自由能
在固体力学中，自由能 \(\psi(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon}^{p},\boldsymbol{\alpha})\) 是描述材料内部能量状态的标量函数，它依赖于总应变 \(\boldsymbol{\varepsilon}\)、塑性应变 \(\boldsymbol{\varepsilon}^{p}\)，以及内部硬化变量 \(\boldsymbol{\alpha}\)
Note
\(\boldsymbol{\alpha}\) 用于描述材料硬化状态，比如屈服面的位置、大小、形状等。常见的有各向同性硬化变量、运动硬化变量；对于各向同性硬化，\(\boldsymbol{\alpha}\) 通常为累计塑性应变 \(\bar{\varepsilon}^{p}\)；对于运动硬化，\(\boldsymbol{\alpha}\) 为背应力 \(\boldsymbol{\xi}\)，代表屈服面位置的移动
该自由能的定义不仅反映了材料由于弹性变形所储存的可逆能量，还涵盖与不可逆过程相关的能量（如塑性变形和硬化效应），该定义方式保证了热力学的一致性（能量守恒和熵增）
以下考虑单位体积的情形，将自由能分解为可逆的弹性部分和与硬化相关的塑性部分
\[\begin{split}
\begin{equation}
\begin{aligned}
\psi(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon}^{p},\boldsymbol{\alpha})&=\psi^{e}(\boldsymbol{\varepsilon}-\boldsymbol{\varepsilon}^{p})+\psi^{p}(\boldsymbol{\alpha})\\
&=\psi^{e}(\boldsymbol{\varepsilon}^{e})+\psi^{p}(\boldsymbol{\alpha})
\end{aligned}
\end{equation}
\end{split}\]
计算
(87)\[
\begin{equation}
\dot{\psi} = \frac{\partial \psi}{\partial \boldsymbol{\varepsilon}}:\dot{\boldsymbol{\varepsilon}} + \frac{\partial \psi}{\partial \boldsymbol{\varepsilon}^{p}}:\dot{\boldsymbol{\varepsilon}}^{p} + \frac{\partial \psi}{\partial \boldsymbol{\alpha}}:\dot{\boldsymbol{\alpha}}
\end{equation}
\]
结合 \(\boldsymbol{\varepsilon}^{e} = \boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}^{p}\)，得到
\[\begin{split}
\begin{equation}
\begin{aligned}
\frac{\partial \psi}{\partial \boldsymbol{\varepsilon}}&=\frac{\partial \psi^e}{\partial \boldsymbol{\varepsilon}} = \frac{\partial \psi^e}{\partial \boldsymbol{\varepsilon}^e}\frac{\partial \boldsymbol{\varepsilon}^e}{\partial \boldsymbol{\varepsilon}} = \frac{\partial \psi^e}{\partial \boldsymbol{\varepsilon}^e},\\
\frac{\partial \psi}{\partial \boldsymbol{\varepsilon}^p}&=\frac{\partial \psi^e}{\partial \boldsymbol{\varepsilon}^p}=\frac{\partial \psi^e}{\partial \boldsymbol{\varepsilon}^e}\frac{\partial \boldsymbol{\varepsilon}^e}{\partial \boldsymbol{\varepsilon}^p} = -\frac{\partial \psi^e}{\partial \boldsymbol{\varepsilon}^e},\\
\frac{\partial \psi}{\partial \boldsymbol{\alpha}}&=\frac{\partial \psi^p}{\partial \boldsymbol{\alpha}},
\end{aligned}
\end{equation}
\end{split}\]
故
\[\begin{split}
\begin{equation}
\begin{aligned}
\dot{\psi} &= \frac{\partial \psi^e}{\partial \boldsymbol{\varepsilon}^e}:\dot{\boldsymbol{\varepsilon}}-\frac{\partial \psi^e}{\partial \boldsymbol{\varepsilon}^e}:\dot{\boldsymbol{\varepsilon}}_{p} + \frac{\partial \psi}{\partial \boldsymbol{\alpha}}:\dot{\boldsymbol{\alpha}}\\
&=\frac{\partial \psi^e}{\partial \boldsymbol{\varepsilon}^e}:\dot{\boldsymbol{\varepsilon}}_{e}+ \frac{\partial \psi^p}{\partial \boldsymbol{\alpha}}:\dot{\boldsymbol{\alpha}},
\end{aligned}
\end{equation}
\end{split}\]
代入到 Clausius-Duhem 不等式
\[
\boldsymbol{\sigma}:\dot{\boldsymbol{\varepsilon}}-\dot{\psi}\geq0,
\]
有
\[
\boldsymbol{\sigma}:\dot{\boldsymbol{\varepsilon}}-\frac{\partial \psi^e}{\partial \boldsymbol{\varepsilon}^e}:\dot{\boldsymbol{\varepsilon}}_{e}- \frac{\partial \psi}{\partial \boldsymbol{\alpha}}:\dot{\boldsymbol{\alpha}}\geq0,
