Strong Ellipticity Research Articles

Conditions for strong ellipticity of 4th-order elasticity tensors

Conditions for strong ellipticity of 4th-order elasticity tensors

Twinning in porous elastomers

It has been known for some time that constitutive models for the large-deformation response of porous elastomers can develop ‘macroscopic’ instabilities as a consequence of loss of strong ellipticity. Indeed, constitutive models obtained by homogenization methods for porous elastomers with periodic and random microstructures can lose strong ellipticity under appropriate loading conditions. For periodic microstructures, it has been shown theoretically, and verified experimentally and numerically, that ‘microscopic’ instabilities consisting in solutions that are periodic on ensembles of unit cells tend to occur before the long-wavelength ‘macroscopic’ instabilities that are captured by the loss of strong ellipticity of the homogenized response. But what about the response of porous elastomers with random microstructures? In this case, microscopic instabilities can be excluded, and the question then arises as to what happens after the onset of a macroscopic instability. Building on earlier work for reinforced elastomers, it is shown here that porous elastomers can undergo twinning after the onset of a macroscopic instability. For this purpose, use is made of a generalized Maxwell-type construction arising from the theory of relaxation and applied to linear comparison variational homogenization estimates for a certain class of two-dimensional porous elastomers consisting of aligned cylindrical pores in a rubber matrix subjected to plane strain loadings. It is shown that such porous elastomers recover Legendre-Hadamard stability by twinning after the onset of a macroscopic instability. Interestingly, the porous elastomers behave like elastic fluids in the twinned region, losing their ability to support shear stresses.

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

Tectonically assisted emplacement of oceanic intraplate volcanoes: The Bathymetrists Seamounts, central Atlantic

The origin of broad intraplate volcanic provinces has been related to deep mantle plumes or shallow lithospheric processes, e.g., underlying large fault systems. One example of an understudied intraplate volcanic province is the Bathymetrists Seamounts (BSM) in the central-eastern Atlantic, an area with dense and extended fracture zones, but the BSM has been associated with a mantle plume origin. Extensive bathymetric mapping and seafloor sampling show that most BSM-seamounts resemble flat-topped guyots capped by carbonate platforms. Vesicular, volcanoclastic samples imply that their tops formed near sea level, followed by reef formation during cessation of volcanism and crustal subsidence. Erosion determines the seamount irregularity proportional to their sizes. Strong ellipticity of some seamounts is related to multiple vents that erupted along fractures. The orientation of the volcanoes, carbonate platforms, and morphological lineaments of the BSM show particular trends that reveal information on their origin and formation mechanisms. Geomorphological analyses indicate a structural control on volcano emplacement related to underlying lithospheric faults resembling a Riedel shear pattern. The stress field corresponding to their orientations is related to a NE-SW tensional setting, fitting to the prevalent tectonic setting ~56–38 Ma years ago that coincides with the BSM formation and an increase in spreading rates. A change in movement of the African plate during this time, together with the reactivation of fracture zones of the strongly sheared equatorial Atlantic, created pathways in the lithosphere and possibly enhanced magmatism. The seamounts do not show distinct differences in erosion state, morphology, carbonate platform depth, or Mn-crust thickness, in contrast to what would be expected for an age progression within the seamount chain. Our observations, therefore, do not support a plume mantle source. While the magmatic source remains undefined, we show that tectonic pathways determined the shape of the seamounts and enabled the broad emplacement of the Bathymetrists seamount chain.

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.

Ellipticity of gradient poroelasticity

We discuss the ellipticity properties of an enhanced model of poroelastic continua called dilatational strain gradient elasticity. Within the theory there exists a deformation energy density given as a function of strains and gradient of dilatation. We show that the equilibrium equations are elliptic in the sense of Douglis–Nirenberg. These conditions are more general than the ordinary and strong ellipticity but keep almost all necessary properties of equilibrium equations. In particular, the loss of the ellipticity could be considered as a criterion of a strain localization or material instability.

