Brittle Failure Criterion Research Articles

Multi-axial brittle failure criterion using Weibull stress for open Kelvin cell foams

Foam structures have found an increasingly wide utilization in modern industry because of their porous geometric characteristic and favorable mechanical properties like lightness, high strength and so on. Elastic buckling, plastic yielding, brittle fracture and crushing are typical failure mechanisms of foam structures. The influences of foam geometry and morphology on the failure mechanisms are still not fully understand. In this paper the Kelvin cell model is used to simulate the foam structure and to study the influence of relative density and geometric properties on failure behavior. By extensive FEM analyses of representative volume elements under multi-axial loading with periodic boundary conditions the macroscopic failure limit surface is calculated in stress space. Locally at the microstructure the probabilistic Weibull model and the maximum principal stress criterion are used as failure criterion.

A Criterion for Brittle Failure of Rocks Using the Theory of Critical Distances

This paper presents a new analytical criterion for brittle failure of rocks and heavily over-consolidated soils. Griffith’s model of a randomly oriented defect under a biaxial stress state is used to keep the criterion simple. The Griffith’s criterion is improved because the maximum tensile strength is not evaluated at the boundary of the defect but at a certain distance from the boundary, known as half of the critical distance. This fracture criterion is known as the point method, and is part of the theory of critical distances, which is utilised in fracture mechanics. The proposed failure criterion has two parameters: the inherent tensile strength, σ 0, and the ratio of the half-length of the initial crack/flaw to the critical distance, a/L. These parameters are difficult to measure but they may be correlated with the uniaxial compressive and tensile strengths, σ c and σ t. The proposed criterion is able to reproduce the common range of strength ratios for rocks and heavily overconsolidated soils (σ c/σ t = 3–50) and the influence of several microstructural rock properties, such as texture and porosity. Good agreement with laboratory tests reported in the literature is found for tensile and low-confining stresses.

Computational Modeling of Failure Patterns of Glass Panels with Imperfections Subjected to Air Blast

Glass fragments are a prime source of injury to occupants of buildings subjected to explosive events during which window glass breaks into either flying shards (annealed glass windows) or fragments (fully tempered glass). These account for injuries ranging from minor cuts to severe wounds. A successful blast-resistant glazing design requires balancing of the safety and security of the window panels with physical appearance and cost-effectiveness. Moreover, it should consider the principles of a reasonable degree of protection against explosive threats based on the proposed level of security and previous lessons learned. In this paper, explicit finite element method (FEM) with is used for modeling the window failure and investigate the failure patterns of annealed and fully tempered glass windows panels with embedded imperfections across the panel. For this purpose, the element-removal technique is utilized in accordance with the fracture micromechanics' principles and the brittle failure criteria of displacements; strains and stresses. The results of the analysis demonstrate that the response of the conventionally-framed window panels to blast loading cases depends upon the glazing technologies; blast load intensities; the glass types; and the location of imperfections. Through the application of element removal technique and stress redistributions at each infinitesimal time increment, the results of this paper further demonstrates that the existence of glass imperfections in the window panels will make them more vulnerable against the air blast overpressures for both annealed and fully-tempered glass panel cases: For fully tempered glass panels, the vulnerability arises from greater stress concentrations at the location of glass defects as well as higher brittleness of the panel itself when compared to the annealed glass panels. The results of this study show that the existence of defects along the edges will help in redistribution of the stresses in a way that the yield lines will pass through these imperfection locations. The general failure pattern for defective annealed glass panels subjected to small intensity air blast will be very similar to the failure pattern of homogeneous window panels yet with a faster pace. However, when subjected to a higher intensity air blast, the severity of the load will cause excessive rotation angles in the panel and subsequent computational divergence and instability. The panel failure will occur much faster and with different patterns compared to the case of annealed glass panels subjected to smaller intensity air blasts. KeywordsGlass; Air Blast; Safety; Window Panels; Fracture Mechanics; Crack Propagation

Numerical modeling of deformation and stress fields around a magma chamber: Constraints on failure conditions and rheology