\]
整理得到
\[
(\boldsymbol{\sigma}-\frac{\partial \psi^e}{\partial \boldsymbol{\varepsilon}^e}):\dot{\boldsymbol{\varepsilon}}_{e} + \boldsymbol{\sigma}\dot{\boldsymbol{\varepsilon}}_{p} - \frac{\partial \psi}{\partial \boldsymbol{\alpha}}:\dot{\boldsymbol{\alpha}}\geq0,
\]
由于弹性应变速率 \(\dot{\boldsymbol{\varepsilon}}_{e}\) 可以取任意方向和任意大小，而 \(\boldsymbol{\sigma}-\frac{\partial \psi^e}{\partial \boldsymbol{\varepsilon}^e}\) 在弹性阶段不受速率影响，因此必须有
\[
\boldsymbol{\sigma}=\frac{\partial \psi^e}{\partial \boldsymbol{\varepsilon}^e},
\]
此外
\[
\mathbf{A} \equiv \frac{\partial \psi^p}{\partial \boldsymbol{\alpha}}
\]
为硬化热力学力，
\[
\Upsilon^p(\boldsymbol{\sigma},\mathbf{A};\dot{\boldsymbol{\varepsilon}}^{p},\dot{\boldsymbol{\alpha}}) \equiv \boldsymbol{\sigma}\dot{\boldsymbol{\varepsilon}}_{p} - \frac{\partial \psi}{\partial \boldsymbol{\alpha}}:\dot{\boldsymbol{\alpha}}
\]
为塑性耗散函数
对于各向同性的线弹性材料，自由能形式为
\[
\begin{equation}
\psi^e = \frac{1}{2}\boldsymbol{\varepsilon}^{e}:\mathbf{D}^{e}:\boldsymbol{\varepsilon}^{e}=G\boldsymbol{\varepsilon}^{e}_{d}:\boldsymbol{\varepsilon}^{e}_{d}+\frac{K}{2}\text{tr}(\boldsymbol{\varepsilon}^{e})^2,
\end{equation}
\]
其中，\(\boldsymbol{\varepsilon}^{e}_{d} = \text{dev}(\boldsymbol{\varepsilon}^{e})\) 是 \(\boldsymbol{\varepsilon}^{e}\) 的偏张量部分
Note
\[
\boldsymbol{\sigma}=\mathbf{D}^e:\boldsymbol{\varepsilon}^{e} = 2G\ \text{dev}(\boldsymbol{\varepsilon}^{e})+K\text{tr}(\boldsymbol{\varepsilon}^{e})\mathbf{I}
\]
例如，通常各向同性硬化自由能为
\[
\psi = \frac{1}{2} \boldsymbol{\varepsilon}^{e} : \mathbf{D}^{e} : \boldsymbol{\varepsilon}^{e} + \frac{1}{2} \mathbf{H} \boldsymbol{\alpha}^2,
\]
运动硬化自由能为
\[
\psi = \frac{1}{2} \boldsymbol{\varepsilon}^{e} : \mathbf{D}^{e} : \boldsymbol{\varepsilon}^{e} + \frac{1}{2} c (\boldsymbol{\alpha} - \boldsymbol{\alpha}_0)^2.
\]
关联塑性流动理论
关联塑性流动模型假定
\[
\Psi = \Phi,
\]
塑性应变和硬化变量的演化方程如下
\[\begin{split}
\begin{equation}
\begin{aligned}
\dot{\boldsymbol{\varepsilon}}^{p}&=\dot{\gamma}\frac{\partial \Phi}{\partial \boldsymbol{\sigma}},\\
\dot{\boldsymbol{\alpha}}&=-\dot{\gamma}\frac{\partial \Phi}{\partial \mathbf{A}},
\end{aligned}
\end{equation}
\end{split}\]
其中，\(\dot{\gamma}\) 是塑性乘子
最大塑性耗散原理
任何可行的应力状态 \((\boldsymbol{\sigma},\mathbf{A})\) 应满足 \(\Phi(\boldsymbol{\sigma},\mathbf{A}) \leq 0\)，把它们形成的集合记为 \(\mathcal{A}\)，最大塑性耗散原理揭示了，对于任意给定的塑性应变率 \(\dot{\boldsymbol{\varepsilon}}^{p}\) 和 硬化内变量变化速率 \(\dot{\boldsymbol{\alpha}}\)，真实应力状态应满足
\[
\Upsilon^p(\boldsymbol{\sigma},\mathbf{A};\dot{\boldsymbol{\varepsilon}}^{p},\dot{\boldsymbol{\alpha}}) \geq \Upsilon^p(\boldsymbol{\sigma}^{*},\mathbf{A}^{*};\dot{\boldsymbol{\varepsilon}}^{p},\dot{\boldsymbol{\alpha}}),\quad \forall\ (\boldsymbol{\sigma}^{*},\mathbf{A}^{*})\in\mathcal{A},
\]
同时，Kuhn-Tucker 最优性条件对应为
\[
\Phi(\boldsymbol{\sigma},\mathbf{A}) \leq 0,\quad \dot{\gamma}\geq0,\quad \Phi(\boldsymbol{\sigma},\mathbf{A})\dot{\gamma}=0.
\]
最大塑性耗散原理及其关联性定律并非普遍适用。虽然该原理在晶体塑性和金属材料中得到了验证，但对于土壤等颗粒材料，实验结果常常与关联性定律不符。在这些情况下，需采用非关联性定律来更准确地描述材料行为
