Incompressible Isotropic Material Research Articles

First integrals of the equilibrium equations in one-dimensional problems of nonlinear elasticity theory

The first integrals of a system of ordinary nonlinear differential equations describing the bending, tension, and torsion of an elastic anisotropic layer have been derived using the conservation laws of the theory of finite strains in simple materials and materials with couple stresses. These integrals can be used for reducing the order of such systems of differential equations, obtaining exact solutions, and checking the calculation accuracy in numerical solution of the differential equations of nonlinear elasticity theory. For example, an exact solution of the problem of large strains under pure bending of an elastic plate is found in the class of isotropic incompressible micropolar materials.

On three-dimensional edge waves in semi-infinite isotropic plates subject to mixed face boundary conditions

This paper is concerned with the propagation of three-dimensional waves localized near the edge of a semi-infinite elastic plate subject to mixed face boundary conditions. In the linear isotropic case it is shown that the problem is closely related to that of Rayleigh surface wave propagation along the free surface of the corresponding half-space. The cut-off frequencies of the analyzed edge waves coincide with the natural frequencies of the associated cross-sectional semi-infinite strip. It is also demonstrated that the eigenspectrum of a rectangular rod can be expressed in terms of the considered three-dimensional waves. The results are then generalized to a prestressed isotropic incompressible material. It is noted that the density of the edge wave spectrum is strongly influenced by the prestress. It is illustrated that the areas of negative group velocity may exist for large primary deformation. Long-wave asymptotic expansions in the vicinity of the cut-off frequencies are presented.

Small Amplitude Plane Waves in Deformed Mooney—Rivlin Viscoelastic Solids

A homogeneous isotropic incompressible viscoelastic material of differential type whose elastic part of the stress tensor is modelled through the use of the Mooney—Rivlin from of the stored energy function, and whose dissipative part is modelled as in a Newtonian fluid, is maintained in a state of finite static homogeneous biaxial deformation. The equations of superimposed infinitesimal motions on the finite deformation are derived. All propagating homogeneous and inhomogeneous solutions are given, as well as the conditions for linear, elliptical, or circular polarization.

Qualitative analysis of dynamical behavior for an imperfect incompressible neo-Hookean spherical shell

The radial symmetric motion problem was examined for a spherical shell composed of a class of imperfect incompressible hyper-elastic materials, in which the materials may be viewed as the homogeneous incompressible isotropic neo-Hookean material with radial perturbations. A second-order nonlinear ordinary differential equation that describes the radial motion of the inner surface of the shell was obtained. And the first integral of the equation was then carried out. Via analyzing the dynamical properties of the solution of the differential equation, the effects of the prescribed imperfection parameter of the material and the ratio of the inner and the outer radii of the underformed shell on the motion of the inner surface of the shell were discussed, and the corresponding numerical examples were carried out simultaneously. In particular, for some given parameters, it was proved that, there exists a positive critical value, and the motion of the inner surface with respect to time will present a nonlinear periodic oscillation as the difference between the inner and the outer presses does not exceed the critical value. However, as the difference exceeds the critical value, the motion of the inner surface with respect to time will increase infinitely. That is to say, the shell will be destroyed ultimately.

A Lie Group Analysis of the Axisymmetric Equations of Finite Elastostatics for Compressible Materials

Many boundary-value problems in nonlinear elastostatics for incompressible isotropic materials have been solved analytically. The situation for compressible materials is quite different. The main difficulty is the absence of the simplified kinematics arising from the incompressibility constraint. Considerable progress in the analytic solution of boundary-value problems for classes of compressible materials has however taken place in recent years and further developments in this direction are considered in this paper. A very special topic in the equilibrium theory for finite deformations of homogeneous isotropic compressible materials is investigated, namely that of spherically symmetric or cylindrically symmetric deformations. Since only axisymmetric deformations are considered, the equilibrium equations reduce to a single second-order nonlinear ordinary differential equation for the radial deformation field. In previous work, two main methods for obtaining analytic solutions for these equations have been developed, the first based on a change of variable that allows one to transform the second-order ODE to a pair of first-order ODEs that may be solved in parametric form. In some cases, the parameter can be eliminated to yield an explicit closed form expression for the deformation. The second method is based on the observation that the second-order ODE is invariant under a stretching transformation of the independent and dependent variables and thus may always be transformed to a first-order ODE. The purpose of this paper is to show how the powerful techniques of Lie group analysis (LGA) for ODEs can be used to unify both the above approaches and how LGA is an invaluable guide in the development of analytic solution methods. It is shown that both of the approaches can be derived from Lie algebra invariance criteria and that the second approach is, in fact, a special case of the first. Results from LGA are then applied to investigate the structure and solution set of the governing ODEs. A systematic procedure for solving the axisymmetric equations of compressible finite elastostatics is thereby proposed. The results are illustrated by application to several specific strain-energy density functions.

