Mark E. Mear

Symmetric weak-form integral equation method for three-dimensional fracture analysis

A symmetric Galerkin boundary element method is developed for the analysis of linearly elastic, isotropic three-dimensional solids containing fractures. The formulation is based upon a weak-form displacement integral equation and a weak-form traction integral equation recently developed by Li and Mear (1997). These integral equations are only weakly singular, and their validity requires only that the boundary displacement data be continuous, hence, allowing standard Co elements to be employed. As part of the numerical implementation a special crack-tip element is developed which has a novel feature in that there exist degrees of freedom associated with the nodes at the crack front. As a result, a higher degree of approximation is achieved for the relevant displacement data on the crack and, further, the stress intensity factors are obtained directly in terms of the crack-front nodal data. Various examples are treated for cracks in unbounded domains and for cracks in finite domains (including both embedded and surface breaking cracks), and it is demonstrated that highly accurate results can be achieved using relatively coarse meshes.

Axisymmetric deformation of power-law solids containing a dilute concentration of aligned spheroidal voids

Abstract T he macroscopic response of an incompressible power-law matrix containing a dispersion of aligned, spheroidal voids is investigated. Attention is restricted to dilute concentrations of voids and to axisymmetric deformation of the solid. The essential step in the analysis is the solution of a kernel problem for an isolated void, and this solution is obtained accurately and efficiently using a Ritz procedure developed for this purpose. Results for macroscopic strain-rates are presented for void shapes ranging from penny-shaped cracks to infinitely long circular cylinders and for a wide range of triaxialities and matrix hardening exponents. These results are used to assess the role of void shape on the overall response of porous solids.

Cold rolling of foil

A theory of cold rolling of thin gauge strip is presented which, within the idealizations of homogeneous deformation and a constant coefficient of Coulomb friction, rigorously models the elastic deformation of the rolls and the frictional traction at the interface. In contrast with classical theories (3) it is shown that, for gauges less than a critical value, plastic reduction takes place in two zones, at entry and exit, which are separated by a neutral zone in which the rolls are compressed fiat and there is no slip between the rolls and the strip. Roll load and torque are governed by five independent non-dimensional parameters which express the influence of gauge, reduction, friction and front and back tensions. Values of load and torque have been computed (for zero front and back tensions) for a wide range of thickness, reduction and friction and have been found to collapse approximately on to a single master curve.

A boundary element method for two dimensional linear elastic fracture analysis

A boundary element method is developed for the analysis of fractures in two-dimensional solids. The solids are assumed to be linearly elastic and isotropic, and both bounded and unbounded domains are treated. The development of the boundary integral equations exploits (as usual) Somiglianas identity, but a special manipulation is carried out to ‘regularize’ certain integrals associated with the crack line. The resulting integral equations consist of the conventional ordinary boundary terms and two additional terms which can be identified as a distribution of concentrated forces and a distribution of dislocations along each crack line. The strategy for establishing the integral equations is first outlined in terms of real variables, after which complex variable techniques are adopted for the detailed development. In the numerical implementation of the formulation, the ordinary boundary integrals are treated with standard boundary element techniques, while a novel numerical procedure is developed to treat the crack line integrals. The resulting numerical procedure is used to solve several sample problems for both embedded and surface-breaking cracks, and it is shown that the technique is both accurate and efficient. The utility of the method for simulating curvilinear crack propagation is also demonstrated.

Effect of inclusion shape on the stiffness of nonlinear two-phase composites

Abstract constitutive relations are established for the axisymmetric deformation of an incompressible power-law matrix containing aligned, rigid spheroidal inclusions. The range of inclusion shapes considered is from thin disks with aspect ratios of 100: 1 to whiskers with aspect ratios of 50: 1. Results are presented for several matrix hardening exponents between n = 1 (linearly elastic) and n = 10. It is found that at a fixed matrix hardening exponent and a fixed volume fraction of inclusions, slender prolate inclusions give rise to a greater increase in stiffness than any other inclusion shape with the same or smaller aspect ratio. It is also found that the stiffening which results from a fixed volume fraction of inclusions is a strong function of matrix nonlinearity; the stiffness of the composite relative to the matrix material is much greater for a highly nonlinear matrix than for a linear matrix.

Singularity-reduced integral equations for displacement discontinuities in three-dimensional linear elastic media

A systematic procedure is followed to develop singularity-reduced integral equations for displacement discontinuities in homogeneous linear elastic media. The procedure readily reproduces and generalizes, in a unified manner, various integral equations previously developed by other means, and it leads to a new stress relation from which a general weakly-singular, weak-form traction integral equation is established. An isolated discontinuity is treated first (including, as special cases, cracks and dislocations) after which singularity-reduced integral equations are obtained for cracks in a finite domain. The first step in the development is to regularize Somigliana’s identity by utilizing a stress function for the stress fundamental solution to effect an integration by parts. The resulting integral equation is valid irrespective of the choice of stress function (as guaranteed by a certain ‘closure condition’ established for the integral operator), but certain particular forms of the stress function are introduced and discussed, including one which admits an interpretation as a ‘line discontinuity’. A singularity-reduced integral equation for the displacement gradients is then obtained by utilizing a relation between the stress function and the stress fundamental solution along with the closure condition. This construction does not rely upon a particular choice of stress function, and the final integral equation (which is a generalization of Mura’s (1963) formula) has a kernel which is a simple function of the stress fundamental solution. From this relation, singularity-reduced integral equations for the stress and traction are easily obtained. The key step in the further development is the construction of an alternative stress integral equation for which a differential operator has been ‘factored out’ of the integral. This is accomplished by, in essence, establishing a stress function for the stress field induced by the discontinuity. A weak-form traction integral equation is then readily obtained and involves a kernel which is only weakly-singular. The nonuniqueness of this kernel is discussed in detail and it is shown that, at least in a certain sense, the kernel which is given is the simplest possible. The results for an isolated discontinuity are then adapted to treat cracks in a finite domain. In doing so, emphasis is given to the development of weakly-singular, weak-form displacement and traction integral equations since these form the basis of an effective numerical procedure for fracture analysis (Li et al., 1998), and such equations are presented for both elastostatics and elastodynamics. A noteworthy aspect of the development is that there is no need to introduce Cauchy principal value integrals much less Hadamard finite part integrals. Finally, the utility of the systematic procedure presented here for use in obtaining singularity-reduced integral equations for other unbounded media (viz. the half-space and bi-material) is indicated.

