Showing papers in "Journal of The Mechanics and Physics of Solids in 1990"

A critical evaluation for a class of micro-mechanics models

Abstract: New results are derived for the effective properties of composite materials composed of a continuous matrix phase containing a highly concentrated suspension of rigid spherical inclusions. The resulting analytical forms from several different theoretical micro-mechanics models are found to vary widely and they are assessed with respect to physical significance.

An analysis of tensile decohesion along an interface

Abstract: A COHESIVE zone type interface model, taking full account of finite geometry changes, is used to study the decohesion of a viscoplastic block from a rigid substrate. Dimensional considerations introduce a characteristic length into the formulation. The specific boundary value problem analysed is one of plane strain tension with a superposed hydrostatic stress. For a perfect interface, if the maximum traction that the viscoplastic block can support is greater than the interfacial strength, decohesion takes place in a primarily tensile mode. If this maximum traction is lower than the interfacial strength, a shear dominated decohesion initiates at the block edge. Imperfections in the form of a non-bonded portion of the interface are considered. The effects of imposed stress triaxiality, size scale, loading rate and interfacial properties on the course of defect dominated decohesion are illustrated. The characterization of decohesion initiation and propagation in terms of rice's (J. appl. Mech. 35, 379, 1968) J-integral is investigated for a variety of interface descriptions and values of the superposed hydrostatic stress.

Application of variational concepts to size effects in elastic heterogeneous bodies

Abstract: H eterogeneous materials that are not statistically uniform are considered. The basic assumptions and conditions underlying the concept of effective properties are reviewed. A general procedure for the experimental or numerical evaluation of the overall properties of the material when the concept of a representative volume does not apply or cannot be used is proposed and then applied to the elastic case as a first example. Three kinds of results are obtained: relationships between experimental results obtained on a large specimen and on an appropriate set of smaller ones, hierarchies between families of specimens of various sizes, absolute bounds for the two limiting cases where the size goes to zero, or to the representative volume when it exists. Possible applications and extensions are mentioned.

Constitutive modelling of orthotropic plasticity in sheet metals

Abstract: T he classical quadratic yield criterion for orthotropic metals is known not to be sufficiently flexible in practice. By the simple expedient of admitting non-integer exponents, however, an improved criterion was devised for sheet with in-plane isotropy (so-called normal anisotropy). On the other hand, an acceptable proposal has not been forthcoming for sheet with in-plane anisotropy (so-called planar anisotropy). It is suggested here that improvement should be sought by incorporating a compatible dependence on orientation in a homogeneous yield function of arbitrary degree. In so doing, the practicalities of forming technology are respected by keeping the number of arbitrary parameters as small as possible. A new criterion is constructed along these lines and its implications are explored in detail. Additionally, a simple means of representing anisotropic yield criteria of any kind is presented with supporting general theorems.

On the driving traction acting on a surface of strain discontinuity in a continuum

Abstract: The notion of the driving traction on a surface of strain discontinuity in a continuum undergoing a general thermomechanical process is defined and discussed. In addition, the associated constitutive notion of a kinetic relation, in which the normal velocity of propagation of the surface of discontinuity may be a given function of the driving traction and temperature, is introduced for the special case of a thermoelastic material.

A theory of mechanical behavior of elastic media with growing damage and other changes in structure

Abstract: S train energy-like potentials are used to model the mechanical behavior of linear and nonlinear elastic media with changing structure, such as micro- and macrocrack growth in monolithic and composite materials. Theory and experiment show that the applied work for processes in which changes in structure occur is in certain cases independent of some of the deformation history. Consequences of this limited path-independence are investigated, and various relationships for stable mechanical response are derived. For example, it is shown that work is at a minimum during stable changes in structure, which should be useful for developing approximate solutions by variational methods. Some final remarks indicate how the theory may be extended to include thermal, viscoelastic and fatigue effects.

