Continuum Fracture Mechanics Research Articles

Development of the Damage Concept in Mechanics of Materials

The review highlights the current state of research in the field of continuum fracture mechanics and dispersed fracture mechanics, including the main approaches to problem formulation, specific results and areas of their practical use. The article is aimed at specialists in creep, long-term strength and fracture mechanics, and may also be of interest to researchers in the field of issues of strength and fracture of materials and structures at high temperatures.

Atomistic and continuum Ascertainment of The crack tip stress fields in anisotropic elastic cubic media

The article provides a comparative analysis of stresses associated with the crack tip in an anisotropic elastic medium with cubic crystalline symmetry of elastic properties under mixed loading conditions obtained by two fundamentally different approaches: atomistic simulations and continuum fracture mechanics methods. The atomistic modelling approach is premised on the application of the molecular dynamics (MD) method implemented in the open source code program Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). The continuum mechanics approach is based on the classical elasticity theory of anisotropic media, in which the mechanical fields associated with the crack tip are represented by the crack tip stress and displacement series expansions generalizing the well-known classical asymptotical presentation of M. Williams to the case of anisotropic media. Using the MD method, a large series of computations of the loading of monocrystalline copper and aluminum plates with a face-centered cubic crystal lattice weakened by a central crack applying the embedded atom potential (EAM) was implemented. MD modeling is aimed at determining the atomic stresses and strains near the crack tip. The computed atomic stresses were compared with the stress field determined by the continuum elasticity theory of anisotropic media for crystal lattices with cubic symmetry of elastic properties. A comparative analysis was carried out for the angular dependencies of the stress and strain tensor components at various selected distances from the crack tip for the entire range of mixed deformation forms starting from pure tensile loading and ending with forms close to pure transverse shear. It is ascertained that the crack tip fields determined on the basis of two fundamentally different approaches, precisely, discrete and continuum, are completely consistent with each other. It is established that mathematical methods of continuum fracture mechanics can be applied to describe stress, strain and displacement fields at the atomic level.

Fracture of Epoxy Networks Using Atomistic Simulations.

Predicting fracture properties through all-atomistic simulations poses challenges due to classical force field limitations in breaking covalent bonds and the computational demands of reactive force fields like ReaxFF. In addressing this, we propose a scale-bridging method for forecasting the fracture behavior of highly cross-linked epoxy combining classical force fields, the LAMMPS package REACTER, and for bond breaking a parameter based on experimental distance criterion. In our analysis, we anticipate the macroscopic fracture energy GC of the epoxy network through the application of a continuum fracture mechanics model developed for fibrils. In addition, we extract the value of the stress intensity factor KI. This modeling approach is specifically implemented for a frequently used epoxy system that consists of bisphenol F and DETDA hardener. Notably, our results demonstrate a robust correlation with existing literature and experimental studies. Moreover, our approach boasts a substantial computational time advantage, facilitating calculations that are significantly faster compared to those performed using reactive force fields.

Numerical investigation to the effect of wear on fretting fatigue life using CDM and fracture mechanics

Fretting fatigue is a common type of problem in the engineering fields. Due to its multiaxial characteristics, it leads to a shorter overall fatigue life compared to plain fatigue. Fretting fatigue is caused by the micro slip at the contact surface, which is always accompanied by interface wear. This paper presents a refined elastic numerical simulation method combining interfacial wear with continuum damage mechanics (CDM) and fracture mechanics to predict fatigue life. The crack initiation life is determined by damage evolution law and the initiation position is determined by the total dissipated energy. Crack propagation life and path can be predicted using stress intensity factor. The stress redistribution and geometric change of contact surface caused by wear are considered in the whole fatigue process. Results indicate that the fretting fatigue simulation method incorporating wear has higher accuracy compared with the experimental results in the literature. It is noteworthy that wear retards nucleation damage process and crack propagation. As stress amplitude decreases, fatigue life increases, and delay effect of wear becomes more obvious. Additionally, wear causes the crack initiation position to move out, but crack initiation region is still near the slipping zone and propagation path does not change.

