Energy–momentum conserving algorithm for the nonlinear dynamics of plane strain solids in strain gradient elasticity

Abstract 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 $$C^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> -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.