Diffusivity of ultrasound in polycrystals

Abstract: T he diffusivity of ultrasound in an untextured aggregate of cubic crystallites is studied theoretically with a view towards nondestructive characterization of microstructures. Multiple scattering formalisms for the mean Green's dyadic and for the covariance of the Green's dyadic (and therefore for the energy density) based upon the method of smoothing are presented. The first-order smoothing approximation used is accurate to leading order in the anisotropy of the constituent crystallites. A further, Born, approximation is invoked which limits the validity of the calculation to frequencies below the geometrical optics regime. Known result for the mean field attenuations are recovered. The covariance is found to obey an equation of radiative transfer for which a diffusion limit is taken. The resulting diffusivity is found to vary inversely with the fourth power of frequency in the Rayleigh, long wavelength, regime and inversely with the logarithm of frequency on the short wavelength, stochastic, asymptote. The results are found to fit the experimental data.

The overall elastoplastic stress-strain relations of dual-phase metals

Abstract: T wo simple , albeit approximate, theories are developed to estimate the stress-strain relations of dual-phase metals of the inclusion-matrix type, where both phases are capable of undergoing plastic flow. The first one is based upon Hill's recognition of a weakening constraint power in a plastically deforming matrix, whereas the second one is based on Kroner's elastic constraint in the treatment of the single inclusion-matrix interaction. The inclusion-inclusion interaction at finite concentration is accounted for by the Mori-Tanaka method in both cases. Consistent with the known elastic behavior, the first theory discloses that the geometrical arrangement of the constituents has a significant influence on the overall elastoplastic response. When the harder phase takes the position of the matrix the composite is far Stiffer than that when it takes the position of inclusions. The strong elastic constraint associated with the second theory tends to provide an upper-bound type of estimate regardless of whether the matrix is the harder phase or the softer, and, therefore, it is suggested that this theory be used only for the class of composites whose matrix is the harder phase. Both theories are finally applied to predict the stress-strain relations of dual-phase stainless steels, and the results are found to be in satisfactory agreement with the test data.

The effect of substrate on the elastic properties of films determined by the indentation test — axisymmetric boussinesq problem

Abstract: T he elastic solution of an axisymmetric mixed boundary value problem is considered. An elastic layer is assumed to be either in frictionless contact or perfectly bonded to a semi-infinite elastic half-space. The elastic field caused by the indentation of the elastic layer, by a rigid indenter is solved for spherical, conical and flat-ended cylindrical indenters. The results are obtained by solving a Fredholm integral equation of the second kind with a continuous symmetrical kernel which depends on the bonding conditions. Numerical results are given for several combinations of film and substrate elastic moduli and film thicknesses. These results provide a guideline for selecting the appropriate film thickness and substrate in order to determine the elastic constants of thin films.

Residual stresses and stored elastic energy of composites and polycrystals

Abstract: R esidual Stresses in heterogeneous materials may arise because of differential or anisotropic thermal expansion of constituents. The paper is concerned with thermoelastic solids whose material properties fluctuate on the microscopic scale. Rigorous general relations between stored elastic energy and statistical averages (mean values and fluctuations) of residual stresses are derived. These results are applied to two-phase composites and to materials where the fluctuations of elastic constants can be neglected. One obtains exactly the stored energy, certain conditional mean values and the covariance matrix of the residual stresses. Under the assumptions of statistical homogeneity and isotropy, the results hold for any type of heterogeneous microstructure.

Polycrystalline plastic deformation and texture evolution for crystals lacking five independent slip systems

Abstract: W e clearly elucidate the kinematic constraints, and the corresponding kinematic indeterminacy of part of the deviatoric stress tensor, in a rigid-viscoplastic single crystal lacking five independent slip systems. The indeterminate stress component is a Lagrange multiplier enforcing the kinematic constraint, and it must be determined from equilibrium considerations. A simple polycrystalline model is constructed which precisely satisfies local kinematic constraints as well as global compatibility. Volume-average global stresses are obtained by approximating the local constraint stress as the corresponding projection of the (to-be-determined) global stress. Applications of the model to hexagonal crystals without pyramidal slip, and to large deformation and texturing of orthorhombic polycrystalline materials (olivine; HDPE) are made.

