## Abstract

Capture of a prey by spider orb webs is a dynamic process with energy dissipation. The dynamic response of spider orb webs under prey impact requires a multi-scale modeling by considering the material microstructures and the assembly of spider silks in the macro-scale. To better understand the prey capture process, this paper addresses a multi-scale approach to uncover the underlying energy dissipation mechanisms. Simulation results show that the microstructures of spider dragline silk play a significant role on energy absorption during prey capture. The alteration of the microstructures, material internal friction, and plastic deformation lead to energy dissipation, which is called material damping. In addition to the material damping in the micro-scale modeling, the energy dissipation due to drag force on the prey is also taken into consideration in the macro-scale modeling. The results indicate that aerodynamic drag, i.e., aero-damping, plays a significant role when the prey size is larger than a critical size.

## 1 Introduction

The primary function of spider webs is prey capture, which is critical for spiders’ survival in nature [1]. Among various spider webs, 2D circular webs built by orb-weaving spiders are known as orb webs. As an introduction, consider the two-step construction of a spider orb web [2]: dragline silk threads are first used to build the main body of the orb web (e.g., moorings, frames, and radials); the main body is then spirally covered by sticky capture silk threads. It is well known that capture silk threads possess totally different composition from that of dragline silk threads [2]. Therefore, their functions are distinct too. According to Gosline et al. [3], capture silk threads are intended to glue preys and they can merely support little force. In contrast, dragline silk fibers have excellent mechanical properties (i.e., high strength and unmatched toughness) and therefore bear almost all structural loads [4,5]. Previous studies [68] show that spider web behavior not only depends on the complex architectures of spider webs but also on its remarkable properties.

The prey capture process (i.e., sticking and interception) consists of two important steps. In order to subdue a prey, a nearly blind spider (i.e., an orb weaving spider) relies on the vibrational signals induced by prey and propagated via the web. In order to localize and discriminate different events, spiders have special information-acquisition strategies where their webs act as an extended cognition system [15]. The mechanical perturbations triggered by the contiguous environmental stimuli (e.g., prey impacting and wind blowing) result in different web response. Through vibration propagation and attenuation in the webs, spiders use their eight legs [16] to perceive the locations and discriminate the types of the vibration sources. Although different species of spiders develop their own specific information-acquisition strategies [17], the basic physical principle is the same [18]. The radial threads of the webs mechanically transmit vibration source signals [19] (e.g., amplitude, frequency spectrum of vibration, and timing) to the legs of spiders where the signals are filtered [20]. Through the characteristics of the signals, spiders are able to match the information to that in their memory or memorize the signal at the first time occurrence [15]. For example, spiders discriminate biotic source (e.g., insect) and abiotic source (e.g., wind) by the frequency of the vibration signal, where biotic source has a much higher vibration frequency [20]. However, there could be overlaps in the frequency response from biotic and abiotic depending on the prey size, mass, and web geometries, thus not all vibration sources could be distinguished by spiders [17].

While there have been many excellent research work on the modeling of spider webs, there is a need of further understanding the dynamic response of such structures. Such a study not only provides scientific information on these amazing structures but may also provide an approach toward developing bio-inspired structures, such as sensors [21] and dampers [22]. Since the cross-sectional diameters of silk threads are very small (i.e., 0.03–4 μm), they are generally modeled as tensile structural members in analysis. From the structural engineering point of view, spider webs are often used as a good example of tensegrity. Many interesting problems arise from the interplay between geometry and mechanics in a tensegrity such as how the tensegrity topology affects the performance of a tensegrity (e.g., impact energy absorption and vibration attenuation). Thus, the studies may provide inspirations for the design of lightweight multifunctional structures [23]. There have been many finite element models for mimicking the behavior of spider webs with complex geometry subjected to various loadings (e.g., wind load and impact load) [11,12,14,17,2325]. Without loss of generality, Tietsh et al. [26] and Yu et al. [14] investigate the dynamic response of an orb web by considering a single dragline fiber and modeling it as a one-degree-of-freedom system in their simulation study. While Tietsh et al. consider damping effects in their model from both material and drag, the model neglects material damage effects. For the model proposed by Yu et al., while large plastic deformations are accounted for, their constitutive model does not consider viscous effect and Mullins effect. In addition, their model does not consider the effect of drag force in their dynamic analyses.

The remainder of this paper is organized as follows. In Sec. 2, a brief review of our developed constitutive model for spider dragline silk is presented. Section 3 presents a theoretical investigation on the dynamic response of an orb web subjected to an impact load. Thereby, the energy absorption process is investigated and discussed. Paper is summarized after, by presenting concluding remarks.

