Matched asymptotic expansions in nonlinear fracture mechanics—III. In-plane loading of an elastic perfectly-plastic symmetric specimen

The matched expansion method, introduced by the authors in two earlier papers (1976) devoted to mode III loading, is applied to the practically important case of mode I loading of a symmetric specimen. The method allows the linear elastic far-field to be considered separately from the elasto-plastic near-tip field, except for coupling through a set of parameters that are determined explicitly in the matching. The effects of the plasticity are thus found, once and for all, from the solution of a set of standard elasto-plastic problems for a semi-infinite crack in an infinite body, whose properties may be tabulated. The solution for any particular specimen geometry and loading then follows from a small set of linear elastic solutions for the specimen, which define, through coefficients γ_ij appearing in their near-tip expansions, all the parameters in the “inner” and “outer” solutions. The effects of plasticity appear in these parameters only through a set of constants C_ti that define the far-field expansions of the “inner” (near-tip) solutions: they are material constants, depending upon the constitutive relation for the material, but not upon specimen geometry and loading. The J-integral, being obtainable from the far-field, is expressed as an explicit asymptotic series in the loading parameter ε, whose coefficients are given as functions of the “elastic” parameters γ_ij and the material constants C_i. It is demonstrated that a plastic-zone correction term, r_y, can be chosen to yield a two-term asymptotic expansion for J; the value of r_y depends upon the yielding model only through the constant C₁.The Dugdale (1960) model of yielding is treated, as a simple example for which all calculations can be performed analytically, and for which exact solutions are available for comparison.Finally, the near-tip solutions are constructed for a material obeying the Mises yield criterion and associated flow-rule, using a specially developed finite element program. The first eight of the constants C_i are tabulated, which suffice to define the J-integral up to terms of order ε⁶ (where ε is a loading parameter) and some representative near-tip features are displayed graphically. The computed value of C₁ shows that the conventionally adopted value for the plastic-zone correction r_y is too large by a factor of roughly 2.8, if it is to yield a genuine asymptotic estimate for J. As an example, the “elastic” parameters γ_ij are found, from a boundary collocation program, for a centre-cracked square plate subjected to tensile loading; and a plot of J versus load, and the plastic-zone shape at a particular load level, are displayed.     

Analysis of the strain field in the vicinity of a crack-tip in an in-plane isotropic paper material

Strains, computed by the finite element method, are evaluated and compared to an experimentally determined strain field. The analyzed low-density paper has been designed to ensure bond–breakage as the dominating damage mechanism and the paper material is approximately in-plane isotropic. An optical non-contact displacement measuring system has been used in fracture tests to determine the strain field in the crack-tip region of a pre-fabricated crack. Additionally, acoustic emission monitored tensile tests have been conducted to determine onset and evolution of damage processes and thereby enabling calibration of required constitutive parameters. The results suggest that the investigated paper material can tolerate significantly higher strains than what is predicted by a classic elastic–plastic J₂-flow theory. Immediately before onset of the final fracture (i.e., localization), the experimental measured normal strain in the near-tip region is around 60% higher than the computed strain when using exclusively an elastic–plastic theory for the corresponding load while the strain computed utilizing a non-local damage theory is of the same order of magnitude as the experimentally measured strain. Hence, it seems essential to include a non-local continuum theory to describe strains in the near-tip region quantitatively correct for paper materials. It is demonstrated that path independence of the well-known J-integral does not prevail for this class of material models. Only for the special situation of a homogenous damage field in the crack-tip region may the stress and strain fields be described by the well-known HRR-solutions.     

Using M-integral for multi-cracked problems subjected to nonconservative and nonuniform crack surface tractions

In this paper, an energy parameter based on the concept of the M-integral is proposed for describing the fracture behavior of a multi-cracked solid subjected to nonconservative and nonuniform crack surface tractions. By using the M-integral with a suitably chosen closed contour, one can evaluate the ‘surface creation energy’ (SCE) required for creation of the stressed cracks. Also, it is demonstrated that the property of path-independence holds even under the action of crack surface tractions. Therefore, the singular stress field in the near-tip areas is not directly involved in the calculation so that a complicated finite element model around the crack tips is not required in evaluation of the M-integral.     

