Non-slipping JKR model for transversely isotropic materials

A generalized plane strain JKR model is established for non-slipping adhesive contact between an elastic transversely isotropic cylinder and a dissimilar elastic transversely isotropic half plane, in which a pulling force acts on the cylinder with the pulling direction at an angle inclined to the contact interface. Full-coupled solutions are obtained through the Griffith energy balance between elastic and surface energies. The analysis shows that, for a special case, i.e., the direction of pulling normal to the contact interface, the full-coupled solution can be approximated by a non-oscillatory one, in which the critical pull-off force, pull-off contact half-width and adhesion strength can be expressed explicitly. For the other cases, i.e., the direction of pulling inclined to the contact interface, tangential tractions have significant effects on the pull-off process, it should be described by an exact full-coupled solution. The elastic anisotropy leads to an orientation-dependent pull-off force and adhesion strength. This study could not only supply an exact solution to the generalized JKR model of transversely isotropic materials, but also suggest a reversible adhesion sensor designed by transversely isotropic materials, such as PZT or fiber-reinforced materials with parallel fibers.     

Finite strain solutions in compressible isotropic elasticity

Three classes of compressible isotropic elastic solids are introduced, for each of which the strain energy, expressed as a function of the three principal invariants of the stretch tensors, is linear in two of its arguments and nonlinear in the third argument. One of these is the class of harmonic materials. Several deformation fields are examined, for which the governing equations, for general compressible isotropic elastic response, reduce to a nonlinear ordinary differential equation. For the three special classes of materials, this differential equation may be solved in closed form, giving a deformation field which is independent of the material response function, or its solution may be reduced to a single quadrature, involving the nonlinear material response function.     

A class of universal relations in isotropic elasticity theory

A class of universal relations for isotropic elastic materials is described by the tensor equationTB = BT. This simple rule yields at most three component relations which are the generators of many known universal relations for isotropic elasticity theory, including the well-known universal rule for a simple shear. Universal relations for four families of nonhomogeneous deformations known to be controllable in every incompressible, homogeneous and isotropic elastic material are exhibited. These same universal relations may hold for special compressible materials. New universal relations for a homogeneous controllable shear, a nonhomogeneous shear, and a variable extension are derived. The general universal relation for an arbitrary isotropic tensor function of a symmetric tensor also is noted.     

Elastic study on singularities interacting with interfaces using alternating technique: Part II. Isotropic trimaterial

Singularity problems in an isotropic trimaterial are analyzed by the same procedure as in an anisotropic trimaterial of Part I [Int. J. Solids Struct. 39, 943–957]. `Trimaterial' denotes an infinite body composed of three dissimilar materials bonded along two parallel interfaces. Linear elastic isotropic materials under plane deformations are assumed, in which the plane of deformation is perpendicular to the two parallel interface planes, and thus Muskhelishvili's complex potentials are used. The method of analytic continuation is alternatively applied across the two parallel interfaces in order to derive the trimaterial solution in a series form from the corresponding homogeneous solution. A variety of problems, e.g. a bimaterial (including a half-plane problem), a finite thin film on semi-infinite substrate, and a finite strip of thin film, etc, can be analyzed as special cases of the present study. A film/substrate structure with a dislocation is exemplified to verify the usefulness of the solutions obtained.     

Constitutive Models for Almost Incompressible Isotropic Elastic Rubber-like Materials

A simple constitutive model is proposed for slightly compressible (or almost incompressible) non-linearly elastic materials that are homogeneous and isotropic. Experimental data for simple tension suggest that there is a power-law kinematic relationship between the stretches for large classes of such materials. It is shown that a common constitutive model for these materials does not, in general, capture this effect. The most general constitutive model giving rise to such a power-law relationship is then obtained. A special case yields the well-known Blatz–Ko model for compressible rubber. The behavior in biaxial tension and pure shear is also discussed.     

Plane-Strain Problems for a Class of Gradient Elasticity Models��A Stress Function Approach

The plane strain problem is analyzed in detail for a class of isotropic, compressible, linearly elastic materials with a strain energy density function that depends on both the strain tensor ?? and its spatial gradient ???. The appropriate Airy stress-functions and double-stress-functions are identified and the corresponding boundary value problem is formulated. The problem of an annulus loaded by an internal and an external pressure is solved.     

A family of solutions describing plane strain cylindrical inflation in finite compressible elasticity

Within the context of finite, compressible, isotropic elasticity, a family of solutions describing plane strain cylindrical inflation of cylindrical shells is obtained for a class of materials that includes both the harmonic and Varga materials. Additionally it is shown that the class of materials chsen is the largest class of materials for which the family of solutions is possible.     