A Numerical Method for Unstable Propagation of Damage in Fiber-Reinforced Plastics with an Implicit Static FE Solver

Finite element analyses of the propagation of damage such as fiber compressive failure and delamination have greatly contributed to the understanding of failure mechanisms of fiber-reinforced plastics owing to extensive studies on methodologies using Continuum Damage Mechanics and Fracture Mechanics. Problems without the need for consideration of inertia, such as Double-Cantilever Beam tests, are usually solved by implicit FE solvers, and explicit FE solvers are appropriate for phenomena that progress with very high velocity such as impact problems. However, quasi-static problems with unstable damage propagation observed in experiments such as Open-Hole Compression tests are still not easy to solve for both types of solvers. We propose a method to enable the static FE solver to solve problems with unstable propagation of damage. In the present method, an additional process of convergence checks on the averaged energy release rate of damaged elements is incorporated in a conventional Newton–Raphson scheme. The feasibility of the present method was validated by two numerical examples consisting of analyses of Open-Hole Compression tests and Double-Cantilever Beam tests. The results of the analyses of OHC tests showed that the present method was applicable to problems with unstable damage propagation. In addition, the results from the analyses of DCB tests with the present method indicated that mesh density and loading history are not significantly influential to the solution.

ATOMISTIC AND CONTINUAL DETERMINATION OF STRESS FIELDS AT A CRACK TIP IN ANISOTROPIC ENVIRONMENT WITH CUBIC SYSTEM

The article presents a comparative analysis of stresses associated with a crack tip in an anisotropic medium with cubic symmetry of elastic properties under mixed mode loading conditions, using two different approaches: atomistic and continuum approaches. The atomistic approach is based on the use of the molecular dynamics method implemented in the open source Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). The continuum approach is based on the classical theory of elasticity of anisotropic media, in which the mechanical fields associated with the crack tip are represented using series that generalize the well-known representation of M. Williams to the case of anisotropic media. Using the method of molecular dynamics, a large series of calculations of loading of single-crystal copper and aluminum plates with a face-centered crystal lattice, weakened by a central crack, was carried out using the embedded atom potential (EAM). Molecular dynamics modeling is aimed at determining atomistic stresses and strains near the crack tip. The calculated atomistic stresses were compared with the stress field determined by the continuum theory of elasticity of anisotropic media for crystal lattices with cubic symmetry of elastic properties. A comparative analysis was carried out for the angular dependences of the stress and strain tensor components at different distances from the crack tip for the entire range of mixed strain modes: from perfect normal tension to loadings close to perfect transverse shear. It is established that the fields found on the basis of two fundamentally different approaches (discrete and continuum) are fully consistent with each other. It is demonstrated that the mathematical methods of continuum fracture mechanics can be used to describe stress, strain, and displacement fields at the atomistic level.

Comparison of the continuum damage and fracture mechanics in fatigue assessment of components containing residual stresses

In the present study, the fatigue crack growth life of standard and smooth notched specimens containing residual stresses are evaluated based on the continuum damage and fracture mechanics approaches. It is indicated that fatigue crack growth lives assessed by the continuum damage model are closer to the experimental results, with a difference of <10% in the specimens containing residual stresses. The total stress intensity factor range is used to consider the effects of residual stresses in the estimation of fatigue crack growth life based on the fracture mechanics. It is found that in existing residual stresses, the stress intensity factor K is not a proper parameter to evaluate crack tip zone. Fatigue crack initiation life is experimentally obtained based on assessing variations of the slope of unloading curve in the force-displacement diagram of fatigue experiments. The cycle including a change in the slope is considered as fatigue crack initiation event. Isotropic and kinematic hardening parameters of ASTM A516 pressure vessel steel are considered in finite element analyses. Tensile residual stresses are introduced into the notched specimens by employing four-point-bending with a magnitude of around 90% of the yield stress of the material. The results indicate that at least 65% of the fatigue life of the specimens is decreased due to the presence of tensile residual stresses. Tensile residual stresses are measured using the hole-drilling technique.

