I} \bar{\otimes} \overset{\land}{I} + \overset{\land}{I} \underline{\otimes} \overset{\land}{I}]$ is the surface fourth-order symmetric identity tensor. The surface fourth-order volumetric, or rather spherical⁵ identity tensor is defined by ${\overset{\land}{I}}^{vol} : = \frac{1}{2} \overset{\land}{I} \otimes \overset{\land}{I}$ ⁠. Furthermore, ${\overset{\land}{•}}^{vol} : = {\overset{\land}{•}} : {\overset{\land}{I}}^{vol}$ denotes the volumetric part of a surface second-order tensor ${\overset{\land}{•}}$ ⁠.

The linearization of the surface Piola stress

\overset{\land}{P}

given in Eq. at the material configuration reads

\begin{matrix} Lin \overset{\land}{P} |_{\overset{\land}{F} = \overset{\land}{I}} = \overset{\land}{P} |_{\overset{\land}{F} = \overset{\land}{I}} + \overset{\land}{A} |_{\overset{\land}{F} = \overset{\land}{I}} : [\overset{\land}{F} - \overset{\land}{I}] = \overset{\land}{γ} \overset{\land}{I} + [\overset{\land}{γ} I \bar{\otimes} \overset{\land}{I} + 2 [\overset{\land}{λ} + \overset{\land}{γ}] {\overset{\land}{I}}^{vol} + 2 [\overset{\land}{μ} - \overset{\land}{γ}] {\overset{\land}{I}}^{sym}] : \overset{\land}{Grad} \overset{\land}{u} = \overset{\land}{γ} \overset{\land}{I} + \overset{\land}{γ} \overset{\land}{Grad} \overset{\land}{u} + 2 [\overset{\land}{λ} + \overset{\land}{γ}] {\overset{\land}{ɛ}}^{vol} + 2 [\overset{\land}{μ} - \overset{\land}{γ}] \overset{\land}{ɛ} (73) \end{matrix}

(73)

where

\begin{matrix} \overset{\land}{A} : = \partial_{\overset{\land}{F}} P = \overset{\land}{γ} \overset{\land}{J} [{\overset{\land}{F}}^{- t} \otimes {\overset{\land}{F}}^{- t} + \overset{\land}{D}] + \overset{\land}{λ} [{\overset{\land}{F}}^{- t} \otimes {\overset{\land}{F}}^{- t} + \ln \overset{\land}{J} \overset{\land}{D}] + \overset{\land}{μ} [\overset{\land}{I} - \overset{\land}{D}] \end{matrix}

with

\begin{matrix} \overset{\land}{D} : = \partial_{\overset{\land}{F}} {\overset{\land}{F}}^{- t} = - {\overset{\land}{F}}^{- t} \underline{\otimes} {\overset{\land}{F}}^{- 1} + [i - \overset{\land}{i}] \bar{\otimes} [{\overset{\land}{F}}^{- 1} \cdot {\overset{\land}{F}}^{- t}] and \overset{\land}{I} : = \partial_{\overset{\land}{F}} \overset{\land}{F} = i \bar{\otimes} \overset{\land}{I} \end{matrix}

Next, the surface Piola–Kirchho stress

\overset{\land}{S}

given in Eq. is linearized at the material configuration

Lin \overset{\land}{S} |_{\overset{\land}{C} = \overset{\land}{I}} = \overset{\land}{S} |_{\overset{\land}{C} = \overset{\land}{I}} + \frac{1}{2} \overset{\land}{C} |_{\overset{\land}{C} = \overset{\land}{I}} : [\overset{\land}{C} - \overset{\land}{I}] = \overset{\land}{γ} \overset{\land}{I} + 2 [\overset{\land}{λ} + \overset{\land}{γ}] {\overset{\land}{ɛ}}^{vol} + 2 [\overset{\land}{μ} - \overset{\land}{γ}] \overset{\land}{ɛ}

(74)

where

\begin{matrix} \overset{\land}{C} : = 2 \partial_{\overset{\land}{C}} \overset{\land}{S} = \overset{\land}{γ} \overset{\land}{J} [\frac{1}{2} {\overset{\land}{C}}^{- 1} \otimes {\overset{\land}{C}}^{- 1} + \overset{\land}{H}] + \overset{\land}{λ} [\frac{1}{2} {\overset{\land}{C}}^{- 1} \otimes {\overset{\land}{C}}^{- 1} + \ln \overset{\land}{J} \overset{\land}{H}] - \overset{\land}{μ} \overset{\land}{H} \end{matrix}

with

\overset{\land}{H} : = \partial_{\overset{\land}{C}} {\overset{\land}{C}}^{- 1} = - \frac{1}{2} [{\overset{\land}{C}}^{- 1} \bar{\otimes} {\overset{\land}{C}}^{- 1} + {\overset{\land}{C}}^{- 1} \underline{\otimes} {\overset{\land}{C}}^{- 1}]

Comparing the Eqs. and , it is obvious that on the surface, and in contrast to the bulk,

Lin \overset{\land}{P} |_{\overset{\land}{F} = \overset{\land}{I}} \neq Lin \overset{\land}{S} |_{\overset{\land}{C} = \overset{\land}{I}}

but rather

Lin \overset{\land}{P} |_{\overset{\land}{F} = \overset{\land}{I}} = Lin \overset{\land}{S} |_{\overset{\land}{C} = \overset{\land}{I}} + \overset{\land}{γ} \overset{\land}{Grad} \overset{\land}{u}

Therefore, the two surface stress measures corresponding to the small-strain assumption are

\begin{matrix} _{\overset{\land}{P}} {\overset{\land}{σ}}_{Lin} : = Lin \overset{\land}{P} |_{\overset{\land}{F} = \overset{\land}{I}} = \overset{\land}{γ} \overset{\land}{I} + \overset{\land}{E} : \overset{\land}{ɛ} + \overset{\land}{γ} \overset{\land}{Grad} \overset{\land}{u} (nonsymmetric)_{\overset{\land}{S}} {\overset{\land}{σ}}_{Lin} : = Lin \overset{\land}{S} |_{\overset{\land}{C} = \overset{\land}{I}} = \overset{\land}{γ} \overset{\land}{I} + \overset{\land}{E} : \overset{\land}{ɛ} (symmetric) \end{matrix}

(75)

\begin{matrix}  \end{matrix}

with

\begin{matrix} \overset{\land}{E} : = [2 {\overset{\land}{λ}}_{eff} {\overset{\land}{I}}^{vol} + 2 {\overset{\land}{μ}}_{eff} {\overset{\land}{I}}^{sym}] & and & {\overset{\land}{λ}}_{eff} : = \overset{\land}{λ} + \overset{\land}{γ} \begin{matrix} where & {\overset{\land}{μ}}_{eff} : = \overset{\land}{λ} - \overset{\land}{γ} \end{matrix} \end{matrix}

The mistake made by Gurtin and Murdoch [32] in their widely cited paper was to omit the surface tension in the definition of the effective quantities ${\overset{\land}{λ}}_{eff}$ and ${\overset{\land}{μ}}_{eff}$ ⁠. This error was rectified in an addendum to their original work [116]. Nonetheless, the error has been reproduced in numerous subsequent publications by many others.

Both of the linearized stress measures given in Eq. (75) are used in the literature. In the absence of surface tension they are obviously equivalent. In the presence of surface tension care must be taken to ensure that they are used in combination with their corresponding balance equation. In the authors' opinion the nonsymmetric version should be used in conjunction with Eq. (21)₂.

Remark. Note that

\overset{\land}{Div} (_{\overset{\land}{P}} {\overset{\land}{σ}}_{Lin}) = \overset{\land}{Div} (_{\overset{\land}{S}} {\overset{\land}{σ}}_{Lin}) + \overset{\land}{γ} \overset{\land}{Div} (\overset{\land}{Grad} \overset{\land}{u})

which shall be compared with Eq. (13) of Ref. [117] where the author explains geometrically the displacement-gradient term. Ru [117] argues that the controversial displacement-gradient term appears due to the fact that the infinitesimal elastic deformation is indeed an incremental deformation superposed on the initial (finite) deformation due to the surface tension. In this manuscript, the reference configuration is assumed to be strain-free as well as stress-free. The surface tension $\overset{\land}{γ}$ introduces a residual stress in the body and deforms it into a state from which the infinitesimal deformation is externally applied. In this sense, the final deformed configuration may be understood as a hybrid formulation combining the linearized deformation of the bulk and finite (second-order) deformation of the surface (see Ref. [117] for further details of this approach). Such justifications seem to be unavoidable when explaining Eq. (75), nevertheless, the mathematical derivations of this section show rigorously the nature of each term.□

For certain solids the influence of surface tension is negligible when compared to that of surface elasticity (see, e.g., Ref. [122]). Thus, in order to study the elastic effects of the surface as compared to the bulk, it is assumed, for the majority of the remainder of this work, that $\overset{\land}{γ} \equiv 0$ ⁠. The role of surface tension will be explained further in the subsequent Young–Laplace example.

In analogy to the bulk, the symmetric linearized surface stress denoted as

{\overset{\land}{σ}}_{Lin} =_{\overset{\land}{S}} {\overset{\land}{σ}}_{Lin}

and infinitesimal surface strain

\overset{\land}{ɛ}

can be decomposed into their volumetric and deviatoric contributions via

\begin{matrix} {\overset{\land}{σ}}_{Lin} = {\overset{\land}{σ}}_{Lin}^{vol} + {\overset{\land}{σ}}_{Lin}^{vol} & and & \overset{\land}{ɛ} = {\overset{\land}{ɛ}}^{vol} + {\overset{\land}{ɛ}}^{dev} \end{matrix}

It follows from Eq. ₂ that

\begin{matrix} {\overset{\land}{σ}}_{Lin}^{vol} = 2 \overset{\land}{κ} {\overset{\land}{ɛ}}^{vol} & and & {\overset{\land}{σ}}_{Lin}^{dev} = 2 \overset{\land}{μ} {\overset{\land}{ɛ}}^{dev} \end{matrix}

with $\overset{\land}{κ} : = \overset{\land}{λ} + \overset{\land}{μ}$ as the surface compression modulus. Note that the definition of the surface modulus $\overset{\land}{κ}$ follows in the spirit of κ in the bulk. This parameter has been defined differently in the literature by incorporating a factor of two (see, e.g., Refs. [58,123]). Nevertheless, it seems that the definition here is more stringent. For the sake of completeness, the surface elastic modulus and surface Poisson ratio are denoted as $\overset{\land}{E}$ and $\overset{\land}{v}$ ⁠, respectively. The relation between the different surface material parameters are summarized in Table 4 (see Javili and Steinmann [92] for further details). Note that the surface incompressibility limit, i.e., the upper bound for the surface Poisson ratio $\overset{\land}{v}$ ⁠, takes the value of 1 and not $\frac{1}{2}$ as in the bulk. This can be justified geometrically.