Effect of constraint induced by crack depth on creep crack-tip stress field in CT specimens

In this paper, the effect of constraint induced by the crack depth on creep crack-tip stress field in compact tension (CT) specimens is examined by finite element analysis, and the effect of creep deformation and damage on the Hutchinson–Rice–Rosengren (HRR) singularity stress field are discussed. The results show the constraint induced by crack depth causes the difference in crack-tip opening stress distributions between the specimens with different crack depth at the same C*. The maximum opening stress appears at a distance from crack tips, and the stress singularity near the crack tips does not exist due to the crack-tip blunting caused by the large creep deformation in the vicinity of the crack tips. The actual stress calculated by the finite element method (FEM) in front of crack tip is significantly lower than that predicted by the HRR field. Based on the reference stress field in the deep crack CT specimen with high constraint, a new constraint parameter R is defined and the constraint effect in the shallow crack specimen is examined at different distances ahead of the crack tip from transient to steady-state creep conditions. During the early stages of creep constraint increases with time, and then approaches a steady state value as time increases. With increasing the distance from crack tips and applied load, the negative R increases and the constraint decreases.     

A reexamination of plasticity-induced crack closure in fatigue crack propagation

The crack closure concept is often used to consider the R-ratio and overload effects on fatigue crack growth. The presumption is that when the crack is closed, the external load produces negligible fatigue damage in the cracked component. The current investigation provides a reassessment of the frequently used concept with an emphasis on the plasticity-induced crack closure. A center cracked specimen made of 1070 steel was investigated. The specimen was subjected to plane-stress mode I loading. An elastic–plastic stress analysis was conducted for the cracked specimens using the finite element method. By applying the commonly used one-node-per-cycle debonding scheme for the crack closure simulations, it was shown that the predicted crack opening load did not stabilize when the extended crack was less than four times of the plastic zone size. The predicted opening load was strongly influenced by the plasticity model used. When the elastic–perfectly plastic (EPP) stress–strain relationship was used together with the kinematic hardening plasticity theory, the predicted crack opening load was found to be critically dependent on the element size of the finite element mesh model. For R = 0, the predicted crack opening load was greatly reduced when the finite element size became very fine. The kinematic hardening rule with the bilinear (BL) stress–strain relationship predicted crack closure with less dependence on the element size. When a recently developed cyclic plasticity model was used, the element size effect on the predicted crack opening level was insignificant. While crack closure may occur, it was demonstrated that cyclic plasticity persisted in the material near the crack tip. The cyclic plasticity was reduced but not negligible when the crack was closed. The traditional approaches may have overestimated the effect of crack closure in fatigue crack growth predictions.     

Analytical and experimental determination of dynamic impact stress intensity factor for 40 Cr steel

A linear elastic three-dimensional finite element analysis is made to analyze the near field stress behavior of an edge cracked rectangular bar simply supported and subjected to central impact at the back side of the crack. The material is made of 40 Cr steel. Determined numerically are the local time histories of the stress wave, displacement near load point, crack tip strain, and dynamic stress intensity factor K_(d)¹. The above quantities were also measured experimentally by performing impact tests; they agreed well with the analytical results and determine the load at fracture initiation and hence the critical dynamic stress intensity factor K_(d)^1c. The interaction effect between the loading bar and specimen appears to be negligible.     

Matched asymptotic expansions in nonlinear fracture mechanics—II. Longitudinal shear of an elastic work-hardening plastic specimen

Earlier analysis given by T.M. Edmunds and J.R. Willis (1976) is extended to deal with cracks in elastic work-hardening plastic specimens subjected to longitudinal shear loads. Solutions are expressed in terms of a set of parameters that are determined from linear elastic solutions alone. It is proved, for any specimen geometry and any loading symmetric about the plane of the crack, that a ‘plastic-zone correction’, obtained by solving a linear elastic problem for a crack which is a length r_y longer than the actual crack, provides a two-term asymptotic expansion for the J-integral, if r_y is defined suitably in terms of the linear elastic stress concentration factor and the initial slope of the work-hardening curve. The general method is applied in detail to a strip of finite width containing an edge crack, for which the effect of the work-hardening on the maximum extent of the plastic zone and on the J-integral is summarized graphically.     

