Fracture Of Brittle Materials Research Articles

Shack–Hartmann wavefront sensing: A new approach to time-resolved measurement of the stress intensity factor during dynamic fracture

The stress intensity factor describes the stress state around a crack tip in a solid material and is important for understanding crack initiation and propagation. Because stresses cannot be measured directly, the characterization of the stress intensity factor relies on the measurement of deformation around a crack tip. Such measurements are challenging for dynamic fracture of brittle materials where the deformation is small and the crack tip velocity can be high. Digital gradient sensing (DGS) is capable of full-field measurement of surface deformation with a sub-micrometer sensitivity and a sub-microsecond temporal resolution, but it has only been demonstrated on centimeter-scale specimens with a spatial resolution of ∼ 1 mm. This makes it challenging to measure deformations close to the crack tip. Here, we demonstrate the potential of Shack–Hartmann wavefront sensing (SHWFS), as an alternative to DGS, for measuring surface deformation during dynamic brittle fracture of millimeter-scale specimens. Using an opaque commercial glass ceramic as an example material, we demonstrate the capability of SHWFS to measure the surface slope evolution induced by a propagating crack with a micrometer spatial resolution and a sub-microsecond temporal resolution. The SHWFS apparatus has the additional advantage of being physically more compact than a typical DGS apparatus. We interpret our SHWFS measurements of the surface slope by comparing them with 2D analytical predictions and phase-field simulations as well as 3D linear-elastic FEM simulations, based on which we discuss the relevance of 3D effects around the crack tip. Then, we introduce our procedure for extracting the apparent stress intensity factor associated with the propagating crack tip using SHWFS measurements. We conclude by discussing potential future enhancements of this technique and how its compactness could enable the integration of SHWFS with other characterization techniques including x-ray phase-contrast imaging (XPCI) for multi-modal characterization of dynamic fracture.

Three-dimensional modeling of impact fractures in brittle materials via peridynamics

Accurate prediction of the impact fracture behavior of brittle materials is crucial in various industries. The paper employs the peridynamic method to model the penetration and damage in brittle materials during impact. The proposed three-dimensional peridynamic impact fracture model (3D-PDIFM) overpasses other numerical models by adopting a new computational technique to couple the impact collisions of different types of projectiles with the peridynamic method. Moreover, the 3D-PDIFM is highly efficient in impact fracture simulations by focusing on the effects of the projectile without simulating its particles, thereby saving more computational efforts. Validation through comparison with experimental measurements and observations ensures that the 3D-PDIFM accurately represents the real-world impact fracture behavior of brittle materials. The paper also highlights the efficiency of the 3D-PDIFM in comparison to traditional mesh-based numerical simulations. Therefore, the 3D-PDIFM can be significant for practical engineering applications as efficient simulations can save time and computational resources. Using the validated model, the impact fracture behavior of PMMA under various conditions, such as changes in panel thickness, impact area, and projectile velocity, is investigated. This analysis gains more insights into how these factors affect the brittle material's response to impact. Moreover, the impact fracture behaviors of tempered glass and float glass panels are examined under different conditions of impact loadings. This study also explains the penetration and impact fracture phenomena in brittle materials. Overall, using the 3D-PDIFM as a modeling and simulation tool can help engineers and researchers analyze the behavior of glass panels under various impact scenarios, thus allowing for optimization of design parameters to improve impact resistance.

Utilizing X-ray Computed Tomography for Lithium Slag: A Guide to Analyzing Microstructure and Its Potential Influence on Liberation

Slag containing lithium aluminate is analyzed for its microstructure. This refers to the mineralogical composition, shape and size of the target and matrix phase, orientation of the target phase, and porosity. To investigate the microstructure of the target phase, a representative sample is taken from the block and measured using the XCT. With the help of the two- and three-dimensional analysis, an insight into the complex structure can be gained. The target phase, in this case, lithium aluminate, has a dendritic structure with different orientations and thus also influences the microstructure of the matrix phase. This is composed of a mix of several minerals and amorphous components. Fine pores with a radius of 10–100 µm are found throughout the sample volume. The fracture behavior of the slag is estimated on the basis of the influencing factors that can initiate fracture in brittle materials or divert the path of the fracture. Since the mineralogical and thus also mechanical properties of the slag can be influenced by its production process, suggestions are given as to how slags should be structured in order to ensure a non-random fracture, which is required for the efficient liberation of the target phase in mechanical processing.

