General expressions for the stress intensity factor of a one-point bend beam

The aim of this work is to find general expressions to determine the stress intensity factor of a one-point bend beam-like specimen, whether from the measurement of the applied load or the crack mouth opening displacement. The expressions, obtained by applying the superposition principle, involve the decomposition of the general case into three auxiliary problems. The solution of two of them (pure bending and three-point bending) is well known, while the solution of the third (one-point bending) is developed in the present work. The proposed expressions are compared to numerical results obtained by the finite element method and their accuracy is equal to or better than available expressions published elsewhere.     

SIMPLE Jc, ESTIMATION FORMULAE FOR THREE-POINT BEND SPECIMENS

This report reviews some simple methods for determining J_c at the onset of cleavage fracture in a three-point bend crack tip opening displacement test. It is concluded that acceptable estimates of J, can be made from specimen mouth opening displacement without precise measurement of crack length.     

Effects of crack depth on elastic-plastic fracture toughness

Short crack test specimens (a/W 0.50) are frequently employed when conventional deep crack specimens are either inappropriate or impossible to obtain, for example, in testing of particular microstructures in weldments and in-service structures containing shallow surface flaws. Values of elastic-plastic fracture toughness, here characterized by the crack tip opening displacement (CTOD), are presented for square (cross-section) three-point bend specimens with a/W ratios of 0.15 and 0.50 throughout the lower-shelf and lower-transition regions. Three dimensional, finite-element analyses are employed to correlate the measured load and crack mouth opening displacement (CMOD) values to the corresponding CTOD values, thus eliminating a major source of experimental difficulty in previous studies of shallow crack specimens. In the lower-transition region, where extensive plasticity (but no ductile crack growth) precedes brittle fracture, critical CTOD values for short crack specimens are significantly larger (factor of 2–3) than the CTOD values for deep crack specimens at identical temperatures. Short crack specimens are shown to exhibit increased toughness at the initiation of ductile tearing and decreased brittle-to-ductile transition temperatures. Numerical analyses for the two a/W ratios reveal large differences in stress fields ahead of the crack tip at identical CTOD levels which verify the experimentally observed differences in critical CTOD values. Correlations of the predicted stresses with measured critical CTOD values demonstrate the limitations of single-parameter fracture mechanics (as currently developed) to characterize the response.     

Influence of crack depth and strength matching on deformation and plastic zone for welded joint specimen

By using the 2-dimensional elasto-plastic finite element method, the effects of crack depth and strength matching properties of welds on the deformation and plastic zone have been studied for three point bend specimens. The results indicate that, in the loading process of a welded joint specimen, the influence of crack depth and strength matching properties on the deformation parameters such as loading point displacement, crack mouth opening displacement, plastic rotational factor, and development of plastic zone is evident. The boundary between shallow crack and deep crack specimens is also influenced by the strength matching of welded joints. Moreover, the crack depth and strength matching of the welded joints have an important effect on the fracture-resistant capability of the specimens. The fracture-resistant capabilities of shallow crack specimens and overmatched joint specimens are better than those of deep crack specimens and undermatched joint specimens, respectively.     

聚丙烯纤维水泥稳定碎石断裂性能试验研究

通过30个尺寸为100mm×100mm×515mm的聚丙烯纤维水泥稳定碎石和普通水泥稳定碎石三点弯曲试件断裂试验,探讨了聚丙烯纤维对水泥稳定碎石断裂韧度(KIC)、断裂能(GF)、临界裂缝嘴张开位移(CMODC)、临界裂缝尖端张开位移(CTODC)、极限裂缝嘴张开位移(CMODmax)和极限裂缝尖端张开位移(CTODmax)的影响。试验结果表明:聚丙烯纤维的掺入可增大水泥稳定碎石的断裂韧度、断裂能、临界裂缝嘴张开位移、极限裂缝嘴张开位移、临界裂缝尖端张开位移和极限裂缝尖端张开位移;随着聚丙烯纤维体积掺量的增加,断裂韧度、临界裂缝嘴张开位移和临界裂缝尖端位移的变化无明显规律,但断裂能、极限裂缝嘴张开位移和极限裂缝尖端位移基本上呈线性增加的。     