## 2 Constitutive Model for Spider Dragline Silk

As a special biological material [5,25,27], spider dragline silk contains proteins in the form of α-helix (80–85% in volume) and β-sheet (15–20% in volume). α-helix proteins are flexible chains. β-sheet proteins are in the form of rigid plates formed by folded protein chains and bonded by hydrogen bonds. While the volume fraction of β-sheet proteins in spider dragline silk is small, they play a significant role like the carbon-blacks in volcanized rubbers [28] that crosslink polymer networks [2932]. In addition to α-helix and β-sheet, it is found that some segments of α-helix proteins could be folded and bonded by weak inner-monomer hydrogen bonds, resulting in a structure similar to β-sheet [27,29]. Thus, it is called β-turn or β-spiral. It is also found that β-spirals are easy to unfold under small tension, which accounts for the high extensibility of spider dragline silk [33]. Based on the knowledge about the micro structures of spider dragline silk, a new constitutive model is developed as follows.

Proteins are biological macro-molecules. Their mechanical behaviors can be understood in virtue of polymer physics. In spider dragline silk, the flexible α-helix proteins can be modeled as polymer chains, which consist of a number of monomers. In the micro-scale, the number of monomers nc, its length lc, and the bond angle θ between neighboring monomers are used to describe a polymer chain. By using statistical mechanics and corresponding assumptions, various force-extension relations for a polymer chain have been derived. Herein, the well-known worm-like chain (WLC) model is adopted [34,35] since it is suitable for modeling stiff polymer chains with a fixed bond angle. The force-extension for the WLC model is defined as
$fW=kBT4lp(4RRmax+1(1−RRmax)2−1)$
(1)
where lp = 2lc/θ2 is the persistent length, Rmax = nclc is the contour length of the chain, R is the mean end-to-end distance of the chain, representing the chain extension, kB is the Boltzmann constant, and T is temperature in Kelvin, which is assumed to be constant (i.e., isothermal). An integration of Eq. (1) with respective to R gives rise to the Helmholtz free energy FW [36,37]
$FW=kBTN2(2Λ~2+11−Λ~−Λ~)+F0$
(2)
where F0 is an integration constant, $Λ~=R/Rmax$ is the average chain elongation, and N is the Kuhn number defined as [35,38], which reflects the deformability of a WLC.
$N=Rmax2lp$
(3)
Inspired by the work [39], we rewrite the average chain elongation in the following form:
$Λ~=RRmax=R0RmaxRR0=ΛΛloc$
(4)
where the root-mean-square end-to-end distance R0 is introduced as [35]
$R0=2lp2[Rmaxlp−1+exp(−Rmaxlp)]$
(5)
Combining Eqs. (4) and (5), the locking stretch Λloc of a WLC is defined as
$Λloc=RmaxR0=L^(N)=2N2N−1+exp(−2N)$
(6)
which is a monotonically increasing function of Kuhn number N. Λloc is therefore another parameter reflecting the stretchability of a WLC.
The stretch Λ of a WLC can be defined as
$Λ=RR0$
(7)
whose range is determined by the locking stretch as Λ ∈ (0, Λloc).
The effective Helmholtz free energy of a WLC chain, Eq. (2), now can be presented as
$FW=kBT2J^(Λ,N)$
(8)
where the constant term F0 is neglected, and $J^(Λ,N)$ is a function of stretch Λ and Kuhn number N as
$J^(Λ,N)=N(2Λ2Λloc2+11−Λ/Λloc−ΛΛloc)$
(9)
For convenience of discussion, the first-order partial derivative of $J^$ function with respective to Λ is also explicitly given as
$Y^(Λ,N)=∂J^∂Λ=N(4ΛΛloc2+1(1−Λ/Λloc)21Λloc−1Λloc)$
(10)

So far, we have modeled the α-helix proteins as WLCs, where the force-stretch relation is obtained by multiplying Eq. (10) with kBT/2. The β-sheet proteins are simply modeled as rigid crosslinks connecting α-helix proteins [40].

Now, we turn our attention to the macro-scale deformation (force-stretch relation) where a spider dragline silk fiber with a very small cross section is subjected to tension. Since fibers with large aspect ratio (length/diameter) can easily buckle under compression, here we are interested in its tensile behavior. It is observed that α-helix proteins are highly oriented in the axial direction of the spider dragline silk fiber [27], indicating that spider dragline silk is anisotropic material. Thus, in the macro-scale, the spider dragline silk can be modeled as an anisotropic string. In order to bridge the gap between the micro-scale WLC chain model and the macro-scale string model, a meso-scale model or network model is required, where α-helix chains are interconnected by β-sheet crosslinks in the meso-scale.