Table 4

Relations between material parameters at the surface where $\overset{\land}{Δ}$ is defined by $\overset{\land}{Δ}$ := ${\overset{\land}{E}}^{2} + 4 {\overset{\land}{λ}}^{2}$

	$\overset{\land}{λ}$	$\overset{\land}{μ} = \overset{\land}{G}$	$\overset{\land}{E}$	$\overset{\land}{ν}$	$\overset{\land}{κ}$
$\begin{matrix} \overset{\land}{λ} & \overset{\land}{μ} \end{matrix}$	$\overset{\land}{λ}$	$\overset{\land}{μ}$	$\frac{4 \overset{\land}{μ} (\overset{\land}{λ} + \overset{\land}{μ})}{\overset{\land}{λ} + 2 \overset{\land}{μ}}$	$\frac{\overset{\land}{λ}}{\overset{\land}{λ} + 2 \overset{\land}{μ}}$	$\overset{\land}{λ} + \overset{\land}{μ}$
$\begin{matrix} \overset{\land}{λ} & \overset{\land}{E} \end{matrix}$	$\overset{\land}{λ}$	$\frac{\overset{\land}{E} - 2 \overset{\land}{λ} \pm \sqrt{\overset{\land}{Δ}}}{4}$	$\overset{\land}{E}$	$\frac{2 \overset{\land}{λ}}{\overset{\land}{E} \pm \sqrt{\overset{\land}{Δ}}}$	$\frac{\overset{\land}{E} + 2 \overset{\land}{λ} \pm \sqrt{\overset{\land}{Δ}}}{4}$
$\begin{matrix} \overset{\land}{λ} & \overset{\land}{ν} \end{matrix}$	$\overset{\land}{λ}$	$\frac{\overset{\land}{λ} (1 - \overset{\land}{ν})}{2 \overset{\land}{ν}}$	$\frac{\overset{\land}{λ} (1 - {\overset{\land}{ν}}^{2})}{\overset{\land}{ν}}$	$\overset{\land}{ν}$	$\frac{\overset{\land}{λ} (1 + \overset{\land}{ν})}{2 \overset{\land}{ν}}$
$\begin{matrix} \overset{\land}{λ} & \overset{\land}{κ} \end{matrix}$	$\overset{\land}{λ}$	$\overset{\land}{κ} - \overset{\land}{λ}$	$\frac{4 \overset{\land}{κ} (\overset{\land}{κ} - \overset{\land}{λ})}{2 \overset{\land}{κ} - \overset{\land}{λ}}$	$\frac{\overset{\land}{λ}}{2 \overset{\land}{κ} - \overset{\land}{λ}}$	$\overset{\land}{κ}$
$\begin{matrix} \overset{\land}{μ} & \overset{\land}{E} \end{matrix}$	$\frac{2 \overset{\land}{μ} (\overset{\land}{E} - 2 \overset{\land}{μ})}{4 \overset{\land}{μ} - \overset{\land}{E}}$	$\overset{\land}{μ}$	$\overset{\land}{E}$	$\frac{(\overset{\land}{E} - 2 \overset{\land}{μ})}{2 \overset{\land}{μ}}$	$\frac{\overset{\land}{μ} \overset{\land}{E}}{4 \overset{\land}{μ} - \overset{\land}{E}}$
$\begin{matrix} \overset{\land}{μ} & \overset{\land}{ν} \end{matrix}$	$\frac{2 \overset{\land}{μ} \overset{\land}{ν}}{1 - \overset{\land}{ν}}$	$\overset{\land}{μ}$	$2 \overset{\land}{μ} (1 + \overset{\land}{ν})$	$\overset{\land}{ν}$	$\frac{\overset{\land}{μ} (1 + \overset{\land}{ν})}{1 - \overset{\land}{ν}}$
$\begin{matrix} \overset{\land}{μ} & \overset{\land}{κ} \end{matrix}$	$\overset{\land}{κ} - \overset{\land}{μ}$	$\overset{\land}{μ}$	$\frac{4 \overset{\land}{μ} \overset{\land}{κ}}{\overset{\land}{μ} + \overset{\land}{κ}}$	$\frac{\overset{\land}{κ} - \overset{\land}{μ}}{\overset{\land}{κ} + \overset{\land}{μ}}$	$\overset{\land}{κ}$
$\begin{matrix} \overset{\land}{E} & \overset{\land}{ν} \end{matrix}$	$\frac{\overset{\land}{E} \overset{\land}{ν}}{1 - {\overset{\land}{ν}}^{2}}$	$\frac{\overset{\land}{E}}{2 (1 + \overset{\land}{ν})}$	$\overset{\land}{E}$	$\overset{\land}{ν}$	$\frac{\overset{\land}{E}}{2 (1 - \overset{⌢}{v})}$
$\begin{matrix} \overset{\land}{E} & \overset{\land}{κ} \end{matrix}$	$\frac{2 \overset{\land}{κ} (2 \overset{\land}{κ} - \overset{\land}{E})}{4 \overset{\land}{κ} - \overset{\land}{E}}$	$\frac{\overset{\land}{E} \overset{\land}{κ}}{4 \overset{\land}{κ} - \overset{\land}{E}}$	$\overset{\land}{E}$	$\frac{2 \overset{\land}{κ} - \overset{\land}{E}}{2 \overset{\land}{κ}}$	$\overset{\land}{κ}$
$\begin{matrix} \overset{\land}{ν} & \overset{\land}{κ} \end{matrix}$	$\frac{2 \overset{\land}{κ} \overset{\land}{ν}}{1 + \overset{\land}{ν}}$	$\frac{\overset{\land}{κ} (1 - \overset{\land}{ν})}{1 + \overset{\land}{ν}}$	$2 \overset{\land}{κ} (1 - \overset{\land}{ν})$	$\overset{\land}{ν}$	$\overset{\land}{κ}$

Finally, the admissible range for the surface material parameters is considered. This range can be determined by analyzing the well-posedness of the boundary value problem. The necessary and sufficient conditions for the loss of well-posedness of the boundary value problem governing linear elastic, homogeneous solids are well known (see, e.g., Refs. [125–134]). They are the loss of ellipticity of the governing equations and the boundary complementing condition, respectively.

Javili et al. [113] show that the sufficient conditions for the well-posedness of the boundary value problem governing a bulk surrounded by a surface are pointwise stability both in the bulk and on the surface. The results are summarized in Table 5, which shows that the strong ellipticity of the surface is necessary but not sufficient for the well-posedness. Shenoy [134] argues that, within the context of an atomistic model, the calculated surface elasticity tensor need not be pointwise stable as the “surface cannot exist independent of the bulk” and that it is the “total energy (bulk + surface)” that needs to “satisfy the positive definiteness condition,” where positive definiteness refers to pointwise stability. This is not entirely correct. The definition of pointwise stability is a local concept. An argument similar to that of Shenoy [134] is applicable when discussing the stability of the weak approximation to the governing equations, for example, as the basis for finite element computations (see Ref. [113] for further details).

Table 5

Summary of the conditions for the strong ellipticity and pointwise stability of the elasticity tensors in the bulk and on the surface in terms of the Lamé moduli

Strong ellipticity

μ > 0 and λ + 2μ > 0

\overset{\land}{μ} > 0

and

\overset{\land}{λ} + 2 \overset{\land}{μ} > 0

Pointwise stability

μ > 0 and 3λ + 2μ > 0

\overset{\land}{μ} > 0

and

\overset{\land}{λ} + \overset{\land}{μ} > 0

Example: Young–Laplace Equation

In order to further clarify the theory of surface elasticity and to demonstrate its generality, the classical example of the Young–Laplace equation is investigated. This well-known equation states that for static fluids the balance equation on the surface is given by

p = - \overset{\land}{γ} \overset{\land}{c}

(76)

where p is the fluid pressure,⁶ and

\overset{\land}{c}

is twice the mean curvature⁷ in the spatial configuration defined in Eq. . The spatial balance of linear momentum on the surface ₂ in the absence of external tractions reads

\overset{\land}{div} \overset{\land}{σ} = σ \cdot n

(77)

Using the definition of the hydrostatic pressure p in terms of the Cauchy stress σ for inviscid fluids σ = − pi on the right-hand side of the previous equation, gives

\overset{\land}{div} \overset{\land}{σ} = - p n

(78)

In order to compare Eq. with the Young–Laplace Eq. , the surface material model proposed in Eq. with vanishing elastic parameters is selected, i.e.,

\overset{\land}{Ψ} = \overset{\land}{γ} \overset{\land}{J} \Rightarrow \overset{\land}{P} = \partial_{\overset{\land}{F}} \overset{\land}{Ψ} = \overset{\land}{γ} \overset{\land}{J} {\overset{\land}{F}}^{- t} \Rightarrow \overset{\land}{σ} = \overset{\land}{j} \overset{\land}{P} \cdot {\overset{\land}{F}}^{t} = \overset{\land}{γ} \overset{\land}{i}

(79)

Inserting the

\overset{\land}{σ} = \overset{\land}{γ} \overset{\land}{i}

in Eq. yields

\overset{\land}{div} (\overset{\land}{γ} \overset{\land}{i}) = - p n \Rightarrow \overset{\land}{γ} \overset{\land}{div} \overset{\land}{i} = - p n

(80)

and finally using the relation $\overset{\land}{div} \overset{\land}{i} = \overset{\land}{c} n$ yields $\overset{\land}{γ} \overset{\land}{c} = - p$ ⁠. Thus, it is clear that the curvature is embedded in the surface divergence operator.

Thermohyperelastic Material

The material models proposed in Secs. 6.1 and 6.2 describes the mechanical response of materials. They do not, however, possess enough structure to capture the thermal behavior and thermomechanically coupled phenomena. The objective of this section is to extend the discussion in the previous sections to account for thermomechanical contributions.

Bulk

For the simple case of thermohyperelasticity, the Helmholtz energy of the bulk is chosen as (see, e.g., Ref. [115])

\begin{matrix} Ψ (F, Θ) : = \frac{1}{2} λ \ln^{2} J + \frac{1}{2} μ [F : F - 3 - 2 \ln J] - 3 α κ [Θ - Θ_{0}] \frac{\ln J}{J} + c_{F} [Θ - Θ_{0} - Θ \ln \frac{Θ}{Θ_{0}}] - Ξ_{0} [Θ - Θ_{0}] \end{matrix}

(81)

The term $\frac{1}{2} λ \ln^{2} J + \frac{1}{2} μ [F : F - 3 - 2 \ln J]$ is the classical Helmholtz energy function of neo-Hookean-type presented in Eq. (64). Thermomechanical coupling is captured by the term −3αk $[Θ - Θ_{0}]$ ln J/J, where the thermal expansion coefficient α weights the product of the bulk compression modulus k and the difference between the current and the reference temperatures $[Θ - Θ_{0}]$ ⁠. The term c_F $[Θ - Θ_{0} - Θ 1 n Θ / Θ_{0}]$ represents the purely thermal behavior in terms of the specific heat capacity c_F. The final term $Ξ$ ₀ accounts for the material specific absolute entropy.