New Damage Accumulation Model for Spall Propagation Mechanism in Bearing Raceways

The aim of this study was to investigate the spall propagation mechanism in ball bearing raceways using physics-based models. Spalling is one of the most common types of bearing failures that can lead to catastrophic failure. This research takes a step forward toward developing a prognostic tool for ball bearings. It is first necessary to understand the spall progression process in order to formulate a constitutive law of spall deterioration and to estimate the amount of remaining useful life. Fragment formation in the vicinity of the spall edge was found to consist of surface and sub-surface cracks that eventually coalesce, and a fragment is released from the raceway, based on naturally-developed spalls. Here, we describe a physics-based model, integrating a dynamic model with a finite element one to simulate this process. A continuum damage mechanics (CDM) approach and fracture mechanics tools were embedded into the finite element model to simulate the damage propagation. The formation of cracks in the vicinity of the spall (surface and sub-surface cracks) were studied using this effective stress CDM model, and the propagation of the cracks was examined using two approaches: a fracture mechanics approach and an accumulated inelastic hysteresis energy CDM approach. The latter also predicts the overall process of a single fragment release. The simulation results of the spall propagation models are supported by experimental results of spalls from both laboratory experimental bearings and an in-service Sikorsky CH-53 helicopter swashplate bearing. The results obtained show that the impact of the ball on the spall edge affects the crack propagation and the appearance of the surface and sub-surface cracks. Both release the residual stresses and cause crack propagation until a fragment is released.

Identification of Linear Elastic Fracture Mechanics Parameters Through the Atomistic Simulation Based Analysis

The contribution presents atomistic modeling of the proximate crack tip and notch stress fields through molecular dynamics method. The study is aspired to comparing atomistic stress distributions for monocrystalline fcc copper and aluminum with continuous stress fields obtained on the basis of classical continuum fracture mechanics. Atomistic simulations were implemented for pure tensile loadings and mixed (I+II mixed loadings) mode loadings in Large-scale Atomistic/ Molecular Massively Parallel Simulator (LAMMPS). In atomistic computations specimens with central cracks and single edge notches made of crystalline fcc copper and aluminum under mixed mode loadings are considered. The elastic tensor components of the selected crystalline materials are obtained via molecular dynamics simulations and the circumferential distributions of stress fields in the straight away proximity of the crack and notch tips in aluminum and copper are constructed. The main regularities arising from the comparative analysis of the atomistic and continuum approaches are revealed. The major conclusion that can be reached after atomistic modeling is that the stress apportionments given by classical linear fracture mechanics qualitatively and quantitatively describe the molecular dynamics stress distributions.

What Is the Internal Pressure That Initiates Damage in Cementitious Materials during Freezing and Thawing? A Micromechanical Analysis

Damage induced by repetitive freezing and thawing processes is one of the critical factors that affect concrete durability in cold climates. This deterioration process manifests as surface scaling and internal damage. The damage processes are governed by physicochemical mechanisms that are active across multiple scales. In this contribution, we present a novel multiscale theoretical framework for estimating the critical pressure required for microcrack initiation during freezing and thawing of cementitious mortar. Continuum micromechanics and fracture mechanics is used to model the phenomena of microcrack initiation and growth. Damage at the microscale is upscaled to the level of the specimen using multilevel homogenization. The critical pressure is estimated using poromechanics at the microscopic scale. A theoretical analysis shows that in the frozen state, the material can resist higher pressures. As a consequence, the material is more susceptible to damage during thawing. The micromechanical predictions are within the range of the predictions obtained by electrokinetic theory.