First, we consider the elastic extension of the spider dragline silk fiber. Inspired by Diani et al. [41], we focus on an anisotropic network model with three material directions $u1=(±1,0,0)$, $u2=(0,±1,0)$, and $u3=(0,0,±1)$(see Fig. 1). Considering the number of α-helix chains per unit volume in each directions as $ni,i=1,2,3$, and total α-helix chain density as n = n1 + n2 + n3. The chain volume fraction in each direction can be defined as ωi = ni/n. Furthermore, consider the stretch along the fiber direction as λ1 (λ2 and λ3 denote the stretches along the other two orthogonal transverse directions). Note that the identity λ1λ2λ3 = 1 holds due to volume conservation [42]. By assuming affine deformation (i.e., Λ1 = λ1, Λ2 = λ2, and Λ3 = λ3), the effective Helmholtz free energy density of the fiber is given as follows:
$W=μω1J^(λ1,N1)+μω2J^(λ2,N2)+μω3J^(λ3,N3)$
(11)
where μ = nkBT/2 is the shear modulus. The Cauchy stresses are therefore derived as
$σ11=∂W∂λ1λ1−p=μω1Y^(λ1,N1)λ1−p$
(12)
$σ22=∂W∂λ2λ2−p=μω2Y^(λ2,N2)λ2−p$
(13)
and
$σ33=∂W∂λ3λ3−p=μω3Y^(λ3,N3)λ3−p$
(14)
where p is the hydrostatic pressure. Note that, for uniaxial extension, the free boundary conditions imply σ22 = 0 and σ33 = 0. By using the boundary conditions, hydrostatic pressure can be eliminated. This results in
$σ11=σ11−σ22=μω1Y^(λ1,N1)λ1−μω2Y^(λ2,N2)λ2$
(15)
and
$σ22−σ33=μω2Y^(λ2,N2)λ2−μω3Y^(λ3,N3)λ3=0$
(16)
Fig. 1
Fig. 1
Close modal
In addition, note that σ11 = 0 when λ1 = 1. Introducing this condition into Eq. (15) results in
$μω1Y^(1,N1)−μω2Y^(1,N2)=0$
(17)
For simplicity, we assume that the chain densities in the second and third directions are the same, i.e., ω2 = ω3. Thus, we denote them as
$z=ω2=ω3=1−ω2$
(18)
where ω indicates the chain density in the fiber direction (ω1 = ω). Note that ω is a material parameter reflecting anisotropic distribution of chains. Herein, it is assumed that ω is a constant through the deformation. Denote σ11 as σ, λ1 as λ, λ2 as x, and N1 as N. Equation (15) can be presented as
$σ=μωY^(λ,N)λ−μzY^(x,N2)x$
(19)
where the unknown x is determined by solving the following nonlinear equation, obtained from Eq. (16)
$Y^(x,N2)λx2−Y^(1λx,N3)=0$
(20)
where the volume conservation assumption λ3 = 1/λx is applied. Furthermore, from the initial condition λ = x = 1, Eq. (20) yields
$Y^(1,N2)−Y^(1,N3)=0$
(21)
Thus, it can be concluded that N2 = N3. Furthermore, the value of N2 can be obtained from Eq. (19) in terms of N
$σ0=ωY^(1,N)−zY^(1,N2)=0$
(22)

Here, we define N2 as y for simplicity.

The two implicit functions based on Eqs. (20) and (22) are used to find x and y
$x=f(λ,y)$
(23)
where
$y=g(ω,N)$
(24)
Both x and y can be evaluated numerically. It is worth mentioning that $x=1/λ$ can also be explicitly evaluated since it was assumed ω2 = ω3. However, here we have presented a more general case. y is a material parameter only requiring evaluation once based on ω and N at the beginning of the calculation. However, the value of x must be updated as the deformation λ changes. Upon the fulfillment of the evaluation of x and y, the axial stress can be numerically determined as
$σ=σ^(λ,N)=μωY^(λ,N)λ−μzY^(x,y)x$
(25)
It is noted that currently derived model is an anisotropic elastic spring model, i.e., stress σ–stretch λ relationship. We call it J-spring model, whose Helmholtz free energy density can be written as
$WJ=W^J(λ,μ,N)=μωJ^(λ,N)+μzJ^(x,y)+μzJ^(1/λx,y)−μωJ^(1,N)−2μzJ^(1,y)$
(26)

Next, we expand the model to account for the viscous damping effect of spider dragline silk.

