MODE I AND MIXED-MODE STABLE CRACK EXTENSIONS THROUGH STIFFENED THREE-POINT BEND SPECIMENS

Abstract Both experimental and theoretical results on stable crack extensions are presented for symmetrically stiffened and unstiffened three point bend specimens subjected to mode I or mixed mode loading systems. Results concerning load-displacement variation, growth of plastic zones, instantaneous crack edge profiles, tunnelling, etc., are presented. There is good agreement between the experimental and theoretical results. The whole of stable crack growth can be characterized by the crack opening angle criterion. The stiffening helps to increase substantially both the initiation load and the maximum fracture load.     

Experimental and finite element studies on mode I and mixed mode (I and II) stable crack growth—II. finite element analysis

Finite element studies are presented on both mode I and mixed mode stable crack growth under static loadings through an aluminium (D16AT) alloy. A COD based criterion has been used to predict the load-displacement diagram from initiation to instability. The theoretical predictions are compared with experimental results presented in Part I. Results on computed crack profiles, stress-strain distribution ahead of the crack tip, J integrals, J resistance curves, plastic zones, etc., are included. The study indicates that the load-displacement diagram associated with a mixed mode stable crack growth in a compact tension type of specimen geometry can be predicted reasonably accurately using the criterion of a fixed crack opening displacement at a finite distance behind the crack tip provided the crack is allowed to grow in the direction of initial growth in the finite element analysis. The crack assumes a more blunted profile in a mixed mode than in the mode I at all the stages of stable extension. The distributions of normal stress and strain in the direction perpendicular to the crack extension line, ahead of the current crack tip, have similarities between the mode I and mixed mode, irrespective of loading angle. Both the stress and strain levels increase as the crack extension proceeds. In a mixed mode, the J integral at the onset of crack extension is the lowest compared with the values at the later stages of the extension. Further, the tearing modulus associated with initial kinking is very small; it becomes close to the mode I values at the later stages. The tearing modulus remained approximately constant during the whole mode I stable growth and it had a similar trend subsequent to kinking in a mixed mode. The specific work of crack extension is zero as Δa → 0 and it increases gradually with Δa irrespective of the mode of loading; the actual variation depends on the loading angle. The plastic zone size grows as the stable extension progresses; the growth is approximately the maximum along the crack extension line.     

Experimental and finite element study on stable crack growth in three point bending

Experimental and finite element results are presented on mode I and mixed mode (involving I and II only) stable crack growth under static loading through an aircraft grade aluminium alloy (D16AT) in three point bending. The results include load-displacement diagrams, J-integrals, plastic zones, tunneling (or crack front curving), etc. During experiment a substantial amount of tunneling is observed, the extent of which increases as the extension progresses in both mode I and mixed mode. The tunneling reduces as a_o/w increases. The crack extends initially almost along a straight line at an angle with the initial crack in a mixed mode. The maximum load is observed to be as high as 1.6 times the initiation load in the whole range examined. From the finite element study it is seen that, in a mixed mode, the J-integral at the onset of extension is the lowest compared with the values at the later stages. The plastic zone size grows as the stable extension progresses; the growth is approximately the maximum along the crack extension line. The direction of initial crack extension in a mixed mode can be predicted through an elastic finite element analysis and using the criterion of maximum tangential principal stress. The study also indicates that the load-displacement diagram associated with a mixed mode stable crack growth can be predicted reasonably accurately using the criterion of crack opening angle.     

Crack Tip Opening Angle Measurement Methods and Crack Tunnelling in 2024‐T351 Aluminium Alloy

Abstract: The crack tip opening angle (CTOA) fracture criterion is one of the most promising fracture criterion used to characterise the stable tearing process in metallic materials. Traditional methods used for the experimental characterisation of the CTOA involve accurate identification of the crack tip at each tearing event. Recently alternative methods have been proposed that reduce the necessity of accurately defining the current crack and rely more on the shape of the crack flanks to define the CTOA. In addition, these methods define an ‘apparent crack tip’, which may be different from the actual surface crack tip and may provide insight into the amount of crack‐front tunnelling that is occurring. In the current research, compact tension specimens fabricated from 6.35 mm thick 2024‐T351 aluminium alloy plate were evaluated to investigate different CTOA measurement methods and their potential for estimating crack‐front tunnelling. In addition to characterizing the CTOA, fatigue marker bands were employed to map the evolution of crack‐front tunnelling. The experimental critical CTOA values obtained from the alternative methods were noticeably lower than that obtained from the traditional approach and showed noticeably more scatter. When compared to the experimentally obtained marker bands, the alternative methods indicated limited potential for predicting crack‐front tunnelling.     

