Browsing by Subject "Gamma convergence"

Now showing 1 - 6 of 6
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    The Foppl-von Karman equations for plates with incompatible strains
    (2010-02-24) Lewicka, Marta; Mahadevan, L.; Pakzad, Reza
    We provide a derivation of the Foppl-von Karman equations for the shape of and stresses in an elastic plate with residual strains. These might arise from a range of causes: inhomogeneous growth, plastic deformation, swelling or shrinkage driven by solvent absorption. Our analysis gives rigorous bounds on the convergence of the three dimensional equations of elasticity to the low-dimensional description embodied in the plate-like description of laminae and thus justifies a recent formulation of the problem to the shape of growing leaves. It also formalizes a procedure that can be used to derive other low-dimensional descriptions of active materials.
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    The infinite hierarchy of elastic shell models: some recent results and a conjecture
    (2009-07-09) Lewicka, Marta; Pakzad, Reza
    We summarize some recent results of the authors and their collaborators, regarding the derivation of thin elastic shell models (for shells with mid-surface of arbitrary geometry) from the variational theory of $3$d nonlinear elasticity. We also formulate a conjecture on the form and validity of infinitely many limiting $2$d models, each corresponding to its proper scaling range of the body forces in terms of the shell thickness
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    The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells
    (2008-10-14) Lewicka, Marta; Mora, Maria Giovanna; Pakzad, Mohammad Reza
    Using the notion of \Gamma-convergence, we discuss the limiting behavior of the 3d nonlinear elastic energy for thin elliptic shells, as their thickness h converges to zero, under the assumption that the elastic energy of deformations scales like h^\beta with 2 < \beta < 4. We establish that, for the given scaling regime, the limiting theory reduces to the linear pure bending. Two major ingredients of the proofs are: the density of smooth infinitesimal isometries in the space of W^{2,2} first order infinitesimal isometries, and a result on matching smooth infinitesimal isometries with exact isometric immersions on smooth elliptic surfaces.
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    A nonlinear theory for shells with slowly varying thickness
    (University of Minnesota. Institute for Mathematics and Its Applications, 2008-07) Lewicka, Marta; Mora, Maria Giovanna; Pakzad, Mohammad Reza
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    Scaling laws for non-Euclidean plates and the W^{2,2} isometric immersions of Riemannian metrics
    (2009-07-09) Lewicka, Marta; Pakzad, Reza
    This paper concerns the elastic structures which exhibit non-zero strain at free equilibria. Many growing tissues (leaves, flowers or marine invertebrates) attain complicated configurations during their free growth. Our study departs from the 3d incompatible elasticity theory, conjectured to explain the mechanism for the spontaneous formation of non-Euclidean metrics. Recall that a smooth Riemannian metric on a simply connected domain can be realized as the pull-back metric of an orientation preserving deformation if and only if the associated Riemann curvature tensor vanishes identically. When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem by introducing the non-Euclidean version of the nonlinear elasticity functional, and establish its $\Gamma$-convergence under the proper scaling. As a corollary, we obtain new necessary and sufficient conditions for existence of a $W^{2,2}$ isometric immersion of a given $2$d metric into $\mathbb R^3$.
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    Shell theories arising as low-energy Γ-limit of 3D nonlinar elasticity
    (University of Minnesota. Institute for Mathematics and Its Applications, 2008-07) Lewicka, Marta; Mora, Maria Giovanna; Pakzad, Mohammad Reza