Fracture Mechanics Research Articles

AI-driven data fusion modeling for enhanced prediction of mixed-mode I/III fracture toughness

This research evaluates the use of artificial intelligence to enhance the accuracy of predictions for mixed-mode I/III fracture toughness in polymethyl methacrylate . Traditionally, assessing fracture toughness relies heavily on destructive testing methods, specifically using edge-notch disc bend specimens subjected to three-point bending tests. These established methods are not only expensive and time-consuming but also frequently limited by the availability of data. To address these challenges, this study introduces a data fusion source modeling approach. This approach integrates primary fracture toughness test data with secondary predictive data derived from the maximum tangential stress criterion and the local strain energy density criterion. By employing adaptive boosting and general regression neural network algorithms, the models developed in this research demonstrate a marked improvement in predictive performance compared to traditional primary source models. Additionally, a feature importance analysis using Shapley Additive exPlanations values reveals that the mode mixity parameter, specimen thickness, and radius are critical factors influencing fracture toughness. The study highlights that while mode mixity emerges as the most significant factor, a reduction in specimen thickness generally leads to decreased fracture toughness, whereas an increase in radius has a more complex, often negative, effect. The results of this study indicate that AI-powered models using data fusion can overcome limitations related to data scarcity, enabling more accurate predictions in fracture mechanics. Furthermore, this approach provides a pathway for utilizing AI in other engineering domains where data sets are limited.

Comprehensive analysis of mode-I cracking in ice: Exploring full-range rate dependency

Ice, as a crystalline material, demonstrates a unique response in its mode-I fracture toughness to variations in loading rates, characterized by an initial decrease and subsequent increase in toughness. This phenomenon has been explored in limited studies. In this work, an extensive numerical analysis of mode-I ice cracking is conducted by using the distinct lattice spring model (DLSM). Norton-Bailey Drucker-Prager DLSM (NB-DP-DLSM) is initially employed to investigate the classical explanations of creep and stress relaxation for the anomaly observed in ice’s fracture toughness at lower loading rates, but this approach does not successfully replicate the experimentally observed strain rate dependency. Then, two rate-dependent constitutive models are introduced to further examine the mode-I fracture mechanics of ice. Our numerical simulations of three-point bending tests show that the relationship between fracture toughness and strain rate at lower levels is more accurately captured by rate-dependent models. For higher strain rates, our numerical modeling of notched semi-circular bending tests indicates that a rate-independent constitutive model can replicate the loading rate dependency of ice’s mode-I fracture toughness. In conclusion, these observations suggest that a reverse-stage-like dynamic constitutive model for DLSM can potentially capture the full range of loading rate dependencies observed in ice.

Effect of filler loading on the frictional, thermal and mechanical properties of ABS/boron nitride (h-BN) nanocomposites

The effect of hexagonal boron nitride (h-BN) nanoparticles in acrylonitrile butadiene styrene (ABS) polymer matrix is investigated. ABS/h-BN nanocomposites were prepared with h-BN content ranging from 0.5 to 5 wt% and their frictional, thermal and mechanical properties were evaluated. XRD analysis showed that the 'd' spacing in h-BN stacks increased in the ABS nanocomposite due to the interpenetration of ABS polymer chains. The tensile properties and thermal stability of ABS matrix showed better improvement with 0.5 wt% addition of h-BN nanoparticles. The tensile fracture mechanism in ABS/h-BN nanocomposites was predicted using tensile fracture surface analysis. Coats-Redfern approach was applied to support the thermal stability analysis results. Significant enhancement (28 %) in frictional property of ABS was observed in the nanocomposite with h-BN. Wettability and flame retardancy of the ABS/h-BN nanocomposites were also investigated.

Investigation of scale effects of rock bridges based on Multi-Physical field monitoring

The presence of rock bridges is crucial for ensuring the stability of rock slopes. However, evaluating the mechanical properties of rock bridges solely based on the joint persistence coefficient (K) raises doubts due to potential variations in the scales of rock bridges within a rock mass with a constant K value. In this study, direct shear tests with different scales were carried out on sandstone specimens. Acoustic Emission (AE) and Digital Image Correlation (DIC) techniques were utilized to monitor the damage procession of the specimens. The evolution process of the specimen was divided into stages and three threshold points were determined based on AE parameters. As the scale decreased, the Joint Roughness Coefficient (JRC) increased, the tensile failure increased, and the shear failure decreased. As the normal stress increased, the JRC with the same scale decreased, the tensile failure decreased, and the shear failure increased. The change of fracture mechanism was the primary reason for the strength deterioration with decreasing scales. The decrease of the scale was not conducive to the failure prediction. By improving the rock bridge failure potential (RBP) index, the proposed RBPn index could appropriately characterize scale effects under different normal stresses.

