Strongly Elliptic Research Articles

Smoothness of Generalized Solutions to the Dirichlet Problem for Strongly Elliptic Functional Differential Equations with Orthotropic Contractions on the Boundary of Adjacent Subdomains

Smoothness of Generalized Solutions to the Dirichlet Problem for Strongly Elliptic Functional Differential Equations with Orthotropic Contractions on the Boundary of Adjacent Subdomains

Anomaly inflow for local boundary conditions

We study the η-invariant of a Dirac operator on a manifold with boundary subject to local boundary conditions with the help of heat kernel methods. In even dimensions, we relate this invariant to η-invariants of a boundary Dirac operator, while in odd dimension, it is expressed through the index of boundary operators. We stress the necessity of the strong ellipticity condition for the applicability of our methods. We show that the Witten-Yonekura boundary conditions are not strongly elliptic, though they are very close to strongly elliptic ones.

Potentials and transmission problems in weighted Sobolev spaces for anisotropic Stokes and Navier–Stokes systems with L∞ strongly elliptic coefficient tensor

ABSTRACTWe obtain well-posedness results in -based weighted Sobolev spaces for a transmission problem for anisotropic Stokes and Navier–Stokes systems with strongly elliptic coefficient tensor, in complementary Lipschitz domains of , . The strong ellipticity allows to explore the associated pseudostress setting. First, we use a variational approach that reduces the anisotropic Stokes system in the whole to an equivalent mixed variational formulation with data in -based weighted Sobolev spaces. We show that such mixed variational formulation is well-posed in the space , , for any p in an open interval containing 2. Then similar well-posedness results are obtained for two linear transmission problems. These results are used to define the Newtonian and layer potential operators for the considered anisotropic Stokes system and to obtain mapping properties of these operators. The potentials are employed to show the well-posedness of some linear transmission problems, which then is combined with a fixed point theorem in order to show the well-posedness of a nonlinear transmission problem for the anisotropic Stokes and Navier–Stokes systems in -based weighted Sobolev spaces, whenever the given data are small enough.

Smoothness of Generalized Solutions of the Dirichlet Problem for Strongly Elliptic Functional Differential Equations with Orthotropic Contractions

In the disk, we consider the first boundary-value problem for a functional differential equation containing transformations of orthotropic contractions of independent variables of the unknown function. We study the smoothness of generalized solutions inside special-type subdomains and near their boundaries and pose strong ellipticity conditions.

Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

The well-posedness of the boundary value problems for second gradient elasticity has been studied under the assumption of strong ellipticity of the dependence on the second placement gradients (see, e.g., Chambon and Moullet in Comput. Methods Appl. Mech. Eng. 193:2771–2796, 2004 and Mareno and Healey in SIAM J. Math. Anal. 38:103–115, 2006).The study of the equilibrium of planar pantographic lattices has been approached in two different ways: in dell’Isola et al. (Proc. R. Soc. Lond. Ser. A 472:20150, 2016) a discrete model was introduced involving extensional and rotational springs which is also valid in large deformations regimes while in Boutin et al. (Math. Mech. Complex Syst. 5:127–162, 2017) the lattice has been modelled as a set of beam elements interconnected by internal pivots, but the analysis was restricted to the linear case. In both papers a homogenized second gradient deformation energy, quadratic in the neighbourhood of non deformed configuration, is obtained via perturbative methods and the predictions obtained with the obtained continuum model are successfully compared with experiments. This energy is not strongly elliptic in its dependence on second gradients. We consider in this paper also the important particular case of pantographic lattices whose first gradient energy does not depend on shear deformation: this could be considered either a pathological case or an important exceptional case (see Stillwell et al. in Am. Math. Mon. 105:850–858, 1998 and Turro in Angew. Chem., Int. Ed. Engl. 39:2255–2259, 2000). In both cases we believe that such a particular case deserves some attention because of what we can understand by studying it (see Dyson in Science 200:677–678, 1978). This circumstance motivates the present paper, where we address the well-posedness of the planar linearized equilibrium problem for homogenized pantographic lattices. To do so: (i) we introduce a class of subsets of anisotropic Sobolev’s space as the most suitable energy space E relative to assigned boundary conditions; (ii) we prove that the considered strain energy density is coercive and positive definite in E; (iii) we prove that the set of placements for which the strain energy is vanishing (the so-called floppy modes) must strictly include rigid motions; (iv) we determine the restrictions on displacement boundary conditions which assure existence and uniqueness of linear static problems. The presented results represent one of the first mechanical applications of the concept of Anisotropic Sobolev space, initially introduced only on the basis of purely abstract mathematical considerations.

