We present mixed formulations and structure-preserving discretization strategies for nonlinear continuum (thermo)-elastic systems within port-Hamiltonian and GENERIC frameworks. First, a port-Hamiltonian (pH) formulation for finite elasticity is developed based on a mixed polyconvex Hu–Washizu-type framework. The underlying polyconvex representation of the stored energy introduces three strain-type fields as independent energy variables, enabling a consistent pH formulation. By extending Livens’ variational principle, the continuous pH structure is derived and subsequently discretized in space in a structure-preserving manner. It is shown that the finite element discretization of the resulting mixed formulation naturally yields a discrete port-Hamiltonian system. Moreover, a time integration scheme based on discrete gradients is proposed, resulting in an energy–momentum consistent framework that accommodates alternative mixed finite element formulations, including those tailored to quasi-incompressible material behavior. In the second part of the talk, the same mixed polyconvex Hu–Washizu-type formulation is embedded via the extended Livens' variational principle into the thermodynamically consistent GENERIC (General Equation for Non-Equilibrium Reversible–Irreversible Coupling) framework to address coupled nonlinear thermo-mechanical problems. Employing a special operator form of GENERIC allows for different choices of thermodynamic variables while preserving the underlying structural properties. Building on this formulation, a discrete gradient-based time integrator is constructed that preserves energy and momentum and, in addition, enforces a non-decreasing entropy-evolution. The resulting Energy–Momentum–Entropy (EME) integrator thus provides a fully structure-preserving time discretization for nonlinear thermo-mechanical systems. Representative numerical examples are presented to illustrate the robustness, accuracy, and structural consistency of the proposed formulations and integrators.