Quasiconvex Envelope Research Articles

Solid Phase Transitions in the Liquid Limit

We address the fundamental difference between solid-solid and liquid-liquid phase transitions within the Ericksen's nonlinear elasticity paradigm. To highlight ideas, we consider the simplest nontrivial 2D problem and work with a prototypical two-phase Hadamard material which allows one to weaken the rigidity and explore the nature of solid-solid phase transitions in a ``near-liquid'' limit. In the language of calculus of variations we probe limits of quasiconvexity in an ``almost liquid'' solid by comparing the thresholds for cooperative (laminate based) and non-cooperative (inclusion based) nucleation. Using these two types of nucleation tests we obtain for our model material surprisingly tight two-sided bounds on the elastic binodal without directly computing the quasiconvex envelope.

Multidimensional rank-one convexification of incremental damage models at finite strains

This paper presents computationally feasible rank-one relaxation algorithms for the efficient simulation of a time-incremental damage model with nonconvex incremental stress potentials in multiple spatial dimensions. While the standard model suffers from numerical issues due to the lack of convexity, our experiments showed that the relaxation by rank-one convexification delivering an approximation to the quasiconvex envelope prevents mesh dependence of the solutions of finite element discretizations. By the combination, modification and parallelization of the underlying convexification algorithms, the novel approach becomes computationally feasible. A descent method and a Newton scheme enhanced by step-size control prevent stability issues related to local minima in the energy landscape and the computation of derivatives. Numerical techniques for the construction of continuous derivatives of the approximated rank-one convex envelope are discussed. A series of numerical experiments demonstrates the ability of the computationally relaxed model to capture softening effects and the mesh independence of the computed approximations. An interpretation in terms of microstructural damage evolution is given, based on the rank-one lamination process.

Analytical and numerical relaxation results for models in soil mechanics

A variational model of pressure-dependent plasticity employing a time-incremental setting is introduced. A novel formulation of the dissipation potential allows one to construct the condensed energy in a variationally consistent manner. For a one-dimensional model problem, an explicit expression for the quasiconvex envelope can be found which turns out to be essentially independent of the original pressure-dependent yield surface. The model problem can be extended to higher dimensions in an empirical manner. Numerical simulation exhibits well-posed behavior showing mesh-independent results.

Extending linear growth functionals to functions of bounded fractional variation

In this paper we consider the minimization of a novel class of fractional linear growth functionals involving the Riesz fractional gradient. These functionals lack the coercivity properties in the fractional Sobolev spaces needed to apply the direct method. We therefore utilize the recently introduced spaces of bounded fractional variation and study the extension of the linear growth functional to these spaces through relaxation with respect to the weak* convergence. Our main result establishes an explicit representation for this relaxation, which includes an integral term accounting for the singular part of the fractional variation and features the quasiconvex envelope of the integrand. The role of quasiconvexity in this fractional framework is explained by a technique to switch between the fractional and classical settings. We complement the relaxation result with an existence theory for minimizers of the extended functional.

Horizontally quasiconvex envelope in the Heisenberg group

This paper is concerned with a PDE-based approach to the horizontally quasiconvex (h-quasiconvex, for short) envelope of a given continuous function in the Heisenberg group.We provide a characterization for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to a first-order nonlocal Hamilton– Jacobi equation. We also construct the corresponding envelope of a continuous function by iterating the nonlocal operator. One important step in our arguments is to prove the uniqueness and existence of viscosity solutions to the Dirichlet boundary problem for the nonlocal Hamilton–Jacobi equation. Applications of our approach to the h-convex hull of a given set in the Heisenberg group are discussed as well.

