HRR Solution Research Articles

Crack-tip constraint analysis of bending specimens for considering higher order terms in asymptotic solution

In this research, a higher-order asymptotic J-A2-A4 solution was introduced for studying the effect of the bending moment on the crack-tip constraint of a specimen. The solution was applied to TPB and SENB specimens with varying bending moment loads, and compared to other methods such as FEA, HRR solution, J-A2 solution and J-A2-M solution. The results were analyzed and compared with the FEA, J-A2 theory and J-A2-M theory solutions obtained from references. The impact of global bending on the calculation of crack-tip opening stress was also investigated. The findings showed that the J-A2-A4 solution closely matched the FEA numerical results, and better accounted for the influence of global bending compared to the J-A2 method. Therefore, there is no need to modify the J-A2 method by adding a bending moment term, as suggested in the J-A2-M theory.

Mixed-mode crack growth analysis using a cyclic plasticity model

The effects of two singularity fields, namely, the HRR solution and modified form of the nonlinear kinematic hardening (NKH) solution, on the pure mode I and mixed mode I/II fatigue crack propagation were investigated. Experiments and computations were performed on compact tension shear (CTS) specimens of P2M steel, aluminium 7050, and Ti-6Al-4V with different monotonic and cyclic elastic–plastic properties. The analysis of the experimental data on the crack growth rate for all tested materials is given in terms of plastic stress intensity factors (SIFs) according to the HRR and NKH models. The elastic–plastic crack tip fields were obtained by FE computation as a function of the position along the curvilinear crack path for the monotonic condition and the first several cycles of fatigue deformation considering the Bauschinger effect. As a complement to the FE modelling, digital image correlation (DIC) measurements of the displacement and strain fields around the crack tip in the CTS specimens were performed. This study aimed to obtain more detailed information on the local cyclic strain ahead of the mixed mode crack tip by using DIC. From the sensitivity analysis, it was established that the DIC value depended significantly on the location along the crack direction of the pair of points selected before and behind the crack tip. In this study, acceleration or retardation of the crack growth rate in terms of the NKH plastic SIF with respect to the HRR model was observed because of the coupling effect of the mode mixity and cyclic plastic properties of the tested P2M steel, aluminium 7050, and Ti-6Al-4V.

Characterization of crack-tip fields for elastoplastic fatigue crack growth Part II: Effects of crack closure and in-plane constraint

The crack-tip fields of elastoplastic fatigue crack growth with crack closure are studied and correlated to the ΔJ-integral. Computational results confirmed that the singularity of the effective stress ranges of the opening process reduces with increasing crack closure, whereas the effective stress ranges from the closing process possess the known HRR solution form. The far-field ΔJeff-integral can be used to characterize the crack-tip field stress in the closing process, which is not affected by the loading ratio and by the path dependence of the ΔJ-integral. Based on extensive FEM computations, an estimation formula for the effective ΔJeff-integral was proposed for various loading ratios and plastic hardening materials. The in-plane constraint effects in fatigue crack growth can be quantified based on the J−Q concept of elastoplastic fracture mechanics, within the frame of ΔJeff-Q characterization. The relationship between Q and the transverse stress defined in the closing processes was established for a quantitative description of the crack tip constraint.

The Improved Irwin’s Model for Crack Problems in Power-Law Hardening Materials

In this paper, an improved Irwin’s model for power-law hardening materials is established through the static equivalence between the linear elastic stress solution and the stress redistribution in the plasticity-affected zone (PAZ) under small-scale yielding (SSY) conditions. HRR solution is adopted to describe the stress field within the crack-tip plastic zone (CTPZ). Analytical expressions of the effective stress intensity factor (SIF) and the size of PAZ are derived to illustrate the effect of CTPZ on the fracture behavior for power-law hardening materials. Influences of external load and hardening exponent on fracture behavior are explored. The proposed model can assess the toughening effect of CTPZ accurately using the SIF ratio, which is always smaller than 1. This model can be applied to various materials obeying power-law hardening constitutive equations. It will reduce to the extended Irwin’s model for elastic-perfectly plastic materials in the limiting case.

