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.     

Nonlinear mode II crack-tip fields for some Hookean materials

This paper presents a modified nonlinear Mode II crack model which is shown to satisfy the nonpenetrating crack surface boundary condition for homogeneous isotropic Hookean materials taking into account finite deformations. A recent investigation of the problem by Knowles [1] reveals apparent interpenetration of the crack surfaces which is considered nonphysical and therefore invalid. This observation is confirmed when a general solution based on Knowles's perturbation boundary layer method to characterize the finite deformation effects on Mode II crack-tip fields for the materials is derived. By deducing complete 2nd order solutions of the problem, the Poynting nonlinear effect becomes self-evident at the crack tip and for Hookean materials with there always exists the penetrating phenomenon between upper and lower crack surfaces.     

Finite deformation of elastic membranes with application to the stability of an inflated and extended tube

A theory is formulated for the finite deformation of a thin membrane composed of homogeneous elastic material which is isotropic in its undeformed state. The theory is then extended to the case of a small deformation superposed on a known finite deformation of the membrane. As an example, small deformations of a circular cylindrical tube which has been subjected to a finite homogeneous extension and inflation are considered and the equations governing these small deformations are obtained for an incompressible material. By means of a static analysis the stability of cylindrically symmetric modes for the inflated and extended cylinder with fixed ends is determined and the results are verified by a dynamic analysis. The stability is considered in detail for a Mooney material. Methods are developed to obtain the natural frequencies for axially symmetric free vibrations of the extended and inflated cylindrical membrane. Some of the lower natural frequencies are calculated for a Mooney material and the methods are compared.     

A comparative study of kinematic hardening rules at finite deformations

Kinematic hardening models describe a specific kind of plastic anisotropy which evolves with the deformation process. It is well known that the extension of constitutive relations from small to finite deformations is not unique. This applies also to well-established kinematic hardening rules like that of Armstrong-Frederick or Chaboche. However, the second law of thermodynamics offers some possibilities for generalizing constitutive equations so that this ambiguity may, in some extent, be moderated. The present paper is concerned with three possible extensions, from small to finite deformations, of the Armstrong-Frederick rule, which are derived as sufficient conditions for the validity of the second law. All three models rely upon the multiplicative decomposition of the deformation gradient tensor into elastic and plastic parts and make use of a yield function expressed in terms of the so-called Mandel stress tensor. In conformity with this approach, the back-stress tensor is defined to be of Mandel stress type as well. In order to compare the properties of the three models, predicted responses for processes with homogeneous and inhomogeneous deformations are discussed. To this end, the models are implemented in a finite element code (ABAQUS).     

Extremal paths of plastic work and deformation

Optimal paths of deformation between homogeneous states of finite strain are identified on the basis that the work expended should be an absolute minimum. The materials considered are classically rigid/plastic, isotropic and nonhardening, with any convex yield surface. A related minimum principle is derived for a class of functionals of the strain history alone, without reference to work or to material properties. Plane isochoric deformations are subsequently treated in particular detail, supported by simple parametric representations of the competing paths and associated kinematic data.     

Localized shear discontinuities near the tip of a mode I crack

In this paper we study the deformation and stress fields near the tip of a crack under plane strain mode I conditions. A fully nonlinear theory of finite deformations is used and the material, which is assumed to be homogeneous, isotropic, incompressible and elastic, is characterized by its stress-strain behavior in simple shear. For the class of materials considered the governing system of differential equations may lose ellipticity at sufficiently severe strains. The analysis is based on a direct asymptotic calculation. The results involve two curves, issuing from each crack-tip, across which the deformation gradient, the effective shear and the stresses are discontinuous.     

Energetic bounds in finite elasticity

Summary We study the conditions under which the internal work of deformation in an elastic isotropic body in finite deformations may be bounded by results obtained from a suitably defined linear infinitesimal problem. The values of the constants appearing in the principal inequalities are calculated and discussed for a certain class of extensional deformations.     