The Quasiconvex Envelope of Conformally Invariant Planar Energy Functions in Isotropic Hyperelasticity

We consider conformally invariant energies W on the group {{,mathrm{GL},}}^{!+}(2) of 2times 2-matrices with positive determinant, i.e., W:{{,mathrm{GL},}}^{!+}(2)rightarrow {mathbb {R}} such that W(AFB)=W(F)for allA,B∈{aR∈GL+(2)|a∈(0,∞),R∈SO(2)},\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} W(A\\, F\\, B) = W(F) \\quad \\text {for all }\\; A,B\\in \\{a\\, R\\in {{\\,\\mathrm{GL}\\,}}^{\\!+}(2) \\,|\\,a\\in (0,\\infty ),\\; R\\in {{\\,\\mathrm{SO}\\,}}(2)\\}, \\end{aligned}$$\\end{document}where {{,mathrm{SO},}}(2) denotes the special orthogonal group and provides an explicit formula for the (notoriously difficult to compute) quasiconvex envelope of these functions. Our results, which are based on the representation W(F)=hbigl (frac{lambda _1}{lambda _2}bigr ) of W in terms of the singular values lambda _1,lambda _2 of F, are applied to a number of example energies in order to demonstrate the convenience of the singular-value-based expression compared to the more common representation in terms of the distortion {mathbb {K}}:=frac{1}{2}frac{Vert F Vert ^2}{det F}. Applying our results, we answer a conjecture by Adamowicz (in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni, vol 18(2), pp 163, 2007) and discuss a connection between polyconvexity and the Grötzsch free boundary value problem. Special cases of our results can also be obtained from earlier works by Astala et al. (Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton University Press, Princeton, 2008) and Yan (Trans Am Math Soc 355(12):4755–4765, 2003). Since the restricted domain of the energy functions in question poses additional difficulties with respect to the notion of quasiconvexity compared to the case of globally defined real-valued functions, we also discuss more general properties related to the W^{1,p}-quasiconvex envelope on the domain {{,mathrm{GL},}}^{!+}(n) which, in particular, ensure that a stricter version of Dacorogna’s formula is applicable to conformally invariant energies on {{,mathrm{GL},}}^{!+}(2).

A convergence result related to the geometric flow of motion by principal negative curvature

In a recent paper (Carlier et al. in ESAIM Control Optim Calc Var 18(3):611–620, 2012), an interpolation flow between an evolution by convexity and the geometric flow of motion by principal negative curvature was informally proposed. It is also expected that the geometric flow will eventually convexify the sub-level sets of the initial function $$u_0$$, yielding the quasiconvex envelope of $$u_0$$. In this note, we establish existence and uniqueness of the interpolation flow under appropriate conditions and provide a rigorous proof for its limit behaviour. In addition, we show by example that, contrary to intuition, the proposed geometric flow does not always convexify the sub-level sets of $$u_0$$.

Quasiconvex envelope for a model of finite elastoplasticity with one active slip system and linear hardening

An explicit characterization of the quasiconvex envelope of the condensed energy in a model for finite elastoplasticity is presented, both in two and in three spatial dimensions. A variational formulation of plasticity, which is appropriate for the first time step in a time discrete formulation of the evolution problem, is used, and it is assumed that only one slip system is active. The model includes a nonlinear elastic energy, which is invariant under SO(n), and an effective plastic contribution which is quadratic in the slip parameter. The quasiconvex envelope arises via the formation of first-order laminates.

Symmetric Div-Quasiconvexity and the Relaxation of Static Problems

We consider problems of static equilibrium in which the primary unknown is the stress field and the solutions maximize a complementary energy subject to equilibrium constraints. A necessary and sufficient condition for the sequential lower-semicontinuity of such functionals is symmetric ${\rm div}$-quasiconvexity, a special case of Fonseca and M\"uller's $A$-quasiconvexity with $A = {\rm div}$ acting on $R^{n\times n}_{sym}$. We specifically consider the example of the static problem of plastic limit analysis and seek to characterize its relaxation in the non-standard case of a non-convex elastic domain. We show that the symmetric ${\rm div}$-quasiconvex envelope of the elastic domain can be characterized explicitly for isotropic materials whose elastic domain depends on pressure $p$ and Mises effective shear stress $q$. The envelope then follows from a rank-$2$ hull construction in the $(p,q)$-plane. Remarkably, owing to the equilibrium constraint the relaxed elastic domain can still be strongly non-convex, which shows that convexity of the elastic domain is not a requirement for existence in plasticity.

