Quasi-static Fracture Research Articles

Fracture as an Emergent Phenomenon

The mechanics of fracture propagation provides essential knowledge for the risk tolerance design of devices, structures, and vehicles. Techniques of free energy minimization provide guidance, but have limited applicability to material systems evolving away from equilibrium. Experimental evidence shows that the material response depends on driving forces arising from mechanical fields. Recent years have witnessed the development of new methods for modeling complex dynamic and quasistatic fracture. New approaches may differ remarkably from previous ones, as they involve implicit coupling between damaged and undamaged states, allowing fracture to be modeled as emergent phenomena.The focus of this workshop is on the most advanced techniques for modeling fracture, represented by eigenerosion methods, variational approaches, phase field fracture models, and non-local approaches. Technical progress is contingent on the further development of the mathematical framework underlying these techniques. This is necessary for accurate and reliable computational modeling of fracture for multiple freely propagating cracks. The objective of this workshop is to mathematically identify and discuss open issues related to fracture modeling and to highlight recent advances. Addressing fundamental issues will foster exchange between the different communities, essential for advancing the field.

Evaluating the effects of nonlocality and numerical discretization in peridynamic solutions for quasi-static elasticity and fracture

The difference between the computational peridynamic results and the corresponding exact classical solutions is contributed by the error induced by numerical discretization and the nonlocality-induced difference. To evaluate and compare these contributions and investigate their dependence on the peridynamic influence functions, in this paper, we apply different peridynamic influence functions and different horizon factors (m, the ratio between the horizon size and the grid spacing) to conduct static (or quasi-static) tensile numerical tests. We calculate the difference between the peridynamic solutions and the corresponding classical solutions for static uniaxial tension of thin plates with or without hole, the J-integral of a single crack under Mode I loading condition, and the quasi-static fracture in a perforated thin plate. For the case of uniaxial tension in a thin, homogeneous plate, we separate the effects of nonlocality and numerical discretization by implementing the boundary conditions in different ways. The numerical results show that both the effects of nonlocality and numerical discretization correspond to the nonlocal constants of the influence functions (with few exceptions). For problems with the presence of the peridynamic surface effect, such as holes, influence function with weaker nonlocality is a better choice to obtain more accurate results.

Spatiotemporal Characterization of Microstructure Morphology, Mechanical Properties and Bone Remodeling of Rat Tibia Under Uniaxial Compressive Overload Loading.

Bone tissue is subjected to increased mechanical stress during high-intensity work. Inadequate bone remodeling reparability can result in the continuous accumulation of microdamage, leading to stress fractures. The aim of this work was to investigate the characteristics and repair mechanisms of tibial microdamage under several degrees of overload. Also, we aimed at better understanding the effects of overload on the multi-scale structure and mechanical properties of bone. Sixty 5-month female rats were divided into three groups with different time points. Micro-CT was used to evaluate the three-dimensional microstructure, and three-point bending, quasi-static fracture toughness and creep mechanical test were carried out to evaluate the mechanical properties. SEM was used to observe the morphological characteristics of fracture surfaces. Section staining was used to count the microdamage parameters and numbers of osteoblasts and osteoclasts. The microarchitectures of cancellous and cortical bones in the three overload groups showed different degrees of damage. Overload led to a messy crystal structure of cortical bone, with slender microcracks mixed in, and a large number of broken fibers of cancellous bone. The properties associated with the elastic plasticity, fracture toughness, and viscoelasticity of cortical bone reduced in three groups, with that corresponding to day 30 presenting the highest damage. The accumulation of microdamage mainly occurred in the first 14days, that is, the crack density peaked on day 14. Peak-targeted bone remodeling of cortical and cancellous bones occurred mainly between days 14 and 30. The influence of overload mechanical environment on bone quality at different time points was deeply investigated, which is of great significance for the etiology and treatment of stress fractures.

