Numerical study of compact shear (Mode II) type test specimen geometry

The Iosipescu shear test specimen geometry has been investigated by a number of research workers in recent years with conflicting results. The paper describes a numerical study of a compact shear test specimen, based on the Iosipescu geometry, which is proposed to investigate size effects in shear failure. A range of geometries has been studied and the extreme cases are reported. Results are presented for the largest absolute principal stresses together with a detailed study of the stresses between and around the roots of the two notches introduced in the test specimens. The results for the largest absolute principal stresses show that tensile stresses are created at the roots of the two notches. These tensile stresses may result in Mode I failure and probably account for the Mode I or mixed mode fracture observed in tests using the Iosipescu geometry. The results for the distribution of stresses between the roots of the two notches show that deep notches increase the likelihood of shear fracture prior to tensile failure. Shallow notches give a stress distribution similar to that developed in the indirect tensile test and hence tensile failure is likely to precede shear failure in such cases. Further numerical and experimental work is proposed.      

Size effect of concrete members applied with flexural compressive stresses

In this study, two types of special experiments are carried out to understand flexural compressive strength size effect of concrete members. The first type is an ordinary cylindrical specimen (CS) with a fully penetrated and vertically standing plate type notch at the mid-height of the specimen, which is loaded in compression at the top surface (e.g., in the parallel direction to the notch length). The second type is a general double cantilever beam (DCB), which is compression loaded in axial direction (e.g., in the parallel direction of the notch). For CS, an adequate notch length is taken from the experimental results obtained from the compressive strength experiment of various initial notch lengths. The trial tests to select the effective initial notch length show that CS with an initial notch length approximately greater than four times the maximum aggregate size fails without an additional increased load and in stable manner under Mode I failure mechanism. Therefore, the initial notch length to the maximum aggregate size ratio of 4.0 is used for all size specimens. For DCB, the eccentricity of loading points with respect to the axial axis of each cantilever and the initial notch length are varied. In both specimens, the compressive loads apply flexural compressive stresses on the crack tip region of the specimens. These two types of specimens fail by Mode I crack opening mechanism. By testing 3 geometrically proportional size specimens for CS and DCB, the experimental datum for flexural compression size effect of concrete are obtained. Using the obtained flexural compressive strength size effect datum, regression analyses are performed using Levenberg-Marquardt's least square method (LSM) to suggest new parameters for the modified size effect law (MSEL). The analysis results show that size effect is apparent for flexural compressive strength of specimens with an initial notch. For CS, the effect of initial notch length on flexural compressive strength size effect is apparent. For DCB, flexural compressive size effect is dependent on the eccentricity of loading points with respect to the axial axis of the cantilever beam. In other words, if DCB specimen is applied with greater tensile stress at the crack tip, the size effect of concrete becomes more distinct. The results show that the flexural compressive strength size effect of initial notch length variation of DCB exists but directly dependent on the loading location. This is due to the fact that the sizes of fracture process zone (FPZ) of all DCB specimens are similar regardless of the differences in the specimen slenderness ratio, but the flexural compressive and tensile stress combinations resulting in stress concentration at the crack tip region has direct effect on size effect of concrete members.     

Size effect on strength of laminate-foam sandwich plates: Finite element analysis with interface fracture

Recent three-point bend tests of size effect on the strength of geometrically scaled sandwich beams of three types – with no notches, and with notches at the upper or lower skin–foam interface, which were previously evaluated using simplified sandwich beam theory and equivalent linear elastic fracture mechanics, are now reanalyzed more accurately by finite elements. Zero-thickness interface elements with a softening cohesive law are used to model fractures at the skin–foam interface, in the fiber composite skins, and in the foam. The fracture energy and fracture process zone length of a shear crack in foam near the interface are deduced by fitting an analytical expression for size effect to the test data. Numerical simulations reveal that small-size specimens with notches just under the top skin develop plastic zones in the foam core near the edges of the loading platen, and that small-size specimens with notches just above the bottom skin develop distributed quasibrittle fracture in the foam core under tension. Both phenomena, though, are found to reduce the maximum load by less than 6%. Further it is shown that, in notch-less beams, the interface shear fracture is coupled with compression crushing of the fiber–polymer composite skin. For small specimens this mechanism is important because, when it is blocked in simulations, the maximum load increases. The size effect law for notch-less beams is calibrated such that beams of all sizes fail solely by interface shear fracture.     

Stress analysis of a kinked crack initiating from a rigid line inclusion. Part II: Direction of propagation

The propagation direction of a kinked crack initiating from a rigid line inclusion is investigated. The criteria of maximum energy release rate, maximum Mode I stress intensity factor, and zero Mode II stress intensity factor are used as predictors for the kinked crack direction. Uniform farfield tension and pure shear loading conditions are used for illustration.     

