Fracture Theory Research Articles

Numerical simulation to determine the fracture aperture in a typical basin of China

Fracture aperture is the main source of uncertainty in reservoir fracture modeling; this uncertainty is due to the difficulty of accurately quantifying underground fracture aperture and the limited understanding of the main factors controlling fracture aperture and the mechanical deformation process. The theoretical evaluation model of underground fracture aperture is a key aspect of fracture characterization and modeling of tight sandstone reservoirs. Based on the self-similarity theory of rock fractures and considering the 3D morphology of a fracture surface, a finite element geomechanical model of a fracture surface is established. A combination of physical experiments and numerical simulations is used to determine the fracture apertures in underground reservoir conditions. Furthermore, a theoretical model of fracture aperture is proposed under the constraints of the in situ stress, rock mechanics parameters, fracture occurrence, fracture scale, fracture filling characteristics and fracture surface characteristics. According to the vertical distribution model of in situ stress, the theoretical average fracture aperture in rock at different depths is determined. With increasing burial depth, the fracture aperture decreases. When the horizontal stress difference is greater than 10 MPa, the effect of the in situ stress difference on the average fracture aperture decreases. Among the three fracture filling patterns considered, the average effective aperture of fractures with uniform filling is the largest. When the fracture spacing is less than 2 m, the fracture aperture increases with increasing fracture spacing; when the fracture spacing is greater than 2 m, the fracture aperture is basically unaffected by the fracture spacing.

Nondestructive Evaluation of Solids Based on Deformation Wave Theory

The application of a recent field theory of deformation and fracture to nondestructive testing (NDT) is discussed. Based on the principle known as the symmetry of physical laws, the present field theory formulates all stages of deformation including the fracturing stage on the same theoretical basis. The formalism derives wave equations that govern the spatiotemporal characteristics of the differential displacement field of solids under deformation. The evolution from the elastic to the plastic stage of deformation is characterized by a transition from longitudinal (compression) wave to decaying longitudinal/transverse wave characteristics. The evolution from the plastic to the fracturing stage is characterized by transition from continuous wave to solitary wave characteristics. Further, the evolution from the pre-fracturing to the final fracturing stage is characterized by transition from the traveling solitary wave to stationary solitary wave characteristics. In accordance with these transitions, the criterion for deformation stage is defined as specific spatiotemporal characteristics of the differential displacement field. The optical interferometric technique, known as Electronic Speckle-Pattern Interferometry (ESPI), is discussed as an experimental tool to visualize those wave characteristics and the associated deformation-stage criteria. The wave equations are numerically solved for the elastoplastic stages, and the resultant spatiotemporal behavior of the differential displacement field is compared with the experimental results obtained by ESPI. Agreement between the experimental and numerical results validates the present methodology at least for the elastoplastic stages. The solitary wave characteristics in the fracturing stages is discussed based on the experimental results and dislocation theory.

Investigation of the 12 orientations variants of nanoscale Al precipitates in eutectic Si of Al-7Si-0.6Mg alloy

Various orientations and diffraction patterns from nanoscale Al precipitates in eutectic Si were investigated by high-resolution transmission electron microscopy combined with transition matrix and stereographic projection. It was found that the Al precipitates had 12 variants, all orientation relationships can be expressed as: (001)Al//{111}Si, [110]Al//<1¯10>Si. Further, a new diffraction pattern model from Al precipitates was established under [111]Si zone axis, which was in good agreement with the experiment data. The microstructure, adhesion strength and electronic structure of the interface between Al precipitate and Si matrix were studied by first-principles calculation and experimental observation. The results show that the covalent bonds are formed between interfacial Al and Si atoms, which play a key role in interfacial bind strength. Based on the Griffith fracture theory, the cracks tend to form and expand in the interior of Al precipitates firstly, and the interfaces can act as a protective layer to prevent crack propagation. Therefore, the nanoscale Al precipitates will improve the toughness of eutectic Si particles by releasing part of stress through lattice distortion. In addition, the stretched nanoscale Al precipitates can act as effective heterogeneous nucleation sites for high density deformation nanotwins in eutectic Si during deformation, which significantly improved the deformability of eutectic Si.

