Isotropic Brittle Materials Research Articles

Revisiting Failure of Brittle Materials

Abstract Failures of isotropic brittle materials were revisited to find out whether there are unified failure criteria, which can be applied to a load-carrying structural component made of a brittle material regardless of whether the component has a crack, a circular notch, or neither of these. To this end, a set of failure criteria were considered to determine failure of a local point which may be located at the crack tip, a notch tip, or any other location. The proposed criteria consist of two conditions, of which both must be satisfied for failure to occur. The first condition is that the local stress must not be lower than the failure strength of the material. The second condition is the stress-gradient condition. Even if the stress at a local point far exceeds the failure strength of the material, failure will not initiate until the stress gradient condition is satisfied. Four different examples cases were presented to explain the proposed failure concept.

Analytical prediction for depth of subsurface damage in silicon wafer due to self-rotating grinding process

Subsurface damage (SSD) induced by silicon wafer grinding process is an unavoidable problem in semiconductor manufacturing. Although experimental attempts have been made on investigation of the influential factors on the SSD depth, however, few theoretical studies have been conducted to obtain SSD depth through grinding parameters. To fill the gap, an analytical model is developed to predict the SSD depth in silicon wafer due to self-rotating grinding process, which can reveal the relationship among SSD depth and the grinding parameters, the size of the abrasive grains and the radial distance from the wafer center. The establishment of the proposed model is based on scratch theory and fracture mechanics of isotropic brittle materials, and we further consider the effects of elastic recovery, cleavage plane and crystalline orientation on SSD formation. To validate the applicability of the proposed predictive model, grinding experiments with varied grinding parameters are performed and the depths of SSD along the <110> and <100> crystal directions are also measured and analyzed. The results given by the proposed model present reasonable accuracy of less than 20% deviation with experimental results. Effects of grinding parameters, wafer radial distance, crystalline orientation, and abrasive grain size on SSD depth are discussed in detail.

Modification of Mohr's criterion in order to consider the effect of the intermediate principal stress

Mohr-Coulomb (M-C) criterion is widely used for isotropic brittle materials, although it has some disadvantages which limit its wider application. One important disadvantage of this theory is that it ignores the effect of the intermediate principal stress, although it has an important influence on materials behaviour. Therefore, Mohr-Coulomb theory is actually used only for materials in biaxial state of stress. In this paper, a modified form of Mohr's criterion has been suggested, in order to overcome this limitation. The modified criterion takes into account the influence of intermediate principal stress in the case of materials under triaxial loading conditions. On the basis of two new hypotheses, a real triaxial state of stress is replaced by a virtual biaxial state of stress. The above mentioned stress states are considered to be equivalent for a material when the failure is imminent in both cases. Both Mohr's theory and the concept of equivalence of stress states are combined in order to obtain the virtual biaxial state of stress, which is determined using all three principal stresses. The modified Mohr's theory can be used only for triaxial state of stress. For biaxial and uniaxial state of stress it is reduced to the classical one. The proposed hypotheses have an important advantage because does not require additional material coefficients. The calculated values using the virtual biaxial state of stress and the linear M-C criterion are in good agreement with experimental data for grey cast iron. The article also presents a compact statistical analysis for both stresses and errors, in order to objectively assess the effectiveness of the formulated hypotheses.

Elastic stress field model and micro-crack evolution for isotropic brittle materials during single grit scratching

Abstract An analytical model for the elastic stress field in isotropic hard and brittle materials during scratching is presented. The model considers the entire elastic stress field and the effect of material densification that was ignored in past studies, and is developed under a cylindrical coordinate system to make the modeling process simpler. Based on the model's predictions, the location and sequence of crack nucleation are estimated and the associated mechanisms are discussed. A single grit scratching experiment with an increasing scratch depth up to 2 µm is conducted for two types of optical glasses representing isotropic brittle materials: fused silica and BK7 glasses. It is found that the model's predictions correlate well with experimental data. Median cracks are found to be formed first during scratching, and the corresponding depth of the scratch sets the basis for determining the critical depth for brittle to ductile machining. Lateral cracks are initiated in the plastic yielding region and deflect to the work surface to cause material removal, while Hertzian cracks interact with lateral cracks to help remove lateral-cracked material. Furthermore, it is found that, owing to its open network molecular structure, fused silica has a much worse ductile machinability than the BK7 glass.

