Strong Ellipticity Condition Research Articles

Conditions for strong ellipticity of 4th-order elasticity tensors

Conditions for strong ellipticity of 4th-order elasticity tensors

On the existence of time-periodic solutions of nonlinear parabolic differential equations with nonlocal boundary conditions of the Bitsadze-Samarskii type

We study a nonlinear parabolic differential equation in a bounded multidimensional domain with nonlocal boundary conditions of the Bitsadze-Samarskii type. We prove existence theorems for a periodic in time generalized solution. Su cient conditions for the existence of generalized solutions contain either an algebraic ellipticity condition or an algebraic strong ellipticity condition for the auxiliary differential-difference operator.

Bifurcations and exact solutions in the model of nonlinear elastodynamics of materials with strong ellipticity condition

Bifurcations and exact solutions in the model of nonlinear elastodynamics of materials with strong ellipticity condition

Variant of strain gradient elasticity with simplified formulation of traction boundary value problems

AbstractIn this paper, we show that within the class of isotropic Mindlin‐Toupin gradient theories there exist a particular variant of the theory, which allows a completely simplified form of traction boundary value problems. This theory can be obtained assuming that the high‐grade part of the strain energy density depends only on the vector‐type quantities, that are the gradient of dilatation and the curl of small rotation. Such incomplete gradient theory becomes positive semi‐definite and at the same time it obeys the strong ellipticity conditions of the general Mindlin‐Toupin first strain gradient elasticity. Based on the variational approach it is shown that the equilibrium equations of the developed theory and its definition for the surface traction can be given only in terms of the total stresses (like in classical elasticity). Such formulation can be useful for derivation of the closed form solutions for the problems with traction‐type boundary conditions. Examples of the solutions for the problem of cylindrical bending and for the inplane crack tip fields are presented. It is shown that considered theory allows to obtain a regularized solution for the crack problems and at the same time it does not predict a non‐physical infinite increase of the material's rigidity under bending.

Conditions of strong ellipticity and calculations of M-eigenvalues for a partially symmetric tensor

Strong ellipticity condition (SE-condition) of the equilibrium equations for general nonlinear elastic materials can be equivalently transformed into the SE-condition of a partially symmetric tensor. Qi et al. in 2009 proved that the SE-condition of a partially symmetric tensor holds if and only if its smallest M-eigenvalue is positive. In this paper, a criterion for the SE-condition of a partially symmetric tensor is first given. And then, an alternative method to compute all M-eigentriples of a partially symmetric tensor is presented. Finally, two numerical examples show the effectiveness of the proposed criterion and the calculation of M-eigenvalues in judging the SE-condition of the equilibrium equations.

On ellipticity of static equations of strain gradient elasticity and infinitesimal stability

Within the framework of strain gradient elasticity under finite deformations we formulate the strong ellipticity conditions of equilibrium equations. Within the model a strain energy density is a function of the first and second deformation gradients. Ellipticity involves certain constraints on the tangent elastic moduli. It is also closely related to infinitesimal stability which is defined as the positive definiteness of the second variation of the potential energy functional. Here we consider the first boundary-value problem, that is with Dirichlettype boundary conditions. For one-dimensional deformations we determine necessary and sufficient conditions of infinitesimal instability. The latter constitute two inequalities for elastic moduli.

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.

Impact of mathematical requirements on the invariant-based anisotropic constitutive models for non-linear biomaterials

Finite element simulations are widely used to study the non-linear mechanical behavior of various biomaterials. Constructing an anisotropic strain energy function within the framework of hyperelasticity is an effective approach to describe the material constitutive behavior. The constructed strain energy function needs to satisfy several mathematical requirements to ensure the framework indifference, the material (near) incompressibility, and the material stability. While the framework indifference, or objectivity, can be naturally satisfied by the invariant-based constitutive formulation, how to enforce the material incompressibility and stability requires detailed discussions and careful treatments. The scope of this paper is to examine in detail the impacts of the mathematical requirements on the constitutive formulation for non-linear anisotropic biomaterials. Particularly, theoretical analyses and numerical simulations are combined to investigate the influences of the material incompressibility as a mathematical constraint and various convexity conditions on the invariant-based constitutive modeling. Through a constructed boundary value problem, analytical solutions are derived via the energy minimization and further used to quantify the influences of the material anisotropic and isotropic components on the material responses. The impact of the volumetric–deviatoric split is demonstrated separately for the strictly incompressible and nearly incompressible materials. In order to ensure the material stability, two commonly used convexity conditions, including the strong ellipticity condition and the polyconvexity condition, are discussed in detail. Several numerical examples are provided to demonstrate their impacts on the material stability under different loading conditions. These discussions are particularly relevant to model biomaterials that exhibit non-linear and anisotropic behaviors under complex loading conditions.