Investigation of crack tunneling in ductile materials

Crack tunneling has been commonly observed in crack growth experiments on specimens made of ductile materials such as steel and aluminum alloys. The objective of this study is to investigate the crack tunneling phenomenon and study the effects of crack tunneling on the distribution of several mechanics parameters controlling ductile fracture. Three-dimensional (3D) elastic-plastic finite element analyses of stable tearing experiments involving tunneling fracture are carried out. Two model problems based on stable tearing experiments are considered. The first model problem involves a plate specimen containing a stationary, single-edge crack with a straight or tunneled crack front, under remote mode I loading. In the numerical analyses, the crack tip opening displacement, the von Mises effective stress, the mean stress, the stress constraint and the effective plastic strain around straight and tunneled crack fronts are obtained and compared. It is found that crack tunneling produces significant changes in the stress and deformation fields around the crack front. The second model problem involves a specimen containing a stably growing single-edge crack with a straight or tunneled crack front, under remote mode I loading. Crack growth events with a straight or tunneled crack front are simulated using the finite element method, and the effect of crack tunneling on the prediction of the load-crack-extension response based on a CTOD fracture criterion is investigated.     

基于裂缝扩展准则的K_R阻力曲线研究

根据混凝土I型裂缝扩展准则,将起裂断裂韧度作为材料参数,提出了一种新的计算混凝土结构裂缝扩展过程KR阻力曲线模型。该模型认为,裂缝扩展阻力由混凝土材料固有的抗裂能力和粘聚力提供的阻力组成并等于裂缝扩展的驱动力;应用该模型并借助有限元法计算了混凝土三点弯曲梁、楔入劈拉试件的裂缝扩展量、断裂过程区（FPZ）长度,在此基础上...     

Prediction of stable crack growth in surface cracked plate of Aluminum 7050 alloy

To simulate stable crack growth, three-dimensional finite element analysis using the constant Crack Tip Opening Angle (CTOA) fracture criterion was performed for a thin plate made of Aluminum 7050 alloy. The critical CTOA value was experimentally obtained by the Rubber Impression Method, which directly measures the three-dimensional crack profiles by inserting the gel-state silicon rubber into the crack, and taking out the solid-state rubber later. From the microscopic observation from the broken specimen, it was found that, as the crack extends, the amount of crack growth near the free surface is more than that in the depth/thickness direction, which creates the special tunneling, e.g., canoe-shaped crack extension. For the numerical simulation, the surface-cracked plate was analyzed by quasi-static elastic-plastic finite strain analysis with the node release and the reloading technique. Consistent with experimental observations, numerical simulation with constant CTOA fracture criteria produced tunneling of a surface crack, but the shape of the crack front deviated from the experimental crack front as the free surface was approached. To address the local crack tip constraint effect on the stable crack growth, various fracture parameters - crack tip triaxiality, equivalent plastic strain, and void growth ratio as the crack extends – were also investigated.     

Fracture assessment of damaged square hollow section (SHS) K-joint using BS7910:2005

A pre-cracked square hollow section K-joint was tested under static loads up to failure. It is found that the load-displacement curves are in good agreement with the finite element results. Ductile tearing was observed to initiate from the crack front parallel to the chord side wall where fracture toughness is smaller. Using plastic collapse load obtained via twice elastic compliance technique and fracture toughness obtained from crack tip opening displacement, the two fracture parameters K_r and L_r are plotted on the standard failure assessment diagram. It shows a conservative assessment for the cracked K-joint subjected to brace end axial loads.     

Study of slant fracture in ductile materials

Slant fracture is widely observed during crack growth in thin sheet specimens made of ductile materials, providing a good case for investigating three-dimensional criteria for mixed-mode ductile fracture. To gain an understanding of slant fracture events and to provide insight for establishing a slant fracture criterion, stable tearing fracture experiments on combined tension-torsion (nominal mixed-mode I/III) specimens and nominal Mode I Arcan specimens made of Al 2024-T3 are analyzed using the finite element method under three-dimensional conditions. Two types of finite element models are considered for the study of slant fracture: (a) combined tension-torsion specimens containing stationary, flat and slant cracks subject to loads corresponding to the onset of crack growth, and (b) stable tearing crack growth with slanting in a nominal Mode I Arcan specimen. Analysis results reveal that there exists a strong correlation between certain features of the crack-front effective plastic strain field and the orientation of the slant fracture surface. In particular, it is observed that (a) at the onset of crack growth in the combined tension-torsion experiments, the angular position of the maximum effective plastic strain around the crack front serves as a good indicator for the slant fracture surface orientation during subsequent crack growth; and (b) during stable tearing crack growth in the Mode I Arcan specimen, which experiences a flat-to-slant fracture surface transition, the crack growth path on each section plane through the thickness of the specimen coincides with the angular position of the maximum effective plastic strain around the crack front.     