Inspired by the theory of finite viscoelasticity [43] and double network theory [44], a new standard rheological model which incorporates two J-springs and a Newtonian dashpot is proposed. The deformation of the first J-spring is measured by λ. The deformation of the dashpot is measured by ξ. The deformation of the second J-spring is derived based on the multiplicative split of the deformation gradient [43] as
$λe=λ/ξ$
(27)
Therefore, the Cauchy stress contributed by the two J-springs are given, respectively, as
$σα=ζμ[ωY^(λ,N)λ−zY^(x,y)x]$
(28)
and
$σβ=(1−ζ)μ[ωY^(λe,N)λe−zY^(xe,y)xe]$
(29)
where ζ is called the modulus ratio, which is a material parameter, and xe = f(λe, y). The overall Cauchy stress is therefore obtained as
$σ=σα+σβ$
(30)
The Helmholtz free energy density for double J-spring network can therefore be written as
$WDJ=W^DJ(λ,λe,μ,N,ζ)=ζW^J(λ,μ,N)+(1−ζ)W^J(λe,μ,N)$
(31)
In addition, the evolution rule (Newtonian fluid law) is introduced as
$ξ˙=ξησβ$
(32)
where η is the dynamic viscosity, which is a material parameter. The above material model incorporates viscoelasticity; however, it is not yet a complete model and lacks contribution of damage (i.e., α-helix chains rupture, and unfolding of β-spirals sheets). Under a tensile test, chains attached to the rigid β-sheets will break when stretch reaches a threshold. Since a shorter chain has a smaller threshold, the breakage leads to the decline of chain density and the increase of the mean chain length. We call it type-1 damage, which is also known as Mullins effect. The second type of damage (type-2) is attributed to the unfolding of β-spirals. Under a very small tensile force, nearly zero, they are unfolded due to the breakage of weak hydrogen bonds. Motivated by these two damage mechanisms, we further modified our model.
Our current model can naturally account for type-1 damage by choosing N as an internal variable according to the theory of network alteration [45,46]. The evolution of N is presented as
$N˙={Kbλ˙,λ=λmaxandλ˙>00,others$
(33)
where Kb is a material parameter indicating how fast the network alters and λmax denotes the maximum stretch in a loading history. The λmax evolves according to the following rule
$λ˙max={λ˙,λ=λmaxandλ˙>00,others$
(34)
Note that its value is given as λmax|t=0 = λ|t=0 = 1 when no damage exists. Furthermore, the evolution of shear modulus can be presented as
$μ˙=−μN˙/N$
(35)
since μN is constant due to the conservation of mass [45,46]. Note that μ|t=0 = μ0 and N|t=0 = N0 both parameters become time dependent with the deformation.
To account for the type-2 damage or the unfolding of β-spirals, we modified our model by introducing a new rheological model (see Fig. 2), where a plastic element was added to the previously introduced standard model. Remarkably, the plastic element resists compression but yields to extension. The stretch δ of the plastic element is defined as an internal damage variable. Therefore, the total stretch λtot is re-defined based on the multiplication decomposition of deformation gradient as
$λtot=λδ$
(36)
where λ is the stretch of the standard model as previously presented. The evolution of δ is defined as
$δ˙={Kpλ˙tot,λ=λmaxandλ˙tot>00,others$
(37)
Fig. 2
Fig. 2
Close modal

Note that there is no type-2 damage when δ = 1.

As a final remark, a constitutive model for spider dragline silk/fibers is presented by considering its viscoelastic behavior and material degradation. The model contains seven physically based parameters, which could be readily evaluated by fitting to experimental data. To characterize these parameters, not only monotonic tensile test but also cyclic tests should be adopted. As a typical example, the experimental data [47] is adopted for this evaluation. The simulation result (see Fig. 3) shows that our model mimics the cyclic behavior of the spider dragline silk well. Our model can also be used to predict compressive behavior of dragline silk. However, there is no experimental data in this regime since fibers buckle under compression. For this reason, we are just interested in tensile behavior of spider dragline silk. The effective stress σeff is defined as
$σeff={σ,σ>00,σ≤0$
(38)
Fig. 3
Fig. 3
Close modal

Using the experimental data in Ref. [47] and the least square optimization technique, the material constants for our model are found as μ0 = 0.19 (GPa), N0 = 0.35, ω = 0.40, ζ = 0.29, η = 0.25 (GPa·min), Kb = 0.37, and Kp = 0.36.

## 3 Energy Dissipation in Prey Capture

The previous section presents a constitutive model for simulating spider dragline silk fibers under cyclic loadings. Based on the model and the material parameters, we will investigate the energy dissipation phenomena in the prey capture process. Various full web models can be found in the literature. However, to understand how various mechanisms contribute to the energy dissipation, a single spider dragline silk fiber as a representative model of the web is used to evaluate energy dissipation.