Experimental and simulation study on quasi-static and dynamic fracture toughness of 2A12-T4 aluminum alloy using CTS specimen

Based on the study of dynamic fracture toughness of mixed mode I/II fracture crack, using compact tensile shear (CTS) specimen and the method of split Hopkinson tensile bar (SHTB) combined with special fixture, which complete the experiments of complex fracture modes of 2A12-T4 aluminum alloy material under quasi-static and dynamic tensile tests. By combining the results of finite element simulation with experimental–numerical method, the feasibility of Richard’s theory formula in dynamic fracture experiment is discussed. The variation law of fracture toughness under quasi-static and dynamic complex fracture under mixed-mode was analyzed by finite element simulation and experiment. The results show that, in the process of quasi-static and dynamic fracture, the fracture toughness of CTS specimen under mixed mode I/II crack loading is the highest. The conclusion is proved by Richard theorical formula and experimental–numerical methods jointly. When the specimen is subjected to pure mode I fracture, the quasi-static fracture toughness is consistent with that is in condition of dynamic experiment. However, when the specimen is subjected to mode II and mixed mode I/II fracture, the dynamic fracture toughness changes greatly compared with the quasi-static fracture toughness. In addition, in the dynamic fracture experiment, the dynamic fracture toughness of mixed mode I/II depends on the initiation time, and the quasi-static fracture criterion is no longer applicable.

GPU-accelerated approach for 2D fracture analysis of structures combining finite particle method and cohesive zone model

The fracture analysis of structures typically involves strong discontinuity and nonlinear behaviors, and it requires fine meshing and small time steps, making it time-consuming. This study proposes an approach accelerated by graphics processing units (GPU) for two-dimensional (2D) fracture analysis of structures combining finite particle method (FPM) and cohesive zone model (CZM). Specifically, the equations of motion of particles are presented, and the internal forces of planar triangular elements are derived based on FPM. Subsequently, the internal forces of four-particle cohesive element are derived to describe the CZM, and the linear elastic stage, softening stage, and failure stage are depicted according to the traction-separation criteria. An explicit GPU-based FPM analysis strategy for structural fracture analysis is then developed, and the parallel solvers for particles, triangular elements, and cohesive elements are implemented. A quasi-static fracture analysis of a concrete beam and a dynamic fracture analysis of the Kalthoff problem is conducted to verify the effectiveness of the proposed approach. The obtained crack propagation path, load vs. displacement curves, and crack propagation speed are in good agreement with the experimental results and the results in the literature. The efficiency of the proposed approach is also investigated. The achieved maximum speedup ratio of the GPU-accelerated approach to the central processing unit (CPU)-based approach reaches around 24, and the achieved speedup ratio relative to the commercial finite element software Abaqus/Explicit reaches around 11.5, demonstrating the computational efficiency of the proposed approach.

Estimation of mode I quasi-static fracture of notched aluminum–lithium AW2099-T83 alloy using local approaches and machine learning

Aluminum–lithium (Al–Li) alloys offer superior performance under different conditions that involve mechanical loading, high temperature and corrosive environment. Therefore, Al–Li alloys are being used in the defense, aerospace, and aircraft industries, specifically in structural parts such as fuselage, empennage, and wings. By modifying the chemical composition, a third generation of Al–Li alloys has been introduced to overcome the mechanical and thermal shortcomings of previous Al–Li generations. As structural parts usually contain notches, it is of paramount importance to study the strength of the newly introduced alloys in the presence of such geometrical discontinuities to select the suitable alloy for a particular application. The aim of this work is to analyze the strength of U and V-notched specimens machined from extruded AW2099-T83 Al–Li alloys under quasi-static loading. The fracture stress of specimens with various notch radii and angles were estimated using strain energy density (SED) and the theory of critical distances (TCD) methods. A support vector machine (SVM) regression model was also implemented to assess the applicability of estimating fracture stress using machine learning approaches. The results show an average absolute discrepancy of 6.6 %, 7.8 % and 2.8 % for SED, TCD and SVM methods, respectively.

