Nonlinear strain gradient and micromorphic one-dimensional elastic continua: Comparison through strong ellipticity conditions

We discuss the strong ellipticity (SE) conditions for strain gradient and micromorphic continua considering them as an enhancement of a simple nonlinearly elastic material called in the following primary material. Recently both models are widely used for description of material behavior of beam-lattice metamaterials which may possess various types of material instabilities. We analyze how a possible loss of SE results in the behavior of enhanced models. We shown that SE conditions for a micromorphic medium is more restrictive than for its gradient counterpart. On the other hand we see that a violation of SE for a primary material affects solutions within enhanced models even if the SE conditions are fulfilled for them. •Strong ellipticity (SE) conditions are compared for nonlinear strain gradient (SG) and micromorphic (MM) elasticity.•Relations between SE of enhanced models and of simple nonlinear elastic (primary) material are clarified.•SE within SG approach is independent on SE of primary material, whereas SE of MM model elasticity inherits it partially.•Both models regularize primary material behavior, so non-existence of solutions is avoided.•SE conditions bring information on material instabilities within enhanced models of continua.