Second-order elastic constants of a solid under stress

For an unstressed solid a set of second-order elastic constants c αβστ can be chosen such that the strain-energy density, the first-order stress-strain relations and the equations of motion all have simple forms; the c αβστ are symmetric with respect to the interchanges (α, β), (σ, τ), (αβ, στ). In the present paper different ways of generalizing the c αβστ when the solid is under stress are discussed and compared, both for a general stress and for the special case of isotropic pressure. It is suggested that the symbols c αβστ (Voigt: c ij ) should be used for the elastic constants defined from the stress-strain relations and C αβστ (Voigt: C ij ) for the elastic constants defined by expressing the strain-energy density as a function of the Lagrangian strain coordinates. The relation of these elastic constants to the dependence of strain energy on homogeneous strain is discussed in detail, and explicit expressions given for cubic crystals under isotropic pressure.