Loss Of Strong Ellipticity Research Articles

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.

Dynamics of prestressed elastic lattices: Homogenization, instabilities, and strain localization

A lattice (or ‘grillage’) of elastic Rayleigh rods (possessing a distributed mass density, together with rotational inertia) organized in a parallelepiped geometry can be axially loaded up to an arbitrary amount without distortion and then be subject to incremental time-harmonic dynamic motion. At certain threshold levels of axial load, the grillage manifests instabilities and displays non-trivial axial and flexural incremental vibrations. Including every possible structural geometry and for an arbitrary amount of axial stretching, Floquet–Bloch wave asymptotics is used to homogenize the in-plane mechanical response, so to obtain an equivalent prestressed elastic solid subject to incremental time-harmonic vibration, which includes, as a particular case, the incremental quasi-static response. The equivalent elastic solid is obtained from its acoustic tensor, directly derived from homogenization and shown to be independent of the rods’ rotational inertia. Loss of strong ellipticity in the equivalent continuum coincides with macro-bifurcation in the lattice, while micro-bifurcation remains undetected in the continuum and corresponds to a vibration of vanishing frequency of the lowest dispersion branch of the lattice, occurring at finite wavelength. Dynamic homogenization reveals the structure of the acoustic branches close to ellipticity loss and the analysis of forced vibrations (both in physical space and Fourier space) shows low-frequency wave localizations. A perturbative approach based on dynamic Green’s function is applied to both the lattice and its equivalent continuum. This shows that only macro-instability corresponds to localization of incremental strain, while micro-instabilities occur in modes which spread throughout the whole lattice with an ‘explosive’ character. In particular, extremely localized mechanical responses are found both in the lattice and in the solid, with the advantage that the former can be easily realized, for instance via 3D printing. In this way, features such as shear band inclination, or the emergence of a single shear band, or competition between micro and macro instabilities become all designable features. The comparison between the mechanics of the lattice and its equivalent solid shows that the homogenization technique allows an almost perfect representation, except when micro-bifurcation is the first manifestation of instability. Therefore, the presented results pave the way for the design of architected cellular materials to be used in applications where extreme deformations are involved.

Formability prediction using bifurcation criteria and GTN damage model

In this paper, four plastic instability criteria, which are based on the bifurcation theory, are coupled with the GTN damage model for the prediction of diffuse and localized necking. General bifurcation (GB) criterion and limit-point bifurcation (LPB) criterion are used to predict diffuse necking, while loss of ellipticity (LOE) criterion and loss of strong ellipticity (LOSE) criterion are used to predict localized necking. The resulting constitutive equations and instability criteria are implemented into the finite element code ABAQUS/Standard. The constitutive equations are formulated within the framework of large deformations and fully three-dimensional approach. Since the developed numerical tools have intended applications mainly for thin sheet metals; therefore, the plane-stress conditions are considered within the instability criteria. The present contribution focuses on the effect of destabilizing mechanisms, due to non-associative plasticity and non-normal plastic flow rule, on the prediction of forming limit diagrams (FLDs). Theoretical classification of the bifurcation criteria, in terms of their order of prediction of critical necking strains, is first presented. Then, several variants of the GTN model are combined with the bifurcation criteria for the prediction of FLDs for fictitious materials. It is shown that the hierarchical prediction order of the different instability criteria is consistent with the theoretical classification, for all the considered variants of the GTN model. More specifically, it is shown that the GB criterion provides a lower bound to all bifurcation criteria, in terms of necking prediction, while the LOE criterion represents an upper bound.

Reinforced elastomers: Homogenization, macroscopic stability and relaxation

This paper deals with the stability and post-bifurcation response of reinforced hyperelastic composites under general loading conditions. It has long been known that these types of materials can undergo both microscopic and macroscopic instabilities. In the latter case, when the instability does not result in material failure, the behavior of the composite after the onset of the instability is less well understood. Recent work (Avazmohammadi and Ponte Castañeda, 2016) indicates that it is possible for the “principal” solution, i.e. the solution before the onset of any instability, to bifurcate into a lower energy solution via the formation of domains. These domains form on a scale much larger than that of the heterogeneity, but still smaller than that of the macroscopic specimen. This work is concerned with such domain formation in neo-Hookean laminates under general three-dimensional loading. In order to obtain the post-bifurcation behavior, the quasiconvexification or relaxation of the principal solution is computed explicitly. In addition, it is shown that the macroscopic instabilities are triggered not by the loss of strong ellipticity, but rather by the loss of global rank-1 convexity of the principal solution, which, in general, happens first. The calculation also reveals that the relaxation requires, in general, a rank-2 “laminate-within-a-laminate” microstructure and allows for multiple “perfectly soft” modes of deformation.

