Browsing by Author "Lewicka, Marta"

Now showing 1 - 11 of 11
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    Existence and stability of viscoelastic shock profiles
    (2010-06-09) Barker, Blake; Lewicka, Marta; Zumbrun, Kevin
    We investigate existence and stability of viscoelastic shock profiles for a class of planar models including the incompressible shear case studied by Antman and Malek-Madani. We establish that the resulting equations fall into the class of symmetrizable hyperbolic--parabolic systems, hence spectral stability implies linearized and nonlinear stability with sharp rates of decay. The new contributions are treatment of the compressible case, formulation of a rigorous nonlinear stability theory, including verification of stability of small-amplitude Lax shocks, and the systematic incorporation in our investigations of numerical Evans function computations determining stability of large-amplitude and or nonclassical type shock profiles.
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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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    A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry
    (2008-11-26) Lewicka, Marta
    We prove that the critical points of the $3$d nonlinear elasticity functional over a thin shell of arbitrary geometry and of thickness $h$, as well as the weak solutions to the static equilibrium equations (formally the Euler Lagrange equations associated to the elasticity functional) converge, in the limit of vanishing thickness $h$, to the critical points of the generalized von Karman functional on the mid-surface, recently derived in [14]. This holds provided the elastic energy of the $3$d deformations scale like $h^4$ and the magnitude of the body forces scale like $h^3$.
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    On the existence of traveling waves in the 3d Boussinesq system
    (2008-10-14) Lewicka, Marta; Mucha, Piotr B.
    We extend an earlier work on traveling waves in premixed ames in a gravitationally stratified medium, subject to the Boussinesq approximation. For three-dimensional channels not aligned with the gravity direction, and under the Dirichlet boundary conditions in the fluid velocity, it is shown that a non-planar traveling wave, corresponding to a non-zero reaction, exists, under an explicit condition relating the geometry of the crossection of the channel to the magnitude of the Prandtl and Rayleigh numbers, or when the advection term in the flow equations is neglected.
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    Reduced theories in nonlinear elasticity
    (2010-06-09) Lewicka, Marta
    The purpose of this note is to report on the recent development concerning the analysis and the rigorous derivation of thin film models for structures with nontrivial geometry. This includes: (i) shells with mid-surface of arbitrary curvature, and (ii) plates exhibiting residual stress at free equilibria. In the former setting, we derive a full range of models, some of them previously absent from the physics and engineering literature. The latter phenomenon has been observed in different contexts: growing leaves, torn plastic sheets and specifically engineered polymer gels. After reviewing available results, we list open problems with a promising angle of approach.
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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 \Gamma-limit of 3d nonlinear elasticity
    (2008-10-02) Lewicka, Marta; Mora, Maria Giovanna; Pakzad, Mohammad Reza
    We discuss the limiting behavior (using the notion of \Gamma-limit) of the 3d nonlinear elasticity for thin shells around an arbitrary smooth 2d surface. In particular, under the assumption that the elastic energy of deformations scales like h4, h being the thickness of a shell, we derive a limiting theory which is a generalization of the von Karman theory for plates.
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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