Baker-Ericksen Inequalities Research Articles

Constitutive framework of a new hyperelastic model for isotropic rubber-like materials for finite element implementation

The strain energy function of hyperelastic material models must fulfill some mathematical conditions to satisfy requirements such as numerical stability and physically plausible mechanical behavior. In the framework of computer simulations with Newton-type methods, numerical stability is assured by the positive definiteness of the tangent operator. The Baker-Ericksen inequalities, on the other hand, are sufficient and necessary conditions in order to guarantee that the material behaves in a physically plausible way, although they are rarely taken into account during parameter identification. The present work proposes a modification in the strain energy function of a previously developed model for isotropic rubber-like materials. The new expression for W allows the satisfaction of both of the aforementioned requirements. The complete constitutive framework for its implementation in a Finite Element code is provided. Representative examples are analyzed and to show the superior performance when compared to well-known models found in the specialized literature both for homogeneous and non-homogeneous cases of deformation.

A consistent isotropic spectral constitutive equation: The infinitesimal strain depends nonlinearly on the stress

Recently, there have been several developments in nonlinear infinitesimal-strain stress constitutive relations to describe the response of elastic bodies. However, some of the nonlinear constitutive equations presented in the literature are unable to model both compressible and incompressible bodies. In this paper, we develop a spectral constitutive equation, where the infinitesimal strain depends nonlinearly on the stresses, and the incompressible behaviour is simply described as a special case obtained from the compressible elastic model, by just letting the value of the Poisson’s ratio equal to a half. The constitutive equation satisfies the Baker-Ericksen inequality and a specific expression for the constitutive equation is proposed to fit experimental data for the compression of a sample of rock. Several boundary value problems with homogeneous deformations and stresses are analysed. Spectral components for the fourth order tensor that is required for the analysis of the propagation of P- and S-waves are given.

An analysis driven construction of distortional-mode-dependent and Hill-Stable elastic potential with application to human brain tissue

We propose an innovative procedure by exploiting the physical meaning of natural strain or Lode invariants with the following salient contributions: 1) Uniaxial data for human brain tissue is used to stipulate the mathematical structure of the potential in terms of the Lode invariant that quantifies the magnitude of distortion along with the modulus term being an unknown function of the Lode angle that quantifies the mode or type of distortion. 2) By a priori analysis using the Baker-Ericksen inequalities, the mathematical form of the modulus function is determined in a novel manner. 3) The derived modulus function is corrected by adding a constant, which in turn is determined using analysis involving sufficient conditions of the stronger Hill inequality. 4) In addition, we also prove that any potential that satisfies Hill inequality also satisfies true-stress-true-strain monotonicity condition in plane stress. Compared to Mihai-Ogden model, besides excellent quantitative agreement with data for human brain tissue (see Mihai et al., 2017), the constructed model also emulates the observed non-linear behavior of shear stress with respect to the amount of shear as opposed to the nearly linear response predicted by the antecedent model. Additionally, when only tension-compression data is available for determining material parameters, the predicted combined tension and shear response associated with the proposed constitutive relation shows monotone decreasing Poynting stress (compressive), while the former predicts an unexpected non-monotone response for certain levels of tension.

Do we need Truesdell’s empirical inequalities? On the coaxiality of stress and stretch

Truesdell’s empirical inequalities are considered essential in various fields of nonlinear elasticity. However, they are often used merely as a sufficient criterion for semi-invertibility of the isotropic stress strain-relation, even though weaker and much less restricting constitutive requirements like the strict Baker–Ericksen inequalities are available for this purpose. We elaborate the relations between such constitutive conditions, including a weakened version of the empirical inequalities, and their connection to bi-coaxiality and related matrix properties. In particular, we discuss a number of issues arising from the seemingly ubiquitous use of the phrase “X,Y have the same eigenvectors” when referring to commuting symmetric tensors X,Y.