Loss of Strong Ellipticity Through Homogenization in 2D Linear Elasticity: A Phase Diagram

Since the seminal contribution of Geymonat, M\"uller, and Triantafyllidis, it is known that strong ellipticity is not necessarily conserved through periodic homogenization in linear elasticity. This phenomenon is related to microscopic buckling of composite materials. Consider a mixture of two isotropic phases which leads to loss of strong ellipticity when arranged in a laminate manner, as considered by Guti\'errez and by Briane and Francfort. In this contribution we prove that the laminate structure is essentially the only microstructure which leads to such a loss of strong ellipticity. We perform a more general analysis in the stationary, ergodic setting.

Electromechanical instabilities in fiber-constrained, dielectric-elastomer composites subjected to all-around dead-loading

In this work we investigate the possible development of instabilities in a certain class of dielectric-elastomer composites (DECs) subjected to all-around dead electromechanical loading. The DECs consist of a dielectric elastomer matrix phase constrained to plane strain deformations by means of aligned, long, rigid dielectric fibers of elliptical cross-section that are also aligned but randomly distributed in the transverse plane. Two types of instabilities are considered: loss of positive definiteness (LPD), and loss of strong ellipticity (LE). LPD simply corresponds to the loss of local convexity of the homogenized electroelastic stored-energy function for the DECs and can be of two types depending on the resulting instability modes. When the modes are aligned with the ‘principal’ solution, the instability corresponds to a maximum in the nominal electric field, possibly followed by snapping behavior. Alternatively, when the modes are orthogonal to the principal solution, the instability corresponds to a bifurcation from the principal solution. The LE, on the other hand, corresponds to LPD of the electromechanical acoustic tensor and manifests itself by the onset of highly localized shear band instabilities. Our results show that the stability, as well as the type of instability (when stability is lost), of the DECs depends sensitively on the loading conditions and is also affected by the microstructure of the DECs (i.e. the volume fraction of the fibers and their aspect ratio). In particular, it is found that the non-aligned LPD instabilities typically precede aligned LPD and LE instabilities, especially for in-plane isotropic microstructures, but aligned (limit load) LPD and LE instabilities are also possible for composites with anisotropic microstructures. In this context, it should be noted that the possible development of non-aligned LPD bifurcation instabilities appears to have been ignored by prior work. Fortunately, however, such instabilities can usually be avoided by orienting the long (in-plane) axis of the fibers parallel to the tensile stress direction and orthogonal to the applied electric field.

Fiber-constrained, dielectric-elastomer composites: Finite-strain response and stability analysis

This paper presents homogenization estimates for the finite-strain effective response of a certain class of dielectric elastomer composites (DECs) subjected to electromechanical loading conditions. The DECs consist of a dielectric elastomer matrix phase constrained to undergo plane strain deformations by means of aligned, long, rigid-dielectric fibers of elliptical cross section that are also aligned but randomly distributed in the transverse plane. The estimates for the effective electro-active response are obtained by means of available estimates for the purely mechanical response of such composites, together with a partial decoupling strategy/approximation. Such homogenization estimates can then be used to assess the effect of various microstructural parameters, such as the concentration and cross-sectional shape of the fibers, on the overall electromechanical response of the DECs, when subjected to an electric potential difference across suitably positioned soft electrodes. In addition, three different instability and failure mechanisms are investigated: loss of positive definiteness, loss of strong ellipticity and dielectric breakdown, with the objective of finding an optimal design of the microstructure and constituent properties for maximal electrostriction before failure.

Marginal Material Stability

Marginal stability plays an important role in nonlinear elasticity because the associated minimally stable states usually delineate failure thresholds. In this paper we study the local (material) aspect of marginal stability. The weak notion of marginal stability at a point, associated with the loss of strong ellipticity, is classical. States that are marginally stable in the strong sense are located at the boundary of the quasi-convexity domain and their characterization is the main goal of this paper. We formulate a set of bounds for such states in terms of solvability conditions for an auxiliary nucleation problem formulated in the whole space and present nontrivial examples where the obtained bounds are tight.

