一阶变分（G\(\mathrm{\hat{a}}\)teaux 导数）
设 \(\mathcal{F}\) 对 \(\psi\) 可微，则其一阶变分定义为
(32)\[
\delta\mathcal{F}(\psi;\eta)\equiv\left.\frac{\mathrm{d}}{\mathrm{d}\epsilon}\mathcal{F}(\psi+\epsilon\eta)\right|_{\epsilon=0}
\]
其中，\(\psi\) 和 \(\eta\) 同属于一个向量空间，一阶变分也称为 G\(\mathrm{\hat{a}}\)teaux 导数，\(\delta\) 被称为变分算子
通常，把 \(\eta\) 记为 \(\delta\psi\) 因此，式 (32) 通常也写为
\[
\delta\mathcal{F}(\psi;\delta\psi)\equiv\left.\frac{\mathrm{d}}{\mathrm{d}\epsilon}\mathcal{F}(\psi+\epsilon\delta\psi)\right|_{\epsilon=0}
\]
令 \(\mathcal{F} = \psi\) 时，代入上式，是自洽的
对构型的扰动，称为 Lagrangian 微分，此时 \(\mathbf{X}\) 固定；对物理场的扰动，称为 Eulerian 微分，此时 \(\mathbf{x}\) 固定
变分算子与微分算子在形式上类似,但二者的本质区别在于:
微分算子作用于函数,描述函数值随自变量变化的变化率(函数空间→数值)
变分算子作用于泛函,描述泛函值随函数(作为自变量)变化的变化率(泛函空间→数值)
变分算子在和，积，比，幂的运算与微分算子一致
类似于偏导数计算，对于多变量泛函，变分运算也可以指定变分变量，并默认其它变量保持不变
线性性
\[
\begin{aligned}
\delta\mathcal{F}(\psi;\alpha\delta\psi_{1}+\beta\delta\psi_{2})=\left.\frac{\mathrm{d}}{\mathrm{d}\epsilon}\mathcal{F}(\psi+\epsilon(\alpha\delta\psi_{1}+\beta\delta\psi_{2}))\right|_{\epsilon=0}
\end{aligned}
\]
记 \(g(\epsilon)=\mathcal{F}(\psi+\epsilon(\alpha\delta\psi_{1}+\beta\delta\psi_{2}))\)，故
\[\begin{split}
\begin{aligned}
\left.\frac{\mathrm{d}g}{\mathrm{d}\epsilon}\right|_{\epsilon=0} &= \frac{\partial\mathcal{F}}{\partial\psi}\cdot(\alpha\delta\psi_{1}+\beta\delta\psi_{2})\\
&=\frac{\partial\mathcal{F}}{\partial\psi}\cdot\alpha\delta\psi_{1}+\frac{\partial\mathcal{F}}{\partial\psi}\cdot\beta\delta\psi_{2}\\
&=\left.\frac{\mathrm{d}}{\mathrm{d}\epsilon}\mathcal{F}(\psi+\epsilon\alpha\delta\psi_{1})\right|_{\epsilon=0} + \left.\frac{\mathrm{d}}{\mathrm{d}\epsilon}\mathcal{F}(\psi+\epsilon\beta\delta\psi_{2})\right|_{\epsilon=0}\\
&=\delta\mathcal{F}(\psi;\alpha\delta\psi_{1})+\delta\mathcal{F}(\psi;\beta\delta\psi_{2})
\end{aligned}
\end{split}\]
变分运算的可交换性
对于 Lagrangian 变分和初始构型的运算，显然有
\[\begin{split}
\begin{aligned}
\delta(\nabla_{0}\mathbf{u})&=\left[\frac{\mathrm{d}}{\mathrm{d}\epsilon}\nabla_{0}(\mathbf{u}+\epsilon\delta\mathbf{u})\right]_{\epsilon=0}=\nabla_{0}(\delta\mathbf{u})\\
\delta\left(\int_{\Omega_{0}}\mathbf{u}\mathrm{d}V\right)&=\left[\frac{\mathrm{d}}{\mathrm{d}\epsilon}\int_{\Omega_{0}}(\mathbf{u}+\epsilon\delta\mathbf{u})\mathrm{d}V\right]_{\epsilon=0}=\int_{\Omega_{0}}\delta\mathbf{u}\mathrm{d}V
\end{aligned}
\end{split}\]
一个例子
考虑 \(a(x) = x^{2}\)，给定一个材料扰动
\[
x_{\epsilon}=x+\epsilon\eta(x)
\]
于是
\[\begin{split}
\begin{aligned}
\delta\nabla a &= \delta (2x) = 2\eta\\