Void growth and constitutive softening in a periodically voided solid

Abstract: A three-dimensional finite element model of a periodically voided elastic-plastic solid is used to predict void growth and constitutive softening under tensile plastic deformation. The analysis considers an infinite block of material containing a three-dimensional periodic array of voids, subjected to a remote deformation or stress field. The predicted initial dilatational and extensional void growth rate, as a function of stress triaxiality and material work-hardening rate, agrees well with the analytical results of Budiansky et al. (Mechanics of Solids, The Rodney Hill 60th Anniversary Volume, edited by H.G. H opkins and M.J. Sewell , p. 13, Pergamon Press, Oxford, 1982) for an isolated void in a viscous solid. Increased initial void volume fraction (⨍) has little effect on the dilatational growth rate, but strongly affects the extensional growth rate at high levels of triaxiality. The effect of void aspect ratio on void growth rate was seen during uniaxial tensile deformation, during which extensive void elongation caused a reduction in the final or asymptotic void growth rates. Constitutive softening is shown to be primarily a function of porosity and stress triaxiality.

The driving force for glide of a threading dislocation in a strained epitaxial layer on a substrate

Abstract: T he process of epitaxial growth of a very thin layer onto a substrate crystal is considered for the particular situation in which the layer and substrate materials have the same crystal structure and orientation but different lattice parameters. Under these conditions, the layer grows with an intrinsic elastic strain determined by the mismatch in lattice parameters. The associated stress in the crystalline layer provides a driving force for the nucleation and motion of defects, primarily dislocations. The focus here is on the glide of a dislocation extending from the free surface of the layer to the layer-substrate interface, the so-called threading dislocation. A general definition of driving force for glide of a threading dislocation is introduced on the basis of work arguments. The definition is then applied to calculate the driving force for steady motion of an isolated threading dislocation in a strained layer, and the result includes Matthews' critical thickness concept as one of its features. Next, a kinetic equation for glide of a dislocation in semiconductor materials is proposed to estimate the glide rate of a threading dislocation in these low mobility materials. Finally, for the case of cubic materials, the general definition of driving force is applied to estimate the additional driving force on a threading dislocation due to an encounter with a dislocation on an intersecting glide plane. The results indicate that this effect is significant in blocking the glide of a threading dislocation for large mismatch strains and for layer thicknesses near the critical thickness.

Three-dimensional crack front fields in a thin ductile plate

Abstract: Based on detailed finite element solutions, various aspects of the 3-D fields near a crack front in a thin ductile plate are analysed. In particular, the stress field in the immediate crack front vicinity is carefully investigated. The existence and size of local J-dominated fields are determined from a comparison of the complete near tip stress field with the plane strain HRR solutions using a dominance parameter. In conjunction with an estimated size of the finite deformation zone, the loss of HRR-dominance along the crack front is also studied. Physically, the loss of HRR-dominance at higher load occurs when the finite deformation region outgrows the (local) plane strain region, the size of which is essentially limited by the geometry of a thin plate. Alternatively, the existence (at the mid-plane) of plane strain HRR-dominance in a low hardening material under (in-plane) small scale yielding requires the thickness of a thin plate to exceed 0.5(K1/σ0)2, thus limiting the maximum in-plane extent of the plastic zone to a fractio of plate thickness. Furthermore, such a restriction on the plastic zone size precludes the coexistence of local plane strain and surrounding plane stress HRR-fields within a thin plate.

