By composite elastic structures, we think of a material where the (3D) bulk contains thin reinforcements (plates or beams) or weaknesses (cracks). In particular, we are interested in the case where the thin elements are desireable to be described by lower-dimensional (2D or 1D) mathematical models. The problem is then that of the appropriate structure of the composite model as a whole, in particular in view of coupling conditions between models of different dimension and complexity.
We consider this problem from the perspective of the mixed-dimensional extension of the de Rham complex introduce in Boon, Nordbotten and Vatne (2021). With this as a building block, we follow the programmatic approach of constructing a "mixed-dimensional Cosserat complex" as a twisted complex based on two times three copies of the de Rham complex, using the non-commutativity of the Kozul operator and the exterior derivative as linking maps. From there, a generalization of the BGG reduction allows for the statement of a "mixed-dimensional elasticity complex", in the sense elaborated in e.g. Arnold and Hu (2021).
Unpacking the abstractions, the mixed-dimensional Cosserat complex gives rise to both contact mechanical models (Hodge-Laplace problem for k=0) and models for reinforced materials (Hodge-Laplace problem for k=n), with Cosserat-type mathematical models in all dimensions. In particular, Cosserat plates (closely related to Mindlin-Reissner plates) and Cosserat beams (closely related to Timoshenko beams) are included.
Conversely, when considering the mixed-dimensional elasticity complex, contact mechanical models and models for reinforced materials appear with simplified models in all dimensions. In the bulk, this means standard (Cauchy) elasticity, while in the lower dimensions, Kirchhoff-Love plates and Euler-Bernoulli beams are included.
In both the Cosserat and elasticity case, the appropriate coupling conditions between models in different dimensions arise naturally as part of the development.
This is joint work with Wietse Boon, Daniel F. Holmen, Kaibo Hu and Jon Eivind Vatne.