Mixed Mode Brittle Fracture Research Articles

Mixed mode brittle fracture prediction in cracks under frictional condition

Neglecting frictional forces between crack surfaces in the brittle fracture of bodies subjected to compressive-shear loading conditions can result in significant fracture load underestimations when using the fracture criteria. The present research suggests a new trial-and-error procedure for predicting the fracture load of cracked components in the presence of frictional forces, which depends on the imposed compressive stress. For validating the proposed model, the results derived from the numerical method are compared with the empirical fracture loads over a wide range of loading and geometry configurations. The experiments are carried out for two different specimens, namely the center cracked Brazilian disk (CCBD) and a redesigned test specimen called angled-edge asymmetric semi-circular bending (AASCB), both made of Polymethyl-methacrylate (PMMA). The accordance between the numerical outcomes and the experimental data from the compression-shear tests demonstrates the effectiveness of the method.

Mixed mode brittle fracture of stereolithographic 3D-printed parts

Technical advances in additive manufacturing (AM), also known as three-dimensional (3D) printing, have led to applications of this technology in creation of end-use items. Consequently, performance and the mechanical strength of AMed parts have become of significant importance. In this research, fracture behavior and crack propagation of AMed cracked plates are investigated. To this aim, the stereolithography (SLA) technique is used to fabricate square plate specimens with a hole in the center and radial cracks that started at the perimeter of the central hole. Here, full range of mixed-mode fracture (from pure mode I to pure mode II) are obtained by altering the angle between the crack and the applied load. We used the finite element method to determine stress intensity factors. This study deals with a series of experiments on 3D-printed cracked plates to study mixed-mode fracture and crack propagation in brittle fracture of SLA 3D-printed components. Additionally, the digital image correlation technique was used to determine strain field on the surface of the specimens. As SLA is one of the most commonly used concepts in polymer 3D printing and has garnered significant attention for fabrication of complex structural elements, the outcomes of this study are useful for next developments and innovative designs of 3D-printed polymeric components.

Study on mixed-mode fracture behavior of TC4 titanium alloy

Mixed mode fracture behaviors of TC4 titanium alloy are studied by finite element method, testing and theoretical analysis systematically. Results show that there exists II-III coupling effect ahead of crack tip for mixed mode crack, based on which three-dimensional stress intensity factors are defined and calculated; The crack under I-II mixed mode loading exhibits a planar propagation state, the deflection angle increases with mode II component increasing; I-II mixed mode fracture of TC4 shows the characterization of brittle fracture mechanism, and brittle fracture criteria are in agreement with test results with some conservative. While the crack under I-III mixed mode loading exhibits a spatial propagation state, the deflection angle firstly increases then decreases with mode III component increasing; It is due to that the fracture mechanism is from brittle to brittle-ductile mixed fracture mode, as mode III component increases. The brittle criterion has a good prediction on I-III mixed mode brittle fracture behavior, with mode III component lower enough. The conclusions in this paper are of great significance to the mixed mode fracture evaluation of TC4 titanium alloys.

A robust Moore–Penrose pseudoinverse-based static finite-element solver for simulating non-local fracture in solids

In this work, we develop a pseudoinverse-based static finite-element solver to model the elastic deformation and non-local brittle fracture of solids. The pseudoinverse of the finite-element stiffness matrix is calculated using a QR decomposition-based method which is faster than using the traditional approach based on the singular value decomposition method. This new finite-element framework has two advantages: (1) the finite-element equations can still be solved even when the finite-element stiffness matrix is singular, and (2) there is no need for introducing an artificial elastic rest energy or viscous regularization for the sole purpose of keeping the finite-element stiffness matrix non-singular in order to solve the finite-element equations. We also show that the proposed method is robust in solving chosen boundary value problems involving the elastic deformation and abrupt non-local mode I and mixed-mode brittle fracture of solids when compared to the traditional element kill method where a small but finite residual stiffness is maintained at a material in order to prevent the finite-element stiffness matrix from being singular. Hence, this proposed method allows the modeling of separation/fragmentation of solids when fracture occurs.Finally, we use the new computational framework to model the fracture of PMMA beam samples at room temperature. By calibrating the material parameters in the constitutive theory using analytical methods and fitting to a Mode I fracture experiment force–displacement response, we show that our newly-proposed computational method is able to predict the experimental fracture loci and crack propagation characteristics in PMMA beam samples undergoing mixed-mode fracture conditions to good accord.

