GENERALIZED LINEAR RELATIONSHIP

In linearized theory, stress quantities and strain quantities obey a linear law. The components of the stress variables are linear combinations of the strain variables. The quantities that relate those components are tensors. We conveniently use matrix notation (Nye, 1985), which allows us to express the relationships in a comprehensive form. Therefore the generalized ``Hooke's Law'' is

 

$$\pmatrix{ {\bf \sigma_1 } \cr {\bf \sigma_2 } \cr {\bf \si... ...bf \epsilon_1} \cr {\bf \epsilon_2} \cr {\bf \epsilon_3} \cr}$$ (1)

In this equation we identify ${\bf \sigma_1}$ with the elastic stress tensor, ${\bf \sigma_2}$ with the electric stress (field strength), and ${\bf \sigma_3}$ with the change in thermal stress (temperature). The term $\epsilon_1$ corresponds to the elastic strain tensor, ${\bf \epsilon_2}$ to the electrical displacement and ${\bf \epsilon_3}$ to the change in thermal flow (entropy). The coefficients of the block matrices $$\mathop{\mbox{${\bf c_{ij} }$}}_{\mbox{$\sim$}} ~(i,j=1,2,3)$$ relate generalized ``stress'' vector ${\bf \sigma_i}~(i=1,2,3)$with the generalized ``strain'' vector ${\bf \epsilon_j }~(j=1,2,3)$. The generalized stiffness c is a matrix of complex numbers, which thus accounts for absorption effects. Equation 1 can be viewed as generated by partial derivatives of a more general state function, which takes the form

$$\Phi~=~U~ -~{\bf \epsilon_1}~{\bf \sigma_1}~-~{\bf \epsilon_2}~{\bf \sigma_2}~-~{\bf \epsilon_3}~{\bf \sigma_3}$$ (2)

where U is the internal energy. This gives the block matrix coefficients in terms of the partial derivatives

$$\begin{eqnarray} \displaystyle \mathop{\mbox{${\bf c_{ii}}$}}_{\mbox{$\sim$}} & ... ...ial {\bf \sigma_i} \partial{\bf \sigma_j}}}\vert _{{\bf \sigma_k}}\end{eqnarray}$$ (3) (4)

The important point is that all stresses not used in the partial derivatives are kept constant. Doing so requires that in actual experiments the boundary conditions be properly applied and maintained during measurements.