Application of Numerical—Analytical Methods to Static and Dynamic Problems for Heterogeneous Half-Space

Abstract

Static and dynamic problems for heterogeneous half-spaces may be solved by using the integral representations given in Chapter 1, where the integrands are expressed by fundamental solutions of equations (1.29)—(1.31), or equations (1.29), (1.126)—(1.129). In the case of a force acting within the half-space, besides two solutions of the homogeneous equations satisfying the condition of absence of sources of disturbance at infinity (as z →∞), an additional solution accounting for the specified force is introduced (a function having index a entering equations of the form (1.109), (1.120), (1.121), (1.157), (1.164), (1.165)). This additional solution has been assumed to vanish for the points of the half-space located below the force; a condition of crossing the horizontal plane in which the force is applied enables us to formulate initial conditions for the additional solution and to construct it in the part of the half-space above the point of application of the force. The construction of analytical representations for the solutions of specified systems of equations may be done only for some particular cases of heterogeneity for isotropic half-spaces. A number of such cases are considered in previous chapters. Here, we point to yet another case: an isotropic half-space with shear modulus increasing with depth by the square law; at Poisson’s ratio v = 0.25, the equation of motion may be separated out and solved by employing modified Bessel’s functions. This possibility has been stated for the plane problem [46] and for the axially symmetrical problem [47]. In [31, 33], the results reported by Gupta [46] were used to study some problems dealing with the dynamics of soil foundations.