ABAQUS材料子程序学习（线性各向同性硬化塑性）

前言

记录自己学习abaqus软件umat子程序的t过程，本文主要参考了《非线性本构关系在ABAQUS中的实现》第四章和技术邻的视频课程“非线性各向同性强化弹塑性umat子程序教程”

塑性力学增量形式实现

计算过程中，体应变和体应力是弹性关系$$\Delta {\sigma }_v=K\cdot \Delta {\epsilon }_v\text{\hspace{1em}(1)}$$

$K$为体积模量，$K=\frac{E}{3(1-\nu )}$ , $\Delta {\epsilon }_v=\Delta {\epsilon }_{11}+\Delta {\epsilon }_{22}+\Delta {\epsilon }_{33}$

所以在下面的讨论中只考虑偏应力张量和偏应力张量

试应力： $${\mathbf{\sigma }}^{\mathbf{t}{\mathbf{r}}^{\prime }}(\mathbf{t})={\mathbf{\sigma }}^{\prime }(\mathbf{t})+{\mathbb{C}}^{\prime }:\mathbf{\Delta }{\mathbf{\epsilon }}^{\prime }\text{\hspace{1em}(2)}$$

viogt表记：
$$\left\{\begin{array}{c}{\sigma }_{11}^{t{r}^{\prime }}\\ {\sigma }_{22}^{t{r}^{\prime }}\\ {\sigma }_{33}^{t{r}^{\prime }}\\ {\sigma }_{12}^{t{r}^{\prime }}\\ {\sigma }_{23}^{t{r}^{\prime }}\\ {\sigma }_{13}^{t{r}^{\prime }}\end{array}\right\}=\left\{\begin{array}{c}{{\sigma }^{\prime }}_{11}(t)\\ {{\sigma }^{\prime }}_{22}(t)\\ {{\sigma }^{\prime }}_{33}(t)\\ {{\sigma }^{\prime }}_{12}(t)\\ {{\sigma }^{\prime }}_{23}(t)\\ {{\sigma }^{\prime }}_{13}(t)\end{array}\right\}+\left[\begin{array}{cccccc}2G& 0& 0& 0& 0& 0\\ 0& 2G& 0& 0& 0& 0\\ 0& 0& 2G& 0& 0& 0\\ 0& 0& 0& G& 0& 0\\ 0& 0& 0& 0& G& 0\\ 0& 0& 0& 0& 0& G\end{array}\right]\cdot \left\{\begin{array}{c}{\Delta {\epsilon }^{\prime }}_{11}(t)\\ {\Delta {\epsilon }^{\prime }}_{22}(t)\\ {\Delta {\epsilon }^{\prime }}_{33}(t)\\ {\Delta \gamma }_{12}(t)\\ {\Delta \gamma }_{23}(t)\\ {\Delta \gamma }_{13}(t)\end{array}\right\}$$
静态分析umat用的工程应变：$\left\{\begin{array}{c}{\Delta {\epsilon }^{\prime }}_{11}(t)\\ {\Delta {\epsilon }^{\prime }}_{22}(t)\\ {\Delta {\epsilon }^{\prime }}_{33}(t)\\ {\Delta \gamma }_{12}(t)\\ {\Delta \gamma }_{23}(t)\\ {\Delta \gamma }_{13}(t)\end{array}\right\}=\left\{\begin{array}{c}{\Delta {\epsilon }^{\prime }}_{11}(t)\\ {\Delta {\epsilon }^{\prime }}_{22}(t)\\ {\Delta {\epsilon }^{\prime }}_{33}(t)\\ {2\Delta {\epsilon }^{\prime }}_{12}(t)\\ {2\Delta {\epsilon }^{\prime }}_{23}(t)\\ {2\Delta {\epsilon }^{\prime }}_{13}(t)\end{array}\right\}$

Mises等效应力：

主应力形式：
$${\sigma }_e={\left\{\frac{1}{2}\left[{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2\right]\right\}}^{\frac{1}{2}}$$

偏应力形式:
$${\sigma }_e={\left[\frac{3}{2}\left({{\sigma }_{11}^{\prime }}^2+{{\sigma }_{22}^{\prime }}^2+{{\sigma }_{33}^{\prime }}^2+2{{\sigma }_{12}}^2+2{{\sigma }_{23}}^2+2{{\sigma }_{13}}^2\right)\right]}^{\frac{1}{2}}\text{\hspace{1em}(3)}$$

其中: ${\sigma }_{11}^{\prime }={\sigma }_{11}-{\sigma }_v$ ，${\sigma }_{22}^{\prime }={\sigma }_{22}-{\sigma }_v$， ${\sigma }_{33}^{\prime }={\sigma }_{33}-{\sigma }_v$； ${\sigma }_v=\frac{1}{3}\left[{\sigma }_{11}+{\sigma }_{22}+{\sigma }_{33}\right]$

Mises等效塑性应变增量：
$$\Delta {\bar{\epsilon }}^p=\frac{\sqrt{2}}{3}{\left[{\left(\Delta {\epsilon }_1^p-\Delta {\epsilon }_2^p\right)}^2+{\left(\Delta {\epsilon }_2^p-\Delta {\epsilon }_3^p\right)}^2+{\left(\Delta {\epsilon }_3^p-\Delta {\epsilon }_1^p\right)}^2\right]}^{\frac{1}{2}}$$

线性硬化塑性$${\sigma }_y={\sigma }_{y0}+h{\bar{\epsilon }}^p\text{\hspace{1em}(4)}$$
初始屈服加上等效塑性应变乘以硬化系数。
等效塑性变形作为状态变量存在STATEV(NSTATV)中，提取 ${\bar{\epsilon }}^p$ 并计算当前的屈服应力 ${\sigma }_y$

(2)代入(3)，计算试应力的等效Mises应力${\sigma }_{Mises}^{tr}$

判断 ${\sigma }_{Mises}^{tr}$ 与 ${\sigma }_y$ 关系:

若 ${\sigma }_{Mises}^{tr}＜{\sigma }_y$ ： $\mathbf{\sigma }(\mathbf{t}+\mathbf{\Delta }\mathbf{t})={\mathbf{\sigma }}^{\mathbf{t}\mathbf{r}}(\mathbf{t})$

一致切线刚度矩阵，为弹性刚度矩阵
$$\mathbf{D}\mathbf{D}\mathbf{S}\mathbf{D}$$