A non-uniform lattice design method for lightweight structures in 3D printing

Lightweight design is an essential topic in aerospace, automotive and other fields. In automobile manufacturing, the engine connecting rod is one of the main components; its lightweight design has a high reference value. And 3D printing provides a feasible solution for designing and manufacturing lightweight structures. Unlike the traditional geometrically homogeneous point design, this study offers a non-homogeneous point design method based on the spatial stress state of additively manufactured components. After numerical simulation of quasi-static stresses on a model of an engine connecting rod, finite element grid cells with different stress values are replaced by lattice cells with different specific stiffnesses at similar local stress levels. The overall specific stiffness of the structure is further improved by continuing the optimized design with the corresponding gradient-type reinforcement of the non-uniform lattice structure. The basic idea of this design is to perform non-uniform adaptive filling of solid parts under localized loading by employing different types of unit cells. Stereolithography 3D printing technology prepares the engine lattice structural parts for quasi-static compression comparison experiments and fracture analysis after failure. The conclusions show that the engine connecting rod members with non-homogeneous lattice have more excellent overall mechanical properties than homogeneous lattice members. This work demonstrates the feasibility of such design methods for 3D printing lightweight structures and optimization.

Quasi-static compressive fracture behavior of three-period minimum surface Al2O3 / Al composites fabricated by stereolithography

Extensive research has been conducted on the structures and preparation methods of ceramic/metal composites to achieve materials with superior strength and toughness. Among these, the triply periodic minimal surface (TPMS) structure stands out due to its remarkable advantages. In this study, we employed SLA additive manufacturing to fabricate a Gyroid-structured ceramic skeleton, and utilized metal infiltration to prepare interpenetrating Al2O3/Al composites. A comparison was made between the compressive properties of Al2O3/Al interpenetrating composites with Gyroid curved surface structure and those with honeycomb structure. The experimental outcomes indicate that compressive failure in both structural composites primarily occurs at the bond interface between the ceramic framework and the metal. These composites exhibit brittle fracture and experience shear failure. Furthermore, the compressive failure mechanism involves a mixed fracture composed of cleavage and micropore polymerization fractures. Gyroid curved structural composites exhibit lower local stress concentration due to their smooth and regular topology, rendering them with higher strength and ductility compared to honeycomb structural composites. The ultimate compressive strength of the Gyroid curved composite is 264 MPa, representing a 3.8-fold increase over the honeycomb structure.

A coupled high-accuracy phase-field fluid–structure interaction framework for Stokes fluid-filled fracture surrounded by an elastic medium

In this work, we couple a high-accuracy phase-field fracture reconstruction approach iteratively to fluid–structure interaction. The key motivation is to utilise phase-field modelling to compute the fracture path. A mesh reconstruction allows a switch from interface-capturing to interface-tracking in which the coupling conditions can be realised in a highly accurate fashion. Consequently, inside the fracture, a Stokes flow can be modelled that is coupled to the surrounding elastic medium. A fully coupled approach is obtained by iterating between the phase-field and the fluid–structure interaction model. The resulting algorithm is demonstrated for several numerical examples of quasi-static brittle fractures. We consider both stationary and quasi-stationary problems. In the latter, the dynamics arise through an incrementally increasing given pressure.

A thermodynamically consistent phase field model for brittle fracture in graded coatings under thermo-mechanical loading