Characterization of crack-tip fields for elastoplastic fatigue crack growth Part I: Analysis of the [formula omitted]-integral

In the present paper, the crack-tip field of elastoplastic fatigue crack growth without crack closure is computationally confirmed to possess the known HRR solution form, where the stress distributions are controlled by the ΔJ-integral. However, the plasticity-induced crack closure from fatigue crack growth changes the crack characterization dramatically and destroys the known singular crack-tip field form. While ΔJ shows the significant path dependency in the plastic zone, only the effective stress ranges in the unloading process can be controlled by the far-field effective ΔJ-integral. Fatigue crack growth can be characterized by the effective ΔJ-integral, with a smaller dominance zone than the conventional HRR zone.

On [formula omitted] characterization of elastic–plastic crack-tip fields under fatigue loading conditions

In the present paper, the applicability of the ΔJ-integral is computationally investigated under the Masing postulate. It confirms that the path-dependency of ΔJ is more severe than the J-integral under incremental plasticity. ΔJ from loading is up to 20% larger than ΔJ from unloading, which is independent of loading amplitudes but due to non-proportional stress variations. The stress range field can be characterized by ΔJ and shows an analogous mathematical structure as the known HRR solution, but not for the strain field under fatigue loading. The ΔJ dominant zone is much smaller than the HRR zone and further decreases with applied loading.

Finite Element Examination and Validation of Stress Field around Ductile Mode I Crack Tip in Elastic Plastic Fracture

The paper presents the stress field, obtained with the help of finite element method, around Mode I crack tip in ductile steel under EPFM regime. Process, plastic and non-K compliant elastic zones develop. Void nucleation ahead of the crack tip is observed in the process zone. The stress solution in the plastic zone is validated with theoretical HRR solution. In the elastic zone, as expected, substantial error is observed between K dominated LEFM solution and numerical EPFM values.Nomenclature

Applications of the element-free Galerkin method for singular stress analysis under strain gradient plasticity theories

Abstract It is known that the plasticity models affect characterization of the crack tip fields. To predict failure one has to understand the crack tip stress field and control the crack. In the present work the element-free Galerkin methods for gradient plasticity theories have been developed and implemented into the commercial finite element code ABAQUS and used to analyze crack tip fields. Based on the modified boundary layer formulation it is confirmed that the stress singularity in the gradient plasticity theories is significantly higher than the known HRR solution and seems numerically to equal to 0.78, independently of the strain-hardening exponent. The strain singularity is much lower than the known HRR one. The crack field in gradient plasticity under small-scale yielding condition consists of three zones: The elastic K-field, the plastic HRR-field dominated by the J-integral and the hyper-singular stress field. Even under gradient plasticity there exists an HRR-zone described by the known J-integral, whereas the hyper-singular zone cannot be characterized by J. The hyper-singular zone is very small (r ≪ J/σ0) and contained by the HRR zone in the infinitesimal deformation framework. The finite strains under the gradient plasticity will not eliminate the stress singularity as r → 0, in contrast to the known finite strain results under the Mises plasticity. Numerically no significant changes in characterization of the stress field were found in comparison with the infinitesimal deformation theory. Since the hyper-singular stress field is much smaller than the HRR zone and in the same size as the fracture process zone, one may still use the known J concept to control the crack in the gradient plasticities. In this sense the gradient plasticity will not change characterization of the crack.

Three-parameter approach for elastic–plastic stress field of an embedded elliptical crack

A three-parameter approach for an embedded center-elliptical crack under tension is presented by systematic three-dimensional (3D) elastic–plastic finite element. It is shown that even in the well embedded condition, the conventional two-dimensional HRR solution and the extended J– Q description which considers the in-plane constraint modification can hardly provide satisfied description for the crack front fields with the increase of radial distances ( r/( J/ σ 0)) and strain hardening exponent n. Thus, a consideration of the out-of-plane constraint and use of the three-parameter description is necessary and efficient to predict the 3D stress fields in the whole plastic zone.

