Limit point instability in pressurization of anisotropic finitely extensible hyperelastic thin-walled tube

Mechanical responses of materials undergoing large elastic deformations can exhibit a loss of stability in several ways. Such a situation can occur when a thin-walled cylinder is inflated by an internal pressure. The loss of stability is manifested by a non-monotonic relationship between the inflating pressure and internal volume of the tube. This is often called limit point instability. The results, known from the literature, show that isotropic hyperelastic materials with limiting chain extensibility property always exhibit a stable response if the extensibility parameter of the Gent model satisfies J_m<18.2. Our study investigates the same phenomenon but for tubes with anisotropic form of the Gent model (finite extensibility of fibers). Anisotropy, used in our study, increases the number of material parameters the consequence of which is to increase degree of freedom of the problem. It will be shown that, in stark contrast to isotropic material, the unstable response is predicted not only for large values of J_m but also for J_m≈1 and smaller, and that the existence of limit point instability significantly depends on the orientation of preferred directions and on the ratio of linear parameters in the strain energy density function (this ratio can be interpreted as the ratio of weights by which fibers and matrix contribute to the strain energy density). Especially tubes reinforced with fibers oriented closely to the longitudinal direction are susceptible to a loss of monotony during pressurization.     

Elastic instabilities for strain-stiffening rubber-like spherical and cylindrical thin shells under inflation

This paper is concerned with investigation of the effects of strain-stiffening on the classical limit point instability that is well-known to occur in the inflation of internally pressurized rubber-like spherical thin shells (balloons) and circular cylindrical thin tubes composed of incompressible isotropic non-linearly elastic materials. For a variety of specific strain-energy densities that give rise to strain-stiffening in the stress-stretch response, the inflation pressure versus stretch relations are given explicitly and the non-monotonic character of the inflation curves is examined. While such results are known for constitutive models that exhibit a gradual stiffening (e.g. exponential and power-law models), our primary focus is on materials that undergo severe strain-stiffening in the stress-stretch response. In particular, we consider two phenomenological constitutive models that reflect limiting chain extensibility at the molecular level. It is shown that for materials with sufficiently low extensibility no limit point instability occurs and so stable inflation is then predicted for such materials. Potential applications of the results to the biomechanics of soft tissues are indicated.     

Finite extension and torsion of fiber-reinforced non-linearly elastic circular cylinders

In the context of the theory of non-linear elasticity for rubber-like materials, the problem of finite extension and torsion of a circular bar or tube has been widely investigated. More recently, this problem has attracted considerable attention in studies on the biomechanics of soft tissues and has been applied, for example, to examine the mechanical behavior of passive papillary muscles of the heart. A recent study in non-linear elasticity was concerned specifically with the effects of strain-stiffening on the response of solid circular cylinders in the combined deformation of torsion superimposed on axial extension. The cylinders are composed of incompressible isotropic non-linearly elastic materials that undergo severe strain-stiffening in the stress–stretch response. For two specific material models that reflect limiting chain extensibility at the molecular level, it was shown that, in the absence of an additional axial force, a transition value γ=γ_t of the axial stretch exists such that for γ<γ_t, the stretched cylinder tends to elongate on twisting whereas for γ>γ_t, the stretched cylinder tends to shorten on twisting. These results are in sharp contrast with those for classical models for rubber such as the Mooney–Rivlin (and neo-Hookean) models that predict that the stretched circular cylinder always tends to further elongate on twisting. Here we investigate similar issues for fiber-reinforced transversely isotropic circular cylinders. We consider a class of incompressible anisotropic materials with strain-energy densities that are of logarithmic form in the anisotropic invariant. These models reflect limited fiber extensibility and in the biomechanics context model the stretch induced strain-stiffening of collagen fibers on loading. They have been shown to model the mechanical behavior of fiber-reinforced rubber and many fibrous soft biological tissues. The consideration of anisotropy leads to a more elaborate mechanical response than was found for isotropic strain-stiffening materials. The results obtained here have important implications for extension–torsion tests for fiber-reinforced materials, for example in the development of accurate extension–torsion test protocols for determination of material properties of soft tissues.     

