Compressible Hyperelastic Materials Research Articles

Uniformity of stresses inside a parabolic inhomogeneity in finite plane elastostatics

Using complex variable techniques, we establish the uniformity of the internal Piola stresses within a parabolic inhomogeneity embedded in a matrix from a particular class of compressible hyperelastic materials of harmonic type subjected to uniform remote Piola stresses. In addition, two Piola stress components inside the inhomogeneity simply coincide with their remotely prescribed counterparts in the matrix, while the two other internal Piola stress components can be determined via the solution of a quadratic equation, and are found to be independent of the single geometric parameter characterizing the inhomogeneity–matrix system. We derive a complete solution to the corresponding boundary value problem describing the mechanical behavior of the composite.

Equilibrium paths of a three-bar truss in finite elasticity with an application to graphene

This paper presents the formulation of the equilibrium problem of a three-bar truss in the nonlinear context of finite elasticity. The bars are composed of a homogeneous, isotropic, and compressible hyperelastic material. The equilibrium equations in the deformed configuration are derived under the assumption of homogeneous deformations and the stability of the solutions is assessed through the energy criterion. The general formulation is then specialized for a compressible Mooney–Rivlin material. The results for both vertical and horizontal load cases show unexpected post-critical behaviors involving several branches, stable asymmetrical configurations, bifurcation, and snap-through. The three-bar truss studied here is not only a benchmark test for the numerical analysis of nonlinear truss structures, but also a representative system for the unit cell of the graphene hexagonal lattice. Therefore, an application to graphene is performed by simulating the covalent bonds between carbon atoms as the bars of the truss, characterized by the modified Morse potential. The results provide insights on the internal mechanisms that take place when graphene undergoes large in-plane deformations, whose influence should be considered when developing molecular mechanics and continuum models in nonlinear elasticity.

Serendipity virtual element formulation for nonlinear elasticity

In this paper, the novel concept of Serendipity Virtual Elements is extended to the case of compressible hyper-elastic materials in 2D. Second order polynomials are adopted for the projection involved in the consistency part, while different stabilization techniques are proposed and critically compared with each other. Various numerical examples, involving very severe loading conditions, indicate that the proposed elements are very accurate and robust with respect to distortion. Furthermore the proposed virtual elements are inherently able to capture higher order deformation modes, such as bending states.

A plane strain analysis of the elastostatic fields near the notch-tip of a Blatz-Ko material

An asymptotic plane strain analysis is carried out in order to study the notch problem in a homogeneous isotropic compressible hyperelastic material whose behavior is governed by the Blatz-Ko constitutive law. The work is realized within the framework of the fully nonlinear fracture mechanics at large strain. Close to the notch tip, a stress-free state is assumed in the notch edges, while at infinity (sufficiently far to the notch tip) fracture modes I and II are considered. The results emphasizes many discrepancies with linear theory predictions like the emergence of a logarithmic singularity and show that solution depends on the notch initial opening angle.

Hyperelastic characterization oriented to finite element applications using genetic algorithms

In this work we highlight the capabilities of genetic algorithms to characterize the behavior shown by hyperelastic materials. We use three fitting tools, two of them included in commercial Finite Element packages and one developed using genetic algorithm methods. Four classic hyperelastic models are employed to test the different tools and to characterize a rubbery material. We show that automatic tools provide accurate data in limited ranges depending in the models. Indeed, not always can fit large strain ranges for the models. However, the dedicated genetic algorithm method performed correctly with all models and wide strain ranges, providing a robust and efficient fitting method. The results are included in a finite element model to study the differences appearing for complex stress states produced only by the selection of the material model. Compressibility effects are also studied to show the large differences in the results if only uniaxial tensile test data is used to characterize compressible hyper-elastic materials, independently of the efficiency of the fitting tools and the selection of the material model.