A probabilistic-phase field model for the fracture of brittle materials

We develop a computational method to determine the failure probability of brittle materials under general mechanical loading conditions. The method is a combination of two parts: (1) numerical simulations of materials with multiple cracks using phase field theory, where the complete fracture process is viewed as ‘damage percolation’ along critical paths or clusters of cracks, rather than the traditional weak-link failure mechanism of Weibull, and (2) an extension of the Batdorf statistical theory of fracture to finite domains, where it is implemented within the finite element framework. The results of phase-field simulations at the ‘percolation threshold’ are used as failure data in the Batdorf theory to determine the overall probability of failure. The input to this approach is the size distribution of cracks in a pristine material. An example is shown, where alumina samples that were previously tested by Abe and coworkers (Abe et al 2003 J. Am. Ceram. Soc. 86 1019–21) in four-point loading are compared to the results of our numerical simulations. The approach developed here has the advantage of being extendable to more complex thermomechanical loading.

Influence of surface effect correction on peridynamic simulation of dynamic fractures in brittle materials

Influence of surface effect correction on peridynamic simulation of dynamic fractures in brittle materials

Experimental and numerical simulation of three-point bending fracture of brittle materials with inverted T-shaped obstacle cracks

In rock materials, the interaction and coalescence of three-dimensional internal cracks play a fundamental role in material failure and instability. This study focused on the generation of three-dimensional internal cracks inside specimens using the three-dimensional internal laser-engraved crack (3D-ILC) method, without causing surface damage. Experimental and numerical investigations were conducted to examine the fracture behavior of brittle solid materials with and without obstacles during three-point bending tests. The experimental results revealed that obstacle cracks positioned horizontally in an inverted T configuration exhibited a shielding effect on the vertical cracks, resulting in reduced propagation velocity and decreased curvature at the lower tip of the vertical cracks. Crack initiation and propagation were promoted in the presence of obstacle cracks, forming distinct fracture patterns, such as pear-shaped and hook-shaped configurations. While the overall failure mode of the specimens was not significantly altered by the presence of obstacle cracks, dynamic fracture dual sources, and new Mode III cracks were observed on the fracture surfaces. The 3D-ILC method provided an effective means for studying the integration and penetration of pre-existing three-dimensional internal cracks. The findings of this study contribute to the theoretical understanding of fracture propagation through obstacles in brittle materials and provide a basis for further investigations using physical experiments.

A coupling algorithm of ordinary and non-ordinary state-based peridynamic models for fracture analysis in brittle and ductile materials

In this work, a novel coupling algorithm of the ordinary state-based peridynamic (OSB-PD) model and non-ordinary state-based peridynamic (NOSB-PD) model is provided under the peridynamic framework. The salient advantages of both OSB-PD and NOSB-PD models can be simultaneously inherited in this coupling model and approach. The feasibility of coupling these two models is first analyzed by demonstrating the numerical equivalence of their constitutive formulations in isotropic elastic and elastoplastic problems, and the detailed numerical implementation strategy of coupling formulation is given as well. With the coupling model in this study, the main calculation procedure of numerical simulations is accomplished in OSB-PD, while the damage and failure analysis is conducted in NOSB-PD. Moreover, different failure criteria are introduced in this coupling model to describe the damage and failure of brittle and metallic materials. Several typical examples, including the mode I and mixed mode I/II fracture of brittle materials, and the ductile fracture and impact failure of metallic materials, are conducted to demonstrate the robustness and accuracy of the presented coupling model and approach.

A local surface roughness mapping method for post-failure interpretation of brittle fracture propagation

Tensile fractures in brittle materials may exhibit striation features running along the surface that indicate the trajectory of fracture propagation. Because of the directional persistence of these features, directional roughness characterization methods can be used on subsampled fracture surface data to map localized roughness features that follow these striations. These maps can aid in the interpretation of fracture initiation sites and crack front positions with the use of post-processing visualization techniques such as vector maps and streamlines. Tensile fractures in rock and synthetic rock analogues are used to demonstrate the various fracture propagation patterns that emerge following this analysis. This mapping method and streamline analysis can be further used to analyze elevation profiles that follow fracture propagation along non-linear paths which may inform about the mechanisms that can associate surface features with the fracture process.

