Maximum Tangential Stress Criterion Research Articles

Three-Dimensional analysis of mixed mode Compact-Tension-Shear (CTS) Specimens: Stress intensity Factors, T-stresses and crack initiation angles

In order to characterize the crack tip stress field in three-dimensional fatigue and fracture problems, stress intensity factor and T-stress solutions are usually necessary. In this paper, extensive three-dimensional (3D) finite element models are constructed for Compact-Tension-Shear (CTS) specimen, which is one of the most widely used specimens for fatigue and fracture tests under mixed-mode I&II loading. In-plane and out-of-plane geometric dimensions including varying crack depths ratios (a/W = 0.3, 0.5 and 0.7) and thicknesses ratios (B/W = 0.1, 0.3 and 0.5) are considered under different loading angles (β = 0°, 15°, 30°, 45°, 60°, 75° and 90°). The results of stress intensity factors (KI,KII and KIII) and T-stresses (T11 and T33) along the crack front are computed and analyzed. The combined effects of geometric dimensions and loading angles on these fracture parameters have been illustrated. The solutions of the parameters are then described using empirical formulae by fitting numerical results with the least-squares method. Solutions obtained have been used to predict the crack initiation angles for CTS specimens based on mixed-mode fracture criteria including the maximum tangential stress (MTS) criterion, the generalized MTS (GMTS) criterion, the strain energy density (SED) criterion and the generalized SED (GSED) criterion. Solutions are useful for fracture and fatigue problems of mixed-mode I&II, such as analysis of the constraint effects on fracture toughness and propagation path of fatigue crack.

Modified Generalized Maximum Tangential Stress Criterion for Simulation of Crack Propagation and Its Application in Discontinuous Deformation Analysis

This work modified the generalized maximum tangential stress (MTS) criterion by taking parallel and perpendicular nonsingular stresses, i.e., Tx and Ty, into account. When applied to the traditional discontinuous deformation analysis (DDA) procedure, the modified generalized MTS criterion was influential in simulating open crack propagation and modeling closed crack propagation. The generalized J integrals (J1 and J2 integrals) were used to calculate the stress intensity factors (SIFs) and the T-stresses, the basic parameters of the modified generalized MTS criterion. The open crack models, namely the center cracked circular disc (CCCD) model and the edge cracked semicircular bend (SCB) model, were simulated by the DDA method with the modified generalized MTS criterion. Moreover, the closed crack models dealing with the specimens with a center crack subjected to compression stress were simulated. The simulation results of the open and closed crack models were all in good agreement with the experimental data.

Assessment of mode I fracture of rock-type sharp V-notched samples considering the size effect

In the present paper, the pure mode I fracture resistance of rock-type sharp V-notched (SV-notch) specimens with different sizes was investigated utilizing the MTS-FEM criterion. This criterion is a modified form of the maximum tangential stress (MTS) criterion and based on finite element method (FEM). According to the MTS criterion, when the maximum tangential stress at the critical distance from the notch border reaches a critical value, the onset of fracture occurs. As the critical distance in rock is relatively large, it is proposed in the MTS-FEM criterion that the tangential stress component at the critical distance is determined directly from finite element analysis. Moreover, it is well established that the critical distance in rock depends on the specimen size. Therefore, the size-dependency of the critical distance is considered in the present study. To evaluate the MTS-FEM criterion by considering size effect, some Brazilian disks with four various radii and central SV-notches were tested under pure mode I loading. It is revealed that, similar to cracked samples, the fracture resistance of SV-notched samples increases for large samples. Furthermore, it is shown that the MTS-FEM criterion is able to predict the notch fracture toughness of SV-notched rock samples by taking into account the size effect very well.