Constitutive modeling and the trousers test for fracture of rubber-like materials

The classical constitutive modeling of incompressible hyperelastic materials such as vulcanized rubber involves strain-energy densities that depend on the first two invariants of the strain tensor. The most well-known of these is the Mooney-Rivlin model and its specialization to the neo-Hookean form. While each of these models accurately predicts the mechanical behavior of rubber at moderate stretches, they fail to reflect the severe strain-stiffening and effects of limiting chain extensibility observed in experiments at large stretch. In recent years, several constitutive models that capture the effects of limiting chain extensibility have been proposed. Here we confine attention to two such phenomenological models. The first, proposed by Gent in 1996, depends only on the first invariant and involves just two material parameters. Its mathematical simplicity has facilitated the analytic solution of a wide variety of basic boundary-value problems. A modification of this model that reflects dependence on the second invariant has been proposed recently by Horgan and Saccomandi. Here we discuss the stress response of the Gent and HS models for some homogeneous deformations and apply the results to the fracture of rubber-like materials. Attention is focused on a particular fracture test, namely the trousers test where two legs of a cut specimen are pulled horizontally apart. It is shown that the cut position plays a key role in the fracture analysis, and that the effect of the cut position depends crucially on the constitutive model employed. For stiff rubber-like or biological materials, it is shown that the influence of the cut position is diminished. In fact, for linearly elastic materials, the critical driving force for fracture is independent of the cut position. It is also shown that the limiting chain extensibility models predict finite fracture toughness as the cut position approaches the edge of the specimen whereas classical hyperelastic models predict unbounded toughness in this limit. The results are relevant to the structural integrity of rubber components such as vibration isolators, vehicle tires, earthquake bearings, seals and flexible joints.     

A Molecular-Statistical Basis for the Gent Constitutive Model of Rubber Elasticity

Molecular constitutive models for rubber based on non-Gaussian statistics generally involve the inverse Langevin function. Such models are widely used since they successfully capture the typical strain-hardening at large strains. Limiting chain extensibility constitutive models have also been developed on using phenomenological continuum mechanics approaches. One such model, the Gent model for incompressible isotropic hyperelastic materials, is particularly simple. The strain-energy density in the Gent model depends only on the first invariant I ₁ of the Cauchy–Green strain tensor, is a simple logarithmic function of I ₁ and involves just two material parameters, the shear modulus μ and a parameter J _m which measures a limiting value for I ₁−3 reflecting limiting chain extensibility. In this note, we show that the Gent phenomenological model is a very accurate approximation to a molecular based stretch averaged full-network model involving the inverse Langevin function. It is shown that the Gent model is closely related to that obtained by using a Padè approximant for this function. The constants μ and J _m in the Gent model are given in terms of microscopic properties. Since the Gent model is remarkably simple, and since analytic closed-form solutions to several benchmark boundary-value problems have been obtained recently on using this model, it is thus an attractive alternative to the comparatively complicated molecular models for incompressible rubber involving the inverse Langevin function. This revised version was published online in June 2006 with corrections to the Cover Date.     

Plane strain bending of strain-stiffening rubber-like rectangular beams

This paper is concerned with investigation of the effects of strain-stiffening for the classical problem of plane strain bending by an end moment of a rectangular beam composed of an incompressible isotropic nonlinearly elastic material. For a variety of specific strain-energy densities that give rise to strain-stiffening in the stress–stretch response, the stresses and resultant moments are obtained explicitly. While such results are well known for classical constitutive models such as the Mooney-Rivlin and neo-Hookean models, our primary focus is on materials that undergo severe strain-stiffening in the stress–stretch response. In particular, we consider in detail two phenomenological constitutive models that reflect limiting chain extensibility at the molecular level and involve constraints on the deformation. The amount of bending that beams composed of such materials can sustain is limited by the constraint. Potential applications of the results to the biomechanics of soft tissues are indicated.     

