Family of crack-tip fields characterized by a triaxiality parameter—II. Fracture applications
TL;DR: In this paper, the J-dominance is used to define 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.
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 material's 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.
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TL;DR: In this paper, the first overview of failure of metals is presented, focusing on brittle and ductile failure under monotonic loadings, where the focus is on linking microstructure, physical mechanisms and overall fracture properties.
639 citations
Cites methods from "Family of crack-tip fields characte..."
...In the framework of fracture mechanics, the constraint effect can be quantified through either the so-called T-stress (for small scale yielding only) or the so-called Q-stress, which measures the departure from the HRR solution [146,147]....
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TL;DR: In this article, a technical review of fracture toughness testing, evaluation and standardization for metallic materials in terms of the linear elastic fracture mechanics as well as the elastic-plastic fracture mechanics is given.
594 citations
TL;DR: In this article, a phase-field model for ductile fracture of elasto-plastic solids in the quasi-static kinematically linear regime is proposed, which captures the entire range of behavior of a ductile material exhibiting $$J_2$$J2-PLasticity, encompassing plasticization, crack initiation, propagation and failure.
Abstract: Phase-field modeling of brittle fracture in elastic solids is a well-established framework that overcomes the limitations of the classical Griffith theory in the prediction of crack nucleation and in the identification of complicated crack paths including branching and merging. We propose a novel phase-field model for ductile fracture of elasto-plastic solids in the quasi-static kinematically linear regime. The formulation is shown to capture the entire range of behavior of a ductile material exhibiting $$J_2$$J2-plasticity, encompassing plasticization, crack initiation, propagation and failure. Several examples demonstrate the ability of the model to reproduce some important phenomenological features of ductile fracture as reported in the experimental literature.
522 citations
TL;DR: In this paper, the authors present a review of the material constitutive equations and computational tools which have been recently developed to simulate ductile rupture and fracture, which are used in structural computations.
Abstract: The past 20 years have seen substantial work on the modeling of ductile damage and fracture. Several factors explain this interest. (i) There is a growing demand to provide tools which allow to increase the efficiency of structures (reduce weight, increase service temperature or load, etc.) while keeping or increasing safety. This goal is indeed first achieved by using better materials but also by improving design tools. Better tools have been provided which consist (ii) of material constitutive equations integrating a physically-based description of damage processes and (iii) of better numerical tools which allow to use the improved constitutive equations in structural computations which become more and more realistic. This article reviews the material constitutive equations and computational tools, which have been recently developed to simulate ductile rupture.
471 citations
Cites methods from "Family of crack-tip fields characte..."
...Q two parameters approach (O’Dowd and Shih, 1991, 1992)....
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TL;DR: In this paper, the Gurson constitutive relation for dilatant plasticity describes the hole growth in a cell resulting in material softening and, ultimately, loss of stress carrying capacity.
Abstract: Many metals which fail by a void growth mechanism display a macroscopically planar fracture process zone of one or two void spacings in thickness characterized by intense plastic flow in the ligaments between the voids; outside this region, the voids exhibit little or no growth. To model this process a material layer containing a pre-existing population of similar sized voids is assumed. The thickness of the layer, D, can be identified with the mean spacing between the voids. This layer is represented by an aggregate of computational cells of linear dimension D. Each cell contains a single void of some initial volume. The Gurson constitutive relation for dilatant plasticity describes the hole growth in a cell resulting in material softening and, ultimately, loss of stress carrying capacity. The collection of cells softened by hole growth constitutes the fracture process zone of length l1. Two fracture mechanism regimes can be identified corresponding to l1 ≈ D and l1 ⪢ D. The connection between these mechanisms and fracture resistance is discussed.
Finite element calculations have been carried out to determine crack growth resistance curves for plane strain, mode I crack growth under small scale yielding. A row of voided cells is placed on the symmetry plane ahead of the initial crack. These cell elements are embedded within a conventional elastic-plastic continuum. Under increasing load, the voids in the cells grow and coalesce to form a new crack surface thereby advancing the crack. Resistance curves are calculated for crack growth exceeding many multiples of D. The parameters affecting fracture resistance are discussed emphasizing the roles of microstructural parameters and continuum properties of the material. The effect of crack tip constraint on fracture resistance is examined under small scale yielding by way of the T-stress. As a final application, resistance curves for a deep and a shallow crack bend bar are computed. These are compared with experimental data.
