Plane-stress crack-tip fields for pressure-sensitive dilatant materials

In this paper we present plane-stress crack-tip stress and strain fields for pressure-sensitive dilatant materials. A hydrostatic stress-dependent yield criterion and the normality flow rule are used to account for pressure-sensitive yielding and plastic dilatancy. The material hardening response is specified by a power-law relation. The plane-stress mode I singular fields are found in a separable form similar to the HRR fields (Hutchinson, J. Mech. Phys. Solids 16, 13–31 and 337–347, 1968; Rice and Rosengren, J. Mech. Phys. Solids 16, 1–12, 1968). The angular variations of the fields depend on the material hardening exponent and the pressure sensitivity parameter. Our low-hardening solutions for different degrees of pressure sensitivity agree well with the corresponding perfectly plastic solutions. An important aspect of the effects of pressure-sensitive yielding and plastic dilatancy on crack-tip fields is the lowering of the opening stress and the hydrostatic stress directly ahead of the crack tip. This effect, similar to that under plane-strain conditions (Li and Pan, to appear in J. Appl. Mech. 1989), has implications in the material toughening observed in some ceramic and polymeric composites.     

Analysis of temperature distribution near the crack tip under constant amplitude loading

An analytical/numerical method has been developed to find the temperature rise near the crack tip under fatigue loading. The cyclic plastic zone ahead of the crack tip is assumed to be the shape of the source of heat generation and some fraction of plastic work done in cyclic plastic zone as heat generation. Plastic work during fatigue load was found by obtaining stress and strain distribution within the plastic zone by Hutchinson, Rice and Rosengren (HRR) crack tip singularity fields applied to small scale yielding on the cyclic stress strain curve. A two‐dimensional conduction heat transfer equation, in moving co‐ordinates, was used to obtain temperature distribution around the crack tip. Temperature rise was found to be a function of frequency of loading, applied stress intensity factor and thermal properties of the material. A power–law relation was found between the rise in temperature at a fixed point near the crack tip and range of stress intensity factor.     

Steady-state crack growth and fracture work based on the theory of mechanism-based strain gradient plasticity

Mode I steady-state crack growth is analyzed under plane strain conditions in small scale yielding. The elastic-plastic solid is characterized by the mechanism-based strain gradient (MSG) plasticity theory [J. Mech. Phys. Solids 47 (1999) 1239, J. Mech. Phys. Solids 48 (2000) 99]. The distributions of the normal separation stress and the effective stress along the plane ahead of the crack tip are computed using a special finite element method based on the steady-state fundamental relations and the MSG flow theory. The results show that during the steady-state crack growth, the normal separation stress on the plane ahead of the crack tip can achieve considerably high value within the MSG strain gradient sensitive zone. The results also show that the crack tip fields are insensitive to the cell size parameter in the MSG theory. Moreover, in the present research, the steady-state fracture toughness is computed by adopting the embedded process zone (EPZ) model. The results display that the steady-state fracture toughness strongly depends on the separation strength parameter of the EPZ model and the length scale parameter in the MSG theory. Furthermore, in order for the results of steady crack growth to be comparable, an approximate relation between the length scale parameters in the MSG theory and in the Fleck-Hutchinson strain gradient plasticity theory is obtained.     

Fatigue crack growth model for constant amplitude loading

A fatigue crack growth model under constant amplitude loading has been developed considering energy balance during growth of the crack. The plastic energy dissipated during growth of a crack within cyclic plastic zone and area below cyclic stress–strain curve was used in the energy balance. The near crack tip elastic–plastic stress and strain were calculated on the basis of Hutchinson, Rice and Rosengren (HRR) formulations. Fatigue crack growth rate in linear and near threshold region of da/dN versus ΔK curve can be determined on the basis of the proposed model in terms of low cycle fatigue (LCF) properties determined on smooth specimen. The predictions of the model have been compared with the experimental and theoretical results available in the literature using mechanical and fatigue properties. The model compares well in the threshold and intermediate region of the da/dN versus ΔK curve for wide range of material tested.     

