On determination of mode II fracture toughness using semi-circular bend specimen

The cracked semi-circular specimen subjected to three-point bending has been recognized as an appropriate test specimen for conducting mode I, mode II and mixed mode I/II fracture tests in brittle materials. The manufacturing and pre-cracking of the specimen are simple. No complicated loading fixture is also required for a fracture test. However, almost all of the theoretical criteria available for mixed mode brittle fracture fail to predict the experimentally determined mode II fracture toughness obtained from the semi-circular bend (SCB) specimen. In this paper, a modified maximum tangential stress criterion is used for calculating mode II fracture toughness K_IIc in terms of mode I fracture toughness K_Ic. The modified criterion is used for predicting the reported values of mode II fracture toughness for two brittle materials: a rock material (Johnstone) and a brittle polymer (PMMA). It is shown that the modified criterion provides very good predictions for experimental results.     

On the circumferential strain factor criterion of mixed mode brittle fracture

In this paper, the criteria of mixed mode brittle fracture are carefully examined. It has been shown that, the circumferential strain factor criterion is rational and safe.With the exception of opening mode, mixed mode plane strain fracture of comparatively ductile materials(metals), in general, does not follow the theory of linear elastic fracture mechanics.Like the stress intensity factor that which is concerned is playing an important role in pure opening mode crack problems. We believe that, in mixed mode crack problems, the circumferential strain factor will become a parameter to determine the rate of fatigue crack propagation per cycle, and of stress corrosion cracking per unit time.First published in Nanking Aeronautical Institute Report of Science and Technology, No. 680, Jan., 1980 (in Chinese).     

Mixed mode fracture in soda lime glass analyzed by using the generalized MTS criterion

Glasses are known to be vulnerable to tensile stresses particularly in the presence of pre-existing cracks. Since cracks in glass components are very often subjected to mixed mode loading, several researchers have studied mixed mode fracture in soda lime glass using different test specimens. Among these specimens, the cracked Brazilian disc specimen has been used most frequently by investigators. However, it is shown in this paper that the previously reported experimental results obtained from the cracked Brazilian disc specimen for several glasses are always underestimated by fracture theories like the maximum tangential stress criterion. A generalized maximum tangential stress criterion is then employed for predicting the mixed mode fracture test results. It is shown that the experimental results obtained from the cracked Brazilian disc specimen and reported in the literature for soda lime glass can be estimated very well when the generalized criterion is used. It is also shown that the same criterion can be used for predicting the test results available in the literature for brittle fracture in a glass plate containing an angled center-crack.     

A nonlocal theory for brittle fracture

In this paper, a nonlocal theory of fracture for brittle materials has been systematically developed, which is composed of the nonlocal elastic stress fields of Griffith cracks of mode-I, II and III, the asymptotic forms of the stress fields at the neighborhood of the crack tips, and the maximum tensile stress criterion for brittle fracture. As an application of the theory, the fracture criteria of cracks of mode-I, II, III and mixed mode I–II, I–III are given in detail and compared with some experimental data and the theoretical results of minimum strain energy density factor.     

The circumferential stress-strain product criterion of mixed mode brittle fracture

This paper presents a new criterion of mixed mode brittle fracture, i.e. the circumferential stress-strain product criterion. This criterion is shown to be in good agreement with known experimental data.     

Determination of mode II fracture toughness for U-shaped notches using Brazilian disc specimen

A brittle fracture criterion is proposed for predicting fracture toughness of U-shaped notches under pure mode II loading. The criterion, called UMTS, is developed based on the maximum tangential stress (MTS) criterion. The UMTS criterion can be generally used for determining the mode II fracture toughness of U-notched components as well as the fracture initiation angle in U-shaped notches under pure mode II loading. To verify the validity of the proposed criterion, a set of experiments were carried out on the U-notched Brazilian disc (UNBD) specimens made of PMMA and also soda-lime glass. It is shown that there is a good agreement between the results of the UMTS criterion and the experimental results both for fracture toughness and for the fracture initiation angle under pure mode II conditions.     