Modeling fracture in brittle materials by higher-order phase field method using C1 non-Sibsonian interpolants

PurposeThis study implements the fourth-order phase field method (PFM) for modeling fracture in brittle materials. The weak form of the fourth-order PFM requires C1 basis functions for the crack evolution scalar field in a finite element framework. To address this, non-Sibsonian type shape functions that are nonpolynomial types based on distance measures, are used in the context of natural neighbor shape functions. The capability and efficiency of this method are studied for modeling cracks.Design/methodology/approachThe weak form of the fourth-order PFM is derived from two governing equations for finite element modeling. C0 non-Sibsonian shape functions are derived using distance measures on a generalized quad element. Then these shape functions are degree elevated with Bernstein-Bezier (BB) patch to get higher-order continuity (C1) in the shape function. The quad element is divided into several background triangular elements to apply the Gauss-quadrature rule for numerical integration. Both fourth-order and second-order PFMs are implemented in a finite element framework. The efficiency of the interpolation function is studied in terms of convergence and accuracy for capturing crack topology in the fourth-order PFM.FindingsIt is observed that fourth-order PFM has higher accuracy and convergence than second-order PFM using non-Sibsonian type interpolants. The former predicts higher failure loads and failure displacements compared to the second-order model due to the addition of higher-order terms in the energy equation. The fracture pattern is realistic when only the tensile part of the strain energy is taken for fracture evolution. The fracture pattern is also observed in the compressive region when both tensile and compressive energy for crack evolution are taken into account, which is unrealistic. Length scale has a certain specific effect on the failure load of the specimen.Originality/valueFourth-order PFM is implemented using C1 non-Sibsonian type of shape functions. The derivation and implementation are carried out for both the second-order and fourth-order PFM. The length scale effect on both models is shown. The better accuracy and convergence rate of the fourth-order PFM over second-order PFM are studied using the current approach. The critical difference between the isotropic phase field and the hybrid phase field approach is also presented to showcase the importance of strain energy decomposition in PFM.

Differential Energy Criterion for Brittle Fracture: Conceptualization and Application to the Analysis of Axial and Lateral Deformation in Uniaxial Compression of Rocks.

This paper discusses modeling the behavior and prediction of fracture of brittle materials. Numerous publications show that progress in this area is characterized by the emergence of a number of new models that meet the requirements of the mining industry, construction and other engineering practices. The authors focus only on one class of models, paying special attention to the compromise between simplicity of solution and versatility of the model. A new version of the model is proposed, taking into account the advantages of previous models. We present a differential energy criterion for brittle fracture substantiated, according to which, fracture occurs at a certain ratio of dissipated and stored (elastic) energy. Fracture is considered as the end of the deformation process with a virtual transformation of the initial material almost without cracks into a real material with cracks. The highest and lowest elastic moduli are analytically determined, respectively, on the ascending and descending branches of the stress-strain curve. A graphical version of the algorithm for determining the brittle fracture point on the post-peak branch of the stress-strain curve is proposed. The modeling results are consistent with the experimental data known from the literature.

ENERGY MODEL OF SOILS AND ROCKS CUTTING DURING WELL DRILLING

The article deals with the process of cutting soils and rocks. Experimental studies show that a coneshaped wedge is formed on the cutting surface. The compacted core of the deformed material contributes to the formation of tensile forces, under the action of which the material is destroyed along the sliding surfaces at a shear angle. An energy model for cutting soils and rocks while drilling wells, consisting of four parts, is proposed. The model is aimed at minimizing the energy consumption of the cutting process during well drilling. It is proposed to determine the chisel rotation speed by the length of the crack that occurs during brittle fracture of materials.

Fracture of brittle material with two 3D parallel internal cracks under thermal stress: Experimental and numerical analysis

Many defects such as cracks are often present inside brittle solids, and most of the cracks do not appear in a single form. The fracture and failure of brittle solids are basically caused by the interaction and expansion of multiple cracks. Most of the traditional studies on this topic have focused on surface or penetration crack interaction under tensile and compressive loads, while there have been very few studies on the expansion characteristics and interaction of parallel internal cracks under temperature loading. This study applies the 3D internal laser-engraved crack (3D-ILC) method to fabricate parallel internal cracks inside the 3D cube. Physical tests and numerical simulations of parallel internal cracks of different sizes and single crack propagation under temperature fields are carried out, and the propagation characteristics of internal cracks and the interaction between parallel internal cracks of unequal size under temperature loading are investigated. The study results show that the big crack will inhibit and shield the expansion of the small crack. In addition, there are typical “anti-binary tree” mode III features on both sides of the parallel large crack and single large and small crack propagation surface, which are classified as mixed I-II-III crack, while the parallel small cracks are mode I-II. Compared with the quasi-static crack growth criterion, the simulation results of the subcritical crack growth criterion are more in line with the physical test results, and there is a subcritical growth phenomenon in the parallel crack propagation under the temperature load. The results provide experimental and theoretical references for the study of 3D bi-parallel cracks interaction of different sizes and the fracture of mode I-II-III crack under the temperature field.