A modified effective crack-length formulation in elastic-plastic fracture mechanics

In examining the performance of standard effective crack-length formulations, the authors noted quantitative accuracy up to “high” fractions of limit load under loading conditions for which the elastic T-stress was non-negative, while a pronounced deviation from the corresponding continuum elastic-plastic plane-strain finite-element solutions was seen in shallow-cracked geometries having negative T-stress. This trend can be rationalized by noting that, under modified boundary layer (K_I and T) loading, the maximum plastic zone radius strongly increases as the T-stress decreases from zero (J.R. Rice (1974), J. Mech. Phys. Solids 22, 17–26; S.G. Larsson and A.J. Carlsson (1973), J. Mech. Phys. Solids 21, 263–277; N.P. O'Dowd and C.F. Shih (1991), J. Mech. Phys. Solids 39(8), 989–1015.) Accordingly, we formulate a modified effective crack length to account for the effects of the elastic T-stress.

The new formulation consistently extends the load range for which accurate predictions of compliance, J-integral, and crack-tip constraint are obtained in several plane strain specimen geometries. The magnitude of influence of the T-stress varies with specimen type and relative crack depth. The greatest “improvement” to standard effective crack length approximations occurs in specimens of “moderately” negative T-stress.     

Mode II intersonic crack propagation in poroelastic media

A crack is steadily running in an elastic isotropic fluid-saturated porous solid at an intersonic constant speed c. The crack tip speeds of interest are bounded below by the slower between the slow longitudinal wave-speed and the shear wave-speed, and above by the fast longitudinal wave-speed. Biot’s theory of poroelasticity with inertia forces governs the motion of the mixture. The poroelastic moduli depend on the porosity, and the complete range of porosities n ∈ [0, 1] is investigated. Solids are obtained as the limit case n = 0, and the continuity of the energy release rate as the porosity vanishes is addressed. Three characteristic regions in the plane (n, c) are delineated, depending on the relative order of the body wave-speeds. Mode II loading conditions are considered, with a permeable crack surface. Cracks with and without process zones are envisaged. In each region, the analytical solution to a Riemann–Hilbert problem provides the stress, pore pressure and velocity fields near the tip of the crack. For subsonic propagation, the asymptotic crack tip fields are known to be continuous in the body [Loret and Radi (2001) J Mech Phys Solids 49(5):995–1020]. In contrast, for intersonic crack propagation without a process zone, the asymptotic stress and pore pressure might display a discontinuity across two or four symmetric rays emanating from the moving crack tip. Under Mode II loading condition, the singularity exponent for energetically admissible tip speeds turns out to be weaker than 1/2, except at a special point and along special curves of the (n, c)-plane. The introduction of a finite length process zone is required so that 1. the energy release rate at the crack tip is strictly positive and finite; 2. the relative sliding of the crack surfaces has the same direction as the applied loading. The presence of the process zone is shown to wipe out possible first order discontinuities.     

Families of asymptotic tensile crack tip stress fields in elastic-perfectly plastic single crystals

In this work, two families of asymptotic near-tip stress fields are constructed in an elastic-ideally plastic FCC single crystal under mode I plane strain conditions. A crack is taken to lie on the (010) plane and its front is aligned along the direction. Finite element analysis is first used to systematically examine the stress distributions corresponding to different constraint levels. The general framework developed by Rice (Mech Mater 6:317–335, 1987) and Drugan (J Mech Phys Solids 49:2155–2176, 2001) is then adopted to generate low triaxiality solutions by introducing an elastic sector near the crack tip. The two families of stress fields are parameterized by the normalized opening stress prevailing in the plastic sector in front of the tip and by the coordinates of a point where elastic unloading commences in stress space. It is found that the angular stress variations obtained from the analytical solutions show good agreement with finite element analysis.     