Review of brittle fracture criteria in case of static and cyclic mixed mode loading

The paper gives in the first part in pressed form a survey of brittle fracture criteria using a reference intensity factor in case of static mixed mode loading. Criteria (expressed in terms of different quantities such as stress, deformation and strain energy) usually refer to a parameter that is characteristic of the material response at fracture. Criteria include information on two basic hypotheses (crack propagation direction and unstable crack growth). In the second part a generalized method is suggested for application of cyclic reference intensity factor in case of cyclic mixed mode loading. Three basic hypotheses describe crack growth direction, stable crack growth steps and unstable crack growth.     

电磁加载下7075铝环的膨胀断裂模式转变研究

利用电磁膨胀环实验技术,研究了7075铝环在2700～8700 s?1拉伸加载应变率下的断裂模式转变现象。实验结果表明:在低应变率下,铝环的断裂受最大正应力控制,发生拉伸断裂;在高应变率下,铝环的断裂受最大剪应力控制,发生剪切断裂;在中应变率下,铝环的断裂同时受最大正应力和最大剪应力控制,为拉伸和剪切同时存在的拉剪混合断裂模式;随着应变率的增加,铝环的破片数量呈先增加、再减小、最后增加的趋势,破片数量变化拐点与断裂模式转变点基本吻合。     

A new criterion for mixed mode fracture initiation based on the crack tip plastic core region

In this work, we propose a new criterion for mixed mode I-II crack initiation angles based on the characteristics of the plastic core region surrounding the crack tip. The shape and size of the plastic core region are thoroughly analyzed under different loading conditions and a new formulation for the non-dimensional variable radius of the core region is presented for mixed mode (K_I−K_II) fracture. The proposed criterion states that the crack extends in the direction of the local or global minimum of the plastic core region boundary depending on the resultant stress state at the crack tip. The results show a well-defined correlation between the plastic core region characteristics and crack extension angles predicted by other criteria. The proposed criterion is formulated for various loading conditions and is compared with other available criteria against the limited available experimental data. It is shown that the proposed criterion provides a better agreement with the experimental data.     

Fracture analysis of U-notched disc-type graphite specimens under mixed mode loading

Fracture phenomenon was investigated both experimentally and theoretically for a type of coarse-grained polycrystalline graphite weakened by a U-shaped notch under mixed mode loading. First, 36 disc-type graphite specimens containing a central U-notch, so called in literature as the U-notched Brazilian disc (UNBD), were prepared for four different notch tip radii and the fracture tests were performed under mode I and mixed mode I/II loading conditions. Then, the experimentally obtained fracture loads and the fracture initiation angles were predicted by using the U-notched maximum tangential stress (UMTS) and the newly formulated U-notched mean stress (UMS) fracture criteria. Both the criteria were developed in the form of the fracture curves and the curves of fracture initiation angle, in terms of the notch stress intensity factors (NSIFs). The results showed that while the criteria could predict successfully the experimental notch fracture toughness values, the UMS criterion provides slightly better predictions than the UMTS criterion, particularly for shear-dominant deformations. Also, found in this research was that the curves of fracture initiation angle were almost identical for the two criteria which both could predict well the experimental results.     

脆性断裂的非局部力学理论

本文提出一种脆性材料断裂的非局部力学理论,内容包括:Ⅰ、Ⅱ、Ⅲ型Griffith裂纹的非局部弹性应力场,裂纹尖端邻域非局部弹性应力场的渐近形式,脆性开裂的最大拉应力准则。文中给出了这种理论应用于三种基本型裂纹和Ⅰ-Ⅱ、Ⅰ-Ⅲ复合型裂纹临界开裂条件的计算结果,并把它们与一些试验资料和最小应变能密度因子理论进行了对比。     

Brittle fracture: Variation of fracture toughness with constraint and crack curving under mode I conditions