A phase-field model for electro-mechanical fracture with an open-source implementation of it using Gridap in Julia

In this paper, we propose a phase-field model for electro-mechanical fracture in brittle materials and provide an open-source implementation of the proposed model using a newly developed finite element (FE) toolbox, Gridap in Julia. Here, we have considered electric potential as an additional kinematic descriptor along with the displacement and phase-field variable. Using the virtual power principle, we have derived force balances for the electro-mechanical forces and a force balance associated with the damage. To incorporate the strong electro-mechanical coupling effect into the model, we have considered the Helmholtz free energy as a function of the electric field vector, strain tensor, phase-field variable, and the gradient of the phase-field variable. The proposed model ensures that the constitutive relations of the thermodynamic fluxes are on the satisfaction of the thermodynamic laws and can readily accommodate dissipative energy effects whenever needed. The proposed model provides complex crack paths as a solution to the governing partial differential equations (PDEs), and thus bypasses the need for re-meshing, enrichment of FE shape functions, and an explicit tracking of the crack surfaces. The use of Gridap makes the FE implementation of the proposed model exceedingly compact and user-friendly requiring very low memory usage and providing a high degree of flexibility to the users in defining weak forms of the governing PDEs. We have validated the proposed model and its implementation against a compact tension test on a single edge notched plate and a set of three-point bending tests on a notched beam made of lead zirconium titanate (PZT)-4 piezoelectric ceramics.

EFFECT OF LOCAL STRESSES ON THE FRACTURE OF METALLIC MATERIALS WITH DIFFERENT LATTICES UNDER SINGLE LOADS

The study discusses the effect of the local stress field at the crack tip on the fracture behavior of coarseand ultrafine-grained materials with bcc, fcc, and hcp lattices under single loads (impact and static). The local stresses at the crack tip under impact and static loading were assessed by the hmax/t ratio, where hmax is the maximum depth of the plastic zone under the fracture surface and t is the specimen thickness. The depth of plastic zones under the fracture surface was determined using layer-by-layer etching of the surface followed by X-ray diffraction analysis. The study results showed that it is not always possible to establish an unambiguous relationship between the fracture mechanisms of metallic materials and the local stress field at the crack tip. Nevertheless, some particular relationships were established: (i) the cleavage, quasi-cleavage, or intergranular brittle fracture of materials, regardless of lattice type, is indicative of plane strain deformation, (ii) under plane stress, all materials, regardless of lattice type, exhibit ductile fracture with micropitting, and (iii) in the transition from plane strain to plane stress, most fcc materials fracture by a mixed mechanism.

The concept of quasi-brittle rock fracture as applied to extraction and processing operations

Research objective is to substantiate the method for calculating the value of specific surface energy as the work irreversibly expended on fracture surfaces formation in the course of rock fracture in mining flow processes. Methods of research. The scheme of a uniformly expanded thin brittle plate with an embedded crack perpendicular to the load (Griffith plate) was taken as the basis for analytical calculations. The statement of the problem was based on building the energy balance of the work irreversibly expended on fracture surfaces formation through crack creation and propagation. The general solution analysis revealed that the well-known Griffith formula refers to a particular case of a perfectly brittle material fracture, when a sample in the form of a plate has specific dimensions with respect to the length of the embedded crack. Griffith theory postulates the presence of embedded structural cracks in a fractured body and does not explain the mechanism of crack creation and propagation up to the critical value. Therefore, it has not yet found practical application in analytical calculations of mining operations. Results. To extend application of the obtained design dependence of the specific surface energy for the tough fracture of rocks as well, a modified Irwin criterion has been proposed that determines the condition for the local fracture of crack-like flaws in the material’s structure. Therefore, by introducing a new constant in the form of an integrated structural-textural constant of the rock material, it has become possible to obtain the final answer to the question of what the value of specific energy consumption is in the flow processes of rock fracture by tensile cracks propagation. The theoretical novelty of the obtained regularity, along with its clear, understandable physical sense, is represented in the specific energy irreversibly spent for through tensile cracks creation by substantiated introduction of energy efficiency in the course of fracture, which substantially increases the value of the design energy consumption.

