Effect of elastomer slight compressibility

This paper is concerned with the constitutive equation for slightly compressible elastic material under finite deformations. We show that material slight compressibility can be effectively taken into account in the case of high hydrostatic pressure or highly confined material. In all other situations the application of the incompressible and nearly incompressible material theories gives practically the same results. Therefore it is of interest to consider the problem in which allowing for material slight compressibility leads to results essentially different from those obtained with help of the incompressible material model. In the present paper this difference has been demonstrated for the problem of high hydrostatic pressure causing an increase of the ‘bulk’ and ‘shear’ material moduli. The behavior of a long hollow cylinder of real material under finite deformations is analyzed. The cylinder is subjected to internal pressure and axial and circular displacements at the outer surface. This problem has been solved analytically using the small parameter method. The solution obtained predicts a decrease of the axial and circular displacements of the outer surface under fixed shear (axial and circular) forces when the internal pressure is applied.     

On the Normal Stresses in Simple Shearing of Fiber-Reinforced Nonlinearly Elastic Materials

We are concerned with a particular aspect of the simple shear problem within the framework of nonlinear elasticity for a class of incompressible transversely-isotropic fiber-reinforced materials. It is well known that, for isotropic hyperelastic materials, the normal stress effect characteristic of nonlinear elasticity is crucial in order to maintain a homogeneous deformation state in the bulk of the specimen. For the fiber-reinforced materials of concern here, we show that the confining traction that needs to be applied to the top and bottom faces of a block in order to maintain simple shear can be compressive or tensile depending on the degree of anisotropy and on the angle of orientation of the fibers. Inclusion of the second invariant in the isotropic part of the strain-energy used is shown to be of crucial importance in assessing the nature of the confining traction. In the absence of such an applied traction, an unconfined sample tends to bulge outwards or contract inwards perpendicular to the direction of shear. The character of the normal component of traction on the inclined faces is also investigated. The results are relevant to the development of accurate shear test protocols for the determination of constitutive properties of fiber-reinforced rubber-like materials and fibrous biological soft tissues.     

Finite strain––isotropic hyperelasticity

This paper presents a strain energy density for isotropic hyperelastic materials. The strain energy density is decomposed into a compressible and incompressible component. The incompressible component is the same as the generalized Mooney expression while the compressible component is shown to be a function of the volume invariant J only. The strain energy density proposed is used to investigate problems involving incompressible isotropic materials such as rubber under homogeneous strain, compressible isotropic materials under high hydrostatic pressure and volume change under uniaxial tension. Comparison with experimental data is good. The formulation is also used to derive a strain energy density expression for compressible isotropic neo-Hookean materials. The constitutive relationship for the second Piola–Kirchhoff stress tensor and its physical counterpart, involves the contravariant Almansi strain tensor. The stress stretch relationship comprises of a component associated with volume constrained distortion and a hydrostatic pressure which results in volumetric dilation. An important property of this constitutive relationship is that the hydrostatic pressure component of the stress vector which is associated with volumetric dilation will have no shear component on any surface in any configuration. This same property is not true for a neo-Hookean Green’s strain–second Piola–Kirchhoff stress tensor formulation.     

超弹性橡胶材料的改进Rivlin模型

讨论了不可压缩橡胶材料的超弹性唯象本构模型。针对典型实验,给出选择应变能函数的原则。从物理机理上,分析了Neo-Hookean模型、Mooney模型、三阶Rivlin模型及Ogden模型的优缺点。在此基础上,将Rivlin模型改进成为 ,这种新形式具有三个优点：①若取前三项（N=1）,则其结果与不可压缩线弹性的应变能相等,能够近似满足剪切的线性关系,但拉伸及压缩的线性关系是精确满足的。②当N≥2时,简单剪切中的应变能及剪应力τxy在小应变情况下是以剪应变γxy为等比的多项式展开;而Rivlin模型只能保证简单剪切实验中的应变能及剪应力τxy是以(γxy)2为等比的级数展开的形式,当取前两项的情况下,Rivlin模型只能精确保证常剪切,拉伸及压缩的线性关系无法得到保证。针对典型实验数据,若取同阶次多项式,本文模型的同类实验数据预测及不同类实验数据间相互预测的精度都比Rivlin模型的高。     

Fluid flow and fluid shear stress in canaliculi induced by external mechanical loading and blood pressure oscillation

The paper studies the problem of fluid flow and fluid shear stress in canaliculi when the osteon is subject to external mechanical loading and blood pressure oscillation. The single osteon is modeled as a saturated poroelastic cylinder. Solid skeleton is regarded as a poroelastic transversely isotropic material. To get near-realistic results, both the interstitial fluid and the solid matrix are regarded as compressible. Blood pressure oscillation in the Haverian canal is considered. Using the poroelasticity theory, an analytical solution of the pore fluid pressure is obtained. Assuming the fluid in canaliculi is incompressible, analytical solutions of fluid flow velocity and fluid shear stress with the Navier-Stokes equations of incompressible fluid are obtained. The effect of various parameters on the fluid flow velocity and fluid shear stress is studied.     

