弹塑性切线模量

弹塑性切线模量#

塑性乘子#

根据塑性流动法则和硬化法则，有

\[\begin{split} \begin{equation} \begin{aligned} \dot{\boldsymbol{\varepsilon}}^{p} &= \dot{\gamma}\mathbf{N}(\boldsymbol{\sigma},\mathbf{A}),\\ \dot{\boldsymbol{\alpha}} &= \dot{\gamma}\mathbf{H}(\boldsymbol{\sigma},\mathbf{A}), \end{aligned} \end{equation} \end{split}\]

其中，\(\mathbf{H}\) 是一般化的硬化模量，决定了硬化参数 \(\boldsymbol{\alpha}\) 的演化

若材料属于塑性屈服状态，则

(88)#\[ \dot{\Phi} = \frac{\partial \Phi}{\partial \boldsymbol{\sigma}}:\dot{\boldsymbol{\sigma}} + \frac{\partial \Phi}{\partial \mathbf{A}}\dot{\mathbf{A}}, \]

代入

(89)#\[ \dot{\boldsymbol{\sigma}} = \mathbf{D}^{e}:(\dot{\boldsymbol{\varepsilon}}-\dot{\boldsymbol{\varepsilon}}^p) = \mathbf{D}^{e}:(\dot{\boldsymbol{\varepsilon}}-\dot{\gamma}\mathbf{N}) \]

和

\[ \dot{\mathbf{A}} = \frac{\mathrm{d} \frac{\partial \psi^p}{\partial \boldsymbol{\alpha}}}{\mathrm{d} t} = \frac{\partial^2 \psi^p}{\partial \boldsymbol{\alpha}^2}\dot{\boldsymbol{\alpha}} = \frac{\partial^2 \psi^p}{\partial \boldsymbol{\alpha}^2}\dot{\gamma}\mathbf{H} \]

到式 (88) 得到

\[ \dot{\Phi} = \frac{\partial \Phi}{\partial \boldsymbol{\sigma}}:\mathbf{D}^{e}:(\dot{\boldsymbol{\varepsilon}}-\dot{\gamma}\mathbf{N}) + \dot{\gamma}\frac{\partial \Phi}{\partial \mathbf{A}}\frac{\partial^2 \psi^p}{\partial \boldsymbol{\alpha}^2}\mathbf{H} \]

于是得到

(90)#\[ \dot{\gamma} = \frac{\frac{\partial \Phi}{\partial \boldsymbol{\sigma}}:\mathbf{D}^{e}:\dot{\boldsymbol{\varepsilon}}}{\frac{\partial \Phi}{\partial \boldsymbol{\sigma}}:\mathbf{D}^{e}:\mathbf{N}-\frac{\partial \Phi}{\partial \mathbf{A}}\frac{\partial^2 \psi^p}{\partial \boldsymbol{\alpha}^2}\mathbf{H}}, \]