Experimental investigation on hydraulic fracturing in cement mortar with tensile stress

A novel device is proposed to test hydraulic fracturing in specimen with tensile stress. The device is composed of splitting apparatus and sealing apparatus. The tensile stress in specimen is realized via splitting load. It is confirmed that the sealing apparatus has a less effect on peak splitting load and the Young’s modulus of specimen. Some hydraulic fracturing experiments of cement mortar specimens with tensile stress are conducted. Critical water pressure of hydraulic fracturing will decrease with increase of the tensile stress. Hydraulic fracturing mechanism of cement mortar with tensile stress is revealed based on the Biot’s theory and fracture process zone. Applied high water pressure on initial crack surface, water pressure in water flow zone, and drag force along initial crack surface and water flow zone have a promotional effect on fracture propagation. The T stress in the test and pore water pressure caused by tensile strain near crack tip will restrain fracture propagation. A critical water pressure model (CWP) is proposed based on general maximum tangential stress criterion, and it is verified that the CWP model is rational and accurate. The T stress and pore water pressure gradient along polar angle direction have an effect on direction of crack initiation.

Geometry Effects on Mode I Brittle Fracture in VO-Notched PMMA Specimens.

This paper gathers experimental and theoretical investigations about both the geometry-dependent fracture initiation angle and the fracture strength in VO-notched polymethyl methacrylate (PMMA) specimens under mode I loading conditions. The numerical analyses revealed that despite the application of pure mode I loading on the geometrically symmetric VO-notched samples, the maximum tangential stress occurs at two points symmetrically placed on either side of the notch bisector line. The experimental tests performed on some specimens showed that a crack does not necessarily propagate along the notch bisector line. Stress-based theoretical studies were then carried out to justify the experimental findings. The conventional maximum tangential stress (MTS) criterion gave weak predictions of the fracture. Therefore, the predictions were checked with the generalized MTS (GMTS) criterion by taking into consideration the higher-order stress terms. It was demonstrated that the GMTS criterion predictions have satisfactory consistency with the experimental results of the crack initiation angle and the fracture strength.

Research on the anisotropic fracture behavior and the corresponding fracture surface roughness of shale

Understanding the anisotropic fracture mechanism of shale is of significance to assess the development of a stimulated reservoir volume. In this study, three-point bending tests were carried out using notched deep beam specimens of shale with 7 different bedding dip angles to investigate the anisotropic fracture behavior. The anisotropic characteristics of the fracture toughness, energy release rate, inelastic zone and crack propagation path were comprehensively analyzed and highlighted. The results showed that the fracture toughness decreases with increasing bedding dip angle. When the bedding dip angle is 0° and 90°, the fracture toughness reaches the maximum and minimum values, respectively, with a ratio of 1.60. The variation tendency of the energy release rate with the bedding inclination angle is consistent with that of the fracture toughness, but the ratio of the maximum value to the minimum value of the energy release rate is about 3.24. The size of the inelastic zone first decreases and then increases at different bedding inclination angles from 0° to 90° and reaches the maximum size of 1.51 mm when the bedding plane inclination angle is 45°. The crack propagates along the bedding plane rather than straight along the initial crack direction when the bedding dip angle is higher than 45°. As the bedding plane dip angle increases from 0° to 45°, the crack surface roughness of the tested shale first increases, then decreases and finally reaches the maximum. The discussion indicated that the presence of the Mode II fracture has a considerable influence on the crack surface roughness and crack propagation path, although it has little effect on the failure load. Compared with the maximum energy release rate (MERR) criterion, the maximum tangential stress (MTS) criterion seems more accurate in predicting the anisotropic Mode I fracture toughness. The applicability of these two classical crack initiation criteria of fracture mechanics has been analyzed based on our experimental results, which demonstrates the validity of the MTS criterion to assess the anisotropic fracture toughness of shales.