We present a stress–strain analysis using the Finite Element Method to investigate failure conditions of pressured magma chambers embedded in an inelastic domain. The pressure build-up induces variations in the stress field until failure conditions are reached. Therefore, the definition of the failure conditions could have a significant impact on the volcano hazard assessment. Using a numerical approach, we analyze the stresses in a gravitationally loaded model assuming a brittle failure criterion, to determine the favorable conditions for magma chamber failure in different source geometries, reference stress states, pore fluid pressures, rock rheologies and topographic profiles. The numerical results allow us to pinpoint the conditions promoting seismicity near the magma chamber. The methodology places a limit on the pressure that a magma chamber can sustain before failing and provides a quantitative estimate of the uplift expected at the ground surface. Thermally-activated ductile regimes, which may develop in the region surrounding a heated magma chamber, are also investigated. The stress relaxation in a ductile shell may prevent the wall rupture, favoring the growth of large overpressured chambers, which could lead to considerable deformation at the ground surface without significant seismicity. The numerical results suggest that a spherical source, compressive regime, gentle edifice topography, and growth of a ductile shell are important factors for the initial formation and the mechanical stability of magma storage systems. On the other hand, an elongated ellipsoidal source, extensional regime, steep volcano topography and high pore fluid pressure lower the overpressure necessary for inducing failure. These findings could help in gaining insights on the internal state of the volcano and, hence, in advancing the assessment of the likelihood of volcano unrest.

Evaluation of Ultimate Torsional Strength of Reinforcement Concrete Beams Using Finite Element Analysis and Artificial Neural Network

Due to lack of theory of elasticity, estimation of ultimate torsional strength of reinforcement concrete beams is a difficult task. Therefore, the finite element methods could be applied for determination of strength of concrete beams. Furthermore, for complicated, highly nonlinear and ambiguous status, artificial neural networks are appropriate tools for prediction of behavior of such states. The main purpose of this paper is evaluation of ultimate torsional strength of rectangular concrete beams using two different approaches: nonlinear finite element modeling and artificial neural network prediction. This aim is achieved by creating a three-dimensional finite element models that employing the brittle failure criterion of concrete and an artificial neural network which training is carried out through experimental data. Numerical modeling is accomplished using the commercial software and its validation is exhibited by selected data from experimental tests. The test specimens were solid rectangular beams that were subjected to pure torsion. All of the experimental data for training of the network has been comprised. Data are divided into three categories: training, testing and validating. Three-layer perceptron network with a back propagation error algorithm is used for training. This study shows that, the results of numerical models are more accurate than neural network models to predict torsional strength of reinforced concrete beams.

RKPM MESHLESS ANALYSIS OF BRITTLE-TO-DUCTILE FAILURE

We study the failure mode transition in double-notched and single-notched steel plates under impact loading. For low impact velocities, brittle crack growths is observed while shear bands occur when the speed of the impactor is increased. We employ the meshless reproducing Kernel particle method (RKPM) that is well suited for problems involving large deformation and localization. The key feature of the proposed methodology is that it (1) treats brittle and ductile failure in the same manner and (2) employs the same constitutive model and failure criterion for both brittle and ductile failure. Therefore, cracks and shear bands are both treated as strong discontinuity in the context of the RKPM. Material stability analysis performed at every sampling point determines automatically the failure mode. We demonstrate the validity of the method by comparison to experimental data.

Mechanical Deformation Characteristics of Hard Rock in Deep Excavation

In the process of deep tunnel excavation, the surrounding rock mass, which often endures high initial geo-stress, is in the state of typical unloading. As the adjustment of stress induced by excavation, rock mass behaves a peculiar kind of mechanical deformation characteristics, different from those in loading conditions. As thus, a series of representative unloading tests on hard granite from DaGangShan power station in China, which is being excavation at 1500 meters deep below earth surface, are carried out to simulate dynamically excavation process for studying the peculiar mechanical deformation characteristics. On the basis of unloading tests, a series of research results are acquired as follows： (1) Under unloading condition, lateral deformation occurs in the unloading direction and volumetric strain changes from compression deformation to dilation deformation. (2) Under unloading condition, hard granites behave brittle failure and Mogi-coulomb criterion can well describe the failure characteristics. (3) According to curves of unloading tests, the constitutive model is deduced for simulation analysis.