Quasiconvex relaxation of isotropic functions in incompressible planar hyperelasticity

AbstractIn this note, we provide an explicit formula for computing the quasiconvex envelope of any real-valued functionW; SL(2) → ℝ withW(RF) =W(FR) =W(F) for allF∈ SL(2) and allR∈ SO(2), where SL(2) and SO(2) denote the special linear group and the special orthogonal group, respectively. In order to obtain our result, we combine earlier work by Dacorogna and Koshigoe on the relaxation of certain conformal planar energy functions with a recent result on the equivalence between polyconvexity and rank-one convexity for objective and isotropic energies in planar incompressible nonlinear elasticity.

Quasiconvexity and Relaxation in Optimal Transportation of Closed Differential Forms

This manuscript extends the relaxation theory from nonlinear elasticity to electromagnetism and to actions defined on paths of differential forms. The introduction of a gauge allows for a reformulation of the notion of quasiconvexity in Bandyopadhyay et al. (J Eur Math Soc 17:1009–1039, 2015), from the static to the dynamic case. These gauges drastically simplify our analysis. Any non-negative coercive Borel cost function admits a quasiconvex envelope for which a representation formula is provided. The action induced by the envelope not only has the same infimum as the original action, but has the virtue to admit minimizers. This completes our relaxation theory program.

Explicit Relaxation of a Two-Well Hadamard Energy

We compute an explicit quasiconvex envelope for a subclass of double-well Hadamard energies which model materials undergoing isotropic-to-isotropic elastic phase transitions. The construction becomes possible because of stability of the entire jump set, representing points that can coexist at phase boundaries. To prove stability we apply a recently developed technique for establishing polyconvexity of points on the jump set.

A note on relaxation with constraints on the determinant

We consider multiple integrals of the Calculus of Variations of the form E(u) = ∫ W(x, u(x), Du(x)) dx where W is a Carathéodory function finite on matrices satisfying an orientation preserving or an incompressibility constraint of the type, det Du > 0 or det Du = 1, respectively. Under suitable growth and lower semicontinuity assumptions in the u variable we prove that the functional ∫ Wqc(x, u(x), Du(x)) dx is an upper bound for the relaxation of E and coincides with the relaxation if the quasiconvex envelope Wqc of W is polyconvex and satisfies p growth from below for p bigger then the ambient dimension. Our result generalises a previous one by Conti and Dolzmann [Arch. Rational Mech. Anal. 217 (2015) 413–437] relative to the case where W depends only on the gradient variable.

When Rank-One Convexity Meets Polyconvexity: An Algebraic Approach to Elastic Binodal

In the variational problems involving non-convex integral functionals, finding the binodal, the boundary of validity of the quasiconvexity of the energy density, is of central importance. We develop a systematic methodology for identifying a part of the binodal corresponding to simple laminates by showing that in this case the supporting null-Lagrangians, establishing polyconvexity, can be constructed explicitly. We present a nontrivial example from nonlinear elasticity where this approach allows one to obtain the entire quasiconvex envelope.

Isotropic polyconvex electromagnetoelastic bodies

The recent renewal of interest in nonlinear electromagnetoelastic interactions comes from the technological importance of electro- or magnetosensitive elastomers, smart materials whose mechanical properties change instantly on the application of an electric or magnetic field. We consider materials with free energy functions of the form , where F is the deformation gradient, d is the electric displacement, and b is the magnetic induction. It was recently shown by the author that such an energy function is polyconvex if and only if it is of the form (*) ψ ( F , d , b ) = Φ ( F , cof F , det F , d , b , F d , F b ) where is a convex function (of 31 scalar variables). Moreover, an existence theorem was proved for the equilibrium solution for a system consisting of a polyconvex electromagnetoelastic solid plus the vacuum electromagnetic field outside the body. The condition (8), is not just the combination of Ball’s polyconvexity of elastomers ψ ( F ) = Φ ( F , cof F , det F ) with the convexity in the electromagnetic variables. The differential constraints div , div allow for the cross mechanical–electric and mechanical–magnetic terms Fd and Fb which substantially enlarge the class of energies covered by the theory. The result (*), applies to a material of any symmetry; this paper analyzes the condition in the case of isotropic materials. A broad sufficient condition for the polyconvexity is given in that case. Further, it is shown that the commonly used isotropic electroelastic or magnetoelastic invariants are polyconvex except for the biquadratic ones; the paper explicitly determines their quasiconvex envelopes and shows that they are polyconvex.