Validation of a 3-D adaptive stable generalized/eXtended finite element method for mixed-mode brittle fracture propagation

In this paper, a Stable Generalized/eXtended Finite Element Method (SGFEM) is combined with mesh adaptivity for the robust and computationally efficient simulation of mixed-mode brittle fracture propagation. Both h-refinement around the fracture front and p-enrichment of the analysis domain are used to control discretization errors. A Linear Elastic Fracture Mechanics (LEFM) model based on Griffith’s criterion is adopted. LEFM scaling relations are used at each fracture propagation step to back calculate SIFs that meet Griffith’s criterion. As a result, no iterations are necessary to find loading scaling parameters or fracture size that meets Griffith’s criterion. The method is validated against several experimental data sets for mode I and mode I+II fracture propagation problems. Very good agreement between SGFEM and experimental results is observed. These include fracture path, Crack Opening Displacement (COD), and load and fracture length versus COD curves. The computational efficiency of the method is also assessed.

A length scale insensitive phase-field damage model for brittle fracture

Being able to model complex nucleation, propagation, branching and merging of cracks in solids within a unified framework, the classical phase-field models for brittle fracture fail in predicting length scale independent global responses for a solid lacking elastic singularities (e.g., corners, notches, etc.). Motivated from Barenblatt’s approximation of Griffith’s brittle fracture with a vanishing Irwin’s internal length, this paper extends our recent work in quasi-brittle failure (Wu, 2017, 2018a) and presents for the first time a length scale insensitive phase-field damage model for brittle fracture. More specifically, with a set of optimal characteristic functions, a phase-field regularized cohesive zone model (CZM) with linear softening law is addressed and applied to brittle fracture. Both the failure strength and the traction – separation law are independent of the incorporated length scale parameter. Compared to other phase-field models and CZM based discontinuous approaches for brittle fracture, the proposed phase-field regularized CZM is of several merits. On the one hand, being theoretically equivalent to Barenblatt’s CZM (at least in the 1-D case), it needs neither the explicit crack representation/tracking nor the elastic penalty stiffness which both are necessary but cumbersome for discontinuous approaches. On the other hand, it gives length scale independent global responses for problems with or without elastic singularities while preserving the expected Γ-convergence property of phase-field models. Representative numerical examples of several well-known benchmark tests support the above conclusions, validating its capability of modeling both mode-I and mixed-mode brittle fracture.

Static Multiaxial Fracture Behavior of Graphite Components: A Review of Recent Results

The problem of mixed mode (I+III) brittle fracture of polycrystalline graphite is investigated systematically here for the first time. The present study considers cylindrical specimens weakened by circumferential notches characterized by different acuities. A new complete set of experimental data is provided considering different geometrical configurations by varying the notch opening angle and the notch tip radius. The multiaxial static tests have been performed considering different values of the mode mixity ratio. A criterion based on the local Strain Energy Density previously applied by the same authors only to pure modes of loading is extended here to the case of tension and torsion loadings applied in combination.

Application of an average strain energy density criterion to obtain the mixed mode fracture load of granite rock tested with the cracked asymmetric four-point bend specimens

Mixed mode brittle fracture behaviour of granite rock is studied experimentally and theoretically using Asymmetric Four Point Bend (AFPB) specimens containing pre-cracks subjected to different mixed mode loading conditions, ranging from pure mode I to pure mode II. The main aim of this paper is twofold. First, to present a complete set of experimental results on fracture of pre-cracked granite samples under various in-plane loading mixities, and second, to predict the fracture loads of the tested rock samples under mixed mode I/II conditions using an energy-based criterion, namely the Average Strain Energy Density (ASED) criterion. Good agreement is found between the experimentally obtained fracture loads and the theoretical predictions based on the constancy of the mean strain energy density over the material volume. It is shown that the ASED criterion is able to provide well predictions for the fracture loads of the investigated rock material containing a pre-crack.