\nabla\delta a &= \nabla(2x\eta) = 2\eta+2x\nabla\eta
\end{aligned}
\end{split}\]
前者是梯度场的变分，后者是变分后场的梯度，得到 \(\delta\nabla a \neq \nabla\delta a\)，其原因在于变分与梯度所使用的描述不同
如果都使用 Lagrangian 描述，记初始状态为 \(x = X\)，变形后为 \(x = X + \epsilon\eta(X)\)，此时
\[
A(X) = a(x(X)) = (X + \epsilon\eta(X))^{2}
\]
于是
\[\begin{split}
\begin{aligned}
\delta\nabla_{0} a &= \frac{\mathrm{d}}{\mathrm{d}\epsilon} \left[2(X + \epsilon\eta)(1+\epsilon\nabla\eta)\right]_{\epsilon=0}\\
&=2\frac{\mathrm{d}}{\mathrm{d}\epsilon}\left[X+\epsilon\eta + \epsilon X\nabla\eta + \epsilon^{2}\eta\nabla\eta\right]_{\epsilon=0}\\
&=2\eta+2X\nabla\eta
\end{aligned}
\end{split}\]
另一方面
\[\begin{split}
\begin{aligned}
\nabla_{0} \delta a &= \nabla_{0}\frac{\mathrm{d}}{\mathrm{d}\epsilon} \left[(X + \epsilon\eta(X))^{2}\right]_{\epsilon=0}\\
&=\nabla_{0}(2X\eta)\\
&=2\eta+2X\nabla\eta
\end{aligned}
\end{split}\]
此时有，\(\delta\nabla_{0} a = \nabla_{0} \delta a\)
交换公式
\[
\delta (\nabla \mathbf{u}) = \nabla (\delta \mathbf{u}) - (\nabla \mathbf{u})(\nabla \delta \mathbf{u})
\]
证明
\[\begin{split}
\begin{aligned}
\delta (\nabla \mathbf{u}) = \delta (\nabla_{0} \mathbf{u}\ \mathbf{F}^{-1}) &= \delta(\nabla_{0} \mathbf{u})\mathbf{F}^{-1} + \nabla_{0} \mathbf{u}\delta\mathbf{F}^{-1}\\
&=\nabla_{0}\delta\mathbf{u}\ \mathbf{F}^{-1}-\nabla_{0} \mathbf{u}\mathbf{F}^{-1}\nabla\delta\mathbf{u}\\
&=\nabla\delta\mathbf{u}-\nabla\mathbf{u}\nabla\delta\mathbf{u}
\end{aligned}
\end{split}\]
其中用到了两条重要的公式
\[\begin{split}
\begin{aligned}
\nabla \mathbf{w} &= \nabla_{0} \mathbf{w}\ \mathbf{F}^{-1}\\
\delta\mathbf{F}^{-1} &=-\mathbf{F}^{-1}\nabla\delta\mathbf{u}
\end{aligned}
\end{split}\]
前者是显然的，对于后者，由于 \(\mathbf{F}\mathbf{F}^{-1} = \mathbf{I}\)，故
\[
\begin{aligned}
\mathbf{0} = \delta(\mathbf{F}\mathbf{F}^{-1}) = \delta\mathbf{F}\mathbf{F}^{-1}+\mathbf{F}\delta\mathbf{F}^{-1}
\end{aligned}
\]
于是
\[
\delta\mathbf{F}^{-1} = -\mathbf{F}^{-1}\delta\mathbf{F}\mathbf{F}^{-1}=-\mathbf{F}^{-1}\nabla_0\delta\mathbf{u}\mathbf{F}^{-1}=-\mathbf{F}^{-1}\nabla\delta\mathbf{u}
\]
一些重要的变分运算
虽然都统一使用 \(\delta\) 来表示变分，但也区分为对不同变量的微分，例如 \(\mathbf{u}\) 的变分和对 \(\mathbf{v}\) 的变分，可以通过右端的变分量进行区分
一般来说，时间无关量的变分量是 \(\mathbf{u}\)，如 \(\mathbf{F}\) 和 \(\mathbf{E}\)，时间，时间相关量的变分量是 \(\dot{\mathbf{F}}\) 和 \(\dot{\mathbf{E}}\)
\[
\nabla\delta\mathbf{u} = \nabla_0 \delta\mathbf{u}\ \mathbf{F}^{-1}; \quad \nabla\delta\mathbf{v} = \nabla_0 \delta\mathbf{v}\ \mathbf{F}^{-1}
\]
证明