Strong ellipticity within the Toupin–Mindlin first strain gradient elasticity theory

We discuss the strong ellipticity (SE) condition within the Toupin–Mindlin first strain gradient elasticity theory. SE condition is closely related to certain material instabilities and describes mathematical properties of corresponding boundary-value problems. For isotropic solids, SE condition transforms into two inequalities in terms of five gradient-elastic moduli.

Criteria for the Strong Ellipticity Condition of a Partially Symmetric Tensor

Criteria for the Strong Ellipticity Condition of a Partially Symmetric Tensor

Nonlinear strain gradient and micromorphic one-dimensional elastic continua: Comparison through strong ellipticity conditions

We discuss the strong ellipticity (SE) conditions for strain gradient and micromorphic continua considering them as an enhancement of a simple nonlinearly elastic material called in the following primary material. Recently both models are widely used for description of material behavior of beam-lattice metamaterials which may possess various types of material instabilities. We analyze how a possible loss of SE results in the behavior of enhanced models. We shown that SE conditions for a micromorphic medium is more restrictive than for its gradient counterpart. On the other hand we see that a violation of SE for a primary material affects solutions within enhanced models even if the SE conditions are fulfilled for them.

Strong ellipticity conditions and infinitesimal stability within nonlinear strain gradient elasticity

We discuss connections between the strong ellipticity condition and the infinitesimal instability within the nonlinear strain gradient elasticity. The strong ellipticity (SE) condition describes the property of equations of statics whereas the infinitesimal stability is introduced as the positive definiteness of the second variation of an energy functional. Here we establish few implications which simplify the further analysis of stability using formulated SE conditions. The results could be useful for the analysis of solutions of homogenized models of beam-lattice materials at different scales.

An exact solution to the Lame problem for a hollow sphere for new types of nonlinear elastic materials in the case of large deformations

Constitutive relations of two classes are proposed for nonlinear elastic isotropic materials, which, in case of purely volumetric deformation, are reduced to the Murnaghan’s equation of state. Exact analytical solution of the Lame problem of the radially symmetric deformation of a hollow sphere is obtained for one of these material classes. Nonlinear effects are studied. The non-uniqueness of solution is obtained for the case in which the sphere radii are specified in the initial configuration. It is shown for this case that there is a limiting pressure, above which the problem has no solution. The strong ellipticity conditions are tested. The obtained results can be used in geomechanics for modeling the recrystallization of metamorphic rocks.

Macro- and micro-instabilities in incompressible bioinspired composite materials with nacre-like microstructure

In this work, we study the macroscopic and microscopic instabilities in 2D incompressible nacre-like composite materials induced by uniaxial loadings. The interchange between macro- and micro-instabilities in bioinspired composite materials has been investigated by examining different combinations of platelets volume fraction, aspect ratio, and shear modulus contrast between the stiff (platelets) and soft (matrix) phases. It has been highlighted that the critical instability mode shapes, together with their critical load factors and wavelengths, are highly influenced by the microscopic geometrical arrangement and the material composition. For a wide range of geometrical and material parameters, the instabilities are found to be global with an in-phase mode shape. It has been shown that the out-of-phase instability mode shapes occur at a significantly higher magnitude of the homogenized energy compared to the in-phase ones. The results indicate that, adopting a small unit cell assembly, the microscopic stability analysis provides in most of cases strong underestimates of the critical stretch ratios, and that an accurate and efficient instability prediction can be instead obtained based on the loss of strong ellipticity condition of the homogenized incremental moduli tensor, except for few cases in which local modes occur.