Failure prediction in toughened epoxy resins

In order to improve the damage tolerance of composites and the performance of adhesives, one of the methods being considered is toughened or modified epoxy resins. The modifiers which are commonly used are CTBN rubber and inorganic fillers. A major toughening mechanism causing the increased toughness is the shear deformation process occurring near the crack tip. The effect of such a deformation process is to blunt the crack tip and increase the size of the plastic zone. Several models are available to predict the toughness on the basis of plastic zone size, crack tip opening displacement or crack tip radius, but these are only applicable to Mode I crack extension. Also, most of these approaches use only one stress component which is normal to the crack plane to predict the fracture toughness. The present paper reviews the existing models and suggests a criterion based on the phenomenological approach to failure in order to study the yielding and fracture toughness behavior of both unmodified and modified epoxies. The proposed yield and fracture criteria give predictions in good agreement with experimental results.     

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.     

Strain and stress fields near the blunted tip of a crack under mixed mode loading and the implications for fracture

The strain distribution in the vacinity of a blunted crack-tip is analysed by slip line theory under the conditions of plane-strain, small-scale yielding, and mixed-mode loading of Modes I and II. A generalized crack-tip opening displacement is introduced by which the strain and stress fields near the blunted crack-tip are determined uniquely over a wide range of Mode I and II combinations. Also, coupled experimental and finite-element analyses under the condition of large-scale yielding reveal that the initiation of stable crack growth occurs when the generalized crack-tip opening displacement attains a critical value which is constant for the material tested. The finite-element analysis is based on the finite deformation theory of elastic-plastic materials. The generalized crack-tip opening displacement criterion is found to be superior to the J-integral and the usual COD for the characterization of the initiation of stable crack growth. The plastic work in a small circular region at the crack-tip is found to be equivalent to the generalized crack-tip opening displacement, as a fracture criterion.     

Three-dimensional stress and deformation fields around flat and slant cracks under remote Mode I loading conditions

This paper investigates the phenomenon of slant fracture observed in stable tearing tests of many ductile materials, where an initially flat crack, loaded under remote Mode I conditions, tends to grow into a slant crack and stay in the slant configuration until final fracture. In an effort to identify potential reasons why cracks prefer to grow in a slant manner, three-dimensional finite element analyses of crack-front stress and deformation fields in Arcan-type specimens containing a flat or slant crack are performed under elastic–plastic and remote Mode-I loading conditions. In particular, the crack-tip opening displacement (COD) at a position behind the crack tip, the mean stress, the effective stress, and a constraint factor (defined as the ratio of the mean stress and effective stress) are studied and compared for the two types of cracks. Analysis results reveal several stress/deformation field variations around flat and slant cracks under identical remote loading conditions. First, close to the crack front, the COD of a slant crack is greater than that of a flat crack. Second, at the specimen’s mid-plane, a flat crack leads to a higher constraint value ahead of the crack than a slant crack. Third, the effective stress ahead of a slant crack is greater than that ahead of a flat crack, especially close to the crack front. The above results seem to suggest that slant fracture may be preferred because a slant crack enhances the driving force in the form of a higher near-tip COD value and because a shearing type of failure is promoted in the case of a slant crack compared to a tensile type of failure in the case of a flat crack.     

混凝土I型裂缝扩展准则比较研究

针对混凝土I型裂缝扩展问题,分别采用以起裂韧度为参数的裂缝扩展准则、最大拉应力准则以及裂尖处应力强度因子为零的裂缝扩展准则,数值模拟了强度等级C20、C40、C60、C80和C100的混凝土三点弯曲梁裂缝扩展全过程,获取了试件的荷载-裂缝口张开位移(P-CMOD)曲线并与试验结果进行了比较。结果表明,三种准则中以起裂韧度为参数的裂缝扩展准则计算得到的峰值荷载及P-CMOD全曲线与试验结果差别最小。随着混凝土强度等级的提高,最大拉应力准则以及裂尖处应力强度因子为零的裂缝扩展准则计算出的P-CMOD曲线与试验结果相比均有较为明显的偏离,但以起裂韧度为参数的裂缝扩展准则计算结果与试验曲线更为吻合。试验与计算结果表明,以起裂韧度为参数的裂缝扩展准则更适用于不同强度混凝土材料的断裂分析。     

Unsteady growth of mode III crack in elastic perfectly-plastic material

In this paper the assembly of the near-tip fields given by J. R. Rice is completed for the mode III crack growing quasi-statically and unsteadily in elastic perfectly-plastic material. The obtained results provide a particular example for the general theoretical relations between the steady state and unsteady state crack growth. Further, the general expression of the rate of crack opening displacement is obtained, which is similar to one by J.R. Rice and co-workers for mode I crack growing in elastic perfectly-plastic material. The fracture criterion of the critical opening displacement at a prescribed distance behind the crack tip is discussed. As a result, the theoretical J-resistance curves are given.     