Elastic-plastic fracture behavior of an aluminum alloy under mixed mode loading

Abstract: M ixed mode elastic-plastic fracture tests were carried out by using A5083-O aluminum alloy compacttension-shear specimens, in which the angle between the loading axis and the crack surface was varied from 90° (mode I) to 0° (mode II) by employing a special loading device. A fractographic investigation was made at various stages of crack tip blunting and stable crack growth. Under mixed mode loadings, opposite deformations appeared at the crack tip: sharpening and blunting. With increasing mode II component, cracks due to shear type fracture initiated at the sharpened corner of the crack tip near the surfaces of the specimen, and then another crack due to dimple type fracture occurred at the blunted corner of the crack tip near the midthickness of the specimen. These experimental results were explained qualitatively by taking account of the stress and strain singularities and triaxial stress distributions near the surface and midthickness of the specimen. Fracture mechanics parameters appropriate for mixed mode elastic-plastic fractures were also studied based on the experimental data.

Effect of viscoplastic flow rules on the initiation and growth of shear bands at high strain rates

Abstract: MARCHAND and Duffy have reported detailed measurements of the temperature and strain as a shear band develops in a HY-100 steel. Assuming their torsional tests in thin-wall tubes can be adequately modeled by a viscoplastic block undergoing overall adiabatic simple shearing deformations. we investigate the effect of modeling the viscoplastic response of the material by a-power law. and flow rules proposed by Litonski, Bodner and Partom, and Johnson and Cook. Each of these flow rules is first calibrated by using the test data at a nominal strain-rate of 3300 s-‘. Then predictions from the use of these flow rules at nominal strain-rates of 1400 s-’ and 1600 s-’ are compared with the experimental findings. It is found that the Bodner-Partom law and the dipolar theory proposed by Wright and Batra predict reasonably well the main features of the shear band formation in a HY-100 steel.

Solution of three-dimensional crack problems by a finite perturbation method

Abstract: An incremental method is described for calculating stress intensity factors in arbitrarily shaped planar cracks subjected to a uniform remote stress. The method is based on recent work by Rice (J. appl. Mech. 52, 571, 1985; Fracture Mechanics : Perspectives and Directions (20th Symp.), ASTM-STP-1020. to appear. 1987), and gao and rice (Int. J. Fracture 33, 115, 1987a; J. appl. Mech. 54, 627, 1987b). who have developed a procedure for computing the variation in stress intensity factor caused by small changes in crack geometry. To date, this technique has only been used to calculate the effects of first-order perturbations in the shape of a crack. In this paper, the method is extended to arbitrarily large perturbations in geometry. Stress intensity factors are calculated by applying a succession of perturbations to a crack of some convenient initial geometry, such as a circular or a half-plane crack. Since this procedure reduces the analysis to evaluating repeatedly two integral equations defined only on the crack front, it has distinct advantages over other existing techniques. The accuracy of the method is demonstrated by calculating stress intensity factors for two test cases : an elliptical crack and a half-plane crack deforming into a prescribed sinusoidal shape of finite amplitude. As further examples of application of the method, solutions to the following problems are presented : a semi-infinite fatigue crack propagating through a particle; a semi-infinite crack trapped by a periodic array of tough particles ; and the unstable growth of a semiinfinite crack through material of decreasing toughness.

A rate-dependent bounding surface model with a generalized image point for cyclic nonproportional viscoplasticity

Abstract: A phenomenologically based time- and rate-dependent bounding surface model is introduced which can be classified among the unified creep-plasticity theories. The model is motivated chiefly by experimental behaviors over a wide range of strain rates with particular emphasis on the behavior of metals under nonproportional loading where bounding surface theories have found success in modeling rate-independent behavior. In addition, a micromechanical interpretation is given for two kinematic hardening variables which lead to the rate-dependent bounding surface interpretation. The definition of the image point for the directional indices of kinematic hardening is left unspecified to maintain generality. The distinction between instantaneous rate sensitivity and rate sensitivity of material hardening is discussed and a framework for partitioning material rate sensitivity is presented. The relationship of the model to previously proposed formulations is discussed. Asymptotic and parametric behaviors of the model are examined with reference to experimentally observed behavior. A rate-independent idealization of the theory is obtained as a limiting, special case of the more general rate-dependent bounding surface framework.