Consider a fiber with initial cross section A0 and initial length L0 (note that A0L0 is a constant). The two ends of the fiber are fixed to a rigid frame after a pre-stretch λpr. Due to volume conservation, the pre-stretch results in a new cross-sectional area of the fiber denoted as $A=λpr−1Ao$ and a new length L = λprL0 (see Fig. 4(a)). The pre-stretch leads to tension and damage to the fiber prior to prey impact. It is assumed the pre-stretch is applied at the moment of impact (i.e., the stress relaxation is negligible if the pre-stretch is applied much before the impact). In addition, we assume bending energy of the fiber can be neglected since the radius of the fiber is very small compared with its length. The inertial effect of the fiber is also neglected because of its lightweight. This can be justified by considering that the mass of 1 m long silk fiber with 4 µm diameter is about 1.57 × 10−5 g [25]. However, the mass of prey considered in this paper is 0.5 g which is much higher than the dragline silk fiber. Furthermore, we assume the fiber buckles when subjected to compressive loads and does not break, however can be damaged. Next, we consider a prey with mass m impacts the fiber with a normal velocity v0 (see Fig. 4(b)). The prey is modeled as a spherical particle and it sticks to the fiber after impact. Consider that several orb weavers monitor web vibrations with a signaling thread by staying off their webs [17]. In addition, consider that most of the previous studies model the web without the spider. For this part of research, we do not consider the mass of spider and its location. Therefore, the dynamic response of the prey is chosen as the object of the study in this section. A full model of the web which includes both spider and prey is currently being considered for our next extension of this investigation.

Fig. 4
Fig. 4
Close modal
To describe the equations of motion of the prey, a 2D Cartesian coordinate system is established to record its position (see Fig. 4(a))
$u=[u1u2]$
(39)

Fiber is separated into two segments from the impact position (see Fig. 4(b)). The length of each segments is denoted as L1 and L2. To indicate the impact position, a nondimensional number φ = L1/L is introduced, where L = L1 + L2. After impact, the two segments deform to new length l1 and l2, and new stretch as λ1 = l1/L1 and λ2 = l2/L2, respectively. As shown later, the deformation of the fiber influences the motion of the prey significantly. The resultant axial forces of each segment of the fiber are presented as $F1=λ1−1Aσ1eff$ and $F2=λ2−1Aσ2eff$, where the effective Cauchy stresses σ1eff and σ2eff for each segment are evaluated based on the constitutive model.

In addition to these forces, we considered the effects of gravity and drag on the motion of the prey. By drawing free body and mass-acceleration diagrams, the equations of motion can be established as
$mu¨=Fweb+Fgravity+Fair$
(40)
From Fig. (4), the web force $Fweb$ can be presented as
$Fweb=[F2cosθ2−F1cosθ1−F2sinθ2−F1sinθ1]$
(41)
where θ1 = a sin(u2/l1) and θ2 = a sin(u2/l2) are shown in Fig. 4(c). The gravity $Fgravity$ is presented as
$Fgravity=[0mg]$
(42)
where g is the gravity constant. The aerodynamic drag force $Fair$ on the prey is evaluated by considering prey size and its velocity at any instant. For simplicity, we model the prey as a sphere with diameter d in a range of 10−5–10−2 m (Note that d < < L). The fluid media is air at room temperature (i.e., 300 K), whose density is ρa = 1.225 kg/m3 and the kinematic viscosity is κ = 1.6 × 10−5 m2/s. The dynamic viscosity is therefore calculated as ηa = ρaκ. For air flow velocity va, the Reynolds number is defined as
$Re=vadκ$
(43)
For the velocity range of va = 100–101 m/s, Reynolds number is in the range of Re = 100–104. For this range of Reynolds number, the drag coefficient can be expressed as
$CD=C1×Re−C2$
(44)
where C1 = 685.84 and C2 = 0.1065 are coefficients obtained from experimental data [48] (see Fig. 5). The aerodynamic drag force is then defined as
$FD=−sign(va)12ρava2CDπd24=−101.78sign(va)d1.8935va1.8935$
(45)
Fig. 5
Fig. 5
Close modal
The contribution of the drag force on the dynamic response of the prey can therefore be evaluated at any time as
$Fair=−101.78d1.8935[sign(u˙1)u˙11.8935sign(u˙2)u˙21.8935]$
(46)
To better understand the underlying physics and also facilitate numerical calculation, we normalize the equations of motion. The normalized time is defined as $t~=t/τ$, where t is the time and τ is the relaxation characteristic time for the dragline silk defined as τ = η/μ0. The normalized spatial coordinates are defined as
$u~=[u~1u~2]$
(47)
where $u~1=u1/L$ and $u~2=u2/L$. In addition, the normalized mass is introduced as $m~=mL/τ2Aμ0$. The normalized web force $F~web$ is defined as
$F~web=[F~2cosθ2−F~1cosθ1−F~2sinθ2−F~1sinθ1]$
(48)
where $F~1=F1/Aμ0=λ1−1σ~1eff$ and $F~2=F2/Aμ0=λ2−1σ~2eff$ are determined by the constitutive model. To be specific, the normalized stress $σ~1$ and $σ~2$ are presented as
$σ~1=σ1μ0=ζμ~[ωY^(λ1,N)λ1−zY^(x1,y)x1]+(1−ζ)μ~[ωY^(λ1e,N)λ1e−zY^(x1e,y)x1e]$
(49)
and
$σ~2=σ2μ0=ζμ~[ωY^(λ2,N)λ2−zY^(x2,y)x2]+(1−ζ)μ~[ωY^(λ2e,N)λ2e−zY^(x2e,y)x2e]$
(50)
where the normalized shear modulus is introduced as $μ~=μ/μ0$. The normalized gravity $F~gravity$ is defined as
$F~gravity=[0m~g~]$
(51)
where the normalized gravity constant $g~=gτ2/L$. Furthermore, the normalized aerodynamic drag force is defined as
$F~air=−c~[sign(u~˙1)u~˙11.8935sign(u~˙2)u~˙21.8935]$
(52)
where the normalized aero-damping coefficient is introduced as $c~=101.78/Aμ0(dL/τ)1.8935$. Now, one can write the nondimensional equations of motion as
$m~u~¨=F~web+F~gravity+F~air$
(53)
Table 1 shows all physical and geometrical parameters used to evaluate our model for prey response and energy dissipation. Based on these data, the normalized parameters for the nondimensional equations of motion are presented in Table 2. Furthermore, the normalized Newtonian evolution rule is presented as
$dξξdt~=(1−ζ)μ~[ωY^(λe,N)λe−zY^(xe,y)xe]$
(54)
Table 1