Studies of the Growth and Collapse of Voids in Viscous Solids

The growth and collapse of isolated voids in power-law viscous matrix materials are investigated. The study is restricted to axisymmetric remote stressing and to voids which are initially spheroidal with the axis of symmetry of the voids coincident with the axis of symmetry of the remote loading. Particular attention is given to the evolution of initially spherical voids, but the effect of initial void shape on subsequent void evolution is also investigated. For linearly viscous matrix materials, the voids evolve through spheroidal shapes and the work of Budiansky et al. (1982) provides the desired information about the history of void shape and volume. For nonlinear matrix materials, the void evolution is idealized as proceeding through a sequence of spheroidal shapes, and the rate of deformation for a given instant is evaluated using a Ritz procedure developed by Lee and Mear (1992). The results of the study demonstrate that the history of void volume and void shape is influenced significantly by the material nonlinearity, the remote stress state and the initial void aspect ratio.

Stress concentration induced by an elastic spheroidal particle in a plastically deforming solid

The fracture of ductile solids occurs by the nucleation, growth and coalescence of microscopic voids. The nucleation stage usually involves the fracture of hard second-phase particles or their decohesion from the surrounding matrix material. In this study, the role of particle shape on the nucleation of microvoids is investigated by considering the axisymmetric deformation of a single isolated linearly elastic, prolate spheroidal particle embedded in a plastically deforming matrix. The local stress concentration factors at the particle–matrix interface and within the particle are evaluated using a Ritz procedure which combines an exact solution within the elastic spheroidal particle with a spectral representation for the plastically deforming matrix. Accurate solutions for the stress concentration factors are provided for several particle aspect ratios, for various sets of material properties and for several remote stress histories. It is shown that the particle shape has a strong effect upon the stress concentration factors, with the degree of influence dependent upon the material properties and the triaxiality of the remote loading. In particular, as the particle aspect ratio is increased with other variables held fixed, the normal stress concentration within the particle rises at a greater rate than does that at the interface (despite the high curvature developed at the particle pole) and the distribution of normal traction on the boundary tends to become less tensile. These findings suggest that there is a critical aspect ratio at which the dominant mechanism of microvoid nucleation changes from one of interface debonding to one of particle fracture. The remote stress triaxiality has a significant effect upon this transition: as the stress triaxiality is increased, the relative magnitudes of the stress concentration factors, as well as the distribution of normal traction on the interface, are altered such that there tends to be a larger range of particle aspect ratios for which interfacial bonding is favored over particle fracture.

Effect of void shape on the macroscopic response of non-linear porous solids

Abstract The macroscopic response of an incompressible power-law matrix containing aligned spheroidal voids is investigated. The voids are assumed to be arranged in a uniform array, and the response of the solid is evaluated by isolating a typical block of the material containing a single void. The requisite boundary value problem for this “unit cell” is solved using a spectral method which is an adaption of that used by Lee and Mear “Axisymmetric Deformation of Power-law Solids containing Elliptical Inhomogeneities. Part I: Rigid Inclusions”, J. Mech. Phys. Solids , (1992) 8 , 1805. Attention is restricted to axisymmetric deformation, and results for the macroscopic strain-rates (or strains) are presented for a range of void shape, void volume concentrations, hardening exponents and remote stress triaxilities.

Effective properties of power-law solids containing elliptical inhomogeneities. Part I: Rigid inclusions

Abstract The plane strain deformation of a power-law material containing a dispersion of rigid elliptical inclusions is investigated. Accurate constitutive relations are established for dilute concentrations of inclusions over a wide range of martrix hardening exponents and inclusion aspect ratios. The essential step in the analysis is the solution of the kernel problem for an isolated inclusion, and this is obtained using a Ritz procedure in which trial displacements are derived from a displacement potential in elliptic-cylindrical coordinates. Approximate constitutive relations for nondilute concentrations of inclusions are then established using a differential self-consistent scheme. The study is primarily concerned with randomly oriented inclusions, but limited results are also presented for aligned inclusions. As part of the investigation, the procedure introduced by Ponte Castaneda (J. Mech. Phys. Solids (1991)) is evaluated. This is a variational procedure which exploits information about a linear solid to obtain bounds or estimates for the overall response of nonlinear solids with the same microgeometry of second phase. Finally, connection is made with recent results for spheroidal inclusions given by Lee and Mear (J. Mech. Phys. Solids (1991); Int. J. Solids Struct. (1991)), and an ad hoc estimate for the overall response of solids containing randomly oriented rigid, spheroidal inclusions is proposed.