Mixed-mode Fracture Research Articles

Tensile Strength Property with PdCo-Cu Multi-Layer Lamination and Heat Treatment of MEMS Probe for Probe Card

Cantilever wire probe card has limited to transport high-speed signal because of long wire but it is suitable for inspecting high-current and high voltage semiconductor such as wide band gap (WBG) power semiconductors.Superior mechanical property of a probe card wire is in demand because it is respectively used wafer inspection with in and out pad on the top surface of the power semiconductor wafer. Therefore, in this study, tensile specimens of micro electro-mechanical systems (MEMS) probe alternatively layered and electroplated with palladium (Pd)-cobalt (Co) alloy layers and copper (Cu) layers. MEMS probe was also measured tensile strength and elongation with multi-lamination and heat treatment conditions. Additionally, fractography of fracture surface after tensile test were analyzed using a scanning electron microscope (SEM). As the number of PdCo-Cu multi-layer increased, 11, 15 and 21 layered specimens showed higher tensile strength and longer elongation than 5 layered specimens because of the increase of laminated interface of PdCo and Cu. 7 layered specimen after isothermal aging had higher tensile strength and longer elongation than before isothermal aging. In 5, 11, 15 and 21 layered specimens, the fracture surfaces of tensile test specimens without aging treatment showed a ductile-brittle mixed fracture mode. Fracture surfaces of 7-layered specimen before and after isothermal aging showed a ductile-brittle mixed fracture mode.

Fracture toughness of alkali-activated concrete subjected to aggressive environment

This study investigates the fracture toughness of alkali-activated granulated blast-furnace slag (GBFS)-based geopolymer concrete (GPC), non-activated slag cement concrete, and ordinary Portland cement (OPC) concrete in acidic, alkaline, and chloride solutions over six months. A total of 580 centrally straight-notched Brazilian disc (CSNBD) specimens were cast to assess pure mode-I (0°), pure mode-II (28.7°), and mixed-mode (15° and 45°) fractures under these conditions. Microstructural analyses (SEM) were performed to evaluate microstructural changes after six months of exposure. Results showed that GPC exhibited the lowest initial fracture toughness but the smallest reduction (7 %-11 % depending on the mode) across all environments, with slag cement concrete and OPC experiencing reductions of 11 %-14 % and 14 %-21 %, respectively, in acidic conditions. In alkaline environments, GPC showed a 3 %-6 % decrease in fracture toughness, while OPC and slag cement concrete showed reductions of 6 %-13 %. The gradient boosting regressor accurately predicted the failure load of CSNBD specimens, achieving an R² score of 0.948. Finally, SHAP sensitivity analysis identified concrete type, exposure time, and environment as critical input features, consistent with experimental results.

Failure Prediction of an Airfoil-Type Crack in Thermoelectric Coupling Materials

Airfoil-type cracks with curved surfaces and sharp tips are frequently encountered in modern engineering applications and the aerospace industry. However, such cracks significantly threaten structures due to stress concentration and further cause damage. This study investigates the theoretical analysis and failure prediction of airfoil-type cracks within thermoelectric coupling materials under the interaction of remote mechanical loads, heat flux and electric current density. By employing complex variable theory, conformal mapping methods, and the analytic continuation theorem, the full-field stress and stress intensity factors of an airfoil-type crack were determined under three different loading conditions. The results indicate that when the airfoil-shaped crack tip is subjected to horizontal compression, vertical tensile mechanical load, and both heat flux and current density parallel to the crack, the crack tip would reach a dangerous state. Additionally, the full-field stress distribution reveals stress concentration around the crack tip, which explains variations in SIFs under different external forces and shape factors. Four critical situations suppressing crack propagation under mixed loading conditions are proposed. Furthermore, using an acrylic plate and bismuth telluride alloy as examples, the critical values of external loading for the mixed-mode fracture are determined based on the strain energy density criterion. These critical values can be used to predict failure initiation, thus reducing the risk of structural failure due to airfoil cracks within thermoelectric coupling materials.