Some Inverse Problems of Deformation and Fracture of Physically Nonlinear Inhomogeneous Media

The following two types of physically nonlinear inhomogeneous media are considered: linear-elastic plane with nonlinear-elastic elliptic inclusions and linear-viscous plane with elliptic inclusions from a material that possesses nonlinear-creep properties. The problem is to determine infinitely distant loads that produce a required value of the principal shear stress (in the first case) or principal shear-strain rate (in the second case) for two arbitrary inclusions. Conditions for the existence of solutions of these problems for incompressible media under plane strains are obtained.     

On Structured Surfaces with Defects: Geometry,Strain Incompatibility,Stress Field,and Natural Shapes

In this paper we study the stress and deformation fields generated by nonlinear inclusions with finite eigenstrains in anisotropic solids. In particular, we consider finite eigenstrains in transversely isotropic spherical balls and orthotropic cylindrical bars made of both compressible and incompressible solids. We show that the stress field in a spherical inclusion with uniform pure dilatational eigenstrain in a spherical ball made of an incompressible transversely isotropic solid such that the material preferred direction is radial at any point is uniform and hydrostatic. Similarly, the stress in a cylindrical inclusion contained in an incompressible orthotropic cylindrical bar is uniform hydrostatic if the radial and circumferential eigenstrains are equal and the axial stretch is equal to a value determined by the axial eigenstrain. We also prove that for a compressible isotropic spherical ball and a cylindrical bar containing a spherical and a cylindrical inclusion, respectively, with uniform eigenstrains the stress in the inclusion is uniform (and hydrostatic for the spherical inclusion) if the radial and circumferential eigenstrains are equal. For compressible transversely isotropic and orthotropic solids, we show that the stress field in an inclusion with uniform eigenstrain is not uniform, in general. Nevertheless, in some special cases the material can be designed in order to maintain a uniform stress field in the inclusion. As particular examples to investigate such special cases, we consider compressible Mooney-Rivlin and Blatz-Ko reinforced models and find analytical expressions for the stress field in the inclusion.     

Extension,torsion and expansion of an incompressible,hemitropic Cosserat circular cylinder

Constitutive equations for the stress and couple stres on an incompressible, hemitropic, constrained Cosserat material are derived, and the theory is applied to study the problem of finite extension, torsion and expansion of a circular cylinder. As in the theory of isotropic simple elastic materials, it is found that the deformation is controllable by application of only a normal force and a tosional moment at the cylinder ends. It is shown that in general the well known universal relation between the torsional stiffness and the axial force for incompressible, isotropic simple materials in the limit as the twist goes to zero does not exist for incompressible, hemitropic Cosserat materials. However, for a special and unusual class of hemitropic materials, the same universal formula is found to hold for a certain reduced torsional stiffness. The main problem is solved completely for incompressible, hemitropic, linearly elastic, Cosserat materials; and certain additional special features of the Kelvin-Poynting type, which here appear to the first order in the amount of twist of the cylinder, are derived and discussed in relation to experimentally observed composite material behavior.     

Deformations of an elastic,internally constrained material. Part 1: Homogeneous deformations

