A new mathematical formulation of the equations of perfect elasto-plasticity

Abstract

A new mathematical formulation for the constitutive laws governing elastic perfectly plastic materials is proposed here. In particular, it is shown that the elastic strain rate and the plastic strain rate form an orthogonal decomposition with respect to the tangent cone and the normal cone of the yield domain. It is also shown that the stress rate can be seen as the projection on the tangent cone of the elastic stress tensor. This approach leads to a coherent mathematical formulation of the elasto-plastic laws and simplifies the resulting system for the associated flow evolution equations. The cases of one or two yield functions are treated in detail. The practical examples of the von Mises and Tresca yield criteria are worked out in detail to demonstrate the usefulness of the new formalism in applications.

Appendix A: Proof of Lemma 4.7

Obviously, $W_{m, i} \in G_m({\varvec{\sigma }})$ for $1 \leqslant i \leqslant 3$. Furthermore, one can easily show that for $i \ne m$ and $j \ne m$

$$\begin{aligned} W_{m, i} : W_{m, j} = \delta _{i, j}, \; W_{m, i} : W_{m, m} = 0, \; W_{m, m} : W_{m, m} = 1. \end{aligned}$$

Furthermore, let ${\varvec{\kappa }}\in G_m({\varvec{\sigma }})$. Since ${\varvec{\kappa }}v_m({\varvec{\sigma }})= 0$, 0 is an eigenvalue of ${\varvec{\kappa }}$. Let $\{u_1, u_2, v_3\}$ be an orthonormal basis of eigenvectors of ${\varvec{\kappa }}$. We have the spectral decomposition

$$\begin{aligned} {\varvec{\kappa }}= \mu _1 u_1 \otimes u_1 + \mu _2 u_2 \otimes u_2 + 0 u_3 \otimes u_3 = \mu _1 u_1 \mu _1 u_1 \otimes u_1 + \mu _2 u_2 \otimes u_2. \end{aligned}$$

On the other hand,

$$\begin{aligned} u_i \in (\mathrm{{span}}\{v_3\})^\perp = \mathrm{{span}}\{v_1, v_2\} \text{ for } i = 1, 2. \end{aligned}$$

Thus, for each $i \leqslant 2$ there exist real numbers $\alpha _i, \beta _i$ such that $u_i = \alpha _i v_1 + \beta _i v_2$. It follows that

$$\begin{aligned} A =( \mu _1 \alpha _1^2 + \mu _2 \alpha _2^2 ) W_{m, 1} + ( \mu _1 \beta _1^2 + \mu _2 \beta _2^2 ) W_{m, 2} + \sqrt{2} ( \mu _1 \alpha _1 \beta _1 + \mu _2 \alpha _2 \beta _2 ) W_{m, 3}. \end{aligned}$$

We conclude that $\{W_1, W_2, W_3\}$ is an orthonormal basis of $G_m({\varvec{\sigma }})$.