The anisotropic R-criterion for crack initiation

The anisotropic nature of mixed modes I-II crack tip plastic core region and crack initiation is investigated in this study using an angled crack plate problem under various loading conditions. Hill’s anisotropic yield criterion along with singular elastic stress field at the crack tip is employed to obtain the non-dimensional variable-radius crack tip plastic core region. In addition, the R-criterion for crack initiation proposed by the authors for isotropic materials is also extended to include anisotropy. The effect of Hill’s anisotropic constants on the shape and size of the crack tip plastic core region and crack initiation angle is presented for both plane stress and plane strain conditions at the crack tip. The study shows a significant effect of anisotropy on the crack tip core region and crack initiation angle and calls for further development of anisotropic crack initiation theory.     

混凝土结构I型裂纹裂尖塑性区研究

应用双剪统一强度理论,研究了I型裂纹的塑性变形问题。给出了包含反映材料拉压性能差异的参数拉压比及反映中间主应力效应的参数b的I型裂纹裂尖塑性区形状和大小的统一解。已有的Tresca准则、Mises准则和Mohr-Coulomb准则解均是本文的特例或线性逼近。针对混凝土结构,画出了不同参数b情况下的裂尖塑性区半径变化图。得出了材料拉压比对I型裂纹裂尖塑性区影响很大。b对I型裂纹裂尖塑性区影响随拉压比的不同而不同,拉压比较大时,b对塑性区影响大,拉压比较小时,b对塑性区影响小的结论。该统一解可以适应于各种不同材料,能充分发挥材料潜力,具有普遍性和广泛的适应性,有一定的工程应用价值。结论对于研究各种材料的断裂问题有参考作用。     

The influence of aluminium alloy anisotropic texture on crack-tip plasticity

Crack‐tip plasticity in textured aluminium alloys was numerically analysed using an anisotropic yield function and an isotropic hardening law as the material constitutive response. Four real textures obtained from extrusions of rectangular and square shapes as well as five ideal textures typical of rolled products were considered and predicted crack‐tip plastic zones were compared with those obtained using the isotropic von Mises yield criterion. The use of the anisotropic yield function revealed important differences in crack‐tip plasticity compared with the isotropic case. The plastic zone features obtained for different textures were compared to single crystal results published in the open literature and similar crack‐tip plastic strain localization was observed.     

An estimation of the plastic zone size for a bimaterial interfacial crack

This note deals with the size of the plastic zone ahead of an interfacial crack between two dissimilar isotropic elastic materials. Dugdale's concept of finite stress at the tip of the crack is used in the study. The plastic zone size is determined for plane stress problems under the von Mises yield condition.     

The effect of load ratio on the threshold stresses for fatigue crack growth in medium carbon steels

A study of the effect of load ratio, R, on the low crack growth rates and the threshold stress conditions exhibited by five medium carbon steels has been conducted. It was found that decreased values of R retarded crack growth to an increasing degree as a defined threshold for crack growth was approached, such that this threshold showed a marked dependence on K max rather than ΔK. Comparisons of the five materials showed that the threshold increased as yield strength increased for a given R, but this effect could be normalised in terms of a constant value of the maximum plastic zone size at the crack tip.     

A continuously distributed dislocation model for fatigue cracks in anisotropic crystalline materials

An elastic-plastic crack model is developed for anisotropic crystalline materials exhibiting elastic-perfect plastic behavior, using the continuously distributed dislocation theory (CDDT). The crack and its associated plastic zone are represented by an inverted pile-up of dislocations. The plastic zone size, the crack-tip opening displacement (CTOD) and the energy release rate of the crack are obtained in closed forms, in a fashion similar to the Bilby–Cottrell–Swinden–Dugdale model for isotropic materials, but in terms of material constants that are orientation dependent. Application of the model to the case of small-scale yielding, e.g., long fatigue cracks in Ni-base single crystal superalloys, is discussed.     

Closed-Form Solutions for the Mode ii Crack tip Plastic Zone Shape

Analytical closed-form solutions for the Mode II crack tip plastic zone shape have been derived for a semi-infinite crack in an isotropic elastic-plastic solid under both plane stress and plane strain conditions. Two yield criteria have been applied: the Von Mises and Tresca yield criteria. The results indicate that the Tresca zone is larger in size than the Von Mises zone. The results also reveal an intricate dependence on Poisson's ratio in plane strain conditions.     