Image singularities of Green’s functions for isotropic elastic bimaterials subjected to concentrated forces and dislocations

Green’s functions for isotropic materials in the two-dimensional problem for elastic bimaterials with perfectly bonded interface are reexamined in the present study. Although the Green’s function for an isotropic elastic bimaterial subjected to a line force or a line dislocation has been discussed by many authors, the physical meaning and the structure of the solution are not clear. In this investigation, the Green’s function for an elastic bimaterial is shown to consist of eight Green’s functions for a homogeneous infinite plane. One of the novel features is that Green’s functions for bimaterials can be expressed directly by knowing Green’s functions for the infinite plane. If the applied load is located in material 1, the solution for the half-plane of material 1 is constructed with the help of five Green’s functions corresponding to the infinite plane. However, the solution for the half-plane of material 2 only consists of three Green’s functions for the infinite plane. One of the five Green’s functions of material 1 and all the three Green’s functions of material 2 have their singularities located in the half-plane where the load is applied, and the other four image singularities of material 1 are located outside the half-plane at the same distance from the interface as that of the applied load. The nature and magnitude of the image singularities for both materials are presented explicitly from the principle of superposition, and classified according to different loads. It is known that for the problem of anisotropic bimaterials subjected to concentrated forces and dislocations, the image singularities are simply concentrated forces and dislocations with the stress singularity of order O(1/r). However, higher orders (O(1/r²) and O(1/r³)) of stress singularities are found to exist in this study for isotropic bimaterials. The highest order of the stress singularity is O(1/r³) for the image singularities of material 1, and is O(1/r²) for material 2. Using the present solution, Green’s functions associated with the problems of elastic half-plane with free and rigidly fixed boundaries, for homogeneous isotropic elastic solid, are obtained as special cases.     

Micromechanical analysis of friction anisotropy in rough elastic contacts

Computational contact homogenization approach is applied to study friction anisotropy resulting from asperity interaction in elastic contacts. Contact of rough surfaces with anisotropic roughness is considered with asperity contact at the micro scale being governed by the isotropic Coulomb friction model. Application of a micro-to-macro scale transition scheme yields a macroscopic friction model with orientation- and pressure-dependent macroscopic friction coefficient. The macroscopic slip rule is found to exhibit a weak non-associativity in the tangential plane, although the slip rule at the microscale is associated in the tangential plane. Counterintuitive effects are observed for compressible materials, in particular, for auxetic materials.     

The stability of the interface between two bodies compressed along interface cracks. 3. Exact solutions for the combined case of equal and unequal roots

The problem of the stability of the interface between two bodies is considered for the case where several plane cracks are located in the interface, and the bodies are compressed along them (along the interface of two different materials). The study is carried out for a plane problem by using the three-dimensional linearized theory of stability of deformable bodies. Complex variables and potentials of the above-mentioned linearized theory are used. This problem is reduced to the problem of linear conjugation of two analytical functions of complex variable. The exact solution of the above-mentioned problem is derived for the case where the basic equation has unequal roots for the first material and equal roots for the second material. In earlier authors' publications, the exact solutions were obtained for the cases where both materials have either equal or unequal roots. Some mechanical effects are analyzed for the general formulation of the problem (elastic, elastoplastic compressible and incompressible isotropic and orthotropic bodies). It is pointed out that, in accordance with the exact solutions, the main result and conclusions have a general form for the above-mentioned cases of roots. S.P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 6, pp. 67–77, June, 2000     

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-amplitude shear horizontal waves propagating in a pre-stressed layer between two half-spaces

The propagation of finite-amplitude time-harmonic shear horizontal waves, in a pre-stressed compressible elastic layer of finite thickness embedded between two identical compressible elastic half-spaces, is investigated. This is accomplished by combining finite-amplitude linearly polarized inhomogeneous transverse plane wave solutions in the half-spaces and finite-amplitude linearly polarized unattenuated transverse plane wave solutions in the layer. The layer and half-spaces are made of different pre-stressed compressible neo-Hookean materials. The dispersion relation which relates wave speed and wavenumber is obtained in explicit form. The special case where the interfaces between the layer and the half-spaces are principal planes of the left Cauchy–Green deformation tensor is also investigated. Numerical results are presented showing the variation of the shear horizontal wave speed with the pre-stress and the propagation angle.     