The nonlinear elastic response of a class of materials for which the deformation is subject to an internal material constraint described in experiments by James F. Bell on the finite deformation of a variety of metals is investigated. The purely kinematical consequences of the Bell constraint are discussed, and restrictions on the full range of compatible deformations are presented in geometrical terms. Then various forms of the constitutive equation relating the stress and stretch tensors for an isotropic elastic Bell material are presented. Inequalities on the mechanical response functions are introduced. The importance of these in applications is demonstrated in several examples throughout the paper.This paper focuses on homogeneous deformations. In a simple illustration of the theory, a generalized form of Bell's empirical rule for uniaxial loading is derived, and some peculiarities in the response under all-around compressive loading are discussed. General formulae for universal relations possible in an isotropic elastic, Bell constrained material are presented. A simple method for the determination of the left stretch tensor for essentially plane problems is illustrated in the solution of the problem of pure shear of a materially uniform rectangular block. A general formula which includes the empirical rule found in pure shear experiments by Bell is derived as a special case. The whole apparatus is then applied in the solution of the general problem of a homogeneous simple shear superimposed on a uniform triaxial stretch; and the great variety of results possible in an isotropic, elastic Bell material is illustrated. The problem of the finite torsion and extension of a thin-walled cylindrical tube is investigated. The results are shown to be consistent with Bell's data for which the rigid body rotation is found to be quite small compared with the gross deformation of the tube. Several universal formulas relating various kinds of stress components to the deformation independently of the material response functions are derived, including a universal rule relating the axial force to the torque.Constitutive equations for hyperelastic Bell materials are derived. The empirical work function studied by Bell is introduced; and a new constitutive equation is derived, which we name Bell's law. On the basis of this law, we then derive exactly Bell's parabolic laws for uniaxial loading and for pure shear. Also, form Bell's law, a simple constitutive equation relating Bell's deviatoric stress tensor to his finite deviatoric strain tensor is obtained. We thereby derive Bell's invariant parabolic law relating the deviatoric stress intensity to the corresponding strain intensity; and, finally, Bell's fundamental law for the work function expressed in these terms is recovered. This rule is the foundation for all of Bell's own theoretical study of the isotropic materials cataloged in his finite strain experiments on metals, all consistent with the internal material constraint studied here.     

Plane Deformations in Incompressible Nonlinear Elasticity

The substantially general class of plane deformation fields, whose only restriction requires that the angular deformation not vary radially, is considered in the context of isotropic incompressible nonlinear elasticity. Analysis to determine the types of deformations possible, that is, solutions of the governing systems of nonlinear partial differential equations and constraint of incompressibility, is developed in general. The Mooney-Rivlin material model is then considered as an example and all possible solutions to the equations of equilibrium are determined. One of these is interpreted in the context of nonradially symmetric cavitation, i.e., deformation of an intact cylinder to one with a double-cylindrical cavity. Results for general incompressible hyperelastic materials are then discussed. The novel approach taken here requires the derivation and use of a material formulation of the governing equations; the traditional approach employing a spatial formulation in which the governing equations hold on an unknown region of space is not conducive to the study of deformation fields containing more than one independent variable. The derivation of the cylindrical polar coordinate form of the equilibrium equations for the nominal stress tensor (material formulation) for a general hyperelastic solid and a fully arbitrary cylindrical deformation field is also given. This revised version was published online in August 2006 with corrections to the Cover Date.     

近似不可压软材料动力分析的完全拉格朗日物质点法

物质点法(MPM)在模拟非线性动力问题时具有很好的效果,其已被广泛应用于许多大变形动力问题的分析中.然而传统的MPM在模拟不可压或近似不可压材料的动力学行为时会产生体积自锁,极大地影响模拟精度和收敛性.本文针对近似不可压软材料的大变形动力学行为,提出一种混合格式的显式完全拉格朗日物质点法(TLMPM).首先基于近似不可压软材料的体积部分应变能密度,引入关于静水压力的方程;之后将该方程与动量方程基于显式物质点法框架进行离散,并采用完全拉格朗日格式消除物质点跨网格产生的误差,提升大变形问题的模拟精度;对位移和压强场采用不同阶次的B样条插值函数并通过引入针对体积变形的重映射技术改进了算法,提升算法的准确性.此外,算法通过实施一种交错求解格式在每个时间步对位移场和压强场依次进行求解.最后,给出几个典型数值算例来验证本文所提出的混合格式TLMPM的有效性和准确性,计算结果表明该方法可以有效处理体积自锁,准确地模拟近似不可压软材料的大变形动力学行为.     

Surface instability of elastic half spaces with hydrostatic loading on their surfaces

In this paper, the plane-strain buckling of compressible and incompressible elastichalf-spaces, whose surfaces are loaded by constant hydrostatic pressures, is studied byusing a small-deformation-superposed-on-large-deformation analysis, and the bucklingcondition for each case is obtained. For Blatz-Ko and harmonic compressible materials aswell as Mooney incompressible material, the influence of the surface hydrostatic pressureon the critical buckling condition is discussed in detail.     