Microstructural characteristics and mechanical response of transient liquid-phase diffusion bonding AlCoCrFeNi2.1 high entropy alloy utilizing BNi-5 interlayer

This study focuses on the microstructural characteristics and mechanical response of the transient liquid phase (TLP) diffusion-bonded joints of AlCoCrFeNi2.1 high entropy alloy. Bonging parameters were chosen and optimized for the AlCoCrFeNi2.1 alloy and the TLP joining mechanism. The relatively complete isothermal solidification zone (ISZ) ensured a reliable connection of the base metal. As the bonding temperature increased from 1060 °C to 1160 °C, the athermal solidification zone (ASZ) disappeared. The homogenization degree of elements in the ISZ continuously increased, the matrix γ phase became fully ordered, and the B2 phase was precipitated, achieving high entropy state in the ISZ of the joint. Additionally, semi-coherent relationships of [0, 1, 1] L12, [0, 1, 1] B2 and (1, 1̅,1) L12, (2̅, 3, 1̅) B2 were detected between the B2 precipitate phase and the L12 matrix. The phase distribution in the ISZ significantly influenced the mechanical properties of the joint. The complete ordering of the matrix phase at 1160 °C eliminated deformation incoordination between the ISZ and the diffusion-affected zone (DAZ). The B2 phase transitioned from precipitating at the grain boundary at lower temperatures to precipitating uniformly in the ISZ, eliminating the weakening effect of the grain boundary and resulting in tensile strength and elongation comparable to the base material. The fracture morphology indicated a mixed fracture mode.

A peridynamic compensated critical energy density criterion for mixed-mode fracturing in quasi-brittle materials

This study integrates a compensated critical energy density criterion (CCED) into the ordinary state-based peridynamics framework to enable detailed analysis of mode I, mode II, and mixed-mode fracturing. The proposed bond failure criterion builds upon the advantages of the traditional critical energy density (CED) to enable precise calculation of strain energy density. Additionally, the critical bond rotation criterion (CR) is employed to account for shear bonds that the CED might overlook. Using the peridynamic differential operator (PDDO) format in formulating the entire computational framework enhances both accuracy and numerical stability. The model’s validation is conducted through benchmark examples, including the mode I double cantilever beam test, the mode II compact shear test, and a mixed-mode mechanical fracturing test in pre-notched specimens with central borehole loading. Convergence studies, comparative analyses with finite element method simulations and theoretical solutions thoroughly examine the proposed model’s performance and accuracy.

A novel mixed mode fracture criterion for functionally graded materials considering fracture process zone

A new mixed mode I/II fracture criterion based on strain energy release rate theory is proposed for investigating fracture behavior in cracked functionally graded materials. In this study, the crack is assumed to be located along the material gradation and within the brittle side of the material. For the first time in this type of material, the fracture process zone and the energy dissipated in this region are considered to enhance the prediction accuracy of the proposed criterion. The mechanical and fracture properties of the damage zone are characterized using the microcrack density parameter. A novel method for manufacturing functionally graded materials is introduced, offering numerous advantages by controlling the gel time of the resin. The critical mixed mode and pure mode II fracture toughness of the fabricated specimens are determined through three-point and four-point bending tests. A comparison of the experimental data with the fracture limit curve demonstrates the effectiveness of the newly proposed criterion for investigating mixed mode fracture in functionally graded materials

Mixed-mode fracture: Combination of Arcan fixture and stereo-DIC

To characterise mixed-mode fracture, including mode III, a combination of Arcan fixture, that offers both in-plane and out-of-plane loading, and stereo digital image correlation (DIC) has proven valuable for measuring in-plane and out-of-plane surface displacement and deformation fields while allowing for direct stress intensity factor (SIF) extraction throughout the tests, but it comes with many caveats. Using stereo DIC, the mixed mode fracture behaviour of PMMA is analysed using a novel mode decomposition technique that combines experimental and analytical approaches. This technique divides the measured displacement field from digital image correlation into three distinct components: a symmetric field (uI), an in-plane anti-symmetric field (uII), and an out-of-plane antisymmetric field (uIII), decomposed by a difference operation on the reflected and non-reflected fields about the crack plane. Then, the strain energy release rate of the crack is calculated for each loading mode using finite elements without knowledge of the sample geometry or nominal loading condition. Our work revealed parasitic loading modes induced by the Archan fixture plus other intrinsic sources related to load accommodation. Nevertheless, the calculated SIFs measured during stable crack growth were normalised using different fracture toughness (KIC) values with KIC of 1.82 ± 0.32 MPa m0.5, providing the best fit to the fracture loci with an R2 of 0.95. This value agrees well with the ASTM-obtained value of 1.7 MPa m0.5.