Quantitative Characterization of Mechanical Stress Field and Fracture Strength in Isotropic Brittle Materials During Crack Tip Propagation

A generalized method for characterizing mechanical stresses in brittle materials during crack tip propagation is presented. This approach was derived from classical fracture mechanics and is therefore mechanism independent and applies to any isotropic brittle material, including glasses, fine grained ceramics or metals, and high stiffness polymers. A practical implementation demonstrates the merits of this technique: the fracture strength can be determined by characterizing the angle between the free surface of a flexural overload fracture and stress intensity factor loci. The accuracy of this method was phenomenologically validated using flexural strength tests on glass as a model material system. In addition, such fractographic measurements can also be used to characterize an inhomogeneous internal stress field, and thereby, for example, help discriminate whether the sample failed due to pure bending loads alone, or whether membrane stresses were also present.

Dynamic crack growth modeling technique based upon the SGBEM in the Laplace domain

This paper presents a new dynamic crack growth prediction tool based upon the symmetric Galerkin boundary element method (SGBEM) in the Laplace domain, combined with the maximum hoop stress (MHS) criterion. The fast Laplace inverse transform developed by Durbin is employed to obtain time-domain solutions of the stress intensity factors required by the MHS criterion. In this work, the proposed Laplace SGBEM tool is used to predict crack trajectories in homogeneous and non-homogeneous isotropic brittle materials under dynamic loading conditions. The effect of the elastic constant mismatches is investigated for the influence of crack–inclusion interaction on crack growth.

Discrete element approach in brittle fracture mechanics

PurposeThe purpose of this paper is to use the discrete element method (DEM) to model the fracture behaviour of brittle materials in 2D.Design/methodology/approachThe material consists of a set of particles in contact with a close‐packed structure. It allows the derivation of an expression for the stress intensity factor as a function of the contact forces near the crack tip. A classical failure criterion, based on the material's toughness, is then adopted for the analysis of crack propagation, represented by the contact loss between particles.FindingsThe DEM approach is compared to two tensile cases (mode I); both presenting a monotonous convergence towards classical solutions for more precise discretization.Originality/valueThe paper proposes a DEM approach in fracture mechanics of isotropic brittle materials entirely compatible with continuous classical theory. Hence the toughness value is directly introduced as a parameter of the material without any previous calibration of the DEM.

3D modeling of fracture in brittle isotropic materials using a novel algorithm for the determination of the fracture plane orientation and crack surface area

A user defined material model for simulation of brittle fracture in isotropic materials was implemented into the explicit FE code Abaqus/Explicit. The model uses a methodology for calculating the orientation of the fracture plane in isotropic materials in 3D space using an analytical approach. Knowing the orientation of the fracture plane, the cross-sectional area of the fracture surface can in a hexagonal element can be calculated. Regularization is then achieved using the size of the fracture plane. The model is first tested on single-element tests and then applied to more complex test cases such as a short bar with chevron notch or micromechanical analyses of composite materials. The numerical predictions correlate well with experimental evidence from the literature.

Deformation of Damaged Elastic-Brittle Isotropic Materials with a Fixed Concentration of Microdefects in a Complex Stress State

The character of deformation of elastic—brittle isotropic materials weakened by flat microdefects in the form of circular or elliptic microcracks randomly dispersed over volume is studied in the complex stress state. It is assumed that concentration of the microcracks under loading remains constant. Equations of state for such materials are derived depending on the stress state mode and sign of applied stresses. The damaged material is simulated by a linear elastic isotropic medium under all-round compression or tension and under biaxial compression, by a linear elastic transversally isotropic medium under biaxial tension, and by a non-linear orthotropic medium under compression along one axis and tension along another axis. To determine compliance characteristics entering into the equations of state, the continual model of a cracked medium and a method based on the equivalence principle of deformation energy of this medium are used. A numerical example is presented.

Enhanced numerical study of infinitesimal non-linear Cosserat theory based on the grain size length scale assumption

A non-linear Cosserat theory involving the Arbitrary Lagrangian–Eulerian (ALE) method has been introduced into the brittle isotropic materials (amorphous glass and cement mortar) using the Modified Brazilian Disk (MBD) under an uni-axial compressive loading. These numerical experiments shed light on the nature of the Cosserat-based media and material moduli determination which are difficult to acquire using the most well-known experimental viewpoints. By using the identical micro-rotation constants α = β = γ = μ L G 2 and Ψ = 2 / 3 , the Cosserat moduli reduce to only four constants for the 3D cases. According to the results obtained in this paper, the present methodology substantiates that the Cosserat theory would be readily applied to the wide range of materials from the full amorphous materials to the heterogeneous materials by changing the length scale parameter. Some fresh routes and new outlooks are discussed afterwards.

