We study the non-Euclidean (incompatible) elastic energy functionals in the description of prestressed thin films, at their singular limits ( @math -limits) as @math in the film's thickness @math . Firstly, we extend the prior results [Lewicka-Pakzad, Bhattacharya-Lewicka-Schaffner, Lewicka-Raoult-Ricciotti] to arbitrary incompatibility metrics that depend on both the midplate and the transversal variables (the "non-oscillatory" case). Secondly, we analyze a more general class of incompatibilities, where the transversal dependence of the lower order terms is not necessarily linear (the "oscillatory" case), extending the results of [Agostiniani-Lucic-Lucantonio, Schmidt] to arbitrary metrics and higher order scalings. We exhibit connections between the two cases via projections of appropriate curvature forms on the polynomial tensor spaces. We also show the effective energy quantisation in terms of scalings as a power of @math and discuss the scaling regimes @math (Kirchhoff), @math (von Karman) in the general case, as well as all possible (even powers) regimes for conformal metrics, thus paving the way to the subsequent complete analysis of the non-oscillatory setting in [Lewicka]. Thirdly, we prove the coercivity inequalities for the singular limits at @math - and @math - scaling orders, while disproving the full coercivity of the classical von Karman energy functional at scaling @math .
Alternative geometrically nonlinear thin plate theories can be used to analyze the self-similar structures with metric asymptotically flat at infinity @cite_51 , a disk with edge-localized growth @cite_27 , the shape of a long leaf @cite_19 , or torn plastic sheets @cite_4 . In @cite_7 @cite_24 a variant of the F "oppl-von K 'arm 'an equilibrium equations has been formally derived from finite incompressible elasticity, via the multiplicative decomposition of deformation gradient @cite_17 similar to ours. See also models related to cockling of paper, grass blades, sympatelous flowers and movement of euglenids DCB, BMT, D . The forward and inverse problems in the study of self-folding in thin sheets of patterned hydrogel bilayers are discussed in the forthcoming paper @cite_44 .
@cite_4 @cite_7 @cite_24 @cite_19 @cite_27 @cite_44 @cite_51 @cite_17
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A flat plate will bend into a curved shell if it experiences an inhomogeneous growth field or if constrained appropriately at a boundary. While the forward problem associated with this process is well studied, the inverse problem of designing the boundary conditions or growth fields to achieve a particular shape is much less understood. We use ideas from variational optimization theory to formulate a well posed version of this inverse problem to determine the optimal growth field or boundary condition that will give rise to an arbitrary target shape, optimizing for both closeness to the target shape and for smoothness of the growth field. We solve the resulting system of PDE numerically using finite element methods with examples for both the fully non-symmetric case as well as for simplified one-dimensional and axisymmetric geometries. We also show that the system can also be solved semi-analytically by positing an ansatz for the deformation and growth fields in a circular disk with given thickness profile, leading to paraboloidal, cylindrical and saddle-shaped target shapes, and show how a soft mode can arise from a non-axisymmetric deformation of a structure with axisymmetric material properties. We study the buckling of thin elastic plates caused by residual strains concentrated near a free edge. This is a model for plant leaves and torn plastic sheet morphologies. We derive new governing equations explaining self-similar patterns reported earlier in experiments. We reveal the cascade mechanism, determine the bounds for its wavelengths, and predict a similarity factor of 3 in agreement with experiments. This is confirmed by numerical solutions with up to five generations of wrinkles. We deduce a macroscopic strain gradient theory for plasticity from a model of discrete dislocations. We restrict our analysis to the case of a cylindrical symmetry for the crystal under study, so that the mathematical formulation will involve a two-dimensional variational problem. The dislocations are introduced as point topological defects of the strain fields, for which we compute the elastic energy stored outside the so-called core region.We show that the G-limit of this energy (suitably rescaled), as the core radius tends to zero and the number of dislocations tends to infinity, takes the form @PARASPLIT E = fO (W(se) + f (Curl se)) dx, @PARASPLIT where e represents the elastic part of the macroscopic strain, and Curl se represents the geometrically necessary dislocation density. The plastic energy density f is defined explicitly through an asymptotic cell formula, depending only on the elastic tensor and the class of the admissible Burgers vectors, accounting for the crystalline structure. It turns out to be positively 1-homogeneous, so that concentration on lines is permitted, accounting for the presence of pattern formations observed in crystals such as dislocation walls. This study develops a small-deformation theory of strain-gradient plasticity for isotropic materials in the absence of plastic rotation. The theory is based on a system of microstresses consistent with a microforce balance; a mechanical version of the second law that includes, via microstresses, work performed during viscoplastic flow; a constitutive theory that allows: • the microstresses to depend on ∇E˙p, the gradient of the plastic strain-rate, and • the free energy ψ to depend on the Burgers tensor G=curlEp. The microforce balance when augmented by constitutive relations for the microstresses results in a nonlocal flow rule in the form of a tensorial second-order partial differential equation for the plastic strain. The microstresses are strictly dissipative when ψ is independent of the Burgers tensor, but when ψ depends on G the gradient microstress is partially energetic, and this, in turn, leads to a back stress and (hence) to Bauschinger-effects in the flow rule. It is further shown that dependencies of the microstresses on ∇E˙p lead to strengthening and weakening effects in the flow rule. Typical macroscopic boundary conditions are supplemented by nonstandard microscopic boundary conditions associated with flow, and, as an aid to numerical solutions, a weak (virtual power) formulation of the nonlocal flow rule is derived.