Classical Fracture Criteria Research Articles

A review of mixed mode I-II fracture criteria and their applications in brittle or quasi-brittle fracture analysis

Fracture criteria play a very important role in predicting crack propagation behavior and life evaluation of cracked bodies. In this paper, mixed mode I-II fracture criteria in the framework of linear elastic fracture mechanics and their applications in brittle or quasi-brittle fracture analysis are systematically reviewed. Meanwhile, the advantages and shortcomings of these criteria are briefly discussed. Furthermore, the effects of T-stresses on brittle fracture characteristics are also studied and some suggestions are given to select appropriate fracture criteria. The results show that different types of fracture criteria differ greatly in predicting the crack propagation behavior, and a suitable criterion can be chosen according to the crack types, loading modes and failure characteristics. Tension-based fracture criteria are applicable for the opening cracks under tensile-based failure, but not for the closed cracks under shear-based failure. T-stresses around the crack tip have significant effects on crack initiation angle, fracture resistance and crack tip plastic zone, which should be introduced into the classical fracture criteria only determined by singular terms for providing more accurate predictions. Moreover, the variable fracture process zone radius can be further considered in these modified criteria considering stress intensity factors and T-stresses, which are not only suitable for brittle materials, but also for plastic materials under small scale yielding. This review will contribute to the development of mixed mode fracture criteria.

Experimental and theoretical assessment of crack kinking in marble stones in the presence of a V-notch under mode I loading

Marble has been used extensively in the historical monuments as a building material. Notches and sharp corners in these buildings are potential locations for crack formation, crack extension and in some cases their eventual failure. In this study, some fracture tests are accomplished on a number of V-notched marble specimens under mode I loading condition. The specimens are in the configurations of double cantilever beam (DCB) and compact tension (CT) each embedding two various notch opening angles. It is substantiated that although both the loading conditions and geometry were symmetric, the crack initiation angle is not necessarily along the notch bisector line. It is also shown that the classical fracture criteria are not able to predict crack kinking in the tested marble specimens. Thus, the conventional strain energy density and maximum tangential stress criteria used for V-notches, V-SED and V-MTS respectively, are generalized by introducing the first non-singular term of notch stress field into the calculations. Finally, it is demonstrated that the generalized criteria, V-GSED and V-GMTS, can predict the fracture strength and the crack initiation angle of the tested marble specimens well.

A new mixed-mode fracture criterion of anisotropic rock

Fracture mechanism of hydraulic fracturing for anisotropic shale is of very importance in shale gas mining technology. The classical fracture criteria can better predict tensile (Mode I) fracture under arbitrary loading condition (pure tensile, pure shear and mixed-mode), but have difficultly in predicting shear (Mode II) fracture. In this paper, a new mixed-mode fracture criterion (modified K-ratio criterion) of anisotropic rock is established based on the ratio of stress intense factors (SIFs) and fracture toughness of arbitrary plane to predict both the crack initiation angle and fracture mode. New physical influencing factors are proposed to describe effects of both the geometry and the material parameters of anisotropic rock on the SIFs of the original crack plane, KI(0) and KII(0), in order to calculate the SIFs on arbitrary crack plane, KI(θ) and KII(θ). Prediction results show that under the pure tension, pure shear and tension-shear loads applied onto the original plane, Mode I fracture can occur on the sedimentary plane or on the plane of KI(θ)max. Mode I or Mode II fracture can occur under compression-shear load. The semi-circular bend (SCB) test results of anisotropic shale specimens with different ψ (crack inclined angle) and β (sedimentary plane angle) are in good agreement with the predicted results and can verify the validity of the modified K-ratio criterion.