基于横观各向同性超弹性理论的短纤维增强橡胶本构模型的建立与应用Establishment and application of short fiber reinforced rubber constitutive model based on transversely isotropic hyperelastic theory

提出了一种能够表征短纤维增强橡胶的横观各向同性超弹性本构模型,并结合试验体系,对其在数值分析中的应用方法和效果进行了研究。基于连续介质力学理论,建立了横观各向同性材料的应变能函数,推导得到不同变形形式下的应力应变关系,给出材料参数辨识试验方法,并成功应用于某短纤维增强橡胶测试中,得到表征其超弹性特性的相关材料参数。利用有限元软件ANSYS对不同纤维排布方向的单轴拉伸和平行纤维方向的平面拉伸进行仿真计算,并对比相应试验数据,以验证材料参数的可靠性。最后基于已验证的本构模型,建立了某铣槽装备减振环仿真模型,并对其进行了校核计算。研究结果表明,本文提出的本构模型能够有效表征短纤维增强橡胶的静态力学特性并且方便嵌入现有的有限元软件中,具有材料参数少、测试简便和结果准确等特点,工程实用性强。     

Constitutive Models for Compressible Nonlinearly Elastic Materials with Limiting Chain Extensibility

Constitutive models are proposed for compressible isotropic hyperelastic materials that reflect limiting chain extensibility. These are generalizations of the model proposed by Gent for incompressible materials. The goal is to understand the effects of limiting chain extensibility when the compressibility of polymeric materials is taken into account. The basic homogeneous deformation of simple tension is considered and simple closed-form relations for the deformation characteristics are obtained for slightly compressible materials. An explicit first-order approximation is obtained for the lateral contraction and for the Poisson function in terms of the axial extension which is shown to be valid for each of two specific compressible versions of the Gent model. One of the main results obtained is that the effect of limiting chain extensibility is to stiffen the material relative to the neo-Hookean compressible case. Mathematics Subject Classifications (2000) 74B20, 74G55.     

Instabilities and loss of ellipticity in fiber-reinforced compressible non-linearly elastic solids under plane deformation

In a recent paper we examined the loss of ellipticity and its interpretation in terms of fiber kinking and other instability phenomena in respect of a fiber-reinforced incompressible elastic material. Here we provide a corresponding analysis for fiber-reinforced compressible elastic materials. The analysis concerns a material model which consists of an isotropic base material augmented by a reinforcement dependent on the fiber direction. The assessment of loss of ellipticity can be cast in terms of the eigenvalues of the acoustic tensors associated with the isotropic and anisotropic parts of the strain-energy function. For the anisotropic part, two different reinforcing models are examined and it is shown that, depending on the choice of model and whether the fiber is under compression or extension, loss of ellipticity may be associated with, in particular, a weak surface of discontinuity normal to or parallel to the deformed fiber direction or at an intermediate angle. Under compression the associated failure interpretations include fiber kinking and fiber splitting, while under extension fiber de-bonding and matrix failure are included.     

Inhomogeneous shearing of strain-stiffening rubber-like hollow circular cylinders

The purpose of this paper is to investigate the effects of strain-stiffening for the classical problems of axial and azimuthal shearing of a hollow circular cylinder composed of an incompressible isotropic non-linearly elastic material. For some specific strain-energy densities that give rise to strain-stiffening in the stress–stretch response, the stresses and resultant axial forces are obtained in explicit closed form. While such results are well known for classical constitutive models such as the Mooney–Rivlin and neo-Hookean models, our main focus is on materials that undergo severe strain-stiffening in the stress–stretch response. In particular, we consider in detail two phenomenological constitutive models that reflect limiting chain extensibility at the molecular level and involve constraints on the deformation. The amount of shearing that tubes composed of such materials can sustain is limited by the constraint. Numerical results are also obtained for an exponential strain-energy that exhibits a less abrupt strain-stiffening effect. Potential applications of the results to the biomechanics of soft tissues are indicated.     

On the modeling of fiber dispersion in fiber-reinforced elastic materials

When an isotropic material is subject to a uniaxial tension, the principal strain transverse to the direction of applied load is always negative. However, in fiber reinforced materials the transverse principal strain can change its sign as the load increases, passing through the zero-points, known as perversions. We investigate how the number of perversions in a material reinforced by two symmetrically aligned families of distributed fibers depends both on the degree of fiber dispersion and the model used for fiber dispersion. Angular integration and three variants of the generalized structure tensor approach are considered and discussed. The study of perversions clearly demonstrates the qualitative difference between these approaches in the case of high dispersion of fibers. The results suggest that this difference is primarily due to the way compressive fibers are modeled.     