A numeric-symbolic approach to the problem of localization of plastic flow

A method combining numerical and symbolic computations is presented for deciding whether or not a mechanical state given numerically in terms of its fourth-order tensor of moduli satisfies the localization condition, and more generally, any constitutive restriction on the tensor of moduli having a polynomial form. The idea underlying the method consists in posing the problem not as an optimization problem (minimizing the determinant of the acoustic tensor with respect to direction), but as a polynomial inequality. The polynomiality of the problem is then fully taken advantage of by using a powerful algorithm to solve polynomial inequalities: (a simplified version of) Collins' cylindrical algebraic decomposition algorithm. The method was implemented using the computer algebra system Mathematica, which provides accuracy-controlled symbolic and numerical computations and many built-in functions to handle polynomials. These features make it possible to determine the onset of localization more accurately and more reliably than with the usual optimization-based approaches, which are not guaranteed to converge to the global minimum. The potential of the method is illustrated by addressing loss of ellipticity and loss of strong ellipticity in the case of Gurson's porous material.

Comments on loss of strong ellipticity in elastoplasticity

The loss of strong ellipticity is analyzed for a rate independent infinitesimal elastoplastic model. This local stability condition corresponds to the loss of positive definiteness of the symmetric part of the acoustic tensor. First, in the case of multisurface plasticity some expressions for the plastic hardening moduli are obtained for various bifurcation criteria. Next, explicit expressions for the critical plastic hardening modulus and the critical orientation are obtained in the case of single-surface plasticity (Hill type comparison solid). The analysis is based on a geometric method. Linear, isotropic elasticity, and a general nonassociative flow rule are assumed. However, the principal axes of the second order tensors of the plastic potential and yield surfaces gradients are coaxial. It is shown that, similar to the loss of ellipticity, the direction of the critical orientation is identical to one of the principal directions, except in the particular case where the gradient of the plastic potential and yield surfaces each have a double eigenvalue. In particular, explicit expression for the plastic hardening modulus, using the same geometric method, is also presented for the Raniecki type comparison solid. As an illustrative example, the critical orientation for the loss of strong ellipticity and the classical shear band localization (loss of ellipticity) are compared for axially-symmetric compression and tension.

On the Eigenvalues of the Fourth-Order Constitutive Tensor and Loss of Strong Ellipticity in Elastoplasticity

The eigenvalues of the fourth-order constitutive tangent modulus and the corresponding acoustic tensors are analyzed. Explicit expressions of the eigenvalues are made for the nonsymmetric tangent modulus tensor, and in the case of the deviatoric associative rule for the symmetric part of the tangent modulus and its acoustic tensor. In this context, a rate independent infinitesimal elastoplastic model is considered. The expressions of the plastic hardening modulus are summarized for the different local stability criteria (loss of second order work positiveness, loss of ellipticity, and loss of strong ellipticity). The critical hardening modulus and orientation are discussed in detail in the case of loss of ellipticity and loss of strong ellipticity. This analysis is based on the geometric method and linear, isotropic elasticity and deviatoric associative flow rule. In particular, the critical orientation for the loss of strong ellipticity and the classical shear band localization are compared.

Loss of strong ellipticity in non-associative elastoplasticity

Loss of strong ellipticity is considered for clastoplastic solids in the presence of non-associative flow laws. Reference is made to the two comparison solids introduced by Ramiecki and Bruhns. The loss of strong ellipticity is expressed in terms of a critical value of the hardening modulus. In the context of the infinitesimal theory, the loss of strong ellipticity is shown to occur simultaneously in the two comparison solids. Finally, an explicit form for the critical hardening modulus is given and applications are performed for the Drucker- Prager and Schleicher yield functions.

Strong ellipticity of comparison solids in elastoplasticity with volumetric non-associativity

Reference is made to non-associative, time independent infinitesimal elastoplasticity with non-associative flow rule. The non-associativity is restricted to the volumetric component of the plastic flow. Loss of strong ellipticity is shown to occur before the snap-back modulus is reached. An analytical solution for the condition of loss of strong ellipticity is obtained for the best chosen comparison solid of the family introduced by Raniecki (1979, Bull. Acad. Polon. Sci. XXVII, 391–399). Finally, this solution is shown to coincide with the loss of strong ellipticity in the comparison solid “in loading”. Therefore, for volumetric non-associative flow-rules, strong ellipticity is lost simultaneously in the best chosen comparison solid of Raniecki and in the comparison solid “in loading”.