Gradient Elasticity Based on Laplacians of Stress and Strain

Using Mindlin's general microstructural elasticity theory we derive models pertaining to a gradient elasticity based on both Laplacians of stress and strain. We prove that such Laplacian-based gradient elasticity models can also be derived on the basis of a thermodynamically consistent gradient elasticity. The proof relies upon a non-conventional thermodynamic framework. It is shown that a general analogy between linear viscoelastic solids based on spring and dashpot elements and specific gradient elasticity models can be established. The governing differential equations of motion of resulting gradient elasticity models are then formulated. By employing energy related arguments, conditions on the material parameters are derived and the appropriate concomitant boundary conditions are specified for both approaches. Finally, the considered models are compared to each other with reference to one-dimensional dispersion relations.