This paper presents a thermodynamically consistent phase field formulation designed for investigating quasi-static brittle fracture under thermo-mechanical loading conditions in functionally graded materials. The governing equations for linear momentum and crack evolution are derived by minimizing the free energy functional, which includes elastic strain energy and surface energy, with respect to displacement field and the phase field variable, respectively. A thermodynamically consistent formulation is employed to establish the governing equation for heat conduction. The proposed formulation is converted into a set of finite element equations using the Ritz method and implemented in COMSOL Multiphysics, a commercial finite element software which allows for writing governing equations in their strong form. A segregated solver is utilized to solve iteratively the coupled problem. The proposed model is verified against existing mechanical and thermo-mechanical phase field models, demonstrating good agreement with the literature. To explore the model’s capabilities, a graded layer composed of Zirconia and a Titanium alloy with a surface crack undergoing thermo-mechanical loading is considered as a case study for this investigation. Additionally, the model incorporates a thermal conductivity degradation proposed. A parametric study is performed to investigate the effect of varying parameters like volume fraction, thermal loading rate, and the thermal conductivity degradation on the load–displacement curve of the graded layer. These findings contribute to the understanding of thermo-mechanical behavior of surface graded layers and serve as a foundation for addressing more complex thermo-mechanical problems.

An elastoplastic phase-field model for quasi-static fracture of nickel-based super-alloys

In the present study, an elastoplastic phase-field model of quasi-static fracture in ductile materials is proposed in the variational framework for J2 plasticity with isotropic hardening, which is suitable to describe the quasi-static behavior of metals as investigated in the performed experiments. These contributions include: (1) the free energy functions for coupling elastic response, plastic yielding and damage evolution are established. (2) The new elastic and plastic energy degradation functions are constructed to quantitatively describe the relationship between energy release and phase-field evolution of elastoplastic materials. (3) Damage evolution and plastic yielding criteria are derived. (4) From a numerical point of view, we derive the governing equations and the corresponding weak forms and the overall solution procedure for the phase-field model is given via the use of a return-mapping algorithm. This phase-field model was validated by a series of tensile experiments on Inconel 718 nickel-based super-alloys standard specimens. In order to compare the simulation results with the experimental results more comprehensively, the digital image correlation (DIC) technique is applied to experimentally investigate the specimen deformation information. In addition, to verify the potential of the model to capture complex cracks, we performed Nooru-Mohamed tests. The numerical simulation results are in good agreements with the results of previous experimental work.

Virtual clustering analysis for phase field model of quasi-static brittle fracture

Virtual clustering analysis for phase field model of quasi-static brittle fracture

Structural and nonlinear J-integral fracture toughness for nanoclay toughened ternary HDPE/LDPE-g-MA/ABS blend nanocomposites

High-density polyethylene (HDPE) has a higher strength-to-density ratio and stiffness but a low branching degree for the packed linear chains that restrict the ability to bond and resist cracking. This study conducts the role of inserting rigid nanoclay (NC) and soft acrylonitrile butadiene styrene (ABS) on the structural, nonlinear fracture toughness and crack resistance of a ternary HDPE/low-density polyethylene-grafted maleic anhydrite (LDPE-g-MA)/ABS blend. Varying additions of 1, 3, 5, and 7 % NC and 5, 10, 15 wt. % ABS were inserted into neat HDPE and HDPE90/LDPE-g-MA10. All materials were hand-mixed before feeding into a single screw extruder and directly melt-blended twice to achieve a good dispersion of nanofiller in the matrix. The structural characteristics and the fracture surfaces of NC/HDPE/LDPE-g-MA and NC/HDPE/LDPE-g-MA/ABS were investigated by TEM, XRD, SEM, and FTIR spectra. Tensile strength and the critical dissipated energy (JIc) determined by quasi-static J-integral fracture mechanic revealed a higher absorbing fracture energy of 75 KJ/m2 for the binary and 85 KJ/m2 for the ternary nanocomposites. The synergistic percolated role of the NC particles and ABS copolymer in front of the crack tip region hinders crack growth for the presence of micro-void coalescence and massive shear-yielding toughening mechanisms.

