Elastic Brittle Model Research Articles

Computational Analysis of the Influence of Residual Stress on the Strength of Composites with Different Aluminum Matrices and Carbide Particles

A numerical study of the mechanical behavior of aluminum matrix–carbide particle composites subjected to combined thermomechanical loading is carried out. The composite structure, corresponding to that observed experimentally, is explicitly taken into account in the calculations. The mechanical response of the aluminum matrix and carbide particles is described using the isotropic elastic–plastic and elastic–brittle models. A fracture criterion of the maximum equivalent stress acting in the local regions of volumetric tension is used to study the crack initiation and propagation in the particles. The dynamic plane stress boundary value problems of cooling and tension of the composites are solved by the finite element method ABAQUS/Explicit. The influence of the cooling-induced residual stress and thermomechanical properties of the matrix and particle materials on the strength of the composites is investigated. A positive or negative effect of the residual stress is found to depend on the ratio between the particle strength and the matrix yield stress. Compressive residual stress formed in the particle after the cooling increases the strength of composites with hard matrices and low-strength particles. A decrease in the matrix–particle interfacial curvature results in a change in the fracture mechanism from in-particle cracking to debonding, which increases the composite strength. Composite elongation upon the fracture onset decreases with the volume fraction of the particles.

Numerical simulation of deformation and fracture of metal-matrix composites with considering residual stresses

Thermomechanical behavior of metal-matrix composite materials is investigated. Boron carbide B4C and high-strength aluminum alloy 6061-T6 are used as strengthening particle and matrix materials, respectively. Microstructure of the metal-matrix composite takes into account the complex shape of particles explicitly. Isotropic elastoplastic and elastic-brittle models were used to simulate the mechanical response of the aluminum matrix and ceramic particles, respectively. To investigate the crack initiation and propagation in ceramic particles, a Huber type fracture criterion was chosen that takes into account the type of the local stress state in ceramic materials: bulk tension or compression. The composite material with a single particle of both the really observed in the experiment and ideally round shapes is considered. The influence of the residual thermal stresses arising during cooling of the composite material from the temperature of aluminum recrystallization to the room temperature on the character of plastic strain localization in the aluminum matrix and fracture of carbide particles and on the macroscopic strength of the composite under external tension or compression is studied numerically. Two-dimensional dynamic boundary value problems in the plane-stress and plane-strain formulations were solved numerically by the finite element method using the Explicit module of the Abaqus software package. VUMAT subroutine procedures incorporating the constitutive models were developed and integrated into the Abaqus solver. Based on the results of the numerical simulation, it was concluded that the residual thermal stresses arising during cooling lead to the change in the mechanism of the particle fracture from in-particle cracking to debonding and increase the strength of the composite subjected to tension after the cooling.

Numerical modeling of logged wellbore breakouts using cohesion-weakening frictional-strengthening models

The wellbore stability analysis based on the simplified models such as the elastic-perfect plastic or the elastic-brittle models results in the unrealistic prediction of the wellbore damage zone characteristics and consequently underestimate or overestimate the required mud pressure respectively. To overcome this weakness, the advanced elastic-plastic models considering the full rock failure mechanisms in the post-peak region are reviewed briefly in this paper. The damage zone surrounding a wellbore located in a carbonate oil field in a southwest of Iran is analyzed based on the different advanced models. The objective is to study how well the numerical simulation results based on the advanced models agree with the logged fallouts with respect to location, depth, and shape of damage zone around the wellbore. An evaluation of which advanced model can best capture the actual rock behavior especially in the post-peak region is carried out by comparing the simulation results with the logged fallouts around the wellbore.The depth of the damage zone and transverse extension angle around the studied wellbore predicted well based on a kind of the cohesion-weakening frictional-strengthening (CWFS) model are 3 cm and 84.3° respectively. The simulation results of this model derived from the tests on the carbonate rocks are in more agreement with the logged breakouts around the studied wellbore drilled in the carbonate rocks like limestone. Moreover, according to the key input parameters sensitivity study carried out in this paper, decrease in the residual cohesion and critical plastic strain parameters result in the larger simulated damage zone. On the contrary, increment of the initial friction angle and drilling mud pressure leads to the smaller simulated breakout around the wellbore. As higher dilation angle is utilized in the numerical model as input parameter, the larger simulated breakout is created around the wellbore.

