Class Of Compressible Hyperelastic Materials Research Articles

Finite deformation of a blunt crack represented by a parabolic cavity in a harmonic solid

We study the finite plane deformation of a blunt crack represented by a parabolic cavity in a particular class of compressible hyperelastic materials of harmonic type. A complete solution to the blunt crack problem is derived using complex variable techniques. Elementary expressions of the stresses along the real axis ahead of the vertex of the parabola and the finite deformation of the parabola are obtained. When the nominal stress intensity factors approach zero, the stress distributions along the real axis are identical to those in linear elasticity. When the nominal stress intensity factors approach infinity, the tractions along the real axis are inversely proportional to the square root of the horizontal coordinate. A finite and non-zero mode II nominal stress intensity factor will induce both shear and normal stresses along the real axis, a phenomenon quite different from that for linear elasticity.

Uniformity of stresses inside a parabolic inhomogeneity in finite plane elastostatics

Using complex variable techniques, we establish the uniformity of the internal Piola stresses within a parabolic inhomogeneity embedded in a matrix from a particular class of compressible hyperelastic materials of harmonic type subjected to uniform remote Piola stresses. In addition, two Piola stress components inside the inhomogeneity simply coincide with their remotely prescribed counterparts in the matrix, while the two other internal Piola stress components can be determined via the solution of a quadratic equation, and are found to be independent of the single geometric parameter characterizing the inhomogeneity–matrix system. We derive a complete solution to the corresponding boundary value problem describing the mechanical behavior of the composite.

A harmonic rigid inclusion loaded by a couple in finite plane elasticity

We consider a rigid inclusion embedded in a matrix from a particular class of compressible hyperelastic materials (so-called harmonic materials) subjected to uniform remote stresses. The inclusion is loaded by a couple, and the inclusion–matrix system undergoes finite plane deformations. We examine whether it is possible to design the shape of the inclusion to achieve the well-known ‘harmonic field condition’ in which the sum of the normal stresses in the uncut matrix remains undisturbed after the introduction of the inclusion. Indeed, we find that the only such rigid inclusion is elliptical. Moreover, we show that when the remote loading is symmetric, the harmonic field condition is satisfied only when two further conditions on prescribed geometric and material parameters are met: One condition requires a relatively simple linear relationship between the moment of the couple and the remote shear stress; the second condition involves a rather complex nonlinear relationship between the two remote normal stresses and the moment of the couple. Once these conditions are satisfied, the interfacial normal and tangential stresses as well as the hoop stress are uniformly distributed along the boundary of the rigid inclusion. Furthermore, the interfacial normal and hoop stresses are dependent only on the remote normal stresses and the aspect ratio of the elliptical inclusion, whereas the interfacial tangential stress depends only on the moment of the couple and the area of the inclusion.

Surface Green’s functions in finite plane elastostatics of harmonic materials

The closed-form representations of surface Green’s functions corresponding to the action of a concentrated force applied at the boundary of a region occupied by a particular class of compressible hyperelastic materials of harmonic type, has been derived. In our analysis, we consider both a bounded region in the form of a circular disk and an unbounded region with either an elliptical hole or a parabolic boundary.

A concentrated couple inside or outside a circular inclusion in finite plane elastostatics

We consider a circular elastic inclusion embedded in an infinite matrix each from a particular class of compressible hyperelastic materials of harmonic type. A concentrated couple is applied either inside the circular inclusion or in the matrix. Closed-form solutions of the corresponding boundary value problems are obtained using complex variable methods, in particular the principle of analytic continuation. Our analysis reveals several interesting conclusions including: the sum σ11+σ22 of the normal stresses inside the inclusion remains constant when the concentrated couple is located in the surrounding matrix; the sum of the normal stresses is zero everywhere in the matrix when the concentrated couple is located inside the inclusion.

On constitutive models of finite elasticity with possible zero apparent Poisson’s ratio

The idea in this paper is to build a class of constitutive equations for highly compressible isotropic materials that, among others, are capable to describe a zero apparent Poisson’s ratio in the whole finite strain range, not only for moderate straining. This remarkable property is, for instance, observed in many soft materials with micro-structures such as sponges and polymeric foams with high porosities. It would then be suitable to describe their behavior within a macroscopic modeling framework. More specifically, herein by means of elementary considerations, we deduce adequate forms of strain-energy functions that are a priori decomposed into purely volumetric and volume-preserving parts. A class of compressible hyperelastic materials of the general Odgen type is obtained. It can consequently be specialized, for instance, to neo-Hookean, Mooney–Rivlin, and Varga’s model types as well. Furthermore, for the elastic parameters, a connection with the limiting case of linear elasticity is made whenever possible, in particular with the classical Poisson’s ratio, and with the bulk to shear moduli ratio.