Interface cracks in anisotropic bimaterials

Abstract: It is known that the displacement at the interface crack surface in a bimaterial under a two-dimensional deformation may be oscillatory. The existence of oscillation depends on the material properties and the orientations of the two materials on both sides of the crack-interface. We show in this paper that the existence of oscillation depends on the material properties only and is otherwise independent of the individual orientation of the two materials. This means that. if the crack surface displacement is oscillatory for one choice of orientations of the two materials, no other orientations obtained by rotating about the x3-axis can alleviate the oscillation. Conversely, if the crack surface displacement is not oscillatory for one choice of orientations of the two materials, no other orientations can generate oscillation. An exception is the Type B bimaterials defined in the paper for which the crack surface displacement is always oscillatory except for a particular Type B bimaterial at a particular choice of relative orientation. For bimaterials which consist of two different orthotropic materials, it is shown that whether the crack surface displacement is oscillatory or not depends on whether C 11 C 22 +C 12 of the two materials are different or not.

Approximate analysis for the formation of adiabatic shear bands

Abstract: Parametric solutions are given for the formation of adiabatic shear bands in the context of the onedimensional nonlinear theory where inertia and elasticity are ignored. When heat conduction is also ignored, the exact solution reduces completely to a sequence of quadratures. For a perfectly plastic material with heat conduction, an implicit parametric solution is also constructed. This is similar to the previous one in many ways, but now it involves two quadratures, a single nonautonomous first-order ODE. and two functions that obey heat equations. This solution appears to be very accurate (compared to the full finite element solution) until the time of stress collapse. Results indicate that for weak rate hardening of the power law type, intense localization depends strongly on the initial characteristics of thermal softening and not at all on the high temperature characteristics. Within the context of rigid/perfect plasticity, a scaling law for the critical strain is given, and a figure of merit is defined that ranks materials according to their tendency to form adiabatic shear bands.

The interaction of cyclic hardening and ratchetting for AISI type 304 stainless steel at room temperature—I. Experiments

Abstract: Uniaxial zero-to-tension load controlled tests at stress rates differing by three orders of magnitude were performed on annealed AISI Type 304 stainless steel. The maximum stress was determined either as the stress reached in displacement control at one per cent strain (History I), or as the stress reached after a 1050 s relaxation test at one per cent strain (History II). The tests of History III were identical to those of History I, except that the specimens were subjected to cyclic hardening prior to the start of the ratchetting test. The ratchet strain accumulations in Histories I and II are significant and depend on stress rate. A stress rate decrease increases the accumulated ratchet strain. The final ratchet strain accumulation is independent of stress rate sequence. In History II, no ratchet strain was found at any stress rate. If the steel had behaved in a rate-independent fashion, no ratchet strain accumulation should have been found in any of the tests. They were performed with an MTS servohydraulic, computer controlled testing machine, a clip on-extensometer and digitized data acquisition.

Experimental and theoretical strain distributions for stationary and growing cracks

Abstract: Experimental strain distributions are determined very near the crack tip in Fe-3wt.%Si single crystals. Both in situ stereoimaging and electron channeling techniques give reasonably reproducible distributions. By growing fatigue cracks on a {100} cleavage plane, the singularity strengths have been determined for both growing and stationary cracks under relatively plane stress and plane strain conditions. This has allowed a comparison to existing theoretical models. It is shown that the HRR singularity (Hutchinson, Rice and Rosengren, 1968) for stationary cracks is very good to within I μm of the crack tip and a hardening model for the growing crack (gao and hwang, Advances in Fracture Research, edited by D. Francois. 5th Int. Conf. on Fracture, Cannes, France, 2, 669, 1981) is surprisingly good. Other issues such as fracture criteria are discussed since strains greater than unity were measured at the crack tip in this relatively brittle material.