Numerical simulation of the mechanical behavior of rock mass with different scale flaws and its application

The rock mass contains the macroscopic and mesoscopic flaws at the same time, and therefore, how to accurately assess their co-effect on the rock mass mechanical behavior is still an important and difficult issue. The elastic-brittle model and Null model in FLAC3D code are adopted to describe the mechanical behavior of the intact rock and pre-crack, respectively, and the superfine element division method is also adopted to mesh the numerical model to simulate the rock mass failure. Meanwhile, a new method to generate the mesoscopic flaws is proposed which can be determined with the rock mass void ratio. Then, the effect of the mesoscopic flaws on the rock mass mechanical behavior is studied, and at the same time, the effect of the pre-crack dip angle and length on the rock mass mechanical behavior is also studied. Finally, the proposed method is adopted to study these two kinds of flaws on the factor of safety and critical failure surface of the rock mass slope. The results indicate that the macroscopic flaws play a control role in the failure mode and compressive strength of the rock mass under uniaxial compression. They also control the failure mode and the factor of safety of the rock mass slope. Meanwhile, although the mesoscopic flaws do not change the control role of the macroscopic ones on the mechanical behavior of the rock mass, they do have some effect on this control role. Finally, the physical test is done to verify the numerical results.

Constitutive model for tailing soils subjected to freeze–thaw cycles based on meso-mechanics and homogenization theory

A constitutive model is proposed for tailing soils subjected to freeze–thaw cycles based on the meso-mechanics and homogenization theory. The evolution of meso-structure upon loading was analyzed within the framework of breakage mechanism. When the new model is formed, tailing soils are idealized as composite materials composed of bonded elements described by an elastic brittle model and frictional elements described by a double hardening model. Based on meso-mechanics and homogenization theory, the nonuniform distributions of stress and strain within the representative volume element are given by introducing a structure parameter of breakage ratio with the derivation of the strain coefficient tensor, which connects the strains of the bonded elements and the representative volume element. The methods for determining model parameters are suggested based on the available tested results. The model proposed here can predict the deformation properties of tailing soils experiencing freeze–thaw cycles with acceptable accuracy. The strain-hardening and post-peak strain-softening behaviors of tailing soils under various confining pressures as well as different numbers of freeze–thaw cycles are well captured, and the dilatancy and contraction features are also adequately represented.

A meso-macroscopic constitutive model of frozen saline sandy soil based on homogenization theory

As an effective multiscale method, homogenization theory is widely applied to the composite material research field. The saturated frozen saline sandy soil can be regarded as a typical kind of composite material composed of sandy soil matrix, ice, salt crystals and unfrozen water inclusions. A meso-macroscopic constitutive model of saturated frozen saline sandy soil is developed based on the homogenization theory. In the proposed model, the constitutive response of equivalent inclusion phase is presented by an elastic-brittle model while that of sandy soil matrix phase is described by a critical state elastoplastic model. Based on the three-phase sphere model and Mori-Tanaka method, the expressions of elastic parameters of equivalent inclusion phase are given. The influences of inclusion volume contents on elastic parameters of equivalent inclusion phase are analyzed. On the other hand, the plastic parameters of sandy soil matrix phase can have considerable effects on the macroscopic deformation features. Compared with the experimental results, the proposed model can predict the deformation of frozen saline sandy soil with different salt contents at various confining pressures. The strain hardening/softening, high dilation and pressure melting features can be well reproduced.

Forecasting Volcanic Eruptions: Beyond the Failure Forecast Method

Volcano-tectonic seismicity and ground movement are the most reliable precursors to eruptions after extended intervals of repose, as well as to flank eruptions from frequently-active volcanoes. Their behavior is consistent with elastic-brittle failure of the crust before a new pathway is opened to allow magma ascent. A modified physical model shows that precursory time series are governed by a parent relation between faulting and elastic deformation in extension, subject to independent constraints on the rate of crustal loading with time. The results yield deterministic criteria that can be incorporated into existing operational procedures for evaluating the probability of crustal failure and, hence, levels of alert during an emergency. They also suggest that the popular Failure Forecasting Method for forecasting eruptions is a particular form of the parent elastic-brittle model, and that magma transport and crustal fracturing during unrest tend towards conditions for minimizing rates of energy loss.

The effects of nearby fractures on hydraulically induced fracture propagation and permeability changes

