Mixed variational principles for dynamic response of thermoelastic and poroelastic continua

Abstract

The primary objective of the present work is to make further connections between variational methods on the one hand and reversible and irreversible thermodynamics on the other. This begins with the development of a new stationary principle, involving mixed field variables, for continuum problems in infinitesimal dynamic thermoelasticity. By defining Lagrangian and dissipation functions in terms of physically-relevant contributions and invoking the Rayleigh formalism for damped systems, we are able to recover the governing equations of thermoelasticity as the Euler–Lagrange equations. This includes the balance laws of linear momentum and entropy-energy, the constitutive models for elastic response and heat conduction, and the natural boundary conditions. By including energy contributions associated with second sound phenomena, one eliminates the paradox of infinite thermal propagation speeds and the resulting set of governing equations has an elegant symmetry, which is most easily seen in the Fourier wave number domain. A related formulation for dynamic poroelasticity yields two new stationary mixed variational principles. Depending upon the selection of primary field variables, these governing equations can also exhibit an elegant structure, which can deepen our understanding of the underlying phenomena and the thermoelastic–poroelastic analogy. In addition to the theoretical significance, the variational formulations developed here can provide the basis for a class of optimization-based methods for computational mechanics.