Elastic Properties

The material properties of a medium determine how stress and strain are related: we define an elastic constant tensor c_ijkl by

$$\sigma_{ij}^{}$$ = c_ijkl$$\epsilon_{kl}^{}$$.

This has, in principle, 81 different entries. This number is reduced by symmetry. First, the symmetries of $\epsilon$ and $\sigma$ reduce the number of components from 9x9 to 6*6, because

c_ijkl = c_jikl = c_ijlk = c_jilk,

and the fact that the strain energy density

U = $${\textstyle\frac{1}{2}}$$c_ijkle_ije_kl

must be positive definite imposes the further restriction

c_ijkl = c_klij,

reducing the number of elements to 21.

These symmetries of c are often used to write stresses, strains and elastic constants in a compressed (Voight) notation, with the stresses and strains labelled by

1 = 11, 2 = 22 3 = 33 4 = 23 5 = 31 6 = 12

and with the fourth order tensor c replaced by the 6 by 6 matrix C .

Further reductions in the numbers of elements of c follow if the system has some symmetry. These may be deduced (see, for example, Nye) by demanding that the tensor be invariant under the transformations of which leave the system invariant.

In the cubic system, for example, there are only three independent elements c₁₁₁₁ , c₁₁₂₂ and c₁₂₁₂ . In an isotropic system only two constants are needed fully to describe the elastic properties, and

c_ijkl = $$\lambda$$$$\delta_{ij}^{}$$$$\delta_{kl}^{}$$ + $$\mu$$$$\delta_{ik}^{}$$$$\delta_{jl}^{}$$ + $$\mu$$$$\delta_{il}^{}$$$$\delta_{jk}^{}$$

which leads to the simplified stress-strain relationship

$$\sigma_{ij}^{}$$ = $$\lambda$$$$\epsilon_{kk}^{}$$$$\delta_{ij}^{}$$ + 2$$\mu$$$$\epsilon_{ij}^{}$$.

These Lame constants are related to more familiar constants (Young's modulus Y , bulk modulus B , Poisson's ratio $\nu$ ): $\mu$ is the normal shear constant.

$$\lambda {\textstyle\frac{2}{3}} \mu$$ B =

$${\mu (3 \lambda + 2 \mu) \over \lambda + \mu}$$ Y =

$$\nu {\lambda \over 2 (\lambda + \mu)}$$ =

Poisson's ratio is a measure of how much the material contracts sideways as it is extended - physical constraints give - 1 < $\nu$ < ${\frac{1}{2}}$ , but for most materials $\nu$ lies between a quarter and a third.