A coupled approach for identification of nonlinear and compressible material models for soft tissue based on different experimental setups – Exemplified and detailed for lung parenchyma

In this paper, a coupled inverse analysis is proposed to identify nonlinear compressible hyperelastic material models described by two sets of experiments. While the overall approach is applicable for different materials, here it will be presented for viable lung parenchyma. Characterizing the material properties of lung parenchyma is essential to describe and predict the mechanical behavior of the respiratory system in health and disease. During breathing and mechanical ventilation, lung parenchyma is mainly subjected to volumetric deformations along with isochoric and asymmetric deformations that occur especially in diseased heterogeneous lungs. Notwithstanding, most studies examine lung tissue in predominantly isochoric tension tests. In this paper, we investigate the volumetric material behavior as well as the isochoric deformations in two sets of experiments: namely, volume–pressure-change experiments (performed with 287 samples of 26 rats) and uniaxial tension tests (performed with 30 samples of 5 rats). Based on these sets of experiments, we propose a coupled inverse analysis, which simultaneously incorporates both measurement sets to optimize the material parameters. Accordingly, we determine a suitable material model using the experimental results of both sets of experiments in one coupled identification process. The identified strain energy function with the corresponding material parameters Ψ=356.7Pa(I1−3)+331.7Pa(I3−1.075−1)+278.2Pa(I3−13I1−3)3+5.766Pa(I313−1)6 is validated to model both sets of experiments precisely. Hence, this constitutive model describes the complex volumetric and isochoric nonlinear material behavior of lung parenchyma. This derived material model can be used for nonlinear finite element simulations of lung parenchyma and will help to quantify the stresses and strains of lung tissue during spontaneous and artificial breathing; thus, allowing new insights into lung function and biology.

High-order mortar-based contact element using NURBS for the mapping of contact curved surfaces

In this paper, we present a high-order mortar-based contact element for the solution of two and three-dimensional frictional contact problems, considering small and large deformations. The Neo-Hookean isotropic compressible hyperelastic material model is considered. The mapping of curved surfaces of elements is performed with Non-Uniform Rational B-Splines. The behavior of the element in small and large deformations is verified by comparing it with solutions available in the literature, presenting studies of accuracy and processing time exhibited by the contact elements considering the h- and p-refinements. The comparative results show that the high-order interpolation is a strategy which has a better performance for the contact problems analysed, while improving solution accuracy of the contact stresses and forces with a lower processing time.

Strong Ellipticity Conditions for Orthotropic Bodies in Finite Plane Strain

Necessary and sufficient conditions for strong ellipticity of the equilibrium equations governing finite plane deformations are established for the class of orthotropic and compressible hyperelastic materials. The conditions are then specialized to the case of an orthotropic material having the symmetry axes parallel to the principal directions of deformation. This case is relevant in experimental protocols employed in the characterization of constitutive relations of rubber-like materials and biological tissues. The classical conditions for infinitesimal linear elasticity and isotropic finite elasticity are recovered as particular cases.

A harmonic rigid inclusion loaded by a couple in finite plane elasticity

We consider a rigid inclusion embedded in a matrix from a particular class of compressible hyperelastic materials (so-called harmonic materials) subjected to uniform remote stresses. The inclusion is loaded by a couple, and the inclusion–matrix system undergoes finite plane deformations. We examine whether it is possible to design the shape of the inclusion to achieve the well-known ‘harmonic field condition’ in which the sum of the normal stresses in the uncut matrix remains undisturbed after the introduction of the inclusion. Indeed, we find that the only such rigid inclusion is elliptical. Moreover, we show that when the remote loading is symmetric, the harmonic field condition is satisfied only when two further conditions on prescribed geometric and material parameters are met: One condition requires a relatively simple linear relationship between the moment of the couple and the remote shear stress; the second condition involves a rather complex nonlinear relationship between the two remote normal stresses and the moment of the couple. Once these conditions are satisfied, the interfacial normal and tangential stresses as well as the hoop stress are uniformly distributed along the boundary of the rigid inclusion. Furthermore, the interfacial normal and hoop stresses are dependent only on the remote normal stresses and the aspect ratio of the elliptical inclusion, whereas the interfacial tangential stress depends only on the moment of the couple and the area of the inclusion.

Surface Green’s functions in finite plane elastostatics of harmonic materials

The closed-form representations of surface Green’s functions corresponding to the action of a concentrated force applied at the boundary of a region occupied by a particular class of compressible hyperelastic materials of harmonic type, has been derived. In our analysis, we consider both a bounded region in the form of a circular disk and an unbounded region with either an elliptical hole or a parabolic boundary.

A concentrated couple inside or outside a circular inclusion in finite plane elastostatics

We consider a circular elastic inclusion embedded in an infinite matrix each from a particular class of compressible hyperelastic materials of harmonic type. A concentrated couple is applied either inside the circular inclusion or in the matrix. Closed-form solutions of the corresponding boundary value problems are obtained using complex variable methods, in particular the principle of analytic continuation. Our analysis reveals several interesting conclusions including: the sum σ11+σ22 of the normal stresses inside the inclusion remains constant when the concentrated couple is located in the surrounding matrix; the sum of the normal stresses is zero everywhere in the matrix when the concentrated couple is located inside the inclusion.