Inelastic Deformation of Flexible Spherical Shells with Two Circular Openings

Elastoplastic analysis of thin-walled spherical shells with two identical circular openings is carried out with allowance for finite deflections. The shells are made of an isotropic homogeneous material and subjected to internal pressure of known intensity. The distributions of stresses (strains or displacements) along the contours of the openings and in the zone of their concentration are studied by solving doubly nonlinear boundary-value problems. The solution obtained is compared with the solutions that account for only physical nonlinearity (plastic deformations) and only geometrical nonlinearity (finite deflections) and with a numerical solution of the linearly elastic problem. The stress–strain state near the two openings is analyzed depending on the distance between the openings and the nonlinear factors accounted for     

Stability and asymmetric vibrations of pressurized compressible hyperelastic cylindrical shells

Cylindrical shells of arbitrary wall thickness subjected to uniform radial tensile or compressive dead-load traction are investigated. The material of the shell is assumed to be homogeneous, isotropic, compressible and hyperelastic. The stability of the finitely deformed state and small, free, radial vibrations about this state are investigated using the theory of small deformations superposed on large elastic deformations. The governing equations are solved numerically using both the multiple shooting method and the finite element method. For the finite element method the commercial program ABAQUS is used.¹ The loss of stability occurs when the motions cease to be periodic. The effects of several geometric and material properties on the stress and the deformation fields are investigated.     

Nonlinear vibrations and instabilities of a stretched hyperelastic annular membrane

The mathematical modeling for the nonlinear vibration analysis of a pre-stretched hyperelastic annular membrane under finite deformations is presented. The membrane is initially fixed along the inner boundary and then subjected to a uniform radial traction along its outer circumference and fixed along the outer boundary. The pre-stretched membrane in then subjected to a transversal harmonic pressure. The membrane material is assumed to be homogeneous, isotropic, and neo-Hookean. First, the solution of the radially stretched membrane is obtained analytically and numerically by the shooting method. The equations of motion of the stretched membrane are then obtained. By analytically and numerically solving the linearized equations of motion, the vibration modes and frequencies of the hyperelastic membrane are obtained, and these normal modes are used, together with the Galerkin method, to obtain reduced order models for the nonlinear dynamic analysis. A parametric analysis of the nonlinear frequency-amplitude relations, resonance curves, bifurcation diagrams and basins of attraction show the influence of the initial stretching ratio and membrane geometry on the type and degree of nonlinearity of the hyperelastic membrane under large amplitude vibrations. To check the accuracy of the reduced order models and the influence of the simplifying hypotheses on the results, the same problem is also analyzed using the finite element method. Excellent agreement is observed.     

Damage in edge cracking of rolled metal slabs

Materials get damaged under shear deformations. Edge cracking is one of the most serious damage to the metal rolling industry, which is caused by the shear damage process and the evolution of anisotropy. To investigate the physics of the edge cracking process, simulations of a shear deformation for an orthotropic plastic material are performed. To perform the simulation, this paper proposes an elasto-aniso-plastic constitutive model that takes into account the evolution of the orthotropic axes by using a bases rotation formula, which is based upon the slip process in the plastic deformation. It is found through the shear simulation that the void can grow in shear deformations due to the evolution of anisotropy and that stress triaxiality in shear deformations of (induced) anisotropic metals can develop as high as in the uniaxial tension deformation of isotropic materials, which increases void volume. This echoes the same physics found through a crystal plasticity based damage model that porosity evolves due to the grain-to-grain interaction. The evolution of stress components, stress triaxiality and the direction of the orthotropic axes in shear deformations are discussed.     

An endochronic plasticity formulation for filled rubber

The nature of elastomeric material demands the consideration of finite deformations, nonlinear elasticity including damage as well as rate-dependent and rate-independent dissipative properties. While many models accounting for these effects have been refined over time to do better justice to the real behavior of rubber-like materials, the realistic simulation of the elastoplastic characteristics for filled rubber remains challenging.The classical elastic-ideal-plastic formulation exhibits a distinct yield-surface, whereas the elastoplastic material behavior of filled rubber components shows a yield-surface free plasticity. In order to describe this elastoplastic deformation of a material point adequately, a physically based endochronic plasticity model was developed and implemented into a Finite Element code. The formulation of the ground state elastic characteristics is based on Arruda and Boyce (1993) eight-chain model. The evolution of the constitutive equations for the nonlinear endochronic elastoplastic response are derived in analogy to the Bergström–Boyce finite viscoelasticity model discussed by Dal and Kaliske (2009).     

两种晶体塑性损伤模型在有限变形条件下的比较

对建立在连续损伤力学和内变量理论基础上的两种晶体塑性损伤模型进行了比较,旨在比较两种模型在描述材料物理性能方面的适用性以及由加载而引起变形响应的不同之处.模拟结果显示,两种模型均能反映出塑性各向异性和损伤的演化;由加载而导致有限变形的响应不仅依赖于变形,而且也依赖于晶格取向;尽管两种模型在揭示单晶体的物理性能方面是不同的,但是在预测材料力学性能方面有着相同的预测趋势.     

