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.     

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.     

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.     

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.     

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.     

Torsion of Incompressible Fiber-Reinforced Nonlinearly Elastic Circular Cylinders

Torsion of solid cylinders in the context of nonlinear elasticity theory has been widely investigated with application to the behavior of rubber-like materials. More recently, this problem has attracted attention in investigations of 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 nonlinear elasticity was concerned specifically with the effects of strain-stiffening on the torsional response of solid circular cylinders. The cylinders are composed of incompressible isotropic nonlinearly elastic materials that undergo severe strain-stiffening in the stress-stretch response. 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 stretch induced strain-stiffening of collagen fibers on loading and have been shown to model the mechanical behavior of 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 classic Poynting effect found for rubber-like materials where torsion induces elongation of the cylinder is shown to be significantly different for the transversely-isotropic materials considered here. For sufficiently large anisotropy and under certain conditions on the amount of twist, a reverse-Poynting effect is demonstrated where the cylinder tends to shorten on twisting The results obtained here have important implications for the development of accurate torsion test protocols for determination of material properties of soft tissues.     

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.     

动载下韧性撕裂在三种管线钢材中传播阻力的研究

阐述了用实验确定在常温及静载和动载条件下韧性撕裂在三种管线钢材中传播时的断裂比能值。实验中采用了销钉加载双面开槽的双悬脊梁(DCB)试件。应用能量平衡法对实验结果进行了分析。考察了加载速率和试件厚度对撕裂韧性的影响。结果表明,管线钢材对韧性撕裂传播的阻力在动载条件下增大,对较薄的试件及应变率敏感性较高的材料其增大更为明显。对于纯剪切断裂的传播来说,撕裂韧性一般随试件厚度的增大而增加。     

基于单轴数据决定橡胶类材料多轴刚性化有界超弹性势

指出表征橡胶类材料应变刚化效应的现有超弹性模型涉及应变能无穷发散困难,提出新方法解决该困难。基于对数应变不变量的多轴扩张和多轴匹配步骤,建议直接构造橡胶类材料大变形弹性势的显式直接方法。该方法从单轴应力-应变关系直接得到多轴弹性势,所得结果避免了现有各方法决定待定参数组的复杂数值计算,能够准确描述应变刚性化效应,且给出有界弹性应变能,从而避免了前述发散困难。数值结果表明,从单轴数据所得到的弹性势可同时很好的拟合平面应变拉伸(剪切)数据以及等双轴拉伸数据。     

An alternative form of energy density demonstrating the severe strain-stiffening in thin spherical and cylindrical shells

The present article investigates an elastic instability phenomenon for internally pressurized spherical thin balloons and thin cylindrical tubes composed of incompressible hyperelastic material. A mathematical model is formulated by proposing a new strain energy density function. In the family of limited elastic materials, many material models exhibit strain-stiffening. However, they fail to predict severe strain-stiffening in a moderate range of deformations in the stress-strain relations. The proposed energy function contains three material parameters and shows substantially improved stain stiffening properties than the limited elastic material models. The model is further applied to explore the elastic instability phenomenon in spherical and cylindrical shells. The findings are compared with other existing models and validated with experimental results. The model shows better agreement with experimental results and exhibits a substantial strain-stiffening effect than the current models.     

On determination of mode II fracture toughness using semi-circular bend specimen

The cracked semi-circular specimen subjected to three-point bending has been recognized as an appropriate test specimen for conducting mode I, mode II and mixed mode I/II fracture tests in brittle materials. The manufacturing and pre-cracking of the specimen are simple. No complicated loading fixture is also required for a fracture test. However, almost all of the theoretical criteria available for mixed mode brittle fracture fail to predict the experimentally determined mode II fracture toughness obtained from the semi-circular bend (SCB) specimen. In this paper, a modified maximum tangential stress criterion is used for calculating mode II fracture toughness K_IIc in terms of mode I fracture toughness K_Ic. The modified criterion is used for predicting the reported values of mode II fracture toughness for two brittle materials: a rock material (Johnstone) and a brittle polymer (PMMA). It is shown that the modified criterion provides very good predictions for experimental results.     

