2017/06/02 by Virginia Agostiniani, Agostiniani, Virginia, Alessandro Lucantonio +3 · 2 citations
Engineering · #Advanced Materials and Mechanics #Analysis of PDEs (math.AP) #Elasticity and Material Modeling #FOS: Mathematics #FOS: Physical sciences #Materials Science (cond-mat.mtrl-sci) #Structural Analysis and Optimization
paper · pdf · doi:10.48550/arxiv.1706.00629
openalex publication_date 2017/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We rigorously derive a Kirchhoff plate theory, via \Γ-convergence, from\na three-di -men -sio -nal model that describes the finite elasticity of an\nelastically heterogeneous, thin sheet. The heterogeneity in the elastic\nproperties of the material results in a spontaneous strain that depends on both\nthe thickness and the plane variables x'. At the same time, the spontaneous\nstrain is h-close to the identity, where h is the small parameter\nquantifying the thickness. The 2D Kirchhoff limiting model is constrained to\nthe set of isometric immersions of the mid-plane of the plate into\n\ℝ3, with a corresponding energy that penalizes deviations of the\ncurvature tensor associated with a deformation from a x'-dependent target\ncurvature tensor. A discussion on the 2D minimizers is provided in the case\nwhere the target curvature tensor is piecewise constant. Finally, we apply the\nderived plate theory to the modeling of swelling-induced shape changes in\nheterogeneous thin gel sheets.\n