Likely Cavitation in Stochastic Elasticity

We revisit the classic problem of elastic cavitation within the framework of stochastic elasticity. For the deterministic elastic problem, involving homogeneous isotropic incompressible hyperelastic spheres under radially symmetric tension, there is a critical dead-load traction at which cavitation can occur for some materials. In addition to the well-known case of stable cavitation post-bifurcation at the critical dead load, we show the existence of unstable snap cavitation for some isotropic materials satisfying Baker-Ericksen inequalities. For the stochastic problem, we derive the probability distribution of the deformations after bifurcation. In this case, we find that, due to the probabilistic nature of the material parameters, there is always a competition between the stable and unstable states. Therefore, at a critical load, stable or unstable cavitation occurs with a given probability, and there is also a probability that the cavity may form under smaller or greater loads than the expected critical value. We refer to these phenomena as ‘likely cavitation’. Moreover, we provide examples of homogeneous isotropic incompressible materials exhibiting stable or unstable cavitation together with their stochastic equivalent.

Loss of ellipticity in elasticity with energy limiters

Abstract Traditional hyperelastic models usually obey requirement of material stability in various forms: Baker-Ericksen inequalities, strong ellipticity, polyconvexity etc. It is reasonable, of course, to require stability for the intact behavior of materials. However, all materials fail and a description of failure should be incorporated in the constitutive law. A simple version of hyperelasticity with failure can be formulated based on the introduction of the limiter in the strain energy density. The limited strain energy bounds maximum achievable stress automatically. Evidently, the elasticity with the energy limiter should exhibit material instability. This work addresses two practically interesting calculations concerning the onset of material instability via the loss of ellipticity. First, we consider simple shear of natural rubber. We find the direction of failure localization, which is in perfect qualitative correspondence with fracture observations in rubber bearings after earthquakes. Interestingly, the direction of failure localization is different from the one predicted by the criterion of maximum tension stress or stretch. Second, we consider equibiaxial tension of a sheet of aneurysm material. We find that the isotropic aneurysm material exhibits infinitely many possible directions of failure localization in equibiaxial tension. The latter means that the random direction of cracks in equibiaxial tension experiment, e.g. membrane inflation, can be an indicator of material isotropy. Accordingly, a preferable direction of the crack alignment can be interpreted as an indicator of the aneurysm anisotropy.

Positive or negative Poynting effect? The role of adscititious inequalities in hyperelastic materials

Motivated by recent experiments on biopolymer gels whereby the reverse of the usual (positive) Poynting effect was observed, we investigate the effect of the so-called ‘adscititious inequalities’ on the behaviour of hyperelastic materials subject to shear. We first demonstrate that for homogeneous isotropic materials subject to pure shear, the resulting deformation consists of a triaxial stretch combined with a simple shear in the direction of the shear force if and only if the Baker–Ericksen inequalities hold. Then for a cube deformed under pure shear, the positive Poynting effect occurs if the ‘sheared faces spread apart’, whereas the negative Poynting effect is obtained if the ‘sheared faces draw together’. Similarly, under simple shear deformation, the positive Poynting effect is obtained if the ‘sheared faces tend to spread apart’, whereas the negative Poynting effect occurs if the ‘sheared faces tend to draw together’. When the Poynting effect occurs under simple shear, it is reasonable to assume that the same sign Poynting effect is obtained also under pure shear. Since the observation of the negative Poynting effect in semiflexible biopolymers implies that the (stronger) empirical inequalities may not hold, we conclude that these inequalities must not be imposed when such materials are described.

Constitutive structure in coupled non-linear electro-elasticity: Invariant descriptions and constitutive restrictions

A constitutive framework for electro-sensitive materials in the context of non-linear elasticity is analyzed. Constitutive equations are given in terms of energy functions that depend on several invariants. The study includes both the analysis of the invariants, which are present in the energy functions, and the analysis of constitutive restrictions that have to be obeyed by the constitutive functions. Isotropic as well as non-isotropic electro-sensitive elastomers are studied. The set of invariants that describe each material model is analyzed under two homogeneous deformations: (i) an uniaxial elongation and (ii) a simple shear deformation. These deformations are chosen since they are relevant to specific experiments, from which one may try to fit constitutive equations. The constitutive restrictions developed are based on classical ones used for isotropic non-linear elastic materials, in particular, are based on the Baker–Ericksen inequality and the ellipticity condition.