Atomic mechanisms for the fracture of AlMo0.5NbTa0.5TiZr refractory high entropy superalloy

Refractory high entropy superalloys (RHESs), known for their excellent high temperature performance, exhibit promising characteristics but are challenged by significant brittleness. Efforts to enhance plasticity through microstructure regulation have achieved only limited success, largely due to the unclear underlying fracture mechanisms of the superstructure. In this study, we systematically investigate the fracture mechanisms of the AlMo0.5NbTa0.5TiZr RHES from microscopic to electronic scales. Interestingly, both experimental and simulation results reveal that the ordered B2 phase demonstrates non-negligible plastic deformation capabilities during fracture, including deformation twinning and amorphization. Despite this, the fracture resistance of the B2 phase is lower compared to the A2/B2 interface and disordered A2 phase, even though the A2 phase shows less twinning and amorphization. Ab initio molecular dynamics simulations, combined with electronic behavior analysis, indicate that bonds involving Al and Zr in the B2 phase often exist in an anti-bonding state, making them more prone to breaking under load. This study provides deeper insights into the fracture mechanisms of the A2/B2 superstructure and its constituent phases at both atomic and electronic levels, offering a systematic approach to improving the fracture properties of such RHESs.

Investigation of fracture performance of asphalt mixtures considering size effect law and equivalent crack propagation law

The integration of the Size Effect Law (SEL) and the equivalent Linear Elastic Fracture Mechanics (LEFM) in characterizing the fracture behavior of asphalt mixtures has received limited attention. This study presents a preliminary exploration into the application of equivalent LEFM on the fracture characteristics of two asphalt mixtures, AC10 and AC16, accounting for variations in specimen size. Semi-circular Bending (SCB) tests were conducted on specimens with four different diameters (D = 76 mm, 100 mm, 125 mm, and 150 mm) and three notch-to-radius ratios (α = 0.2, 0.4, and 0.6) at a temperature of −10 °C. The equivalent crack propagation (ECP) length, crack growth resistance, and the size of the Fracture Process Zone (FPZ) were determined and subjected to analysis. The findings revealed a size effect on the ECP length, crack growth resistance, and FPZ size, with an increase in specimen size leading to a corresponding increase in these parameters. Conversely, an increase in α resulted in a decrease in all three types of data. A power law relationship was established between the ECP length and crack growth resistance. It was observed that AC16 demonstrated superior fracture resistance compared to AC10, particularly in terms of crack growth resistance. The initial notch length, ECP length, and FPZ size were found to be of the same order of magnitude, suggesting that the equivalent LEFM can produce more accurate fracture results under low temperatures than the traditional LEFM.

Designing brittle fracture-resistant structures:A tensile strain energy-minimized topology optimization

This research proposes a novel method for designing fracture-resistant structures. By analyzing the relationship between tensile strain energy and phase field brittle fracture, it has been found that minimizing tensile strain energy can delay fracture and enhance resistance. Capitalizing on this insight, a new topology optimization method is proposed. This method focuses on minimizing tensile strain energy to suppress the well-documented tension-dominated fracture behavior observed in phase field brittle fracture analysis. In contrast to traditional topology optimization methods based on von Mises stress, this method generates more robust structures under tension. Furthermore, the method can incorporate stress constraints to mitigate the potential for stress concentrations arising from geometric discontinuities. Numerical results demonstrate the effectiveness of the proposed method. Using phase-field modeling, the mechanical fracture properties of the optimized structures, including peak load, failure displacement, and absorbed elastic energy before fracture, are quantified. Furthermore, experimental tests are also conducted. Both numerical simulations and experimental results are consistently show that structures designed with minimized tensile strain energy exhibit superior fracture toughness. Furthermore, the method offers significant computational efficiency compared to conventional approaches due to its reliance solely on linear elasticity analysis within the optimization process.

