C.F. Shih

Family of crack-tip fields characterized by a triaxiality parameter—I. Structure of fields

Abstract Central to the J-based fracture mechanics approach is the existence of a HRR near-tip field which dominates the actual field over size scales comparable to those over which the micro-separation processes are active. There is now general agreement that the applicability of the J-approach is limited to so-called high-constraint crack geometries. We review the J-annulus concept and then develop the idea of a J-Q annulus. Within the J-Q annulus, the full range of high- and low-triaxiality fields are shown to be members of a family of solutions parameterized by Q when distances are measured in terms of J σ 0 , where σ0 is the yield stress. The stress distribution and the maximum stress depend on Q alone while J sets the size scale over which large stresses and strains develop. Full-field solutions show that the Q-family of fields exists near the crack tip of different crack geometries at large-scale yielding. The Q-family provides a framework for quantifying the evolution of constraint as plastic flow progresses from small-scale yielding to fully yielded conditions, and the limiting (steady-state) constraint when it exist. The Q value of a crack geometry can be used to rank its constraint, thus giving a precise meaning to the term crack-tip constraints, a term which is widely used in the fracture literature but has heretofore been unquantified. A two-parameter fracture mechanics approach for tensile mode crack tip states in which the fracture toughness and the resistance curve depend on Q, i. JC(Q) and JR(Δa, Q), is proposed.

Family of crack-tip fields characterized by a triaxiality parameter—II. Fracture applications

Abstract C entral to the J-based fracture mechanics approach is the concept of J-dominance whereby J alone sets the stress level as well as the size scale of the zone of high stresses and strains. In Part I the idea of a J Q annulus was developed. Within the annulus, the plane strain plastic near-tip fields are members of a family of solutions parameterized by Q when distances are normalized by J σ 0 , where σ0is the yield stress, J and Q have distinct roles: J sets the size scale over which large stresses and strains develop while Q scales the near-tip stress distribution and the stress triaxiality achieved ahead of the crack. Specifically, negative (positive) Q values mean that the hydrostatic stress is reduced (increased) by Qσ0 from the Q = 0 plane strain reference state. Therefore Q provides a quantitative measure of crack-tip constraint, a term widely used in the literature concerning geometry and size effects on a materials resistance to fracture. These developments are discussed further in this paper. It is shown that the J Q approach considerably extends the range of applicability of fracture mechanics for shallow-crack geometries loaded in tension and bending, and deep-crack geometries loaded in tension. The J Q theory provides a framework to organize toughness data as a function of constraint and to utilize such data in engineering applications. Two methods for estimating Q at fully yielded conditions and an interpolation scheme are discussed. The effects of crack size and specimen type on fracture toughness are addressed.

A COMPARISON OF METHODS FOR CALCULATING ENERGY RELEASE RATES

Abstract We compare two methods for calculating the energy released during quasi-static crack advance. One method is based on a surface integral (line integral in two dimensions) expression for the energy release rate, whereas the other method is based on a volume (area) integral representation. A concise derivation of the volume (area) integral expression is given using a virtual-work-type identity for Eshelbys energy momentum tensor. The finite-element implementation of the volume (area) integral formulation corresponds to the virtual crack extension method. Within the context of this formulation, we outline a procedure for calculating pointwise values of the energy release rate along a three-dimensional crack front. For illustrative purposes, numerical examples are presented for a fully plastic, plane strain, edge-cracked panel subject to combined tension and bending.

Crack tip and associated domain integrals from momentum and energy balance

Abstract A unified derivation of crack tip flux integrals and their associated domain representations is laid out in this paper. Using a general balance statement as the starting point, crack tip integrals and complementary integrals which are valid for general material response and arbitrary crack tip motion are obtained. Our derivation emphasizes the viewpoint that crack tip integrals are direct consequences of momentum balance. Invoking appropriate restrictions on material response and crack tip motion leads directly to integrals which are in use in crack analysis. Additional crack tip integrals which are direct consequences of total energy and momentum balance are obtained in a similar manner. Some results on dual (or complementary) integrals are discussed. The study provides a framework for the derivation of crack tip integrals and allows them to be viewed from a common perspective. In fact, it will be easy to recognize that every crack tip integral under discussion can be obtained immediately from the general result by appropriately identifying the terms in the general flux tensor. The evaluation of crack tip contour integrals in numerical studies is a potential source of inaccuracy. With the help of weighting functions these integrals are recast into finite domain integrals. The latter integrals are naturally compatible with the finite element method and can be shown to be ideally suited for numerical studies of cracked bodies and the accurate calculation of pointwise energy release rates along a curvilinear three-dimensional crack front. The value of the domain integral does not depend on domain size and shape — this property provides an independent check on the consistency and quality of the numerical calculation. The success of the J -based fracture mechanics approach has led to much literature on pathindependent integrals. It will be shown that various so-called path-independent integrals (including path and area integrals) are but alternate forms of the general result referred to above and do not provide any additional information which is not already contained in the general result. Recent attempts to apply these ‘newer’ integrals to crack growth problems are discussed.

Cracks on bimaterial interfaces: elasticity and plasticity aspects

Substantial progress has been made on the mechanics of interface fracture. An engineering program has emerged which allows the fracture resistance of interfaces to be measured and utilized. This recent development, assessed in an Acta-Scripta Metallurgica Proceedings, is discussed. Several results have been obtained on the plasticity aspects of interface cracks in which one (or both) of the constituent materials can deform plastically. The crack tip fields are members of a family parametrized by plastic mode mixity parameter ξ. The J integral scales each member field. Analyses of finite width crack geometries loaded by remote tension show that the effects of load, ligament plasticity and geometry on the near-tip fields are adequately accounted for by the J integral. The progress is summarized.

