We discuss how the Reissner-Mindlin plate model [3,4] can be derived from nonlinear three-dimensional finite elasticity in terms of Γ-convergence. The presence of transverse shear effects in the model requires to scale different components of the three-dimensional elastic strain differently, unlike the approach used in classical papers for Kirchhoff–Love elasticity [1,2].
The Reissner–Mindlin plate theory extends the classical Kirchhoff–Love plate theory by allowing transverse shear deformation and independent rotations of material fibers. It becomes relevant when, for instance, layered or sandwich structures make shear effects significant. In such settings, experiments show noticeable deviations from classical theories, especially in bending and vibration.

References.

[1] G. Friesecke, R.D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., 55(11) (2002), pp. 1461--1506. https://doi.org/10.1002/cpa.10048

[2] G. Friesecke, R.D. James, S. Müller, A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence, Arch. Rational Mech. Anal., 180 (2006), pp. 183--236. https://doi.org/10.1007/s00205-005-0400-7

[3] R. D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, J. Appl. Mech., 18 (1951), pp. 31--38. https://doi.org/10.1115/1.4010217
[4] E. Reissner, The effect of transverse shear deformation on the bending of elastic plates, J. Appl. Mech., 12 (1945), pp. 68-–77. https://doi.org/10.1115/1.4009435

Concentrated forces acting at the tip of a two-dimensional wedge give rise to the classical Flamant solution to linear elasticity, whose displacement and strain are singular at the tip of the wedge. Starting from nonlinear elasticity, we prove that the Flamant solution gives the leading order response of a slightly truncated wedge to small boundary displacements or loads. This asymptotic result holds for general hyperelastic energies with super-quadratic growth at infinity; it also holds in the borderline case of quadratic growth at infinity, so long as the tip of the wedge is subjected to small enough displacements or loads. A main point of the proof is to restore compactness to low-energy sequences. We do so by applying a logarithmic change of variables sufficiently far from the tip. To justify this change of variables, we prove a geometric rigidity inequality in L^p for truncated wedge domains with a constant that is uniform in the truncation length. This follows from the bi-Lipschitz invariance of the constant in the L^p Friesecke--James-Müller inequality. Using this change of variables, we derive an asymptotic variational principle characterizing the Flamant solution in the singular limit of an ideal wedge. This is joint work with Paul Plucinsky and Ian Tobasco.

We investigate variational models for thin crystalline films on a rigid substrate under heteroepitaxial growth. The key feature of these models is a competition between the elastic energy, coming from a lattice mismatch, and the surface energy. Understanding the dependence and stability of energy minimisers on the underlying model parameters is of interest, one particular example being the wetting regime in which a completely flat configuration is favourable.

So far these models have mostly been studied under the assumption that the geometry of the film can be described as the subgraph of a function over the substrate. We relax this restriction to allow for a larger class of competitors and are still able to establish concrete parameter thresholds for which the flat configuration is energetically stable. Based on joint work with Barbara Zwicknagl.

Fourier-based localization methods can be used to obtain lower scaling bounds for singular perturbation models. To illustrate this approach, we begin with a three-well problem within the theory of geometrically linearized elasticity. We then consider settings with more wells and discuss the singular perturbed Tartar square, revisiting the results by Rüland and Tribuzio (2022). We also briefly compare the technique with the results by Chan and Conti (2015), which rely on real-space localization methods, and discuss the extent to which the Fourier-based approach can be applied in the geometrically nonlinear case. The talk is based on joint works with Angkana Rüland, Antonio Tribuzio, and Timo Hofmann.