代入 \(\mathbf{w} = \mathbf{\delta u}\) 和 \(\mathbf{w} = \mathbf{\delta v}\) 到
\[
\nabla\mathbf{w} = \nabla_{0}\mathbf{w}\ \mathbf{F}^{-1}
\]
立证
\[
\delta\mathbf{F} = \delta(\nabla_0 \mathbf{u}) = \nabla_0\delta\mathbf{u} = \nabla(\delta\mathbf{u})\mathbf{F}
\]
\[
\delta\dot{\mathbf{F}} = \delta(\nabla_0 \mathbf{v}) = \nabla_0\delta\mathbf{v} = \nabla(\delta\mathbf{v})\mathbf{F}
\]
\[
\delta\mathbf{F}^{-1} = -\delta\nabla\mathbf{u}= -\mathbf{F}^{-1}\nabla\delta\mathbf{u}
\]
证明
一方面
\[
\delta\mathbf{F}^{-1} = \delta \nabla(\mathbf{x} - \mathbf{u}) = \delta(\mathbf{I} - \nabla\mathbf{u}) = -\delta\nabla\mathbf{u}
\]
另一方面
\[
\delta\mathbf{F}^{-1} = -\mathbf{F}^{-1}\nabla\delta\mathbf{u}
\]
通常 \(\delta\nabla\mathbf{u}\neq\nabla\delta\mathbf{u}\)，接下来证明 \(\delta\nabla\mathbf{u} = \mathbf{F}^{-1}\nabla\delta\mathbf{u}\)，根据交换公式
\[
\delta \nabla \mathbf{u} = \nabla \delta \mathbf{u} - (\nabla \mathbf{u}) \cdot (\nabla \delta \mathbf{u}) = (\mathbf{I} - \nabla \mathbf{u})\nabla \delta \mathbf{u} = \mathbf{F}^{-1}\nabla \delta \mathbf{u}
\]
\[
\delta\boldsymbol{\varepsilon} = \frac{1}{2}\left[(\nabla_0 \delta\mathbf{u})^{\mathrm{T}} + \nabla_0 \delta\mathbf{u}\right]
\]
由 \(\nabla_{0}\) 和 \(\delta\) 的交换性立证
\[\begin{split}
\begin{aligned}
\delta\mathbf{E} &= \frac{1}{2}\mathbf{F}^{\mathrm{T}} \left[(\nabla\delta\mathbf{u})^{\mathrm{T}} + \nabla\delta\mathbf{u}\right] \mathbf{F}\\
&\approx \mathbf{F}^{\mathrm{T}} \delta\boldsymbol{\varepsilon} \mathbf{F}
\end{aligned}
\end{split}\]
对于小位移小应变的情形，第二行近似成立
证明
\[\begin{split}
\begin{aligned}
\delta\mathbf{E} = \frac{1}{2}\delta (\mathbf{F}^{T}\mathbf{F}-\mathbf{I})
&=\frac{1}{2}(\delta\mathbf{F}^{T}\mathbf{F}+\mathbf{F}^{T}\delta\mathbf{F})\\
&=\frac{1}{2}\left((\nabla_{0}\delta\mathbf{u})^{T}\mathbf{F}+\mathbf{F}^{T}\nabla_0\delta\mathbf{u}\right)\\
&=\frac{1}{2}\mathbf{F}^{T}\left((\nabla_{0}\delta\mathbf{u}\ \mathbf{F}^{-1})^{T}+\nabla_{0}\delta\mathbf{u}\ \mathbf{F}^{-1}\right)\mathbf{F}\\
&=\frac{1}{2}\mathbf{F}^{T}\left((\nabla\delta\mathbf{u})^{T}+\nabla\delta\mathbf{u}\right)\mathbf{F}\\
\end{aligned}
\end{split}\]
\[
\delta\mathbf{d} = \frac{1}{2}\left[(\nabla\delta\mathbf{v})^{\mathrm{T}} + \nabla\delta\mathbf{v}\right] = \text{sym}(\nabla\delta\mathbf{v}) = \text{sym}(\delta\dot{\mathbf{F}}\mathbf{F}^{-1})
\]
对 \(\mathbf{d}\) 来说，变分变量为 \(\mathbf{v}\)，而 \(\mathbf{u}\) 保持不变
证明
\[
\nabla\delta\mathbf{v}=\nabla_{0}\delta\mathbf{v}\ \mathbf{F}^{-1}=\delta\dot{\mathbf{F}}\mathbf{F}^{-1}
\]