Elastic and plastic stress analysis of an interface crack between two dissimilar layers

In the present paper, the mixed-mode Dugdale model is applied to investigate the plastic zone size and the crack tip opening displacement of an interface crack between two dissimilar layers. In the analysis, both normal and shear stresses are assumed to exist in the plastic zones and satisfy the Von Mises yield criterion. The plastic zone sizes can be determined on condition that the stress intensity factors caused by the stresses in the plastic zones and applied loading are zero. Then, the crack tip opening displacement can be obtained by dislocation theories. In numerical examples, the plane stress condition is considered. The plastic zone size and the crack tip opening displacement of an interface crack between two dissimilar layers under a uniform load are examined. The effects of Dundurs’ parameters and the thickness of materials on the plastic zone size and the crack tip opening displacement are investigated in detail. Numerical results show that in the case of small thickness, the values of the normalized plastic zone size and the normalized crack tip opening displacement decrease with increasing Dundurs’ parameters, α and β, while, in the case of infinite thickness, the value of the normalized plastic zone size is independent of α, and the value is symmetric about the axis on which β = 0.     

The form of crack tip plastic zones

The “radius” of the plastic zone at a crack tip is a parameter that has numerous applications in fracture mechanics. However, attention is drawn here to the confusion that is apparent, even in text-books, concerning the calculation of the plastic zone “radius” under plane strain conditions. The aim of this work has been to resolve this point, to determine the actual shape and size of the zone and to investigate the influence of stress state and other factors.The plastic zone dimensions have been simply calculated, over a range of values of Poisson's ratio, for isotropic materials subjected to loading under plane stress and plane strain conditions; the analysis has been further extended to cover some effects of anisotropy. It has been demonstrated that, for isotropic materials, the maximum extent of the plastic zone directly ahead of, and in the plane of, a crack is ($\text{K}{}_{\mathrm{I}}\text{/Y)}{}^2\text{18π}$ under plane stress loading and is ($\text{K}{}_{\mathrm{I}}\text{/Y)}{}^2\text{18π}$ under plane strain loading. This latter result is smaller, by a factor of $\text{1}\text{3}$ than the plastic zone “radius” under plane strain conditions that is widely quoted in fracture mechanics texts. That “radius”, ($\text{K}{}_{\mathrm{I}}\text{/Y)}{}^2\text{6π}$ is, in fact, the maximum size of the zone parallel to, but not in, the plane of the crack, if Poisson's ratio is taken to be $\text{1}\text{3}$.A lower value of Poisson's ratio or an increased material anisotropy can lead to an enlarged plastic zone; this latter conclusion suggests that test-pieces for valid fracture toughness measurements on anisotropic materials could be required to be larger than defined in the relevant British Standard.     

A finite element analysis of plane strain dynamic crack growth in materials displaying the Bauschinger effect

In this paper dynamic crack growth in an elastic-plastic material is analyzed under mode I plane strain small-scale yielding conditions using a finite element procedure. The main objective of this paper is to investigate the influence of anisotropic strain hardening on the material resistance to rapid crack growth. To this end, materials that obey an incremental plasticity theory with linear isotropic or kinematic hardening are considered. A detailed study of the near-tip stress and deformation fields is conducted for various crack speeds. The results demonstrate that kinematic hardening does not oppose the role of inertia in decreasing the plastic strains and stresses near the crack tip with increase in crack speed to the same extent as isotropic strain hardening. A ductile crack growth criterion based on the attainment of a critical crack opening displacement at a small micro-structural distance behind the tip is used to obtain the dependence of the theoretical dynamic fracture toughness with crack speed. It is found that for any given level of strain hardening, the dynamic fracture toughness displays a much more steep increase with crack speed over the quasi-static toughness for the kinematic hardening material as compared to the isotropic hardening case.     

On the Plastic Zone Size and the Crack Tip Opening Displacement of an Interface Crack Between two Dissimilar Materials

In the paper, the elastic-plastic fracture behavior of an interface crack between two dissimilar materials is investigated. The mixed-mode Dugdale model is applied to examine the plastic zone size and the crack tip opening displacement. In numerical examples, the plastic zone size and the crack tip opening displacement of an interface crack under uniform loads are studied in detail. Two formulae are proposed to calculate the plastic zone size and the crack tip opening displacement of an interface crack under small scale yielding conditions.     

Plastic zone formation and fatigue crack extension during high cyclic bending of steels

In repeated high cyclic bending, with constant load amplitude, the size and the shape of the plastic zone preceding the propagating crack is controlled by local structural conditions near the tip rather than by stress intensity. No significant correlations were found between the experimentally determined sizes of plastic zone and the theoretically predicted values of Liu and Rice. The plastic zone sizes ahead of the propagating crack cannot be simply expressed as proportional to the rate of fatigue crack propagation, though a simple relationship exists between the rate and the stress intensity factor. The relationship given by Paris, dl/dN = QΔKⁿ, describes the rate of crack propagation only in a limited range of relative crack length, x < 0.5. The extent of this range depends on the structure and on the level of applied cyclic stress. Beyond this range, the Paris equation could not be applied and the crack propagation cannot be related to the stress intensity factor.     