Ideally soft nematic elastomers

The paper examines a class of energies $W$ of nematic elastomers that exhibit ideally soft behavior. These are generalizations of the neo-classical energy function proposed by Bladon, Terentjev & Warner [7]. The effective energy (quasiconvexification) of $W$ is calculated for a large subclass of considered energies. Within the subclass, the rank 1 convex, quasiconvex, and polyconvex envelopes coincide and reduce to the largest function below $W$ that satisfies the Baker–Ericksen inequalities. Compressible cases are included. The effective energy displays three regimes: one fluid-like, one partially fluid-like and one hard, as established by DeSimone & Dolzmann [20] for the energy function of Bladon, Terentjev & Warner. Ideally soft deformation modes are shown to arise.

On Certain Deformation Classes of Compressible Hencky Materials

For the case of plane strain and in the absence of body forces an exact solution that describes the straightening of annular cylindrical sectors composed of compressible Hencky materials is obtained and, using the energy criterion, its stability is investigated. The most general class of isotropic, compressible, hyperelastic materials for which a solution of this type is possible is also determined. Under certain restrictions, an inequality that may be regarded as a universal relation is shown to hold for all materials in this class as well as for all isotropic, compressible, hyperelastic materials that satisfy the Baker-Ericksen inequality. Some other inequalities, showing that certain deformations of bodies composed of Hencky materials are necessarily accompanied by an overall volume increase (which may also be viewed as universal relations for the Hencky materials), are established.

Generalized Baker–Ericksen Inequalities

It is well known that isotropic, nonlinearly elastic materials satisfy the Baker–Ericksen inequalities as a consequence of the strong ellipticity or rank 1 convexity. Here we present a generalization to a non-isotropic elastic material which posseses a preferred element in the symmetry group.

On a class of controllable deformations of isotropic incompressible elastic solids with simple material inhomogeneity

In a preceding paper, a class of deformations involving one or two unknown (shear) functions was proposed for consideration as statically possible in all incompressible, isotropic elastic materials, which are heat conducting and may also have a specific form of latent material inhomogeneity due to an infinitesimally thin layered structure. Within three well-defined subclasses Cl, C2 and C3, the existence of controllable deformations was established, if the free energy of the material is smooth as a function of the invariants of the deformation tensor and satisfies the constraints of the Baker-Ericksen (B-E) inequalities. A deformation of a body is said to be controllable if the boundary tractions necessary to maintain it in static equilibrium are known. Here the subclass C4 of deformations, which is the set complementary to those previously demonstrated to be controllable in all materials of the general type, is considered. The (controllable) existence of the elements of this subclass is established in the main if (i) the free energy of the material is twice continuously differentiable with respect to the first two invariants of the left Cauchy-Green deformation tensor when the two unknown shear functions take the values zero and (ii) the B-E inequalities apply. For the exceptional cases (controllable) existence is established if the E-inequalities replace the B-E inequalities. The results are achieved by use of the implicit function theorem. The smoothness assumptions are stronger than those of the preceding paper.

The Baker–Ericksen inequalities for hyperelastic models using a novel set of invariants of Hencky strain

We study a class of models that describes isotropic hyperelastic materials and that is defined using a novel set of invariants for the Hencky strain. For these models, we derive the appropriate form of the Baker–Ericksen inequalities. To illustrate an application, we then use this form of the Baker–Ericksen inequalities to develop a set of specific constitutive restrictions for a model of rubber-like materials proposed in [Criscione, J.C., Humphrey, J.D., Douglas, A.S., Hunter, W.C., 2000. J. Mech. Phys. Solids 48, 2445–2465].

Monotonicity of rotationally invariant convex and rank 1 convex functions

Let f : Mn×n → R ∪ {∞} be a function on the space Mn×n of n by n matrices that is invariant with respect to the left and right multiplication by proper orthogonal tensors. It is shown that f(A) ≤ f(Ā) if f is convex and the partial sums of the singular values of A, Ā ∈ Mn×n satisfy certain ordering inequalities. The same holds if f is rank 1 convex and the partial products of the singular values satisfy analogous inequalities. The proofs emphasize the roles of the ordered-forces inequalities and the Baker-Ericksen inequalities for invariant convex and rank 1 convex functions. As an application, the evaluation of the convex and lamination convex hulls of fully rotationally invariant sets by Dacorogna and Tanteri is simplified and similar results are given for sets invariant only with respect to the proper orthogonal group.

