General solution and Green function with bifurcation for nonlinear plane stress deformation of a compressible wedge

Exact stress, strain and displacement fields with closed form are determined for a nonlinear boundary value wedge problem. The compressible wedge experiencing small plane stress deformation is loaded by a concentrated force at its apex and the material is assumed to satisfy the power-law a~ = E06 where E0 and n (0 < n ~< 1) are positive constants, aE and EE are the stress and strain intensities, respectively. The results show that bifurcation with three branches occurs when the value of n is close to v/(1 +v) where v is the Poisson ratio. The discontinuity of displacement components and their gradients proves to exist if the solution pertaining to one branch characterized by n = v/(1 + v) is required to satisfy the symmetric and 0-dependence conditions. These phenomena can be ascribed to the property conversion of the governing equation from the elliptic (n > v/(l + v)) to parabolic (n = v/(l +v)) or hyperbolic type (n < v/(1 +v)). As illustrative examples, the Green functions are ascertained for a half plane subjected to a normal and a shear forces, respectively. It is found that the stress distribution for symmetric problems of the three branches fundamentally differs from each other. For deformation of the elliptic type, two fanlike tensile zones are discovered near the boundary of the half plane loaded by a compressive normal force. Copyright © 1996 Elsevier Science Ltd.