A New Method for Predicting Double-Crack Propagation Trajectories of Brittle Rock

Multi-crack propagation is investigated mainly by experimental measurement and little by theoretical prediction. The classical fracture criteria can better predict tensile fracture under arbitrary loading conditions (pure tensile, pure shear and mixed-mode), but have difficulty in predicting shear fracture. In this paper, Mode I and Mode II SIFs of branch-cracks initiated by the original cracks were calculated by the complex function and superposition method, and a new theory of multi-crack propagation was established based on the criterion of maximum tensile-shear SIF ratio. Theoretical results of two collinear cracks under uniaxial compression show that the cracks initiate more easily at [Formula: see text] (the crack inclination angle) than other angles. Coalescence of branch-crack only occurs at [Formula: see text] with the maximum crack propagation length. Peak stress [Formula: see text] reaches minimum when [Formula: see text] (inner friction angle of rock), and the larger the [Formula: see text], the closer to the compressive strength of rock the [Formula: see text]. Mechanism of the crack initiation and propagation are all Mode I under uniaxial compression. Uniaxial compressive test results of red sandstone (the rock material is assumed to be homogeneous) pre-cracked specimens agree well with predicted results of the crack initiation, stable and unstable propagation, which can prove the validity of the new multi-crack propagation prediction method.

Prediction of ductile fracture in skew rolling processes

This paper begins with a review of the literature regarding the modelling of skew rolling processes, with a special focus on the problem of material fracture. Numerical modelling is performed to compare stresses and strains in the cross sections of bars produced by skew rolling with the use of two and three rolls. It has been found that the two-roll skew-rolled bars are more prone to the occurrence of fracture in their axial zone. It is proposed that cracking in skew rolling processes be predicted with the use of classical fracture criteria. The accuracy of crack prediction by these criteria depends on the correct determination of the critical damage. The critical damage in skew rolling processes can be determined by two tests: rotary compression of a tapered specimen (when a process is performed with two tools) or rotary compression of a disc-shaped specimen in a cavity between the tools (when a process is performed with three tools). Six fracture criteria and a new test of rotary compression of a tapered specimen are employed to investigate fracture in a two-roll skew rolling process for producing cylindrical bars made of 100Cr6 grade steel. It has been found that stress-based criteria are more suitable for modelling fracture due to their lower sensitivity to temperature variations in the axial zone of rolled bars.

Frictional crack initiation and propagation in rocks under compressive loading

Crack initiation and propagation in a brittle material such as a rock are affected by the friction between crack surfaces. Choosing an appropriate fracture criterion for frictional crack surfaces plays an important role in determining fracture parameters. To this end, a two-dimensional crack propagation code (TDCPC) was developed in this study based on the displacement discontinuity method (DDM) to predict the crack propagation path under the effect of friction. In this study, three classes of fracture criteria were defined: (1) classical fracture criteria (maximum tangential stress criterion, minimum strain energy density criterion, and maximum strain energy release rate criterion) under mixed mode loading without the effect of friction (CFC-I-II), (2) classical fracture criteria under the pure mode II (shear fracturing) without the effect of friction (CFC-II), and (3) the Swedlow criterion which takes into account the effect of friction between crack surfaces (SW-II). A special element (Swedlow-element) was developed and used in the displacement discontinuity method to predict crack trajectories under the effect of friction. The Swedlow criterion was developed to study the fracture process under bi-axial loading. Results indicated that CFC-I-II are not valid for frictional crack surfaces. Crack propagation paths predicted by CFC-II and SW-II under uniaxial loading proved that friction coefficient has a significant effect on the crack initiation angle (the first stages of the crack propagation path), and away from the crack tip, CFC-II and SW-II almost have the same path. For bi-axial loading, crack propagation paths predicted by SW-II deviate away from the original crack as friction coefficient is increased. The stress ratio (Ko) and friction coefficient have opposite effects on the crack propagation path. Crack trajectory deviates away from the original crack under the effect of friction coefficient and moves toward the original crack under the effect of the stress ratio. The present study indicated that the results predicted using TDCPC are in good agreement with the experimental and numerical results of other studies.

Applicability of the classical fracture mechanics criteria to predict the crack propagation path in rock under compression

A higher order displacement discontinuity method (HDDM) with quadratic elements is used in this study to evaluate the validity of classical fracture criteria in order to predict crack propagation path under mixed mode loading I–II in the case of a crack in an infinite body. Crack propagation path is predicted here by defining three stress states, uniaxial tensile stress (UTS), uniaxial compressive stress (UCS) and biaxial compressive stress (BCS). Comparing with experimental results, it was found that by increasing the crack inclination angle, the length of crack propagation path increases and the slope of this path decreases. For BCS state, G-criterion almost loses the validity and can be replaced with σ-criterion. S-criterion is strongly dependent on Poisson’s ratio and it is recommended to study fracture under brittle and ductile behaviour. Due to the complexity of crack tip element equations, the effect of this element on the results has been studied in detail and it was found that by considering the mid-point of the last element as a reference point to evaluate displacement discontinuities (Ds and Dn), there is no need to use special crack tip element. This procedure facilitates the modelling and shortens the solution.