2010 X-FEMによる三次元弾塑性破壊解析(S04-1 FEM/メッシュフリー法/粒子法とその関連技術(1),S04 FEM/メッシュフリー法/粒子法とその関連技術)

In this study, numerical procedures for thee-dimensional elastic-plastic X-FEM analysis including selections of the near-tip enrichment functions are examined. The utilized enrichment functions approximate the asymptotic displacement field near the crack tip obtained by the HRR solution. As a numerical example, a surface crack problem of elastic-plastic materials is solved to evaluate the crack opening displacement and the J-integrals along the crack front, which is evaluated using the domain integral method. Obtained results are compared with those obtained using conventional finite element analysis.

Mixed mode fracture in power law hardening materials near Mode I

Mixed mode fracture in power law hardening materials near mode I loading conditions is investigated for the case of plane strain. It is demonstrated by application of full-field finite element analysis and two-parameter asymptotic analysis that the asymptotic mixed mode solution of Shih (1974), where the effects of arbitrary loading combine to enter the leading term of the asymptotic series, does not apply close to mode I. Instead, a `mode I dominant' higher order asymptotic form applies, where the leading term is the pure mode I symmetric HRR solution and the second term is antisymmetric. The significant difference between the two asymptotic solutions is the singular nature of the antisymmetric part of the stresses and strains. Therefore, near mode I the antisymmetric part of the local fields does not contribute to the J-integral. The transition from the mixed mode asymptotic solution to this two-term solution occurs when the ratio of the radial stress on the upper crack surface to that on the lower crack surface, i.e., σrr(r,+π)/σrr(r,−π), for the limit as r approaches zero, switches from −1 to +1. Budiansky and Rice (1973) predicted this change in sign somewhere between mode I and mode II. This sign change corresponds to the singular nature of the antisymmetric fields switching from the leading HRR eigenvalue to a weaker second term eigenvalue. Based on full-field finite element results, the far-field loading for which this `jump' occurs is very difficult to identify, as it originates deep within the plastic zone at unrealistically small distances from the crack tip. At the physical length scale on the order of the crack tip opening displacement, the transition from the mode II dominant asymptotic solution to the mode I dominant solution appears to be continuous, not abrupt. Identification of this transition is estimated from the full-field results by making use of the two asymptotic solutions. These results identify a breakdown in the HRR theory for the representation of the antisymmetric part of the stresses for mixed mode fracture near mode I. This breakdown is explained by the switching from a mixed mode solution to a higher order solution. Unfortunately, the practical application of this higher-order solution is limited to hardening powers of approximately 2<n<3.

Crack-tip constraint in mode II deformation

Studies of cracked specimens loaded in mode I have shown that the stresses near the crack tip depend significantly on the level of constraint. The stresses can be determined near the crack tip using the HRR solution, but only for high constraint specimens. For other levels of constraint, O'Dowd and Shih's Q parameter may be used to adjust the stresses derived from the HRR solution. Only limited research has been carried out to study the effect of constraint in mode II. In this paper a mode II boundary layer formulation is used to study the effect of far field elastic stresses on the size and shape of the plastic zone around the crack tip and on the stresses inside the plastic zone. It is shown that in mode II, both positive and negative values of remote T-stress influence the tangential stress along the direction of maximum tangential stress. In the spirit of O'Dowd and Shih, a dimensionless parameter QII is introduced to quantify the constraint for mode II specimens failing by brittle fracture. The relation between QII and T/σ0 is determined for different values of the strain hardening coefficient n. To investigate the range of validity of the Q−T diagram for real specimens, the constraint parameter QII is calculated directly from finite element analysis for three mode II specimens and compared with the evaluation using the Q−T diagram.