Numerical simulation of elastic linear micropolar media based on the pore space length scale assumption

The 3D micropolar theory numerical simulations have been performed on the brittle isotropic materials (amorphous glass, brittle rock and two different lightweight concretes) with different pore sizes using the cylindrical models under uniaxial compressive loading. To pursue this goal, it is assumed that first, second and third microrotation constants (α, β, and γ), which appear in the couple stress equilibrium equation, are proportional to the square of average pore diameter or so called characteristic length. Unexpectedly such an assumption leads to a constant polar ratio and consequently, the polar ratio cannot be accounted for as a material constant. The present phenomenon substantiates the existence of a redundant material constant for the 3D micropolar media. Accordingly, the micropolar shear constant κ is a material constant. Different coupling numbers N, with relevant domain are numerically investigated to explore the characteristic features of the micropolar shear constant κ. According to the results obtained, the present methodology shows a very good convergence and is consistent with the physically accepted results for the heterogeneous and homogeneous materials including nano-and microscale pores, whereas several unconverted or discontinuous stress fields are found out when using mesoscale pores. The latter disadvantage is believed to be caused by the impact of voids’ ratio variation under quasistatic loading.

Fundamental Analysis of the New Version of the New Proposed Fracture Theory

New version of new proposed fracture theory called N-N was introduced as concepts and hypotheses. In this research analytical discussions and applications in addition to detailed explanation of physical meaning and conceptual formation of is introduced. The current theory depends mainly on energy approach in LEFM for isotropic brittle materials and can be applied for ductile materials and composites regarding their different properties. This theory represents an extension of the continued efforts to obtain a unified view for all ideas and proposals of the majority of the researchers in the field of development of the fracture theories to predict the mechanism and mechanical aspects of cracking process. Fundamental new concepts and analytical hypotheses are introduced.

Strength/size effect for anisotropic brittle materials under a random stress state

A simple analytical method for predicting the strength/size effect is proposed for anisotropic brittle materials under 2- or 3-D random stress states. Statistical characteristics of the stress state (mean values and all possible correlations) along with statistical characteristics of random strength properties are assumed to be known and are used as initial data. The anisotropy of the random strength response is also taken into account. The method is a generalization of a previously proposed approach allowing one to predict the probability of failure or reliability by means of an exact analytical solution. The new method is developed for ideally brittle materials and accounts for the size effect and non-uniformity of the random stress state, when all statistical parameters of loading are functions of the coordinates. Examples are provided to illustrate the use of the method and to show effects of size, variability of strength properties, and variability of loading for typical composites. Although the method is proposed for anisotropic composite materials, it may also be used for brittle isotropic materials (glass, ceramic, etc.).

Superficial Plastic Response Determination of Hard Isotropic Materials Using Ball Indentations and a FEM Optimization Technique

This article introduces a novel method for determining the superficial relation between stresses and elastoplastic strains for hard and commonly brittle isotropic materials. An experimental-computational procedure is developed based on a finite element method simulation of successive ball indentations with increasing loads onto machine elements, which are in their final geometry and heat treatment. The imprint profiles, measured by means of profilometry, are used as input data to a finite element method-supported optimization algorithm. The goal of the optimization procedure is to determine the constitutive stress strain curve, expressed as a multilinear hardening law. The proposed method yields superficial plastic properties accurately, extending forward already existing models that determine bulk-flow material data using ball indentations. The developed procedure is demonstrated with the aid of 100Cr6 through hardened bearing steel, annealed at different temperatures.

Indentation cracks in superplastically deformed silicon nitride consisting of strongly aligned rod-shaped grains

Vickers indentation in silicon nitride, which was produced by superplastic tensile deformation of 150%, was conducted. The specimen had a highly anisotropic microstructure, where rod-like grains aligned along the tensile direction, and showed unusual crack systems. In the case of the indentation whose diagonals point to 0°/90° (0° direction is the tensile direction), two V-shaped surface cracks were propagated from each 90° edge and one crack from each 0° edge. In the case of the 45°/45° indentation, surface cracks emanated from the edges of the indent and deflected towards the tensile direction. Subsurface crack systems also showed unique shapes, differing from a cross-shaped crack system in isotropic brittle materials. Three-dimensional crack systems were estimated on the basis of the observation, and the crack evolution process was considered. Furthermore, the mechanical properties of the anisotropic silicon nitride were measured, and showed superior bending strength and fracture toughness in specific directions, compared to the undeformed material.