From the bulk Helmholtz energy in Eq. , the Piola stress and the entropy in the bulk follow in the familiar format of thermohyperelasticity from the constitutive laws ₁ as

P = \frac{\partial Ψ}{\partial F} = [λ \ln J - μ] F^{- t} + μ F - 3 α κ \frac{1}{J} [Θ - Θ_{0}] [1 - \ln J] F^{- t}

and

Ξ = - \frac{\partial Ψ}{\partial Θ} = 3 α κ \frac{\ln J}{J} + c_{F} \ln \frac{Θ}{Θ_{0}} + Ξ_{0}

Next the corresponding contributions to the tangents required for a nonlinear finite element implementation using the Newton–Raphson scheme are derived. The partial derivatives of the Piola stress P with respect to deformation gradient F and temperature

Θ

are given by

\begin{matrix} \frac{\partial P}{\partial F} = λ F^{- t} \otimes F^{- t} + μ I + [λ \ln J - μ] D + 3 α κ [Θ - Θ_{0}] [\frac{2 - \ln J}{J} F^{- t} \otimes F^{- t} - \frac{1 - \ln J}{J} D], \frac{\partial P}{\partial Θ} = - 3 α κ \frac{1 - \ln J}{J} F^{- t} \end{matrix}

The partial derivatives of the heat flux vector Q with respect to the temperature gradient Grad

Θ

and the deformation gradient F are, respectively, given by

\begin{matrix} \frac{\partial Q}{\partial Grad Θ} = - Jk G & and & \frac{\partial Q}{\partial F} = - Jk G \underline{\cdot} Grad Θ \end{matrix}

where, for the case of isotropic thermal conduction in the spatial configuration,⁸

\begin{matrix} G \equiv F^{- 1} \cdot F^{- t} & and & G \equiv F^{- 1} \cdot F^{- t} \otimes F^{- t} + B \end{matrix}

with

\begin{matrix} B : = \frac{\partial F^{- 1} \cdot F^{- t}}{\partial F} =^{t} [F^{- 1} \cdot D] + F^{- 1} \cdot D = - [F^{- 1} \bar{\otimes} F^{- 1}] \cdot F^{- t} - F^{- 1} \cdot [F^{- t} \underline{\otimes} F^{- 1}] \end{matrix}

Finally, for the partial derivatives of the Gough–Joule contribution, one makes use of the identity

F^{- t} : D_{t} F = F^{- t} : Grad V = div V

to obtain

\begin{matrix} \frac{\partial (Θ P_{, Θ} : D_{t} F)}{\partial Θ} = - 3 α κ \frac{1 - \ln J}{J} div V, \frac{\partial (Θ P_{, Θ} : D_{t} F)}{\partial F} = - 3 α κ Θ [\frac{\ln J - 2}{J} div V F^{- t} + \frac{1 - \ln J}{J} D] \end{matrix}

where

D : = \frac{\partial div V}{\partial F} = \frac{\partial F^{- t} : D_{t} F}{\partial F} = D_{t} F : \frac{\partial F^{- t}}{\partial F} + F^{- t} : \frac{\partial D_{t} F}{\partial F}

Surface

As in the bulk, a simple thermohyperelastic surface Helmholtz energy of the following form is chosen:

\begin{matrix} \overset{\land}{Ψ} (\overset{\land}{F}, \overset{\land}{Θ}) : = \frac{1}{2} \overset{\land}{λ} \ln^{2} \overset{\land}{J} + \frac{1}{2} \overset{\land}{μ} [\overset{\land}{F} : \overset{\land}{F} - 2 - 2 \ln \overset{\land}{J}] + \overset{\land}{γ} \overset{\land}{J} - 2 \overset{\land}{α} \overset{\land}{κ} [\overset{\land}{Θ} - {\overset{\land}{Θ}}_{0}] \frac{\ln \overset{\land}{J}}{J} + {\overset{\land}{c}}_{\overset{\land}{F}} [\overset{\land}{Θ} - {\overset{\land}{Θ}}_{0} - \overset{\land}{Θ} \ln \frac{\overset{\land}{Θ}}{{\overset{\land}{Θ}}_{0}}] - {\overset{\land}{Ξ}}_{0} [\overset{\land}{Θ} - {\overset{\land}{Θ}}_{0}] \end{matrix}

(82)

The term

\frac{1}{2} \overset{\land}{λ} \ln^{2} \overset{\land}{J} + \frac{1}{2} \overset{\land}{μ} [\overset{\land}{F} : \overset{\land}{F} - 2 - 2 \ln \overset{\land}{J}]

is the classical Helmholtz energy function of neo-Hookean-type given in Eq. . The thermomechanical coupling is captured by the term

- 2 \overset{\land}{α} \overset{\land}{κ} [\overset{\land}{Θ} - {\overset{\land}{Θ}}_{0}] \ln \overset{\land}{J} / J

where

\overset{\land}{α}

is the surface thermal expansion coefficient. The surface temperatures in the reference and current configurations are denoted as

{\overset{\land}{Θ}}_{0}

and

\overset{\land}{Θ}

⁠, respectively. The surface specific heat capacity

{\overset{\land}{c}}_{\overset{\land}{F}}

represents the purely thermal behavior. Finally, the surface specific entropy

{\overset{\land}{Ξ}}_{0}

is also included. The surface Piola stress and entropy corresponding to the surface Helmholtz energy given in Eq. follow from the constitutive laws ₂ as

\overset{\land}{P} = \frac{\partial \overset{\land}{Ψ}}{\partial \overset{\land}{F}} = [\overset{\land}{λ} \ln \overset{\land}{J} - \overset{\land}{μ} + \overset{\land}{γ} \overset{\land}{J}] {\overset{\land}{F}}^{- t} + \overset{\land}{μ} \overset{\land}{F} - 2 \overset{\land}{α} \overset{\land}{κ} \frac{1}{\overset{\land}{J}} [\overset{\land}{Θ} - {\overset{\land}{Θ}}_{0}] [1 - \ln \overset{\land}{J}] {\overset{\land}{F}}^{- t}

and

\overset{\land}{Ξ} = - \frac{\partial \overset{\land}{Ψ}}{\partial \overset{\land}{Θ}} = 2 \overset{\land}{α} \overset{\land}{κ} \frac{\ln \overset{\land}{J}}{\overset{\land}{J}} + {\overset{\land}{c}}_{\overset{\land}{F}} \ln \frac{\overset{\land}{Θ}}{{\overset{\land}{Θ}}_{0}} + {\overset{\land}{Ξ}}_{0}

The surfaces contributions to the tangent are formally identical to those in the bulk. However, due to the rank-deficiency, subtle modifications need to be taken into account. Thus,

\begin{matrix} \frac{\partial \overset{\land}{P}}{\partial \overset{\land}{F}} = [\overset{\land}{λ} + \overset{\land}{γ} \overset{\land}{J}] {\overset{\land}{F}}^{- t} \otimes {\overset{\land}{F}}^{- t} + \overset{\land}{μ} \overset{\land}{I} + [\overset{\land}{λ} \ln \overset{\land}{J} - \overset{\land}{μ} + \overset{\land}{γ} \overset{\land}{J}] \overset{\land}{D} + 2 \overset{\land}{α} \overset{\land}{κ} [\overset{\land}{Θ} - {\overset{\land}{Θ}}_{0}] [\frac{2 - \ln \overset{\land}{J}}{\overset{\land}{J}} {\overset{\land}{F}}^{- t} \otimes {\overset{\land}{F}}^{- t} - \frac{1 - \ln \overset{\land}{J}}{\overset{\land}{J}} \overset{\land}{D}], \frac{\partial \overset{\land}{P}}{\partial \overset{\land}{Θ}} = - 2 \overset{\land}{α} \overset{\land}{κ} \frac{1 - \ln \overset{\land}{J}}{\overset{\land}{J}} {\overset{\land}{F}}^{- t} \end{matrix}

The partial derivatives of the interface heat flux vector

\overset{\land}{Q}

with respect to the temperature gradient

\overset{\land}{Grad} \overset{\land}{Θ}

and the interface deformation gradient

\overset{\land}{F}

are

\begin{matrix} \frac{\partial \overset{\land}{Q}}{\partial \overset{\land}{Grad} \overset{\land}{Θ}} = - \overset{\land}{J} \overset{\land}{k} \overset{\land}{G} & and & \frac{\partial \overset{\land}{Q}}{\partial \overset{\land}{F}} = - \overset{\land}{J} \overset{\land}{k} \overset{\land}{G} \underline{\cdot} \overset{\land}{Grad} \overset{\land}{Θ} \end{matrix}

where for isotropic thermal conduction along the interface in the spatial configuration⁹

\begin{matrix} \overset{\land}{G} \equiv {\overset{\land}{F}}^{- 1} \cdot {\overset{\land}{F}}^{- t} & and & \overset{\land}{G} \equiv {\overset{\land}{F}}^{- 1} \cdot {\overset{\land}{F}}^{- t} \otimes {\overset{\land}{F}}^{- t} + \overset{\land}{B} \end{matrix}

with

\overset{\land}{B} : = \frac{\partial {\overset{\land}{F}}^{- 1} \cdot {\overset{\land}{F}}^{- t}}{\partial \overset{\land}{F}} =^{t} [{\overset{\land}{F}}^{- 1} \cdot \overset{\land}{D}] + {\overset{\land}{F}}^{- 1} \cdot \overset{\land}{D}

Finally, for the partial derivatives of the Gough–Joule contribution, one makes use of the identity

{\overset{\land}{F}}^{- t} : D_{t} \overset{\land}{F} = {\overset{\land}{F}}^{- t} : \overset{\land}{Grad} \overset{\land}{V} = \overset{\land}{div} \overset{\land}{V}

to obtain

\begin{matrix} \frac{\partial (\overset{\land}{Θ} {\overset{\land}{P}}_{, \overset{\land}{Θ}} : D_{t} \overset{\land}{F})}{\partial \overset{\land}{Θ}} = - 2 \overset{\land}{α} \overset{\land}{κ} \frac{1 - \ln \overset{\land}{J}}{\overset{\land}{J}} \overset{\land}{div} \overset{\land}{V}, \frac{\partial (\overset{\land}{Θ} {\overset{\land}{P}}_{, \overset{\land}{Θ}} : D_{t} \overset{\land}{F})}{\partial \overset{\land}{F}} = - 2 \overset{\land}{α} \overset{\land}{κ} \overset{\land}{Θ} [\frac{\ln \overset{\land}{J} - 2}{\overset{\land}{J}} \overset{\land}{div} \overset{\land}{V} {\overset{\land}{F}}^{- t} + \frac{1 - \ln \overset{\land}{J}}{\overset{\land}{J}} \overset{\land}{D}] \end{matrix}

where

\overset{\land}{D} : = \frac{\partial \overset{\land}{div} \overset{\land}{V}}{\partial \overset{\land}{F}} = \frac{\partial {\overset{\land}{F}}^{- t} : D_{t} \overset{\land}{F}}{\partial \overset{\land}{F}}