Unloading Characteristics of Hard Rock at Depth

DaGangShan power station in China is being excavation at 1500 meters deep below earth surface, where the surrounding rock is affected by high initial geo-stress. Deep excavation of rock tunnels causes release of high initial geo-stress and produces unloading action. As thus, a series of representative unloading tests on hard granite from DaGangShan power station are carried out to simulate dynamically excavation process, and in unloading tests, the stress path of decreasing confining pressure along with increasing axial pressure is accepted. The results of tests show as follows: (1) Under unloading condition, lateral deformation occurs in the unloading direction and volumetric strain changes from compression deformation to dilation deformation. (2) Under unloading condition, crack deformation appears, which results in Poisson’s ration rise and deformation modulus decrease. (3) Under unloading condition, hard granites behave brittle failure and Mogi-coulomb criterion can well describe the failure characteristics.

Conjugate faulting along pre-existing fractures in Bundelkhand granite, Hirapur, central India

Re-examination of the outcrop of conjugate of strike-slip faults mapped by Roday et al. (1989) near forest rest house at Hirapur reveals that the main dextral strike-slip fault that strikes N35°E and is a manifestation of the earliest NE-SW trending subhorizontal σ1 that produced extensional reef system in the Bundelkhand massif. Although the change in the stress system though 90° rotation of the principal compressive stress σ1 and σ3 (with σ2 maintaining near vertically) is correct, another point of interest is that the σ1 for the system of faults bisects the obtuse angle between the two sets and not an acute one as required by the brittle failure criterion. The sinistral strike-slip faults were probably formed by rejuvenation of the initial dextral strike-slip faults that were generated when the maximum principal compressive stress was oriented NS. The reversal of fault displacement is seen on all scales in the Bundelkhand massif. The dextral strike-slip fault related to the late stress system was preferentially produced along pre-existing tensile fractures that were generated under NE-SW directed subhorizontal σ1. Some of these fractures were converted into sinistral strike-slip faults under NS directed maximum principal compression acting subhorizontally.

Optimized thick-wall cylinders by virtue of Poisson's ratio selection: Part 1: Isochoric pressure application

The principal stress distributions in thick-wall cylinders due to variation in the Poisson's ratio are predicted using analytical and finite element methods. Analyses of appropriate brittle and ductile failure criteria show that under the isochoric pressure conditions investigated that auextic (i.e. those possessing a negative Poisson's ratio) materials act as stress concentrators; hence they are predicted to fail before their conventional (i.e. possessing a positive Poisson's ratio) material counterparts. The key finding of the work presented shows that for constrained thick-wall cylinders the maximum tensile principal stress can vanish at a particular Poisson's ratio and aspect ratio. This phenomenon is exploited in order to present an optimized design criterion for thick-wall cylinders. Moreover, via the use of a cogent finite element model, this criterion is also shown to be applicable for the design of micro-porous materials.

Finite element modelling of wooden structures at large deformations and brittle failure prediction

This paper presents a constitutive wood model that accounts for both hardening associated with material densification at large compressive deformations and brittle failure modes. The model is adapted from previous work by the authors and has been modified to deal with wood behaviours. The main novelty of the model is the coupling between the anisotropic plasticity and the ductile densification. The model developed is successfully implemented in the commercial ABAQUS software. Validation was made for uniaxial compressive loadings and an application on a three-points bending test. The results obtained, for the uniaxial compressive loadings, demonstrate the capability of the model to simulate the wood behaviour at large compressive deformations and show clearly the effect of the densification on the plastic behaviour. The result obtained for the three-points bending test shows a good implementation of the brittle failure criterion and demonstrates the suitability of the developed model to analyse and design wooden structures.

