The consistent coupling boundary condition for the classical micromorphic model: existence, uniqueness and interpretation of parameters

Abstract

We consider the classical Mindlin-Eringen linear micromorphic model with a new strictly weaker set of displacement boundary conditions. The new consistent coupling condition aims at minimizing spurious influences from arbitrary boundary prescription for the additional microdistortion field P . In effect, P is now only required to match the tangential derivative of the classical displacement u which is known at the Dirichlet-part of the boundary.We derive the full boundary condition, in adding the missing Neumann condition on the Dirichletpart. We show existence and uniqueness of the static problem for this weaker boundary condition. These results are based on new coercive inequalities for incompatible tensor fields with prescribed tangential part.Finally, we show that compared to classical Dirichlet conditions on u and P , the new boundary condition modifies the interpretation of the constitutive parameters.

Lagrange and $$H({\text {curl}},{{\mathcal {B}}})$$ based finite element formulations for the relaxed micromorphic model

“…where ∂B P ∩∂B m = ∅ and ∂B P ∪∂B m = ∂B. A dependency between the displacement field and the micro-distortion field on the boundary was proposed by [40] and subsequently considered in [10,45,46,50]. This so-called consistent coupling condition is defined by…”