A double scalar elastic damage constitutive model of ice materials

The constitutive model of material is the basis of structural analysis and is extremely important. In order to improve the analysis accuracy of the novel ice structure, a double scalar elastic damage constitutive model of ice materials including plain ice and fiber-reinforced ice (FRI) based on continuum damage theory and fracture mechanics is established in this study. The modified Ottosen equivalent strain criterion which can consider the comprehensive mechanical properties of compression, tension and shear is used as the damage initiation criterion and the internal variable of damage evolution. Based on the uniaxial compressive and tensile experiments, the two damage evolution laws of tensile and compressive damage with the modified Ottosen equivalent strain criterion as the internal variables are established. A unified weighted damage evolution function is constructed by the strain decomposition in the principal strain space. Based on the strain equivalent assumption, a complete damage constitutive model is established, which can consider the properties of ice materials at different temperatures and fiber contents. The energy regularization of crack bond model in fracture mechanics is adopted and the softening behavior of materials is described through fracture energy and stress-crack width relationship instead of the relationship between stress and strain, which solves the problem of mesh dependence in finite element analysis. The damage constitutive model is integrated into the large-scale finite element analysis software ABAQUS through the user-defined material subroutine (UMAT), and the establishment of the numerical model is realized, which can be conveniently used for numerical simulation of ice structures. Finally, the proposed damage constitutive model is evaluated according to the comparison between the results of finite element analysis and the results of uniaxial compression, uniaxial tension and four-point bending beam experiments. The results show that the damage constitutive model can well describe the stress-strain nonlinear behavior and capture the basic failure mode of ice materials. This study will lay a foundation for the refined analysis of ice structures.

A novel atomic J-integral concept beyond conventional fracture mechanics

The J-integral, which is crucial to the nonlinear fracture analysis within the framework of continuum fracture mechanics, fails at the atomic scale due to the atomic discreteness emerging with decreasing size. In this paper, a novel atomic J-integral calculation method is proposed for atomically sharp cracks by developing a new integral scheme to describe the local displacement gradient and stress field for discrete models. The applicability of the atomic J-integral is verified by comparing with traditional FEM and stress-based method, and its path-independency is elucidated by evaluating the geometric nonlinearity at the atomic scale. Furthermore, it is corroborated that the atomic J-integral method is dimension-independent even at the ultra-small scale below the dimensional limit of continuum mechanics.

General principles and criteria of failure of solids on different scale-structural levels under static and alternating loading

Experimental and theoretical studies of the fracture regularities under monotonic and alternating loadings, including stress-corrosion fracture, revealed the main scale-structural levels of the brittle fracture (formation of a strip structure, nucleation of microcracks, microstructurally short non-propagating cracks and short propagating cracks, merging of short cracks with a macrocrack nucleation) and viscous fracture (an evolution of dislocation in slip bands, cellular structure formation, nucleation of a pit relief with micropores, merging of pores with a formation of meso-shear bands and initiation of a viscous crack, growing of a viscous crack under inelastic deformation) of crystal bodies. A transition from one level to another is determined by changes in the fracture mechanisms and occurs with varying degrees of probability. The necessity of considering the destruction as a hierarchical random step-by-step process on all scale-structural levels is substantiated. The fracture of plastic materials is considered as the sum of independent or dependent events, namely the initiation and growing of brittle cracks and evolution of pores through various mechanisms. Attaining of the limit states is determined by the statistics of the distribution of inhomogeneities throughout the entire body volume. The results of physical studies (using electron microscopy, X-ray structural analysis, etc.) provide determination of the geometry and density of defects, dislocations, pores, cracks at different levels. Macrocharacteristics of static, long-term and fatigue strength are determined using methods of mechanical tests. The regularities of the fracture evolution of engineering structures are determined using methods of non-destructive testing (ultrasonic analysis, acoustic emission analysis, magnetic flaw detection, etc.). A brief analytical review of the main known physical approaches — dislocation and energy structural theories, dislocation continuous theories, dilaton-frustron models, stochastic physical theories, and phenomenological approaches, including continuum damage theory and fracture mechanics — is presented. The analysis of gained data leads to the necessity of describing fracture in the framework of stochastic multilevel models. We consider the approach according to which the probability of reaching limit sates at each level under arbitrary multiaxial loading is determined by linear functionals of the loading process, kernels of the functionals being random functions of time. For simple proportional loading, the theory of brittle fatigue scale-structural failure is presented.