Crack trajectories under mixed mode and non-proportional loading

A novel testing procedure for mixed mode crack propagation is presented; it offers the possibility of non-proportional loading and of changing the crack trajectory during testing, features which allow non-traditional mixed mode tests which, in turn, may help in discriminating mixed mode fracture criteria. Detailed experimental stable crack trajectories and corresponding load-CMOD (or load-displacement) curves from these non-traditional tests are proposed as benchmarks for numerical programs of mixed mode crack propagation.     

A model of brittle fracture propagation part I—continuum aspects

The micromechanism of crack propagation in steel is described and analyzed in continuum terms and related to the macroscopic fracture behavior. It is proposed that propagation of cleavage microcracks through favorably oriented grains ahead of the main crack tip is the principal weakening mode in brittle fracture. This easy cleavage process proceeds in the Griffith manner and follows a continuous, multiply connected, nearly planar path with a very irregular front which spreads both forward and laterally and leaves behind disconnected links which span the prospective fracture surface. A discrete crack zone which extends over many grains thus exists at the tip of a running brittle crack. Final separation of the links is preceeded by plastic straining within the crack zone and occurs gradually with the increasing crack opening displacement. It is suggested that in low stress fracture, straining of the links is the only deformation mode. However, it is recognized that under certain conditions plastic enclaves may adjoin the crack zone. This deformation mode is associated with high stress fracture, energy transition and eventually with crack arrest.

Energy dissipation resulting from the two deformation mechanisms is related to crack velocity, applied load and temperature and the crack velocity in a given material is expressed as a function of the external conditions. Fracture initiation and crack arrest are then discussed in terms of the conditions which are necessary to maintain the propagation process. Finally, the dimensions of a small scale crack tip zone for a steady state, plane strain crack are evaluated as functions of material properties and the elastic stress intensity factor.

The microstructural aspects of brittle fracture will be discussed in a separate Part 2 [1].     

Can pure mode III fatigue loading contribute to crack propagation in metallic materials?

The possibility of pure mode III crack growth is analysed on the background of theoretical and experimental results obtained in the last 20 years. Unlike for modes I and II, there is no plausible micromechanistic model explaining a pure mode III crack growth in ductile metals. In order to realize 'plain' mode III fracture surface, we propose the propagation of a series of pure mode II cracks along the crack front. Fractographical observations on crack initiation and propagation in a low alloy steel under cyclic torsion support such a model. The authors have not seen any clear indication of a pure mode III crack growth micromechanism in metals until now.     

Characteristics of brittle fracture under general combined modes including those under bi-axial tensile loads

In this paper, the brittle fracture initiation characteristics under general combination of the opening mode (Mode I), sliding mode (Mode II) and tearing mode (Mode III) were investigated both theoretically and experimentally.

First, the perfectly brittle fracture tests were conducted on specimens of PMMA (Polymethylmethacrylate) for all possible combinations of the fracture modes including respective pure modes. The experimental fracture strengths were compared with those predicted by the fracture criteria which are represented in terms of: (1) maximum tangential stress, [σ_gq]_max, extended to general combined modes, (2) maximum energy release rate at the propagation of a small kinked crack, [G^k(γ)]_max, and (3) newly derived maximum energy release rate at the initiation of a small kinked crack, [G(γ)]_max. It was found that the [G^k(γ)]_max or [G(γ)]_max criterion was very effective to predict both the direction of initial crack propagation and the fracture strength. These energy release rates are expressed in closed forms, and the interaction curves of the brittle fracture strength under arbitrary combinations of Modes I, II and III were derived.

Next, for fracture accompanied by plastic deformation, tests were carried out on specimens of mild steel (SM 41) imposing bi-axial tensile loads at various low temperatures. Then, brittle fracture with plastic deformation occurs under a combination of Modes I and II. In the case of brittle fracture with small scale yielding, the [G(γ)]_max criterion predicts well the direction of initial crack propagation but estimates only lower fracture strength than the experimental one. In the cases of brittle fracture with large scale yielding and under general yielding, it was found from the fracture tests that the direction of initial crack propagation was nearly normal to the resultant vector of the crack opening displacements in the opening and sliding modes at the notch tip. To this type of fracture, the modified COD criterion predicts well the direction of initial crack propagation, but lower fracture strength.

When brittle fracture occurs under the influence of plastic deformation, in such cases as the last three mentioned above, the actual fracture strength is higher than what the most reliable criterion predicts and it increases as deformation in Mode II becomes larger.