短纤维增强三元乙丙橡胶横观各向同性黏-超弹性本构模型

短纤维增强三元乙丙橡胶包覆薄膜,是一种应用于固体火箭发动机缠绕包覆装药的新型复合材料.为了描述其在工作过程中受振动、冲击等载荷作用时的力学行为,基于黏弹性理论和纤维增强连续介质力学理论,提出了一种考虑应变率强化效应的横观各向同性黏-超弹本构模型.模型中应变能函数被分解为超弹性应变能和黏性应变能,其中超弹性应变能包括表征各向同性的橡胶基体应变能和表征各向异性的纤维拉伸应变能,黏性应变能采用表征橡胶和纤维黏性响应的宏观唯象模型.选取表征各应变能的函数形式,经过数学变换、替代、叠加,求解确定最终的应力应变形式,明确模型参数获取的具体步骤,将预测结果与实验结果对比分析,准确性较高.研究表明:该模型能有效预测材料在低应变率下纤维方向为0?～45?的非线性率相关力学特性;模型形式易于实现有限元开发,对固体火箭发动机装药结构完整性分析具有参考价值.     

The remarkable Gent constitutive model for hyperelastic materials

In 1996, Alan Gent published a short paper that proposed the use of a very simple two parameter phenomenological constitutive model for hyperelastic isotropic incompressible materials. The model is empirical but has the advantages of mathematical simplicity, reflects the severe strain-stiffening at large strains observed experimentally, reduces to the classic neo-Hookean model for small strains and involves just two material parameters namely the shear modulus for infinitesimal deformations and a parameter that measures a maximum allowable value of strain. The model reflects the limiting chain extensibility characteristic of non-Gaussian molecular models for rubber. Here we review some of the numerous developments, extensions and widespread applications that have resulted from that groundbreaking paper not only in rubber elasticity but also in the area of biomechanics of soft biomaterials. The Gent model is remarkably robust: its mathematical simplicity combined with physical basis has ensured that it has reached status as a fundamental canonical phenomenological constitutive model for hyperelastic materials.     

Propagation of electroacoustic waves in the transversely isotropic piezoelectric medium reinforced by randomly distributed cylindrical inhomogeneities

The propagation of electroacoustic waves in a piezoelectric medium containing a statistical ensemble of cylindrical fibers is considered. Both the matrix and the fibers consist of piezoelectric transversely isotropic material with symmetry axis parallel to the fiber axes. Special emphasis is given on the propagation of an electroacoustic axial shear wave polarized parallel to the axis of symmetry propagating in the direction normal to the fiber axis.The scattering problem of one isolated continuous fiber (“one-particle scattering problem”) is considered. By means of a Green’s function approach a system of coupled integral equations for the electroelastic field in the medium containing a single inhomogeneity (fiber) is solved in closed form in the long-wave approximation. The total scattering cross-section of this problem is obtained in closed form and is in accordance with the electroacoustic analogue of the optical theorem.The solution of the one-particle scattering problem is used to solve the homogenization problem for a random set of fibers by means of the self-consistent scheme of effective field method. Closed form expressions for the dynamic characteristics such as total cross-section, effective wave velocity and attenuation factor are obtained in the long-wave approximation.     

Finite thermoelasticity with limiting chain extensibility

Rubber-like materials and soft tissues exhibit a significant stiffening or hardening in their stress-strain curves at large strains. Considerable progress has been made recently in the phenomenological modeling of this effect within the context of isotropic hyperelasticity. In particular, constitutive models reflecting limiting chain extensibility at the molecular level have been used to accurately capture strain-hardening. Here we generalize such models to isotropic thermoelasticity. We also show that specific non-polynomial strain-energies for both hyperelastic and thermoelastic materials can be obtained on using a modification of a systematic scheme of Rivlin and Signorini. The Rivlin-Signorini method was based on approximation of the strain-energy density function by polynomials whereas here we use the more general class of rational functions to approximate the material response functions. We then propose a simple generalization to thermoelasticity of a constitutive model for incompressible hyperelastic materials reflecting limiting chain extensibility due to Gent (Rubber Chem. Technol. 69 (1996) 59-61). For this new thermoelastic constitutive model we investigate the inhomogeneous deformation problem of axial shear of a circular cylindrical tube.     