Interface

In an identical fashion to the surface, a simple thermohyperelastic interface Helmholtz energy of the following form is chosen:

\begin{matrix} \bar{Ψ} (\bar{F}, \bar{Θ}) : = \frac{1}{2} \bar{λ} \ln^{2} \bar{J} + \frac{1}{2} \bar{μ} [\bar{F} : \bar{F} - 2 - 2 \ln \bar{J}] + \bar{γ} \bar{J} - 2 \bar{α} \bar{κ} [\bar{Θ} - {\bar{Θ}}_{0}] \frac{\ln \bar{J}}{J} + {\bar{c}}_{\bar{F}} [\bar{Θ} - {\bar{Θ}}_{0} - \bar{Θ} \ln \frac{\bar{Θ}}{{\bar{Θ}}_{0}}] - {\bar{Ξ}}_{0} [\bar{Θ} - {\bar{Θ}}_{0}] \end{matrix}

(83)

The term $\frac{1}{2} \bar{λ} \ln^{2} \bar{J} + \frac{1}{2} \bar{μ} [\bar{F} : \bar{F} - 2 - 2 \ln \bar{J}]$ mimics the classical Helmholtz-energy function of neo-Hookean-type characterized through the two interface Lamé constants $\bar{λ}$ and $\bar{μ}$ ⁠. The term $\bar{γ}$ represents surface-tension-like behavior on the interface. Thermomechanical coupling is captured by the term −2 $\bar{α} \bar{κ} [\bar{Θ} - {\bar{Θ}}_{0}] \ln \bar{J} / J$ where $\bar{α}$ is the interface thermal expansion coefficient and $\bar{κ} : = \bar{μ} + \bar{λ}$ the interface compression modulus. The interface temperatures in the reference and current configurations are denoted as ${\bar{Θ}}_{0}$ and $\bar{Θ}$ ⁠, respectively. The interface specific heat capacity ${\bar{c}}_{\bar{F}}$ represents the purely thermal behavior. Finally, the interface specific entropy ${\bar{Ξ}}_{0}$ is included.

From the constitutive laws ₃ the interface Piola stress and entropy corresponding to the interface Helmholtz energy in Eq. are obtained as

\begin{matrix} \bar{P} = \frac{\partial \bar{Ψ}}{\partial \bar{F}} = [\bar{λ} \ln \bar{J} - \bar{μ} + \bar{γ} \bar{J}] {\bar{F}}^{- t} + \bar{μ} \bar{F} - 2 \bar{α} \bar{κ} \frac{1}{\bar{J}} [\bar{Θ} - {\bar{Θ}}_{0}] [1 - \ln \bar{J}] {\bar{F}}^{- t} \end{matrix}

and

\bar{Ξ} = - \frac{\partial \bar{Ψ}}{\partial \bar{Θ}} = 2 \bar{α} \bar{κ} \frac{\ln \bar{J}}{\bar{J}} + {\bar{c}}_{\bar{F}} \ln \frac{\bar{Θ}}{{\bar{Θ}}_{0}} + {\bar{Ξ}}_{0}

Remark. The interface Helmholtz energy given in Eq. accounts for thermomechanical coupling and the interface heat capacity. For a Kapitza interface these two contributions, as well as the heat conduction along the interface and the interface prescribed heat source, should vanish to satisfy the Kapitza assumption of continuity of the normal heat flux across the interface (see Eq. ₃). Therefore, the form of the Helmholtz energy for the generalized Kapitza interface simplifies to

\bar{Ψ} (\bar{F}, \bar{Θ}) = \frac{1}{2} \bar{λ} \ln^{2} \bar{J} + \frac{1}{2} \bar{μ} [\bar{F} : \bar{F} - 2 - 2 \ln \bar{J}] + \bar{γ} \bar{J} - {\bar{Ξ}}_{0} [\bar{Θ} - {\bar{Θ}}_{0}]

(84)

□

Curve

Finally, for the sake of completeness, and in a very similar fashion to the surface and interface, a simple thermohyperelastic curve Helmholtz energy of the following form is chosen:

\begin{matrix} \tilde{Ψ} (\tilde{F}, \tilde{Θ}) : = \frac{1}{2} \tilde{μ} [\tilde{F} : \tilde{F} - 1 - 2 \ln \tilde{J}] + \tilde{γ} \tilde{J} - \tilde{α} \tilde{μ} [\tilde{Θ} - {\tilde{Θ}}_{0}] \frac{\ln \tilde{J}}{J} + {\tilde{c}}_{\tilde{F}} [\tilde{Θ} - {\tilde{Θ}}_{0} - \tilde{Θ} \ln \frac{\tilde{Θ}}{{\tilde{Θ}}_{0}}] - {\tilde{Ξ}}_{0} [\tilde{Θ} - {\tilde{Θ}}_{0}] \end{matrix}

(85)

The term $\frac{1}{2} \tilde{μ} [\tilde{F} : \tilde{F} - 1 - 2 \ln \tilde{J}]$ mimics the classical Helmholtz-energy function of neo-Hookean-type characterized through a constant $\tilde{μ}$ ⁠. Due to the one-dimensional nature of the curve, one mechanical parameter suffices to model the elastic behavior of the curve. In this sense, the curve parameter $\tilde{μ}$ can be understood as the elastic-modulus or the compression modulus of the curve and shear modulus is no longer meaningful. A surface-tension-like term is introduced by $\tilde{γ}$ ⁠. The thermomechanical coupling is captured by the term $- \tilde{α} \tilde{μ} [\tilde{Θ} - {\tilde{Θ}}_{0}] \ln \tilde{J} / J$ where $\tilde{α}$ is the curve thermal expansion coefficient. The curve temperatures in the reference and current configurations are denoted as ${\tilde{Θ}}_{0}$ and $\tilde{Θ}$ ⁠, respectively. The curve specific heat capacity ${\tilde{c}}_{\tilde{F}}$ represents the purely thermal behavior. Finally, the curve specific entropy ${\tilde{Ξ}}_{0}$ is included.

From the constitutive laws ₄ the curve Piola stress and entropy corresponding to the curve Helmholtz energy Eq. are obtained as

\tilde{P} = \frac{\partial \tilde{Ψ}}{\partial \tilde{F}} = [- \tilde{μ} + \tilde{γ} \tilde{J}] {\tilde{F}}^{- t} + \tilde{μ} \tilde{F} - \tilde{α} \tilde{μ} \frac{1}{\tilde{J}} [\tilde{Θ} - {\tilde{Θ}}_{0}] [1 - \ln \tilde{J}] {\tilde{F}}^{- t}

and

\tilde{Ξ} = - \frac{\partial \tilde{Ψ}}{\partial \tilde{Θ}} = \tilde{α} \tilde{μ} \frac{\ln \tilde{J}}{\tilde{J}} + {\tilde{c}}_{\tilde{F}} \ln \frac{\tilde{Θ}}{{\tilde{Θ}}_{0}} + {\tilde{Ξ}}_{0}

Numerical Examples

The objective of this section is to briefly elucidate the theory presented in Secs. 2–6 using a series of numerical examples. The roles of the surface and interface on the overall response of a body are studied.¹⁰ Specific attention is paid to the influence of the material parameters.

These examples are of particular relevance to problems involving materials at the nanoscale. The influence of decreasing specimen size on the overall response can be captured by fixing the bulk material parameters and increasing those of the surface and interface. This is equivalent to fixing the bulk Helmholtz energy and increasing the Helmholtz energy of the surface and interface.

The material behavior in the bulk is characterized by the thermohyperelastic Helmholtz energy function given in Eq. (81). Heat conduction in the bulk is assumed to be spatially isotropic, i.e., k = k i.

The material parameters for the bulk are stated in Table 6 and hold unless specified otherwise.

Table 6

Bulk material properties

Lamé constant

μ

80193.8

N/mm²

Lamé constant

λ

110743.5

N/mm²

Thermal expansion coefficient

α

1.0 × 10⁻⁵

1/K

Conductivity

k

45.0

N/(s K)

Specific capacity

c_F

3.588

N/(mm² K)

Reference temperature

Θ

₀

298.0

K

Surface

The following set of examples illustrate the role of the surface on the overall response. Thus, the surface Helmholtz energy presented in Sec. 6.3.2 is chosen. The role of the various terms in the energy is then studied by varying their corresponding constitutive coefficients. In Sec. 7.1.1 the influence of surface tension $\overset{\land}{γ}$ is demonstrated. In contrast to all of the other terms in the surface Helmholtz energy, it does not vanish at the reference (strain-free) configuration and acts in a fashion similar to an external force field. In this sense, it should be treated as a prescribed force and, therefore, is applied incrementally [90].

The surface theory presented here is not restricted to the isotropic case and anisotropic behavior can be captured if the form of the energy includes the anisotropy via a dependence on the spatial surface normal; see Eq. (38). For the case of surface tension a simple anisotropic model is proposed. The numerical results demonstrate the flexibility and generality of the proposed scheme.

In Sec. 7.1.2 the focus is on the thermomechanical behavior of the surface and its coupling with the bulk. The influence of surface tension is ignored. Furthermore heat conduction on the surface is assumed to be spatially isotropic, i.e., $\overset{\land}{k} = \overset{\land}{k} \overset{\land}{i} .$

Surface Tension

Consider the unit-cube depicted in Fig. 4 (top-left). The cube is discretized using 1728 trilinear hexahedral elements and 2197 nodes. The cube is covered by an energetic surface with a surface Helmholtz energy containing only the surface tension terms in Eq. (82). The surface tension of $\overset{\land}{γ}$ = μ is prescribed in 20 equal steps.

Fig. 4

Transformation of a cube into a sphere due to an incrementally prescribed surface tension of γ∧/μ = 1.0 mm

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Transformation of a cube into a sphere due to an incrementally prescribed surface tension of $\overset{\land}{γ} /$ μ = 1.0 mm

For a nonvanishing surface tension the reference configuration does not coincide with that of an equilibrium state. Thus, the cube deforms in order to reach an equilibrium state, i.e., to minimize the total energy of the bulk and the surface. Clearly, any deformation introduces an increase in the bulk energy as it is zero at the reference configuration. Nevertheless, the total energy of the system can still decrease by decreasing the surface energy and increasing the bulk energy. Recall that surface tension is equivalent to a constant energy per unit deformed area. Therefore, the surface energy can decrease by decreasing the surface area. As a result, the cube tends to transform into a shape where the surface-to-volume ratio is minimal, i.e., a sphere. The surface also tends to shrink. The transformation of the cube into a spherical shape is illustrated in Fig. 4.

