Higher-order solution for near-tip fields of cracks growing in elastic perfectly-plastic compressible materials

The higher-order asymptotic solution of a quasi-static steadily propagating mode-I crack under the plane strain condition in an elastic perfectly-plastic compressible material is studied. In order to statisfy the higher-order compatibility equation for the rate of deformation in the centered fan sector, the stress near the crack tip is expanded asymptotically as an irregular logarithmic power series. The higher order terms near the crack tip were successfully derived. These higher order solutions are distinctly different from those for a stationary crack. The present solution for a growing crack is a one-parameter near-tip field based on a characteristic length A, through which the influence of loading and crack geometry enter into the near-tip field. This feature is substantiated by the numerical solution obtained by A.G. Varias and C.F. Shih. Comparisons between the analytic solution and the numerical results are presented.Presented at the Far East Fracture Group (FEFG) International Symposium of Fracture and Strength of Solids, 4–7 July 1994 in Xi'an, China.     

Asymptotic crack-tip fields in anisotropic power law material under antiplane shear

A hodograph transformation in conjunction with an appropriate affine transformation are both used to investigate the strain and stress fields near the crack tip in an anisotropic power law material under antiplane shear. Stress and strain exponents as well as angular distributions for the asymptotic stress and strain fields are obtained analytically. All the stress strain exponents are independent of material anisotropy, and the effect of material anisotropy on the asymptotic stress and strain field is discussed including higher order terms.     

Elastic-plastic analysis of an arbitrarily-oriented mode III crack touching an interface

In this paper we investigate a semi-infinite crack terminating at an arbitrarily oriented interface between two elastic-plastic materials under an anti-plane shear loading. An analytical solution is first developed for general power-law hardening materials under a mode III loading. If both materials have the same hardening exponent, the formulation results in a nonlinear eigenequation which can be solved numerically. The stress singularities are determined as a function of two material constants: the hardening exponent n and parameter G which represents the relative resistance of the two materials. In addition to the power of the singularity, the stress, strain and displacement asymptotic fields are also determined. If the hardening exponents are not the same, the leading order terms of an expansion model ensure the stress continuity across the interface. The results show that the stress singularity mainly depends upon the material having the larger hardening exponent, with the highest stresses in the material having the smaller hardening exponent. By taking the hardening exponent n , the perfectly plastic bimaterial problem is studied. It has been found that if the crack lies in the less stiff material, the entirely plastic asymptotic fields around the crack tip can be determined. On the other hand, if the crack lies in the stiffer material, the crack-tip fields are partially elastic and partially plastic. For both cases, unique asymptotic fields can be determined explicitly. For those cases when the materials present a strain hardening property, different mathematical models are established.     

Elastoplastic crack analysis for pressure-sensitive dilatant materials — Part I: Higher-order solutions and two-parameter characterization

An elastoplastic solution with higher-order terms for cracks in materials exhibiting pressure-sensitive yielding and plastic volumetric deformation is presented in this paper. Two-term expansions of the plane strain and plane stress solutions for a crack in a homogeneous material are obtained. It is shown that a variable-separable solution form under plane strain conditions exists only for weakly pressure-sensitive materials and the limit values of the pressure-sensitivity factor depend on the strain-hardening exponent. The second-order plane strain terms have to be solved as an eigenvalue problem and the elastic terms enter the second-order solutions only when the material has substantial strain-hardening. It follows that the second stress amplitude factor must be determined by the applied load. The values of the second exponents in the stress expansion are slightly larger than zero for most hardening materials and behave as an increasing function of the pressure-sensitivity factor. The finite element computations confirm that the second-order terms under plane strain conditions will increase dominance of the asymptotic solution remarkably. The plane stress analysis shows that the amplitudes of both leading-order and second-order solution are determined by the J-integral for most pressure-sensitive dilatant materials. The variable-separable asymptotic solution exists for all available values of the pressure-sensitivity factor. Because of rapid changes in leading-order terms of the stress component 295-1 at 160° the second-order solution will not significantly improve the prediction of the asymptotic solution in the whole tip field. Numerical results based on the incremental theory of plasticity show that the asymptotic solution characterizes the near-tip fields. Finite strains dominate in the region 295-2 under plane strain conditions. The two-parameter boundary layer formulation with different T-stresses predicts that the higher-order terms are only weakly dependent on the distance to the crack tip and vary significantly with in the forward sector.     