A lubrication contact pair model for simulating gear micro-pitting damage characteristics based on contour integral

A new two-dimensional finite element model of a lubricated contact pair, based on a contour integral, is proposed to investigate the formation of micro-pitting on gear tooth surfaces. Meanwhile, the contact properties and elasto-hydrodynamic lubrication (EHL) conditions of the gears are considered in the lubricated contact pair model. Then, the stress intensity factors (SIFs) KI and KII and the propagation angle θ C at the crack tip are analyzed by ABAQUS software. Next, the equivalent SIF Kσ can be calculated according to the maximum tangential stress (MTS) criterion, which is often used as the criterion for crack propagation. Considering the effect of a moving contact, the crack more easily propagates under the load x0/ b = −0.895. Furthermore, the pit shapes and variation of stress intensity factor are determined for various combinations of initial crack length a0 and angle β. The results show that longer germinated cracks propagate in areas that are deeper below the tooth surface. And the total length of final crack increases with the initial length and germination angle. These research results provide theoretical support for contact fatigue life analysis and meshing stiffness calculations of micro-pitting gears.

Loading rate effect on mixed mode I/II brittle fracture behavior of PMMA using inclined cracked SBB specimen

This paper investigates the effect of loading rate on the mixed-mode I/II fracture behavior of PolymethylMethacrylate (PMMA). Experimental studies are performed using a novel fracture test specimen which has recently been developed, called short bend beam (SBB), in different loading rates of 1 mm/min, 20 mm/min, and 200 mm/min. Fracture test data are presented for pure mode I, pure mode II, and mixed-mode I/II conditions, thanks to the geometrical configuration offered by the specimen covering a full spectrum of mode mixities. Two fracture criteria including maximum tangential stress (MTS), and generalized maximum tangential stress (GMTS) are employed to predict the onset of fracture and the fracture initiation angle in different loading rates. The MTS criterion only considers the singular crack tip terms associated with the stress intensity factors (SIFs) in the solution while the GMTS includes the T-stress effect as well as the SIF-related terms. While enhanced predictions provided by the GMTS criterion, as compared to MTS results, reveals the significance of including the T-stress term in the theoretical calculations, the critical distance (rc) is also found as another major parameter playing an important role in the fracture behavior. It is shown that the size of rc decreases by increasing the loading rate. Therefore, for the first time, a variable and loading rate dependent rc value is utilized for employment of the GMTS criterion and it is demonstrated that considering variable critical distance can provide acceptable theoretical predictions for different loading rates.

Numerical study on fatigue crack propagation behaviors in lubricated rolling contact

The surface-initiated Rolling Contact Fatigue (RCF) including pitting and micro-pitting is one of the key issues affecting the reliability of tribological components such as gears and bearings used in various devices. In this work, a surface-initiated crack Finite Element (FE) model which considers the effect of lubricant on crack faces was developed to investigate surface-initiated RCF using an automatic crack propagating Python script. Different lubricating states, initial crack parameters and loading conditions were simulated to analyze the evolution of crack propagation and the Stress Intensity Factors (SIFs). The RCF crack propagation path and life were predicted by employing the Maximum Tangential Stress (MTS) criterion coupled with the Paris’s law. A typical RCF failure is predicted in the numerical simulation. Results reveal that the lubricating pressurization dominates the surface-initiated RCF. In addition, the initial crack angle has a significant effect on the RCF crack propagation path and the fatigue life.

Laboratory-scale mixed-mode I/II fracture tests on columnar saline ice

Laboratory-scale experiments on mixed-mode I/II (including pure modes I and II) fracture of non-oriented columnar saline ice were performed. In the experiments, obliquely notched, three-point bending specimens were used, with the load applied in the direction perpendicular to the columnar grains. Tests were performed on dry isothermal (−10 °C) specimens and specimens floating in water and having a temperature gradient (mean temperature: −2.5 °C). The fracture loads and crack initiation angles were analyzed. The apparent mode I, mode II and mixed-mode fracture toughness and the flexural strength of the floating and relatively warm specimens were only about thirty percent of those of the dry and cold specimens. The results showed that the T-stress significantly influenced the mixed-mode I/II fracture of ice; good agreement was found between the experimental results and those predicted using the generalized maximum tangential stress criterion. Both trans- and inter-granular crack growth were observed, regardless of the specimen type or mode of loading.