Neutrality of an elliptical inhomogeneity in finite plane elastostatics of harmonic materials

We examine the neutrality of an elliptical inhomogeneity embedded in a particular class of compressible hyperelastic materials of harmonic type when a uniform Piola stress field is prescribed in the surrounding matrix. The present method is based on a spring-type imperfect interface model and on the proper choice of imperfect interface function, which realizes the same degree of imperfection in both normal and tangential directions. The analysis indicates that, in general, the neutral shape and the single imperfect interface function are both dependent on the magnitude of the prescribed stress field.

A nanosized circular inhomogeneity in finite plane elastostatics

We consider the deformation of a nanocomposite by incorporating the effect of surface elasticity on the finite plane deformations of a circular inhomogeneity embedded in a particular class of compressible hyperelastic materials of harmonic type subjected to uniform remote Piola stresses. We incorporate the surface mechanics by using a version of the continuum-based surface/interface model of Gurtin and Murdoch. A complete solution is derived by reducing the original boundary value problem to two coupled first-order differential equations which can be solved analytically. The solution clearly demonstrates that the stress fields in the composite are size dependent and that the stress field inside a circular inhomogeneity is, in general, non-uniform. Two special cases which admit an internal uniform, yet size-dependent stress field, are discussed.

Harmonic three-phase circular inclusions in finite elasticity

We study the exterior stress field in a three-phase circular inclusion which is bonded to the surrounding matrix through an intermediate interphase layer. All three phases belong to a particular class of compressible hyperelastic materials of harmonic type. We focus on the design of a harmonic elastic inclusion which by definition, does not disturb the sum of the normal stresses in the surrounding matrix. We show that in order to make the coated inclusion harmonic, certain inequalities concerning the material and geometric parameters of the three-phase composite must first be satisfied. The corresponding remote loading parameters can then be uniquely determined while keeping the associated phase angles arbitrary. Our results allow for both uniform and non-uniform remote loading. We show that the stress field inside the inclusion is uniform when the remote loading is uniform.

Cylindrical Cavitation in a Class of Compressible Hyper-Elastic Materials

The finite deformation about the radial direction is examined for a solid cylinder composed of a class of compressible hyper-elastic materials. Some interesting conclusions are proposed by using the theoretical solutions and the numerical simulations. There exists a critical stretch such that the cylinder would remain a solid one if the radial stretch does not exceed the critical value and that a cylindrical cavity would form at the axial line of the cylinder if the radial stretch exceeds the critical value. The phenomena of concentration and catastrophe of radial and circumferential stresses further show that cavitation in compressible hyper-elastic cylinder coincides with the actual physical grounds.

Hill’s class of compressible elastic materials and finite bending problems: Exact solutions in unified form

Hill (1978) proposed a natural extension of Hooke’s law to finite deformations. With all Seth-Hill finite strains, Hill’s natural extension presents a broad class of compressible hyperelastic materials over the whole deformation range. We show that a number of known Hookean type finite hyperelasticity models are included as particular cases in Hill’s class and that Bell’s and Ericksen’s constraints may be derived as natural consequences from Hill’s class subjected to internal constraints. Also we present a unified study of finite bending problems for elastic Hill materials. To date exact results are available for certain particular classes of compressible elastic materials, which do not cover Hill’s class. Here, with a novel idea of circumventing the strong nonlinearity we show that it is possible to derive exact solutions in unified form for the whole class of elastic Hill materials. Reduced results are also given for cases subjected to internal constraints.

Finite plane deformations of a three-phase circular inhomogeneity-matrix system

We consider finite plane deformations of a three-phase circular inhomogeneity-matrix system in which the inhomogeneity, the interphase layer and the matrix belong to the same class of compressible hyperelastic materials of harmonic-type but with each phase possessing its own distinct material properties. We obtain the complete solution when the system is subjected to general classes of remote (Piola) stress, specifically, remote stress distributions characterized by stress functions described by general polynomials of order n ⩾ 1 in the corresponding complex variable z used to describe the matrix. As a particular case of the aforementioned analysis, we establish an Eshelby-type result namely that, for this class of harmonic materials, a three-phase circular inhomogeneity under uniform remote stress and eigenstrain, admits an internal uniform stress field when subjected to plane deformations.

Designing an inhomogeneity with uniform interior stress in finite plane elastostatics

We consider an inhomogeneity-matrix system from a particular class of compressible hyperelastic materials of harmonic-type undergoing finite plane deformations. We discuss the idea that by adjusting the remote (Piola) stress we can design the shape of the inhomogeneity in such a way that the interior (Piola) stress distribution remains uniform. In fact, using complex variable techniques, we show that a uniform (Piola) stress distribution can be achieved inside the inhomogeneity when the system is subjected to linear remote loading, in which case, the inhomogeneity is necessarily hypotrochoidal with three cusps. We obtain the complete solution of the corresponding inhomogeneity-matrix system in this case. In addition, we consider the more general case when the system is subjected to nonuniform remote loading characterized by stress functions in the form of nth degree polynomials in the complex variable z describing the matrix. In this case, by finding the complete solution of the system, we show that we can again obtain uniform stress inside a hypotrochoidal inhomogeneity with n + 1 cusps. Finally, we illustrate our results with an example.