Ductile fracture prediction of additively manufactured Ti-6Al-4 V alloy based on void growth and coalescence of a unit-cell model

In this work, we have studied the fracture mechanism and proposed a micromechanical method with a unit cell model to predict the ductile fracture of additively manufactured (AMed) Ti-6Al-4 V alloy. The tensile experiments with smooth round bars and notched round bars were conducted to investigate the mechanical properties and ductile fracture behavior of materials. The proposed unit-cell model was validated by quasi-static tensile fracture tests to predict the ductile failure of titanium alloy manufactured by laser power bed fusion (L-PBF) under different stress states, which include a wide range of stress triaxialities. An equivalent initial void fraction was introduced to evaluate the ductile behavior of the titanium alloy. Our investigation shows that the stress states significantly influence the void coalescence behavior of AMed Ti-6Al-4 V alloy, and the fracture locus of L-PBF fabricated Ti-6Al-4 V alloy under different stress states was successfully predicted by the proposed unit-cell model under proportional loading. Our study provides useful guidance for the future design and application of metal materials for additive manufacturing.

Co-Influence of Restoration Bonding and Inlay Cavity Design on Fracture Load of Restored Tooth.

The aim of this study was to investigate the co-influence of indirect mesio- occlusal-distal (MOD) cavity geometry and inlay restoration bonding on quasi-static fracture load of the restored tooth. Forty-eight intact human molar teeth were selected and prepared for standardized edge-shaped or round-shaped MOD cavities. The resin composite (Cerasmart, GC) inlays were bonded with the state-of-the-art inlay bonding protocol or with intentionally deteriorated bonding using n-hexane-wax solution for preconditioning. Restored teeth were loaded along the long axis of the tooth. Ultimate fracture load was recorded, and the type of fracture was visually determined and classified. Statistical analysis of load values was performed by Kruskal-Wallis test. Round-shaped cavity design with bonded restoration presented the highest fracture load (1658N). Bonding had significant influence on the fracture load of roundshaped cavity design (p=0.0003), whereas cavity design had no influence when the bonding was deteriorated (p=0.8075). In the case of deteriorated bonding, either the inlay or tooth fractured separately whereas in the bonded inlays fractures were commonly found both in the tooth and inlay. According to this study, bonded inlay restoration increased fracture resistance, while cavity design had no statistical difference on fracture resistance of the restored tooth.

Dynamic fracture response of adhesive joints subjected to mode-I impact wedge loading

ABSTRACT This paper aims to examine the behavior of cracked adhesive joints experimentally and numerically when they are subjected to quasi-static and dynamic loads using the tensile and falling-wedge impact tests, respectively. Double cantilever beam (DCB) specimens were manufactured and tested under impact loading with varying impact energies to analyze the adhesive joint mode-I dynamic fracture response. To determine quasi-static and dynamic mode-I fracture energies, the compliance-based beam method (CBBM) was utilized accounting for the axial force exerted by the wedge. It was found that by changing the loading condition from quasi-static to impact, the maximum force and fracture energy of adhesively bonded joints were considerably increased by 275% and 452%, respectively. However, within the impact test conditions, increases of 10% and 22% were obtained in maximum force and fracture energy when the impact energy was tripled. The adhesive joints tested under static loading experienced a mixed interfacial/cohesive failure pattern, whereas by shifting the loading condition to impact, the failure pattern turned fully cohesive. The quasi-static and dynamic fracture responses of adhesive joints were modeled using the cohesive zone model employing a triangular traction-separation law. The numerical and experimental results exhibited reasonable correlation.

On the (lack of) representativeness of quasi-static variational fracture models for unstable crack propagation

The present work is devoted to prove that unstable crack propagation events do not comply with quasi-static hypotheses and thus should be modelled by dynamic approaches. Comprehensive supporting evidence is provided on the basis of three different analyses conducted on multi-ligament unstable fracture conditions, including a simplified Spring-Mass model, detailed quasi-static and dynamic Phase Field fracture models, and bespoke experiments with 3D printed specimens. The obtained results unequivocally show that neglecting the inertial effects can lead to unsafe predictions in the presence of energetic barriers for the development of fracture. Likewise, quasi-static Phase Field fracture models are proven to yield crack patterns that disagree with the experimental evidence because they overlook the progressive diffusion of the mechanical information within the continuum. Moreover, the inability of quasi-static approaches to follow unstable crack propagation is shown to weaken the crucial irreversibility condition of fracture. Overall, these experimentally backed insights should be gravely reckoned with, for they are not exclusive to Phase Field fracture models but common to (almost) any variational approach to fracture, inter alia Cohesive Zone Models or Continuum Damage Mechanics.