Creep behavior of crack near bi-material interface characterized by integral parameters

Creep behavior of crack in dissimilar materials is studied using steady-state C^* path independent integral and ABAQUS finite element code. The specific geometry involves an edge crack parallel to the interface of a bi-material tensile specimen at high temperature. Under extensive creep, the C^* value for the bi-material specimen can be significantly higher than that for the homogeneous specimen. For small-scale creep material mismatch has little influence on the transient integral designated by C_t. The integral parameters C^* or C_t are shown to depend on the inhomogeneity of the system and cannot characterize the creep behavior of cracks.The approach is extended to creep crack growth in a welded compact tension specimen. Modification factors are introduced for different crack and weld interface geometries.     

Finite-deformation analysis of the crack-tip fields under cyclic loading

The plane-strain crack subjected to mode I cyclic loading under small scale yielding was analysed. The influence of the load range, load ratio and overload on the near-tip deformation-, stress- and strain-fields was studied. Although the near-tip zones of appreciable cyclic plastic flow for all loading regimes matched closely one another, when scaled with (ΔK/σ_Y)², the activities of plastic flow within them manifested dependence on K_max and K_min, as well as on overload. Cyclic trajectories of the crack-tip opening displacement (CTOD) converged to stable self-similar loops of the sizes proportional to ΔK² and positions in CTOD-K plane dependent on the maximum K along the whole loading route, including an overload. Computed near-tip deformation evidenced plastic crack advance, this way visualising of the Laird–Smith concept of fatigue cracking. This crack growth by blunting-resharpening accelerated with rising ΔK and was halted by an overload. Crack closure upon unloading had no place. The affinities were revealed between computed near-tip stress–strain variables and the experimental trends of the fatigue crack growth rate, such as its dependence on K_max and K_min (or ΔK and K_max), and retardation by overload. Thus, the effects of loading parameters on fatigue cracking, hitherto associated with crack closure, are attributable to the stress–strain fields in front of it as the direct drives of the key fatigue constituents – damage accumulation and bond breaking.     

On the fracture behavior of inhomogeneous materials––A case study for elastically inhomogeneous bimaterials

This paper presents a case study, examining the influence of a sharp bimaterial interface on the effective crack driving force in a fracture mechanics specimen. The inhomogeneity of the elastic modulus in linear elastic and non-hardening and hardening elastic–plastic bimaterials is considered. The interface is perpendicular to the crack plane. The material properties and the distance between the crack tip and the interface are systematically varied. The effect of the material inhomogeneity is captured in form of a quantity called “material inhomogeneity term”, C_inh. This term can be evaluated either by a simple post-processing procedure, following a conventional finite element stress analysis, or by computing the J-integral along a contour around the interface, J_int. The effective crack driving force, J_tip, can be determined as the sum of C_inh and the nominally applied far-field crack driving force, J_far. The results show that C_inh can be accurately determined by both methods even in cases where J_tip-values are inaccurate. When a crack approaches a stiff/compliant interface, C_inh is positive and J_tip becomes larger than J_far. A compliant/stiff transition leads to a negative C_inh, and J_tip becomes smaller than J_far. The material inhomogeneity term, C_inh, can have the same order of magnitude as J_far. Based on the numerical results, the dependencies of C_inh on the material parameters and the geometry are derived. Simple expressions are obtained to estimate C_inh.     

Development of a multiaxial fatigue theory by considering constraint effects on small mixed-mode cracks

The effects of the transverse strain (the normal strain in the crack-line direction) on the near-tip fields of small shallow surface cracks (Case A cracks) in power-law hardening materials are investigated by finite element analyses. The small Case A cracks are under plane stress, general yielding, and mixed mode I and II conditions. Constant effective stress contours representing the intense straining zones near the tip, deformed crack-tip profiles and near-tip mode mixity factors are presented for different transverse strains in the crack-line direction. Based on the concept of characterization of fatigue crack growth by the cyclic J-integral, the effects of the transverse strain on J are investigated. The results suggest that the fatigue life prediction based on multiaxial fatigue theories and the critical plane approach should include the constraint effects due to the transverse strain. Consequently, the concept of constant fatigue life contour on the Γ-plane in multiaxial fatigue theories is generalized to the constant fatigue life surface in the Γ-space where the shear strain and the two normal strains are the three axes. Finally, a damage parameter as a function of the shear strain and the two normal strains is proposed for evaluation of fatigue damage under multiaxial loading conditions.     