A study of the critical strain of hyperelastic materials: A new kinematic frame and the leading order term

In this paper, an investigation is made on the problem of critical strain of hyperelastic materials. A new kinematic frame, which takes into account the characteristics of the material, is firstly proposed to consider a hyperelastic rectangular layer under compression. The simplified model equations are derived with the aid of virtual principal and asymptotic expansion method. Through linear bifurcation analysis, the critical strain is determined, which is in good agreement with existing results. Comparisons between the classical Eular Beam theory, first-order shear deformation theory and the present frame are also made, and it shows that the present frame can provide much better results than the other two frames. Moreover, under the same framework, several other typical compressible hyperelastic materials are examined. One interesting finding is that their critical strain share the same leading order term which is independent of the form of strain energy functions.

An approach for verification of finite-element analysis in nonlinear elasticity under large strains

An approach to verification of finite-element calculations of stress-strain state of nonlinear elastic bodies under large deformations is suggested. The problems that may be reduced to one-dimensional ones using a semi-inverse method are taken as test problems. An example of such a test problem is the Lame problem for a cylinder. Generally, this problem for compressible hyperelastic materials has no exact analytical solution, but it can be reduced to a boundary value problem for an ordinary second-order nonlinear differential equation, and in some cases - to the Cauchy problem. A numerical solution of this problem can be used as a test one for finite element calculations carried out in three-dimensional statement. Some results of such verification (finite element calculations were performed using the Fidesys CAE-system) are presented.

Stochastic modeling of the Ogden class of stored energy functions for hyperelastic materials: the compressible case

This paper is devoted to the modeling of compressible hyperelastic materials whose response functions exhibit uncertainties at some scale of interest. The construction of parametric probabilistic representations for the Ogden class of stored energy functions is specifically considered and formulated within the framework of Information Theory. The overall methodology relies on the principle of maximum entropy, which is invoked under constraints arising from existence theorems and consistency with linearized elasticity. As for the incompressible case discussed elsewhere, the derivation essentially involves the conditioning of some variables on the stochastic bulk and shear moduli, which are shown to be statistically dependent random variables in the present case. The explicit construction of the probability measures is first addressed in the most general setting. Subsequently, particular results for classical Neo‐Hookean and Mooney‐Rivlin materials are provided. Salient features of the probabilistic representations are finally highlighted through forward Monte‐Carlo simulations. In particular, it is seen that the models allow for the reproduction of typical experimental trends, such as a variance increase at large stretches. A stochastic multiscale analysis, where uncertainties on the constitutive law of the matrix phase are taken into account through the proposed approach, is also presented.

Closed-form solution of the Ogden–Hill’s compressible hyperelastic model for ramp loading

This article deals with the visco-hyperelastic modelling approach for compressible polymer foam materials. Polymer foams can exhibit large elastic strains and displacements in case of volumetric compression. In addition, they often show significant rate-dependent properties. This material behaviour can be accurately modelled using the visco-hyperelastic approach, in which the large strain viscoelastic description is combined with the rate-independent hyperelastic material model. In case of polymer foams, the most widely used compressible hyperelastic material model, the so-called Ogden–Hill’s model, was applied, which is implemented in the commercial finite element (FE) software Abaqus. The visco-hyperelastic model is defined in hereditary integral form, therefore, obtaining a closed-form solution for the stress is not a trivial task. However, the parameter-fitting procedure could be much faster and accurate if closed-form solution exists. In this contribution, exact stress solutions are derived in case of uniaxial, biaxial and volumetric compression loading cases using ramp-loading history. The analytical stress solutions are compared with the stress results in Abaqus using FE analysis. In order to highlight the benefits of the analytical closed-form solution during the parameter-fitting process experimental work has been carried out on a particular open-cell memory foam material. The results of the material identification process shows significant accuracy improvement in the fitting procedure by applying the derived analytical solutions compared to the so-called separated approach applied in the engineering practice.