Fracture analysis of orthotropic functionally graded materials using element-based peridynamics

A fracture model of orthotropic functionally graded materials (FGMs) based on element-based peridynamics (EBPD) is proposed. Two-dimensional (2D) orthotropic FGMs use three-node triangular elements to describe the force density of EBPD. Due to the gradient difference of material properties inside the element, the Gauss integral formula is used to average the node performances to the integral point of the element. The equations of motion for EBPD is established, which is also applied to orthotropic FGMs by the Gauss integral formula. The micromodulus coefficient of the EBPD is derived for orthotropic FGMs. The critical strain energy density criterion is used as the failure criterion to simulate the quasi-static and dynamic fracture problems. A series of numerical examples, including displacement analysis, stress analysis, dynamic crack propagation of FGM beam, and crack growth of orthotropic FGMs under four-point bending load, are used to validate the accuracy of the fracture model of orthotropic FGMs. The proposed model can be helpful to the material gradient design for FGMs satisfying certain requirements.

Quasi-Static Fracture Toughness and Damage Monitoring in Liquid Metal Reinforced Hybrid Composites

An experimental study is performed to investigate the quasi-static fracture toughness and damage monitoring capabilities of liquid metal (75.5% Gallium/24.5% Indium) reinforced intraply glass/carbon hybrid composites. Two different layups (G-0, where glass fibers are along the crack propagation direction; C-0, where carbon fibers are along the crack propagation direction) and two different weight percentages of liquid metal (1% and 2%) are considered in the fabrication of the composites. A novel four-probe technique is employed to determine the piezo-resistive damage response under mode-I fracture loading conditions. The effect of layups and liquid metal concentrations on fracture toughness and changes in piezo-resistance response is discussed. The C-composite without liquid metal demonstrated higher fracture toughness compared to that of the G-composite due to carbon fiber breakage. The addition of liquid metal decreases the fracture initiation toughness of both G- and C-composites. Scanning electron microscopy images show that liquid metal takes the form of large liquid metal pockets and small spherical droplets on the fracture surfaces. In both C- and G-composites, the peak resistance change of composites with 2% liquid metal is substantially lower than that of both no-liquid metal and 1% liquid metal composites.

Adaptive coupling of non-ordinary state-based peridynamics and classical continuum mechanics for fracture analysis

In this study, we proposed a coupled model between non-ordinary state-based peridynamics (NOSPDs) and classical continuum mechanics (CCMs) using the Morphing strategy for both quasi-static and dynamic fractures. We leveraged a novel strategy to approximate the deformation energy of the NOSPD model, which can unify the existing BPD and OSPD cases. We established the unified governing equations for NOSPDs and CCMs and applied the generalized strength criteria in the coupling model to activate the NOSPDs only when necessary. The critical stretch is recalibrated for the NOSPD model, and generalized strength-based bond failure criteria are proposed to determine bond failure. The model was numerically implemented using the finite element method (FEM) supplemented with discrete elements in the NOSPD subdomain. Parallel in-house code is written on graphics processing units (GPU) using the Compute Unified Device Architecture (CUDA). The 1-, 2- and 3-dimensional (D) numerical results demonstrated the accuracy and efficiency of the proposed method. Notably, the relative errors due to coupling during wave propagation in a 1D bar and uniform deformation in a 2D plate were less than 0.0375% and 0.14%, respectively. Further, the observed crack propagation speed aligned with the reference value.