An Experimental Study on the Elastic–Plastic Fracture in a Ductile Material under Mixed-Mode Loading

Abstract:  Stereo vision was used to measure the crack-tip parameters, such as J integral, plastic mixity and elastic mixity of mixed-mode fracture specimens, and to study the applicability of the Shih's plane strain solution to the mixed-mode crack-tip fields. The fracture specimen used in this study was a compact tension shear (CTS) specimen made of 2024-O aluminum. The in-plane strain and stress fields near the mixed-mode crack tip of the CTS specimen were determined using the deformation field measured by the stereo vision. It is observed that the J integral values computed along rectangular contours surrounding the mixed-mode crack-tip approach constant values after r / h  > 0.5. The in-plane strains determined experimentally at several points near the crack tip and at several radial lines emerging from the crack tip are compared with the values calculated using Shih's plane-strain solution and the HRR slope, named after the investigations of Hutchinson, Rice and Rosengren respectively. It is found that the measured values follow the trends of the Shih's plane-strain solution. The elastic mixity evaluated using the measured crack-tip stress fields is close to that obtained from analytical solution. However, the evaluated plastic mixity deviates from the analytical solution.     

A failure criterion for brittle elastic materials under mixed-mode loading

The failure criterion of Leguillon at reentrant corners in brittle elastic materials (Leguillon 2002, Eur J Mech A/Solids 21: 61–72; Leguillon et al. (2003), Eur J Mech A—Solids 22(4): 509–524) validated in (Yosibash et al. 2004, Int J Fract 125(3–4): 307–333) for mode I loading is being extended to mixed mode loading and is being validated by experimental observations. We present an explicit derivation of all quantities involved in the computation of the failure criterion. The failure criterion is validated by predicting the critical load and crack initiation angle of specimens under mixed mode loading and comparison to experimental observations on PMMA (polymer) and Macor (ceramic) V-notched specimens.     

A combined dislocation—cohesive zone model for fracture in a confined ductile layer

The collective dislocation behavior near a crack tip in a ductile layer sandwiched between two brittle solids is analyzed via two-dimensional dislocation dynamics (DD) simulations that incorporate a cohesive zone (CZ) model. The cohesive crack tip is treated as part of a much larger finite crack confined in the ductile layer. The underlying boundary value problem is formulated with a set of boundary integral equations and numerically evaluated with a collocation method. The fracture energy of the layered composite material is shown to be strongly correlated with the layer thickness and is directly influenced by the cohesive strength of the ductile layer (Hsia KJ et al. (1994) J Mech Phys Solids 6 877–896).     

Analysis of in-plane transonically propagating interface crack with a finite contact zone

The report of Lambros and Rosakis [(1995) J Mech Phys Solids 43(2): 169–188] has focused attention on steady-state transonic interfacial crack growth accounting for the phenomenon of crack face contact in elastic/rigid bimaterial but could not handle issues relating to energy transmission across the interface. The present paper attempts to provide a complete explicit expression of the asymptotic fields induced by transonically propagating interfacial crack in elastic/elastic bimaterial for in-plane case. The energy distribution on the contact area, crack tip and two singular characteristic lines is analysed thoroughly and compared with the dynamic separated J-integrals. The length of the contact zone is also discussed briefly by establishing energy fracture criterion that satisfies contact condition. The two-dimensional in-plane asymptotic deformation field surrounding the contact area of a crack propagating transonically along an elastic/elastic bimaterial interface is observed and discussed thoroughly.     

On crack tip stress fields in pseudoelastic shape memory alloys

In a domain of reasonable accuracy around the crack tip, asymptotic equations can provide closed form expressions that can be used in formulation of crack problem. In some studies on shape memory alloys (SMAs), although the pseudoelastic behavior results in a nonlinear stress–strain relation, stress distribution in the vicinity of the crack tip is evaluated using asymptotic equations of linear elastic fracture mechanics (LEFM). In pseudoelastic (SMAs), upon loading, stress increases around the crack tip and martensitic phase transformation occurs in early stages. In this paper, using the similarity in the loading paths of a pseudoelastic SMA and a strain hardening material, the stress–strain relation is represented by nonlinear Ramberg–Osgood equation which is originally proposed for strain hardening materials, and the stress distribution around the crack tip of a pseudoelastic SMA plate is reworked using the Hutchinson, Rice and Rosengren (HRR) solution, first studied by Hutchinson. The size of the transformation region around the crack tip is calculated in closed form using a thermodynamic force that governs the martensitic transformation together with the asymptotic equations of HRR. A UMAT is written to separately describe the thermo-mechanical behavior of pseudoelastic SMAs. The results of the present study are compared to the results of LEFM, computational results from ABAQUS, and experimental results for the case of an edge cracked NiTi plate. Both set of asymptotic equations are shown to have different dominant zones around the crack tip with HRR equations representing the martensitic transformation zone more accurately.     