Torsional Swelling of a Hyperelastic Tube with Helically Wound Reinforcement

This paper examines the combination of radial deformation with torsion for a circular cylindrical tube composed of a transversely isotropic hyperelastic material subject to finite deformation swelling. The stored energy function involves separate matrix and fiber contributions such that the fiber contribution is minimized when the fiber direction is at a natural length. This natural length is not affected by the swelling. Hence swelling preferentially expands directions that are orthogonal to the fiber. The swelling itself is described via a swelling field that prescribes the local free volume at each location in the body. Such a treatment is a relatively simple generalization of the conventional incompressible theory. The direction of transverse isotropy associated with the fiber reinforcement is described by a helical winding about the tube axis. The swelling induced preferential expansion orthogonal to this direction induces the torsional aspect of the deformation. For a specific class of strain energy functions we find that the twist increases with swelling and approaches a limiting asymptotic value as the swelling becomes large. The fibers reorient such that fibers at the inner portion of the tube assume a more circumferential orientation whereas, at least for small and moderate swelling, the fibers in the outer portion of the tube assume a more axial orientation. For large swelling the fibers in the outer portion of the tube reorient beyond the axial orientation, and so are described by helices with orientation in the opposite sense to that in the reference configuration.      

Homogenization estimates for fiber-reinforced elastomers with periodic microstructures

This work presents a homogenization-based constitutive model for the mechanical behavior of elastomers reinforced with aligned cylindrical fibers subjected to finite deformations. The proposed model is derived by making use of the second-order homogenization method [Lopez-Pamies, O., Ponte Castañeda, P., 2006a. On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: I—theory. J. Mech. Phys. Solids 54, 807–830], which is based on suitably designed variational principles utilizing the idea of a “linear comparison composite.” Specific results are generated for the case when the matrix and fiber materials are characterized by generalized Neo-Hookean solids, and the distribution of fibers is periodic. In particular, model predictions are provided and analyzed for fiber-reinforced elastomers with Gent phases and square and hexagonal fiber distributions, subjected to a wide variety of three-dimensional loading conditions. It is found that for compressive loadings in the fiber direction, the derived constitutive model may lose strong ellipticity, indicating the possible development of macroscopic instabilities that may lead to kink band formation. The onset of shear band-type instabilities is also detected for certain in-plane modes of deformation. Furthermore, the subtle influence of the distribution, volume fraction, and stiffness of the fibers on the effective behavior and onset of macroscopic instabilities in these materials is investigated thoroughly.     

Micromechanical design of hierarchical composites using global load sharing theory

Hierarchical composites, embodied by natural materials ranging from bone to bamboo, may offer combinations of material properties inaccessible to conventional composites. Using global load sharing (GLS) theory, a well-established micromechanics model for composites, we develop accurate numerical and analytical predictions for the strength and toughness of hierarchical composites with arbitrary fiber geometries, fiber strengths, interface properties, and number of hierarchical levels, N. The model demonstrates that two key material properties at each hierarchical level—a characteristic strength and a characteristic fiber length—control the scalings of composite properties. One crucial finding is that short- and long-fiber composites behave radically differently. Long-fiber composites are significantly stronger than short-fiber composites, by a factor of 2^N or more; they are also significantly tougher because their fiber breaks are bridged by smaller-scale fibers that dissipate additional energy. Indeed, an “infinite” fiber length appears to be optimal in hierarchical composites. However, at the highest level of the composite, long fibers localize on planes of pre-existing damage, and thus short fibers must be employed instead to achieve notch sensitivity and damage tolerance. We conclude by providing simple guidelines for microstructural design of hierarchical composites, including the selection of N, the fiber lengths, the ratio of length scales at successive hierarchical levels, the fiber volume fractions, and the desired properties of the smallest-scale reinforcement. Our model enables superior hierarchical composites to be designed in a rational way, without resorting either to numerical simulation or trial-and-error-based experimentation.