Strong Ellipticity and Infinitesimal Stability within Nth-Order Gradient Elasticity

We formulate a series of strong ellipticity inequalities for equilibrium equations of the gradient elasticity up to the Nth order. Within this model of a continuum, there exists a deformation energy introduced as an objective function of deformation gradients up to the Nth order. As a result, the equilibrium equations constitute a system of 2N-order nonlinear partial differential equations (PDEs). Using these inequalities for a boundary-value problem with the Dirichlet boundary conditions, we prove the positive definiteness of the second variation of the functional of the total energy. In other words, we establish sufficient conditions for infinitesimal instability. Here, we restrict ourselves to a particular class of deformations which includes affine deformations.

On well‐posedness of the first boundary‐value problem within linear isotropic Toupin–Mindlin strain gradient elasticity and constraints for elastic moduli

AbstractWithin the linear Toupin–Mindlin strain gradient elasticity we discuss the well‐posedness of the first boundary‐value problem, that is, a boundary‐value problem with Dirichlet‐type boundary conditions on the whole boundary. For an isotropic material we formulate the necessary and sufficient conditions which guarantee existence and uniqueness of a weak solution. These conditions include strong ellipticity written in terms of higher‐order elastic moduli and two inequalities for the Lamé moduli. The conditions are less restrictive than those followed from the positive definiteness of the deformation energy.

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.

IDENTIFYING STRONG ELLIPTICITY VIA BOUNDS ON THE MINIMUM <i>M</i>-EIGENVALUE OF ELASTICITY <i>Z</i>-TENSORS

<i>M</i>-eigenvalues of fourth-order partially symmetric tensors play an important role in the nonlinear elastic material analysis. In this paper, we establish sharp upper and lower bounds on the minimum <i>M</i>-eigenvalue via extreme eigenvalue of the symmetric matrices extracted from elasticity <i>Z</i>-tensors without irreducible conditions, which improves some existing results. Based on the lower bound estimations for the minimum <i>M</i>-eigenvalue, we provide some checkable sufficient or necessary conditions for the strong ellipticity of elasticity <i>Z</i>-tensors. Numerical examples are given to demonstrate the proposed results.

On Strong Ellipticity and Infinitesimal Stability in Third-Order Nonlinear Strain Gradient Elasticity

On Strong Ellipticity and Infinitesimal Stability in Third-Order Nonlinear Strain Gradient Elasticity

Z-Eigenvalue Intervals of Even-Order Tensors with Application to Judge the Strong Ellipticity of an Elasticity Tensor

Z-Eigenvalue Intervals of Even-Order Tensors with Application to Judge the Strong Ellipticity of an Elasticity Tensor

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.

Homogenization, macroscopic instabilities and domain formation in magnetoactive composites: Theory and applications

This work deals with the stability and macroscopic response of soft magnetoelastic composites under combined magnetic and mechanical loading. It is known that, as in the purely mechanical setting, soft composites with suitably designed microstructures can undergo macroscopic (long wavelength) instabilities under certain special loading conditions. The mechanism for such instabilities, and their effects on the macroscopic response of the composites, are less well understood. We investigate here both the onset the instabilities, as well as the macroscopic response after the instabilities, by means of a ‘twinning’ or domain formation mechanism. For this purpose, we propose a generalized Maxwell construction, which builds on the work of Ball and James (1987) for shape-memory alloys and properly accounts for mechanical and magnetic compatibility requirements between ‘twins’ or domains. Thus, generalizing the work of Avazmohammadi and Ponte Castañeda (2016) for the purely mechanical response of reinforced elastomers, this is accomplished by means of the relaxation or quasiconvexification of the ‘principal’ homogenized solution, i.e., the solution prior to the onset of an instability. The methodology is then applied to two-phase magnetoelastic laminates under plane-strain loading conditions with a magnetic field applied in the plane of deformation. We find that the principal solution generally loses (strict) global rank-one convexity before it loses strong ellipticity, as the laminate finds stability by forming ‘twinned’ lamellar domains. In addition, we use the estimates to characterize the ‘phase diagrams’ for the laminates and to describe in detail their macroscopic response when the magnetic field is applied either parallel or perpendicular to the layers. We find that the formation of the lamellar domains leads to a softer mechanical response – including perfectly soft modes of deformation – which can be either facilitated or impeded, depending on the direction of the applied magnetic field. This mechanism opens up novel strategies for the design of magnetoactive elastomer sensors and actuators.

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.