Fracture toughness of unidirectional fibre reinforced composites in MODE II

Experimental investigations have been performed on unidirectional glass fibre reinforced/epoxy composites in Mode II (Forward shear) with the presence of crack parallel to the fibres direction through the use of end-cracked beam. A concentrated load at the Centre of the beam produced bending-induced shear deformation at the crack tip. Calibration factors for Mode II have been obtained. The stress-intensity factor at instability $\text{K}$_IIR(INR) is obtained by experiments on a small end cracked beam through a compliance matching procedure. The crack growth resistance at instability and the corresponding critical strain energy release rate are independent of initial crack in the range of crack length investigated. In composite materials, fibre-matrix interfacial shear stress play an important role in load transfer mechanism: hence Mode II study may be very useful to analyse the interfacial mechanisms and to understand the fracture behaviour of unidirectional fibre reinforced composites in Mode I when load is applied in the direction of the fibres.     

Acoustic emission imaging of shear failure in large reinforced concrete structures

The averaged value of the strain energy density over a well-defined volume is used to predict the static strength of U-notched specimens under mixed-mode conditions due to combined bending and shear loads. The volume is centered in relation to the maximum principal stress present on the notch edge, by rigidly rotating the crescent-shaped volume already used in the literature to analyse U- and V-shaped notches subject to mode I loading. The volume size depends on the ultimate tensile strength σ _u and the fracture toughness K _IC of the material. In parallel, an experimental programme was performed. All specimens are made of polymethyl-metacrylate (PMMA), a material which exhibits quasi-brittle behaviour at -60°C. Good agreement is found between experimental data for the critical loads to failure and theoretical predictions based on the constancy of the mean strain energy density over the control volume.     

Mixed mode fracture energy of sprucewood

The characterization of Mixed Mode (Mode I and Mode II) behaviour of wood was concentrated on concepts of linear fracture mechanics in the past. Using an adopted version of the splitting test it was possible to obtain complete load displacement curves under different Mixed Mode loading cases for crack propagation along the grain. Therefore fracture energy concepts (specific fracture energy) could be used to characterize the material behaviour. Additionally strength parameters were used in order to describe crack initation in two crack propagation systems. The values for specific fracture energies as well as the strength values were compared with pure Mode I fracture tests. Moreover, the size effect under Mixed Mode loading was investigated to guarantee size independent material characterizing values for the specific fracture energies.     

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.     

Examination of the Mode II fracture behaviour of wood with a short crack in an asymmetric four-point bending test

Asymmetric four-point bending tests of agathis specimens with a short crack along the neutral axis in a tangential–longitudinal system were conducted onto analyse the failure behaviour of wood with a short crack. The nominal shear strength and Mode II critical stress intensity factors of the specimens with various crack lengths were measured, and the influence of crack length on these properties was examined. The nominal shear strength of the cracked specimens was significantly lower than the strength of a crack-free specimen, even when the crack was extremely short. This finding suggests that the fracture mechanics theory is effective for analysing the failure behaviour of wood with a very short crack in this loading condition. However, the Mode II critical stress intensity factor still depends on the crack length. When the crack length was corrected with considering the formation of fracture process zone ahead of the crack tip, the critical intensity factor could be predicted effectively as well as the nominal shear strength.     

A Finite Fracture Mechanics approach to the asymptotic behaviour of U‐notched structures

A Finite Fracture Mechanics (FFM) criterion is formalized to predict the critical failure loads of brittle U‐notched specimens, subjected to mode I loading. The criterion, recently applied to V‐notched structures, requires the contemporaneous fulfilment of stress requirements and energy conditions for fracture to propagate: the stress field ahead of the notch tip and the stress intensity factor related to a crack stemming from the root are involved. Both the apparent fracture toughness and the critical crack advancement result to be structural parameters. For sufficiently slender notches, the root radius becomes the only relevant geometric dimension. The consistency of the approach is proved by the comparison with experimental data available in the Literature.     

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.     

Fatigue cracks emanating from sharp notches in high-strength aluminium alloys: The effect of loading direction, kinking, notch geometry and microstructure