The effects of shear and near tip deformations on energy release rate and mode mixity of edge-cracked orthotropic layers

In this paper semi-analytical expressions are derived for the energy release rate and the stress intensity factors of edge-cracked homogeneous and orthotropic layers subject to arbitrary generalized end forces. The expressions are accurate for long and short cracks. Following the work of Li et al. [Li S, Wang J, Thouless MD. The effects of shear on delamination in layered materials. J Mech Phys Solids 2004;52(1):193-214] for isotropic bi-material layers, the derivation extends the method proposed by Suo [Suo ZG. Delamination specimens for orthotropic materials. J Appl Mech 1990;57(3):627-34] for axial forces and bending moments in order to include the contribution of the shear forces. The shear contribution to the fracture parameters depends on the shear deformations along the layer and the elastic near tip deformation of the material. Li et al. [Li S, Wang J, Thouless MD. The effects of shear on delamination in layered materials. J Mech Phys Solids 2004;52(1):193-214] derived semi-analytical expressions for the fracture parameters that depend on the crack tip stress resultants, the elastic constants and five numerically-determined constants globally describing the effect of shear. In this paper analogous constants are derived for orthotropic layers and defined by semi-analytical expressions that highlight their physical significance and allow separation of the different contributions. The derivation is based on the assumption that the near tip deformation can be described by means of relative rotations between the cross sections of the different sub-layers at the crack tip (root rotations). The root rotations depend linearly on the crack tip stress resultants through compliance coefficients that are derived numerically in the paper for a wide range of orthotropic materials. Applications to different mixed mode delamination and peeling problems, for which accurate two-dimensional finite element solutions can be found in the literature, highlight the accuracy of the proposed expressions.     

Failure mechanisms in granular media: a discrete element analysis

This paper attempts to numerically validate the concept of diffuse failure using a discrete element method. First, the theoretical background is reviewed, and it is shown how the kinetic energy of a system, initially at rest after a loading history, is likely to abruptly increase under the effect of disturbances. The vanishing of the second-order work thus constitutes a basic ingredient, related to both the pioneering work of Hill (J Mech Phys Solids (6):236–249, 1958) and the notion of bifurcation applied to geomechanics (Vardoulakis and Sulem in Bifurcation analysis in geomechanics, Chapman & Hall Publisher, London, 1995). Discrete numerical simulations were performed on homogeneous three-dimensional specimens, and the three basic conditions that must be satisfied in order to observe a failure mechanism are numerically checked. Finally, this work illustrates the phenomena that are likely to affect in situ slopes, for instance, when the loading (due to weather conditions or human activities) meets the three basic conditions for a failure mechanism to develop.     

Non-Uniform Distribution of Energy Density Near a Crack Tip in the Framework of HRR Model

The energy density distribution in the frame work of HRR model (Hutchinson, 1968a; Rice and Rosengren, 1968) is obtained. The analysis indicates that within a sector ahead of a crack, the hydrostatic stresses are very high while the effective stresses are very small, near to the yield stress, so the elastic strain energy density of volume deformation prevails in the forward sector. However, this sector is nearly a "rigid area" in HRR field (Hutchinson, 1968b). Meanwhile, a large amount of plastic dissipation distributes out of the sector, where the plastic dissipation makes up the main part of the J value. The high constraint means that the high elastic volume energy is stored in the local sector, thus the one-parameter fracture criterion based on J is invalid.     