Use of a Semi-Explicit Probabilistic Numerical Model for Concrete Cracking: From Static to Dynamic Loadings

In this paper, concrete cracking is investigated in dynamics through finite element modeling. A probabilistic semi-explicit model, previously developed and validated for static loading, is extended for dynamic loading. The model in statics is based on two material mechanical parameters: the tensile strength and the critical strain-energy release rate in mode I, GIC, of the Linear Elastic Fracture Mechanics (LEFM) theory. Concerning the dynamic aspects of the model, the tensile strength rate effect is modeled by an empirical dynamic-to-static strength ratio (Dynamic Increase Factor—DIF) and a similar formulation is proposed for GIC. The structural rate effect is naturally captured when mass and damping are included in the equation of motion. For static and dynamic loading, only macroscopic crack propagation is considered. Some numerical simulations in statics and dynamics are presented in the present paper. The main results related to this work can be summarized as follows: the dispersion of the numerical results related to the load–displacement curves decreases with the loading rate. The crack pattern considerably changes with loading rate (numerically and experimentally); the agreement between the experimental and numerical results (load–displacement curves and crack pattern) indicates the model is promising for engineering applications.

Analysis of Fracturing Above Block Caving Back: A Spherical Shell Theory Approach and BEM Numerical Simulation

ABSTRACTExperiments and field monitoring have revealed that in block caving, fractures over the cave crown tend to form in narrow curved bands that are parallel or subparallel to the cave back surface. These fractures delineate curved shells of orebody between the bands and the cave back. The effectiveness of block caving hinges on the subsequent fracturing and fragmentation of these orebody shells. This study adopts a dual approach, combining thin spherical shell theory and full 3D numerical simulations along with principles of linear elastic fracture mechanics, to investigate the fracturing behaviour of these shells. Analytical analysis indicates that under axisymmetric loading, latitudinal tensile fractures predominantly initiate across the most part of the shell, occurring on both the upper and lower surfaces, except at a localised area. Additionally, longitudinal tensile fractures may initiate at the central area of the upper surface, while shear fractures tend to occur around the edge of the shell. Consequently, the shells become susceptible to fracturing, leading to the collapse or cave‐in of the orebody. Numerical simulations agree with these findings, illustrating that fracturing points within the shell region are longitudinally dispersed throughout the entire shell. Most of these fracturing points satisfy the criteria for tensile fracturing, particularly within the middle portion of the shell, aligning with the analytical results. Furthermore, simulations considering nonaxisymmetric loading patterns demonstrate that regions surrounding the caving cavity, aligned with the minimum principal in situ stress, exhibit heightened susceptibility to fracture initiation. This insight holds potential significance for optimising the design of the caving process.

Influence of Specimen Width on Crack Propagation Process in Lightly Reinforced Concrete Beams

Models used to study the fracture process of concrete are often considered 2D, ignoring the influence of specimen width. However, during the fracture process in pre-cracked concrete beams, the crack length varies along the thickness direction, especially in reinforced concrete. To study the influence of specimen width on reinforced concrete fracture behavior, a 3D numerical method was used to simulate the crack propagation processes of lightly reinforced concrete beams based on Fracture Mechanics. Nonlinear spring elements with different stress-displacement constitutive laws were employed to characterize the softening behavior of concrete and the bond-slip behavior between the steel bars and concrete, respectively. It is assumed that the crack begins to propagate when the maximum stress intensity factor at the crack tip along the beam width reaches the initial fracture toughness of concrete. To verify the validity of the proposed method, the completed crack propagation processes of lightly reinforced concrete three-point bending notched beams were simulated, and the calculated load-crack mouth opening displacement curves showed a reasonable agreement with the experimental data. Moreover, the impact of the 2D reinforced concrete beam model on the crack propagation process was analyzed. The results indicate that at the initial loading stage, the external load P obtained from the 2D model is significantly larger than the result from the presented 3D model.

Femoral neck fractures: a cohort comparison of nonunion and complication rates after ballistic versus blunt mechanism fractures.