Bifurcation conditions for a thick elastic plate under thrust

A thick rectangular plate of incompressible isotropic elastic material is subjected to a pure homogeneous deformation by tensile forces or thrusts applied to a pair of opposite faces. The theory of small deformations superposed on finite deformations is applied to determine the critical conditions under which bifurcation solutions (i.e. adjacent equilibrium positions) can exist. The adjacent equilibrium positions considered are those for which the superposed deformation is two-dimensional and is coplanar with the loading force and the thickness direction of the plate, the faces of the plate normal to its thickness being force-free. A number of theorems relating to the critical conditions for superposed deformations of the flexural and barreling types are derived under conditions on the strain-energy function more general than those employed in earlier work. It is also shown how these results can be applied to the determination of the bifurcation conditions corresponding to any specified strain-energy function.     

New universal relations for nonlinear isotropic elastic materials

A nonlinear isotropic elastic block is subjected to a homogeneous deformation consisting of simple shear superposed on triaxial extension. Two new relations are established for this deformation which are valid for all nonlinear elastic isotropic materials, and hence are universal relations. The first is a relation between the stretch ratios in the plane of shear and the amount of shear when the deformation is supported only by shear tractions. The second relation is established for a thin-walled cylinder under combined extension, inflation and torsion. Each material element of the cylinder undergoes the same local homogeneous deformation of shear superposed on triaxial extension. The properties of this deformation are used to establish a relation between pressure, twisting moment, angle of twist and current dimensions when no axial force is applied to the cylinder. It is shown that these relations also apply for a mixture of a nonlinear isotropic solid and a fluid.     

Capturing anisotropic constitutive models with WYPiWYG hyperelasticity; and on consistency with the infinitesimal theory at all deformation levels

The elastic nonlinear behavior of fiber-reinforced materials and soft biological tissues is analyzed using anisotropic hyperelastic models. Frequently, these models are not compatible with the corresponding infinitesimal theory, but some of them may be modified to accommodate that theory in the limit. WYPiWYG hyperelasticity is compatible with the infinitesimal theory at all deformation levels and capable of capturing exactly a complete set of experimental data, which reproduces all deformation modes at every strain level, under homogeneous deformations. In this work we study the relevance of recovering the infinitesimal theory at every deformed configuration and also the performance of the WYPiWYG method in predicting the behavior of anisotropic materials at large strains under nonhomogeneous deformations.     

Time-dependent behavior of passive skeletal muscle

An isotropic three-dimensional nonlinear viscoelastic model is developed to simulate the time-dependent behavior of passive skeletal muscle. The development of the model is stimulated by experimental data that characterize the response during simple uniaxial stress cyclic loading and unloading. Of particular interest is the rate-dependent response, the recovery of muscle properties from the preconditioned to the unconditioned state and stress relaxation at constant stretch during loading and unloading. The model considers the material to be a composite of a nonlinear hyperelastic component in parallel with a nonlinear dissipative component. The strain energy and the corresponding stress measures are separated additively into hyperelastic and dissipative parts. In contrast to standard nonlinear inelastic models, here the dissipative component is modeled using an evolution equation that combines rate-independent and rate-dependent responses smoothly with no finite elastic range. Large deformation evolution equations for the distortional deformations in the elastic and in the dissipative component are presented. A robust, strongly objective numerical integration algorithm is used to model rate-dependent and rate-independent inelastic responses. The constitutive formulation is specialized to simulate the experimental data. The nonlinear viscoelastic model accurately represents the time-dependent passive response of skeletal muscle.     

On the Dirichlet problem for incompressible elastic materials

The linearized equations governing the deformations of incompressible elastic bodies are discussed. The Dirichlet problem is formulated for this system of equations using the theory of elliptic systems due to Douglis and Nirenberg. A uniqueness theorem is proved. Necessary and sufficient conditions for uniqueness of solution to the Dirichlet problem are obtained for small deformations of a Mooney-Rivlin material which has been subjected to a finite homogeneous biaxial deformation.     

Determining the model parameters of nonlinear deformation of isotropic and composite materials from the results of an experimental-computational analysis of impulsively loaded circular plates

A method for identifying the material parameters of the constitutive relations of viscoelastic and elastoplastic deformation for isotropic and homogeneous composite materials is developed based on minimizing the mismatch between the results of numerical and experimental modeling of unsteady deformation of structural elements made of the materials studied. The method is tested and shown to be effective in determining the viscoelastic and elastoplastic characteristics of models for the nonlinear deformation of impulsively loaded isotropic and composite circular plates.