The effect of remote and internal crack stresses on mixed-mode brittle fracture propagation of open cracks under compressive loading

The standard mixed-mode fracture propagation models, based on near-tip stress approximations, have limited application for fractures under far-field compressive loading. Most studies in engineering mechanics have focused on extending or modifying the standard models to solve mixed-mode fracture propagation under tensile loading. However, for geomechanics and geologic applications, the ambient stress is compressive, and internal fluid pressure or frictional stress can play an important role in the near-tip stress field altering the predicted propagation path. In this study, we modify the maximum tangential principal stress (MTPS) criterion to improve fracture propagation path predictions relative to published experimental results under both tensile and compressive external loadings. In addition, new experimental results on open cracks under compressive loading were consistent with our proposed model. Accurate propagation path predictions are independent on whether small scale yielding conditions are met in the experiments. Overall, the modified MTPS-criterion proved capable for predicting propagation paths for open cracks under compressive far-field loading.

Mixed Mode Fracture Mechanics Behaviour of PMMA

SummaryThe mixed mode brittle fracture (mode I/mode II) of poly(methyl methacrylate) (PMMA) was investigated using single‐edge notched tensile (SENT) specimens by applying a mixed mode special loading device which allows varying the loading angle from 0° (pure mode I) to 90° (pure mode II). The experimental results dealing with mixed mode fracture toughness and initial angle of crack propagation are in good agreement with predictions based on a modified criterion of maximum tangential stress (MTS criterion) where non‐singular stresses (T stresses) are considered. These additional T stresses are the reason why distinct fracture by normal stress with crazing is always macroscopically observed in PMMA under the selected loading conditions including pure external mode II loading. Therefore, the length of the craze zone, i.e. the process zone, has been determined to be about 0.5 mm independent of the loading angle. It has to be pointed out that the fracture surface morphology should be kept in mind more often in future for assessing the mixed mode fracture mechanics parameters.

Multiaxial fracture of graphite components: a review of recent results

Abstract: While a large bulk of experimental results from cracked specimens of polycrystalline graphite under pure modes of loading, in particular under mode I loading, can be found in the literature, only a very limited number of tests have been carried out on notches. At the best of authors' knowledge dealing with the specific case of V-notches under mixed mode loading (tension + torsion) no results can be found in the literature. With the aim to fill this lack, the problem of mixed mode (I+III) brittle fracture of polycrystalline graphite is investigated systematically here for the first time. The present study considers cylindrical specimens weakened by circumferential notches characterized by different acuities. A new complete set of experimental data is provided considering different geometrical configurations by varying the notch opening angle and the notch tip radius. The multiaxial static tests have been performed considering different values of the mode mixity ratio (i.e. the ratio between the nominal stress due to tension and that due to torsion loading). A criterion based on the local Strain Energy Density previously applied by the same authors only to pure modes of loading is extended here to the case of tension and torsion loadings applied in combination. The proposed criterion allows a sound assessment of the fracture loads.

The role of first non‐singular stress terms in mixed mode brittle fracture of V‐notched components: an experimental study

AbstractThis paper investigates the effects of the first non‐singular stress terms on the fracture assessment of sharp V‐notches under mixed mode loading. First, numerical studies have been performed on a fracture test configuration called single V‐notched ring (SVR) specimen. Then, the notch stress intensity factors as well as the coefficients of the first non‐singular stress terms, which are vital parameters in brittle fracture of V‐notched components, were calculated via a finite element over‐deterministic algorithm for a wide range of loading and geometry conditions. The obtained results demonstrate that the SVR specimen is able to provide a complete range of mode mixities from pure mode I to pure mode II loading conditions. The numerical results, next, have been converted to dimensionless parameters and are illustrated in several graphs. Indeed, these graphs can be easily employed by the engineers for rapid calculation of the corresponding notch stress intensity factors and the coefficients of the first non‐singular stress terms in the SVR specimen. The obtained fracture parameters are then submitted to the maximum tangential stress criterion to assess the effects of the first non‐singular terms on fracture behaviour of the specimen. Finally, an experimental study has been performed on the SVR specimen made of Nayriz Marble rock for two notch angles with a complete range of mode mixities. The obtained experimental data confirm the significant role of the first non‐singular stress terms. In fact, these results show that considering only the singular stress terms may induce an average error of 38% in the predicted fracture loads, which can be decreased to about 12% just by adding the contribution of the first non‐singular terms to the maximum tangential stress criterion.