This study aims to review complications and union rates of ballistic femoral neck fractures when compared to their blunt counterparts. We hypothesize that gunshot injuries to the femoral neck may have higher rates of complications (nonunion, avascular necrosis [AVN], and implant failure) as a result of increasing comminution and disruption to the already tenuous blood supply through the femoral neck. Following the Institutional Review Board approval, a retrospective chart review was performed identifying femoral neck fractures at a single level 1 trauma center from 2018 to 2023. Inclusion criteria were age 16-45years old with intracapsular femoral neck fractures. Exclusion criteria were absence of radiographic and clinical follow-up beyond 6 weeks, prior femur fractures, pathologic fractures, or primary surgery of total hip arthroplasty. Garden classification and Pauwels angle were reported. Clinical union was defined as the absence of painwith ambulation in the absence of radiographic changes. Radiographic union was defined using the validated Radiographic Union Score for Hip (RUSH) scoring system. Complication rates (deep and superficial infection, failure of fixation, and AVN) and subsequent surgery were recorded. After exclusion, 12ballistic and 16 blunt femoral neck fractures met inclusion criteria and were reviewed. The ballistic cohort had significant more Garden I fractures than the blunt cohort (4 [33%] vs. 0 [0%]), and significantly fewer Garden IV fractures than the blunt cohort (2 [17%] vs. 9 [56%]). Despite different Garden classification types of fracture patterns, the Pauwels angle between the two cohorts was similar (51° ballistic vs. 60° blunt, p = 0.16). The overall nonunion rate was 24.1%. Nonunion rates were similar between the two cohorts (3, 25% ballistic vs. 4, 25% blunt, p = 0.99). The mean final RUSH score between the two cohorts was similar (20.1ballistic vs. 23.4 blunt [p = 0.05]). The total complication rate was 31% (9/28) consisting of 7 nonunions and 2 malunions across both cohorts. The current study found no difference in the union rates between blunt and ballistic femoral neck fractures. Although comparing these fractures using traditional classification systems (Garden and Pauwels) are difficult due to the inherent differences in mechanism and the applicability of these tools, both groups presented with high rates of nonunion and complications. We believe the similar RUSH scores and complication rates in both cohorts show these fractures fundamentally behave similarly and remain challenging injuries to manage with ORIF.

Effect of Thermal Expansion Mismatch on Thermomechanical Behaviour of Compacted Graphite Iron

Compacted graphite iron (CGI) attracts significant attention in the automotive industry thanks to its suitable thermomechanical properties and cost-effectiveness. A primary fracture mechanism at the microscale for CGI involves interfacial damage and debonding between graphite inclusions and its metallic matrix, which can occur under high-temperature service conditions due to a mismatch in the coefficients of thermal expansion between these two phases. Such microscopic interfacial damage can initiate macroscopic fractures in cast-iron components subjected to thermal loading. While this phenomenon was studied in various composites, there remains a lack of detailed information for CGI, especially related to the complex morphology of its graphite inclusions. This study investigates the influence of graphite morphology and type of matrix on the thermomechanical performance of CGI at high temperatures. A set of three-dimensional finite-element models were developed in the form of unit cells with a single graphite inclusion embedded within a cubic domain of the metallic matrix. Elastoplastic behaviour was assumed for both phases in the numerical simulations. The study is focused on the response of the constituents in CGI to pure thermal loading in order to explore the relationship between graphite morphology and fracture mechanisms. The findings aim to enhance understanding of how graphite morphology affects the behaviours of CGI under high-temperature conditions.

Mechanical properties and crack deflection mechanisms in 3D-Printed porous geopolymers with cellular structures

ABSTRACT This study focuses on using helical design patterns via 3D printing to create geopolymer with a highly porous structure in order to enhance their strength-density relationship and fracture properties. In this regard, to create porous structure, different pitch angles and infill densities were chosen, and mechanical and fracture properties were examined. The results of mechanical strength tests revealed that while the pitch angle does not significantly affect compressive strength, flexural strength is improved by implementing a low pitch angle helical structure, which leads to the strength-density relationship improvement. Additionally, the work of fracture results demonstrated an enhancement for samples with low pitch angles, such as α15° and α30°, compared to cast and non-directional printed samples. The digital image correlation and fracture surface analysis showed several fracture mechanisms, predominantly crack deflection and twisting, in samples with low pitch angles, which contributes to the observed improvements in the work of fracture.

