This paper proposes a hybrid (semi-analytic) solution for determining the contact footprint and subsurface stress field in a two-dimensional adhesive problem involving a multi-layered elastic solid loaded normally by a rigid indenter. The subsurface stress field is determined using a semi-analytic solution and the footprint using a fast converging iterative algorithm. The solid to be indented consists of a graded elasticity coating with exponential increase of decay of its shear modulus bonded on a homogeneously elastic substrate. By applying the Fourier Transform to the governing boundary value problem, we formulate expressions for the stresses and displacements induced by the application of line forces acting both normally and tangentially at the origin. The superposition principle is then used to generalize these expressions to the case of distributed normal pressure acting on the solid surface. A pair of coupled integral equations are further derived for the parabolic stamp problem which are easily solved using collocation methods.

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