Homogenization of the Neumann problem for higher order elliptic equations with periodic coefficients

Let be a bounded domain of class . In , we study a self-adjoint strongly elliptic operator of order 2p given by the expression , , with Neumann boundary conditions. Here, is a bounded and positive definite matrix-valued function in , periodic with respect to some lattice; is a differential operator of order p. The symbol is subject to some condition ensuring strong ellipticity of the operator . We find approximations for the resolvent in different operator norms with error estimates depending on and .

Homogenization of weakly coercive integral functionals in three-dimensional linear elasticity

This paper deals with the homogenization through Γ-convergence of weakly coercive integral energies with the oscillating density L(x/e)∇v : ∇v in three-dimensional elasticity. The energies are weakly coercive in the sense where the classical functional coercivity satisfied by the periodic tensor L (using smooth test functions v with compact support in R 3) which reads as Λ(L) > 0, is replaced by the relaxed condition Λ(L) ≥ 0. Surprisingly, we prove that contrary to the two-dimensional case of [2] which seems a priori more constrained, the homogenized tensor L 0 remains strongly elliptic, or equivalently Λ(L 0) > 0, for any tensor L = L(y1) satisfying L(y)M : M + D : Cof(M) ≥ 0, a.e. y ∈ R 3 , ∀ M ∈ R 3×3 , for some matrix D ∈ R 3×3 (which implies Λ(L) ≥ 0), and the periodic functional coercivity (using smooth test functions v with periodic gradients) which reads as Λper(L) > 0. Moreover, we derive the loss of strong ellipticity for the homogenized tensor using a rank-two lamination, which justifies by Γ-convergence the formal procedure of [8].

Analytical and numerical investigation of failure of ellipticity for a class of hyperelastic laminates

Abstract This work concerns the behavior of a two-phase periodic laminate composed of homogeneous, isotropic, and hyperelastic layers in equilibrium in the absence of body force and subjected to a finite deformation on its boundary. The effective behavior of unbounded laminates is studied elsewhere using the tangent second-order homogenization method. In the case of a laminate composed of penalty compressible Neo-Hookean phases and subjected to pure shear deformation on its boundary, analytical results indicate that a combination of loading conditions, material properties, and phase concentrations may lead to a reduction of the overall stiffness of the laminate, which may be related to the loss of overall strong ellipticity of the corresponding effective medium, in spite of the fact that the layers are strongly elliptic. Here, the Asymptotic Homogenization Method (AHM) and the Finite Element Method (FEM) are used to verify and extend these results for laminates subjected to similar boundary conditions and compressible Neo-Hookean phases. Using FEM, the rotation angles of the layers at the center of the laminate are calculated and compared to the rotation angles obtained analytically. For certain loading conditions and material properties, the rotation angles are very close to each other up to a critical shear deformation predicted analytically, after which, the angle obtained via FEM changes abruptly from the angle obtained analytically, suggesting a bifurcation-like behavior. This critical deformation depends strongly on the heterogeneity contrast ratio between the phases. For a contrast ratio close to one, no such behavior is observed. As we increase this ratio, the critical deformation becomes smaller and tends to the identity deformation as the contrast ratio tends to infinity. This work may be of interest to practitioners doing numerical simulation of problems involving composites and to researchers interested in material instabilities of effective nonlinear elastic media.

The first boundary-value problem for strongly elliptic functional-differential equations with orthotropic contractions

We obtain a number of necessary and sufficient strong ellipticity conditions for a functional-differential equation containing, in its leading part, orthotropic contractions of the argument of the unknown function. We establish the unique solvability of the first boundary-value problem and the discreteness, semiboundedness, and sectorial structure of its spectrum.

Gaussian bounds, strong ellipticity and uniqueness criteria

Let h be a quadratic form with domain W 1 , 2 ( R d ) given by h ( φ ) = ∑ i , j = 1 d ( ∂ i φ , c i j ∂ j φ ) , where c i j = c j i are real-valued, locally bounded, measurable functions and C = ( c i j ) ⩾ 0 . If C is strongly elliptic, that is, if there exist λ , μ > 0 such that λ I ⩾ C ⩾ μ I > 0 , then h is closable, the closure determines a positive self-adjoint operator H on L 2 ( R d ) which generates a submarkovian semigroup S with a positive distributional kernel K and the kernel satisfies Gaussian upper and lower bounds. Moreover, S is conservative, that is, S t 1 = 1 for all t > 0 . Our aim is to examine converse statements. First, we establish that C is strongly elliptic if and only if h is closable, the semigroup S is conservative and K satisfies Gaussian bounds. Secondly, we prove that if the coefficients are such that a Tikhonov growth condition is satisfied, then S is conservative. Thus, in this case, strong ellipticity of C is equivalent to closability of h together with Gaussian bounds on K. Finally, we consider coefficients c i j ∈ W loc 1 , ∞ ( R d ) . It follows that h is closable and a growth condition of the Täcklind type is sufficient to establish the equivalence of strong ellipticity of C and Gaussian bounds on K.

