Strains and Displacements in Stochastic Stress Models

The stochastic model of stress distributions in loaded granular half-spaces is extended to describe strains and displacements by postulating a linear constitutive equation with position-dependent moduli, which is consistent with the strain compatibility conditions. The constitutive equation adopted is a particular case of the cross-anistropic form. Expressions for the moduli and strain components are deduced, as they are for the vertical surface displacement. Under a point load, the surface displacement behaves as a power of the inverse distance from the load, similar to the assumed behavior in certain inhomogeneous soil models. The requirement that the strain energy be positive restricts the power to be no less than unity and therefore the deflection basin is as curved or more so than for an elastic material. Deflection data taken on thick granular layers seems to be fitted very well by such a power law. However, the restriction on the power is not obeyed in all cases. This may be a consequence of some cohesiveness or inhomogeneity in the materials.