The role of anisotropy is now investigated. The example is identical to the previous one except that the surface tension

\overset{\land}{γ}

is no longer constant. It depends on the surface normal and the surface Jacobian via the relation

\overset{\land}{γ} = μ [1 + \overset{\land}{β} \overset{\land}{J} {[n \cdot e]}^{2}]

where the material parameter $\overset{\land}{β}$ controls the anisotropic contribution and the vector e is a given unit vector characterizing a preferred direction. The anisotropic effects will be greatest where the normal to the surface is parallel to e and will vanish where the normal to the surface is orthogonal to e. The vector e = [0,0,1] together with the reference and deformed configurations are shown in Fig. 5. The anisotropic effect will be maximum on the top and the bottom of the model and will vanish on the lateral walls of the model (elsewhere the effect is between these two extremes). The model is expected, therefore, to shrink more on the top and the bottom and less at the lateral walls of the cube. Consequently, the cube will transform into an ellipsoidal shape instead of the spherical one shown in Fig. 4. The resulting ellipsoidal shapes for different values of $\overset{\land}{β}$ are shown in Fig. 5.

Fig. 5

Transformation of a unit-cube into an ellipsoid due to a prescribed anisotropic surface tension γ∧/μ = [1 + β∧J∧[n·e]2] mm

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Transformation of a unit-cube into an ellipsoid due to a prescribed anisotropic surface tension $\overset{\land}{γ} /$ μ = [1 + $\overset{\land}{β} \overset{\land}{J} {[n \cdot e]}^{2}$ ] mm

Thermomechanical Surface

Consider the three-dimensional strip shown in Fig. 6. The strip is subjected to plane-strain-like conditions across the thickness. In the middle of the specimen is a hole of radius 0.2 mm. Lateral deformations are prevented, i.e., the width of the strip cannot change. A displacement of 0.5 mm is prescribed at the edges resulting in a tensile loading. Thermally homogeneous Neumann boundary conditions are applied to all boundaries of the bulk. The resulting amount of deformation is clearly in the finite strain regime. The displacement loading is applied in 20 equal steps and the total time is 10 ms. The strip is discretized using 576 trilinear hexahedral elements and 1008 nodes. Assume that the surface of the hole in the specimen is energetic. The surface Helmholtz energy is chosen to be of the form given in Eq. (82) with vanishing $\overset{\land}{γ}$ ⁠. The response of the material without the energetic surface is studied first. Thereafter, mechanical and thermomechanical surface effects are detailed, respectively.

Fig. 6

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Strip with surface: geometry (left) and applied boundary conditions (right). Dimensions are in mm. The thickness is 0.1 mm.

Bulk effect.

Consider first a thermomechanical bulk without an energetic surface. Under the prescribed boundary conditions the tensile stress increases. This increase of the tensile stress, or more precisely the divergence of the strain rate, cools the specimen due to the Gough–Joule effect. The presence of the hole leads to a nonuniform stress distribution in the specimen as illustrated in Fig. 7. Consequently, the nonuniform temperature profile depicted in Fig. 8 is obtained.

Fig. 7

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The stress distribution for a thermomechanical bulk. The results (a)–(f) correspond to the reference configuration and 20%, 40%, 60%, 80%, and 100% of the final deformation, respectively. The stress depicted is the xx-component of the Cauchy stress tensor. The reference configuration is indicated using a dashed (white) line.

Fig. 8

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The temperature distribution for a thermomechanical bulk. The results (a)–(f) correspond to the reference configuration and 20%, 40%, 60%, 80%, and 100% of the final deformation, respectively. The reference configuration is indicated using a dashed (white) line.

Mechanical surface effect.

Consider now a purely mechanical surface. That is $\overset{\land}{k} = {\overset{\land}{c}}_{\overset{\land}{F}} = \overset{\land}{α} = 0 .$ The mechanical material parameters for the surface are assumed to be $\overset{\land}{μ} / μ = \overset{\land}{λ} / λ = 2 mm .$ The mechanical resistance of the surface influences the bulk, resulting in a different stress distribution when compared to that shown Fig. 7 obtained in the absence of surface effects. The evolution of the stress field is illustrated in Fig. 9. The increase in the tensile stress cools the specimen due to the Gough–Joule effect. Thus, the temperature decreases as the applied deformation increases. The nonhomogeneous stress distribution produces the nonhomogeneous temperature distribution shown in Fig. 10.

Fig. 9

The influence a purely mechanical surface on the stress distribution for μ∧/μ=λ∧/λ = 2 mm. The results (a)–(f) correspond to the reference configuration and 20%, 40%, 60%, 80%, and 100% of the final deformation, respectively. The stress depicted is the xx-component of the Cauchy stress tensor. The reference configuration is indicated using a dashed (white) line.

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The influence a purely mechanical surface on the stress distribution for $\overset{\land}{μ} / μ = \overset{\land}{λ} / λ$ = 2 mm. The results (a)–(f) correspond to the reference configuration and 20%, 40%, 60%, 80%, and 100% of the final deformation, respectively. The stress depicted is the xx-component of the Cauchy stress tensor. The reference configuration is indicated using a dashed (white) line.

Fig. 10

The influence of a purely mechanical surface on the temperature distribution for μ∧/μ=λ∧/λ = 2 mm. The results (a)–(f) correspond to the reference configuration and 20%, 40%, 60%, 80%, and 100% of the final deformation, respectively. The reference configuration is indicated using a dashed (white) line.

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The influence of a purely mechanical surface on the temperature distribution for $\overset{\land}{μ} / μ = \overset{\land}{λ} / λ$ = 2 mm. The results (a)–(f) correspond to the reference configuration and 20%, 40%, 60%, 80%, and 100% of the final deformation, respectively. The reference configuration is indicated using a dashed (white) line.

Thermomechanical surface effect.

The surface is now endowed with thermal properties. The mechanical properties of the surface are fixed as before by setting $\overset{\land}{μ} / μ = \overset{\land}{λ} / λ = 2 mm .$ In order to better understand the role of the surface thermal properties, the surface conductivity $\overset{\land}{k},$ heat capacity ${\overset{\land}{c}}_{\overset{\land}{F}}$ ⁠, and heat expansion coefficient $\overset{\land}{α}$ are investigated separately.

The effect of surface conductivity.

The surface heat conduction coefficient $\overset{\land}{k}$ is increased from zero to $\overset{\land}{k} / k$ = 100 mm. The remaining surface thermal parameters are set to zero. Figure 11 shows the temperature distributions at the end of the applied loading for a range of $\overset{\land}{k} / k$ ⁠. Increasing the surface conductivity clearly produces an increasingly uniform temperature distribution along the surface. Note that the result shown in Fig. 11 for $\overset{\land}{k} / k$ = 0 is the same as shown in Fig. 10(f).

Fig. 11

The influence of interface conduction on the temperature distribution for μ∧/μ=λ∧/λ = 2 mm and k∧/k = 0–100 mm

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The influence of interface conduction on the temperature distribution for $\overset{\land}{μ} / μ = \overset{\land}{λ} / λ$ = 2 mm and $\overset{\land}{k} / k$ = 0–100 mm

The effect of surface heat capacity.

The surface heat capacity ${\overset{\land}{c}}_{\overset{\land}{F}}$ is now varied from zero to ${\overset{\land}{c}}_{\overset{\land}{F}} / c_{F} = 1 mm$ ⁠. The remaining surface thermal parameters are set as zero. Increasing the surface heat capacity causes the surface to conserve its initial temperature and resist temperature change. A higher value of the surface heat capacity results in less temperature decrease at the surface. This can be observed in Fig. 12, which shows the final temperature distributions for different surface heat capacities.

Fig. 12

The influence of the interface heat capacity on the temperature distribution for μ∧/μ = λ∧/λ = 2 mm and c∧F∧/cF = 0–1 mm

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The influence of the interface heat capacity on the temperature distribution for $\overset{\land}{μ} / μ = \overset{\land}{λ} / λ$ = 2 mm and ${\overset{\land}{c}}_{\overset{\land}{F}} / c_{F}$ = 0–1 mm

The effect of the surface Gough–Joule contribution.

Finally, the effect of the surface heat expansion coefficient $\overset{\land}{α}$ on the temperature distribution is studied. The surface heat expansion coefficient $\overset{\land}{α}$ is varied from zero to $\overset{\land}{α} / α = 0.75$ while the remaining surface thermal parameters are specified as zero. According to Eq. (82), increasing the surface heat expansion coefficient increases the surface thermomechanical coupling. As a result of the Gough–Joule effect, the surface cools due to the positive strain rate applied to the surface from the surrounding bulk material. The surface cooling is proportional to $\overset{\land}{α} / α$ ⁠. This explains the temperature distributions shown in Fig. 13.

Fig. 13

The influence of the interface heat expansion coefficient on temperature distribution for μ∧/μ=λ∧/λ = 2 mm and α∧/α = 0–0.75

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The influence of the interface heat expansion coefficient on temperature distribution for $\overset{\land}{μ} / μ = \overset{\land}{λ} / λ$ = 2 mm and $\overset{\land}{α} / α$ = 0–0.75

Interface

The “toy” problem is a cube with an internal energetic interface, as illustrated in Fig. 14. The material interior and exterior to the interface is identical. For both the bulk and the interface a thermohyperelastic material model is assumed. The choice of the Kapitza or highly conducting assumption on the interface's behavior is examined.

Fig. 14

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A cube containing an internal energetic interface. The geometry is show on the left and the two different load cases on the right. The first load case corresponds to a thermal load while the second is a mechanical one.

Kapitza Interface

The interface is initially defined to be a Kapitza interface. Hence, while discontinuities in the temperature field are permitted across the interface, the normal component of the heat flux vector is continuous. The domain is loaded according to the thermal load case depicted in Fig. 14. The top and the lateral sides of the cube are thermally isolated, i.e., ${\overset{\land}{Q}}^{p}$ = 0. A temperature increase of ΔΘ^p= 40 K relative to the initial temperature of Θ₀ = 298 K is applied to the lower face of the specimen. The resulting temperature distributions at the same instant in time for different Kapitza resistance coefficients ${\bar{r}}_{Q}$ are illustrated in Fig. 15. As expected, the jump in the temperature across the interface increases with increasing Kapitza resistance.

Fig. 15

The temperature distribution for varying values of the Kapitza resistance r¯Q on the interface

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The temperature distribution for varying values of the Kapitza resistance ${\bar{r}}_{Q}$ on the interface

Highly Conducting Interface

The interface is now assumed to be highly conducting. Hence, while discontinuities in the normal component of the heat flux vector are permitted across the interface, the temperature field is continuous. The domain is subjected to the mechanical loading conditions illustrated in Fig. 14. The cube is extended by 100% by prescribing equal displacements to the top and bottom faces. Lateral contractions are prevented. The loading is applied quasi-statically and the cube is thermally isolated.