A study on fracture of metal plate under the detonation wave interaction

In this paper, the response of metal plates under the interaction of detonation wave is studied experimentally and numerically. Detonation wave interaction experiments were carried out on metal plates of different materials and thicknesses, and different results such as incision, partial fracture and complete fracture were obtained. ALE algorithm and fluid−solid coupling mode were used to simulate the conditions similar to experiments, and the fracture process and pressure distribution of the metal plate were analyzed. The experimental results were in good agreement with the simulation results. Finally, based on the micro-element fracture theory, the theoretical analysis of the fracture law of the metal plate was carried out, and the fracture criterion based on the threshold impulsive velocity was concluded. It has been verified that this criterion is in good agreement with the simulation and experimental results. It can be used as a basis for estimating the fracture problem of metal plates under explosive load.

Institute of Mining SB RAS

Introduction. The article considers a variant of a straight finite fracture modeled by a mathematical cut in the elastic plane. Aim. The new model proposed differs from the existing models by the damage zone bounded by the elastic material at the fracture tip up to the moment of the fracture growth. The process of fracturing is essentially nonlinear. Methodology. The model is based on the full-scale tension experiments with a reference sample of rocks enclosing a fracture and having the characteristic stress points, namely, proportionality limit, elasticity limit, plasticity domain and the domain in the vicinity of destructive stresses. Results. The problem with fracture is considered as an experiment to determine deformation with growing pressure in the fracture. The problem has no correct analytical solution. The problem on hydrofracture 20 "Izvestiya vysshikh uchebnykh zavedenii. Gornyi zhurnal". No. 4. 2020 ISSN 0536-1028 assumes the presence of the initial stress field in rock mass, which is essentially used in formulation of boundary conditions. Conclusions. All such problems belong to the class of Cauchy’s problems with an infinitely distant point in the computational domain. This article proposes the correct formulation of the fracture theory problem in the static, kinematic and dynamic framework.

A phase field model for cohesive fracture in micropolar continua

While crack nucleation and propagation in the brittle or quasi-brittle regime can be predicted via variational or material-force-based phase field fracture models, these models often assume that the underlying elastic response of the material is not size-dependent. Yet, a length scale parameter must be introduced to these models to enable sharp cracks properly represented by a regularized implicit function. However, many materials with internal microstructures that contain surface tension, micro-cracks, micro-fracture, inclusion, cavity or those of particulate nature often exhibit size-dependent behaviors in both the path-independent and path-dependent regimes. This paper is intended to introduce a unified treatment that captures the size effect of the materials in both elastic and damaged states. By introducing a cohesive micropolar phase field fracture theory, along with the computational model and validation exercises, we explore the interacting size-dependent elastic deformation and fracture mechanisms exhibits in materials of complex microstructures. To achieve this goal, we introduce the distinctive degradation functions of the force-stress–strain and couple-stress-micro-rotation energy-conjugated pairs for a given regularization profile such that the macroscopic size-dependent responses of the micropolar continua are insensitive to the length scale parameter of the regularized interface. Then, we apply the variational principle to derive governing equations from the micropolar stored energy and dissipative functionals. Numerical examples are introduced to demonstrate the proper way to identify material parameters and the capacity of the new formulation to simulate complex crack patterns in the quasi-static regime.

A Cutting Force Prediction Model, Experimental Studies, and Optimization of Cutting Parameters for Rotary Ultrasonic Face Milling of C/SiC Composites

Ceramic matrix composites of type C/SiC have great potential because of their excellent properties such as high specific strength, high specific rigidity, high-temperature endurance, and superior wear resistance. However, the machining of C/SiC is still challenging to achieve desired efficiency and quality due to their heterogeneous, anisotropic, and varying thermal properties. Rotary ultrasonic machining (RUM) is considered as a highly feasible technology for advanced materials. Cutting force prediction in RUM can help to optimize input variables and reduce processing defects in composite materials. In this research, a mathematical axial cutting force model has been developed based on the indentation fracture theory of material removal mechanism considering penetration trajectory and energy conservation theorem for rotary ultrasonic face milling (RUFM) of C/SiC composites and validated through designed sets of experiments. Experimental results were found to be in good agreement with theoretically simulated results having less than 15% error. Therefore, this theoretical model can be effectively applied to predict the axial cutting forces during RUFM of C/SiC. The surface roughness of the workpiece materials was investigated after machining. The relationships of axial cutting force and surface roughness with cutting parameters, including spindle speed, feed rate, and cutting depth, were also investigated. In order to identify the influence of cutting parameters on the RUFM process, correlation analysis was applied. In addition, response surface methodology was employed to optimize the cutting parameters.