Novel strategy for the hyperelastic parameter fitting procedure of polymer foam materials

The most widely used approach to model the large strain elastic response of polymer foams in a finite element (FE) simulation is the use of the Ogden–Hill compressible hyperelastic material model. This model is implemented and termed as “hyperfoam” material model in the commercial FE software ABAQUS. The hyperfoam model is able to characterize the large compressibility (in volumetric sense) of the foam material. In order to find the material parameters of the model for a particular foam specimen, we need to fit the simulated responses to the available experimental data. This task is easier for incompressible hyperelastic materials because we can use the incompressibility constraint to eliminate the transverse stretch from the stress solutions. However, this simplification cannot be used for the hyperfoam model, therefore, in the stress-strain relations, the transverse stretch is included, which makes the parameter fitting procedure more complicated. In this paper, a novel strategy is proposed for the parameter fitting task. The performance of the new algorithm is demonstrated by presenting fitted material responses for a particular polymer foam material. The major advantage of the new strategy is that it can be used with any third-party optimization solver and there is no need to write our own code.

On constitutive models of finite elasticity with possible zero apparent Poisson’s ratio

The idea in this paper is to build a class of constitutive equations for highly compressible isotropic materials that, among others, are capable to describe a zero apparent Poisson’s ratio in the whole finite strain range, not only for moderate straining. This remarkable property is, for instance, observed in many soft materials with micro-structures such as sponges and polymeric foams with high porosities. It would then be suitable to describe their behavior within a macroscopic modeling framework. More specifically, herein by means of elementary considerations, we deduce adequate forms of strain-energy functions that are a priori decomposed into purely volumetric and volume-preserving parts. A class of compressible hyperelastic materials of the general Odgen type is obtained. It can consequently be specialized, for instance, to neo-Hookean, Mooney–Rivlin, and Varga’s model types as well. Furthermore, for the elastic parameters, a connection with the limiting case of linear elasticity is made whenever possible, in particular with the classical Poisson’s ratio, and with the bulk to shear moduli ratio.

Neutrality of an elliptical inhomogeneity in finite plane elastostatics of harmonic materials

We examine the neutrality of an elliptical inhomogeneity embedded in a particular class of compressible hyperelastic materials of harmonic type when a uniform Piola stress field is prescribed in the surrounding matrix. The present method is based on a spring-type imperfect interface model and on the proper choice of imperfect interface function, which realizes the same degree of imperfection in both normal and tangential directions. The analysis indicates that, in general, the neutral shape and the single imperfect interface function are both dependent on the magnitude of the prescribed stress field.

On a Consistent Dynamic Finite-Strain Plate Theory and Its Linearization

This paper develops a dynamic finite-strain plate theory consistent with three-dimensional Hamilton’s principle under general loadings with a fourth-order error. Starting from the three-dimensional field equations for a compressible hyperelastic material and by a series expansion about the bottom surface, we deduce a vector dynamic plate equation with three unknowns, which exhibits the local momentum-balance structure. Associated weak formulations are considered, in connection with various boundary conditions. Then, by linearization, we provide a novel linear plate theory for orthotropic materials, which takes into account both stretching and bending effects. For isotropic materials, it is further modified to a refined linear plate theory, leading to higher-order results under certain circumstances. To verify the present plate theory, an exhaustive study of the free vibration and static bending problems is carried out. Comparing with the available exact three-dimensional solutions for these problems, it is shown that the plate theory indeed provides correct asymptotic results for displacements and stresses distributions. The advantages of this linear plate theory include (i) its simplicity (as simple as the Kirchhoff-Love theory), (ii) high accuracy for frequencies and deflections, as well as capability of capturing the distributions of all concerned quantities, and (iii) applicability for general loadings.

Swelling and instability of a gel annulus

We study the swelling of a gel annulus attached to a rigid core when it is immersed in a solvent. For equilibrium states, the free-energy function of the gel can be converted into a strain energy function, and as a result the gel can be treated as a compressible hyperelastic material. Asymptotic methods are used to study the inhomogeneous swelling in order to obtain the leading-order solution. Some analytical insights are then deduced. Because of the compressive hoop stress in this state, at some stage instability can occur, leading to wrinkles in the gel. An incremental deformation theory in nonlinear elasticity is used to conduct a linear bifurcation analysis for understanding such instability. More specifically, the critical loading for the onset of a wrinkled state is obtained. Detailed discussions on the behaviors of various physical quantities in this critical state are given. It is found that the critical mode number, while insensitive to the material parameters, is greatly influenced by the ratio of the inner and outer radii of the gel. Also, an interesting finding is that the critical swelling ratio is an increasing function of this geometrical parameter, which implies that a thin annulus is more likely to be unstable than a thick one.