An anisotropic eigenfracture approach accounting for mixed fracture modes in wooden structures by the Representative Crack Element framework

Finite Element analysis of anisotropic fracture phenomena in wood is a challenging task, particularly when dealing with intricate loading scenarios and mode-specific behavior. The appeal of energetically motivated approaches, such as the eigenfracture method, is that they enable simulation of fracture without prior knowledge of the crack path. The promising eigenfracture method has shown good numerical performance for isotropic materials, and this contribution showcases its application to anisotropic materials. Wood is one such anisotropic material and in this manuscript, the directional dependence of both elasticity and fracture evolution are incorporated into the eigenfracture approach. Further, the eigenfracture approach is used in conjunction with Representative Crack Elements (RCE), which permit accurate modeling of physical crack deformations. The governing equations are systematically derived and implemented into the Finite Element framework. By representative numerical examples, some advantages over the alternative phase-field method are demonstrated. Another highlight of this work is that it is possible to provide a realistic ratio of the energy release rates parallel to and perpendicular to the fiber direction in order to achieve physically accurate crack patterns. Additionally, the calculation effort is reduced, because the unknowns required to determine the crack kinematics can be solved analytically at the material level, a feature that also enables parallelization.

Mixed-mode fracture assessment of largely deformable hyperelastic materials highlighting crack removal phenomenon

Due to the special mechanical characteristics, rubbers are nowadays utilized in various industries. In this research, the effect of an existing crack on the behavior of Nitrile Butadiene Rubber (NBR) filled with Carbon Black (CB), which has shown high deformability, is investigated experimentally, numerically and analytically under mixed-mode (I/II) loading. It is observed experimentally that in the cracked largely deformable rubbers, after the initial crack blunting, the crack is nearly removed and the specimen is deformed in such a way that almost no sign of stress concentration factor exists in the sample. Thus, the rupture of these rubbers is divided into two separate phases; named here as the crack removal phase and the final rupture phase, and for each separate phase, a distinct criterion is developed to assess the fracture of cracked rubbers. The good agreement between the theoretical predictions based on each criterion and the corresponding experimental data confirms the very good accuracy of the presented criteria. The independence of critical J-integral value (Jc) from the mode mixity is also investigated.

A strain-based criterion for predicting size-dependent fracture resistance of quasi-brittle materials under mixed mode loading

In the present study, a strain-based approach called the modified maximum tangential strain criterion has been developed for evaluating size effect on the mixed-mode fracture resistance of quasi-brittle materials. This approach relies on the generalized maximum tangential strain criterion, which considers the singular (K) and the first non-singular (T) terms of Williams series expansion. Furthermore, as an important and size-dependent parameter in the proposed criterion, a new formulation is presented to calculate the critical distance (rc). The predictions of the new criterion are compared not only with the available experimental data, but also with the results estimated by another size effect criterion named the modified maximum tangential stress criterion. It is shown that the proposed strain-based criterion is highly capable of estimating the size-dependent mixed-mode fracture behavior of quasi-brittle materials without requiring to compute the other higher order terms.

A phase field model with modified volumetric-deviatoric decomposition for the mixed-mode fracture of rock

Recent attempts of phase field models to simulate rock fracture problems have been regarded as fruitful, owing to their powerful crack characterization capability. However, existing phase field models still exhibit limitations in simulating quasi-coplanar shear cracks and related coalescence patterns that often occur in rock failure. In this study, we propose a novel phase field model to simulate the mixed-mode fracture, in which a modified volumetric-deviatoric decomposition that combines the advantages of the volumetric-deviatoric decomposition and the spectral decomposition is conducted to distinguish between tensile, tensile-shear, and compressive-shear fractures; to reduce unrealistic damage in the area outside the crack trajectory, a threshold parameter is introduced into the degradation function to control whether the related energy participates in damage evolution; and the hybrid formulation is employed to efficiently and robustly solve the displacement and phase fields alternately. The feasibility of the proposed phase field model is first validated by a benchmark example. Next, the application of this model to the simulations of different rock specimens under uniaxial compression reveals a better agreement with experimental observations than previous simulation results, demonstrating an advancement over existing phase field models.

Bonding Behavior and Quality of Pressureless Ag Sintering on (111)-Oriented Nanotwinned Cu Substrate in Ambient Air.

