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 \[ \mathbf{u} = (u, v, w) \] and let \[ \delta \mathbf{u} = (\delta u, \delta v, \delta w) \] 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: \[ \delta \epsilon_{xx} = \frac{\partial (\delta u)}{\partial x}, \quad \delta \epsilon_{yy} = \frac{\partial (\delta v)}{\partial y}, \quad \delta \epsilon_{zz} = \frac{\partial (\delta w)}{\partial z} \] and the shear strains: \[ \delta \gamma_{xy} = \frac{\partial (\delta u)}{\partial y} + \frac{\partial (\delta v)}{\partial x}, \quad \text{etc.} \]

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 \[ \boxed{ \delta W_B = \int_V \rho \mathbf b \cdot \delta \mathbf u \, dV } \] where \(\mathbf b = [b_x, b_y, b_z]\) is the body force per unit mass.

The work of surface traction is given by \[ \boxed{\delta W_S=\int_s \mathbf{t}\boldsymbol{\cdot}\hat{\mathbf{n}}\ dS} \]

For a surface element with outward normal
\[\mathbf n = [n_x \; n_y \; n_z],\] the traction vector is defined as: \[ \mathbf t = \mathbf n \cdot \boldsymbol{\sigma} \] where \(\boldsymbol{\sigma}\) is the Cauchy stress matrix: \[ \boldsymbol{\sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{bmatrix} \]

and the virtual displacement is the column vector: \[ \delta \mathbf{u} = \begin{bmatrix} \delta u \\ \delta v \\ \delta w \end{bmatrix}. \]

Hence, the virtual work on the surface is: \[ \boxed{ \delta W_S = \int_S \mathbf t \cdot \delta \mathbf u \, dS = \int_S (\mathbf n \, \boldsymbol{\sigma}) \, \delta \mathbf u \, dS } \]

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 \[ \mathbf F = \boldsymbol{\sigma} \, \delta \mathbf u = \begin{bmatrix} \sigma_{xx}\delta u + \sigma_{xy}\delta v + \sigma_{xz}\delta w \\ \sigma_{yx}\delta u + \sigma_{yy}\delta v + \sigma_{yz}\delta w \\ \sigma_{zx}\delta u + \sigma_{zy}\delta v + \sigma_{zz}\delta w \end{bmatrix}, \] then \[ \boxed{ \mathbf t \cdot \delta \mathbf u = n_x F_x + n_y F_y + n_z F_z } \] This shows clearly that the expression \(\mathbf t \cdot \delta \mathbf u\) acts as the dot product of the normal vector \(\mathbf n\) with the vector \(\mathbf F = \boldsymbol{\sigma}\, \delta \mathbf u\).

Using the Divergence Theorem

Apply the divergence theorem to convert the surface integral to a volume integral: \[ \boxed{ \int_S (F_x n_x + F_y n_y + F_z n_z) \, dS = \int_V \left( \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \right) dV } \]

Hence, \[ \delta W_S = \int_V \left( \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \right) dV \]

\[ \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 \[ \delta W = \int_V \left( \sigma_{xx}\,\delta \epsilon_{xx} + \sigma_{yy}\,\delta \epsilon_{yy} + \sigma_{zz}\,\delta \epsilon_{zz} + \sigma_{xy}\,\delta \gamma_{xy} + \sigma_{xz}\,\delta \gamma_{xz} + \sigma_{yz}\,\delta \gamma_{yz} \right) dV. \] ## Strain Energy Density

It follows from the first law of thermodynamics under adiabatic and static conditions (\(\delta W = \delta U\)) that \[ \delta U = \int_V \left( \sigma_{xx}\,\delta \epsilon_{xx} + \sigma_{yy}\,\delta \epsilon_{yy} + \sigma_{zz}\,\delta \epsilon_{zz} + \sigma_{xy}\,\delta \gamma_{xy} + \sigma_{xz}\,\delta \gamma_{xz} + \sigma_{yz}\,\delta \gamma_{yz} \right) dV. \]

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 \[ \delta U_0 = \sigma_{xx}\,\delta \epsilon_{xx} + \sigma_{yy}\,\delta \epsilon_{yy} + \sigma_{zz}\,\delta \epsilon_{zz} + \sigma_{xy}\,\delta \gamma_{xy} + \sigma_{xz}\,\delta \gamma_{xz} + \sigma_{yz}\,\delta \gamma_{yz}. \]

The above equation may be expressed in differential form as \[ \boxed{ dU_0 = \sigma_{xx}\, d\epsilon_{xx} + \sigma_{yy}\, d\epsilon_{yy} + \sigma_{zz}\, d\epsilon_{zz} + \sigma_{xy}\, d\gamma_{xy} + \sigma_{xz}\, d\gamma_{xz} + \sigma_{yz}\, d\gamma_{yz}. } \]

Notice that the terms involving the shear strains can be written as the sum of two components corresponding to tensorial shear strains \(\epsilon_{ij}\).
For example: \[ \sigma_{xy}\gamma_{xy} = \sigma_{xy}\epsilon_{xy} + \sigma_{yx}\epsilon_{yx}. \]

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.