Mixed mode I/II crack propagation in stainless steel 316L sheets by large plastic deformations: Prediction of critical load by combining LEFM with fictitious material concept

In this research, the load carrying capacity (LCC) of O-notched diagonally loaded square plate (DLSP) samples with pre-existing cracks emanating from the notch edge is investigated theoretically and experimentally under mixed mode I/II loading. The specimens are made of the stainless steel 316L, which is highly ductile and has great strain-hardening. In order to determine the LCC experimentally, several fracture tests are performed on DLSP specimens. Experimental observations reveal that these specimens experience significant plastic deformations at the onset of crack propagation. For theoretical prediction of the experimental LCCs, the Fictitious Material Concept (FMC) is first employed. By utilizing FMC, the stainless steel 316L with highly ductile behavior and great strain-hardening can be replaced with a virtual brittle material having linear elastic behavior. Then, by coupling FMC with two well-known stress based brittle fracture criteria in the field of the linear elastic fracture mechanics (LEFM), the LCCs of the cracked DLSP samples are estimated. In fact, by using FMC, complex and time-consuming elastoplastic failure analysis can be avoided and by performing only linear elastic analysis, the LCC of ductile DLSP specimens can be estimated. It is shown that combination of FMC with the maximum tangential stress (MTS) and the generalized MTS (GMTS) criteria is quite successful in predicting the crack growth instance in ductile DLSP specimens.

Notch Fracture in Polymeric Specimens under Compressive Stresses: The Role of the Equivalent Material Concept in Estimating the Critical Stress of Polymers

In this paper, the fracture of notched polymeric specimens under compressive stresses was investigated both experimentally and theoretically. In the experimental section, to determine the load-carrying capacity (LCC) of U-notched specimens made of general-purpose polystyrene (GPPS) and polymethyl-methacrylate (PMMA) polymers, tests were performed on notched square samples under compression, i.e., negative mode I loading. In the observation of the nonlinear behavior of the two polymers in the standard compressive tests, for the first time, the equivalent material concept (EMC) was used under compressive loading to theoretically estimate the critical stresses of the two polymers, which were shown to be significantly different from the ultimate strengths obtained from the standard compression tests. By linking the EMC to the maximum tangential stress (MTS) and mean stress (MS) criteria, the LCC of the notched specimens was predicted. The outcomes are twofold: First, MTS, MS, EMC–MTS, and EMC–MS criteria provide accurate predictions of the experimental critical loads observed in the U-notched polymeric specimens; second, the combination of the EMC with the MTS and MS criteria, allow such predictions to be obtained without any need for experimental calibration.

Influence of basalt fiber on mode I and II fracture properties of asphalt mixture at medium and low temperatures

Basalt fiber could effectively improve the crack resistance of asphalt mixture. However, most cracks in asphalt pavement present complex fracture characteristics. Effect of basalt fiber on the mixed mode fracture resistance of asphalt mixture is still unknown. Thus, the asymmetric semi-circular bend test was used to study the mode I and II fracture properties of the control and basalt fiber reinforced asphalt mixtures (BFRAM) at low and medium temperatures. Influences of basalt fiber on the critical stress intensity factor and critical fracture energy of asphalt mixture were investigated. The applicability of different fracture criterions to asphalt mixture also was discussed. Results indicate that the critical stress intensity factors of BFRAM first decreases and then increases with the rise of Me (which describes different fracture modes) at medium and low temperatures. Basalt fiber soundly improves the critical fracture energy of asphalt mixture at medium and low temperature even though it reduces the critical stress intensity factor at 20 ℃. The reinforcing effect of basalt fiber is optimal for the mixed mode fracture. Generalized maximum tangential stress (GMTS) criterion is suitable for the prediction of fracture initiation angle. The linear empirical fracture criterion has good applicability to the control and basalt fiber reinforced asphalt mixtures. This work can be used to analyze the mixed modes fracture of asphalt layers.