Physical and geometrical parameters

Physical quantityNotationValue (unit)
Fiber lengthL$1(m)$
Fiber diameterdsilk$4(μm)$
Shear modulusμ0$0.19(GPa)$
Kuhn numberN$0.35$
Silk anisotropyω$0.40$
Modulus ratioζ$0.29$
Silk dynamic viscosityη$0.25(GPa⋅min)$
Silk damage gainKb$0.37$
Silk plasticity gainKp$0.36$
Prey massm$0.5(g)$
Prey diameterd$10−5–10−2(m)$
Gravity constantg$9.8(kgm/s2)$
Air densityρa$1.225(kg/m3)$
Air kinematic viscosityκa$1.6×10−5(m2/s)$
Impact velocityv0$0–10(m/s)$
Pre-stretchλp$1.05$
Physical quantityNotationValue (unit)
Fiber lengthL$1(m)$
Fiber diameterdsilk$4(μm)$
Shear modulusμ0$0.19(GPa)$
Kuhn numberN$0.35$
Silk anisotropyω$0.40$
Modulus ratioζ$0.29$
Silk dynamic viscosityη$0.25(GPa⋅min)$
Silk damage gainKb$0.37$
Silk plasticity gainKp$0.36$
Prey massm$0.5(g)$
Prey diameterd$10−5–10−2(m)$
Gravity constantg$9.8(kgm/s2)$
Air densityρa$1.225(kg/m3)$
Air kinematic viscosityκa$1.6×10−5(m2/s)$
Impact velocityv0$0–10(m/s)$
Pre-stretchλp$1.05$
Table 2

Normalized system parameters

NotationValue
$m~$$3.305×10−5$
$g~$$6.2632×104$
$v~0$0–222.067
$c~$0.0018
NotationValue
$m~$$3.305×10−5$
$g~$$6.2632×104$
$v~0$0–222.067
$c~$0.0018

Note that the normalized shear modulus $μ~=μ/μ0$ has an initial value of 1.

The equations of motion of the prey together with the constitutive equations of the fiber segments comprise the highly nonlinear governing equations of the prey-web system. In order to find the dynamic response of the system, numerical techniques are used to find the time-stepping of the system response. It is noted that the complex governing equations can be re-formatted into a system of first-order ordinary differential equations, which can be readily and accurately integrated in time by using fourth-order Runge–Kutta method. The numerical procedure is based on defining a variable $xts$ which presents the state of the specific fiber segment. Note that $xts$ is a vector consisting five components
$xts=[μsNsλmaxsξsδs]$
(55)

The physical meaning of these five components has been introduced in Sec. 2. Here, the superscript s is used to distinguish fiber segments. In addition, the time derivative of the state variable is presented as $x˙ts$, which can be readily defined according to the previously introduced constitutive equations.