Role of stable‐structural yarns and fabrics on the tensile and bending performance of carbon fiber resin composites

AbstractThe component yarns and fabrics effected the formability and stability of preforms, and finally determined the performance of corresponding composites. In this research, four typical fabric structures of carbon fiber resin composites (CFRC) were introduced to conduct the comparative investigation. The stability of component yarns in natural state and in fabrics have been discussed at mesoscopic scale, and the main methods to improve the structural stability of preforms were proposed. Composites with and without stable‐structural fibers and fabrics were manufactured by the modified curing process, and the yarn cross‐sections in specimens were analyzed. The tensile and bending performance of these composite specimens was comparatively conducted to discuss the influence of stable preforms on the elastic properties. The fracture mode and mechanism of specimens were illustrated in meso‐scale. The results depicted that yarns and fabrics with stable‐structural cross‐section could ensure a high fiber volume fraction, reducing the curvature of load‐bearing yarns, and effectively reduce the yarn curvature along bearing direction. The high interfacial bonding strength of composites with stable geometrical fabrics exhibited the excellent in‐plane mechanical properties, and presented the obvious brittle fracture characteristics when the fibers fully exert their load‐bearing capacity.Highlights Stable‐structural fibers result in the prominent performance of composites. Performance of composites with stable preforms show the low dispersion. Stable cross‐section of component yarns can enhance the stability of preforms. Low curvature yarns lead to brittle fracture characteristics of composites.

AFM Nanoindentation of Stiff Inhomogeneous Layer onPolymeric Substrate.

Analysis of indentation data of heterogeneous material, in particular, layer on an elastic substrate requires information about the contact area that is essential for calculating mechanical properties. The actual shape of the AFM-tip is not described by simple body of revolution. In this work, the indentation of a stiff layer on a hyperelastic substrate by a truncated conical tip is studied using finite element methods. The size of the tip, elastic modulus, and thickness of the layer are varied. The obtained model dependences of loading and contact area versus the depth of indentation are used in analysis of the experimental data of AFM indentation of stiff inhomogeneous nanolayers of polyurethane surface. Thickness and elastic modulus of the layers, fracture properties of the surface are studied. The obtained results can find application in the study of mechanical properties and fracture toughness of thin flexible films on an elastic substrate.

Operational control of concrete frost resistance

The analysis of some methods for determining the frost resistance of concrete, including computational and experimental and accelerated ones, has been performed.It is noted that direct methods for determining the frost resistance of concrete are characterized by considerable labor intensity, as well as a long test time, which for concretes of high grades in frost resistance can be several months. Indirect methods do not always make it possible to determine the grade of concrete by frost resistance with a high degree of reliability. The paper proposes an accelerated method for determining the frost resistance of concrete, based on the principles of fracture mechanics, namely the relationship between the frost resistance of concrete and changes in the stress intensity coefficient after a single freeze to a temperature of -50оC. The proposed method allows for operational control of concrete produced monolithic and prefabricated structures. It is shown that the provisions of fracture mechanics describing the process of fatigue failure can be used to describe and analyze the mechanism of frost failure.This will make it possible to develop ways to control the process of frost destruction, slow it down and increase the frost resistance of concrete.

Simulation Study of Gas Seepage in Goaf Based on Fracture–Seepage Coupling Field

In order to solve the problem of gas overrun in the fully mechanized caving face and the upper corner of high gas and extra-thick coal seam, the fracture and caving process of the roof in the goaf is analyzed and studied by using the relevant theories of fracture mechanics and seepage mechanics. The mathematical model of fracture and caving of the immediate roof and main roof in the goaf is established. Combined with ANSYS Fluent 6.3.26, the seepage process of gas in coal and rock accumulation in the goaf under different ventilation modes is simulated. The distribution law of gas concentration in the goaf is obtained, and the application scope of different ventilation modes is determined. In addition, the influence of the tail roadway application and the wind speed size on the gas concentration in the goaf and the upper corner of the fully mechanized caving face is also explored. The results show that, affected by wind speed and rock porosity, along the strike of the goaf, about 30 m near the working face, the gas concentration is low and growth is slow. In the range of 30~160 m, the gas concentration increases rapidly and reaches a higher value. After 160 m, the gas concentration tends to be stable. Along with the tendency of the working face, the gas concentration in the goaf increases gradually from the inlet side to the return side, and the gas concentration increases noticeably near the return air roadway. Along the vertical direction of the goaf, the gas concentration gradually increases, and the concentration of the fracture zone basically reaches 100%. Different ventilation modes have different application scopes. The U-type ventilation mode is suitable for the scenario of less desorption gas in the coal seam, while U + I and U + L-type ventilation modes are suitable for the scenario of more desorption gas in coal seam or higher mining intensity. The application of the tail roadway can reduce the gas concentration in the upper corner to a certain extent, but it has limited influence on the overall gas concentration distribution in the goaf. In addition, when the wind speed of the working face should be controlled at 2.0~3.5 m/s, it is more conducive to the discharge of gas, the method of reducing the gas concentration in the upper corner by increasing the wind speed of the working face is more suitable for the case where the absolute gas emission of the fully mechanized caving face is low, and the effect is limited when the absolute gas emission is high. The above conclusions provide a reference for solving the problem of gas overrun in the goaf and the upper corner of a fully mechanized caving face.