Plastic yielding at the tip of a blunt notch during static and fatigue loading

Rice's analytical Mode III solution for the relationship between anti-plane stress and anti-plane strain was used to determine the small scale plastic yielding at the tip of a two-dimensional blunt notch. The results were applied to fatigue loading. The plastic zone size and crack opening displacement derived in the present analysis were determined as functions of applied stress, geometric factors (notch radius and length) and material properties (yield stress and the work hardening rate). The minimum stress intensity required for plastic yielding at a blunt notch tip was postulated to be the experimentally observed threshold stress intensity for fatigue crack initiation. The threshold stress intensity so determined depends not only on the notch geometry but also on material properties. There is good agreement with calculated and measured values of the threshold stress intensity for fatigue crack initiation.     

A finite element investigation of quasi-static and dynamic asymptotic crack-tip fields in hardening elastic-plastic solids under plane stress

The asymptotic structures of crack-tip stress and deformation fields are investigated numerically for quasi-static and dynamic crack growth in isotropic linear hardening elastic-plastic solids under mode I, plane stress, and small-scale yielding conditions. An Eulerian type finite element scheme is employed. The materials are assumed to obey the von Mises yield criterion and the associated flow rule. The ratio of the crack-tip plastic zone size to that of the element nearest to the crack tip is of the order of 1.6 × 10⁴. The results of this study strongly suggest the existence of crack-tip stress and strain singularities of the type r ^s (s < 0) at r=0, where r is the distance to the crack tip, which confirms the asymptotic solutions of variable-separable type by Amazigo and Hutchinson [1] and Ponte Castañeda [2] for quasi-static crack growth, and by Achenbach, Kanninen and Popelar [3] for dynamic crack propagation. Both the values of the parameter s and the angular stress and velocity field variations from the present full-field finite element analysis agree very well with those from the analytical solutions. It is found that the dominance zone of the r ^s-singularity is quite large compared to the size of the crack-tip active plastic zone. The effects of hardening and inertia on the crack-tip fields as well as on the shape and size of the crack-tip active plastic zone are also studied in detail. It is discovered that as the level of hardening decreases and the crack propagation speed increases, a secondary yield zone emerges along the crack flank, and kinks in stress and velocity angular variations begin to develop. This dynamic phenomenon observable only for rapid crack growth and for low hardening materials may explain the numerical difficulties, in obtaining solutions for such cases, encountered by Achenbach et al. who, in their asymptotic analysis, neglected the existence of reverse yielding zones along the crack surfaces.     

Elastic and plastic fracture analysis of a crack perpendicular to an interface between dissimilar materials

Elastic and plastic fracture analysis of a Mode I crack perpendicular to an interface between dissimilar materials is carried out. Continuously distributed dislocations are used to simulate the crack. The simulation will cause singular integral equations with Cauchy kernel. By solving the singular integral equations numerically, the effects of crack depth (distance from the interface to the crack middle point) and Dundurs’ parameters on the Mode I stress intensity factor are investigated systematically. Then, based on the Dugdale model, the plastic zone size, and the crack tip opening displacement of the crack under uniform loadings are investigated. The effects of uniform loadings, crack depth, and Dundurs’ parameters on the plastic zone size and the crack tip opening displacement are examined. Numerical results show that when the crack is embedded in a stiffer material, the values of both the normalized plastic zone size and the normalized crack tip opening displacement are larger than 1. On the contrary, if the crack is embedded in a softer material, the values of both the normalized plastic zone size and the crack tip opening displacement are less than 1.     

Prediction of fatigue crack growth based on low cycle fatigue properties

Theoretical models of the fatigue crack growth without artificial adjustable parameters were proposed by considering the plastic strain energy and the linear damage accumulation, respectively. The crack was regarded as a sharp notch with a small curvature radius and the process zone was assumed to be the size of cyclic plastic zone. The near crack tip elastic–plastic stress and strain were evaluated in terms of modified Hutchinson, Rice and Rosengren (HRR) formulations. Predicted results from two established models have been soundly compared with open reports for frequently used materials. It is found that experimental results agree well with theoretical solutions.     