Qualitative study of cavitated bifurcation for a class of incompressible generalized neo-Hookean spheres

The problem of spherical cavitated bifurcation was examined for a class of incompressible generalized neo-Hookean materials, in which the materials may be viewed as the homogeneous incompressible isotropic neo-Hookean material with radial perturbations. The condition of void nucleation for this problem was obtained. In contrast to the situation for a homogeneous isotropic neo-Hookean sphere, it is shown that not only there exists a secondary turning bifurcation point on the cavitated bifurcation solution which bifurcates locally to the left from trivial solution, and also the critical load is smaller than that for the material with no perturbations, as the parameters belong to some regions. It is proved that the cavitated bifurcation equation is equivalent to a class of normal forms with single-sided constraints near the critical point by using singularity theory. The stability of solutions and the actual stable equilibrium state were discussed respectively by using the minimal potential energy principle.

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.

An algorithm using co‐ordinate transformation for free vibration analysis of fully incompressible materials

AbstractThis paper presents an algorithm for conducting eigenvalue analysis of fully incompressible isotropic materials (a Poisson's ratio of exactly 0.5). This algorithm is based on the concept of co‐ordinate transformation defined by eigenvectors with positive eigenvalues of the system stiffness matrix. The transformation to the new co‐ordinates of orthogonal bases in the system potential energy can reduce the size of the system equations. The original eigenvalue problem is reduced to a smaller one that can be solved. This algorithm possesses two drawbacks. One is the increase of system degrees of freedom (DOF) due to the mixed method for incompressible problems; the other is due to additional eigenvalue calculation. However, the present algorithm is inherently able to compensate for the above two drawbacks by reducing the DOF. Numerical demonstrations indicate that the exact transformation yields accurate results that converge to analytical results. They also indicate that an approximated transformation retaining only several per cent of all the eigenvectors produces accurate results in the lower frequency range that is generally of interest in engineering analyses. Therefore, the proposed algorithm is judged to be an efficient procedure for eigenvalue problems of fully incompressible materials. This co‐ordinate‐transformation scheme, derived from the system stiffness matrix, will be applicable to static incompressible problems. Copyright © 2003 John Wiley & Sons, Ltd.

Effects of elastic property of the wall on flow characteristics through arterial stenoses

Hemodynamic characteristics of blood flow through arterial stenoses are numerically investigated in this work. The blood is assumed as a Newtonian fluid and the pulsatile nature of flow is modeled by using measured values of the flowrate and pressure for the canine femoral artery. An isotropic elastic and incompressible material is assumed for the wall at each axial section, but a non-uniform distribution of the shear modulus in axial direction is used to model the high stiffness of the wall at the stenosis location. Full Navier equations for a thick wall are used as the governing equations for the wall displacements. A continuous grid extending over the flow field and the wall is considered and governing equations are transformed for use in the computational domain. Discretized forms of the transformed wall and flow equations, which are coupled through the boundary conditions at their interface, are obtained by control volume method and simultaneously solved using the well-known SIMPLER algorithm. To study the effects of wall deformability, solutions are obtained for both rigid and elastic walls. The results indicate that deformability of the wall causes an increase in the time average of pressure drop, but a decrease in the maximum wall shear stress. Displacement and stress distributions in the wall are presented.

Damage Induced Stress-Softening in the Torsion, Extension and Inflation of a Cylindrical Tube

The non-homogeneous deformation of combined torsion, extension and inflation of a stress-softening cylindrical tube is discussed within the framework of a theory of isotropic stress-softening in incompressible isotropic materials. The theory is based on the idea that the stress-softening material is an inelastic material that remembers only the maximum previous deformation to which it has been subjected. This selective memory dependence is incorporated within general material response functions that are monotone decreasing functions of a stress-softening variable; the latter is a monotone increasing function of the maximum previous strain experienced by the material. Results demonstrating the effects of stress-softening are obtained for general stress-softening materials in combined torsion, extension and inflation. A special analytical model is used to illustrate some general results as well as to provide graphical examples. By deforming a cylinder successively, first in simple torsion and then in uniaxial extension, the concept of deformation-induced inhomogeneity is presented. Finally, it is shown that the overall effects of stress-softening in pure torsion are much smaller than the corresponding effects in uniaxial extension.