A stress analytical solution for Mode III crack within modified gradient elasticity

The strain gradient exists near a crack tip may significantly influence the near-tip stress field. In this paper, the strain gradient and the internal length scales are introduced into the basic equations of mode III crack by the modified gradient elasticity (MGE). By using a complex function approach, the analytical solution of stress fields for mode III crack problem is derived within MGE. When the internal length scales vanish, the stress fields can be simplified to the stress fields of classical linear elastic fracture mechanics. The results show that the singularity of the shear stress is made up of two parts, r^−1/2 part and r^−3/2 part, and the sign of the stress σ_yz changes. With the increase of l_x, the peak value of σ_yz decrease and its location moves farther from the fracture vertex. The influence of strain gradient for mode III crack problem cannot be ignored.     

High strain-rate crack growth in rate-dependent plastic solids

At high crack velocities in metallic materials nearly all plastic strain accumulates at very high strain-rates, typically in the range 10³ s^?1 to 10⁵ s^?1. At these rates, dislocation motion is limited by dynamic lattice effects and the plastic strain-rate increases approximately linearly with stress. The problem for a crack growing at high velocity is posed for steady-state, small scale yielding in elastic/rate-dependent plastic solids. A general expression is derived for the near-tip stress intensity factor in terms of the remote intensity factor, or equivalently for the near-tip energy release-rate in terms of the overall release-rate. An approximate calculation of the plastic strain-rates provides this relation in analytical form. Imposition of the condition that the near-tip energy release-rate be maintained at a critical value provides a propagation equation for the growing crack. A single, nondimensional combination of material constants emerges as the controlling parameter. Implications for dynamic crack propagation are discussed.     

Energy dissipation and contour integral characterizing fracture behavior of incremental plasticity

Jep -integral is derived for characterizing the frac- ture behavior of elastic-plastic materials. The J ep -integral differs from Rice’s J-integral in that the free energy density rather than the stress working density is employed to define energy-momentum tensor. The J ep -integral is proved to be path-dependent regardless of incremental plasticity and deformation plasticity. The J epintegral possesses clearly clear physical meaning: (1) the value J ep tip evaluated on the infinitely small contour surrounding the crack tip represents the crack tip energy dissipation; (2) when the global steadystate crack growth condition is approached, the value of J ep farss calculated along the boundary contour equals to the sum of crack tip dissipation and bulk dissipation of plastic zone. The theoretical results are verified by simulating mode I crack problems.     

Finite deformation analysis of crack-tip opening in elastic-plastic materials and implications for fracture

Analyses of the stress and strain fields around smoothly-blunting crack tips in both non-hardening and hardening elastic-plastic materials, under contained plane-strain yielding and subject to mode I opening loads, have been carried out by use of a finite element method suitably formulated to admit large geometry changes. The results include the crack-tip shape and near-tip deformation field, and the crack-tip opening displacement has been related to a parameter of the applied load, the J-integral. The hydrostatic stresses near the crack tip are limited due to the lack of constraint on the blunted tip, limiting achievable stress levels except in a very small region around the crack tip in power-law hardening materials. The J-integral is found to be path-independent except very close to the crack tip in the region affected by the blunted tip. Models for fracture are discussed in the light of these results including one based on the growth of voids. The rate of void-growth near the tip in hardening materials seems to be little different from the rate in non-hardening ones when measured in terms of crack-tip opening displacement, which leads to a prediction of higher toughness in hardening materials. It is suggested that improvement of this model would follow from better understanding of void-void and void-crack coalescence and void nucleation, and some criteria and models for these effects are discussed. The implications of the finite element results for fracture criteria based on critical stress or strain, or both, is discussed with respect to transition of fracture mode and the angle of initial crack-growth. Localization of flow is discussed as a possible fracture model and as a model for void-crack coalescence.     

The surface-forming energy release rate based fracture criterion for elastic–plastic crack propagation

The J-integral based criterion is widely used in elastic–plastic fracture mechanics. However, it is not rigorously applicable when plastic unloading appears during crack propagation. One difficulty is that the energy density with plastic unloading in the J-integral cannot be defined unambiguously. In this paper, we alternatively start from the analysis on the power balance, and propose a surface-forming energy release rate (ERR), which represents the energy available for separating the crack surfaces during the crack propagation and excludes the loading-mode-dependent plastic dissipation. Therefore the surface-forming ERR based fracture criterion has wider applicability, including elastic–plastic crack propagation problems. Several formulae are derived for calculating the surface-forming ERR. From the most concise formula, it is interesting to note that the surface-forming ERR can be computed using only the stress and deformation of the current moment, and the definition of the energy density or work density is avoided. When an infinitesimal contour is chosen, the expression can be further simplified. For any fracture behaviors, the surface-forming ERR is proven to be path-independent, and the path-independence of its constituent term, so-called J_s-integral, is also investigated. The physical meanings and applicability of the proposed surface-forming ERR, traditional ERR, J_s-integral and J-integral are compared and discussed. Besides, we give an interpretation of Rice paradox by comparing the cohesive fracture model and the surface-forming ERR based fracture criterion.     