Analytical forms of higher-order asymptotic elastic-plastic crack-tip fields in a linear hardening material under antiplane shear

An analytical study of the higher-order asymptotic solutions of the stress and strain fields near the traction-free crack tip under antiplane shear in a linear hardening material is investigated. The results show that every term of the asymptotic fields is controlled by both elasticity and plasticity and all the higher-order asymptotic fields are governed by linear nonhomogeneous equations. The first four term solutions are presented analytically and the first four terms are described by two independent parameters J and K ₂. The amplitude of the second order term solution is only dependent on the material properties, but independent of loading and geometry. This paper focuses on the case with traction-free crack surface boundary conditions. The effects of different crack surface boundary conditions, such as clamped and mixed surfaces, on the crack-tip fields are also presented. Comparison of multi-term solution with leading term solution, and finite element solution in an infinite strip with semi-infinite crack under constant displacements along the edges is provided.     

Analytical solutions of fully plastic crack-tip higher order fields under antiplane shear

Analytical solutions of higher order fields in a fully plastic power-law hardening material are presented. By the use of hodograph transformation and asymptotic analysis the stress and strain exponents, angular distributions of shear stresses and strains are analytically determined. Special cases, such as linearly elastic, perfectly plastic materials are discussed. Similar characteristics between mode III and mode I plane strain, and mode II plane stress are examined. Comparison of four-term asymptotic solutions with exact and leading term solutions in an infinite strip with a semi-infinite crack under constant displacements along its edges is provided.     

Interface crack and nonideal interface concept (Mode III)

Asymptotic behaviour of displacements and stresses in a vicinity of the interface crack tip situated on a nonideal interface between two different elastic materials is investigated. The nonideal interface is described by special transmission conditions along the material bonding. The corresponding modelling boundary value problem is reduced to a singular integral equation with fixed point singularities. It is shown from the solution to the problem that asymptotic behaviour of displacement and stresses near the crack tip essentially depends on the model parameters. Some numerical examples are presented and discussed with respect to the stress singularity exponent and the generalized stress intensity factors.     

The mode III full-field solution in elastic materials with strain gradient effects

The near-tip asymptotic field and full-field solution are obtained for a mode III crack in an elastic material with strain gradient effects. The asymptotic analysis shows that, even though the near-tip field is governed by a single parameter B (similar to the mode III stress intensity factor), the near-tip field is very different from the classical K_III field; stresses have r ^-3/2 singularity near the crack tip, and are significantly larger than the classical K _III field within a zone of size l to the crack tip, where l is an intrinsic material length, depending on microstructures in the material. This high-order stress singularity, however, does not violate the boundness of strain energy around a crack tip. The parameter B of the near-tip asymptotic field has been determined for two anti-plane shear loadings: the remotely imposed classical K _III field, and the arbitrary shear stress tractions on crack faces. The mode III full-field solution is obtained analytically for an elastic material with strain gradient effects subjected to remotely imposed classical K _III field. It shows that the near-tip asymptotic field dominates within a zone of size 0.5 l to the crack tip, while strain gradient effects are clearly observed within 5l. It is also shown that the conventional way to evaluate the crack tip energy release rate would lead to an incorrect, infinite value. A new evaluation gives a finite crack tip energy release rate, and is identical to the J-integral. This revised version was published online in August 2006 with corrections to the Cover Date.     