An analytical model for force prediction in micromilling silicon carbide particle–reinforced aluminum matrix composites

In micromilling the silicon carbide particle–reinforced aluminum matrix composites, cutting forces can provide a better insight of the cutting mechanism. In this article, an analytical model for force prediction in micromilling composites is developed considering the size effect of the matrix. In modeling, for the matrix, the cutting area is divided into shearing area and plowing area and the removal forces are established considering chip formation and edge forces; for the particle, the removal forces are established based on Griffith fracture theory. The model is verified by micromilling experiments. The influences of the process parameters (milling width, milling depth, and feed per tooth) on the milling force were studied. It shows that the maximum milling force increased with the increase in the feed per tooth and the milling depth and increases first and then stabilizes with the increase in milling width; the average milling force increases with the increase in the three parameters. In addition, the contribution of the particle fracture force is analyzed, and it is found that the contribution of the particle fracture force is affected by the feed per tooth, which basically accounts for about 23% of the maximum milling force and accounts for 23%–30% of the average milling force.

Extension of Elastic Models to Decagonal Quasicrystals

The main design of this paper is to adopt potential functions for solving plane defect problems originating from two-dimensional decagonal quasicrystals. First, we analyze the strict potential function theory for the plane problems of two-dimensional quasicrystals. To clarify effectiveness of the method, we give some examples and the results which can be precisely determined, including the elasticity and fracture theories of two-dimensional quasicrystals. These results maybe play a positive role in studying the fracture of two-dimensional quasicrystals in the future.

An extended ghost interlayer model in peridynamic theory for high-velocity impact fracture of laminated glass structures

The peridynamic theory is advantageous for problems involving damage since the peridynamic equation of motion is valid everywhere, regardless of existing discontinuities, and an external criterion is not necessary for predicting damage initiation and propagation. However, the current peridynamic modeling methods utilize regular or quasi-regular lattice structures, which poses difficulties in modeling laminated structures with alternating thin interlayers and thick ply layers. This study proposes an extended version of a nonlocal ghost interlayer model for simulating laminated structures with multi-body interactions. Using numerical studies, the proposed numerical schemes are verified. It is further confirmed that the proposed schemes are valid for simulating the high-velocity impact fracturing behavior of laminated glass.

Fracturing Gels as Analogs to Understand Fracture Behavior in Shale Gas Reservoirs

Hydraulic fracturing is widely used in the exploitation of unconventional reservoirs, such as shale gas and tight gas. However, a full understanding of the activation of natural fractures, prediction of fracture growth, distribution of proppant, and network fracture system effectiveness remain unresolved. The onset of fracturing in the media requires energy and this is due to the buildup of pressure within the rock due to continuous injection of fluid. In other words, when the energy associated with the injection fluid reaches the fracture strength of the rock, the fracture initiates and propagates into the formation. Here, we use gelatin in hydraulic fracturing laboratory tests and compare the results to a modified radial hydraulic fracturing theory. The mechanics of the gelatin, procedures to make a testing gelatin block, and procedures to conduct the test are described. The results show that the fracture evolving behaviours from experiments are well matched by the theory. The results are then scaled up to understand fracture growth behaviour in a tight rock reservoir.

A semi-analytical method for forced vibration analysis of cracked laminated composite beam with general boundary condition

In this paper, a semi-analytical method for the forced vibration analysis of cracked laminated composite beam (CLCB) is investigated. One computational model is formulated by Timoshenko beam theory and its dynamic solution is solved using the Jacobi-Ritz method. The boundary conditions (BCs) at both ends of the CLCB are generalized by the application of artificial elastic springs, the CLCB is separated into two elements along the crack, the flexibility coefficient of fracture theory is used to model the essential continuous condition of the connective interface. All the allowable displacement functions used to analyze dynamic characteristics of CLCB are expressed by classical Jacobi orthogonal polynomials in a more general form. The accuracy of the proposed method is verified through the compare with results of the finite element method (software ABAQUS is used in this paper). On this basis, the parametric study for dynamic analysis characteristics of CLCB is performed to provide reference datum for engineers.