Elastic-plastic analysis of combined mode I and III crack-tip fields under small-scale yielding conditions

Abstract: WITHIN THE context of the small-strain approach, combined mode I and III near-tip fields of a stationary crack in an elastic-plastic solid are obtained by finite element analysis under small-scale yielding conditions. To investigate the behavior of the near-tip fields, the normalized stresses ahead of the crack tip are plotted as functions of the normalized radial distance to the tip for several combinations of prescribed mode 1 and III elastic K fields. The angular variations of the normalized stresses at a fixed radial distance deep within the plastic zone are also plotted for several combinations of remote mode I and III elastic K fields. These plots show an unmistakeable pattern : the in-plane stresses are more singular than the out-of-plane shear stresses. Over a certain distance, the near-tip in-plane stresses can be said to be more singular than r- ’ ‘“+ ‘) while the near-tip out-of-plane shear stresses are less singular than r- I’(“+ I), where I is the radial distance to the tip and n is the strain hardening exponent of the material. lmpli~tions of these features as they relate to three-dimensional engineering fracture analyses are discussed.

A theory for the ultimate strength of a brittle-matrix composite

Abstract: A theory for calculating the ultimate strength of an infinite, brittle-matrix fiber composite is presented. At some stress level which is less than the ultimate strength, the composite is presumed to undergo multiple matrix cracking; this results in intact fibers spanning a series of roughly equally spaced matrix cracks. Subsequent damage in the form of fiber breaks is assumed to occur with the fiber strength being statistically distributed. With the use of some plausible assumptions, the spatial distribution of fiber breaks is computed and their effect on the subsequent load-carrying capacity of the composite is determined. The variables which most affect the ultimate strength are the interfacial shear stresses, which control load transfer near the matrix cracks and near the fiber breaks, and the fiber strength variability. A comparison with data in the literature indicates that a reliance upon simple rule-of-mixtures-type estimates of composite strength can be misleading.

Bifurcations of compressible pressure-sensitive materials in plane strain tension and compression

Abstract: B ifurcations , including shear band and diffuse geometric modes, are analysed for plane strain deformation of a rectangular slab subject to a constant lateral confining stress. The response of the slab material is characterized by an incrementally linear constitutive relation that allows for the possibility of volume change accompanying shear deformation, pressure dependence and deviations from plastic “normality”. When the lateral confining stress is zero, the picture of bifurcations is qualitatively similar to that investigated by N eedleman ( J. Mech. Phys. Solids 27 , 231, 1979) for incompressible materials. For example, when normality is satisfied, localization is excluded by a positive uniaxial tangent modulus (to within terms of order stress divided by elastic modulus), deviations from normality promote localization, and the occurrence of a long wavelength symmetric, diffuse bifurcation coincides with the attainment of maximum load in tension. However, increasing the compressibility decreases the value of the uniaxial tangent modulus at which localization modes become possible. When lateral confining stress is non-zero, differences from the analysis of N eedleman (1979) are more dramatic. For example, a finite stress difference is required for the onset of an anti-symmetric, long wavelength bifurcation and, when the lateral stress is compressive, shear band modes become possible prior to the maximum load in tension.

Localization of plastic deformation in shear due to microcracks

Abstract: A constitutive relation for simple shear of an elastic-plastic material containing a periodic array of cracks is developed. The relation is based on finite element analysis and slip-line field solutions for interacting cracks in simple shear. Typical shear stress-strain curves display a peak in the nominal shear stress due to competition between strain hardening of the matrix and material softening due to rotation and stretching of cracks with deformation. The effect of nonuniform crack distributions on localization behavior is studied by determining the critical conditions for which the shear strain in a band of cracks becomes unbounded relative to that in the surrounding, uncracked material. The results show that the strain to localization depends strongly on the ratio of crack length to crack spacing, crack orientation, crack-face friction and matrix hardening. The results are helpful to understanding shear localization under confining pressures, where voids adopt a crack-like morphology.