Plasticity spread in mild steel pre-cracked charpy bars

During dynamic service loading, small fatigue cracks are normally seen to emanate from the root of sharp machined stress concentration region. In a recent authors' paper at ICF4, the fracture strength of a charpy type notched beam has been studied in three different engineering materials, when a small fatigue crack emanates from the notch root. Fracture tests on these medium and high strength materials demonstrate the presence of a large size plastic zone near the crack tip [1–6]. To understand the mechanism of fracture for such complex geometry. it is important to know the size and shape of these plastically yielded regions. The present paper is mainly on the experimental measurement of plasticity spread as well as the stress intensity factor (S.I.F.) for such short cracks. Firstly, the S.I.F. is approximately measured by plain transmission photoelasticity on model castolite specimens. Secondly, plastic zones around crack tip are measured for a wide range of notch root radii and crack-length, by using photo-stress PS-3B coating on mild steel pre-cracked charpy type notched specimens. It is observed that for small scale yielding at the crack tip, the plasticity spread is around 60–65° angle to the line of crack-extension. On the contrary, as the gross applied stress approaches the yield strength of the material, the maximum plasticity spreads around 45° angle. Finally, it is noticed that at high stress level, the plastic zone boundary (for short crack) touches the free machined notch surface. These experimental observations explain the nature and degree of non-linearity in a load—C.O.D. diagram during the fracture test of a short cracked-notched specimen. These data are also useful to predict the crack-extension load for an elastic-plastic material.     

Arbitrarily oriented microcracks near the tip of an interface macrocrack in bonded dissimilar anisotropic materials

It is well known that microcracking in brittle materials results in a reduction of the stress intensity factor (SIF) and energy release rate (ERR). The reduced SIF or ERR represents crack tip shielding which is of significant interest to micromechanics and material science researchers. However, the effect of microcracking on the SIF and ERR is a complicated subject even for isotropic homogeneous materials, and becomes much more formidable in case of interface cracks in bonded dissimilar solids. To unravel the micromechanics of interface crack tip shielding in bonded dissimilar anisotropic solids, an interface crack interacting with arbitrarily oriented subinterface microcracks in bonded dissimilar anisotropic materials is studied. After deducing the fundamental solutions for a subinterface crack under concentrated normal and tangential tractions, the present interaction problem is reduced to a system of integral equations which is then solved numerically. A J‐integral analysis is then performed with special attention focused on the J₂‐integral in a local coordinate system attached to the microcracks. Theoretical and numerical results reassert the conservation law of the J‐integral derived for isotropic materials 1 , 2 also to be valid for bonded dissimilar anisotropic materials. It is further concluded that there is a wastage when the remote J‐integral transmits across the microcracking zone from infinity to the interface macrocrack tip. In order to highlight the influence of microstructure on the interfacial crack tip stress field, the crack tip SIF and ERR in several typical cases are presented. It is interesting to note that the Mode I SIF at the interface crack tip is quite different from the ERR in bonded dissimilar anisotropic materials.     

Analysis of crack tip plasticity for microstructurally small cracks using crystal plasticity theory

Crack tip plastic zone sizes and crack tip opening displacements (CTOD) for stationary microstructurally small cracks are calculated using the finite element method. To simulate the plastic deformation occurring at the crack tip, a two-dimensional small strain constitutive relationship from single crystal plasticity theory is implemented in the finite element code ANSYS as a user-defined plasticity subroutine. Small cracks are modeled in both single grains and multiple grains, and different crystallographic conditions are considered. The computed plastic zone sizes and CTOD are compared with those found using conventional isotropic plasticity theory, and significant differences are observed.     

Effects of crack size and weld metal mismatch on the has cleavage toughness of wide plates

The heat affect zone (HAZ) is in many cases considered to be the most critical part of a weldment. In this paper, the effect of crack size and weld metal mismatch on the HAZ cleavage toughness of wide plate specimens with X-groove has been investigated by the J-Q-M theories and a simple micromechanism for cleavage fracture. Two crack sizes have been studied (a/w = 0.1 and 0.3). In the analyses, the HAZ yield strength is assumed to be higher than the base metal. For each crack size, weld metal local overmatch and local evenmatch with respect to the HAZ are considered. For a given global strain, the results indicate that weld metal overmatch and evenmatch yield the same crack tip loading in terms of J-integral for a/w = 0.3. For a/w = 0.1, overmatch gives lower crack tip loading than evenmatch. For a given crack tip loading, weld metal local evenmatch in general results in less effective crack tip loading than the overmatch. Overmatch is detrimental to HAZ toughness, but this detrimental effect becomes less significant when the crack size decreases.