Strain Energy

Internal Energy and the First Law of Thermodynamics

When external forces are applied to a deformable body, these forces perform work on the body. According to the first law of thermodynamics, the work done on the system by the external forces, $\delta W$, and the heat that flows into the system, $\delta Q$, is equal to the change in its internal energy, $\delta U$, and kinetic energy, $\delta K$: $$\delta W+\delta Q=\delta U+\delta K.$$

Under adiabatic conditions ($\delta Q=0$) and quasi-static equilibrium ($\delta K=0$), this reduces to: $$\delta W=\delta U$$ Hence, the infinitesimal external work done on the body is entirely stored as internal energy.

Virtual Work of External Forces

Let the displacement field in the body be $$𝐮=(u,v,w)$$ and let $$\delta 𝐮=(\delta u,\delta v,\delta w)$$ be infinitesimal virtual displacements, which are arbitrary small variations in the displacement field consistent with boundary conditions.

The corresponding infinitesimal virtual strains are obtained from the virtual displacement gradients: $$\delta {ϵ}_{xx}=\frac{\partial (\delta u)}{\partial x},\delta {ϵ}_{yy}=\frac{\partial (\delta v)}{\partial y},\delta {ϵ}_{zz}=\frac{\partial (\delta w)}{\partial z}$$ and the shear strains: $$\delta {\gamma }_{xy}=\frac{\partial (\delta u)}{\partial y}+\frac{\partial (\delta v)}{\partial x},\text{etc.}$$

The external work done by the external forces consists of two parts: 1. The work of surface tractions, $\delta W_S$, and
2. The work of body forces, $\delta W_B$.

Thus, $$\delta W=\delta W_S+\delta W_B$$

The work of body forces is given by $$\boxed{\delta W_B={\int }_V\rho 𝐛\cdot \delta 𝐮dV}$$ where $𝐛=[b_x,b_y,b_z]$ is the body force per unit mass.

The work of surface traction is given by $$\boxed{\delta W_S={\int }_s𝐭\mathbf{\cdot }\widehat{𝐧}dS}$$

For a surface element with outward normal
$$𝐧=[n_x{n}_y{n}_z],$$ the traction vector is defined as: $$𝐭=𝐧\cdot 𝝈$$ where $𝝈$ is the Cauchy stress matrix:

and the virtual displacement is the column vector: $$\delta 𝐮=\left[\begin{array}{c}\delta u\\ \delta v\\ \delta w\end{array}\right].$$

Hence, the virtual work on the surface is: $$\boxed{\delta W_S={\int }_S𝐭\cdot \delta 𝐮dS={\int }_S(𝐧𝝈)\delta 𝐮dS}$$

Expanding this term explicitly as shown in your derivation: \begin{aligned} \mathbf n \cdot \boldsymbol{\sigma} \cdot \delta \mathbf u &= \begin{bmatrix} n_x & n_y & n_z \end{bmatrix}\begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{bmatrix} \begin{bmatrix} \delta u \\ \delta v \\ \delta w \end{bmatrix}\\ &=n_x(\sigma_{xx}\delta u + \sigma_{xy}\delta v + \sigma_{xz}\delta w) + n_y(\sigma_{yx}\delta u + \sigma_{yy}\delta v + \sigma_{yz}\delta w)\\ &\qquad + n_z(\sigma_{zx}\delta u + \sigma_{zy}\delta v + \sigma_{zz}\delta w) \end{aligned}

Define the vector $$𝐅=𝝈\delta 𝐮=\left[\begin{array}{c}{\sigma }_{xx}\delta u+{\sigma }_{xy}\delta v+{\sigma }_{xz}\delta w\\ {\sigma }_{yx}\delta u+{\sigma }_{yy}\delta v+{\sigma }_{yz}\delta w\\ {\sigma }_{zx}\delta u+{\sigma }_{zy}\delta v+{\sigma }_{zz}\delta w\end{array}\right],$$ then $$\boxed{𝐭\cdot \delta 𝐮=n_x{F}_x+n_y{F}_y+n_z{F}_z}$$ This shows clearly that the expression $𝐭\cdot \delta 𝐮$ acts as the dot product of the normal vector $𝐧$ with the vector $𝐅=𝝈\delta 𝐮$.

Using the Divergence Theorem

Apply the divergence theorem to convert the surface integral to a volume integral: $$\boxed{{\int }_S(F_x{n}_x+F_y{n}_y+F_z{n}_z)dS={\int }_V(\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z})dV}$$

Hence, $$\delta W_S={\int }_V(\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z})dV$$