Thermodynamically-consistent derivation and computation of twinning and fracture in brittle materials by means of phase-field approaches in the finite element method

A theoretical-computational framework is proposed for predicting the failure behavior of two anisotropic brittle materials, namely, single crystal magnesium and boron carbide. Constitutive equations are derived, in both small and large deformations, by using thermodynamics in order to establish a fully coupled and transient twin and crack system. To study the common deformation mechanisms (e.g., twinning and fracture), which can be caused by extreme mechanical loading, a monolithically-solved Ginzburg–Landau-based phase-field theory coupled with the mechanical equilibrium equation is implemented in a finite element simulation framework for the following problems: (i) twin evolution in two-dimensional single crystal magnesium and boron carbide under simple shear deformation; (ii) crack-induced twinning for magnesium under pure mode I and mode II loading; and (iii) study of fracture in homogeneous single crystal boron carbide under biaxial compressive loading. The results are verified by a steady-state phase-field approach and validated by available experimental data in the literature. The success of this computational method relies on using two distinct phase-field (order) parameters related to fracture and twinning. A finite element method-based code is developed within the Python-based open-source platform FEniCS. We make the code publicly available and the developed algorithm may be extended for the study of phase transformations under dynamic loading or thermally-activated mechanisms, where the competition between various deformation mechanisms is accounted for within the current comprehensive modeling approach.

Simulating Brittle Fracture with Material Points

Large-scale topological changes play a key role in capturing the fine debris of fracturing virtual brittle material. Real-world, tough brittle fractures have dynamic branching behaviour but numerical simulation of this phenomena is notoriously challenging. In order to robustly capture these visual characteristics, we simulate brittle fracture by combining elastodynamic continuum mechanical models with rigid-body methods: A continuum damage mechanics (CDM) problem is solved, following rigid-body impact, to simulate crack propagation by tracking a damage field. We combine the result of this elastostatic continuum model with a novel technique to approximate cracks as a non-manifold mid-surface, which enables accurate and robust modelling of material fragment volumes to compliment fast-and-rigid shatter effects. For enhanced realism, we add fracture detail, incorporating particle damage-time to inform localised perturbation of the crack surface with artistic control. We evaluate our method with numerous examples and comparisons, showing that it produces a breadth of brittle material fracture effects and with low simulation resolution to require much less time compared to fully elastodynamic simulations.

An open-source implementation of a phase-field model for brittle fracture using Gridap in Julia

This article proposes an open-source implementation of a phase-field model for brittle fracture using a recently developed finite-element toolbox, Gridap in Julia. This work exploits the advantages of both the phase-field model and Gridap toolbox for simulating fracture in brittle materials. On one hand, the use of the phase-field model, which is a continuum approach and uses a diffuse representation of sharp cracks, enables the proposed implementation to overcome such well-known drawbacks of the discrete approach for predicting complex crack paths as the need for re-meshing, enrichment of finite-element shape functions, and an explicit tracking of the crack surfaces. On the other hand, the use of Gridap makes the proposed implementation very compact and user-friendly that requires low memory usage, and provides a high degree of flexibility to the users in defining weak forms of partial differential equations. Tests on a single-edge notched plate under tension, an L-shaped panel, a notched plate with a hole, a notched beam under symmetric three-point bending and a notched beam with three holes under asymmetric three-point bending are considered to demonstrate how the proposed Gridap-based phase-field Julia code can be used to simulate fracture in brittle materials.

Procedure for determining the constants of JH-2 (Johnson – Holmquist) dynamic fracture model for brittle materials

A simplified method for evaluating the constants in the JH-2 (Johnson – Holmqvist) model of the dynamic fracture for brittle materials is presented. The classical procedure suggests the use of 21 independent parameters describe the material, and the problem of their determination entails a large number of calculations experiments which hamper the use of the JH-2 model. The proposed technique requires fewer complex calculations and experimental data to determine the parameters of the material thus making it more feasible in use. In this work, the technique is used to search for the parameters of aluminum oxide (Al2O3) with a density of 99.5%, which is the material of a ceramic barrier subjected to high-speed interaction with the impactor. We present the results of three tests for penetration of a corundum plate: two of them are used to determine the constants of the model, and the third is used to verify the obtained values. Note that, the results obtained using the presented approach match quite accurately the experimental data, which is demonstrated in the course verification of the procedure.

Modeling fracture in brittle materials with inertia effects using the phase field method

The phase field method uses a length scale parameter to regularize the discrete crack to a diffuse crack, which removes the numerical tracking of the discontinuities in the displacement. The displacement field is coupled with the phase field and both are solved as a sequentially coupled systems using staggered method. The phase field varies between zero and unity (i.e., for intact region and for fully broken region), and it is a scalar. In this study, a new way of implementation is done using ABAQUS software to solve for the two fields. User defined element subroutine (UEL) is used to solve for the phase field variable and user defined material subroutine (UMAT) for the displacement field variable. Phase field model can simulate any complex crack paths and branching even without previously defined cracks. Some benchmark examples of quasi-static brittle fracture and dynamic brittle fracture are solved and verified with the existing numerical results. To account for the rate-dependent effect under high-rate loading, micro-inertia is incorporated into the phase-field model for dynamic fracture as proposed in the literature and verified with one example.