A Lagrangian–Hamiltonian unified formalism for a class of dissipative systems

Entropy production in classical thermomechanical systems is the result of three sources: transfer of heat; dissipative stresses, such as viscosity; and internal variables. In this paper, a variational treatment for dissipative systems due to internal variables is presented. Specifically, in the context of the theory of internal variables, a novel dissipative Lagrangian–Hamiltonian formalism is developed. Two fundamental thermodynamic functions (the free energy and the entropy production rate) form the basis of this formalism. The Hamiltonian formulation reveals a new structure on the phase space, and is applied to prove large-time solutions for the semilinear problem. Finally, the formalism is applied to the problem of dynamic brittle fracture