Simple shear is not so simple

For homogeneous, isotropic, non-linearly elastic materials, the form of the homogeneous deformation consistent with the application of a Cauchy shear stress is derived here for both compressible and incompressible materials. It is shown that this deformation is not simple shear, in contrast to the situation in linear elasticity. Instead, it consists of a triaxial stretch superposed on a classical simple shear deformation, for which the amount of shear cannot be greater than 1. In other words, the faces of a cubic block cannot be slanted by an angle greater than 45° by the application of a pure shear stress alone. The results are illustrated for those materials for which the strain-energy function does not depend on the principal second invariant of strain. For the case of a block deformed into a parallelepiped, the tractions on the inclined faces necessary to maintain the derived deformation are calculated.     

A finite element approach to study cavitation instabilities in non-linear elastic solids under general loading conditions

This paper proposes an effective numerical method to study cavitation instabilities in non-linear elastic solids. The basic idea is to examine—by means of a 3D finite element model—the mechanical response under affine boundary conditions of a block of non-linear elastic material that contains a single infinitesimal defect at its center. The occurrence of cavitation is identified as the event when the initially small defect suddenly grows to a much larger size in response to sufficiently large applied loads. While the method is valid more generally, the emphasis here is on solids that are isotropic and defects that are vacuous and initially spherical in shape. As a first application, the proposed approach is utilized to compute the entire onset-of-cavitation surfaces (namely, the set of all critical Cauchy stress states at which cavitation ensues) for a variety of incompressible materials with different convexity properties and growth conditions. For strictly polyconvex materials, it is found that cavitation occurs only for stress states where the three principal Cauchy stresses are tensile and that the required hydrostatic stress component at cavitation increases with increasing shear components. For a class of materials that are not polyconvex, on the other hand and rather counterintuitively, the hydrostatic stress component at cavitation is found to decrease for a range of increasing shear components. The theoretical and practical implications of these results are discussed.     

Finite amplitude, free vibrations of an axisymmetric load supported by a highly elastic tubular shear spring

The finite amplitude, free vibrational characteristics of a simple mechanical system consisting of an axisymmetric rigid body supported by a highly elastic tubular shear spring subjected to axial, rotational, and coupled shearing motions are studied. Two classes of elastic tube materials are considered: a compressible material whose shear response is constant, and an incompressible material whose shear response is a quadratic function of the total amount of shear. The class of materials with constant shear response includes the incompressible Mooney-Rivlin material and certain compressible Blatz-Ko, Hadamard, and other general kinds of models. For each material class, the quasi-static elasticity problem is solved to determine the telescopic and gyratory shearing deformation functions needed to evaluate the elastic tube restoring force and torque exerted on the body. For all materials with constant shear response, the differential equations of motion are uncoupled equations typical of simple harmonic oscillators. Hence, exact solutions for the forced vibration of the system can be readily obtained; and for this class, engineering design formulae for the load-deflection relations are discussed and compared with experimental results of others'. For the quadratic material, however, the general motion of the body is characterized by a formidable, coupled system of nonlinear equations. The free, coupled shearing motion for which either the axial or the azimuthal shear deformation may be small is governed by a pair of equations of the Duffing and Hill types. On the other hand, the finite amplitude, pure axial and pure rotational motions of the load are described by the classical, nonlinear Duffing equation alone. A variety of problems are solved exactly for these separate free vibrational modes, and a number of physical results are presented throughout.     

Rheological predictions of a transversely isotropic fluid model with extensible microstructure

A continuum extensible director theory was formulated to describe the isothermal, incompressible flow of uniaxial rodlike semiflexible liquid crystalline polymers. The model is strictly restricted to material that flow-align in shear, and that, in the absence of flow, are sufficiently far from the nematic-isotropic phase transition. The microstructure of the continuum is described by a variable length director, but the extensibility is finite. The model is an extension of the TIF (Transversely Isotropic Fluid) model of Ericksen (1960). The thermodynamic restrictions on the model parameters are found using the non-negative definiteness of the entropy production. The rheological material functions predicted by the model are calculated for steady simple shear and steady uniaxial extensional flows. In the rigid rod limit the model predictions agree with those of the TIF model, and for the finite extensibility case the model predictions are in agreement with those associated with flexible isotropic polymers: strong non-Newtonian shear viscosity, positive first normal stress differences, recoverable shear of order one, negative second normal stress differences, and a maximum in the steady uniaxial extensional viscosity.     

Creep transition in torsion

Stresses for a circular cylinder of compressible material subjected to torsion are derived in closed form for steady state creep. It is shown that the asymptotic solution through stress leads from elastic state to plastic and then to creep and through stress difference leads to the creep state. The effect of compressibility is presented graphically. The results indicate that the value of maximum shear stress for a cylinder of compressible material is greater than that for an incompressible material and increases with the increase in a measure index n. For an incompressible material, as a particular case, the results obtained are the same as given by Marin [9].     