Hydrogen Embrittlement of Galvanized Press-Hardened Steels: A Review

Press-hardened steels (PHS), as an alternative to traditional steels and aluminum alloys, combine great mechanical performance with low manufacturing costs. PHS are martensitic steels with ultimate tensile strength (UTS) up to 2000 MPa. These steels are commonly coated with zinc-based coatings (PHS GI) consisting of multiple Zn–Fe phases to enhance corrosion resistance. However, similar to all high-strength steels, PHS are known for their elevated susceptibility to hydrogen embrittlement (HE). Absorption of atomic hydrogen into the steel lattice can lead to a transition from a ductile to a brittle fracture mechanism and decrease the stress necessary for fracture initiation. This review examines the microstructure of PHS GI with a focus on how the manufacturing process influences key parameters of the coating. The material’s susceptibility to HE is discussed in the following sections, along with the potential for hydrogen introduction through corrosion in atmospheric environments. The relationship between the content of hydrogen and its effects on fracture behavior is discussed, along with the corrosion behavior of PHS GI. The potential areas for future research and development of PHS GI with increased HE resistance are proposed.

Research on the Propagation Law of Temperature-Induced Reflective Cracks in Semi-Rigid Base Asphalt Pavement

In order to solve the problem of reflection cracks in semi-rigid base asphalt pavement, a study of temperature-type reflection crack expansion is carried out according to the typical climatic characteristics of Guilin, Guangxi Province. According to the theory of fatigue fracture mechanics, the stress intensity factor analysis at the crack tip is carried out, the reflection crack propagation life under single-factor and multi-factor changes is calculated using the Paris fatigue growth formula, and the prediction model of the propagation life of reflected crack under the action of a temperature gradient is established. The results show that the propagation life of temperature-type reflective cracks increases with an increase in the thickness of the middle course and bottom course, and decreases with an increase in the surface course modulus, base course modulus, base course thickness, and temperature decrease. The prediction model includes the factors that have a great influence on the reflection crack extension and provides a theoretical supplement and reference for the design and calculation of asphalt pavement structure in areas with large temperature differences.

Fracture Characteristics and Tensile Strength Prediction of Rock–Concrete Composite Discs Under Radial Compression

To investigate the fracture mechanism of rock–concrete (R–C) systems with an interface crack, Brazilian splitting tests were conducted, with a focus on understanding the influence of the interface crack angle on failure patterns, energy evolution, and RA/AF characteristics. The study addresses a critical issue in rock–concrete structures, particularly how crack propagation differs with varying crack angles, which has direct implications for structural integrity. The experimental results show that the failure paths in R–C disc specimens are highly dependent on the interface crack angle. For crack angles of 0°, 15°, 30°, and 45°, cracks initiate from the tips of the interface crack and propagate toward the loading ends. However, for angles of 60°, 75°, and 90°, crack initiation shifts away from the interface crack tips. The AE parameters RA (rise time/amplitude) and AF (average frequency) were used to characterize different failure patterns, while energy evolution analysis revealed that the highest percentage of energy consumption occurs at a crack angle of 45°, indicating intense microcrack activity. Moreover, a novel tensile strength prediction model, incorporating macro–micro damage interactions caused by both microcracks and macrocracks, was developed to explain the failure mechanisms in R–C specimens under radial compression. The model was validated through experimental results, demonstrating its potential for predicting failure behavior in R–C systems. This study offers insights into the fracture mechanics of R–C structures, advancing the understanding of their failure mechanisms and providing a reliable model for tensile strength prediction.