Further Restrictions on Material Properties

The considerations above place restrictions on the properties of stable materials. These thermodynamic constraints are independent from, but combine with, any symmetry properties if the material in question. To remind you what you are probably learning in 3.13, restrictions on the symmetry of a material follow from Neumann's principle.

Neumann's Principle

Any observable symmetry of a physical property of a material must include the symmetry elements of the point group of the material.

Consider how this couples to the condition of a positive definite Hessian in a thermodynamic system. In particular, consider a linear elastic material:¹

$$\input{equations/18-2A}$$ (25-1)

or as a matrix equation

$$\input{equations/18-2B}$$ (25-2)

Adding strains to the internal degrees of freedom,

$$\input{equations/18-2C}$$ (25-3)

Therefore the second derivative will contain terms such as,

$$\input{equations/18-2D}$$ (25-4)

or

$$\input{equations/18-3A}$$ (25-5)

The stiffness matrix that must be positive definite for an isotropic material is:

$$\input{equations/c-iso}$$ (25-6)

Where $\lambda$ and $\mu$ are the elastic Lamé coefficients and are related to the isotropic elastic coefficients:

$$\input{equations/c-iso-defs}$$ (25-7)

The matrix in Equation 25-6 has three unique eigenvalues: $E_{el}/[2(1+\nu)]$, $E_{el}/(1+\nu)$, and $E_{el}/(1-2\nu)$.

Therefore for an isotropic elastic material to be thermodynamically stable, the following conditions must be satisfied, if $E > 0$ then $-1 < \nu < 1/2$²Therefore, $\nu$ can be negative. This is weird, but true. Cork has a small or almost negative Poisson's ratio, which makes it easy to push into a bottle and makes a good seal.

For a cubic material, the stiffness tensor is:

$$\input{equations/c-cubic}$$ (25-8)

which has three eigenvalues, $c_{11} - c_{12}$, $c_{11} + 2 c_{12}$, and $c_{44}$.

The positive definite condition for a cubic elastic material is, then

$$\input{equations/c-conds-cubic}$$ (25-9)