Role of plasticity on interface crack initiation from a free edge and propagation in a nano-component

In order to elucidate the role of plasticity on interface crack initiation from a free edge and crack propagation in a nano-component, delamination experiments were conducted by a proposed nano-cantilever bend method using a specimen consisting of ductile Cu and brittle Si and by a modified four-point bend method. The stress fields along the Cu/Si interface at the critical loads of crack initiation and crack propagation were analyzed by the finite element method. The results reveal that intensified elastic stresses in the vicinity of the interface edge and the crack tip are very different, although the Cu/Si interface is identical in both experiments. The plasticity of Cu was then estimated on the basis of the nano-cantilever deflection measured by in situ transmission electron microscopy. The plasticity affects the stress fields; the normal stress near the interface edge is intensified while that near the crack tip is much reduced. Both the elasto-plastic stresses are close to each other in the region of about 10 nm. This suggests that the local interface fracture, namely, the crack initiation at the interface edge and the crack propagation along the interface, is governed by elasto-plastic normal stress on the order of 10 nm.     

Analysis of elastoplastic sharp notches

In this paper the elastoplastic solutions with higher-order terms for apex V-notches in power-law hardening materials have been discussed. Two-term expansions of the plane strain and the plane stress solutions have been obtained. It has been shown that the leading-order singularity approaches the value for a crack when the notch angle is not too large. In plane strain cases the elasticity does not enter the second-order solutions when the notch opening angle is too small. For a large notch angle, the two-term expansions of the plane strain near-tip fields are described by a single amplitude parameter. The plane stress solutions generally contain the elasticity terms. The boundary layer formulations based on the small-strain plasticity theory confirm that a dominance zone exists ahead of the notch tip. Finite element results give good agreement to the asymptotic solutions under both plane strain and plane stress conditions. The second-order terms cannot improve the predictions significantly. The near-tip fields are dominated by a single parameter. Finite element calculations under the finite strain J ₂-flow plasticity theory revealed that the finite strains can only affect local characterization of the asymptotic solution. The asymptotic solution has a large dominance zone around the notch tip. For an apex notch bounded to a rigid substrate the leading-order singularity falls with the notch angle significantly more slowly than in the homogeneous material. It vanishes at the notch angle about 135° for all power-hardening exponents. The elasticity effects enter the second-order solutions when the notch angle becomes large enough. The tip fields are characterized by the hydrostatic stress and the shear stress ahead of the notch.     

Higher order asymptotic solutions of V-notch tip fields for damaged nonlinear materials under antiplane shear loading

The higher order solutions of stress and deformation fields near the tip of a sharp V-notch in a power-law hardening material with continuous damage formation are analytically investigated under antiplane shear loading condition. The interaction between a macroscopic sharp notch and distributed microscopic damage is considered by describing the effect of damage in terms of a damage variable in the framework of damage mechanics. A deformation plasticity theory coupled with damage and a damage evolution law are formulated. A hodograph transformation is employed to determine the solution of damaged nonlinear notch problem in the stress plane. Then, inversion of the stress plane solution to the physical plane is performed. Consequently, higher order terms in the asymptotic solutions of the notch tip fields are obtained. Analytical expressions of the dominant and second order singularity exponents and associated angular distribution functions of notch tip stress and strain are presented. Effects of damage and strain hardening exponents and notch angle on the singular behavior of the notch tip quantities are discussed detailly. It is found that damage can lead to a weaker singularity of the dominant term of stress on one hand, but to stronger singularities of the second order term of stress and the dominant and second order terms of strain compared to that for undamaged case on the other. Also, both hardening exponent and notch angle have important effects on the notch tip quantities. Moreover, reduction of the notch tip solutions to a damaged nonlinear crack problem is carried out, and higher order solutions of the crack tip fields are obtained. Effects of damage and hardening exponents on the dominant and second order terms in the crack tip solutions are detailly discussed. Discussions on some other special cases are also presented, which shows that if damage exponent equals to zero, then the present solutions can be easily reduced to the solutions for undamaged cases. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stress singularities in a periodically layered composite near interface crack tips

The paper deals with the asymptotic analysis of stresses near interface crack tips in the periodically two-layered elastic composites. The problem is investigated for the plane state of strain within the framework of the homogenized model with microlocal parameters. The angular dependence of stresses at the crack tip is presented for different mechanical and geometrical properties of the composite.     