Scaling effect on the fracture toughness of bone materials using MMTS criterion.

The aim of this study is to present a stress based approach for investigating the effect of specimen size on the fracture toughness of bone materials. The proposed approach is a modified form of the classical fracture criterion called maximum tangential stress (MTS). The mechanical properties of bone are different in longitudinal and transverse directions and hence the tangential stress component in the proposed approach should be determined in the orthotropic media. Since only the singular terms of series expansions were obtained in the previous studies, the tangential stress is measured from finite element analysis. In this study, the critical distance is also assumed to be size dependent and a semi-empirical formulation is used for describing the size dependency of the critical distance. By comparing the results predicted by the proposed approach and those reported in the previous studies, it is shown that the proposed approach can predict the fracture resistance of cracked bone by taking into account the effect of specimen size.

Seepage Analysis of The Structure with Cracks Based on XFEM

For the difficulty of applying classical fracture criteria to the actual hydraulic engineering and simulating the process of cracking by conditional FEM, the XFEM was introduced in the analysis of the seepage field in hydraulic structures in this paper. Firstly, the enriched forms of nodes are analyzed in the elements intersecting with cracks, and then the enriched functions were built, which could either reflect the features of conductivity matrix within cracks, or satisfy the condition that osmotic pressure is continuous across the crack. Thus, the XFEM approximation form was obtained. Finally, combining the initial conditions and boundary conditions, the discrete equations and workflow of XFEM for solving the seepage field were established. The case study shows that the method is reasonable and reliable.

Monitoring Methods of Crack Behavior in Hydraulic Concrete Structure Based on Crack Mouth Opening Displacement (CMOD)

For the difficulty of applying classical fracture criteria to the actual hydraulic engineering and silulating the process of cracking by conditional FEM, in this paper, a new method of analyzing and monitoring crack behavior in hydraulic structures under the effect of Hydro-Mechanical (HM) interaction is studied by using the XFEM, in which crack mouth opening displacement (CMOD) is adopted as monitoring index. The core innovation done in this study is that a method of determining macro crack tip is proposed based on cohesive force for the first time, and the critical value of CTOD is investigated and proved to be a material parameter. it is verified by comparing with the concrete tests and numerical calculations of former related literatures. All of the above shows that the present monitoring method based on CMOD provides a practical way to simulate cracking in hydraulic concrete structures.

Finite Fracture Mechanics at elastic interfaces

In the present paper we provide a method to determine the load causing delamination along an interface in a composite structure. The method is based on the elastic interface model, according to which the interface is equivalent to a bed of linear elastic springs, and on Finite Fracture Mechanics, a crack propagation criterion recently proposed for homogeneous structures. The procedure outlined is general. Details are given for the pull–push shear test. For such geometry, the failure load is obtained and compared with the estimates provided by stress concentration analysis and Linear Elastic Fracture Mechanics. It is seen that Finite Fracture Mechanics provides intermediate values. Furthermore, it is shown that the predictions provided by Finite Fracture Mechanics are almost coincident with the ones provided by the Cohesive Crack Model. As far as we are concerned with the determination of the failure load, the advantage of using Finite Fracture Mechanics with respect to the Cohesive Crack Model is evident, since a troublesome analysis of the softening taking place in the fracture process zone is not necessary. A final comparison with classical fracture criteria based on critical distances, such as the average stress criterion, concludes the paper.

Contact strength of two elastic half spaces with circular groove

We present the analysis of the stress-and-strain state of a contact couple formed by two isotropic semi-infinite solids one of which has a small surface groove. Based on the classical fracture criteria, namely, on the criterion of maximal principal stresses and the criterion of maximal shear stresses, we determine the regions of the most possible formation of the zones of crack initiation and plastic zones. Brittle cracking can be induced both by tensile and compressive stresses arising on the interface. For some shapes of the groove considered as an example, our analysis reveals that the fracture process starts from the boundary of contacting solids.