Consider first the response without an interface as shown in Fig. 16(a). In the absence of the interface, the stress state in the cube is homogeneous. The inclusion of the energetic interface results in a change in the distribution of the stress field, as shown in Fig. 16(b). The mechanical resistance of the interface causes the stress to increase above and below the region enclosed by the interface. Half of the midplane of the cube, corresponding to the cases where the interface was omitted and included, is shown alongside one another in Fig. 16(c) to facilitate the comparison of the mechanical response. The mechanical resistance of the energetic interface causes the material surrounded by the interface to deform less than the corresponding region in the absence of an interface (the spatial discretizations are identical).

Fig. 16

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The stress distribution without and with an HC interface are shown in (a) and (b), respectively, and compared in (c). The resulting temperature distribution is shown in (d).

The nonhomogeneous stress distribution that arises due the mechanical resistance of the interface results in a nonhomogeneous temperature field as shown in Fig. 16(d). The Gough–Joule effect causes the temperature in the body to decrease. The temperature field is clearly continuous across the interface.

Conclusion

A procedure for deriving the equations governing the response of a solid bulk, surrounded by a surface, and intersected by an interface, all of which are energetic, has been presented. The interaction between the surface and the interface has also been accounted for. The structure of the constitutive relations in the various parts of the body was made clear by assigning these regions distinct Helmholtz energies and following a Coleman–Noll-like procedure. The remaining reduced dissipation inequalities suggest the structure of the heat and entropy flux vectors and permissible restrictions on the temperature field in the various parts of the body; particular attention was paid to the interface. The weak formulation of the governing equations was then given and compared to six commonly used restrictions thereof. The theory was elucidated using several numerical examples.

The theory proposed here is often motivated by applications in nanomechanics where size-effects play a critical role. The ratio of the bulk Helmholtz energy to those in the remaining parts of the body implicitly accounts for the size-effects. However, the theory lacks a physical length scale related to the thickness of the interface or surface. Steigmann and Ogden [40] incorporate local elastic resistance to flexure into the Gurtin and Murdoch [32] theory, thereby introducing an inherent length scale and regularizing the associated variational problem. Recent work by Forest et al. [135], comparing first and second strain gradient theories for capillary effects in elastic fluids at small length scales, emphasizes the importance of nonlocality and the associated issue of length scale. An extension to account for the flexural resistance of the surface or interface is, therefore, suggested (see also Refs. [41,42,136]).

Acknowledgment

The support of this work by the European Research Council Advanced Grant MOCOPOLY is gratefully acknowledged. The second author thanks the National Research Foundation of South Africa for their support.

Geometry of Surfaces and Curves

It is enlightening to briefly review some basic terminologies and results on surfaces and curves. For further details the reader is referred to Refs. [89,102–104] among others. Here, some technicalities are borrowed from Steinmann [89].

Surfaces

A two-dimensional (smooth) surface

S

in the three-dimensional, embedding Euclidean space with coordinates

x

is parameterized by two surface coordinates

η^{α}

with α = 1, 2 as

x = x (η^{α})

The corresponding tangent vectors

a_{α} \in T S

to the surface coordinate lines

η^{α}

⁠, i.e., the covariant (natural) surface basis vectors are given by

a_{α} = \partial_{η^{α}} x

The associated contravariant (dual) surface basis vectors

a^{α}

are defined by the Kronecker property

δ_{β}^{α} = a^{α} \cdot a_{β}

and are explicitly related to the covariant surface basis vectors

a_{α}

by the co- and contravariant surface metric coefficients a_αβ (first fundamental form of the surface) and a^α^β, respectively, as

\begin{matrix} \begin{matrix} a_{α} = a_{α β} a^{β} & with & a_{α β} = a_{α} \cdot a_{β} = {[a^{α β}]}^{- 1}, \end{matrix} \\ \begin{matrix} a^{α} = a^{α β} a_{β} & with & a^{α β} = a^{α} \cdot a^{β} = {[a_{α β}]}^{- 1} \end{matrix} \end{matrix}

The contra- and covariant base vectors

a^{3}

and

a_{3}

⁠, normal to T

S

⁠, are defined by

a^{3} : = a_{1} \times a_{2} and a_{3} : = {[a^{33}]}^{- 1} a^{3} so that a^{3} \cdot a_{3} = 1

Thereby, the corresponding contra- and covariant metric coefficients, respectively, [a³³] and [a₃₃] follow as

[a^{33}] = | a_{1} \times a_{2} |^{2} = det [a_{α β}] = {[det [a^{α β}]]}^{- 1} = {[a_{33}]}^{- 1}

Accordingly, the surface area element ds and the surface normal

n

are computed as

d s = | a_{1} \times a_{2} | d η^{1} d η^{2} = {[a^{33}]}^{1 / 2} d η^{1} d η^{2} and n = {[a^{33}]}^{1 / 2} a^{3} = {[a^{33}]}^{1 / 2} a_{3}

Moreover, with

i

denoting the ordinary mixed-variant unit tensor of the three-dimensional, embedding Euclidian space, the mixed-variant surface unit tensor

\overset{\land}{i}

is defined as

\overset{\land}{i} : = δ_{β}^{α} a_{α} \otimes a^{β} = a_{α} \otimes a^{α} = i - a_{3} \otimes a^{3} = i - n \otimes n

Clearly the mixed-variant surface unit tensor acts as a surface (idempotent) projection tensor.

The surface gradient and surface divergence operators for vector fields are defined by

{grad}_{s} {•} : = \partial_{η^{α}} {•} \otimes a^{α} and {div}_{s} : = \partial_{η^{α}} {•} \cdot a^{α}

As a consequence, observe that grad_s

{•} \cdot n = 0

holds by definition. For fields that are smooth in a neighborhood of the surface, the surface gradient, and surface divergence operators are alternatively defined as

{grad}_{s} {•} : = grad {•} \cdot \overset{\land}{i} and {div}_{s} : = {grad}_{s} {•} : \overset{\land}{i} = grad {•} : \overset{\land}{i}

Finally, the Gauss formulae for the derivatives of the co- and contravariant surface basis vectors read

\partial_{η^{β}} a_{α} = Γ_{α β}^{γ} a_{γ} + {\overset{\land}{k}}_{α β} n and \partial_{η^{β}} a^{α} = - Γ_{β γ}^{α} a^{γ} + {\overset{\land}{k}}_{β}^{α} n

where

Γ_{α β}^{γ} = \partial_{η^{β}} a_{α} \cdot a^{γ}

denote the surface Christoffel symbols and

{\overset{\land}{k}}_{α β}

are the coefficients of the curvature tensor. The curvature tensor

\overset{\land}{k} = {\overset{\land}{k}}_{α β} a^{α} \otimes a^{β}

and twice the mean curvature

\overset{\land}{k} = {\overset{\land}{k}}_{α}^{α}

of the surface

S

are defined as the negative surface gradient and surface divergence of the surface normal

n

⁠, respectively,

\overset{\land}{k} : = {- grad}_{s} n = - \partial_{η^{β}} n \otimes a^{β} and \overset{\land}{k} : = {- div}_{s} n = - \partial_{η^{β}} n \cdot a^{β}

(A1)

The covariant coefficients of the curvature tensor (second fundamental form of the surface) are computed by ${\overset{\land}{k}}_{α β} = a_{α} \cdot \overset{\land}{k} \cdot a_{β} = - a_{α} \cdot \partial_{η^{β}} n$ ⁠.

Curves

A one-dimensional (smooth) curve

C

in the three-dimensional, embedding Euclidian space with coordinates

x

is parameterized by the arc length η as

x = x (η)

The corresponding tangent vector

t \in T C

to the curve, together with the (principal) normal and binormal vectors

m

and

b

⁠, orthogonal to T

C

⁠, are defined by

t : = \partial_{η} x and m : = \partial_{η} t / | \partial_{η} t | and b : = t \times m

Due to the parametrization of the curve in its arc length η, the tangent vector

t

has unit length and the curve line element dc is computed as

d c = | \partial_{η} x | d η = | t | d η = d η

Moreover, we define the mixed-variant curve unit tensor

\tilde{i}

as

\tilde{i} : = t \otimes t = i - m \otimes m - b \otimes b

Clearly the mixed-variant curve unit tensor acts as a curve (idempotent) projection tensor.

The curve gradient and curve divergence operators for vector fields are defined by

{grad}_{c} {•} : = \partial_{η} {•} \otimes t {and div}_{c} : = \partial_{η} {•} \cdot t

As a consequence, observe that grad_c

{•} \cdot m = 0

and grad_c

{•} \cdot b = 0

hold by definition. For fields that are smooth in a of the curve, the curve gradient and curve divergence operators are alternatively defined as

\begin{matrix} {grad}_{c} {•} : = grad {•} \cdot \tilde{i} {and div}_{c} : = {grad}_{c} {•} : \tilde{i} = grad {•} : \tilde{i} \end{matrix}

Finally, the Frénet–Serret formulae for the derivatives of the curve basis vectors read

\partial_{η} t = \tilde{k} m and \partial_{η} m = - \tilde{k} t + \tilde{k'} b and \partial_{η} b = - \tilde{k'} m

where

\tilde{k}

and

\tilde{k}'

denote the scalar valued curvature and torsion of the curve. Based on the Frénet–Serret formulae, the curve divergence of the curve unit tensor

\tilde{i}

renders

{div}_{c} \tilde{i} = \tilde{k} m

Likewise, the curvature tensors

{\tilde{k}}_{| |}, {\tilde{k}}_{⊥}

⁠, and the scalar valued curvature

\tilde{k}

are defined as the curve gradient of the curve tangent

t

and the negative curve gradient and curve divergence of the curve normal

m

⁠, respectively,

\begin{matrix} {\tilde{k}}_{| |} : = {grad}_{c} t = \partial_{η} t \otimes t and {\tilde{k}}_{⊥} : = - {grad}_{c} m = - \partial_{η} m \otimes t, \tilde{k} : = {- div}_{c} m = - \partial_{η} m \cdot t \end{matrix}

The curvature tensors are computed as ${\tilde{k}}_{| |} = \tilde{k} m \otimes t$ and ${\tilde{k}}_{⊥} = \tilde{k} t \otimes t - \tilde{k'} b \otimes t .$

Useful Relations

In order to derive the balance equations presented in Sec. 3, a number of mathematical identities are needed. They are listed here without proof. For scalars R,