Vibrations of a body supported by shear mountings of incompressible material with memory

Constitutive equations of materials with memory are applied in the study of mechanical response of rubber-like materials in a simple shearing deformation superimposed on a static longitudinal stretch. The Cauchy stress is assumed to be the sum of the contribution of internal elastic forces derived from a strain energy and internal dissipative forces, which depend on previous deformations with an effective memory that is governed by two parameters: the modulus and the time of relaxation. Creep and recovery processes are analyzed for incompressible isotropic materials with various strain energies (Mooney–Rivlin, quadratic, Gent). The finite amplitude vibrations (free and forced with sinusoidal excitation) of a rigid body supported by incompressible simple shearing mountings are investigated. The governing equations have non-linear terms due to the elastic part of the Cauchy stress tensor. Then it is always possible to determine the equivalent discrete model with spring and dashpot. The more relevant features, in the case of cubic non-linearity, which corresponds to the quadratic strain energy, are obtained by numerical integration and perturbation methods. The influence of the relaxation time is shown in the frequency response curves and in the related “backbone curve”, which presents a forked shape. As already observed by Fosdick et al. (Int. J. Solids Struct. 35(5–6) (1998a) 403–420; Int. J. Nonlinear Mech. 33(3) (1998b) 447–459), in their linear analysis of axial vibrations, this behavior suggests potential applications as adaptive damping devices. Indeed, several experimental results show that the values of the internal damping parameters are affected by changes in temperature and by action of electric fields.

Pure azimuthal shear of isotropic, incompressible hyperelastic materials with limiting chain extensibility

The purpose of this research is to investigate the pure azimuthal shear problem for a circular cylindrical tube composed of isotropic hyperelastic incompressible materials with limiting chain extensibility. Three popular models that account for hardening at large deformations are examined. These involve a strain-energy density which depends only on the first invariant of the Cauchy–Green tensor. In the limit as a polymeric chain extensibility tends to infinity, all of these models reduce to the classical neo-Hookean form. The stress fields and axial displacements are characterized for each of these models. Explicit closed-form analytic expressions are obtained for two of the strain-energy densities considered. The results are compared with one another and with the predictions of the neo-Hookean model.

The Mullins effect in equibiaxial deformation

Common experience teaches that the pressure required to inflate a balloon is noticeably reduced by prestretching it several times prior to its primary inflation. This preconditioning phenomenon is known as stress-softening and often is called the Mullins effect. A theory of stress-softening in incompressible isotropic materials is applied to study this effect in an equibiaxial extension for which some general results are presented. It is shown, for example, that effects of stress-softening in a simple uniaxial compression can be adduced from those demonstrated for equibiaxial extension under plane stress. The general equibiaxial results are applied to study the Mullins effect in the inflation of a spherical membrane. The stress-softening phenomenon in cyclic inflation and deflation of the balloon is investigated for Mooney-Rivlin and biotissue parent material models. It is shown analytically that the effect of preconditioning in uniaxial extension is to significantly reduce the pressure required to first inflate a balloon to an equivalent strain intensity. This result characterizes the familiar softening phenomenon associated with balloon inflation.

A theory of stress-softening in incompressible isotropic materials

A general theory of isotropic stress-softening in incompressible isotropic materials is developed. The principal idea is that a stress-softening material is an inelastic material that has selective memory of only the maximum previous deformation to which it is subjected. This memory dependence is incorporated within general material response functions that are monotone decreasing functions of a stress-softening variable, which is a monotone increasing function of the maximum previous strain experienced by the material. A loading criterion is introduced to identify when the material is loaded along its virgin deformation path where the maximum previous strain is its current value, and to identify when it is unloaded to deform subsequently as an ideal isotropic elastic material in both elastic loading and unloading, so long as the maximum previous strain is not exceeded. The effect of loading from a configuration of maximum previous strain is to further stress-soften the material. Results demonstrating the effects of stress-softening are obtained for general isotropic stress-softening materials in simple uniaxial extension and in simple shear. A simplified analytical model together with a special softening function are introduced to illustrate some general results and to provide specific analytical and graphical examples. Both general and model-specific analytical results obtained for simple uniaxial extension are shown to be consistent with the overall ideal phenomenological behavior exhibited in experiments by others on stress-softening in simple tension and compression. Similar but totally new results for simple shear are derived, and their relation to effects in simple tension are discussed. It is demonstrated that the larger effect of softening occurs in the simple uniaxial extension, the effect in even a gross equivalent simple shear being small. All results are obtained from general three-dimensional constitutive equations.

