Strain:thermodynamic force. Stress:state variable. true/false?

[ebarbero]

The mechanics of materials literature considers micro-structural damage to be a thermodynamic variable conjugate to strain energy release rate, not strain, not stress.

The Gibbs energy density is defined as
$$\begin{equation}
\chi = \rho^{-1} \sigma : \epsilon - \psi
\end{equation}$$
where $\psi$ is the Helmholtz Free Energy (HFE).

Form it follows the definition of strain $\epsilon$, conjugate to stress $\sigma$, and thermodynamic damage force $p$, conjugate to damage state variable $s$
$$\begin{equation}
\epsilon = \rho\frac{\partial\chi}{\partial\sigma} \quad ; \quad p = \rho\frac{\partial\chi}{\partial s}
\end{equation}$$

If the thermodynamic damage force $p$ is driving the damage $s$, then strain $\epsilon$ is driving the state of stress $\sigma$, and not the opposite, as it is usually implied in elementary mechanics of materials courses.

Thermodynamic-ally speaking, strain is the "force" driving the "state of stress".

For example, thermal load induces thermal strain, $\alpha\Delta T$. Stress appears only if the strain is somehow restrained, say between rigid walls, or between dissimilar laminas in a laminated composite, etc.

 

[renormalize]

Can you please redo your post to properly display your LaTeX equations? (There is a LaTeX Guide available at the bottom left of the post editing window.)

 

[QuarkyMeson]

Stress and strain are related through constitutive equations, and either can be prescribed depending on the boundary conditions. Your thermal expansion example demonstrates that constrained thermal strain generates stress, not that strain is the cause of stress. Your conclusion does not follow from the equations.

A counterexample is an elastic bar. Apply a tensile load, and the resulting stress produces an elongation. Alternatively, apply a displacement, and the resulting deformation produces stress. The distinction depends on the boundary conditions.

Look at your own eqn 2.