The effect of constraint on brittle fracture of solids under predominantly elastic deformation and mode I loading conditions is studied. Using different cracked specimen geometry, the variation of constraint is achieved in this work. Fracture tests of polymethyl methacrylate were performed using single edge notch, compact tension and double cantilever beam specimens to cover a bread range of constraint. The test data demonstrate that the apparent fracture toughness of the material varies with the specimen geometry or the constraint level. Theory is developed using the critical stress (strain) as the fracture criterion to show that this variation can be interpreted using the critical stress intensity factorK _Cand a second parameterT orA ₃,whereT andA ₃are the amplitudes of the second and the third term in the Williams series solution, respectively. The implication of this constraint effect to the ASTM fracture toughness value, crack tip opening displacement fracture criterion and energy release rateG _Cis discussed. Using the same critical stress (strain) as the fracture criterion, the theory further predicts crack curving or instability under mode I loading conditions. Experimental data are presented and compared with the theory.     

Analysis of fracture initiation angle in some cracked ceramics using the generalized maximum tangential stress criterion

Brittle fracture in ceramics sometimes occurs under combined opening-sliding (or mixed mode I/II) crack deformation. In this paper, a generalized maximum tangential stress criterion is employed for predicting the fracture initiation angle under mixed mode I/II loading in some brittle ceramics including alumina, zirconia, soda lime glass and three silicon based ceramics. The experimental results reported for the fracture angles in these ceramics have been obtained from fracture tests on the centrally cracked circular disc (often called the Brazilian disc). Very good agreement is shown to exist between the experimental results and the theoretical predictions. According to the fracture model, the mixed mode fracture angle is strongly dependent on the elastic T-stress in the tested ceramics. The negative T-stress that exists in the Brazilian disc specimen can be the main influencing parameter for decreasing the fracture initiation angle in the investigated ceramics.     

Experimental and Theoretical Assessment of Brittle Fracture in Engineering Components Containing a Sharp V-Notch

A criterion was proposed to predict brittle fracture in engineering components containing sharp V-shaped notches and subjected to mixed mode I/II loading. The criterion, called SV-MTS, was developed based on the maximum tangential stress (MTS) criterion proposed originally for analyzing crack problems. The curves which are obtained from the SV-MTS criterion could be used conveniently to predict the fracture resistance and also the notch bifurcation angle in sharp V-notched components under pure mode II and also mixed mode loading. To evaluate the validity of the proposed criterion, a set of fracture tests were conducted on a new test specimen, called sharp V-notched Brazilian disc (SV-BD), under mixed mode loading conditions. It is shown that the experimental results obtained from PMMA specimens are in very good agreement with the curves of SV-MTS criterion.     

I-II复合型裂缝断裂角的分层剪滞模型与分析

采用修正的剪滞理论建立了岩石、混凝土等准脆性材料的I-II复合型裂缝在单向拉伸荷载作用下的计算模型,得到了与实验相吻合且优于传统S判据的断裂角。通过对远场应力、斜裂缝区应力以及子层位移的合理简化,得到了求解剪滞分析模型的边界条件,进而得到了含斜裂缝的各子层位移分布函数。引入最大应力集中因子,对I-II复合型裂缝前缘应力场进行简化;基于斜裂缝沿最大应力集中因子方向扩展,得到裂缝的断裂角。根据斜裂缝的应力分布,设置不同的子层分区,得到了更为细化的位移分布模式。通过对计算数据的分析,针对单向拉伸荷载作用下的I-II复合型裂缝,建立了按应力场分区设置子层的分层剪滞模型,得到更为精确的斜裂缝断裂角。     

A MIXED MODE FRACTURE CRITERION BASED ON THE MAXIMUM TANGENTIAL STRESS IN BRITTLE INCLUSION

A closed-form solution for predicting the tangential stress of an inclusion located in mixed mode Ⅰ and Ⅱ crack tip field was developed based on the Eshelby equivalent inclusion theory. Then a mixed mode fracture criterion, including the fracture direction and the critical load, was established based on the maximum tangential stress in the inclusion for brittle inclusioninduced fracture materials. The proposed fracture criterion is a function of the inclusion fracture stress, its size and volume fraction, as well as the elastic constants of the inclusion and the matrix material. The present criterion will reduce to the conventional one as the inclusion having the same elastic behavior as the matrix material. The proposed solutions are in good agreement with detailed finite element analysis and measurement.     