Nonlinear axisymmetric deformations of an elastic tube under external pressure

The problem of the finite axisymmetric deformation of a thick-walled circular cylindrical elastic tube subject to pressure on its external lateral boundaries and zero displacement on its ends is formulated for an incompressible isotropic neo-Hookean material. The formulation is fully nonlinear and can accommodate large strains and large displacements. The governing system of nonlinear partial differential equations is derived and then solved numerically using the C++ based object-oriented finite element library Libmesh. The weighted residual-Galerkin method and the Newton-Krylov nonlinear solver are adopted for solving the governing equations. Since the nonlinear problem is highly sensitive to small changes in the numerical scheme, convergence was obtained only when the analytical Jacobian matrix was used. A Lagrangian mesh is used to discretize the governing partial differential equations. Results are presented for different parameters, such as wall thickness and aspect ratio, and comparison is made with the corresponding linear elasticity formulation of the problem, the results of which agree with those of the nonlinear formulation only for small external pressure. Not surprisingly, the nonlinear results depart significantly from the linear ones for larger values of the pressure and when the strains in the tube wall become large. Typical nonlinear characteristics exhibited are the “corner bulging” of short tubes, and multiple modes of deformation for longer tubes.     

Localization and shear-crippling (kinkband) instability in a thick imperfect laminated composite ring under hydrostatic pressure

A fully nonlinear finite elements analysis for prediction of localization representing shear-crippling (kinkband) instability in a thick laminated composite (plane strain) ring (infinitely long cylindrical shell) under applied hydrostatic pressure is presented. The primary accomplishment of the present investigation is prediction of meso(lamina)-structure-related equilibrium paths, which are often unstable in the presence of local imperfections and/or material nonlinearity, and which are considered to “bifurcate” from the primary equilibrium paths, representing periodic buckling patterns pertaining to global or structural level stability of the thick cross-ply ring with modal or harmonic imperfection. The present nonlinear finite elements solution methodology, based on the total Lagrangian formulation, employs a quasi-three-dimensional hypothesis, known as layerwise linear displacement distribution theory (LLDT) to capture the three-dimensional interlaminar (especially, shear) deformation behavior, associated with the localized interlaminar shear-crippling failure.A thick laminated composite [90/0/90] imperfect (plane strain) ring is investigated with the objective of analytically studying its premature compressive failure behavior. Numerical results suggest that interlaminar shear/normal deformation (especially, the former) is primarily responsible for the appearance of a limit (maximum pressure) point on the post-buckling equilibrium path associated with a periodic (modal or harmonic) buckling pattern, for which a modal imperfection serves as a perturbation. Localization of the buckling pattern results from “bifurcation” at or near this limit point, and can be viewed as a symmetry breaking phenomenon.In order to investigate a localization of the buckling pattern, a local or dimple shaped imperfection superimposed on a fixed modal one is selected. With the increase of local imperfection amplitude, the limit load (hydrostatic pressure) decreases, and also the limit point appears at an increased normalized deflection. Additionally, the load–deflection curves tend to flatten (near-zero slope) to an undetermined lowest pressure level, signaling the onset of “phase transition” in the localized region, and coexistence of two “phases”, i.e., a highly localized band of shear crippled (kinked) phase and its unshear-crippled (unkinked) counterpart along the circumference of the ring. Interlaminar shear-crippling triggered by the combined effect of imperfection, material nonlinearity and interlaminar shear/normal deformation appears to be the dominant compressive failure mode. A three-dimensional or quasi-three-dimensional theory, such as the afore-mentioned LLDT is essential in order to capture the meso-structure-related instability failure such as localization of the interlaminar shear crippling, triggered by the combined presence of local imperfection and material nonlinearity.     

In-plane buckling of circular arches and rings with shear deformations

A finite strain formulation is developed for elastic circular arches and rings in which the effects of shear deformations are included. Timoshenko beam hypothesis is adopted for incorporating shear. Finite strains are defined in terms of the normal and shear component of the longitudinal stretch. The constitutive relations for stress and finite strain are based on a hyperelastic constitutive model. Virtual work and equilibrium equations are derived. Closed-form in-plane buckling solutions are developed for circular rings and high arches under hydrostatic pressure. The effects of axial deformation prior to buckling as well as shear deformations are included in the buckling analysis. The formulation developed is compared with solutions in the literature and to the predictions of the finite element package ANSYS. The importance of including the effects of shear deformations for deep arches is investigated.