Multiphase Finite Element Modeling of Machining Unidirectional Composites: Prediction of Debonding and Fiber Damage

A multiphase finite element model using the commercial finite element package ABAQUS/EXPLICIT is developed for simulating the orthogonal machining of unidirectional fiber reinforced composite materials. The composite materials considered for this study are a glass fiber reinforced epoxy and a tube formed carbon fiber reinforced epoxy. The effects of varying the fiber orientation angle and tool rake angle on the cutting force and damage during machining are considered for the glass fiber reinforced epoxy. In the case of carbon fiber reinforced epoxy, only the effect of fiber orientation on the measured cutting force and damage during machining is considered. Two major damage phenomena are predicted: debonding at the fiber-matrix interface and fiber pullout. In the multiphase approach, the fiber and matrix are modeled as continuum elements with isotropic properties separated by an interfacial layer, while the tool is modeled as a rigid body. The cohesive zone modeling approach is used for the interfacial layer to simulate the extent of debonding below the work surface. Bulk deformation and shear failure are considered in the matrix for both the models and the glass fiber. A brittle failure criterion is used for the carbon fiber specimen and is coded in FORTRAN as a user defined material (VUMAT). The brittle failure of the carbon fibers is modeled using the Marigo model for brittle failure. For validation purposes, simulation results of the multiphase approach are compared with experimental measurements of the cutting force and damage. The model is successful in predicting cutting forces and damage at the front and rear faces with respect to the fiber orientation. A successful prediction of fiber pullout is also demonstrated in this paper.

Micromechanics of crystal interfaces in polycrystalline solid phases of porous media: fundamentals and application to strength of hydroxyapatite biomaterials

Interfaces are often believed to play a role in the mechanical behavior of mineralized biological and biomimetic materials. This motivates the micromechanical description of the elasticity and brittle failure of interfaces between crystals in a (dense) polycrystal, which serves as the skeleton of a porous material defined one observation scale above. Equilibrium and compatibility conditions, together with a suitable matrix-inclusion problem with a compliant interface, yield the homogenized elastic properties of the polycrystal, and of the porous material with polycrystalline solid phase. Incompressibility of single crystals guarantees finite shear stiffness of the polycrystal, even for vanishing interface stiffness, while increasing the latter generally leads to an increase of polycrystal shear stiffness. Corresponding elastic energy expressions give access to effective stresses representing the stress heterogeneities in the microstructures, which induce brittle failure. Thereby, Coulomb-type brittle failure of the crystalline interfaces implies Drucker–Prager-type (brittle, elastic limit-type) failure properties at the scale of the polycrystal. At the even higher scale of the porous material, high interfacial rigidities or low interfacial friction angles may result in closed elastic domains, indicating material failure even under hydrostatic pressure. This micromechanics model can satisfactorily reproduce the experimental strength data of different (brittle) hydroxyapatite biomaterials, across largely variable porosities. Thereby, the brittle failure criteria can be well approximated by micromechanically derived criteria referring to ductile solid matrices, both criteria being even identical if the solid matrix is incompressible.

Explosion Dynamics in Saturated Rocks and Solids

Non-elastic pore deformations and crack propagations are the principal causes of dynamic damage in rocks and soils. In the case of downhole blasting from wellbores, these two mechanisms compete with each other. Therefore, to carry out a mechanical analysis of rock blasting, a sufficiently complete model that takes these various mechanisms into account has to be developed. To address this issue, this paper proposes the use of an elastic–plastic model, which includes a yield condition with a non-associated plastic flow rule, the effects of pore fluid saturation, and a brittle failure criterion under extension. The results presented in this paper describe underground explosions with spherical motion (cavity growth under the internal pressure of detonated gases without leakage into the formation), typical for oil or water reservoirs. The governing equations are written in a Cartesian system of coordinates for the case of spatial dynamic medium deformation. For this case, Cartesian coordinates are more convenient than spherical coordinates because they avoid numerical difficulties connected with the non-divergent terms of the non-linear form of the Biot–Frenkel equations. The numerical method uses the Wilkins approach, which has been generalized for the model described in this paper. The dilatancy of the material during plastic deformation is neglected for simplicity. The numerical results show that, when using typical parameters for relatively “soft” porous skeleton, the plastic flow overcomes the brittle failure. An extension zone only appears near the cavity. The results also show the presence of the two Biot P-waves. The second Biot wave, however, is only seen in the case of an extremely high permeability rock. Furthermore, in the case of the first Biot wave, the saturating liquid and the solid skeleton particles are moving with different velocities in a 100 darcy rock and with the same velocity in a 0.01 darcy rock. Calculated radial particle velocities as a function of the scaled radius are close to measured velocities in rigid dense media but are larger than measured ones in clays. It is suggested that the difference is due to different levels of water saturation, assumed full saturation in the calculation, partial saturation in the experiments.