(111)-oriented nanotwinned Cu ((111)nt-Cu) has shown its high surface diffusion rate and better oxidation resistance over common polycrystalline Cu (C-Cu). The application of (111)nt-Cu as an interface metallization layer in Ag-sintered technology under the role of oxygen was investigated in this work, and its connecting behavior was further clarified by comparing it with C-Cu. As the sintering temperature decreasing from 300 to 200 °C, the shear strength on the (111)nt-Cu substrate was still greater than 55 MPa after sintering for 10 min. The fracture surface correspondingly changed from the interface of Ag/die to mixed fracture mode, involving the interface of the Ag/Cu substrate and Ag/die. The existence of copper oxide provided a tight connection between Ag and the (111)nt-Cu substrate at all of the studied temperatures. Although lots of small dispersed voids were seen at the interface between copper oxide and (111)nt-Cu at 300 °C, these impurity-induced voids would not necessarily be a failure position and could be improved by adjusting the sintering temperature and time; for example, 200 °C/10 min or heating to 300 °C, and then start cooling at the same time. The microstructure of Ag-Cu joint on (111)nt-Cu behaved better than that on C-Cu. The thinner copper oxide layer and the higher connection ratio of the interface between copper oxide and Ag were still found on the (111)nt-Cu connection's structure. The poor connection between copper oxide and Ag on C-Cu easily became the failure interface. By controlling the thickness of copper oxide and the content of impurity-induced voids, the use of (111)nt-Cu in advanced-packaging could be improved to a new level.

Fracture toughness of mixed-mode anticracks in highly porous materials

When porous materials are subjected to compressive loads, localized failure chains, commonly termed anticracks, can occur and cause large-scale structural failure. Similar to tensile and shear cracks, the resistance to anticrack growth is governed by fracture toughness. Yet, nothing is known about the mixed-mode fracture toughness for highly porous materials subjected to shear and compression. We present fracture mechanical field experiments tailored for weak layers in a natural snowpack. Using a mechanical model for interpretation, we calculate the fracture toughness for anticrack growth for the full range of mode interactions, from pure shear to pure collapse. The measurements show that fracture toughness values are significantly larger in shear than in collapse, and suggest a power-law interaction between the anticrack propagation modes. Our results offer insights into the fracture characteristics of anticracks in highly porous materials and provide important benchmarks for computational modeling.

An improved phase-field model for oil–water two-phase flow and mixed-mode fracture propagation in hydraulic fracturing

An improved novel phase-field model is proposed for describing fracture propagation, effectively characterizing the interactions between oil–water two-phase fluids and solids as well as mixed-mode fracturing. This model encompasses equations for two-phase flow stress balance, fluid flow, and a complex mode fracture phase-field specific to two-phase flow. Within the fluid flow equations, the capillary pressure caused by the oil–water interface during the fluid loss process in the matrix is accounted for, and the anisotropic relative permeability during fluid flow is linked to normalized saturation. The driving forces for fracture propagation in the phase-field are categorized into Mode I and Mode II forces, with the contribution from the oil–water two-phase fluid attributed to the Mode I (tensile) driving force. The model uses finite element numerical discretization and the Newton-Raphson iterative method to establish a numerical iteration format, employing an implicit-explicit staggered solution scheme. Additionally, the model is used to investigate several key factors. It assesses the impact of different intersection angles on fracture propagation in naturally porous media. It also studies the effect of different natural fracture permeabilities on hydraulic fracture propagation. Finally, the model analyzes the propagation patterns of hydraulic fractures under different perforation phase angles.

Low-Temperature Mixed Mode (I/II) Fracture Characterization of Polymerized Sulfur Modified AC Mixtures

In this study, asphalt concrete (AC) mixtures were modified with polymerized sulfur, using PG58-22 bitumen, and crushed siliceous aggregate. Modifications involved replacing the base binder with 20%, 30%, and 50% polymerized sulfur, compared to a control mix with no replacement. The mixtures were subjected to Single Edge Notched-Beam (SE(B)) fracture tests under mixed mode (I/II) conditions with notch offset value of 48 mm, with temperatures ranging from 0 °C to -20 °C. These tests, focusing on the mixtures' response to mixed mode loading, provided load-displacement curves, enabling the determination of fracture energy. Results indicated an increase in fracture energy for 20% and 30% sulfur-modified mixtures. However, a trend towards increased embrittlement was also observed, as fractures occurred at lower displacements. Significantly, higher sulfur content correlated with similar or decreased mixed-mode (I/II) fracture energy, suggesting an improved resistance to low-temperature cracking for lower replacement percentages.

