Cornelius O. Horgan

Recent Developments Concerning Saint-Venant's Principle

This chapter provides an overview of the recent developments concerning Saint-Venants principle. The task of determining, within the framework of the linear theory of elasticity, the stresses and displacements in an elastic cylinder in equilibrium, under the action of loads that arise solely from tractions applied to its plane ends has come to be called Saint- Venants problem. Saint-Venants construction does not permit the arbitrary preassignment of the point-by-point variation of the end tractions giving rise to these forces and moments; indeed, this variation is essentially determined as a consequence of the special assumptions made in connection with his so-called semi-inverse procedure. The early work of Saint-Venant and Boussinesq furnished the seeds from which grew a large number of more general assertions, most referring to elastic solids of arbitrary shape and many being rather imprecise, concerning the effect on stresses within the body of replacing the tractions acting over a portion of its surface by statically equivalent ones. Such propositions usually went by the name of Saint-Venunts principle, despite the fact that Saint-Venants original conjecture was intended to apply only to cylinders. This chapter discusses in detail about flow in a cylinder, a representation for the exact solution, and energy decay for other linear elliptic second-order problem. Linear elastostatic problems are also stated in the chapter.

The Pressurized Hollow Cylinder or Disk Problem for Functionally Graded Isotropic Linearly Elastic Materials

The purpose of this research is to investigate the effects of material inhomogeneity on the response of linearly elastic isotropic hollow circular cylinders or disks under uniform internal or external pressure. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e., materials with spatially varying properties tailored to satisfy particular engineering applications. The analog of the classic Lamé problem for a pressurized homogeneous isotropic hollow circular cylinder or disk is considered. The special case of a body with Youngs modulus depending on the radial coordinate only, and with constant Poissons ratio, is examined. It is shown that the stress response of the inhomogeneous cylinder (or disk) is significantly different from that of the homogeneous body. For example, the maximum hoop stress does not, in general, occur on the inner surface in contrast with the situation for the homogeneous material. The results are illustrated using a specific radially inhomogeneous material model for which explicit exact solutions are obtained.

Korn's inequalities and their applications in continuum mechanics

Korn’s inequalities have played a central role in the development of linear elasticity, not only in connection with the basic theoretical issues such as existence and uniqueness, but also in a variety of applications. The Korn inequalities, and other related inequalities for integrals of quadratic functionals, also arise in the analysis of viscous incompressible fluid flow. The dimensionless optimal constants appearing in these inequalities, the Korn constants, depend only on the shape of the domains of concern. Information on the geometric dependence of these constants is essential in applications. In this review article, we summarize the major results on Korn’s inequalities for bounded domains in two and three dimensions, with emphasis on results concerning the Korn constants. Some applications in continuum mechanics are also described.

A bifurcation problem for a compressible nonlinearly elastic medium: growth of a micro-void

In this paper, we carry out an explicit analysis of a bifurcation problem for a solid circular cylinder composed of a particularcompressible nonlinearly elastic material. This problem is concerned with the bifurcation of a solid body into a configuration involving an internal cavity. A discussion of its physical interpretation is then carried out. In particular, it is shown that this model may be used to describe the nucleation of a void from apre-existing micro-void.

Anti-plane shear deformations in linear and nonlinear solid mechanics

The intent of this expository paper is to draw the attention of the applied mathematics community to an interesting two-dimensional mathematical model arising in solid mechanics involving a single second-order linear or quasi-linear partial differential equation. This model has the virtue of relative mathematical simplicity without loss of essential physical relevance. Anti-plane shear deformations are one of the simplest classes of deformations that solids can undergo. In anti-plane shear (or longitudinal shear, generalized shear) of a cylindrical body, the displacement is parallel to the generators of the cylinder and is independent of the axial coordinate. Thus anti-plane shear, with just a single scalar axial displacement field, may be viewed as complementary to the more complicated (yet perhaps more familiar) plane strain deformation, with its two in-plane displacements. In recent years, considerable attention has been paid to the analysis of anti-plane shear deformations within the context of variou...

A theory of stress softening of elastomers based on finite chain extensibility

In this paper we develop a theory to describe the Mullins effect in rubber–like solids, based on the notion of limiting chain extensibility associated with the Gent model of rubber elasticity. We relate the theory to the mechanisms of network alteration and to the pseudo–elasticity theory of the Mullins effect. The inherently anisotropic nature of the Mullins effect is also discussed.

Saint-Venant’s Principle and End Effects in Anisotropic Elasticity

The purpose of this paper is to draw attention to the fact that the routine application of Saint-Venant’s principle in the solution of elasticity problems involving highly anisotropic or composite materials is not justified in general. This is illustrated in the context of the plane problem of elasticity for an anisotropic rectangular strip loaded only on the short ends. For highly anisotropic transversely isotropic materials, the slow decay of end effects is demonstrated using a method involving self-equilibrating eigenfunctions. For a graphite/epoxy composite, for example, the characteristic decay length is shown to be approximately four times that for an isotropic material. The results have implications in the accurate measurement of mechanical properties of anisotropic materials.

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

Molecular constitutive models for rubber based on non-Gaussian statistics generally involve the inverse Langevin function. Such models are widely used since they successfully capture the typical strain-hardening at large strains. Limiting chain extensibility constitutive models have also been developed on using phenomenological continuum mechanics approaches. One such model, the Gent model for incompressible isotropic hyperelastic materials, is particularly simple. The strain-energy density in the Gent model depends only on the first invariant I1 of the Cauchy–Green strain tensor, is a simple logarithmic function of I1 and involves just two material parameters, the shear modulus μ and a parameter Jm which measures a limiting value for I1−3 reflecting limiting chain extensibility. In this note, we show that the Gent phenomenological model is a very accurate approximation to a molecular based stretch averaged full-network model involving the inverse Langevin function. It is shown that the Gent model is closely related to that obtained by using a Padè approximant for this function. The constants μ and Jm in the Gent model are given in terms of microscopic properties. Since the Gent model is remarkably simple, and since analytic closed-form solutions to several benchmark boundary-value problems have been obtained recently on using this model, it is thus an attractive alternative to the comparatively complicated molecular models for incompressible rubber involving the inverse Langevin function.

Phenomenological hyperelastic strain-stiffening constitutive models for rubber

Abstract Many rubber-like materials and soft biological tissues exhibit a significant stiffening or hardening in their stress-strain curves at large strains. The accurate modeling of this phenomeno...

The Stress Response of Functionally Graded Isotropic Linearly Elastic Rotating Disks

The purpose of this research is to investigate the effects of material inhomogeneity on the response of linearly elastic isotropic solid circular disks or cylinders, rotating at constant angular velocity about a central axis. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e., materials with spatially varying properties tailored to satisfy particular engineering applications. The analog of the classic problem for a homogeneous isotropic rotating solid disk or cylinder is considered. The special case of a body with Youngs modulus depending on the radial coordinate only, and with constant Poissons ratio, is examined. For the case when the Youngs modulus has a power-law dependence on the radial coordinate, explicit exact solutions are obtained. It is shown that the stress response of the inhomogeneous disk (or cylinder) is significantly different from that of the homogeneous body. For example, the maximum radial and hoop stresses do not, in general, occur at the center as in the case for the homogeneous material. Furthermore, for the case where the Youngs modulus increases with radial distance from the center, it is shown that radially symmetric solutions exist provided the rate of growth of the Youngs modulus is, at most, cubic in the radial variable. It is also shown for the general inhomogeneous isotropic case how the material inhomogeneity may be tailored so that the radial and hoop stress are identical throughout the disk.