Investigating the effect of confining pressure on fracture toughness of CO2-saturated coals

The present paper investigates the effects of CO2-saturation and confining pressures on the fracture toughness of bituminous coals. Three-point bending experiments were performed with semi-circular coal specimens (SCB) after they were immersed in different CO2 pressures. The experimental results show that the pure modes I and II fracture toughnesses of coal samples gradually decrease with the growth of CO2-saturation pressure. Additionally, the fracture toughness reduces significantly after the CO2-saturation pressure exceeds 7.38 MPa, the threshold value of its supercritical state. When the CO2-saturation pressure arrives at 10 MPa, the degradation degree of the fracture toughness of the coal reaches the maximum. Furthermore, we tested the fracture toughness of coal SCB specimens under N2-saturation and CO2-saturation confining pressure conditions, respectively. As the N2-saturation confining pressure rises, the mode I and II fracture toughnesses increase gradually. The relationship between confining pressure and fracture toughness can be characterized by linear and quadratic functions for the mode I and mode II loading, respectively. The fracture toughness of coals decreases more rapidly under supercritical CO2 (scCO2)-saturation confining pressure than that under N2-saturation confining pressure. Furthermore, the theoretical values based on modified maximum tangential stress (MMTS) criterion that takes into account the effects of the T-stress are found to comply better with experimental results than those predicted by traditional maximum tangential stress (MTS) criterion.

Geometry effects on mode I brittle fracture in U‐notched specimens

AbstractThe effects of geometrical constraints on the fracture initiation location and the fracture strength are evaluated in U‐notched specimens using theoretical and experimental analyses. It is proven that high geometrical constraints in pure mode I loading of geometrically symmetric U‐notched specimens can result in occurrence of the maximum tangential stress (MTS) at two symmetric points on both sides of the notch bisector line. The experiments also indicated that the fracture takes place from a direction that is not along the notch bisector line. The experimental results are then examined theoretically through a stress‐based brittle fracture criterion. Because the conventional MTS criterion was poor to predict the onset of fracture properly, an attempt is made to use the generalized MTS (GMTS) criterion by considering the higher order terms in the fracture model. It is shown that the GMTS criterion gives very good predictions for experimentally obtained values of crack initiation angle and notch fracture resistance.

A Novel Robust Remeshing Finite Element Technique for Fracture Propagation

In this work, a novel and robust remeshing algorithm for crack opening problems is proposed, combined with triangular plane stress finite elements. In the proposed algorithm, the crack tip efficiently propagates until a pre-established maximum crack length is achieved and the crack propagation direction is defined considering the maximum tangential stress criterion. The stress state at the crack tip is obtained using a weighted average of the stresses of the integration points adjacent to the crack tip, to smoothen the stress field near the crack tip. In order to achieve accurate stress fields in the vicinity of the singularity, the proposed algorithm establishes that there is always a fixed number of nodes and elements surrounding the crack tip. To verify the accuracy of the algorithm, three benchmark tests were analyzed and the solutions were compared with results available in the literature. It was observed that the proposed technique allows to maintain the meshes regular during the propagation process, significantly reducing the number of distorted elements, which solves one of the main problems when simulating crack propagation with the finite element method (FEM). Additionally, the obtained results allowed to understand that this algorithm generally leads to accurate crack paths.

Numerical Investigation on Mixed Mode (I-II) Fracture Propagation of CCBD Specimens Under Confining Pressure

A new numerical method, verified by the analytical solution of the weight functions and experimental paths, is developed to evaluate the crack initiation and propagation generally in mixed mode (I-II). This numerical method combining the interaction integral method and the maximum tangential stress (MTS) criterion is based on the finite element method of secondary development. The influence of combined confining pressure and diametric forces on crack propagation trajectories for CCBD specimens are studied. It is indicated that the crack propagation direction independent of the confining pressure keeps the same with the line of original crack as the loading angle is equal to [Formula: see text]. But when the loading angle is greater than [Formula: see text], the curvature of the curve trajectory in the early stage of crack propagation increases with a larger confining pressure. Further, it is found that larger values of the loading angle and relative length will make the effect of confining pressure more significant at the early stage of crack growth.

