7. Conjugate Pairs

Whatever measures are chosen to represent the stress and strain (or rate of strain) couplets, their product provides a measure of the work done (or the power spent). Therefore, it is imperative that the stress and strain tensors be conjugate. The notion of conjugation in this context was introduced in the last century. As an example of conjugate pairs, the mechanical work produced by combining second Piola-Kirchhoff stress with Green-Lagrange strain must match that obtained by combining Cauchy stress with Almansi strain. The matching of conjugate pairs requires that mass in an infinitesimal volume be conserved. Therefore, the initial (Lagrangian) differential volume is multiplied by the Jacobian to convert into the current (Eulerian) differential volume in the matching of conjugate integrals. Another example of conjugate pairs matches the power conjugates of second Piola-Kirchhoff stress and the rate of Green-Lagrange strain with Cauchy stress and the rate of deformation. Conjugation in power is helpful in the development of weak forms; measures of stress and rate which are conjugate in power can be used to construct principles of virtual work or power, i.e., weak forms of the momentum equation.