Harmonic Shapes in Finite Elasticity

We consider the design of harmonic shapes in a particular class of compressible hyperelastic materials of harmonic-type undergoing finite plane deformations. Harmonic shapes are characterized by a `harmonicity condition' imposed on the final stress field. The `harmonicity condition' used in this paper is a generalization of the original condition used in the corresponding problems of linear elasticity: specifically, that the first invariant of the stress tensor (i.e. the sum of the normal stresses) in the original stress field remains unchanged everywhere after the introduction of the harmonic hole or inclusion. Using complex variable techniques, we formulate the general equations for the identification of harmonic shapes in a material of harmonic-type subjected to plane deformations. Under the assumption of uniform biaxial loading, we identify shapes of harmonic rigid inclusions and harmonic free holes. Finally, comparisons are drawn to the analogous cases from linear elasticity.

An Elliptic Inhomogeneity Subjected to a General Class of Nonuniform Remote Loadings in Finite Elasticity

We consider a composite material containing an elastic elliptic inhomogeneity which is assumed to be perfectly bonded to a surrounding matrix of similar elastic material. In fact, both the inhomogeneity and the matrix belong to the same class of compressible hyperelastic materials of harmonic-type but each has its own distinct material properties. We consider finite plane deformations of the inhomogeneity-matrix system and obtain the complete solution when the system is subjected to classes of nonuniform remote (Piola) stress characterized by stress functions described by general polynomials of order n ≥ 1 in the corresponding complex variable z used to describe the matrix.

A circular inhomogeneity subjected to non-uniform remote loading in finite plane elastostatics

We consider an inhomogeneity–matrix system from a particular class of compressible hyperelastic materials of harmonic-type undergoing finite plane deformations. We obtain the complete solution for a perfectly bonded circular inhomogeneity when the system is subjected to non-uniform remote stress characterized by stress functions described by general polynomials of order n ⩾ 1 in the corresponding complex-variable z used to describe the matrix.

The Contact Problem in a Compressible Hyperelastic Material

We consider the contact problem for a particular class of compressible hyperelastic materials of harmonic type undergoing finite plane deformations. Using complex variable techniques, we derive subsidiary results concerning a half-plane problem corresponding to this class of materials. Using these results, we solve the contact problem for a harmonic material in the case of a uniform load acting on a finite area. Finally, we show how we can then deduce the corresponding results for the case of a point load.

A unified exact analysis for the Poynting effects of cylindrical tubes made of Hill’s class of Hookean compressible elastic materials at finite strain

A direct, natural extension of Hooke’s law to finite strain was achieved by R. Hill in 1978, employing the notion of work-conjugate measures of stress and strain. With Seth-Hill (Doyle-Ericksen) class of finite strain measures, this extension actually defines a broad class of compressible hyperelastic materials at finite strain, each of which retains the simple linear structure of Hooke’s law as stress–strain relationship. Several known simple elasticity models at finite strain are included as its particular examples. With a novel idea of utilizing a suitable parametric variable, here we present a unified study of the free-end torsion problem (Poynting effects) of thin-walled cylindrical tubes made of the foregoing Hill’s class of Hookean type hyperelastic materials. We show that it is possible to derive a unified exact solution to the nonlinear coupling equations relating the torque (the shear stress) and the controlling deformation quantities including, in particular, the axial length change. Discussions and comparisons concerning various Hookean type elasticity models are made based on the exact solution obtained.

Harmonic Shapes in Finite Elasticity Under Nonuniform Loading

We investigate the classic (inverse) problem concerned with the design of so-called harmonic shapes for an elastic material undergoing finite plane deformations. In particular, we show how to identify such shapes for a particular class of compressible hyperelastic materials of harmonic type. The “harmonic condition,” in which the sum of the normal stresses in the original stress field remains unchanged everywhere after the introduction of the harmonic hole or inclusion, is imposed on the final stress field. Using complex variable techniques, we identify particular harmonic shapes arising when the material is subjected nonuniform (remote) loading and discuss conditions for the existence of such shapes.

Exact solution for cavitated bifurcation for compressible hyperelastic materials

In this paper, a new exact analytic solution for spherical cavitated bifurcation is presented for a class of compressible hyperelastic materials. The strain energy density of the materials is assumed to be a linear function of three strain invariants, which may be regarded as a first-order approximation to the general strain energy density near the reference configuration, and also may satisfy certain constitutive inequalities of hyperelastic materials. An explicit formula for the critical stretch for the cavity nucleation and a simple bifurcation solution for the deformed cavity radius which describes the cavity growth are obtained. The potential energy associated with the cavitated deformation is examined. It is always lower than that associated with the homogeneous deformation, thus the state of cavitated deformation is relatively stable. On the basis of the presented analytic solutions for the stretches and stresses, the catastrophic transition of deformation and the jumping of stresses for the cavitation are discussed in detail. The boundary layers of the displacements, the strain energy distribution and stresses near the formed cavity wall are observed. These investigations illustrate that cavitation reflects a local behaviour of materials.