Mechanics of Dynamic Debonding

Singular fields around a crack running dynamically along the interface between two anisotropic substrates are examined. Emphasis is placed on extending an established frame work for interface fracture mechanics to include rapidly applied loads, fast crack propagation and strain rate dependent material response. For a crack running at non-uniform speed, the crack tip behaviour is governed by an instantaneous steady-state, two-dimensional singularity. This simplifies the problem, rendering the Stroh techniques applicable. In general, the singularity oscillates, similar to quasi-static cracks. The oscillation index is infinite when the crack runs at the Rayleigh wave speed of the more compliant material, suggesting a large contact zone may exist behind the crack tip at high speeds. In contrast to a crack in homogeneous materials, an interface crack has a finite energy factor at the lower Rayleigh wave speed. Singular fields are presented for isotropic bimaterials, so are the key quantities for orthotropic bimaterials. Implications on crack branching and substrate cracking are discussed. Dynamic stress intensity factors for anisotropic bimaterials are solved for several basic steady state configurations, including the Yoffe, Gol’dshtein and Dugdale problems. Under time-independent loading, the dynamic stress intensity factor can be factorized into its equilibrium counterpart and the universal functions of crack speed.

Continuum and micromechanics treatment of constraint in fracture

Two complementary methodologies are described to quantify the effects of crack-tip stress triaxiality (constraint) on the macroscopic measures of elastic-plastic fracture toughness J and Crack-Tip Opening Displacement (CTOD). In the continuum mechanics methodology, two parameters J and Q suffice to characterize the full range of near-tip environments at the onset of fracture. J sets the size scale of the zone of high stresses and large deformations while Q scales the near-tip stress level relative to a high triaxiality reference stress state. The materials fracture resistance is characterized by a toughness locus Jc(Q) which defines the sequence of J-Q values at fracture determined by experiment from high constraint conditions (Q∼0) to low constraint conditions (Q<0). A micromechanics methodology is described which predicts the toughness locus using crack-tip stress fields and critical J-values from a few fracture toughness tests. A robust micromechanics model for cleavage fracture has evolved from the observations of a strong, spatial self-similarity of crack-tip principal stresses under increased loading and across different fracture specimens. We explore the fundamental concepts of the J-Q description of crack-tip fields, the fracture toughness locus and micromechanics approaches to predict the variability of macroscopic fracture toughness with constraint under elastic-plastic conditions. Computational results are presented for a surface cracked plate containing a 6:1 semielliptical, a=t/4 flaw subjected to remote uniaxial and biaxial tension. Crack-tip stress fields consistent with the J-Q theory are demonstrated to exist at each location along the crack front. The micromechanics model employs the J-Q description of crack-front stresses to interpret fracture toughness values measured on laboratory specimens for fracture assessment of the surface cracked plate.

Fracture normal to a bimaterial interface: Effects of plasticity on crack-tip shielding and amplification

Abstract The problem of a crack approaching a bimaterial interface is considered in this paper. Attention is focused on an interface between two elastoplastic solids whose elastic properties are identical and whose plastic properties are different. For the case of a crack approaching a bimaterial interface perpendicularly, it is shown by recourse to detailed finite element analyses that the near-tip “driving force” for fracture is strongly influenced by whether the crack approaches the interfaces from the lower strength or the higher strength materials. Specifically, it is demonstrated that the crack-tip is “shielded” from the remote loads when it approaches the interface from the weaker material, and that the effective J-integral at the crack tip is greater than the remote J when it approaches the interface from the stronger material. This plasticity effect determines whether a crack approaching the bimaterial interface continues to advance through the interface or is arrested before penetrating the interface. These theoretical findings are substantiated using controlled experiments of fatigue crack growth perpendicular to a ferrite—austenite bimaterial interface. The effect of the non-singular T-stress, acting parallel to the crack plane, on shielding and amplification of the stress fields is also discussed.

A theory for cleavage cracking in the presence of plastic flow

Abstract A theory is proposed for cleavage cracking surrounded by pre-existing dislocations. Dislocations are assumed not to emit from the crack front. It is argued that the pre-existing dislocations, except for occasional interceptions with the crack front, are unlikely to blunt the major portion of the crack front, so that the crack front remains nanoscopically sharp, advancing by atomic decohesion. The fracture process therefore consists of two elements: atomic decohesion and background dislocation motion. An elastic cell, of size comparable to dislocation spacing or dislocation cell size, is postulated to surround the crack tip. This near-tip elasticity accomodates a large stress gradient, matching the nanoscopic, high cohesive strength to the macroscopic, low yield strength. Consequences of this theory are explored in the context of slow cleavage cracking, stress-assisted corrosion, fast running crack, fatigue crack growth, constraint effects, and mixed mode fracture along metal/ceramic interfaces. Computational models and experiments to ascertain the range of validity of this theory are proposed.

Kink band formation and band broadening in fiber composites under compressive loading

Various stages of kink band formation, propagation and band width broadening were recorded by a high resolution video camera. Based on these observations, the easiest modes of deformation have been identified and these form the basis of a new kinematic model for kinking. Theoretical predictions for kink band orientation and compression strength under steady-state band broadening are made. The conditions at incipient kinking and the incipient kinking stress are investigated. The relevance of the incipient kinking stress and the band broadening stress are discussed.