322 citations
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TL;DR: In this paper, an integral is exhibited which has the same value for all paths surrounding a class of notches in two-dimensional deformation fields of linear or non-linear elastic materials.
Abstract: : An integral is exhibited which has the same value for all paths surrounding a class of notches in two-dimensional deformation fields of linear or non-linear elastic materials. The integral may be evaluated almost by inspection for a few notch configurations. Also, for materials of the elastic- plastic type (treated through a deformation rather than incremental formulation) , with a linear response to small stresses followed by non-linear yielding, the integral may be evaluated in terms of Irwin's stress intensity factor when yielding occurs on a scale small in comparison to notch size. On the other hand, the integral may be expressed in terms of the concentrated deformation field in the vicinity of the notch tip. This implies that some information on strain concentrations is obtainable without recourse to detailed non-linear analyses. Such an approach is exploited here. Applications are made to: Approximate estimates of strain concentrations at smooth ended notch tips in elastic and elastic-plastic materials, A general solution for crack tip separation in the Barenblatt-Dugdale crack model, leading to a proof of the identity of the Griffith theory and Barenblatt cohesive theory for elastic brittle fracture and to the inclusion of strain hardening behavior in the Dugdale model for plane stress yielding, and An approximate perfectly plastic plane strain analysis, based on the slip line theory, of contained plastic deformation at a crack tip and of crack blunting.
7,468 citations
TL;DR: In this paper, the authors investigated the C rack-tip strain singularities with the aid of an energy line integral exhibiting path independence for all contours surrounding a crack tip in a two-dimensional deformation field of an elastic material (or elastic/plastic material treated by a deformation theory).
Abstract: C rack-tip strain singularities are investigated with the aid of an energy line integral exhibiting path independence for all contours surrounding a crack tip in a two-dimensional deformation field of an elastic material (or elastic/plastic material treated by a deformation theory). It is argued that the product of stress and strain exhibits a singularity varying inversely with distance from the tip in all materials. Corresponding near crack tip stress and strain fields are obtained for the plane straining of an incompressible elastic/plastic material hardening according to a power law. A noteworthy feature of the solution is the rapid rise of triaxial stress concentration above the flow stress with increasing values of the hardening exponent. Results are presented graphically for a range of hardening exponents, and the interpretation of the solution is aided by a discussion of analogous results in the better understood anti-plane strain case.
2,890 citations
TL;DR: In this paper, a total deformation theory of plasticity, in conjunction with two hardening stress-strain relations, is used to determine the dominant singularity at the tip of a crack in a tension field.
Abstract: D istributions of stress occurring at the tip of a crack in a tension field are presented for both plane stress and plane strain. A total deformation theory of plasticity, in conjunction with two hardening stress-strain relations, is used. For applied stress sufficiently low such that the plastic zone is very small relative to the crack length, the dominant singularity can be completely determined with the aid of a path-independent line integral recently given by rice (1967). The amplitude of the tensile stress singularity ahead of the crack is found to be larger in plane strain than in plane stress.
2,667 citations
TL;DR: In this paper, the critical value of tensile stress (a) for unstable cleavage fracture to the fracture toughness (K,,) for a high-nitrogen mild steel under plane strain conditions.
Abstract: SUMMARY AN ANALYSIS is presented which relates the critical value of tensile stress (a,) for unstable cleavage fracture to the fracture toughness (K,,) for a high-nitrogen mild steel under plane strain conditions. The correlation is based on (i) the model for cleavage cracking developed by E. Smith and (ii) accurate plastic*lastic solutions for the stress distributions ahead of a sharp crack derived by J. R. Rice and co-workers. Unstable fracture is found to be consistent with the attainment of a stress intensification close to the tip such that the maximum principal stress a,, exceeds a, over a characteristic distance, determined as twice the grain size. The model is seen to predict the experimentally determined variation of K,, with temperature over the range -150 to -75°C from a knowledge of the yield stress and hardening properties. It is further shown that the onset of fibrous fracture ahead of the tip can be deduced from the position of the maximum achievable stress intensiiication. The relationship between the model for fracture ahead of a sharp crack, and that ahead of a rounded notch, is discussed in detail.
1,374 citations
TL;DR: In this article, 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.e., JC(Q) and JR(Δa, Q), is proposed.
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.
1,023 citations