The deformation of each fiber segment, i.e., $λtots$ at any instant is determined by the variation of the coordinates of its two end points. The coordinates of the two points at any instant are presented as
$uAs=[uA1suA2s]$
(56)
and
$uBs=[uB1suB2s]$
(57)
The length of the segment can be evaluated as
$ls=‖uAs−uBs‖$
(58)
Considering the initial length of fiber segment as $l0s$. The total stretch can therefore be evaluated as
$λtots=lsl0s$
(59)
Furthermore, the time derivative of $λtots$ can be determined as
$λ˙tots=l˙sl0s$
(60)

The deformation rate of the fiber segment is correlated with the position and velocity of its own two end points.

By using the deformation rate $λ˙tots$ and the effective stretch λs, the rate of the state variable $x˙ts$ can be readily determined according to the constitutive model. It is worth mentioning that the initial value of the state variable is given as
$x0s=[μ0sN0s111]$
(61)

When pre-stretch is accounted for, the initial condition would be updated according to the pre-stretch loading condition.

The numerical integration outlined here is used toward evaluating the prey response corresponding to Fig. 4. This model can be easily expanded for finding system response consisting of the entire web, prey, and spider. However, we restrict our study in such a typical simulation case where the prey dynamic response is evaluated for specific conditions of $m~=3.305×10−5$ impacts at the middle of the spider dragline fiber (i.e., φ = 0.5) which has a pre-stretch λpr = 1.05 and its normalized loading rate $λ~˙pr=1$(i.e., $λ˙pr=1/τ$), and initial prey velocity of $v~0=222.067$ without aerodynamic damping consideration. Figure 6(a) shows prey response after impact. The cyclic history of the vertical displacement is shown in Fig. 6(b). The results show that for the parameters selected, prey oscillates for number of cycles before reaching an equilibrium position. The equilibrium displacement is achieved due to the balance of the gravity and the tension in the spider dragline silk fibers. Furthermore, the results indicate there is no overshoot from the initial impact position. Figure 6(c) shows velocity–displacement phase diagram, where the velocity finally goes to zero at the equilibrium state. In addition, as shown in the tension-velocity phase diagram Fig. 6(d), a final residual tension remains to balance the gravity as the velocity reduces to zero in the equilibrium state. The largest displacement occurs at the moment of the first stop. In this period, both damping mechanism and damage mechanism are active. The extent of type-1 damage and type-2 damage are determined by the largest stretch in the loading history. This implies that during this period most of the kinetic energy of the prey is dissipated since both type-1 damage and type-2 damage mechanisms are active. At the first stop, the elastic restoring force of the fiber is greater than the weight of the prey, which results in a rebounding of the prey. It is noted that only viscous damping mechanism is active during subsequent rebounding process. After repeating several oscillations of dropping and rebounding, the kinetic energy of the prey is completely dissipated.

Fig. 6
Fig. 6
Close modal

In order to perform dynamic analysis of spider orb webs for a more complicated geometry under impact loads, the constitutive model is implemented as a user material subroutine in the commercial finite element analysis software (abaqus/explicit), where truss element is adopted. The finite element code was validated for the case presented in Fig. 6 and the results were in agreement with the numerical results.

Our model is further evaluated in order to understand the effects of prey size and aero-damping on the dynamic response of the prey and its energy dissipations. Since the air drag is related to the prey size, here we considered two typical prey sizes without loss of generality, d/L = 0.003 and d/L = 0.01, where the prey mass is taken as a constant. The time histories of the displacement responses for the three cases are shown in Fig. 7(a), where the amplitude, overshoot, and settling time of the dynamic oscillation are reduced after the introduction of air drag. The extent of the reduction depends on the prey size. Figure 7(b) shows the time histories of the velocity responses. Similar conclusions as those for displacement can be presented for the velocity history case. There are apparently little oscillations when the fiber is impacted with a large prey size and aerodynamic damping is also present. Figure 7(c) shows the velocity–displacement phase diagrams. The final equilibrium state of the prey is marked by a black cross in the Fig. 7(c). The results clearly show that as the prey size increases, the aero-damping effect becomes more significant. This results in less damage (types 1 and 2) to the fiber. Therefore, final equilibrium position is less than those cases with no aerodynamic damping or smaller prey size. The effects of aerodynamic damping can be further manifested by considering the total travel distance of the prey, Fig. 7(d). The total travel distance the prey moved in a specified time period ($t~=0.01$) was larger for the case of no aerodynamic damping and reduced significantly for a larger prey size and with aerodynamic damping active. The histories of air drags versus time and travel distance are presented in Figs. 7(e) and 7(f). The results indicate that while the direction of the drag force changes due to prey oscillation for smaller prey, for a larger prey the direction of drag force does not change due to lack of oscillation. The prey simply stops at its first maximum displacement. This can be further confirmed by the displacement-travel distance diagram shown in Fig. 8(a) and velocity-travel distance diagram Fig. 8(b), where the oscillations can be clearly shown.