Plane strain characteristics theory for soils and granular materials with density dependent yield criteria

Abstract: An Analysis of the quasi-static, plane strain, deformations of a rigid/plastic material whose yield stress depends on the current material density is presented. Both steady and unsteady flows are studied. Such constitutive relations are widely used to model the yielding behaviour of soils and granular media in what is generally known as “critical state soil mechanics”. It is shown that, contrary to a widely held belief, the stress and velocity characteristics for such materials do coincide. The differential relations along the characteristics are derived and shown to have a structure typical of hardening materials. In addition it is shown that a second family of weak discontinuities, embedded in the material, can exist and that they give rise to the familiar “Sokolovskii stress characteristics” in the critical state limit.

The interaction of cyclic hardening and ratchetting for AISI type 304 stainless steel at room temperature—II. Modeling with the viscoplasticity theory based on overstress

Abstract: The viscoplasticity theory based on overstress (VBO) is used to predict ratchetting tests reported by Ruggles and Krempl in Part I (J. Mech. Phys. Solids 38, 575, 1990). The VBO predicts the influence of ratchetting rate correctly. An increase in ratchet stress rate reduces the ratchet strain. Quantitative predictions are reasonable for tests with no prior cyclic hardening. For tests with prior hardening the theory predicts too large a ratchet strain. It is shown that this can be corrected by making the shape function dependent on history. Theory and experiment do not exhibit a stress rate history effect.

Asymptotic analysis of a crack in a power-law material under combined in-plane and out-of-plane shear loading conditions

Abstract: A symptotic crack-tip fields for power-law hardening materials under combined in-plane and out-of-plane shear loading conditions are investigated The governing equations of the asymptotic crack-tip fields are formulated from two stress functions, one for the in-plane stresses and the other for the out-of-plane shear stresses These stress functions are of separable functional forms of r and /gq, which represent the polar coordinates centered at the crack tip Perturbations of the governing equations were carried out The singular behavior and the angular functions of the crack-tip in-plane and the out-of-plane shear stresses obtained from the perturbation analysis agree well with the results of the finite element computations of P an and S hih ( J Mech Phys Solids 38 , 161, 1990; J appl Mech , in press, 1989) In particular, the results from the perturbation analysis indicate that the singularity of the mode I in-plane stresses is slightly stronger than that of the mode III out-of-plane shear stresses under combined mode I and III conditions, whereas the singularity of the mode III out-of-plane shear stresses is slightly stronger than that of the mode II in-plane stresses under combined mode II and III conditions

A crack at the interface between an elastic-perfectly plastic solid and a rigid substrate

Abstract: A crack at the interface between an incompressible, elastic-perfectly plastic solid and a rigid substrate is studied for the stationary and quasi-statically growing cases under the condition of plane strain. A one-parameter family of asymptotic near-tip stress fields is obtained for the case of a stationary crack and compared with the finite element results given by Shih and Asaro for a crack at the interface of a power-law hardening material and a rigid substrate. Shih and Asaro's finite element results for a deformable material of low hardening strongly suggest this one-parameter family of stress distributions as appropriate for an interface crack between an elastic-perfectly plastic solid and a rigid substrate. The one-parameter family of solutions presented here for an interface crack is analogous to Shih's family of solutions at a mixed mode homogeneous crack tip with mixities that are arbitrary as far as the asymptotic analysis is concerned, but determinate from a full-field analysis. A complete asymptotic near-tip stress and deformation field based on the Prandtl-Reuss flow rule and the Huber-Mises yield criterion is obtained for a crack growing quasi-statically along the interface of an incompressible, elastic-perfectly plastic solid and a rigid substrate. Similar to a mode II growing plane stress or strain crack in a homogeneous medium, the near-tip field consists of three different types of angular sectors: a “centered fan” plastically deforming sector near the interface, a “constant stress” plastically deforming sector near the crack free surface and an elastically deforming sector in between. The velocity components are fully continuous, and the strains are found to be singular like (ln r ) 2 as r → 0, where r is distance from the crack tip.