Influence of the root radius of crack-like notches on the fracture load of brittle components

Narrow notches often cause damage that can lead to the destruction of components. The stress field in the vicinity of such crack-like notches in two-dimensional (2D) structures is similar to the stress field around equivalent cracks. Therefore similar investigations are necessary to predict the fracture load for components with cracks or narrow notches. Thus, the asymptotical stress field for a narrow notch with a rounded notch root is deduced from an Airy’s stress function. Based on this stress field a fracture criterion is developed. Comparing the theoretical fracture limit curves derived from the fracture criterion with experimental results it can be shown that for brittle material the local stress state at the fracture initiation point is the same for mode I, mixed-mode and mode II loading.     

Mixed mode (I and II) crack tip fields in bulk metallic glasses

Stationary crack tip fields in bulk metallic glasses under mixed mode (I and II) loading are studied through detailed finite element simulations assuming plane strain, small scale yielding conditions. The influence of internal friction or pressure sensitivity on the plastic zones, notch deformation, stress and plastic strain fields is examined for different mode mixities. Under mixed mode loading, the notch deforms into a shape such that one part of its surface sharpens while the other part blunts. Increase in mode II component of loading dramatically enhances the normalized plastic zone size, lowers the stresses but significantly elevates the plastic strain levels near the notch tip. Higher internal friction reduces the peak tangential stress but increases the plastic strain and stretching near the blunted part of the notch. The simulated shear bands are straight and extend over a long distance ahead of the notch tip under mode II dominant loading. The possible variations of fracture toughness with mode mixity corresponding to failure by brittle micro-cracking and ductile shear banding are predicted employing two simple fracture criteria. The salient results from finite element simulations are validated by comparison with those from mixed mode (I and II) fracture experiments on a Zr-based bulk metallic glass.     

Numerical simulation of fracture mode transition in ductile plates

Fracture mode of ductile solids can vary depending on the history of stress state the material experienced. For example, ductile plates under remote in-plane loading are often found to rupture in mode I or mixed mode I/III. The distinct crack patterns are observed in many different metals and alloys, but until now the underlying physical principles, though highly debated, remain unresolved. Here we show that the existing theories are not capable of capturing the mixed mode I/III due to a missing ingredient in the constitutive equations. We introduce an azimuthal dependent fracture envelope and illustrate that two competing fracture mechanisms, governed by the pressure and the Lode angle of the stress tensor, respectively, exist ahead of the crack tip. Using the continuum damage plasticity model, we demonstrate that the distinctive features of the two crack propagation modes in ductile plates can be reproduced using three dimensional finite element simulations. The magnitude of the tunneling effect and the apparent crack growth resistance are calculated and agree with experimental observations. The finite element mesh size dependences of the fracture mode and the apparent crack growth resistance are also investigated.     

Tensile-shear transition in mixed mode I/III fracture

The propensity of the transition of fracture type in either brittle or ductile cracked solid under mixed-mode I and III loading conditions is investigated. A fracture criterion based on the competition of the maximum normal stress and maximum shear stress is utilized. The prediction of the fracture type is determined by comparing τ_max/σ_max at a critical distance from the crack tip to the material strength ratio τ_C/σ_C, i.e., (τ_max/σ_max)<(τ_C/σ_C) for tensile fracture and (τ_max/σ_max)>(τ_C/σ_C) for shear fracture, where σ_C (τ_C) is the fracture strength of materials in tension (shear). Mixed mode I/III fracture tests were performed using circumferentially notched cylindrical bars made of PMMA and 7050 aluminum alloy. Fracture surface morphology of the specimens reveals that: (1) for the brittle material, PMMA, only tensile type of fracture occurs, and (2) for the ductile material, 7050 aluminum alloy, either tensile or shear type of fracture occurs depending on the mode mixity. The transition (in ductile material) or non-transition (in brittle material) of the fracture type and the fracture path observed in experiments were properly predicted by the theory. Additional test data from open literature are also included to validate the proposed theory.