Numerical investigation on stable crack growth in plane stress

Large deformation finite element analysis has been carried out to investigate the stress-strain fields ahead of a growing crack for compact tension (a/W=0.5) and three-point bend (a/W=0.1 and 0.5) specimens under plane stress condition. The crack growth is controlled by the experimental J-integral resistance curves measured by Sun et al. The results indicate that the distributions of opening stress, equivalent stress and equivalent strain ahead of a growing crack are not sensitive to specimen geometry. For both stationary and growing cracks, similar distributions of opening stress and triaxiality can be found along the ligament. During stable crack growth, the crack- tip opening displacement (CTOD) resistance curve and the cohesive fracture energy in the fracture process zone are independent of specimen geometry and may be suitable criteria for characterizing stable crack growth in plane stress. This revised version was published online in August 2006 with corrections to the Cover Date.     

Determination of K Id at or below NDTT using instrumented drop-weight testing

Two kinds of surface cracked specimens are numerically analyzed by the three dimensional elastic-plastic FEM. Near tip regions are divided into fine elements, and the stress and displacement fields at the crack tip are compared with HRR solutions.At first, surface cracked specimens subjected to bending are analyzed by changing the aspect ratio and depth/thickness ratio. The effect of the loading condition, crack shape and the crack depth on the stress and displacement fields are discussed. Then the pipe with surface crack subjected to bending is analyzed and the availability of the J-integral concept to the LBB analysis is discussed.In every specimen, it is shown that in the regions very near to the crack tip, the displacement field is similar to HRR solutions of plane strain. In the outer regions, however, the stress and displacement fields depend strongly on the shape, thickness, and loading conditions.     

Fully plastic analyses of plane strain single-edge-cracked specimens subject to combined tension and bending

Accurate yield surfaces of plane strain single-edge-cracked specimens having shallow as well as deep cracks are developed using finite element limit analyses and monotonic interpolation functions. Fully plastic shallow crack configurations are classified based on certain aspects of the yield surfaces. Relationships between incremental plastic crack tip and crack mouth opening displacements and incremental load point displacement/rotation are obtained for a wide range of relative crack depths and loading ratios. Fully plastic crack-tip fields for a sufficiently deep crack in a single-edge cracked specimen are examined to provide the stress triaxiality and the angular orientation of flow line at the crack tip in terms of the remotely applied tension-to-bending ratio. Evidence for fully plastic crack-tip stress fields consisting of an incomplete Prandtl fan and a crack plane constant state region is discussed.     

On fractography of shallow and deep HY-100 cracked bend specimens

The influence of shallow cracks on the fracture behavior of structural components has been studied extensively in recent years. Finite element analyses have indicated dramatic differences in the crack-tip stress states between shallow and deep cracked bend specimens. In this study, an experimental program was carried out to investigate the fracture behavior of HY-100 steel containing various initial flow depths. Four a/w ratios ranging from 0.05 to 0.5 were chosen for the notched three-point bend tests. Test results showed that higher fracture toughness values are associated with specimens having shorter surface cracks. Also, fractographic studies indicated that two sets of dimples are present for a/w = 0.5 specimen, one set of equiaxed dimple for a/w = 0.05 specimen near the crack initiation zone. As the crack grows, increases in the volume fraction of the small dimple were observed. Finally, it showed that the characteristic features of surfaces can be correlated with the previous numerical predictions.     

Influence of state of stress on mixed mode stable crack growth through D16AT aluminium alloy

The stable crack growth through three-point bend (TPB) and stiffened and unstiffened compact tension (CT) specimens of D16AT aluminium alloy has been studied both theoretically and experimentally. The specimen thickness is 8 mm. The variation of load with crack opening displacement, the extent of stable crack growth, the cumulative plastically deformed zone and crack edge profiles have been obtained experimentally. These are also predicted theoretically under the assumption of either a state of plane stress or plane strain using a finite element scheme and the COA criterion. Generally, the experimental results agree well with the predictions based on the plane stress condition. There appears to be no significant variation in size of the experimental cumulative plastic zone across the specimen thickness, thereby indicating that the constraint on the plastic zone does not develop near the mid-thickness region.     