On the Existence of a Stretch for a Prescribed Stress in Isotropic, Incompressible Elastic Materials

The authors show that an isotropic, incompressible elastic material can support any system of homogeneous tractions if it satisfies a strengthened form of Truesdell's inequalities. Specifically, if this strengthened condition is satisfied, the Cauchy and first Piola-Kirchhoff stress tensors can assume all values. The strengthened condition is satisfied by certain Mooney-Rivlin and neo-Hookean materials, and implies both the Baker-Ericksen inequalities and a restriction proposed by Carroll and McCarthy. Finally, it also implies two growth conditions, either one of which is sufficient to guarantee the existence result for the Cauchy stress tensor.

On the Ubiquity of Fracture in Nonlinear Elasticity

This paper points out a subtle and little known consequence of assuming rank-one convexity for an isotropic hyperelastic material [e.g., of assuming a common ordering of principal stretches and principal stresses and hence the Baker–Ericksen inequalities]. What is shown is that rank-one convexity necessarily privilages those affine deformations which are dilatations: – the stored-energy associated with a dilatation is smaller than the stored energy associated with any other affine deformation possessing the same determinant. Also pointed out are fracture related consequences of this property that arise when the stored-energy function assigns to the dilatation of determinant δ > 0 a value A(δ∈) which is not an everywhere convex function of (0,∞).

A note on the stability of certain deformations of compressible nonlinearly elastic media

The main stability result established in [15] (according to which a plane strain equilibrium solution of a harmonic material which satisfies the tension-extension condition is globally stable for boundary conditions of place and zero body forces if and only if a weakened form of the Baker-Ericksen inequality is satisfied at this particular solution) is extended in two different directions. Firstly, it is shown that this stability result holds not only for plane strain but also for three-dimensional deformations and secondly, for the case when the sum of principal stretches is constant, it is shown (following [17] where such a result was obtained within the context of a discussion regarding the plane strain radial inflation of hollow cylinders) that if for boundary conditions of place and zero body forces a certain equilibrium solution is stable relative to the harmonic materials, then it is also stable relative to a certain generalization of these materials.

Some New Implications of Certain Constitutive Inequalities in Plane Isotropic Elasticity

In plane isotropic elasticity a strengthened form of the Ordered–Forces inequality is shown to imply that the restriction of the strain-energy function to the class of deformation gradients which share the same average of the principal stretches is bounded from below by the strain energy corresponding to the conformal deformations in this class. For boundary conditions of place, this property (together with a certain version of the Pressure–Compression inequality) is then used (i) to show that the plane radial conformal deformations are stable with respect to all radial variations of class C1 and (ii) to obtain explicit lower bounds for the total energy associated with arbitrary plane radial deformations. For the same type of boundary conditions and together with a different version of the Pressure–Compression inequality, an analogous property in plane isotropic elasticity (established in [3] under the assumption that the material satisfies a strengthened form of the Baker–Ericksen inequality and according to which the restriction of the strain-energy function to the class of deformation gradients which share the same determinant is bounded from below by the strain energy corresponding to the conformal deformations in that class) is used (i) to show that the plane radial conformal deformations are stable with respect to all variations of class C1 and (ii) to obtain explicit lower bounds for the total energy associated with any plane deformation.

On deformations with constant modified stretches

Two plane, non-homogeneous, non-conformal deformations with constant modified stretches (the inflation of a circular cylindrical tube and the bending of a rectangular block into a sector of a circular tube) are shown to be possible in the absence of body forces in isotropic elastic solids that satisfy both the classical pressure-compression inequality and the Baker-Ericksen inequality.

A NOTE ON A GENERALIZED SHEAR DEFORMATION

In the absence of body forces a certain generalized shear deformation is shown to be possible in an unconstrained isotropic elastic solid if and only if the elastic solid satisfies a condition which renders it incapable of obeying the Baker-Ericksen (BE) inequalities. In the case when the elastic solid satisfies the weakened BE inequalities the deformation is shown to be at best neutrally stable