Basic equations for the microscopic theory of plasticity of polycrystalline materials with given texture

An expression for the yield stress of anisotropic materials is applied to the anisotropic strength of hard rolled copper foils whose crystallographic texture is known. We assume that this crystallographic texture is the only cause of the anisotropic plastic behaviour of the material. The model used for the yield stress is also used to deduce:

1. Stress-strain relations for isotropic polycrystalline materials;
2. A formula for the fully plastic strain tensor, applied to anisotropic hard rolled copper foils.

For the anisotropic copper foils considered the calculated curves of the yield stress and of the strain tensor as a function of the angle x between rolling and tensile direction agree qualitatively with the measured values. However, the theory is not complete, since the yield stress and the plastic strain tensor are both a function of a parameter Q, the fraction of the number of available crystallographic slip planes on which the maximum shear stress has reached the critical value τ_a. We assume that for “fully” plastic deformation a certain critical fraction Q _e of the total number of slip planes has to be active. The fraction Q _e is called the critical active quantity. With the parameter Q _e we adjust the calculated curves to the measured ones. The dependence of Q _e on the properties of the material (e.g. the crystallographic texture) is discussed in Appendix I.