三维计算断裂力学

断裂力学理论从1921年Griffith研究玻璃的脆性断裂问题开始,经历了从线弹性体系到弹塑性以及蠕变理论体系、从单参数到多参数体系和从理想的二维平面理论到实际三维含裂纹结构的三维断裂理论的发展历程。针对应力强度因子K和J积分以及C(t)积分的计算方法从理想化模型的理论计算发展到实际复杂工程结构裂纹体计算的各种商业软件平台以及专业的断裂理论分析平台。尤其是随着计算机技术的发展,对三维含裂结构的静态和扩展裂纹的计算模拟已经能够融入计算机辅助设计。结合本研究组近30年来在三维疲劳断裂理论和应用研究方面的体会,简述了三维计算断裂力学从裂纹体应力应变分析和断裂参数计算到三维蠕变断裂和疲劳裂纹扩展模拟的国内外进展,并对涉及的计算方法,包括原子尺度和跨尺度的计算模拟,以及目前面临的挑战性问题作了简要介绍和分析。     

Transient crack growth under creep conditions

Transient crack growth in an elastic/power-law creeping material is investigated under antiplane shear loading and small-scale-yielding conditions. At time t = 0 the solid is suddenly loaded far from the crack by tractions that correspond to the elastic crack-tip stress distribution. At that time the crack begins to propagate at a constant velocity. The stress fields evolve in a complex manner as the crack propagates due to the competing effects of stress relaxation due to constrained creep and stress elevation due to the instantaneous elastic material response to crack growth. From detailed finite element calculations it is shown that these fields can be approximated by a simple matching of three asymptotic singular crack-tip solutions. A characteristic stress, distance and time are defined for this problem which provide a normalization that accounts for any crack velocity, loading and all material properties for a given creep exponent n. Results are presented for crack-tip stresses, strains, crack opening displacements and creep zones.     

Crack separation energy-rates for the DBCS model under biaxial modes of loading

A Modified version of the Dugdale-Bilby-Cottrell-Swinden (DBCS) model simulating the effect of plasticity at the tip of a crack in an infinite region was used by kfouri and rice (1978) to calculate the crack separation energy-rate G^Δ corresponding to a finite crack growth step Δa during plane strain mode I crack extension. The loading consisted of a remote uniaxial tension σp applied normally to the plane of the crack. Using Rice's path-independent integral J to characterize the applied load in the crack tip region, and assuming the length R of the crack tip plastic zone to be small compared with the length of the crack, an analytical expression was derived relating the ratios (J/G^Δ) and (2a/R) for small values of (2a/R). The analytical solution was incomplete in itself in that the value assumed in the plastic region of the DBCS model for the normal stress Y acting on the extending crack surfaces was the product of the yield stress in uniaxial tension σ_Y and an unknown parameter C, the value of which depends on the effect of the local hydrostatic stresses in the case of plane strain conditions. The analytical solution was compared with a numerical solution obtained from a plane strain elastic-plastic finite element analysis on a centre-cracked plate (CCP) of material obeying the von Mises yield criterion. The value used for the yield stress was 310 MN/m² and moderate isotropic linear hardening was applied with a tangent modulus of 4830 MN/m². A uniaxial tension σp was applied on the two appropriate sides of the plate. The comparisons showed that the analytical and finite element solutions were mutually consistent and they enabled the value of C to be established at 1.91. In the present paper similar comparisons are made between the analytical solution and the finite element solutions for the CCP of the same material under different biaxial modes of loading. By assuming further that the form of the analytical solution does not depend on the details of the geometry and of the loading at remote boundaries, a comparison has also been made with the results of a finite element analysis on a compact tension specimen (CTS) made of the same material as the CCP. The different values of C obtained in each case are consistent with investigations by other authors on the effect of load biaxiality on crack tip plasticity.