Comparing the condition of strong ellipticity and the solvability for a purely elastic problem and the corresponding dislocated problem

The aim of this paper is a comparison of a purely elastic problem and the corresponding dislocated one at two levels: (a) strong ellipticity; and (b) solvability. The corresponding dislocated problem is defined through Noll’s theory of materially uniform but inhomogeneous bodies, which is a theory of frozen dislocations. The framework for strong ellipticity is general enough; no specific assumption is made for the elastic field, the field of the defects and the elastic energy. Solvability (existence and uniqueness) in suitable spaces is examined only for a specific distribution of edge dislocations with a particular expression for the energy (neo-Hookean like) and the elastic field (anti-plane shear). The outcome is that strong ellipticity is not affected by the field of the defects. Namely, if one starts from a strongly elliptic elastic material, he will arrive at a dislocated one with the same property. For the dislocated problem we find bounds that the field of the defects has to satisfy in order for solvability to be retained when solvability of the corresponding elastic problem is known. The whole framework is designed for the finite hyperelastostatic case.

Microstructure evolution in hyperelastic laminates and implications for overall behavior and macroscopic stability

In this paper we study the evolution of the underlying microstructure in hyperelastic laminates subjected to finite deformations. To this end, we identify a set of relevant microstructural variables and derive rigorous formulae for their evolution along finite-deformation paths. Making use of these results we then establish connections between the evolution of the microstructure and the overall stress–strain relation and macroscopic stability in hyperelastic laminates. In particular, we show that the rotation of the layers may lead to a reduction in the overall stiffness of the laminates under appropriate loading conditions. Furthermore, we show that this geometric mechanism is intimately related to the possible loss of strong ellipticity of the overall behavior of these materials, which may occur even in the case when the underlying constituents are taken to be strongly elliptic.

Spatial Behavior in the Strongly Elliptic Anisotropic Thermoelastic Materials

In the present article we consider a prismatic semi-infinite cylinder occupied by an anisotropic homogeneous compressible linear thermoelastic material having an elasticity tensor that is strongly elliptic. The cylinder is subjected to zero body force and heat supply, zero displacement–temperature variation on the lateral boundary and pointwise specified displacement–temperature variation over the base. The main purpose of this contribution is to examine how certain measures of the displacement–temperature variation evolve with respect to the axial variable, provided that the strong ellipticity of the elasticity tensor is assumed. To this end, some appropriate measures are associated with the displacement–temperature variation and then an appropriate second-order differential inequality is established under the strong ellipticity condition on the elasticity tensor. The results are specialized for transversely isotropic and rhombic systems of elastic materials.

Homogenization-based constitutive models for porous elastomers and implications for macroscopic instabilities: II—Results

In Part I of this paper, we developed a homogenization-based constitutive model for the effective behavior of isotropic porous elastomers subjected to finite deformations. In this part, we make use of the proposed model to predict the overall response of porous elastomers with (compressible and incompressible) Gent matrix phases under a wide variety of loading conditions and initial values of porosity. The results indicate that the evolution of the underlying microstructure—which results from the finite changes in geometry that are induced by the applied loading—has a significant effect on the overall behavior of porous elastomers. Further, the model is in very good agreement with the exact and numerical results available from the literature for special loading conditions and generally improves on existing models for more general conditions. More specifically, we find that, in spite of the fact that Gent elastomers are strongly elliptic materials, the constitutive models for the porous elastomers are found to lose strong ellipticity at sufficiently large compressive deformations, corresponding to the possible onset of “macroscopic” ( shear band-type) instabilities. This capability of the proposed model appears to be unique among theoretical models to date and is in agreement with numerical simulations and physical experience. The resulting elliptic and non-elliptic domains, which serve to define the macroscopic “failure surfaces” of these materials, are presented and discussed in both strain and stress space.

Homogenization-based constitutive models for porous elastomers and implications for macroscopic instabilities: I—Analysis

The purpose of this paper is to provide homogenization-based constitutive models for the overall, finite-deformation response of isotropic porous rubbers with random microstructures. The proposed model is generated by means of the “second-order” homogenization method, which makes use of suitably designed variational principles utilizing the idea of a “linear comparison composite.” The constitutive model takes into account the evolution of the size, shape, orientation, and distribution of the underlying pores in the material, resulting from the finite changes in geometry that are induced by the applied loading. This point is key, as the evolution of the microstructure provides geometric softening/stiffening mechanisms that can have a very significant effect on the overall behavior and stability of porous rubbers. In this work, explicit results are generated for porous elastomers with isotropic, (in)compressible, strongly elliptic matrix phases. In spite of the strong ellipticity of the matrix phases, the derived constitutive model may lose strong ellipticity, indicating the possible development of shear/compaction band-type instabilities. The general model developed in this paper will be applied in Part II of this work to a special, but representative, class of isotropic porous elastomers with the objective of exploring the complex interplay between geometric and constitutive softening/stiffening in these materials.