Stress Intensity Factors of Continuum Fracture Mechanics at the Nanoscale

The paper presents the analysis of the stress field in the neighborhood of the crack tip by molecular dynamics method implemented in a classical molecular dynamics code LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator). The molecular dynamics simulations are aimed at computing continuum fracture mechanics parameters. Stress intensity factors for a copper plate with a central crack under Mode I and Mixed Mode loadings are evaluated with the help of atomistic modelling. The wide class of the computational experiments in LAMMPS is realized. The atomistic values of stress intensity factors are compared with the values obtained from continuum fracture mechanics considerations. It is shown that the continuum fracture theory successfully describes fracture even at extremely confined singular stress field of only several nanometers.

An Over-deterministic Method Based on Atomistic Stress Fields: Higher Order Terms of the Williams power expansion

The investigation of the atomistic near crack-tip stress fields under Mixed Mode loading via molecular dynamics method implemented in a flexible classical molecular dynamics method software LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) is presented. The molecular dynamics modelling is applied for evaluation of the coefficients of higher-order terms of the Williams’ expansion -based approximation for the near crack tip fields in isotropic linear elastic materials. Fracture mechanics parameters such as stress intensity factors, higher-order coefficients of the Williams power-series for a copper plate with a central crack under Mode I and Mixed Mode (Mode I+Mode II) loadings are obtained by atomistic simulations. All computational experiments are realized in LAMMPS with the Embedded Atom Method (EAM) potential. The atomistic values of higher-order coefficients of the Williams series expansion are computed by the over-deterministic method and liken to the values obtained from classical continuum fracture mechanics solutions. It is demonstrated that the conventional continuum fracture theory successfully describes fracture quantities at extremely limited singular stress field of only several nanometers.

Computation of conventional fracture mechanics parameters via molecular dynamics simulations

The Williams series expansion is the fundamental solution of conventional continuum fracture mechanics for characterizing the stress fields around the crack tip in a homogeneous material in the linear elastic regime. Nowadays stress intensity factors, T-stresses and the higher-order coefficients of the Williams series expansion are proved to be important parameters describing the near crack-tip fields in isotropic linear elastic materials. The central aim of this study is to understand if one can obtain these fracture parameters of conventional fracture mechanics from atomistic simulations based on molecular dynamics simulations. The ability to describe fracture processes at atomic scale via stress intensity factors, T-stresses and higher-order coefficients will provide the opportunity to take into account many effects such as material microstructures, crystallographic orientation, chemical compositions and concentrations and others. In this research the values of stress intensity factors, T-stress and coefficients of the Williams series expansion are obtained using atomistic simulations based on molecular dynamics method with a classical molecular dynamics code Large-scale Atomic/Molecular Massively Parallel Simulator. The classical over-deterministic method is used to determine stress intensity factors, T-stresses and coefficients of higher-order terms of the Williams series expansion of the near crack-tip stress field from molecular dynamics modelling of a plate with a central crack. The accuracy of the proposed approach is tested for this rather simple cracked configuration. There is the theoretical analytical solution with all the coefficients of the higher-order terms in the Williams series expansion. The existing theoretical solution allows us to compare the angular distributions of the stress tensor components for a large plane with the central crack. It is shown that results obtained from molecular dynamics simulations and the theoretical analytical solutions are in good agreement.

Thermodynamically consistent model for poroelastic rocks towards tectonic and volcanic processes and earthquakes

Summary A general-purpose model combining concepts from rational continuum mechanics, fracture and damage mechanics, plasticity, and poromechanics is devised in Eulerian coordinates, involving objective time derivatives. The model complies with mass, momentum, and energy conservation as well as entropy inequality and objectivity. It is devised to cover many diverse phenomena, specifically rupture of existing lithospheric faults, tectonic earthquakes, generation and propagation of seismic waves, birth of new tectonic faults, or volcanic activity, aseismic creep, folding of rocks, aging of rocks, long-distance saturated water transport and flow in poroelastic rocks, melting of rocks and formation of magma chambers, or solidification of magma.