Mechanical heterogeneity and validity of J-dominance in welded joints

Fracture criterion of the J-integral finds wide application in the integrity evaluation of welded components, but there exist some confused problems such as the dependence of the fracture toughness on the strength mis-matching and specimen geometry which need to be clarified. It is rough and unsuitable to attribute the variation of J-integral fracture parameter simply to the effect of mechanical heterogeneity. In the present paper, a two-dimensional finite element method is employed to analyze the distribution and variation of crack tip field of welded joints with different strength mis-matching in four kinds of specimen geometry, and then the validity of J-dominance in welded joints is investigated. It is found that the crack tip field of mis-matched joint is different from that of either the weld metal or base metal of which the joint is composed, but it is situated between those of weld metal and base metal. Under the plane strain, there is obvious difference in stress triaxiality for different strength mis-matched joints. The validity of J-dominance in welded joint can not be obtained by comparing whether the stress triaxiality meets that required by the HRR solution because of the existence of mechanical inhomogeneity. By ascertaining if the stress triaxiality of welded joint near the crack tip is dependent of specimen geometry, the conclusion can be arrived at: for plane stress the validity of J-dominance is valid, whilst for plane strain the validity of J-dominance is lost. Based on the above, attempt has been made to point out that the influence of mechanical heterogeneity on the fracture toughness of weldment arises from the variation of constraint intensity-crack tip stress triaxiality. Compared with the effect of mechanical heterogeneity on the stress triaxiality, the losing of validity of J-dominance in mis-matched joint under plane strain may play a more critical role in the variation of J-integral fracture parameter of weldment.

Dynamic crack initiation in ductile steels

The goal of the work presented here is to study dynamic crack initiation in ductile steels (Ni–Cr steel and 304 stainless steel ) at different loading rates and to establish appropriate dynamic failure criteria. A variety of infrared and visible optical methods and high-speed photography are used in this study. Precracked steel specimens are subjected to dynamic three-point bend loading by impacting them in a drop weight tower. During the dynamic deformation and fracture initiation process the time history of the transient temperature in the vicinity of the crack tip is recorded experimentally using a high-speed infrared detector. The dynamic temperature trace in conjunction with the HRR solution is used to determine the time history of the dynamic J-integral J d ( t), and to establish the dynamic fracture initiation toughness, J d c . The measurements made using high-speed thermography are validated through comparison with determination of J d ( t) by dynamic optical measurements of the crack tip opening displacement (CTOD). Finally, the micromechanisms of fracture initiation are investigated by studying the fracture surface using scanning electron microscopy.

Analytic study on angular fields of HRR solution

By using the constructing function method, a systematic and strict analysis is carried out on the angular distribution field near a crack tip in a power-law hardening material and the analytic solution is provided for HRR problem. In addition, the equivalence of H equation and RR equation is proved. The present analytic solutions for HRR problem can reduce to (he well-known Williams solution in the limit case ofN→1 (orn→1) and Prandtl solution in the limit case ofN→0 (orn→∞). It is particularly interesting that from the deformation theory of plasticity one obtains the Prandtl solution based on the increatment theory of plasticity.

A numerical study of crack-tip plasticity in glassy polymers

This paper reports on a finite element analysis of the crack-tip plastic zone and near-tip fields in viscoplastic glassy amorphous polymers. The constitutive model employed in this study accounts for the typical shear yielding behavior of glassy polymers, i.e., the intrinsic softening upon yielding and the subsequent orientational strain hardening. T'he small scale yielding, boundary layer approach is adopted to model the local finite-strain deformation processes in front of a crack with a blunt notch. Numerical results show that the shape of the plastic zone near the tip of a Mode I crack in a glassy polymer depends sensitively on the combined effect of softening and strain hardening. Softening tends to intensify the plastic deformation, while the subsequent hardening tends to depress plastic flow and gives rise to continuous propagation of the current plastic zone. Thus, the plastic zone in typical amorphous polymers is found to be quite different from the HRR solution for yielding in hardening metals. It is also found that the distribution of hydrostatic stress, which is probably responsible for crazing, is intimately related to the pattern of the plastic deformation in front of the crack tip.

