Derivation of linearized constitutive equations for plane-strain of an elastic-plastic (strain hardening) material

The purpose of this report is to derive several of the basic equations for a paper on folding and folding of idealized rock (Johnson, 1979). That paper is based on constitutive equations for an ideal, strain-hardening material defined by Hill (1950, p. 30) and generalized to include Coulomb behavior and dilatancy by Rudnicki and Rice (1975). The most attractive feature of the theory is that folding and faulting, which are intimately related in nature, are different responses of the same material to different boundary conditions. In the paper by Johnson (1979) it is shown that single layers of sedimentary rocks are unlikely to fold, rather they will fault, because of low contrasts in elasticity and strength properties of sedimentary rocks such as layers of dolomite, limestone, sandstone, or siltstone in media of shale. Further, multi-layers of these same rocks will fault rather than fold if contacts are bonded, but they will fold readily if contacts between layers are frictionless, or have low yield strengths, for example due to high pore-water pressure. These conclusions are based on solutions of the equations presented in the following pages, using experimental stress-strain curves for various rock types.