Rather than using polar coordinates to formulate the inhomogeneous half-space problem of Section 14.4, some researchers have used Cartesian coordinates instead. Using the x, z-coordinates as shown in Fig. 14.14, consider the plane strain case with inhomogeneity only in the shear modulus such that μ = μ(z) and v = constant. First show that combining the strain displacement relations with Hooke’s law gives:

Data from section 14.4

Fig 14.14

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where = = ekk equations yield = du dw + x z du - 24 (2) (3M 4 + 10) x 0x = 0 = 24 (2) (+00) +v*v txz < = A(z) (3-2) + z is the dilatation and v* = x (1-2) du dw vu+(1+2v*) -+h(z) + z x . Next show that the equilibrium = 0 dv vw + (1+2v*) + 2h(z) (3H +v0) = ( 0 z z where h(z) = (-) (d). Explore the simplification of these equations for the special dz inhomogeneous case in which u = uoea, where u and a are constants.