Simple torsion of isotropic, hyperelastic, incompressible materials with limiting chain extensibility

The purpose of this research is to investigate the simple torsion problem for a solid circular cylinder composed of isotropic hyperelastic incompressible materials with limiting chain extensibil- ity. Three popular models that account for hardening at large deformations are examined. These models involve a strain-energy density which depends only on the first invariant of the Cauchy- Green tensor. In the limit as a polymeric chain extensibility tends to infinity, all of these models reduce to the classical neo-Hookean form. The main mechanical quantities of interest in the torsion problem are obtained in closed form. In this way, it is shown that the torsional response of all three materials is similar. While the predictions of the models agree qualitatively with experimental data, the quantitative agreement is poor as is the case for the neo-Hookean material. In fact, by using a global universal relation, it is shown that the experimental data cannot be predicted quantitatively by any strain-energy density which depends solely on the first invariant. It is shown that a modification of the strain energies to include a term linear in the second invariant can be used to remedy this defect. Whether the modified strain-energies, which reflect material hardening, are a feasible alternative to the classic Mooney-Rivlin model remains an open question which can be resolved only by large strain experiments.

Pure Axial Shear of Isotropic, Incompressible Nonlinearly Elastic Materials with Limiting Chain Extensibility

The purpose of this research is to investigate the pure axial shear problem for a circular cylindrical tube composed of isotropic hyperelastic incompressible materials with limiting chain extensibility. Two popular models that account for hardening at large deformations are examined. These involve a strain-energy density which depends only on the first invariant of the Cauchy–Green tensor. In the limit as a polymeric chain extensibility tends to infinity, all of these models reduce to the classical neo-Hookean form. The stress fields and axial displacements are characterized for each of these models. Explicit closed-form analytic expressions are obtained. The results are compared with one another and with the predictions of the neo-Hookean model.

Inflation of a Thick-Walled Shell Which Exhibits Stress Softening

The Mullins effect (Mullins, 1947), also known as stress softening, is exhibited by certain rubberlike materials and refers to changes of the mechanical properties, due to prior deformation. Johnson and Beatty (1995) have investigated the Mullins effect in equibiaxial tension by performing cycles of static inflation and deflation experiments on latex balloons. These experiments show that stress softening results in a decrease in the pressure necessary to inflate a balloon, and in addition, indicate inelastic effects of hysteresis and permanent set. The objective of this paper is to investigate the finite deformation static inflation from the virgin state, followed by quasi-static removal of the internal pressure, of a thick-walled homogeneous spherical shell composed of an incompressible isotropic rubberlike material which exhibits stress softening and permanent set. Since the initial inflation of the shell, due to application of an internal pressure, does not result in a homogeneous deformation, a state of residual stress is present after complete removal of the internal pressure. A procedure is presented for the determination of the response of the shell for the first cycle of inflation and deflation from the virgin state, and the analysis includes strain softening and the inelastic effects of hysteresis and permanent set. It is assumed that, for the initial static inflation of the shell from the virgin state, the internal pressure and stress distribution for a monotonically increasing internal or external radius are the same as for a hyperelastic shell, and also that the magnitude of the permanent set of an element of the material is related monotonically to the deformation at the end of the inflation.

Universal Deformations for a Class of Compressible Isotropic Hyperelastic Materials

Differential conditions are derived for a smooth deformation to be universal for a class of isotropic hyperelastic materials that we regard as a compressible variant (a notion we make precise) of Mooney–Rivlin’s class, and that includes the materials studied originally by Tolotti in 1943 and later, independently, by Blatz. The collection of all universal deformations for an incompressible material class is shown to contain, modulo a uniform dilation, all the universal deformations for its compressible variants. As an application of this result, by searching the known families of universal deformations for all incompressible isotropic materials, a nontrivial universal deformation for Tolotti materials is found.

Stability of thick spherical shells

The pressure-radius relation of spherical rubber balloons has been derived and its stability behavior investigated before. In this work, we show that similar results remain valid for thick spherical shells of Mooney-Rivlin materials. In addition, we show that eversion of a spherical shell is possible for any incompressible isotropic materials if the shell is not too thick.

LARGE ELASTIC COMPRESSION OF FINITE RECTANGULAR BLOCKS OF RUBBER

SUMMARY For the problem of large elastic compression of a finite rectangular rubber block which has bonded metal plates to its upper and lower surfaces, a new load-deflection relation is derived assuming an isotropic incompressible Mooney material and that the pointwise vanishing of the stress on the free faces can be replaced by the vanishing of the force resultant on these faces. The load-deflection relation so obtained is derived from a fully three-dimensional deformation which is a generalization of a plane-strain deformation given previously for infinitely long rectangular rubber pads. The 'finite' and 'infinite' load-deflection relations are compared with experimental results and with that predicted by the conventional engineering approximation deduced from the so-called 'shape-factor' method.