\overset{\land}{R}, \bar{R}

⁠, and

\tilde{R}

vectors Q,

\overset{\land}{Q}, \bar{Q}

⁠, and

\tilde{Q}

⁠, and second-order tensors P,

\overset{\land}{P}, \bar{P,} and \tilde{P}

the following relations hold:

\begin{matrix} Div (Q \times P) = Q \times Div P + ε : [Grad Q \cdot P^{t}] in B_{0} \overset{\land}{Div} (\overset{\land}{Q} \times \overset{\land}{P}) = \overset{\land}{Q} \times \overset{\land}{Div} \overset{\land}{P} + ε : [\overset{\land}{Grad} \overset{\land}{Q} \cdot {\overset{\land}{P}}^{t}] in S_{0} \bar{Div} (\bar{Q} \times \bar{P}) = \bar{Q} \times \bar{Div} \bar{P} + ε : [\bar{Grad} \bar{Q} \cdot {\bar{P}}^{t}] in I_{0} \tilde{Div} (\tilde{Q} \times \tilde{P}) = \tilde{Q} \times \tilde{Div} \tilde{P} + ε : [\tilde{Grad} \tilde{Q} \cdot {\tilde{P}}^{t}] in C_{0} \end{matrix}

(B1)

\begin{matrix} Div (R Q) = R Div Q + Q \cdot Grad R in B_{0} \overset{\land}{Div} (\overset{\land}{R} \overset{\land}{Q}) = \overset{\land}{R} \overset{\land}{Div} \overset{\land}{Q} + \overset{\land}{Q} \cdot \overset{\land}{Grad} \overset{\land}{R} in S_{0} \bar{Div} (\bar{R} \bar{Q}) = \bar{R} \bar{Div} \bar{Q} + \bar{Q} \cdot \bar{Grad} \bar{R} in I_{0} \tilde{Div} (\tilde{R} \tilde{Q}) = \tilde{R} \tilde{Div} \tilde{Q} + \tilde{Q} \cdot \tilde{Grad} \tilde{R} in C_{0} \end{matrix}

(B2)

\begin{matrix} Div (Q \cdot P) = Q \cdot Div P + P : Grad Q in B_{0} \overset{\land}{Div} (\overset{\land}{Q} \cdot \overset{\land}{P}) = \overset{\land}{Q} \cdot \overset{\land}{Div} \overset{\land}{P} + \overset{\land}{P} : \overset{\land}{Grad} \overset{\land}{Q} in S_{0} \bar{Div} (\bar{Q} \cdot \bar{P}) = \bar{Q} \cdot \bar{Div} \bar{P} + \bar{P} : \bar{Grad} \bar{Q} in I_{0} \tilde{Div} (\tilde{Q} \cdot \tilde{P}) = \tilde{Q} \cdot \tilde{Div} \tilde{P} + \tilde{P} : \tilde{Grad} \tilde{Q} in C_{0} \end{matrix}

(B3)

Reformulation of the Global Working in Terms of the Various Stress Measures

The steps to reformulate the expression for the global working in terms of the stress measures in the various parts of the body, as given in Eq. , are as follows. The global working

W

₀ is defined in Eq. by

\begin{matrix} W_{0} : = \int_{B_{0}} V \cdot B^{p} d V + \int_{S_{0}^{ext}} \overset{\land}{V} \cdot {\overset{\land}{B}}^{p} d A + \int_{I_{0}} \bar{V} \cdot {\bar{B}}^{p} d A + \int_{C_{0}^{ext}} \tilde{V} \cdot {\tilde{B}}^{p} d L + \int_{S_{0}^{int}} V \cdot [P \cdot M] d A + \int_{L_{0}} \overset{\land}{V} \cdot [\overset{\land}{P} \cdot \overset{\land}{N}] dL + \int_{C_{0}^{int}} \bar{V} \cdot [\bar{P} \cdot \bar{M}] dL + \sum_{\partial C_{0}^{ext}} \tilde{V} \cdot [\tilde{P} \cdot \tilde{N}] \end{matrix}

Using the balances of linear momentum to express the various stress measures in terms of their associated bulk, surface, interface, and curve forces, denoted

B^{p}, {\overset{\land}{B}}^{p}, {\bar{B}}^{p}, and {\tilde{B}}^{p}

⁠, respectively, the global working becomes

\begin{matrix} W_{0} : = \int_{B_{0}} V \cdot [- Div P] d V + \int_{S_{0}^{ext}} \overset{\land}{V} \cdot [- \overset{\land}{Div} \overset{\land}{P} + P \cdot N] d A + \int_{I_{0}} \bar{V} \cdot [- \bar{Div} \bar{P} - [[P]] \cdot \bar{N}] d A + \int_{C_{0}^{ext}} \tilde{V} \cdot [- \tilde{Div} \tilde{P} + \bar{P} \cdot \tilde{N_{i}} - [[\overset{\land}{P}]] \cdot {\tilde{N}}_{S} \end{matrix}] d L + \int_{S_{0}^{int}} V \cdot [P \cdot M] d A + \int_{L_{0}} \overset{\land}{V} \cdot [\overset{\land}{P} \cdot \overset{\land}{N}] d L + \int_{C_{0}^{int}} \bar{V} \cdot [\bar{P} \cdot \bar{M}] d L + \sum_{\partial C_{0}^{ext}} \tilde{V} \cdot [\tilde{P} \cdot \tilde{N}]

It follows from the definition of the control regions, given in Sec. 2.3, that

\begin{matrix} \int_{S_{0}^{int}} {•} d A = \int_{\partial B_{0}} {•} d A - \int_{S_{0} ext} {•} d A, \int_{C_{0}^{int}} {\bar{•}} d L = \int_{\partial I_{0}} {\bar{•}} d L - \int_{C_{0} ext} {\bar{•}} d L \end{matrix}

Furthermore, the outward normal to

\partial B_{0}

on the surface is the same as the outward normal to the surface, i.e.,

M |_{S_{0}^{ext}}

= N. In an identical fashion to the surface

\bar{M} |_{C_{0}^{ext}} = {\tilde{N}}_{i}

⁠. Due to kinematic slavery on the surface

V |_{S_{0}^{ext}} = \overset{\land}{V}

and

\bar{V} |_{C_{0}^{ext}} = \tilde{V}

⁠. From these observations, the following two terms in the global working can be restated as follows:

\begin{matrix} \int_{S_{0}^{int}} V \cdot [P \cdot M] d A = \int_{\partial B_{0}} V \cdot [P \cdot M] d A - \int_{S_{0}^{ext}} \overset{\land}{V} \cdot [P \cdot N] d A, \int_{L_{0}} \overset{\land}{V} \cdot [\overset{\land}{P} \cdot \overset{\land}{N}] d L = \int_{\partial I_{0}} \bar{V} \cdot [\bar{P} \cdot \bar{M}] d L - \int_{C_{0}^{ext}} \tilde{V} \cdot [\bar{P} \cdot {\tilde{N}}_{i}] d L \end{matrix}

Thus, the global working becomes

\begin{matrix} W_{0} = \int_{B_{0}} V \cdot [- Div P] d V + \int_{S_{0}^{ext}} \overset{\land}{V} \cdot [- \overset{\land}{Div} \overset{\land}{P}] d A + \int_{I_{0}} \bar{V} \cdot [- \bar{Div} \bar{P} - [[P]] \cdot \bar{N}] d A + \int_{C_{0}^{ext}} \tilde{V} \cdot [- \tilde{Div} \tilde{P} - [[\overset{\land}{P}]] \cdot {\tilde{N}}_{S}] d L + \int_{\partial B_{0}} V \cdot [P \cdot M] d A + \int_{\partial I_{0}} \bar{V} \cdot [\bar{P} \cdot \bar{M}] d L + \int_{L_{0}} \overset{\land}{V} \cdot [\overset{\land}{P} \cdot \overset{\land}{N}] d L + \sum_{\partial C_{0}^{ext}} \tilde{V} \cdot [\tilde{P} \cdot \tilde{N}] \end{matrix}

and using the identities yields

\begin{matrix} = \int_{B_{0}} P : Grad V d V - \int_{B_{0}} Div (V \cdot P) d V + \int_{S_{0}^{ext}} \overset{\land}{P} : \overset{\land}{Grad} \overset{\land}{V} d A - \int_{S_{0}^{ext}} \overset{\land}{Div} (\overset{\land}{V} \cdot \overset{\land}{P}) d A + \int_{I_{0}} \bar{P} : \bar{Grad} \bar{V} d A - \int_{I_{0}} \bar{Div} (\bar{V} \cdot \bar{P}) d A - \int_{I_{0}} \bar{V} \cdot [[P]] \cdot \bar{N} d A + \int_{C_{0}^{ext}} \tilde{P} : \tilde{Grad} \tilde{V} d L - \int_{C_{0}^{ext}} \tilde{Div} (\tilde{V} \cdot \tilde{P}) d L - \int_{C_{0}^{ext}} \tilde{V} \cdot [[\overset{\land}{P}]] \cdot {\tilde{N}}_{s} d L + \int_{\partial B_{0}} V \cdot P \cdot M d A + \int_{\partial I_{0}} \bar{V} \cdot \bar{P} \cdot \bar{M} d L + \int_{L_{0}} \overset{\land}{V} \cdot \overset{\land}{P} \cdot \overset{\land}{N} d L + \sum_{\partial C_{0}^{ext}} \tilde{V} \cdot \tilde{P} \cdot \tilde{N} \end{matrix}

Finally, using the extended divergence theorems for the various parts of the body gives

\begin{matrix} W_{0} = \int_{B_{0}} P : Grad V d V + \int_{S_{0}^{ext}} \overset{\land}{P} : \overset{\land}{Grad} \overset{\land}{V} d A + \int_{I_{0}} \bar{P} : \bar{Grad V} d A + \int_{C_{0}^{ext}} \tilde{P} : \tilde{Grad} \tilde{V} d L - \int_{\partial B_{0}} V \cdot P \cdot M d A + \int_{I_{0}} [[V \cdot P]] \cdot \bar{N} d A - \int_{L_{0}} \overset{\land}{V} \cdot \overset{\land}{P} \cdot \overset{\land}{N} d L + \int_{S_{0}^{ext}} \overset{\land}{C} \overset{\land}{V} \cdot \overset{\land}{P} \cdot N d A + \int_{C_{0}^{ext}} [[\overset{\land}{V} \cdot \overset{\land}{P}]] \cdot {\tilde{N}}_{s} d L - \int_{\partial I_{0}} \bar{V} \cdot \bar{P} \cdot \bar{M} d L + \int_{I_{0}} \bar{C} \bar{V} \cdot \bar{P} \cdot \bar{N} d A - \sum_{\partial C_{0}^{ext}} \tilde{V} \cdot \tilde{P} \cdot \tilde{N} + \int_{C_{0}^{ext}} \tilde{C} \tilde{V} \cdot \tilde{P} \cdot {\tilde{N}}_{c} d L - \int_{I_{0}} \bar{V} \cdot [[P]] \cdot \bar{N} d A - \int_{C_{0}^{ext}} \tilde{V} \cdot [[\overset{\land}{P}]] \cdot {\tilde{N}}_{s} d L + \int_{\partial B_{0}} V \cdot P \cdot M d A + \int_{\partial I_{0}} \bar{V} \cdot \bar{P} \cdot \bar{M} d L + \int_{L_{0}} \overset{\land}{V} \cdot \overset{\land}{P} \cdot \overset{\land}{N} d L + \sum_{\partial C_{0}^{ext}} \tilde{V} \cdot \tilde{P} \cdot \tilde{N} = \int_{B_{0}} P : Grad V d V + \int_{S_{0}^{ext}} \overset{\land}{P} : \overset{\land}{Grad} \overset{\land}{V} d A + \int_{I_{0}} \bar{P} : \bar{Grad} \bar{V} d A + \int_{C_{0}^{ext}} \tilde{P} : \tilde{Grad} \tilde{V} d L + \int_{I_{0}} [[V \cdot P]] \cdot \bar{N} d A + \int_{C_{0}^{ext}} [[\overset{\land}{V} \cdot \overset{\land}{P}]] \cdot {\tilde{N}}_{s} d L - \int_{I_{0}} \bar{V} \cdot [[P]] \cdot \bar{N} d A - \int_{C_{0}^{ext}} \tilde{V} \cdot [[\overset{\land}{P}]] \cdot {\tilde{N}}_{s} d L \end{matrix}