On strain and damage interactions during tearing: 3D in situ measurements and simulations for a ductile alloy (AA2139-T3)

Strain and damage interactions during tearing of a ductile Al-alloy with high work hardening are assessed in situ and in 3D combining two recently developed experimental techniques, namely, synchrotron laminography and digital volume correlation. Digital volume correlation consists of registering 3D laminography images. Via simultaneous assessments of 3D strain and damage at a distance of 1-mm ahead of a notch root of a thin Compact Tension-like specimen, it is found that parallel crossing slant strained bands are active from the beginning of loading in a region where the crack will be slanted. These bands have an intermittent activity but are stable in space. Even at late stages of deformation strained bands can stop their activity highlighting the importance of plasticity on the failure process rather than damage softening. One void is followed over the loading history and seen to grow and orient along the slant strained band at very late stages of deformation. Void growth and strain are quantified. Gurson–Tvergaard–Needleman-type simulations using damage nucleation for shear, which is based on the Lode parameter, are performed and capture slant fracture but not the initial strain fields and in particular the experimentally found slant bands. The band formation and strain distribution inside and outside the bands are discussed further using plane strain simulations accounting for plastic material heterogeneity in soft zones.     

Direct determination of multi-axial elastic potentials for incompressible elastomeric solids: an accurate,explicit approach based on rational interpolation

With a novel approach based on certain logarithmic invariants, we demonstrate that a multi-axial elastic potential for incompressible, isotropic rubber-like materials may be obtained directly from two one-dimensional elastic potentials for uniaxial case and simple shear case, in a sense of exactly matching finite strain data for four benchmark tests, including uniaxial extension, simple shear, bi-axial extension, and plane-strain extension. As such, determination of multi-axial elastic potentials may be reduced to that of two one-dimensional elastic potentials. We further demonstrate that the latter two may be obtained by means of rational interpolating procedures for uniaxial data and shear data displaying strain-stiffening effects. Numerical examples are presented in fitting Treloar’s data and other data.     

Mathematical modeling of mechanical properties of metals and alloys at large strains

We discuss problems in mathematical modeling of the mechanical behavior of metals and alloys at large strains. Attention is mainly paid to the analysis of the stress-strain state of specimens and structural fragments made of highly plastic materials with the effect of stability loss under tensile stresses taken into account. We discuss the methods for determining the true property diagram at strains exceeding the ultimate uniform strain. We process experimental data and determine the true property diagrams for AMg6, AMg6M, and 1201 aluminum alloys and BrKh08 alloy. To calculate the load-carrying capacity of structural members, one often uses the conventional ultimate strength σ _b accepted in regulations as a material characteristic. But it follows from the method for experimentally determining this characteristic that it depends on the properties of the specimen viewed as a structure. As a result, a formal use of fracture criteria recommended in regulations leads to a discrepancy between design and experimental values of fracture loads. Nowadays, the finite element method is widely used in practical strength analysis. This method permits one to study the elastoplastic strained state of geometrically complicated structures in detail, take into account physical nonlinearity at large strains, determine damage boundaries, and improve experimental methodology. The wide capabilities of this method allow one to use test results more completely.     

An experimental method for determining dynamic fracture toughness

The boundary and loading conditions in many dynamic fracture test methods are frequently not well defined and, therefore, introduce a degree of uncertainty in the modeling of the experiment to extract the dynamic fracture resistance for a rapidly propagating crack. A new dynamic fracture test method is presented that overcomes many of these difficulties. In this test, a precracked, three-point bend specimen is loaded by a transmitter bar that is impacted by a striker bar fired from a gas gun. Different levels of energy can be imparted to the specimen by varying the speed and length of the striker to induce different crack growth rates in the material. The specimen is instrumented with a crack ladder gage, crack-opening displacement gage and strain gages to develop requisite data to determine toughness. Typical data for AISI 4340 steel specimen are presented. A simple quasi-dynamic analysis model for deducing the fracture toughness for a running crack from these data is presented, and the results are compared with independent measurements.     

A Shear Compression Disk Specimen with Controlled Stress Triaxiality under Quasi-Static Loading

This paper introduces a double shear axisymmetric specimen (Shear Compression Disk) and the methodology to extract flow and fracture properties of ductile materials, under various stress triaxiality levels. A thorough numerical investigation of the experimental set-up is performed, which reveals that the stresses are quite uniformly distributed in the gauge section during all the stages of the test. The attainable level of stress triaxiality (with pressures of up to 1.9 GPa) ranges from −0.1 to 1, which can be adjusted by a proper choice of geometrical parameters of the specimen. The methodology is implemented to quasi-static experiments on 4340 Steel and Aluminum 7075-T651 specimens. The flow properties are compared to those obtained by upsetting cylinders and show a very good agreement. For these materials it is observed that, contrary to the fracture strain, the flow properties are quite insensitive to the level of stress triaxiality. The fracture strain of the aluminum alloy increases with triaxiality and may be fitted with an exponential polynomial of the type suggested by [27]. These examples demonstrate the potential of the new specimen to obtain flow and fracture properties of ductile materials under controlled triaxiality.     