Revisiting nucleation in the phase-field approach to brittle fracture

Twenty years in since their introduction, it is now plain that the regularized formulations dubbed as phase-field of the variational theory of brittle fracture of Francfort and Marigo (1998) provide a powerful macroscopic theory to describe and predict the propagation of cracks in linear elastic brittle materials under arbitrary quasistatic loading conditions. Over the past ten years, the ability of the phase-field approach to also possibly describe and predict crack nucleation has been under intense investigation. The first of two objectives of this paper is to establish that the existing phase-field approach to fracture at large — irrespectively of its particular version — cannot possibly model crack nucleation. This is so because it lacks one essential ingredient: the strength of the material.The second objective is to amend the phase-field theory in a manner such that it can model crack nucleation, be it from large pre-existing cracks, small pre-existing cracks, smooth and non-smooth boundary points, or within the bulk of structures subjected to arbitrary quasistatic loadings, while keeping undisturbed the ability of the standard phase-field formulation to model crack propagation. The central idea is to implicitly account for the presence of the inherent microscopic defects in the material — whose defining macroscopic manifestation is precisely the strength of the material — through the addition of an external driving force in the equation governing the evolution of the phase field. To illustrate the descriptive and predictive capabilities of the proposed theory, the last part of this paper presents sample simulations of experiments spanning the full range of fracture nucleation settings.

Statistics of ceramic strength: Use ordinary Weibull distribution function or Weibull statistical fracture theory?

“Weibull statistics” for strength distribution analysis refers to either the ordinary Weibull distribution function or the Weibull statistical fracture theory. The ordinary Weibull distribution function is an empirical distribution function on an equal footing with other type of classical empirical distributions such as normal and log-normal distributions for fitting the statistical data of various random variables nonexclusive to materials strength. It has no explicit physical meaning and cannot be used for size scaling and prediction of strength. The Weibull statistical fracture theory is a weakest-link statistical fracture model for a solid with the strength distribution of an elemental volume being described by the ordinary Weibull distribution function. It has the capability of size scaling and prediction of strength for specimens with different geometries and different loading modes. The three-parameter Weibull statistical fracture theory in uniaxial flexure of prismatic beams is reformulated and validated by both numerical and real strength experiments of different ceramics.

Phase field model for fracture analysis of functionally graded power-based shell structures

The phase field (PF) approach to fracture has emerged as a promising modeling tool that regularizes the variational fracture theory by Griffith via the introduction of a nonlocal damage-like field variable in the corresponding formulation. In this work, we outline a PF formulation for triggering brittle fracture phenomena in shell structures made of Functionally Graded Materials (FGMs). This model relies on the 6-parameter shell formulation complying with a solid shell kinematic description and incorporates the use of the Enhanced Assumed Strain (EAS) and Assumed Natural Strain (ANS) methods in order to alleviate different locking pathologies. The corresponding multi-field variational formalisms is consistently derived and discretized within the context of the Finite Element Method (FEM). Details regarding the implementation in the general purpose FE packages are outlined. The applicability of this model is demonstrated by means of several numerical applications.

Phase field approach to mode-I fracture by introducing an eigen strain tensor: General theory

A thermodynamically consistent phase field theory for fracture is developed. An alternative approach to describe the reduction in the slope of the stress–strain curve and the reduction of the stresses to zero during damage is utilized by introducing transformation strain instead of the degradation function. The main requirements for the thermodynamic potential are presented and used to present the general energy terms. The analysis of the equilibrium solution is formulated which describes the desired stress–strain curves and for which the infinitesimal damage produces an infinitesimal change in the equilibrium elastic modulus. Analytical analysis of the equations is performed to calibrate phase field parameters for a general case. The explicit form of the Ginzburg-Landau equation is derived based on which the evolution of the order parameter is obtained. The importance of the analysis of the thermodynamic potential in terms of stress–strain curves is shown. The developed theory includes a broad spectrum of the shapes of stress–strain relationships.

On the effect of an out-of-plane constraint on the three-dimensional crack front fields in a thin elastic plate

Two-dimensional theories of fracture are still applied widely today and provide theoretical foundations for solutions to many practical problems. These two-dimensional theories are based on the plane strain or plane stress assumption. However, strictly speaking, for a thin elastic plate with a through-thickness crack under tension, plane strain conditions can be met only at the crack front (except the corn point) and plane stress conditions exist at a distance of about one half of the plate thickness from the crack front in the mid-plane. What are the stress fields in the region where both plane strain and plane stress conditions cannot be met? In this paper, further investigations into the problem are carried out. Three-dimensional Maxwell stress functions, the principle of minimum complementary potential energy and three-dimensional J-integrals are employed to obtain an analytical solution to depict the relationship among out-of-plane constraints, three-dimensional J-integrals and stress intensity factors. Three-dimensional finite element simulations with fine meshes are carried out to verify the analytical results. Compared with the corresponding plane strain solution, the solution proposed is valid in a larger region.