Plastic η factor solutions of homogeneous and bi-material SE(T) specimens for toughness and creep crack growth testing

Based on slip line field analysis and finite element analysis of elastic-perfectly plastic materials, plastic η factor solutions for single edge-cracked specimens in tension (SE(T)) with a wide range of crack lengths are proposed, both for homogeneous specimens and for bi-material specimens with interface cracks. Moreover, two different plastic η factor solutions are given: one based on experimental load–load line displacement records, ηVLLp , and the other based on experimental load–crack mouth opening displacement (CMOD) records, ηCMODp . Comparison with existing finite element results shows good agreement. For deep cracks (a/w > ∼0.45), the ηVLLp solutions are insensitive to the strain hardening, to the specimen length and to the specimen thickness. However, for shallower cracks (a/w < ∼0.45), the ηVLLp solutions are sensitive to the specimen thickness, to the strain hardening and to the specimen length, suggesting difficulties associated with a robust determination of J and C * integrals from experimental data. On the other hand, the ηCMODp solution is not sensitive to the crack length, to the specimen thickness, to strain hardening and to the specimen length, even for shallow cracked specimens. This suggests that the use of CMOD can provide robust J and C * estimation schemes even for shallow crack testing.     

Automatic 3-D crack propagation calculations: a pure hexahedral element approach versus a combined element approach

This article presents an evaluation of two different crack prediction approaches based on a comparison of the stress intensity factor distribution for three example problems. A single edge notch specimen and a quarter circular corner crack specimen subjected to shear displacements and a three point bend specimen with a crack inclined to the mid-plane are examined. The stress intensity factors are determined from the singular stress field close to the crack front. Two different fracture criteria are adopted for the calculation of an equivalent stress intensity factor and crack deflection angle. The stress intensity factor distributions for both numerical methods agree well to available reference solutions. Deviations are recorded at crack front locations near the free surface probably due to global contraction effects and the twisting behaviour of the crack front. Crack propagation calculations for the three point bending specimen give results that satisfy intuitive expectations. The outcome of the study encourages further pursuit of a crack propagation tool based on a combination of elements.     

Numerical simulations of specimen size and mismatch effects in ductile crack growth - Part I: Tearing resistance and crack growth paths

The Gurson-Tvergaard-Needleman (GTN) model has been used for detailed numerical simulations of the effects of specimen size and yield stress mismatch on ductile crack growth behaviour in two different finite specimen geometries. For deep cracked bending specimens the crack growth resistance, expressed through the far-field J, increases as the specimen size is reduced, most strongly seen in case of low hardening. An opposite effect can be seen to some extent for shallow cracked specimens loaded in tension for low and intermediate hardening. For the yield stress mismatch cases low hardening and bend loading are found to promote crack growth deviation away from the initial crack plane.     

J c DETERMINATION FOR SHALLOW NOTCH WELDED BEND SPECIMENS

Abstract— Methods are presented for the determination of J from 3 point bend specimens having 0.1 ≤ a/W ≤ 0.5 using the load versus crack mouth opening displacement trace. It is intended that the procedure be used to determine J_c , J at unstable fracture in the brittle to ductile transition, from specimens and test procedures which conform to the CTOD test standard BS5762. Elastic-plastic finite element computations are used to demonstrate the adequacy of the proposed estimation procedures for J and to show that it is still possible to apply them to specimens containing welded joints. The problems of defining a centre of rotation for plastic deformation in shallow notched bend specimens are discussed.     

小尺寸混凝土试件双K断裂参数试验研究

采用最大尺寸为680mm×160mm×40mm的标准三点弯曲梁试件,利用在初始裂缝两侧粘贴电阻应变片并利用混凝土裂缝扩展到此处时其应变回缩的方法测得了起裂荷载Pini,在此基础上根据Pini及初始缝长a0得到了起裂断裂韧度KIiCni;根据在试验中测得的最大荷载Pmax及对应的裂缝口张开位移CMODC计算了混凝土等效裂缝长度aC,据此计算了失稳断裂韧度KIuCn。结果表明:采用电阻应变片法可准确测定混凝土的起裂荷载Pini,且方法简单。试验结果还表明:在本试验范围内,三点弯曲梁法测得的混凝土双K断裂参数KIiCni、KIuCn与试件高度无关,进一步说明了混凝土双K断裂参数可以作为描述混凝土裂缝扩展的断裂参数。     

On the analysis of a mixed mode bending sandwich specimen for debond fracture characterization

