The current paper proposes a semi-analytical solution method which predicts both the sub-surface stress fields and pressures that result from adhesive contact between a multi-layered functionally graded material (FGM) and a rigid punch. The FGM is deemed to consist of a homogeneously elastic coating and substrate joined together by a transition layer whose shear modulus depends on the depth coordinate in some predetermined way. It is further assumed that the FGM is in a state of plane strain so that we may perform a two-dimensional analysis.

We present analytical solutions for the horizontal and vertical displacements within the solid which hold when the applied surface pressure is known and then discuss how we may use these solutions to solve the contact problem. A selection of numerical results are then presented using this model and it is found that the algorithm proposed in this work is both accurate and computationally cheap to use.

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