The interface is assumed to be coherent; thus $[[V]]$ = 0 and $[[\overset{\land}{V}]]$ = 0. Therefore, $\bar{V} \cdot [[P]]$ = $[[V \cdot P]]$ and $\tilde{V} \cdot [[\overset{\land}{P}]]$ = $[[\overset{\land}{V} \cdot \overset{\land}{P}]]$ and the last four terms vanish. What remains is the global working given in Eq. (22).

Derivation of the Interface Temperature Evolution Equation

It is enlightening to review the procedure to obtain the nonstandard interface temperature evolution Eq. (55)₃.

The balance of internal energy on the interface ₃ reads

\begin{matrix} \bar{P} : \bar{Grad} \bar{V} - \bar{Div} \bar{Q} + \bar{Q} = D_{t} \bar{E} & with & \bar{Q} = {\bar{Q}}^{p} - [[Q]] \cdot \bar{N} & on & I_{0} \end{matrix}

The rate of change of the internal energy on the interface D_t

\bar{E}

is expanded using the definition of the Helmholtz energy ₃ as follows:

D_{t} \bar{E} = D_{t} (\bar{Ψ} + \bar{Θ} \bar{Ξ})

and from the parametrization of the Helmholtz energy given in Eq. _3,

= [\frac{\partial \bar{Ψ}}{\partial \bar{F}} + \frac{\partial \bar{Ψ}}{\partial \bar{n}} \cdot \frac{\partial \bar{n}}{\partial \bar{F}}] : D_{t} \bar{F} + \frac{\partial \bar{Ψ}}{\partial \bar{Θ}} D_{t} \bar{Θ} + \bar{Ξ} D_{t} \bar{Θ} + \bar{Θ} D_{t} \bar{Ξ},

and from the constitutive relations _3,

= \bar{P} : D_{t} \bar{F} + \bar{Θ} D_{t} \bar{Ξ}

Inserting this result into balance of energy on the interface ₃ yields

\bar{P} : \bar{Grad} \bar{V} - \bar{Div} \bar{Q} + \bar{Q} = \bar{P} : D_{t} \bar{F} + \bar{Θ} D_{t} \bar{Ξ}

and recalling that

\bar{Grad} \bar{V} = \frac{\partial}{\partial \bar{X}} (\bar{V}) = \frac{\partial}{\partial \bar{X}} (D_{t} \bar{x}) = D_{t} (\frac{\partial \bar{x}}{\partial \bar{X}}) = D_{t} \bar{F}

allowing the balance of energy to be expressed as

- \bar{Div} \bar{Q} + \bar{Q} = \bar{Θ} D_{t} \bar{Ξ}

(D1)

The right-hand side of Eq. can be simplified as follows:

\bar{Θ} D_{t} \bar{Ξ} = \bar{Θ} \frac{\partial \bar{Ξ}}{\partial \bar{Θ}} D_{t} \bar{Θ} + \bar{Θ} \frac{\partial \bar{Ξ}}{\partial \bar{F}} : D_{t} \bar{F} + \bar{Θ} \frac{\partial \bar{Ξ}}{\partial \bar{n}} \cdot D_{t} \bar{n}

and using the constitutive relation

\bar{Ξ} = - (\partial \bar{Ψ} / \partial \bar{Θ}),

\begin{matrix} = - \bar{Θ} \frac{\partial^{2} \bar{Ψ}}{\partial \bar{Θ} \partial \bar{Θ}} D_{t} \bar{Θ} - \bar{Θ} \frac{\partial^{2} \bar{Ψ}}{\partial \bar{F} \partial \bar{Θ}} : D_{t} \bar{F} + \bar{Θ} \frac{\partial \bar{Ξ}}{\partial \bar{n}} \cdot \frac{\partial \bar{n}}{\partial \bar{F}} : D_{t} \bar{n} = \underset{{\bar{c}}_{\bar{F}}}{\underset{︸}{- \bar{Θ} \frac{\partial^{2} \bar{Ψ}}{\partial \bar{Θ} \partial \bar{Θ}}}} D_{t} \bar{Θ} - \bar{Θ} \frac{\partial}{\partial \bar{Θ}} [\underset{\bar{P}}{\underset{︸}{\frac{\partial \bar{Ψ}}{\partial \bar{F}} + \frac{\partial \bar{Ψ}}{\partial \bar{n}} \cdot \frac{\partial \bar{n}}{\partial \bar{F}}}}] : D_{t} \bar{F} \end{matrix}

where ${\bar{c}}_{\bar{F}}$ is the interface heat capacity. Finally, inserting this result into Eq. (D1) gives the equation for the evolution of the interface temperature as in Eq. (55)₃.

1

Pyotr Leonidovich Kapitza (1894–1984) was a physicist and Nobel laureate who first proposed the presence of thermal interface resistance corresponding to a discontinuous temperature field across an interface for liquid helium; see Kapitza [82].

2

Here and henceforth, the subscripts t and 0 shall designate spatial and material quantities, respectively, unless specified otherwise.

3

Note that inertial forces are omitted. Furthermore, in the absence of mass flux, mass is conserved according to the standard relations given in Table 1.

4

The reduced dissipation inequality on the interface is alternatively expressed in the literature (see, e.g., Ref. [47]) in one of the following forms:

\begin{matrix} {\bar{D}}_{⊥} = \bar{Θ} [[Q \cdot [\frac{1}{Θ} - \frac{1}{\bar{Θ}}]]] \cdot \bar{N} = \bar{Θ} [[\frac{Q}{Θ} \cdot [1 - \frac{Θ}{\bar{Θ}}]]] \cdot \bar{N} = [[\frac{Q}{Θ} \cdot [\bar{Θ} - Θ]]] \cdot \bar{N} \geq 0 \end{matrix}

5

The term volumetric is used in analogy to the bulk. Nevertheless, it has a different meaning on the surface. A volumetric deformation in the bulk is a deformation mode that changes the volume uniformly. A volumetric surface deformation, however, is a deformation mode that changes the area uniformly. In this sense the term spherical seems more appropriate. Nonetheless, for the sake of consistency, the term volumetric is used henceforth.

6

For the sake of simplicity, it is assumed that the surface is an interface between the fluid and an outside vacuum.

7

Often in the literature $\overset{\land}{c}$ is written as sum of the principal curvatures $\overset{\land}{c}$ = −[1/r₁+ 1/r₂] where r₁ and r₂ denote the principal radii of curvature. Based on this definition, the curvature is negative if the surface curves away from the normal.

8

For isotropic thermal conduction in the material configuration $G \equiv I$ and $G \equiv I \otimes F^{- t}$ ⁠.

9

For isotropic thermal conduction in the material configuration $\overset{\land}{G} \equiv \overset{\land}{I} and \overset{\land}{G} \equiv \overset{\land}{I} \otimes {\overset{\land}{F}}^{- t} .$

10

Numerical investigations of a more general case where the surface, interface, and curve are all considered involve additional complications while providing little additional insight into the problem and, therefore, are not studied here.

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Strong ellipticity	μ > 0 and λ + 2μ > 0	$\overset{\land}{μ} > 0$ and $\overset{\land}{λ} + 2 \overset{\land}{μ} > 0$
Pointwise stability	μ > 0 and 3λ + 2μ > 0	$\overset{\land}{μ} > 0$ and $\overset{\land}{λ} + \overset{\land}{μ} > 0$

Thermomechanics of Solids With Lower-Dimensional Energetics: On the Importance of Surface, Interface, and Curve Structures at the Nanoscale. A Unifying Review

Abstract

Introduction

Preliminaries

Notation and Definitions

Kinematics

Deformation Gradients

Control Regions

Key Definitions and Identities

Jump and Average Relations

Integral Relations

Piola Identity and Piola Transform

Superficial and Tangential Tensor Fields

Governing Equations

Mechanical Power

Bulk

Surface

Interface

Curve

Material and Spatial Formulations

Thermal Power

Entropy Power

Thermohyperelastic Helmholtz Energy

Constitutive Equations

Reduced Dissipation Inequalities

Dissipation Inequalities in the Bulk and Along the Surface, Interface, and Curve

Dissipation Inequalities Across the Surface, Interface, and Curve

Surface.

Interface.

Coldness-based discussion.

Temperature-based discussion

Summary.

Curve.

Temperature Evolution Equations

Weak Formulation

Weak Formulation—Mechanical

Weak Formulation—Thermal

Comparison With Several Other Models

Standard Thermomechanical Solids

Highly Conducting Interfaces

Kapitza Interfaces

Thermomechanical Surfaces

Thermomechanical Highly Conducting Interfaces

Generalized Kapitza Interfaces

Material Modeling

Hyperelastic Material in the Bulk

Hyperelastic Material on the Surface

Example: Young–Laplace Equation

Thermohyperelastic Material

Bulk

Surface

Interface

Curve

Numerical Examples

Surface

Surface Tension

Thermomechanical Surface

Bulk effect.

Mechanical surface effect.

Thermomechanical surface effect.

The effect of surface conductivity.

The effect of surface heat capacity.

The effect of the surface Gough–Joule contribution.

Interface

Kapitza Interface

Highly Conducting Interface

Conclusion

Acknowledgment

Geometry of Surfaces and Curves

Surfaces

Curves

Useful Relations

Reformulation of the Global Working in Terms of the Various Stress Measures

Derivation of the Interface Temperature Evolution Equation

References

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