Isotropic Md Simulations Of Dynamic Brittle Fracture

AbstractWe present results obtained by molecular dynamics simulations on the propagation of fast cracks in triangular 2D lattices. Our aim is to simulate Mode I fracture of brittle isotropic materials. We propose a force law that respects the isotropy of the material. The code yields the correct imposed sound c║, shear c┴ and surface VR wave speeds. Different notch lengths are systematically studied. We observe that initially the cracks are linear and always branch at a particular critical velocity c* ≈0.8VR and that this occurs when the crack tip reaches the position of a front emitted from the initial crack tip and propagating at a speed c = 0.6 8VR.

A new fracture criterion for brittle rocks

A general form of a “fracture function” for isotropic brittle materials is expressed in terms of the three invariants of the stress tensor. The coefficients in the function are determined by use of the small number of experimental data under specific conditions. This function is applicable to an estimate of the fracture condition of brittle rocks under a general stress state i.e., σ 1 ≠ σ 2 ≠ σ 3. The application of this function is attempted for the data of three brittle rocks i.e., Dunham dolomite, Mizuho trachyte, and Westerly granite, reported by previous workers. For the first two, this criterion gives a good estimation of the effect of the intermediate principal stress σ 2 on failure. For the last, the fracture strength at high confining pressure is estimated by use of the several data obtained under very low confining pressures, and the agreement with experimental data is also satisfactory.

A Fracture Theory for Brittle Anisotropic Materials

This paper outlines the development of a general macroscopic failure theory. The result is a relatively simple cubic equation where interaction coefficients for tension-tension and compression-compression strengths may be defined separately. Approximations for biaxial and triaxial normal stress interactions are included to reduce the number of experimental data to nine for brittle three-dimensionally orthotropic materials and to two for brittle isotropic materials. A necessary condition that the theory closely describe brittle isotropic material strength is satisfied for a comprehensive set of cast iron biaxial strength data. The surface is graphically illustrated for various materials.

A note on the penetration of a rigid wedge into a nonisotropic brittle material

In the present note an existing wedge penetration theory for isotropic brittle materials is extended to explain some ‘bedding plane’ effects observed in a series of experiments on Indiana limestone. The general features of the theory presented by Paul and Sikarskie [ Trans. AIME 232, 372–383 (1965)] are kept here and in order to extend the analysis to the anisotropic case Jaeger's modification of the Coulomb-Mohr failure criteria is used. Two specific cases of bedding plane orientation are considered, namely those in which the bedding planes are parallel and perpendicular to the direction of penetration. The theory presented gives an insight into the anisotropic effects and predicts, in reasonable agreement with the experimental results, the observed differences in specific energies for the different penetration directions.

Hertzian Fracture in Single Crystals with the Diamond Structure

Extension of an earlier theory of Hertzian fracture in brittle isotropic materials is here made to include the case of brittle single crystals, with particular reference to crystals having the diamond structure. A detailed description is first given of the inhomogeneous stress field in a flat, elastic specimen loaded normally with a hard sphere. The geometry of cracks growing in such a stress field is then discussed, taking into account the anisotropy in surface energy relevant to the diamond-structure crystals. Analyses of the mechanics of crack growth into the crystals subsequently indicate that for a certain range of indenter size, the Hertzian crack passes through four equilibrium stages, as it does in glass, before reaching its fully developed length. As a result Auerbach's law, which states that the critical load on a spherical indenter necessary to produce a fully developed Hertzian fracture is proportional to the radius of the indenter, holds within this certain range of indenter size. This law is experimentally confirmed Hertzian fracture tests on single crystals of silicon. The mechanism of crack initiation and growth outlined in this paper is then discussed in terms of conclusions made by previous workers on the nature of the Hertzian cracks in the diamond structure crystals. Finally, possible application of the Hertzian test to the study of some mechanical properties related to the fracture surface energy in brittle single crystals is indicated.