Mixed mode brittle fracture analysis of high strength cement mortar using strain-based criteria

Strain based fracture criteria are employed to predict the brittle fracture of a cracked high strength cement mortar subjected to combined tensile/shear loading. Theoretical background of a traditional and extended versions of maximum tangential strain criteria are introduced, and the mixed mode fracture toughness and crack propagation angle are calculated for centrally cracked Brazilian disk specimens with various dimensions. In addition to the strain based criteria, a traditional stress based criterion and strain energy based criteria are used to predict the crack propagation conditions of the Brazilian disk specimens. The validity of the fracture criteria are evaluated by comparing the predicted fracture conditions to the previously published experimental data. The comparisons show that the extended version of the maximum tangential strain criterion, which takes into account the effect of both the first nonsingular strain term and the singular strain terms in Williams series expansion for strain field near crack tip, provides highly accurate predictions for the crack propagation conditions. The results indicate that the T-strain, the first non-singular term, plays an important role in the crack propagation of the cement mortar.

A REVIEW ON INVESTIGATION OF MIXED MODE FRACTURE IN BRITTLE MATERIALS

A series of mixed mode fracture tests were conducted on specimens made up of brittle materials to find out fracture parameters. Some of the parameters of interest are stress intensity factor (K), fracture toughness (Kc) and crack growth direction. In order to minimize catastrophic effects of mixed mode brittle fracture, investigation of mixed mode fracture is necessary. A number of researches have been carried out utilizing universal testing machine with suitable fixtures. Some of the experiments conducted used techniques like indentation techniques, tension tests, compression tests, bending and so on. Each one has its own advantages and disadvantages. The experimental results obtained were theoretically verified by many mixed mode fracture criteria. Some researchers also came up with empirical relations to approximate results which wouldn’t have been possible by existing famous fracture criteria. Even numerical methods like FEM analysis were carried out using commercially available softwares like ABACUS.

Strain-based criteria for mixed-mode fracture of polycrystalline graphite

The mixed-mode brittle fracture of two types of commercial graphite is investigated focusing on strain-based fracture criteria. The previously published experiments using centrally-cracked Brazilian disk specimens subjected to mixed-mode loadings are simulated by two strain-based fracture criteria: the traditional maximum tangential strain (MTSN) criterion only considering the singular terms, and the extended maximum tangential strain (EMTSN) criterion, which considers the first nonsingular strain term as well as the singular terms. Numerical simulations on the centrally-cracked Brazilian disk specimen show that the first nonsingular tangential strain term significantly influences the tangential strain distribution around the crack tip. The comparison of the evaluations by the MTSN and EMTSN criteria with the experimental data shows that the EMTSN criterion is more capable of successfully estimating the fracture resistance of graphite materials rather than the traditional MTSN criterion. In addition, when the first nonsingular term is considered, the strain-based fracture criterion provides better predictions for near mode II loadings than the stress-based fracture criterion.

On the necessity of using critical distance model in mixed mode brittle fracture prediction of V-notched Brazilian disk specimens under negative mode І conditions

Fracture load is experimentally measured for round-tip V-notched Brazilian disk (RV-BD) specimen made of PMMA and loaded under mixed mode І/ІІ with negative mode І components. The experimental results are obtained for 90° notch angle and various notch tip radii. The experimental observations show that in some cases, fracture does not initiate from the notch border and it occurs from the edge of the central square hole of the specimen, which is not usually expected. It is shown by the finite element (FE) analysis that although the notch is under mixed mode І/ІІ loading with negative mode І conditions and the notch bisector line experiences compressive stresses, the notch round border is divided into two compressive and tensile sides, suggesting that two critical points may exist on the notch border. Moreover, the FE analysis shows that in some cases, the value of tangential stress on the notch border is larger than that on the edge of the central square hole; but according to the maximum tangential stress (MTS) criterion, which is based on the Theory of Critical Distances (TCD), because the value of tangential stress at the critical distance from the notch border is lower than that on the edge of the hole, fracture is not expected to occur from the notch border. In such cases, the experimental observations are consistent well with predictions of the MTS criterion for the fracture initiation points. The results demonstrate the necessity of using the critical distance in brittle failure prediction of engineering components weakened by blunt V-notches.