Discrete fracture modeling of asphalt concrete

A heterogeneous fracture approach is presented for modeling asphalt concrete that is composed of solid inclusions and a viscous matrix, and is subjected to mode-I loading in the fracture test configuration. A heterogeneous fracture model, based on the discrete element method (DEM), is developed to investigate various fracture toughening mechanisms of asphalt materials using a high-resolution image processing technique. An energy-based bilinear cohesive zone model is used to model the crack initiation and propagation of materials, and is implemented as a user-defined model within the discrete element method. Experimental fracture tests are performed to investigate various fracture behavior of asphalt concrete and obtain material input parameters for numerical models. Also, bulk material properties are necessary for each material phase for heterogeneous numerical models; these properties are determined by uniaxial complex modulus tests and indirect tensile strength tests. The main objective of this study is to integrate the experimental tests and numerical models in order to better understand the fracture mechanisms of asphaltic heterogeneous materials. Experimental results and numerical simulations are compared at different test conditions with excellent agreement. The heterogeneous DEM fracture modeling approach has the potential capability to understand various crack mechanisms of quasi-brittle materials.     

A methodology to assess the rate and pressure sensitivity of polymers over a wide range of strain rates

This paper details a methodology to test the mechanical response of soft, pressure sensitive materials, over a wide range of strain rates. A hybrid experimental-numerical procedure is used to assess the constitutive parameters. The experimental phase involves axial compression of a cylindrical specimen which is confined by a tightly-fit sleeve that is allowed to yield plastically, thus applying a constant confining pressure. The usually neglected frictional effects between the specimen and the sleeve are fully accounted for and characterized in detail. With commercial polycarbonate as a typical example, it is shown that pressure sensitivity and rate sensitivity are not coupled, thus reducing the number of tests needed to characterize a material. The results of numerical simulations indicate that the pressure sensitivity index (angle β in the Drucker-Prager material model) has little influence on the hydrostatic and confining pressures, whereas the equivalent stress sustained by the specimen increases with β, which for commercial polycarbonate is found to be β=15°.     

DETERMINATION OF FRACTURE TOUGHNESS OF BRITTLE MATERIALS BY INDENTATION

Fracture toughness is one of the crucial mechanical properties of brittle materials such as glasses and ceramics which demonstrate catastrophic failure modes. Conventional standardized testing methods adopted for fracture toughness determination require large specimens to satisfy the plane strain condition. As for small specimens, indentation is a popular, sometimes exclusive testing mode to determine fracture toughness for it can be performed on a small flat area of the specimen surface. This review focuses on the development of indentation fracture theories and the representative testing methods. Cracking pattern dependent on indenter geometry and material property plays an important role in modeling, and is the main reason for the diversity of indentation fracture theories and testing methods. Along with the simplicity of specimen requirement is the complexity of modeling and analysis which accounts for the semi-empirical features of indentation fracture tests. Some unresolved issues shaping the gap between indentation fracture tests and standardization are also discussed.     

Magnetoelastic fracture of soft ferromagnetic materials

Effects of magnetic field on fracture toughness of soft ferromagnetic materials were studied using experimental techniques and theoretical models. The manganese–zinc ferrite with a single-edge-notch-beam (SENB) were chosen to be the specimen and the Vickers’ indentation specimen subjected to a magnetic field were chosen to be the specimens. Results indicate that there is no significant variations of the measured fracture toughness of the manganese–zinc ferrite ceramic in the presence of the magnetic field. The theoretical model involves an anti-plane shear crack with finite length in an infinite magnetostrictive body where an in-plane magnetic field prevails at infinity. Magnetoelasticity is used. The crack-tip elastic field is different from that of the classical mode III fracture problem. Furthermore, the magnetoelastic fracture of the soft ferromagnetic material was studied by solving the stress field for a soft ferromagnetic plane with a center-through elliptical crack. The stress field at the tip of a slender elliptical crack is obtained for which only external magnetic field normal to the major axis of the ellipse is applied at infinity. The results indicate that the near field stresses are governed by the magnetostriction and permeability of the soft ferromagnetic material. The induction magnetostrictive modulus is a key parameter for finding whether magnetostriction or magnetic-force-induced deformation is dominant near the front an elliptically-shaped crack. The influence of the magnetic field on the apparent toughness of a soft ferromagnetic material with a crack-like flaw can be regarded approximately in two ways: one possesses a large induction magnetostrictive modulus and the other has a small modulus. Finally, a small-scale magnetic-yielding model was developed on the basis of linear magnetization to interpret the experimental results related to the fracture of the manganese–zinc ferrite ceramics under magnetic field. Studied also is the fracture test of the soft ferromagnetic steel with compact tension specimens published in the existing literature.