Contact interaction of two compressed electroelastic half-spaces

The contact interaction of two compressed electroelastic transversely isotropic half-spaces with different properties is studied. One of the half-spaces has an axisymmetric notch of special shape. The contact plane coincides with the plane of isotropy of the half-spaces. Explicit formulas for the contact pressure, displacements, and the size of the gap between piezoelectric half-spaces are derived. These contact characteristics for transversely isotropic and isotropic elastic half-spaces are obtained as special cases     

Deformations with discontinuous gradients in plane elastostatics of compressible solids

We investigate certain issues pertaining to plane deformations with discontinuous gradients sustained by compressible, isotropic, hyperelastic materials. Conditions on the elastic potential which are necessary and sufficient for the existence of such deformations are derived. An alternative, explicit set of criteria is deduced from these, which involves jump conditions restricting the deformation invariants on either side of the discontinuity. This result, which is expressed in terms of the local amounts of shear and dilatation, characterizes all possible two-phase states sustained by a given elastic potential. Some implications of ellipticity loss on the existence of such states are considered.     

A note on isotropic elastic response

For an important class of incompressible isotropic elastic solids, the response function for the extra stress is a (tensor-valued) function of scalar type. It is shown here that the stress response for compressible isotropic elastic solids cannot be of scalar type.     

The stability of the interface between two bodies compressed along interface cracks. 1. Exact solutions for the case of unequal roots

The buckling of the interface between two bodies is considered in the case where the interface contains several plane cracks and the bodies are compressed along them (along the interface of two different materials). The investigation is carried out for a plane problem using the three-dimensional linearized theory of stability of deformable bodies. Complex variables and potentials of the mentioned linearized theory are applied. This problem is reduced to the problem of linear conjugation of two analytical functions of complex variables. The exact solution is derived for the case of unequal roots of the basic equation. Some mechanical effects are analyzed for the general formulation of problems (elastic, elastoplastic, compressible, incompressible, isotropic, and orthotropic bodies). S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 69–79, April, 2000.     

Void collapse in an elastic solid

Radial deformations of an infinite medium surrounding a traction-free spherical cavity are considered. The body is composed of an isotropic, incompressible elastic material and is subjected to a uniform pressure at infinity. The possibility of void collapse (i.e. the void radius becoming zero at a finite value of the applied stress) is examined. This does not occur in all materials. The class of materials that do exhibit this phenomenon is determined, and for such materials, an explicit expression for the value of the applied pressure at which collapse occurs is derived. The stability of the deformation and the influence of a finite outer radius are also considered. The results are illustrated for a particular class of power-law materials. In certain respects, the present results for void collapse are complementary to Ball (1982)'s results for cavitation in an incompressible elastic material.Some brief observations on void collapse in compressible materials are made. The collapse of a void under non-symmetric conditions is also discussed by utilizing a solution obtained by Varley and Cumberbatch (1977, 1980).The results reported here were obtained in the course of an investigation supported in part by the U.S. Army Research Office.     

Hypoplasticity theory for granular materials—II: Three-dimensional axially symmetric exact solutions

In part I of this paper, we consider the governing equations of hypoplasticity theory for two-dimensional steady quasi-static plane strain compressible gravity flow and determine some exact analytical solutions applying for certain special cases. Similarly, for the three-dimensional situation considered here in part II, we undertake a similar mathematical investigation to determine some simple solutions of the governing equations for three-dimensional steady quasi-static axially symmetric compressible gravity flow for hypoplastic granular materials. We again find that for certain special cases, we are able to determine some exact solutions for the stress and velocity profiles. We comment that hypoplasticity theory generally gives rise to complicated systems of coupled non-linear differential equations, for which the determination of any analytical solutions is not a trivial matter, and that the solutions determined here might be exploited as benchmarks for full numerical schemes.     

Plane Strain Bending of Cylindrical Sectors of Admissible Compressible Hyperelastic Materials

In the theory of nonlinear elasticity of rubber-like materials, if a homogeneous isotropic compressible material is described by a strain–energy function that is a homogeneous function of the principal stretches, then the equations of equilibrium for axisymmetric deformations reduce to a separable first-order ordinary differential equation. For a particular class of such strain–energy functions, this property is used to obtain a general parametric solution to the equilibrium equation for plane strain bending of cylindrical sectors. Specification of the arbitrary function that appears in such strain–energy functions yields some parametric solutions. In some cases, the parameter can be eliminated to yield closed-form solutions in implicit or explicit form. Other possible forms for the arbitrary constitutive function that are likely to yield such solutions are also indicated.     

Axisymmetric displacement boundary value problem for a penny-shaped crack

Summary A solution is derived from equations of equilibrium in an infinite isotropic elastic solid containing a penny-shaped crack where displacements are given. Abel transforms of the second kind stress and displacement components at an arbitrary point of the solid are known in the literature in terms of jumps of stress and displacement components at a crack plane. Limiting values of these expressions at the crack plane together with the boundary conditions lead to Abel-type integral equations, which admit a closed form solution. Explicit expressions for stress and displacement components on the crack plane are obtained in terms of prescribed face displacements of crack surfaces. Some special cases of the crack surface shape functions have been given in the paper.