The effect of notch geometry on the propagation of fatigue cracks emanating from sharp V-shaped notches was investigated. To this purpose, an experimental campaign has been conducted on Al-7075–T651 specimens carrying notches with aperture angles of 45°, 90°, and 135°. In order to investigate the role of microstructure texture, specimens were extracted from the plates with the main axis either in the longitudinal rolling direction (L-samples) or in the transversal direction (T-samples), or 45° inclined with respect to both directions (LT-samples). The effect of stress amplitude was investigated by performing tests at two load levels. Three loading directions θ = 0°, 45° and 90° were considered. Some specimens experienced pure Mode I loading condition, whereas the remaining ones were subjected to combined Mode I and Mode II loading condition. The crack deflection induced by the variation in loading direction was determined by measuring the kinking angle. A linear elastic fracture mechanics approach was adopted for the analysis of experimental results. Stress intensity factors (SIF) of straight cracks were calculated using an appropriate weight function set up for studying inclined edge cracks emanating from sharp V-notches. On the contrary, a finite element model has been built up to derive the SIFs at the tip of the kinked cracks. The influence of K_II on the crack propagation was discussed on the basis of theoretical and semi-empirical models. It has been found that (i) the crack initiation at the notch root occurred in mixed mode conditions, (ii) a decreasing Mode II component with growing crack length was observed under initial loading direction θ₀ = 45° and θ₀ = 90°, (iii) a crack deflection was observed after 45° rotation of the initial loading direction; a good prediction of the kinking angle was obtained using the maximum tangential stress criterion, and (iv) a fairly good rationalization of all the collected crack growth rate data is obtained if the driving force for crack propagation is expressed in terms of K_I.     

ELASTIC-PLASTIC FINITE ELEMENT ANALYSES OF CRACKED NOTCHES IN PLATES UNDER MIXED-MODE LOADING

Plates containing inclined elliptical notches with and without cracks are analysed for five different major axis inclinations, namely, θ equal to 0, 11.25, 22.5, 33.75 and 45?. Short cracks emanating from the roots of such inclined circular and elliptical notches are analysed. Under various mixed-mode loading conditions, when cracks are present, a simple method is described for the evaluation of the Mode I and Mode II stress intensity factors, for the elastic state. Values of the J integral along three different contours are also evaluated in the elastic and elastic-plastic states, as well as the Mode I and Mode II components of the crack tip opening displacenient (CTOD), the maximum principal stresses, the maximum average equivalent plastic strains and the location of the elements in which they occur. Comparisons of the different loading conditions are presented and the implications of the effects of mixed-mode loading on fracture and fatigue crack propagation are discussed.     

紧凑四点剪切砼Ⅱ型断裂实验研究与数值分析

有关Iosipescu四点剪切实验的争论很多,国内外的研究者提出了各种各样的结果。本文对紧凑四点剪切加载下砼Ⅱ型断裂破坏机理进行了大量的实验研究和数值分析。通过对实验中试件裂纹的扩展过程及开裂角cq等的分析,结合有限元法对裂尖应力场进行了研究。分析了加载点与裂纹面的距离对Ⅱ型断裂性能的影响。可以看出,虽然随着加载点与裂纹面距离的缩短,CKⅡ在逐渐增大,但纯剪加载下裂纹扩展并不是沿剪应力所在方向,即破坏并不是剪切破坏。     

Connecting the essential work of fracture, stress intensity factor, Hill’s criterion in mixed mode I/II loading

Ductile sheet structures are frequently subjected to mixed mode loading, resulting that the structure is under the influence of a mixed mode stress field. Instances of interest are when stable crack growth occurs and when the crack-tip is propagating in this complex mixed-mode condition, prior to final fracture. Purposely designed apparatus was built to test thin-sheets of steel (Grade: DX51D) under mixed-mode I/II. These tests, under plane stress conditions, also investigated the effect of thickness on the specific essential work of fracture or the fracture toughness of the material under quasi-static cracking conditions. The fracture toughness is evaluated under incremental mixed-mode loading conditions. The direction of the propagating crack path and fracture type were observed and discussed as the loading mixity was varied. Whilst the specific essential work of fracture or fracture toughness was obtained using the energy approach, the theoretical analysis of the fracture type and direction of crack path were based on the crack tip stresses and fracture criterions of maximum hoop stress and maximum shear stress along with the utilisation of Hill’s theory. For mixed-mode I/II loading, the variation in the fracture toughness contributions ratios are evaluated and used predicatively using the established energy criterion approach to the crack tip stress intensity approach. The comparison between the theoretical directions of the crack path, failure mode propagation are in good agreement with those obtained from experimental testing indicating the definite link between both approaches.     

Fatigue strength of rolled JIS SM50A beams

The fatigue strength of rolled beams with stiffeners welded to the web was examined. The state of residual stress in these beams was varied by different cooling and straightening processes. Parallel series of tests were carried out both on plate specimens with corner-notches, simulating the edge notches in the beam-flange tip, and on fillet welded specimens. These test results were analysed using the fracture mechanics concepts of stable crack growth. A new fracture mechanics model for cracks originating from notches is proposed. This is based on the concept that the cyclic plastic zone size at the root of a notch determines the equivalent size of the notch as a fatigue crack. With this model, the severity of notches as well as their size can be taken into account to describe the initial flaw conditions. With the initial crack size estimated, a theoretical crack-growth equation was derived from the fatigue test data of rolled beams and notched plates. The analysing method together with the derived relationship was applied for the evaluation of the fatigue strength of welded beams and transverse fillet welds. The effect of residual stresses on fatigue behavior of these beams and plate specimens was also estimated by assuming an additional effect of the maximum stress on the theoretical crack-growth equation.     