Determination of K Id at or below NDTT using instrumented drop-weight testing

Two kinds of surface cracked specimens are numerically analyzed by the three dimensional elastic-plastic FEM. Near tip regions are divided into fine elements, and the stress and displacement fields at the crack tip are compared with HRR solutions.At first, surface cracked specimens subjected to bending are analyzed by changing the aspect ratio and depth/thickness ratio. The effect of the loading condition, crack shape and the crack depth on the stress and displacement fields are discussed. Then the pipe with surface crack subjected to bending is analyzed and the availability of the J-integral concept to the LBB analysis is discussed.In every specimen, it is shown that in the regions very near to the crack tip, the displacement field is similar to HRR solutions of plane strain. In the outer regions, however, the stress and displacement fields depend strongly on the shape, thickness, and loading conditions.     

HotQC simulation of nanovoid growth under tension in copper

We apply the HotQC method of Kulkarni et al. (J Mech Phys Solids 56:1417–1449, 2008) to the study of quasistatic void growth in copper single crystals at finite temperature under triaxial expansion. The void is strained to 30% deformation at initial temperatures and nominal strain rates ranging from 150 to 600 K and from 2.5 × 10⁵ to 2.5 × 10¹¹s⁻¹, respectively. The interatomic potential used in the calculations is Johnson’s Embedded-Atom Method potential Johnson (Phys Rev B 37:3924–3931, 1988). The computed pressure versus volumetric strain is in close agreement with that obtained using molecular dynamics, which suggests that inertia effects are not dominant for the void size and conditions considered. Upon the attainment of a critical or cavitation strain of the order of 20%, dislocations are abruptly and profusely emitted from the void and the rate of growth of the void increases precipitously. Prior to cavitation, the crystal cools down due to the thermoelastic effect. Following cavitation dislocation emission causes rapid local heating in the vicinity of the void, which in turn sets up a temperature gradient and results in the conduction of heat away from the void. The cavitation pressure is found to be relatively temperature-insensitive at low temperatures and decreases markedly beyond a transition temperature of the order of 250 K.     

Prediction of fatigue crack growth based on low cycle fatigue properties

Theoretical models of the fatigue crack growth without artificial adjustable parameters were proposed by considering the plastic strain energy and the linear damage accumulation, respectively. The crack was regarded as a sharp notch with a small curvature radius and the process zone was assumed to be the size of cyclic plastic zone. The near crack tip elastic–plastic stress and strain were evaluated in terms of modified Hutchinson, Rice and Rosengren (HRR) formulations. Predicted results from two established models have been soundly compared with open reports for frequently used materials. It is found that experimental results agree well with theoretical solutions.     

Towards a new model of crack tip stress fields

This work introduces a novel mathematical model of the stresses around the tip of a fatigue crack, which considers the effects of plasticity through an analysis of their shielding effects on the applied elastic field. The ability of the model to characterize plasticity-induced effects of cyclic loading on the elastic stress fields is assessed and demonstrated using full-field photoelasticity. The focus is on determining the form of the shielding stress components (induced by compatibility requirements at the elastic–plastic interface along the crack flank and via the crack tip plastic zone) and how they influence the crack tip elastic stress fields during a load cycle. The model is successfully applied to the analysis of a fatigue crack growing in a polycarbonate CT specimen.     

Mode I plane stress crack-tip constraint for elastic-plastic materials with different strain hardening

Finite element analyses and simulations have been undertaken to investigate the triaxial constraint in the crack-tip regions of a stationary crack and a steady-state growing crack under mode I plane stress for elastic-plastic materials with different strain hardening. The results show that the triaxial constraint in the crack-tip region is independent of specimen geometry, and material strain hardening, both for a stationary and an extending crack quasi-statically. The triaxial constraints for the various configurations examined are in better accordance with those required by the HRR solution for a stationary crack, which defines the low and similar constraints in crack-tip regions for different material strain hardening in the plane stress case. Along the entire ligament ahead of a crack tip, the constraint level transites gradually from that defined by the HRR solution within the near tip zone to that characterized by the stress intensity factor K _I in the far field.