On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: II—Application to cylindrical fibers

In Part I of this paper, we presented a general homogenization framework for determining the overall behavior, the evolution of the underlying microstructure, and the possible onset of macroscopic instabilities in fiber-reinforced elastomers subjected to finite deformations. In this work, we make use of this framework to generate specific results for general plane-strain loading of elastomers reinforced with aligned, cylindrical fibers. For the special case of rigid fibers and incompressible behavior for the matrix phase, closed-form, analytical results are obtained. The results suggest that the evolution of the microstructure has a dramatic effect on the effective response of the composite. Furthermore, in spite of the fact that both the matrix and the fibers are assumed to be strongly elliptic, the homogenized behavior is found to lose strong ellipticity at sufficiently large deformations, corresponding to the possible development of macroscopic instabilities [ Geymonat, G., Müller, S., Triantafyllidis, N., 1993. Homogenization of nonlinearly elastic materials, macroscopic bifurcation and macroscopic loss of rank-one convexity. Arch. Rat. Mech. Anal. 122, 231–290]. The connection between the evolution of the microstructure and these macroscopic instabilities is put into evidence. In particular, when the reinforced elastomers are loaded in compression along the long, in-plane axis of the fibers, a certain type of “flopping” instability is detected, corresponding to the composite becoming infinitesimally soft to rotation of the fibers.

Positivity and strong ellipticity

We consider partial differential operators H = − div ⁡ ( C ∇ ) H=-\operatorname {div} (C\nabla ) in divergence form on R d \mathbf {R}^d with a positive-semidefinite, symmetric, matrix C C of real L ∞ L_\infty -coefficients, and establish that H H is strongly elliptic if and only if the associated semigroup kernel satisfies local lower bounds, or, if and only if the kernel satisfies Gaussian upper and lower bounds.

Spectral asymptotics of Euclidean quantum gravity with diff-invariant boundary conditions

A general method is known to exist for studying Abelian and non-Abelian gauge theories, as well as Euclidean quantum gravity, at 1-loop level on manifolds with boundary. In the latter case, boundary conditions on metric perturbations h can be chosen to be completely invariant under infinitesimal diffeomorphisms, to preserve the invariance group of the theory and BRST symmetry. In the de Donder gauge, however, the resulting boundary-value problem for the Laplace-type operator acting on h is known to be self-adjoint but not strongly elliptic. The latter is a technical condition ensuring that a unique smooth solution of the boundary-value problem exists, which implies, in turn, that the global heat-kernel asymptotics yielding 1-loop divergences and 1-loop effective action actually exists. The present paper shows that, on the Euclidean 4-ball, only the scalar part of perturbative modes for quantum gravity is affected by the lack of strong ellipticity. Further evidence for lack of strong ellipticity, from an analytic point of view, is therefore obtained. Interestingly, three sectors of the scalar-perturbation problem remain elliptic, while lack of strong ellipticity is ‘confined’ to the remaining fourth sector. The integral representation of the resulting ζ-function asymptotics on the Euclidean 4-ball is also obtained; this remains regular at the origin by virtue of a spectral identity here obtained for the first time.

Second-Order Estimates for the Macroscopic Response and Loss of Ellipticity in Porous Rubbers at Large Deformations

This work presents the application of a recently proposed “second-order” homogenization method (Ponte Castaneda, 2002) to generate estimates for effective behavior and loss of ellipticity in hyperelastic porous materials with random microstructures that are subjected to finite deformations. The main concept behind the method is the introduction of an optimally selected “linear thermoelastic comparison composite”, which can then be used to convert available linear homogenization estimates into new estimates for the nonlinear hyperelastic composite. In this paper, explicit results are provided for the case where the matrix is taken to be isotropic and strongly elliptic. In spite of the strong ellipticity of the matrix phase, the homogenized “second-order” estimates for the overall behavior are found to lose ellipticity at sufficiently large compressive deformations corresponding to the possible development of shear band-type instabilities (Abeyaratne and Triantafyllidis, 1984). The reasons for this result have been linked to the evolution of the microstructure, which, under appropriate loading conditions, can induce geometric softening leading to overall loss of ellipticity. Furthermore, the “second-order” homogenization method has the merit that it recovers the exact evolution of the porosity under a finite-deformation history in the limit of incompressible behavior for the matrix.

Strong Ellipticity of Transversely Isotropic Elasticity Tensors

The strong ellipticity of the elasticity tensor of a linearly hyperelastic, transversely isotropic material is investigated. The necessary and sufficient conditions for the elasticity tensor to be strongly elliptic are determined for the five constants characterizing it.