Stability analysis of vertical boreholes using the Mogi–Coulomb failure criterion

A main aspect of wellbore stability analysis is the selection of an appropriate rock failure criterion. The most commonly used criterion for brittle failure of rocks is the Mohr–Coulomb criterion. This criterion involves only the maximum and minimum principal stresses, σ 1 and σ 3, and therefore assumes that the intermediate stress σ 2 has no influence on rock strength. As the Mohr–Coulomb criterion ignores the strengthening effect of the intermediate stress, it is expected to be too conservative in estimating the critical mud weight required to maintain wellbore stability. Recently, Al-Ajmi and Zimmerman [Relationship between the parameters of the Mogi and Coulomb failure criterion. Int J Rock Mech Min Sci 2005;42(3):431–39.] developed the Mogi–Coulomb failure criterion, and showed that it is reasonably accurate in modelling polyaxial failure data from a variety of rocks. We then develop a model for the stability of vertical boreholes, using linear elasticity theory to calculate the stresses, and the fully-polyaxial Mogi–Coulomb criterion to predict failure. Our model leads to easily computed expressions for the critical mud weight required to maintain wellbore stability.

Prediction of Brittle Failure of Notched Graphite and Silicon Nitride Bars

High stress gradients occur at metal-to-ceramic joints due to the different thermal and mechanical properties of the materials. In some cases, the magnitude of the highly localized stresses lead to failure thus compromising the structural integrity of such joints. The study of notched ceramic bars with high stress gradients can assist with the prediction of failure of metal ceramic joints. Experiments and fracture mechanics analysis were performed on notched and un-notched POCO E.D.M 3 graphite and AS800 Silicon Nitride bars with different notch parameters. The twoparameter, multi-axial Weibull statistics method and a brittle fracture criterion based on the average stress over an area approach were used to predict the failure of the bars and the results obtained were compared with experimental results. The brittle failure criterion appears to give much better correlation with experimental results than the multi-axial Weibull statistics approach. The findings also appear to highlight the limitations of the Weibull’s statistics method in cases involving very high stress gradients.

A semi-analytical elastic stress–displacement solution for notched circular openings in rocks

A semi-analytical plane elasticity solution of the circular hole with diametrically opposite notches in a homogeneous and isotropic geomaterial is presented. This solution is based on: (i) the evaluation of the conformal mapping function of a hole of prescribed shape by an appropriate numerical scheme and (ii) the closed-form solutions of the Kolosov–Muskhelishvili complex potentials. For the particular case of circular notches––which resemble to the circular cavity breakout in rocks––it is demonstrated that numerical results pertaining to boundary stresses and displacements predicted by the finite differences model FLAC2D, as well as previous analytical results referring to the stress-concentration-factor, are in agreement with analytical results. It is also illustrated that the solution may be easily applied to non-rounded diametrically opposite notch geometries, such as “dog-eared” breakouts by properly selecting the respective conformal mapping function via the methodology presented herein. By employing a stress-mean-value brittle failure criterion that takes into account the stress-gradient effect in the vicinity of the curved surfaces in rock as well as the present semi-analytical solution, it is found that a notched hole, e.g. borehole or tunnel breakout, may exhibit stable propagation. The practical significance of the proposed solution lies in the fact that it can be used as a quick-solver for back-analysis of borehole breakout images obtained in situ via a televiewer for the estimation of the orientation and magnitude of in situ stresses and of strain–stress measurements in laboratory tests.

A Possible Limiting Case Behavior for Brittle Material Fracture

The self-consistent scheme is used to model the state of an elastic material with a very high density of nearly connected cracks. Then fracture mechanics is used to pose the problem of the complete and final failure of the material under uniaxial and eqibiaxial tension. These failure states are taken to be those of the extreme case of brittle fracture. A specific form for the resulting extreme brittle failure criterion is given.