A partial differential equation for the $\epsilon$-uniformly quasiconvex envelope

In a series of papers Barron, Goebel, and Jensen studied Partial Differential Equations (PDE)s for quasiconvex (QC) functions \cite{barron2012functions, barron2012quasiconvex,barron2013quasiconvex,barron2013uniqueness}. To overcome the lack of uniqueness for the QC PDE, they introduced a regularization: a PDE for $\e$-robust QC functions, which is well-posed. Building on this work, we introduce a stronger regularization which is amenable to numerical approximation. We build convergent finite difference approximations, comparing the QC envelope and the two regularization. Solutions of this PDE are strictly convex, and smoother than the robust-QC functions.

Computing the Level Set Convex Hull

Quasiconvex (QC) functions are functions whose level sets are convex. The quasiconvex envelope (QCE) of a given function, g, is the maximal QC function below g. The QCE provides a level set representation for the convex hull of every level set of a given function. We present a nonlocal line solver for computing the QCE of a given function. The algorithm is based on solving the one dimensional problem on lines, which can be done by a fast marching or sweeping method. Convergence of the algorithm is established.

Relaxation in crystal plasticity with three active slip systems

We study a variational model for finite crystal plasticity in the limit of rigid elasticity. We focus on the case of three distinct slip systems whose slip directions lie in one plane and are rotated by 120° with respect to each other, with linear self-hardening and infinite latent hardening, in the sense that each material point has to deform in single slip. Under these conditions, plastic deformation is accompanied by the formation of fine-scale structures, in which activity along the different slip systems localizes in different areas. The quasiconvex envelope of the energy density, which describes the macroscopic material behavior, is determined in a regime from small up to intermediate strains, and upper and lower bounds are provided for large strains. Finally sufficient conditions are given under which the lamination convex envelope of an extended-valued energy density is an upper bound for its quasiconvex envelope.

Upper and lower bounds on the set of recoverable strains and on effective energies in cubic-to-monoclinic martensitic phase transformations

A major open problem in the mathematical analysis of martensitic phase transformations is the derivation of explicit formulae for the set of recoverable strains and for the relaxed energy of the system. These are governed by the mathematical notion of quasiconvexity. Here we focus on bounds on these quasiconvex hulls and envelopes in the setting of geometrically-linear elasticity. Firstly, we will present mathematical results on triples of transformation strains. This yields further insight into the quasiconvex hull of the twelve transformation strains in cubic-to-monoclinic phase transformations. Secondly, we consider bounds on the energy of such materials based on the so-called energy of mixing thus obtaining a lamination upper bound on the quasiconvex envelope of the energy. Here we present a new algorithm that yields improved upper bounds and allows us to relate numerical results for the lamination upper bound on the energy with theoretical inner bounds on the quasiconvex hull of triples of transformation strains.

On the Theory of Relaxation in Nonlinear Elasticity with Constraints on the Determinant

We consider vectorial variational problems in nonlinear elasticity of the form $I[u]=\int W(Du)dx$, where $W$ is continuous on matrices with positive determinant and diverges to infinity long sequences of matrices whose determinant is positive and tends to zero. We show that, under suitable growth assumptions, the functional $\int W^{qc}(Du)dx$ is an upper bound on the relaxation of $I$, and coincides with the relaxation if the quasiconvex envelope $W^{qc}$ of $W$ is polyconvex and has $p$-growth from below with $p\ge n$. This includes several physically relevant examples. We also show how a constraint of incompressibility can be incorporated in our results.