Fracture assessment of graphite components weakened by rounded V-notches and subjected to static multiaxial loading

While a large bulk of experimental results from cracked specimens of polycrystalline graphite under pure modes of loading, in particular under mode I loading, can be found in the literature, only a very limited number of tests have been carried out on notches. At the best of authors’ knowledge dealing with the specific case of V-notches under mixed mode loading (tension + torsion) no results can be found in the literature. With the aim to fill this lack, the problem of mixed mode (I+III) brittle fracture of polycrystalline graphite is investigated systematically here for the first time.The present study considers cylindrical specimens weakened by circumferential notches characterized by different acuities. A new complete set of experimental data is provided considering different geometrical configurations by varying the notch opening angle and the notch tip radius. The multiaxial static tests have been performed considering different values of the mode mixity ratio (i.e. the ratio between the nominal stress due to tension and that due to torsion loading).A criterion based on the local Strain Energy Density previously applied by the same authors only to pure modes of loading is extended here to the case of tension and torsion loadings applied in combination. The proposed criterion allows a sound assessment of the fracture loads.

Crack deflection in brittle materials by Finite Fracture Mechanics

When dealing with mixed-mode brittle fracture of cracked elements, T-stress affects both the stress field and the energy balance. This problem is investigated here through the coupled Finite Fracture Mechanics (FFM) criterion by varying mode mixity of the main crack. Results are presented in terms of the critical stress intensity factors (SIF) and the critical kinking angle. As concerns pure mode I loading conditions, if T >0 is large enough, the crack ceases to propagate collinearly and the critical SIF deviates from the fracture toughness of the material. On the other hand, for mode II loading conditions, if T <0 is sufficiently low, the critical SIF ceases to increase and the critical kinking angle jumps to an infinitesimal value.

Finite Fracture Mechanics: a deeper investigation on negative T-stress effects

In the present work, the coupled stress and energy criterion of Finite Fracture Mechanics (FFM) is applied to investigate negative T-stress effects related to mixed-mode brittle fracture of cracked elements. Only two material parameters are involved in the analysis, the tensile strength and the fracture toughness, which are independent of the mode-mixity. Below a critical T-value, the shear contribution to the strain energy release rate (ERR) starts to prevail in the mode II-dominated zone. This affects FFM results in terms of: (1) the fracture loci, with the critical mode II-stress intensity factor (SIF) never exceeding the fracture toughness; (2) the critical kinking angle and the actual crack advance (which results to be a structural parameter), both decreasing to infinitesimal quantities as mode II-loading conditions are approached. These predictions can be revised by considering a large amount of energy dissipated under mode II loading conditions and by assuming a mode-mixity dependent ERR. A discussion on experimental data for brittle and quasi-brittle materials available in the literature is included.

Brittle Fracture of Rounded V-Notches in Isostatic Graphite under Static Multiaxial Loading

While a large bulk of experimental results from cracked specimens of polycrystalline graphite under pure modes of loading, in particular under mode I loading, can be found in the literature, only a very limited number of tests have been carried out on notches. At the best of the author knowledge dealing with the specific case of V-notches under mixed mode loading (tension + torsion) no results can be found in the literature. With the aim to fill this lack, the problem of mixed mode (I + III) brittle fracture of polycrystalline graphite is investigated systematically here for the first time. The present study considers cylindrical specimens weakened by circumferential notches characterized by different acuities. A new complete set of experimental data is provided considering different geometrical configurations by varying the notch opening angle and the notch tip radius. The multiaxial static tests have been performed considering different values of the mode mixity ratio (i.e. the ratio between the nominal stress due to tension and that due to torsion loading). A criterion based on the local strain energy density previously applied by the same authors only to pure modes of loading is extended here to the case of tension and torsion loadings applied in combination. The proposed criterion allows a sound assessment of the fracture loads.