Hoek-Brown parameters for predicting the depth of brittle failure around tunnels

A review of underground openings, excavated in varying rock masstypes and conditions, indicates that the initiation of brittlefailure occurs when the damage index, Di, expressed as theratio of the maximum tangential boundary stress to the laboratoryunconfined compressive strength exceeds approx0.4. When thedamage index exceeds this value, the depth of brittle failure around a tunnel can be estimated by using a strengthenvelope based solely on cohesion, which in terms of theHoek-Brown parameters implies that m = 0. It is proposed that inthe brittle failure process peak cohesion and friction are notmobilized together, and that around underground openings thebrittle failure process is dominated by a loss of the intrinsiccohesion of the rock mass such that the frictional strengthcomponent can be ignored for estimating the depth of brittlefailure, an essential component in designing support for theopening. Case histories were analyzed using the Hoek-Brownfailure criterion, with traditional frictional parameters, and withthe proposed brittle rock mass parameters: m = 0 and s = 0.11. Theanalyses show that use of a rock mass failure criteria withfrictional parameters (m > 0) significantly underpredicts thedepth of brittle failure while use of the brittle parametersprovides good agreement with field observations. Analyses usingthe brittle parameters also show that in intermediate stressenvironments, where stress-induced brittle failure is localized, atunnel with a flat roof is more stable than a tunnel with anarched roof. This is consistent with field observations. Hence,the Hoek-Brown brittle parameters can be used to estimate thedepth of brittle failure around tunnels, the support demand-loadscaused by stress-induced failure, and the optimum geometry of theopening.Key words: spalling, depth of failure, rock mass strength, brittle failure criterion, cohesion loss, Hoek-Brown brittle parameters

Deformation Fields Around A Fault Embedded In A Non-Linear Ductile Medium

SUMMARY The geological record of deformation is often characterized by a combination of discontinuous deformation, in which strain is concentrated in faults, and continuous deformation, in which strain is distributed through the material. Where slip occurs on a fault that terminates, the surrounding material is deformed. In the lower crust and in cases where large strains occur over long geological time-scales, it is appropriate to model the deformation using a viscous (probably non-linear viscous) rheology. We describe a method for practical finite-element solution of this problem using a dynamically self-consistent formulation for stress and displacement on a fault of arbitrary geometry; the accuracy of the method is tested by comparison with an analytical solution for the linear rheology. We describe here the instantaneous deformation fields around a mode II fault under both plane-strain and plane-stress conditions, and a range of rheological exponents n (where strain rate is proportional to deviatoric stress to the nth power). the distributions of stress and strain rate around the fault tip are controlled primarily by the rheological exponent n. A localized zone of high strain rate projects beyond the end of the fault if n is about 3 or greater, and the degree of localization of deformation increases with the value of n. the zone of high shear-strain rate can be defined in practical terms by considering (1) the region in which the creep velocity differs by more than 20 per cent from the velocity on the nearby external boundary and (2) the region in which the maximum shear-strain rate is greater than about twice the externally imposed shear-strain rate. For n= 1, the volumes so defined differ considerably, but for large values of n, the two definitions both describe the same narrow zone of deformation beyond the end of the fault. Evaluation of the Navier-Coulomb criterion for brittle failure of the medium surrounding the fault tip shows first that brittle failure is much more likely on the extensional side of the fault than the compressional side. It also shows that the volume of material subject to brittle failure decreases rapidly with increasing n because of the relatively weaker stress singularity. We analyse previously published displacement versus distance data for faults terminating in sedimentary rocks at 0.1 to 100 m length-scales under different tectonic conditions, in order to determine the rheological exponent n. These analyses result in n values between approximately 0.85 and 5 for the different faults, with error bounds on n typically pL 1. the variation in n values may result from differences in pressure, temperature and fluid conditions at the time of faulting. More importantly, the analysis demonstrates a new method for the determination of the effective rheological exponent under in situ geological conditions.