Strain energy density decompositions in phase-field fracture theories for orthotropy and anisotropy

In phase-field theories of fracture, decompositions of the strain energy density into tensile and compressive parts are often necessary to avoid interpenetration of cracked surfaces and to select physically trustworthy crack paths. General formulations accounting for orthotropy of the well-known spectral and hydrostatic-deviatoric decompositions of the strain tensor (often referred to as Miehe and Amor decompositions) are presented in this study. Additionally, a new principal energy decomposition based on spectral decomposition of the stiffness tensor is proposed for general anisotropic materials. The decompositions are evaluated numerically in a quadratic specimen with an initial stationary edge crack subject to both tensile and shear global remote loading. It is shown that when an isotropic case is considered, solutions agree well with results reported elsewhere for both the spectral and the hydrostatic-deviatoric approaches. The principal energy decomposition results in similar crack paths as the other approaches, with only subtle differences. When orthotropy is considered, however, significant differences in the resulting crack paths as well as global force-displacement behavior are obtained, especially when the crack is subjected to shear loading. For global tensile loading, the decompositions result in similar crack paths and force-displacement relations. The results provide a step forward when developing phase-field fracture theories for brittle materials with an orthotropic nature and highlight the importance of a proper decomposition of the strain energy density.

Rigorous Estimates for Effective Creep-coefficients of Microcracked Masonry Accounting for Cracks Interactions

Based on the association of finite elements homogenization method and a rigorous homogenization scheme accounting for crack interactions, this paper provides rigorous predictions for the local and effective properties of microcracked viscoelastic masonry with or without creep of bricks. For the sake of simplicity, viscoelastic brick and mortar are assumed to follow the Generalized Maxwell rheological model and to be respectively safe and microcracked. In the mortar, the distribution of microcracks orientations is assumed to be random. Two steps are followed. The first one is based on the identification at the short and long terms of an approximate analytical creep function for the mortar. This step relies on the coupling between the Griffith’s brittle fracture theory and a rigorous homogenization scheme - the Ponte Castañeda &amp; Willis model - accounting for crack interaction instead of the dilute scheme adopted previously in Rekik et al. Two cases are considered: open and closed cracks. The first step allows to avoid recourse to 'heavy' numerical inversion of the Laplace-Carson transform. The second one provides overall creep coefficients of masonry by means of periodic homogenization carried out by finite elements method. For open cracks state, time-dependent crack density is investigated. The proposed model is validated by comparison with an analytical one available for a compressed masonry wall with "standard" viscoelastic mortar joints. Effect induced by microcracks is also highlighted by comparison with uncracked masonry. At last, results provided by the proposed model can be considered to be rigorous solution improving on dilute estimates for the creep behavior of microcracked mortar and demonstrating the interest to not neglect both cracks interactions and creep of bricks units.

Theoretical and Numerical Study on Three-Dimensional Internal Crack Specimen Under Uniaxial Compression

This paper study the three-dimensional stress state of internal ellipse crack tip under uniaxial compression based on fracture theory, and the change law of crack tip deflection angle (θ) and torsion angle (Φ) can be calculated in agreement with the stress concentration factor χ. Instead of existing numerical computation methods (J-integral), the M-integral method was used to calculate the mixed mode fracture parameters (KI, KII, KIII), and achieve 3-D crack propagation depending on different fracture criterion. The results show that the variation trend of KII and KIII under the initial fracture state in keeping with the variation trend of deflection angle (θ) and torsion angle (Φ). It is further verified that the analytical solution of the fracture angle at the long axis end and the short axis end is in good accord with the numerical solution. The numerical simulation path shows that wrapping wing crack propagation occurs on the original crack surface. Along with the propagation, the crack tip gradually deflected towards the compression direction, which indentifies with experimentally obtained propagation path. This numerical simulation method has successfully realized the initiation and propagation of 3-D internal crack under uniaxial compression, and the proposed method can be used to simulate 3-D internal crack propagation path of brittle materials such as rock under uniaxial compression.