\begin{aligned} \delta W = \int_S &\mathbf t \cdot \delta \mathbf u \, dS + \int_V \rho \mathbf b \cdot \delta \mathbf u \, dV\\ = \int \Bigg( &\frac{\partial \sigma_{xx}}{\partial x} \, \delta u + \sigma_{xx} \frac{\partial (\delta u)}{\partial x} + \frac{\partial \sigma_{xy}}{\partial x} \, \delta v + \sigma_{xy} \frac{\partial (\delta v)}{\partial x} + \frac{\partial \sigma_{xz}}{\partial x} \, \delta w + \sigma_{xz} \frac{\partial (\delta w)}{\partial x}\\ &+ \frac{\partial \sigma_{yx}}{\partial y} \, \delta u + \sigma_{yx} \frac{\partial (\delta u)}{\partial y} + \frac{\partial \sigma_{yy}}{\partial y} \, \delta v + \sigma_{yy} \frac{\partial (\delta v)}{\partial y} + \frac{\partial \sigma_{yz}}{\partial y} \, \delta w + \sigma_{yz} \frac{\partial (\delta w)}{\partial y}\\ &+ \frac{\partial \sigma_{zx}}{\partial z} \, \delta u + \sigma_{zx} \frac{\partial (\delta u)}{\partial z} + \frac{\partial \sigma_{zy}}{\partial z} \, \delta v + \sigma_{zy} \frac{\partial (\delta v)}{\partial z} + \frac{\partial \sigma_{zz}}{\partial z} \, \delta w + \sigma_{zz} \frac{\partial (\delta w)}{\partial z} \Bigg) dV\\ &+ \int (\rho b_x \delta u + \rho b_y \delta v + \rho b_z \delta w) \, dV\\ = \int &\Bigg\{ \left( \frac{\partial \sigma_{xx}}{\partial x} + \frac{\partial \sigma_{yx}}{\partial y} + \frac{\partial \sigma_{zx}}{\partial z} + \rho b_x \right) \delta u + \left( \frac{\partial \sigma_{xy}}{\partial x} + \frac{\partial \sigma_{yy}}{\partial y} + \frac{\partial \sigma_{zy}}{\partial z} + \rho b_y \right) \delta v\\ &+ \left( \frac{\partial \sigma_{xz}}{\partial x} + \frac{\partial \sigma_{yz}}{\partial y} + \frac{\partial \sigma_{zz}}{\partial z} + \rho b_z \right) \delta w + \sigma_{xx} \frac{\partial (\delta u)}{\partial x} + \sigma_{yy} \frac{\partial (\delta v)}{\partial y} + \sigma_{zz} \frac{\partial (\delta w)}{\partial z}\\ &+ \sigma_{xy} \left( \frac{\partial (\delta v)}{\partial x} + \frac{\partial (\delta u)}{\partial y} \right) + \sigma_{xz} \left( \frac{\partial (\delta w)}{\partial x} + \frac{\partial (\delta u)}{\partial z} \right) + \sigma_{yz} \left( \frac{\partial (\delta v)}{\partial z} + \frac{\partial (\delta w)}{\partial y} \right) \Bigg\} dV\\ \end{aligned} Since $$\frac{\partial {\sigma }_{xi}}{\partial x}+\frac{\partial {\sigma }_{yi}}{\partial y}+\frac{\partial {\sigma }_{zi}}{\partial z}+\rho b_i=0,$$ we conclude that $$\delta W={\int }_V({\sigma }_{xx}\delta {ϵ}_{xx}+{\sigma }_{yy}\delta {ϵ}_{yy}+{\sigma }_{zz}\delta {ϵ}_{zz}+{\sigma }_{xy}\delta {\gamma }_{xy}+{\sigma }_{xz}\delta {\gamma }_{xz}+{\sigma }_{yz}\delta {\gamma }_{yz})dV.$$ 

Strain Energy Density

It follows from the first law of thermodynamics under adiabatic and static conditions ($\delta W=\delta U$) that $$\delta U={\int }_V({\sigma }_{xx}\delta {ϵ}_{xx}+{\sigma }_{yy}\delta {ϵ}_{yy}+{\sigma }_{zz}\delta {ϵ}_{zz}+{\sigma }_{xy}\delta {\gamma }_{xy}+{\sigma }_{xz}\delta {\gamma }_{xz}+{\sigma }_{yz}\delta {\gamma }_{yz})dV.$$

The change in internal energy (due to mechanical forces) per unit volume is called the strain energy density, denoted by $U_0$: $$\delta U={\int }_V\delta U_{0}dV.$$

By comparing the last two equations, we obtain $$\delta U_0={\sigma }_{xx}\delta {ϵ}_{xx}+{\sigma }_{yy}\delta {ϵ}_{yy}+{\sigma }_{zz}\delta {ϵ}_{zz}+{\sigma }_{xy}\delta {\gamma }_{xy}+{\sigma }_{xz}\delta {\gamma }_{xz}+{\sigma }_{yz}\delta {\gamma }_{yz}.$$

The above equation may be expressed in differential form as $$\boxed{d{U}_0={\sigma }_{xx}d{ϵ}_{xx}+{\sigma }_{yy}d{ϵ}_{yy}+{\sigma }_{zz}d{ϵ}_{zz}+{\sigma }_{xy}d{\gamma }_{xy}+{\sigma }_{xz}d{\gamma }_{xz}+{\sigma }_{yz}d{\gamma }_{yz}.}$$

Notice that the terms involving the shear strains can be written as the sum of two components corresponding to tensorial shear strains ${ϵ}_{ij}$.
For example: $${\sigma }_{xy}{\gamma }_{xy}={\sigma }_{xy}{ϵ}_{xy}+{\sigma }_{yx}{ϵ}_{yx}.$$

Therefore, \bbox[5px,border:1px #f2f2f2;background-color:#f2f2f2]{\begin{aligned} dU_0&=\sigma_{xx}\,d\epsilon_{xx}+\sigma_{xy}\, d\epsilon_{xy}+\sigma_{xz}\,d\epsilon_{xz}+\cdots+\sigma_{zz}d\epsilon_{zz}\\ &=\sum_{i=1}^3\sum_{j=1}^3 \sigma_{ij}\, d\epsilon_{ij} \end{aligned}}

It follows from the above expression that \bbox[5px,border:1px #f2f2f2;background-color:#f2f2f2]{ \frac{\partial U_0}{\partial \epsilon_{ij}} = \sigma_{ij}. }

References

1. Boresi, A. P., Schmidt, R. J., & Sidebottom, O. M. (1993). Advanced mechanics of materials (6th ed.). John Wiley & Sons.
2. Malvern, L. E. (1969). Introduction to the mechanics of a continuous medium. Prentice Hall.
3. Sokolnikoff, I. S. (1956). Mathematical theory of elasticity (2nd ed.). McGraw-Hill.