Plastic strain curl theory

Abstract

The main mechanisms for the hardening of metal materials are the multiplication, accumulation and interaction of dislocations. The dislocation density tensor can be decomposed into two parts: one is the plastic strain curl tensor and the other is the plastic curvature tensor. The influence of the plastic curvature can be characterized by the interaction between the Cauchy's stresses and the couple stresses. The plastic strain curl is supposed to play the most important role for the stress level. Three rotational degrees of freedom x i , named as micro rotation, are introduced besides the displacement components u i . Micro rotations x i have no direct dependence upon u i while the material rotation h ¼ r  u=2. The generalized normality law is used to describe constitutive relations of Cauchy's stresses versus strains and couple stresses versus curvatures. Plastic strain curl is incorporated into the instantaneous tangent modulus. In this way, the generalized equivalent stress is no longer a single-variable function of the generalized equivalent strain. The plastic strain energy density is no longer determined by the generalized equivalent strain solely, too.Based on the present theory, an FEM program is developed to simulate the microindentation tests on Copper and Tungsten. The calculated hardness is observed to elevate as the indent depth decreases. The calculated results agree well with the experimental data. The crack tip field for small scale yielding condition is also studied. The calculated results clearly show that the stress level near the crack tip with plastic strain curl effects is considerably higher than that in the conventional plasticity theory. The singularity of the mean stress near the crack tip is nearly equal to the square-root singularity, and the singularity of the effective stress field is slightly greater than the square-root singularity. Consequently, the singularity of stress components is also slightly greater than the square-root singularity. The J-integral is observed to be essentially path independent.