Reduced Order Multiscale Simulation of Diffuse Damage in Concrete.

Damage in concrete structures initiates as the growth of diffuse microcracks that is followed by damage localisation and eventually leads to structural failure. Weak changes such as diffuse microcracking processes are failure precursors. Identification and characterisation of these failure precursors at an early stage of concrete degradation and application of suitable precautionary measures will considerably reduce the costs of repair and maintenance. To this end, a reduced order multiscale model for simulating microcracking-induced damage in concrete at the mesoscale level is proposed. The model simulates the propagation of microcracks in concrete using a two-scale computational methodology. First, a realistic concrete specimen that explicitly resolves the coarse aggregates in a mortar matrix was generated at the mesoscale. Microcrack growth in the mortar matrix is modelled using a synthesis of continuum micromechanics and fracture mechanics. Model order reduction of the two-scale model is achieved using a clustering technique. Model predictions are calibrated and validated using uniaxial compression tests performed in the laboratory.

Molecular Dynamics-Based Cohesive Law for Epoxy–Graphene Interfaces

Molecular dynamics simulations are used to obtain mode I and mode II fracture energies and cohesive laws for bulk epoxy and interfaces formed between epoxy and single-layer graphene (SLG), multilayer graphene (MLG), and multilayer graphene oxide (MLGO). The elastic moduli and ultimate tensile and shear strengths of epoxy–graphene interfaces are calculated from uniaxial tension and simple shear loadings. The results show that Young’s modulus and the ultimate tensile strength increase relative to bulk epoxy, whereas the shear modulus and ultimate shear strength are reduced. Failure of epoxy–graphene interfaces in tension occurs due to the formation of voids in the epoxy. Failure in shear is due to tangential slipping at the interface. Under mixed-mode conditions, the shear modulus and shear strength decrease with increasing tensile load. The critical energy release rate $$G_{\text{c}}$$ for the studied epoxy–SLG/MLG/MLGO systems is obtained using a continuum fracture mechanics approach and is found to be significantly lower than for bulk epoxy. All of the results are combined to define mode I and II cohesive laws for bulk epoxy and epoxy–SLG/MLG/MLGO interfaces that can be used in theoretical models and numerical methods, such as finite elements, that employ cohesive zones.

Atomistic Determination of Fracture Mechanics Parameters

Stress intensity factors, T-stresses and higher order coefficients in the Williams series expansion are the fundamental concepts of continuum fracture mechanics for characterizing the stress fields around the crack tip in a homogeneous material in the linear regime. It is well-known that critical values of stress intensity factors stipulate the of materials to growth a crack. Nowadays the modern multi-purpose computational tools such as Simulia Abaqus allow us to calculate T-stresses in cracked specimens and structures along with stress intensity factors. The aim of this study is to understand if one can obtain this fracture parameters of conventional fracture mechanics from atomistic simulations based on molecular dynamics method. The ability to describe fracture processes at atomic scale by means of stress intensity factors, T-stresses and higher-order coefficients will provide the opportunity to take into account many effects such as material microstructures, chemical compositions and concentrations and others. In this study the values of stress intensity factors, T-stress and coefficients of the Williams series expansion are determined using atomistic simulations based on molecular dynamics method with a classical molecular dynamics code Large-scale Atomic/Molecular Massively Parallel Simulator. The over-deterministic method is used to determine SIFs, T-stresses and coefficients of higher-order terms from molecular dynamics modelling of a plate with a central crack. The accuracy of the proposed approach is tested for this rather simple cracked configuration. There is the theoretical analytical solution with all the coefficients of the higher-order terms in the Williams series expansion. The existing theoretical solution allows us to compare the angular distributions of the stress tensor components for a large plane with the central crack. It is shown that results obtained from molecular dynamics simulations and the theoretical analytical solutions coincide qualitatively.