An existing multi-scale FE² finite element framework is extended to steady-state, coupled thermo-mechanical analysis of hyperelastic materials undergoing large deformations. Two formulations are considered: a standard first-order formulation and a thermo-mechanical framework combining second-order mechanical homogenization with a first-order thermal model. The latter is introduced to account for size effects arising from microstructural heterogeneities and thermal expansion. The framework is tested under plane strain conditions for composite microstructures containing either soft or stiff inclusions. The results show that, while the second-order formulation provides a more accurate representation of the RVE fields, the overall macroscale improvement over the classical first-order formulation is problem-dependent. The study demonstrates the practicality of the numerical framework for both approaches in multi-scale thermo-mechanical analysis. 

This study concerns the nonlinear dynamics of plane strain solids under Mindlin’s Form II strain gradient elasticity. An energy–momentum conserving algorithm is applied with -continuous finite elements, ensuring the preservation of energy and momentum. Higher-order inertia terms are included to achieve dynamic consistency. Numerical examples demonstrate that the energy–momentum conserving algorithm is stable and yields results similar to Newmark’s trapezoidal rule. The robustness and accuracy of the approach are validated through convergence studies over varying mesh densities and time step sizes. Consistent with previous studies, increasing length scales reduces displacement amplitudes and increases vibration frequencies. Despite being limited to plane strain conditions, the study provides a foundation for future extensions to three-dimensional problems and further comparative analyses with alternative formulations.