## Multiple region contact solutions for a flat indenter on a layered elastic half space |

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2-3 boundary Abramowitz and Stegun annular indenter approximation axisymmetric dashed axisymmetric geometry axisymmetric problem behavior Bessel function boundary conditions calculated Cauchy singularity coefficient method dot-dashed comer components contact dimensions contact pressure contact problem contact region solution coordinate curve defined determined elliptic integral equa Erdogan and Gupta evaluated figure 14 Gauss-Chebyshev integrator geometry parameter space given by equation Gladwell and Gupta Green's function Gupta moment method half-space solution Hankel transform indenter contacting inner radius integrand interface stresses kernel LAYER MODULUS Figure layer thickness linear equations load equation logarithmic singularity method dashed method of solution moduli of 0.25 nondimensional obtained orthogonal polynomials parameter space divisions plane-strain geometry plane-strain problem plane-strain solid Poisson's ratio present solution solid shear modulus shear stress single layer singular integral equation solu solve stress intensity factor substrate surface displacement surface slope symmetric three contact region tion TOP LAYER MODULUS transfer matrix values variable vector