Verification of a non-local stress criterion for mixed mode fracture in wood

An extension of a non-local stress fracture criterion to orthotropic materials based on the damage model of an elastic solid containing growing microcracks was presented in this paper. By taking this approach, a new fracture condition expressed in terms of the mixed mode stress intensity factors for orthotropic materials was proposed and its applicability to predict of a crack initiation and propagation in wood was validated. Predicted values of the stress intensity factors at failure were compared to experimental observations carried out on wood specimens for cracks arbitrarily oriented with respect to the orthotropy axes. Special considerations were applied to the comparison of the non-local stress fracture criterion with some classical fracture criteria for orthotropic materials.

Dynamic cracking resistance of structural materials predicted from impact fracture of an aircraft alloy

A new approach to studying the dynamic strength properties of structural materials is demonstrated with fracture of 2024-T3 aircraft aluminum alloy. The central idea of this approach is the incubation time to failure. In [1], experimental data for dynamic fracture of this alloy were analyzed in terms of the classical fracture criterion, which is based on the principle of maximum critical stress intensity factor [2]. In [1], the dependence of the stress intensity factor limiting value (the dynamic fracture toughness KId, which was assumed to be a functional characteristic of the material) on the loading rate was also measured. The same experimental data were analyzed in terms of an alternative structure-time approach [3]. In this approach, the dynamic fracture toughness KId is considered as an estimable characteristic of the problem, so that determination of limiting loads does not require a priori knowledge of the loading-rate dependence of the dynamic fracture toughness. The incubation time to failure of the aircraft aluminum alloy is calculated. The difference in the loading-rate dependences of the dynamic fracture toughness, which is observed for various structural materials, is explained. The dynamic fracture toughness of the alloy under pulsed threshold loads is calculated.

COMPUTATION of generalized stress intensity factors for bonded elastic structures

We consider coupled structures consisting of two different linear elastic materials bonded along an interface. The material discontinuities combined with geometrical peculiarities of the outer boundary lead to unbounded stresses. The mathematical analysis of the singular behaviour of the elastic fields, especially near points where the interface meets the outer boundary, can be performed by means of asymptotic expansions with respect to the distance from the geometrical and structural singularities. The coefficients in the asymptotics, which are called generalized stress intensity factors, play an important role in classical fracture criteria. In this paper we present several formulas for the generalized stress intensity factors for 2D and 3D coupled elastic structures. The formulas have the form of scalar products or convolution integrais of the given data or the unknown displacement field and the so-called weight functions, similar to Maz'ya/Plamenevsky functionals introduced in (19) for elliptic boundary value problems. The weight functions are non- energetic elastic fields, which admit a decomposition into a known singular part and a more regular one, which is computed by boundary ele- ment domain decomposition methods. Numerical experiments for two-dimensional problems illustrate the theoretical results.

Crack growth in a mixed-mode loading on marble beams under three point bending

A combined theoretical and experimental study of a crack growth in a mixed-mode I–II loading is presented. A 160×40×20 mm marble beam, with an artificial crack 8 mm and 10 mm long each, was subjected to three point bending. The crack was located vertically to the beam's lower longitudinal fiber, through the whole width of the beam. The position of the crack was displaced from the center of the beam to one of the supporting points. The vertical force P, placed on the middle of the upper fiber of the beam, imposed the combination of the opening (mode I) and the sliding (mode II) modes on the crack mouth, creating the mixed-mode I–II loading case. The stress intensity factors K I and K II, which describe the local stress and strain field around the crack tip, were determined by a suitable finite element program. The crack growth was defined by two classical fracture criteria of LEFM; the minimum strain energy density and the maximum circumferential stress criteria. The initial crack growth angle (θ) was calculated from both criteria and the critical load (P c) from the minimum strain energy density (SED) criterion. These theoretical predictions were compared with some experimental results found from three marbles with different elastic constants; the Krystallina of Kavala, the Snow-white of Thassos and the White of Piges Drama. The theoretical results showed the same trend of θ and P c as the experimental ones and they are in good agreement.