Stress singularities for crack normal to and terminating at bimaterial interface on orthotropic half-plates

The problem of a crack normal to and terminating at an interface in two joined orthotropic plates is considered and the eigenequation for the asymptotic behavior of stresses at the crack tip on the interface is given in an explicit form. It is found that the singular stress field around the crack tip can be separated into two independent fields, respectively of the mode I and II. Also it is found that for both the mode I and II deformations the effects of elastic constants on the stress singularity order can be respectively expressed by three material parameters, two of which are the same for both the mode I and mode II deformations.     

Interface crack problems with strain gradient effects

In this paper, the strain gradient theory proposed by Chen and Wang (2001a, 2002b) is used to analyze an interface crack tip field at micron scales. Numerical results show that at a distance much larger than the dislocation spacing the classical continuum plasticity is applicable; but the stress level with the strain gradient effect is significantly higher than that in classical plasticity immediately ahead of the crack tip. The singularity of stresses in the strain gradient theory is higher than that in HRR field and it slightly exceeds or equals to the square root singularity and has no relation with the material hardening exponents. Several kinds of interface crack fields are calculated and compared. The interface crack tip field between an elastic-plastic material and a rigid substrate is different from that between two elastic-plastic solids. This study provides explanations for the crack growth in materials by decohesion at the atomic scale.     

Dynamic growth of anti-plane shear cracks in ideally plastic crystals

A near-tip asymptotic analysis is given for the stress and deformation field near the tip of crack propagating dynamically under anti-plane shear in an ideally plastic single crystal. A paricular class of orientations of the crack relative to the crystals is considered so that the yield is so simple diamond shape (relative to directions along perpendicular to the crack) in the plane of the anti-plane shear stresses. The near-tip solution is shown to consists of sectors which carry constant stresses, at yield levels, corresponding to adjacent vertices on the diamond-shaped yield locus, and which are joined along an elastic-plastic shock discontinuity. All plastic flow in the near-tip region occurs in the shock. Plastic strains and particle velocity are finite at the crack tip. The plastic strain is proportional to the elastic strain at onset of yielding and is inversely proportional to the elastic Mach number associated with the speed of crack growth.     

Two-parameter characterization of the near-tip stress fields for a bi-material elastic-plastic interface crack

A particular case of interface cracks is considered. The materials at each side of the interface are assumed to have different yield strength and plastic strain hardening exponent, while elastic properties are identical. The problem is considered to be a relevant idealization of a crack at the fusion line in a weldment. A systematic investigation of the mismatch effect in this bi-material plane strain mode I dominating interface crack has been performed by finite strain finite element analyses. Results for loading causing small scale yielding at the crack tip are described. It is concluded that the near-tip stress field in the forward sector can be separated, at least approximately, into two parts. The first part is characterized by the homogeneous small scale yielding field controlled by J for one of the interface materials, the reference material. The second part which influences the absolute value of stresses at the crack tip and measures the deviation of the fields from the first part can be characterized by a mismatch constraint parameter M. Results have indicated that the second part is a very weak function of distance from the crack tip in the forward sector, and the angular distribution of the second part is only a function of the plastic hardening property of the reference material.     