Higher order crack tip fields and its implication for fracture of solids under mode II conditions

Detailed results from an asymptotic mathematical analysis for the stress and strain fields at a crack tip embedded in a power-law elastic-plastic material under mode II loading conditions are presented. The asymptotic results include the higher order terms as well as the singular term, the HRR solution. Excellent agreement with the finite element numerical results confirms the analytical findings. It is further demonstrated that the HRR solution alone is able to characterize the stress and strain fields in a large area surrounding the crack tip which encompasses the critical distance controlling the fracture event. Therefore, it is anticipated that a single parameter such as J is sufficient to characterize the fracture event under pure mode II conditions for either plane stress or plane strain case. Mixed mode fracture is also discussed.

An incremental analysis of plane strain fully plastic crack growth in strain-hardening materials under extension

Analyses of inelastic fracture have mainly followed two directions. One is crack-tip-field analysis in strain-hardening materials (e.g., the HRR solution). The other is whole field analysis in non-hardening materials (e.g., McClintock's slip-line approach). In this paper, a theoretical approach that combines the two directions is presented to account for large crack growth. As an example, plane strain mode I fully plastic crack growth in a single-edge-cracked-specimen under extension is analyzed. The incremental analysis based on the deformed configuration is developed for large crack growth in strain-hardening materials. A kinematically admissible displacement increment field with crack-tip singularity is first constructed in a presumed symmetrical triangular deformation zone extending from the crack tip to the back flank of the ligament (whole field). Then the size of the deformation zone is determined by minimizing the total force in each incremental step. The strain histories of all material points in the ligament are traced and a fracture criterion based on the hole growth theory is applied. The comparisons between the present study and the experiments existing in the literature show the validity of the present approach.

Study on Constraint Effect at Crack Tip.

The constraint effect on fracture toughness is studied. Fracture toughness tests are conducted using CT and three-point-bend specimen. Their initial crack lengths are changed from a/W=0.1 to 0.6 in five cases, which means that the constraint at the crack tip changes. It is shown that a larger fracture toughness value is obtained by a short crack specimen than by the long crack one. Two-dimensional finite-element analyses are carried out. The crack tip stress fields are compared with HRR solutions, and it is shown that the crack-tip field deviates from the HRR solution as the crack length decreases. The Q factor is evaluated for these specimens, and the relation between JIC and Q is obtained. The conventional J-integral is compared with that obtained by the contour integral, and the experimental results are corrected. The results show the effect of crack tip constraint on the fracture toughness value clearly. The same problems are also analyzed using Gurson's constitutive equation taking into consideration the void behaviors. The effect of the constraint on void nucleation and growth is studied.

A closed-form small-scale yielding collinear strip yield model for strain hardening materials

A closed-form extension to the Dugdale strip yield model allowing the consideration of strain hardening effects under small-scale yielding conditions has been developed. This model incorporates a nonuniform singular plastic zone cohesive stress which is a function of the material strain hardening exponent n and is an extension of a numerical model developed by M. Hoffmann and T. Seeger [ The Crack Tip Opening Displacement in Elastic-Plastic Fracture Mechanics, pp. 57–77. Workshop on the CTOD Methodology, Geesthacht, Germany (1985)]. The model presented employs an empirical modification of the Hoffmann and Seeger cohesive stress distribution. This modification results in improved crack surface displacement predictions. In addition, the degree of material hardening which may be considered has been extended. Attention is confined to the center cracked panel; however, the model employs weight function techniques and is applicable to arbitrary two-dimensional cracked geometries under plane stress for which a weight function is available. Closed-form analytical expressions for the plastic zone size and crack surface displacement are presented for a center cracked panel, and comparisons are made with displacements from the more rigorous HRR solution and finite element predictions.