Effect of grain morphology and interface on the toughness of nacre-like aluminas

Ceramic composites that are both tough and strong are needed for extreme environments, ranging from aerospace and nuclear, to medical applications. Bioinspired ceramic composites tackle this challenge by mimicking tough natural structures. Nacre-like aluminas (NLAs) are ceramics processed using advanced colloidal techniques to obtain a microstructure resembling the brick-and-mortar structure of nacre. NLAs can achieve flexural strengths up to 600 MPa and fracture toughnesses ranging from 5 MPa.m1/2 up to 16 MPa.m1/2 after crack propagation. These high values mainly rely on deflection mechanisms; these can operate at temperatures up to 1200°C and under impact testing. The fracture properties have been found to be highly dependent on the mortar composition and processing technique used. However, we still don't know how far we can extend the structural properties of NLAs by changing their mortar strength and brick morphology. In this study, we combine mode I wedge splitting testing and in situ synchrotron X-ray microtomography of mixed-mode SENB fracture testing along with a kinked crack analytical formulation to study the microstructure-properties relationship in NLAs having two mortar compositions/morphologies. We compare the toughness values obtained in mode I and II with deflection models and study the effect of mode-mixity on the NLA toughness. Our findings show the grain morphology strongly influences the fracture behaviour, with the longer grain material showing longer stable crack deflection which could be further optimised going forwards. By contrast the mortar properties which provide deflection and local toughening mechanisms may be nearing their optimal strength.

Mixed-mode thermo-mechanical fracture: An adaptive multi-patch isogeometric phase-field cohesive zone model

This work presents an adaptive phase-field cohesive zone model (PF-CZM) for simulating mixed-mode crack nucleation and growth in isotropic rock-like materials subjected to thermo-mechanical interactions. The proposed approach combines an adaptive multi-patch isogeometric analysis (MP-IGA) and length-scale insensitive PF-CZM. The formulation captures the distinct critical energy release rates for Mode-I and Mode-II fractures, which is crucial for predicting mixed-mode thermo-mechanical fracture behavior in isotropic rock-like materials. The PF-CZM governing equations are solved with isogeometric analysis based on locally refined non-uniform rational B-splines (LR NURBS), and the complex structural geometry is exactly described with multiple LR NURBS patches. The field variables, such as displacement, phase-field, and temperature at the interface of adjacent patches, are coupled using Nitsche’s method. To enhance the computational efficiency while maintaining accuracy, a refinement-correction adaptive scheme combined with the structured mesh refinement strategy is developed. The proposed framework is validated against recent numerical and experimental results in the literature, particularly in the context of capturing complex behavior of mixed-mode crack propagation in isotropic rock-like materials subjected to thermo-mechanical loading.

Theoretical analysis of mode I fracture of adhesively bonded bi-material DCB joints

There has been a disputation about how to achieve a pure mode I fracture and thus measure its toughness for an adhesively bonded bi-material double cantilever beam (DCB) specimen. This paper therefore develops a theoretical methodology to calculate the normal and shear stress distributions in the adhesive layer and to perform data reduction and mode partitioning for generic bi-material DCB specimens. The theoretical model is validated using experimental data. The adhesive layer in a generic bi-material DCB joint is loaded both in normal and shear based on the predicted stress distributions in the adhesive layer, resulting in a mixed mode fracture behavior. The prediction results substantiate that a strain-based design principle eliminates the shear stresses and thus leads to pure mode I fracture in bi-material DCB specimens. This paper justifies that the established data reduction method is applicable for measuring mode I fracture toughness when a bi-material DCB is designed according to the strain-based design criterion.

Laser lap joining of R60702 zirconium alloy and 304 stainless steel utilizing CoNiTiV high entropy alloy

In this study, the laser lap welding of R60702 zirconium alloy and 304 stainless steel was successfully achieved by incorporating CoNiTiV high entropy alloy (HEA) interlayer. The addition of HEA interlayer altered elemental diffusion and prevented the formation of Zr-Fe intermetallic compounds. The maximum tensile and shear strength of the joint (998.4 N) is 2.5 times that of the sandwich joint without the HEA layer (392.1 N), with the displacement also increasing by 85 %. The inclusion of the intermediate HEA layer transforms the original brittle fracture mode into a tough-brittle mixed fracture mode. The elimination of Zr-Fe intermetallic compounds is the primary factor contributing to the enhancement of mechanical properties. The main phase composition includes α-Zr, Cr2Zr, CoTi2, Ni7Zr2 and Zr3Co phase.

Improving Mixed-Mode Fracture Properties of Concrete Reinforced with Macrosynthetic Plastic Fibers: An Experimental and Numerical Investigation

Improving Mixed-Mode Fracture Properties of Concrete Reinforced with Macrosynthetic Plastic Fibers: An Experimental and Numerical Investigation