Fig. 7
Fig. 7
Close modal
Fig. 8
Fig. 8
Close modal
Furthermore, we investigated the prey capture process from the energy transformation and dissipation point of view. From the previous study, we showed three basic mechanisms for energy transformation: material damping (i.e., material internal friction, degradation, and plastic deformation), material storage (i.e., elastic deformation), and aero-damping. These three mechanisms account for the change of the mechanical energy of the prey (i.e., kinetic energy plus potential energy) from the initial impact to the final equilibrium. The mechanical energy is denoted as E = T + P, where the normalized kinetic energy T is defined as
$T=12m~u~˙22$
(62)
and the normalized gravity potential energy as
$P=−m~g~u~2$
(63)
At the instant of impact, the gravitational potential energy P0 = 0. Thus, the initial mechanical energy E0 of the prey is determined by its initial kinetic energy $T0=1/2m~v~02$. For illustration, the total absorbed energy is defined as
$Etot=E0−E$
(64)
After impact, the energy dissipated by aero-damping Ea is evaluated according to the work done by air drag force as
$Ea=∫t~0t~1|F~air|⋅|u~˙2|dt~=∫t~0t~1c~|u~˙2|1.8935|u~˙2|dt~$
(65)
The normalized elastic energy stored in the material is determined by Eq. (31) as
$Ee=W^DJ(λ,λe,μ,N,ζ)/μ0$
(66)
Thus, the energy dissipated by material damping and type-1 and type-2 damages is evaluated from
$Em=Etot−Ea−Ee$
(67)

To visualize these energy variations, their time histories are shown in Fig. 9, where Figs. 9(a) and 9(b) are for the case without considering aero-damping, Figs. 9(c) and 9(d) are for the case of prey size d/L = 0.003, and Figs. 9(e) and 9(f) are for the case of prey size d/L = 0.01. For the case of no aero-damping, Fig. 9(a) shows the kinetic energy of the particle initially increases, since there is little energy loss or storage for a short period after impact. The velocity increases due to conversion of potential energy to kinetic energy. However, further movement of the particle results in energy loss due to different mechanisms and stored elastic energy. The results also indicates for the case of no aero-damping, when the tension in the fiber goes to zero after first stop, the total energy is constant since there is no further energy loss or storage active in that period (see Fig. 6(d)). Particle simply decelerates due to gravity once tension goes to zero. Energy is further lost after the second stop due to just viscous contributions. The results for this case also indicate that energy loss due to viscous damping and damages (both type-1 and type-2) is much larger than stored elastic energy (see Fig. 9(b)). For the case when aero-damping is present, Figs. 9(c)9(f) show that kinetic energy decreases immediately after impact. Furthermore, the energy loss due to aero-damping is significantly higher than energy loss from other material damping mechanisms. This is especially more pronounced for larger prey size (see Fig. 9(f)). To elucidate this conclusion, the energy loss percentage at the equilibrium is plotted as a function of prey size. Figure 10 shows that as the prey size increases, the energy dissipation due to aero-damping dominates energy loss when compared with the mechanical energy loss (i.e., viscoelastic, type-1 and type-2 damages).

Fig. 9
Fig. 9
Close modal
Fig. 10
Fig. 10
Close modal

## 4 Conclusions

To better understand the prey capture process of a spider orb web, a multi-scale modeling approach is proposed. The approach consists of development of a new anisotropic constitutive relation for spider dragline silk based on its microstructure, and its dynamic response due to prey impact and considering aerodynamic effects. The special microstructure of the material is found to have amazing energy absorption capability, which is the key factor for a successful prey capture. The results show that the energy absorption by the orb web is affected by several damage mechanisms related to its microstructure, such as its viscoelastic properties, α-helix chain fracture, and unfolding β-sheet spiral. These damage mechanisms are incorporated in developing dynamic response of spider orb webs. The results show that while these damage mechanisms dominate prey energy absorption for small preys, for larger prey size the aerodynamic damping is more dominant. In addition, the results demonstrate that orb web have oscillatory response when prey size is small. However, there is no oscillation when prey size is large. The results also indicate that the orb web may suffer more damage from impact by a smaller size particle than by a larger particle with the same mass.

## Acknowledgment

This work was supported partially by the National Science Foundation and Northeastern University, Department of Mechanical and Industrial Engineering.

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