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, , and the heat that flows into the system, , is equal to the change in its internal energy, , and kinetic energy, :

Under adiabatic conditions () and quasi-static equilibrium (), this reduces to: 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 and let 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: and the shear strains:

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

Thus,

The work of body forces is given by where is the body force per unit mass.

The work of surface traction is given by

For a surface element with outward normal
the traction vector is defined as: where is the Cauchy stress matrix:

and the virtual displacement is the column vector:

Hence, the virtual work on the surface is:

Expanding this term explicitly as shown in your derivation:

Define the vector then This shows clearly that the expression acts as the dot product of the normal vector with the vector .

Using the Divergence Theorem

Apply the divergence theorem to convert the surface integral to a volume integral:

Hence,

Since we conclude that  

Strain Energy Density

It follows from the first law of thermodynamics under adiabatic and static conditions () that

The change in internal energy (due to mechanical forces) per unit volume is called the strain energy density, denoted by :

By comparing the last two equations, we obtain

The above equation may be expressed in differential form as

Notice that the terms involving the shear strains can be written as the sum of two components corresponding to tensorial shear strains .
For example:

Therefore,

It follows from the above expression that

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.