The mixed mode bending specimen originally developed for mixed mode delamination fracture characterization of unidirectional composites has been extended to the study of debond propagation in foam cored sandwich specimens. The compliance and strain energy release rate expressions for the mixed mode bending sandwich specimen are derived based on a superposition analysis of solutions for the double cantilever beam and cracked sandwich beam specimens by applying a proper kinematic relationship for the specimen deformation combined with the loading provided by the test rig. This analysis provides also expressions for the global mode mixities. An extensive parametric analysis to improve the understanding of the influence of loading conditions, specimen geometry and mechanical properties of the face and core materials has been performed using the derived expressions and finite element analysis. The mixed mode bending compliance and energy release rate predictions were in good agreement with finite element results. Furthermore, the numerical crack surface displacement extrapolation method implemented in finite element analysis was applied to determine the local mode mixity at the tip of the debond.     

Implications of test methodology on post-cracking and fracture behaviour of Steel Fibre Reinforced Concrete

It is now universally recognized that the mechanical, cracking and fracture, properties of Steel Fibre Reinforced Concrete (SFRC) are far superior to those of plain concrete. The use of SFRC contributes effectively to preserve the structural stability and structural integrity of concrete elements and improve their ductile behaviour.To optimize the performance of SFRC in structural members it is necessary to establish the mechanical properties very precisely. The best test methodology to evaluate the post-cracking and toughness properties of SFRC is the beam bending test. Design codes recommend one of two bending test configurations: the three-point or the four-point bending test. The results obtained from these two test configurations are not identical.The overall focus of this paper is to evaluate the contributions of fibres to the post-cracking and fracture behaviour of concrete as determined by the two different standard test procedures. To achieve these aims plain and fibre concrete specimens were tested. All the test specimens were extensively instrumented to establish the strength properties, crack tip and crack mouth opening displacement, post-cracking and fracture behaviour. The results of the two types of bending tests were then critically analysed and evaluated to identify the differing effects of the bending load configurations on material and structural behaviour.SFRC specimens subjected to four-point bending test showed higher stress values compared to those obtained from the three-point bending tests. The first crack strength values evaluated following the two standards are close with an improvement of 10% for the European standard.     

COMPATIBLE COMPLIANCE AND STRESS INTENSITY EXPRESSIONS FOR THE STANDARD THREE-POINT BEND SPECIMEN

Available expressions for load-line compliance and crack-mouth opening compliance expressions for standard three-point bend specimens (span over width ratio =4) are assessed. It is concluded that none of the available expressions are able to reproduce the source data (boundary collocation results and theoretical limits for relative crack depths equal to 0 and 1) for the entire range of relative crack depth. New expressions for example are proposed and the accuracy of each is indicated. A by-product of the new expression for load-line compliance is a new, fully compatible stress intensity expression that is, consistent with the load line compliance expression as known from theoretical considerations. Also inverse expressions are presented which predicting the crack length from compliance. To improve the accuracy of such inverse expressions, a correction that results in a considerable error reduction is presented.     

Mode II fracture mechanics properties of wood measured by the asymmetric four-point bending test using a single-edge-notched specimen

Using a single-edge-notched specimen of spruce, an asymmetric four-point bending test was conducted to obtain the mode II fracture toughness G_IIc and critical stress intensity factor K_IIc, and the test method was numerically and experimentally analyzed. A three-point bend end-notched flexure test was also conducted and the results were compared with those of the asymmetric four-point bending tests. The crack length had a small influence on the load/loading-line displacement relationship in the asymmetric four-point bending test, so it was difficult to determine the value of G_IIc, which requires the measurement of loading-line displacement. In contrast, the value of K_IIc obtained by two tests was similar when the initial crack length ranged from 0.7 to 0.85 times the depth of the specimen. These results show that the asymmetric four-point bending test is a promising means of determining K_IIc.     

A method for determining the stress intensity factor of a single edge-notched tensile specimen

A straightforward method for determining the stress intensity factor of a single edge-notched tensile specimen with unknown end conditions is presented. It involves the use of measured crack mouth opening displacements, coupled with existing solutions for stress intensity factors and crack opening displacements for uniform tensile loading and for pure bending. The use of the method is demonstrated through experiments on a porous matrix ceramic composite.