Fracture Behavior of Two Biopolymers Containing Notches: Effects of Notch Tip Plasticity

This paper analyzes the notch effect on the fracture behavior of two biomaterials (a brittle bone cement and a ductile dental material) under mode I loading. U-notched Brazilian disk (UNBD) specimens of both materials were tested under remote compression, determining the corresponding fracture loads and load-displacement curves. Additionally, cracked rectangular and semicircular bend (SCB) specimens were tested under symmetric three-point bending in order to determine the fracture toughness of the two materials. Then, fracture loads were derived theoretically by applying the maximum tangential stress (MTS) and the mean stress (MS) criteria. Due to the brittle linear elastic behavior of the bone cement material, the MTS and MS criteria were directly applied to this material; however, given the significant nonlinear behavior of the dental material, the two fracture criteria were combined with the Equivalent Material Concept (EMC) for the fracture analyses of the dental material specimens. The results reveal a very good accuracy of both the MTS and the MS criteria for the fracture analysis of bone cement notched specimens. In the case of the dental material, very good results are also obtained when combining the MTS and the MS criteria with the EMC. The proposed approach can be useful for the fracture analysis of a wide range of biopolymers, from brittle to ductile behavior.

Influence of chlorides on the fracture toughness and fracture resistance under the mixed mode I/II of high-performance concrete

This experimental study shows the conjunction of fracture mechanics with the ability of high-performance concrete (HPC) to resist an aggressive chloride environment. Moreover, the influence of a chloride-contaminated environment on the fracture toughness and fracture resistance under the mixed mode I/II is presented. The experimental study was performed considering different aggressivity levels of the environment, water, and fully saturated sodium chloride solution (NaCl-). The experimental tests were performed on a Brazilian disc specimens to test the indirect tensile strength ft and on a Brazilian disc with a central notch to evaluate the fracture toughness KIC and the fracture resistance under the mixed mode I/II. The mixed mode I/II fracture resistance was evaluated by the generalised maximum tangential stress (GMTS) criterion. The level of chloride contamination present and its boundary in the concrete was estimated by the colorimetric method using silver nitrate (AgNO3).

Breakdown Pressure and Propagation Surface of a Hydraulically Pressurized Circular Notch Within a Rock Material

Rock masses contain pre-existing cracks. These cracks are not usually considered when predicting the maximum injection pressure, i.e. the breakdown pressure, in hydraulic fracture stimulations. The conventional approach to predict breakdown pressure is to use the maximum tensile stress failure criterion to calculate the pressure when a point on the borehole wall reaches the tensile strength of the rock. In addition, a pre-existing crack intersecting a hydraulically pressurized section of a borehole may produce a non-planar crack propagation surface. It is important to predict these non-planar crack propagation surfaces to design productive hydraulic fracturing stimulations and to mitigate risks associated with uncertainties of the resultant crack propagation. To gain a better understanding of this problem, a series of hydraulic fracturing experiments were conducted to investigate the breakdown pressures and crack propagation surfaces of a pressurized circular crack represented by a thin notch, subjected to different external triaxial stresses. The results show that the breakdown pressures under the shear stress conditions studied can be estimated using only the resultant normal stress on the plane of the crack. As the material properties of the experimental specimens are well defined and the crack propagation surfaces were mapped, the experimental results presented in this study provide a very useful measured dataset for the validation of various modelling approaches. The propagation surfaces from experiments were found to align closely to the computational predictions based on the maximum tangential stress criterion. Finally, this study gives evidence in three-dimensions that via the hydraulic fracturing process, the propagation of an initially arbitrarily oriented crack will eventually realign to be perpendicular to the minor principal stress direction.