Programmable sufficient conditions for the strong ellipticity of partially symmetric tensors

The condition for strong ellipticity of the equilibrium equations plays a significant role in the theory of elasticity. For isotropic elastic materials and anisotropic linearly elastic materials, identification of the strong ellipticity conditions for the corresponding equilibrium equations has been discussed in many references and obtained some equivalent checkable criteria. But for general nonlinearly elastic materials, it is hardly possible to give checkable equivalent criteria for the strong ellipticity condition of the associated equilibrium equations. In 2009, Qi et al. pointed that the strong ellipticity condition of the equilibrium equations can be equivalently transformed into the strong ellipticity condition of partially symmetric tensors. In this paper, using the M-eigenvalues of partially symmetric tensors, we give some easily computable and verifiable sufficient conditions for the strong ellipticity of partially symmetric tensors. Based on these criteria, some direct algorithms for identifying the strong ellipticity condition of partially symmetric tensors are derived. Numerical examples show that the proposed criteria are efficient in identifying the strong ellipticity condition of the equilibrium equations, especially for general nonlinearly elastic materials.

Failure of Fredholm solvability for the Dirichlet problem corresponding to weakly elliptic systems

It is known (cf. Martell et al. in Rev Mat Iberoam 32(3):913–970, 2016) that the $$L^p$$ -Dirichlet problem for constant coefficient second-order systems satisfying the Legendre-Hadamard (strong) ellipticity condition is well posed in the upper half-space. However, this may fail if only weak ellipticity for the system in question is assumed. In this paper we shall show that the aforementioned failure is at a fundamental level, in the sense that there exist systems which are weakly elliptic (i.e., their characteristic matrix is invertible) for which the $$L^p$$ -Dirichlet problem in the upper half-space is not even Fredholm solvable.

The effect of residual stress on the stability of a circular cylindrical tube

Residual stresses in an unloaded configuration of an elastic material have a significant influence on the response of the material from that configuration, but the effect of residual stress on the stability of the material, whether loaded or unloaded, has only been addressed to a limited extent. In this paper we consider the level of residual stress that can be supported in a thick-walled circular cylindrical tube of non-linearly elastic material without loss of stability when subjected to fixed axial stretch and either internal or external pressure. In particular, we consider the tube to have radial and circumferential residual stresses, with a simple form of elastic constitutive law that accommodates the residual stress, and incremental deformations restricted to the cross section of the tube. Results are described for a tube subject to a level of (internal or external) pressure characterized by the internal azimuthal stretch. Subject to restrictions imposed by the strong ellipticity condition, the emergence of bifurcated solutions is detailed for their dependence on the level of residual stress and mode number.

On Ellipticity of Hyperelastic Models Restored by Experimental Data

The condition of ellipticity of the equilibrium equation plays an important role for the correct description of the mechanical behavior of materials and is a necessary condition for new constitutive relations. Earlier, new deformation measures were proposed to remove correlations between the terms, which dramatically simplifies restoration of constitutive relations from experimental data. One of these new deformation measures is based on the QR-expansion of deformation gradient. In this paper, we study the strong ellipticity condition for hyperelastic material using the QR-expansion of deformation gradient.

Bounds of M-eigenvalues and strong ellipticity conditions for elasticity tensors

ABSTRACT Novel upper and lower bounds are derived for the M-eigenvalues of a fourth-order elasticity tensor by using the extreme eigenvalues of the symmetric matrices extracted from the given elasticity tensor. The new upper bound of the M-spectral radius is tighter than that given by S. Li et al. in [J Comput Appl Math. 2019;356]. Based on the new lower bound of the M-eigenvalues, a stronger sufficient condition is presented for the strong ellipticity of an elasticity tensor. Numerical examples are also given to demonstrate the results.

Stability analysis at the micro- and macro-scales in periodic bioinspired composites

This work deals with the onset of instabilities at the microscopic and macroscopic scales in uniaxially compressed bioinspired composite materials with periodic microstructure. particular, the interchange between macro- and micro-instabilities in incompressible nacre-like composite materials has been investigated for different combinations of platelets aspect ratio, platelets volume fraction and shear modulus contrast between platelets and soft matrix. It has been demonstrated that the critical stretch ratios and the critical mode shapes are highly influenced by both the geometrical and materials parameters. The stability analyses highlighted that the microscopic stability analysis provides in most cases strong underestimates of the critical stretch ratios (adopting a small unit cell assembly), and that, by performing a macroscopic stability analysis based on the strong ellipticity condition of the homogenized tangent moduli tensor, a more accurate evaluation of the load associated with the primary instability can be instead obtained with less computational efforts. In addition, for a wide range of the investigated parameters, the instabilities are found to be characterized by long wavelengths with an in-phase critical mode shape.