Applications of normal stress dominated cohesive zone models for mixed-mode crack simulation based on extended finite element methods

In conventional cohesive zone models the traction-separation law starts from zero load, so that the model cannot be applied to predict mixed-mode cracking. In the present work the cohesive zone model with a threshold is introduced and applied for simulating different mixed-mode cracks in combining with the extended finite element method. Computational results of cracked specimens show that the crack initiation and propagation under mixed-mode loading conditions can be characterized by the cohesive zone model for normal stress failure. The contribution of the shear stress is negligible. The maximum principal stress predicts crack direction accurately. Computations based on XFEM agree with known experiments very well. The shear stress becomes, however, important for uncracked specimens to catch the correct crack initiation angle. To study mixed-mode cracks one has to introduce a threshold into the cohesive law and to implement the new cohesive zone based on the fracture criterion. In monotonic loading cases it can be easily realized in the extended finite element formulation. For cyclic loading cases convergence of the inelastic computations can be critical.     

On the influence of rubbing fracture surfaces on fatigue crack propagation in mode III

Fatigue crack growth has been studied under fully reversed torsional loading (R = ?1) using AISI 4340 steel, quenched and tempered at 200°, 400° and 650°C. Only at high stress intensity ranges and short crack lengths are all specimens characterized by a microscopically flat Mode III (anti-plane shear) fracture surface. At lower stress intensities and larger crack lengths, fracture surfaces show a local hill-and-valley morphology with Mode I, 45° branch cracks. Since such surfaces are in sliding contact, friction, abrasion and mutual support of parts of the surface can occur readily during Mode III crack advance. Without significant axial loads superimposed on the torsional loading to minimize this interference, Mode III crack growth rates cannot be uniquely characterized by driving force parameters, such as ΔK_III and ΔCTD_III, computed from applied loads and crack length values. However, for short crack lengths (?0.4 mm), where such crack surface interference is minimal in this steel, it is found that the crack growth rate per cycle in Mode III is only a factor of four smaller than equivalent behaviour in Mode I, for the 650°C temper at $\text{ΔK}{}_{\mathrm{III}}\text{= 45 MPa m}\text{1}\text{2}$.     

Brittle fracture considerations of two external notches under combined loading

The interaction behavior of two external notches, modeled by plane hyperbolas, is analyzed using the strain energy density failure criterion[1,2]. It is shown that the pure shear and normal loadings may be combined to the case of inclined or off-axis loading of this double-notch geometry. The crack trajectory emanating from the notch surface is assumed for most cases to follow a path along which the material elements experience more volume change than shape change. This is determined by taking the minimum values of the strain energy density function. The failure loads are obtained by holding the critical strain energy density function constant and are shown to vary with the angle between the inclined load and the major axis of the notch. The variations will depend on the aspect ratios of the notches. Graphical plots of the crack trajectories for different loading angles are displayed. The double-notch geometry is such that only local trajectory information can be gleaned from the initial state. Numerical results of failure load, fracture angle, etc. are given for notches with different degrees of bluntness.     

基于两种破裂判据的裂隙岩体单轴压缩起裂分析

该文引入Rankine最大拉应力准则和Mohr-coulomb剪切破坏准则分别作为岩石基质的拉伸和压剪破裂判据,分析了单轴压缩下裂隙岩体的起裂机制。根据含单个椭圆裂隙的无限域岩体在单轴压缩下的应力理论解,编制了Matlab程序,计算分析了不同短轴与长轴比k和倾角α(加载轴与裂隙长轴间的夹角)下的岩石基质应力集中系数、两种不同起裂机制的破裂函数值、开裂位置和开裂临界荷载。对多裂隙岩体,采用ABAQUS有限元软件进行了应力计算和起裂机制分析。计算结果表明:1)与单裂隙岩体相比,多裂隙岩体的岩石基质应力集中系数略大、起裂临界荷载略小,但起裂位置相同;2)随着裂隙倾角α的增大,岩石基质的主拉应力集中区由裂隙端部附近很小的区域逐渐变为裂隙中部的大面积区域,而主压应力集中区则反之;3)存在临界裂隙倾角α0,其值在45°附近。当裂隙倾角0<α≤α0时,在裂隙端部同时有拉应力和压剪应力集中,拉破裂临界荷载小于压剪破裂临界荷载,但随着裂隙轴比的增大二者逐渐相等,表明岩体受拉破裂和压剪破裂共同影响越来越明显;当α0<α≤90°时,尽管拉破裂临界荷载大于压剪破裂临界荷载,但首先发生在裂隙端部的压剪破裂区范围很小,而随后将在裂隙中部或端部发生大量的拉伸破裂。上述分析结果与实验现象较为吻合。