A hybrid finite element analysis of interface cracks

A versatile hybrid finite element scheme consisting of special crack-tip elements and crack face contact elements is developed to analyse a partially closed interface crack between two dissimilar anisotropic elastic materials. The crack-tip element incorporates higher-order asymptotic solutions for an interfacial crack tip. These solutions are obtained from complex variable methods in Stroh formalism. For a closed interfacial crack tip, a generalized contact model in which the crack-tip oscillation is eliminated is adopted in the calculation. The hybrid finite element modelling allows the stress singularity at an open and closed crack tip to be accurately treated. The accuracy and convergence of the developed scheme are tested with respect to the known interface crack solutions. Utilizing this numerical scheme, the stress intensity factors and contact zone are calculated for a finite interface crack between a laminated composite material.     

Dynamic asymptotic mode III crack tip field in rate dependent materials

The asymptotic field at a dynamically growing crack tip in strain-rate sensitive elastic-plastic materials is investigated under anti-plane shear loading conditions. In the conventional viscoplasticity theory, the rate sensitivity is included only in the flow stress. However, it is often found that the yield strength is also affected by previous strain rates. The strain rate history effects in metallic solids are observed in strain rate change tests in which the flow stress decreases gradually after a rapid drop in strain rate. This material behavior may be explained by introducing the rate sensitivity in the hardening rule in addition to the flow rule. The strain-rate history effect is pronounced near the propagating crack where the change of strain rates take place. Effects of the rate dependency in the flow rule and the hardening rule on the crack propagation are analyzed. The order of the stress singularity in the asymptotic field is determined in terms of material parameters which characterize the rate sensitivity of the material. The results show that an elastic sector is present in the wake zone when the rate-dependency is considered only in the hardening rule. Terminal crack propagation speed is determined by applying the critical stress fracture criterion and the critical strain criterion to the asymptotic fields under the small scale yielding condition.     

Determination of the asymptotic crack tip fields for a crack perpendicular to an interface between elastic-plastic materials

Summary The singular behavior near a crack tip at the interface between two power-law hardening materials with the crack perpendicular to the interface is studied for both Mode I and Mode II loading under either plane strain or plane stress conditions. The mathematical model developed can be expressed as a fourth order ordinary differential equation with homogeneous boundary condition. A shooting method is applied to obtain the eigenvalues and to solve the differential equation with homogeneous boundary conditions. When both materials have the same hardening exponent,N, another material parameter, , representing the relative resistance of two materials to plastic deformation, is introduced to reflect the joint effect of the two materials on the singularity. Results indicate that if both materials have the sameN, the singularity at the crack tip is reduced as increases; however, when becomes large there appears to be little change in the singularity for a fixedN. When the hardening exponents are not the same, the mathematical model assumes stress continuity across the interface. The results show that the order of the singularity depends largely on the softer material, with the largest stresses in the harder material.     

Effects of non-singular stresses on crack-tip fields for pressure-sensitive materials,Part 1: Plane strain case

Mode I near-tip stress fields for elastic perfectly plastic pressure-sensitive materials under plane strain and small-scale yielding conditions are presented. A Coulomb-type yield criterion described by a linear combination of the effective stress and the hydrostatic stress is adopted in the analysis. The finite element computational results sampled at the distance of a few crack opening displacements from the tip show that, as the pressure sensitivity increases, the magnitudes of the normalized radial and hoop stress ahead of the tip decrease, the total angular span of the singular plastic sectors decreases, and the angular span of the elastic sectors bordering the crack surfaces increases. When non-singular T stresses are considered along the boundary layer of the small-scale yielding model, the near-tip stresses decrease as the T stress decreases. The plastic zone shifts toward the crack surfaces as the T stress increases. When the discontinuities of the radial stress and the out-of-plane normal stress along the border between the plastic sector and the elastic sector are allowed, the angular variations of the asymptotic crack-tip fields agree well with those of the finite element computations. Variation of the Q stresses for pressure-sensitive materials can be found from the asymptotic solutions when the plastic zone size ahead of the tip is relatively larger than the crack opening displacement. In addition the T stress is shown to have strong effects on the plastic zone sizes and shapes which could affect the toughening of pressure-sensitive materials.