Fracture propagation caused by hydraulic fracturing operations can be significantly influenced by adjacent fractures. This paper presents a detailed coupled hydro-mechanical analysis to study the effects of nearby natural fractures on hydraulically induced fracture propagation and changes in fracture permeability. Two rock domains were considered in comparison: FD1, with one fracture, and FD2, with two adjacent parallel or non-parallel fractures. It was assumed that water injection occurred in a borehole that intersected the single fracture in FD1 and one of the two fractures in FD2. Simulations were made for a time period of 3h with an injection period of 2h followed by 1h of shut-in. An elastic-brittle model based on the degradation of material properties was implemented in a 2D finite-difference scheme and used for elements of the intact rock subjected to tension and shear failure. The intact rock was considered to have a low but non-negligible permeability. A verification study against analytical solutions showed that the fracture propagation and stress concentrations due to differential boundary stresses could be accurately represented by our model. Next, a base case was considered, in which the stress ratio (SR) between the magnitudes of the maximum and minimum boundary stresses, the permeability kR of the intact rock and the initial permeability kTF of the tension failure regions were fixed. In FD2, the distance dF between the two natural fractures defined by the closest distance was also fixed. The results showed that in both rock domains, the fracture started to propagate when the pore pressure was approximately 85% of the magnitude of the minimum boundary stress. The propagation of a single fracture was significantly greater than the propagation of a double fracture system because, in the latter case, the pore pressure decreased when the two fractures connected. As a result, changes in permeability in FD2 were smaller than in FD1. At the end of injection, the maximum ratios between the final and initial permeability of the natural fractures were found to be approximately 3 and 2 for rock domains FD1 and FD2, respectively. For non-parallel fractures, the controlling factor for fracture propagation was the separation between the tips of the pressurised fracture and the neighbouring non-pressurised fracture. A sensitivity study was conducted to study the influence of the key parameters dF, SR, kR and kTF on the simulation results. Fracture propagation showed more sensitivity to dF and SR than to the other parameters.

Simulating progressive failure in brittle jointed rock masses using a modified elastic-brittle model and the application

To analyse the damage evolution process of jointed rock masses, a modified numerical model on the basis of secondary development in fast Lagrangian analysis of Continua (FLAC3D) is proposed to simulate the fracture development of jointed rock mass. To validate the feasibility of this numerical model, numerous case studies on 2-D and 3-D cracking problems are conducted. The progressive failure processes of rock specimens with two pre-cracks under uniaxial and biaxial compressions are simulated and compared with the results obtained from the lab experiments, and they are found to be in good agreement. The failure processes of heterogeneous rock specimens with four pre-cracks are studied for further analysis. When applied to 3-D cases, the numerical results consistently match lab test results. Moreover, it is also used to analyse the failure process of a 3-D Weibull distribution specimen with an internal crack. Finally, it is used to investigate the crack propagation and stability of a slope project, presenting an excellent effect. It is concluded that this numerical model is effective and efficient in dealing with 3-D and numerous element cases.

Study of hydraulic fracturing processes in shale formations with complex geological settings

Hydraulic fracturing has been applied to extract gas from shale-gas reservoirs. Complicated geological settings, such as spatial variability of the rock mass properties, local heterogeneities, complex in situ stress field, and pre-existing bedding planes and faults, could make hydraulic fracturing a challenging task. In order to effectively and economically recover gas from such reservoirs, it is crucial to explore how hydraulic fracturing performs in such complex geological settings. For this purpose, numerical modelling plays an important role because such conditions cannot be reproduced by laboratory experiments. This paper focuses on the analysis of the influence of confining formations and pre-existing bedding planes and faults on the hydraulically-induced propagation of a vertical fracture, which will be called injection fracture, in a shale-gas reservoir. An elastic-brittle model based on material property degradation was implemented in a 2D finite-difference scheme and used for rock elements subjected to tension and shear failure. A base case is considered, in which the ratio SR between the magnitudes of the horizontal and vertical stresses, the permeability kc of the confining formations, the elastic modulus Ep and initial permeability kp of the bedding plane and the initial fault permeability kF are fixed at reasonable values. In addition, the influence of multiple bedding planes, is investigated. Changes in pore pressure and permeability due to high pressure injection lasting 2h were analysed. Results show that in our case during the injection period the fracture reaches the confining formations and if the permeability of those layers is significantly larger than that of the shale, the pore pressure at the extended fracture tip decreases and fracture propagation becomes slower. After shut-in, the pore pressure decreases more and the fracture does not propagate any more. For bedding planes oriented perpendicular to the maximum principal stress direction and with the same elastic properties as the shale formation, results were found not to be influenced by their presence. In such a scenario, the impact of multiple bedding planes on fracture propagation is negligible. On the other hand, a bedding plane softer than the surrounding shale formation leads to a fracture propagation asymmetrical vertically with respect to the centre of the injection fracture with a more limited upward fracture propagation. A pre-existing fault leads to a decrease in fracture propagation because of fault reactivation with shear failure. This results in a smaller increase in injection fracture permeability and a slight higher injection pressure than that observed without the fault. Overall, results of a sensitivity analysis show that fracture propagation is influenced by the stress ratio SR, the permeability kc of the confining formations and the initial permeability kp of the bedding plane more than the other major parameters.

Wellbore stability analysis using strain hardening and/or softening plasticity models

In this paper, the general shear strain hardening and/or softening Drucker-Prager models based on the common triaxial compression test results have been introduced to model the mechanical behaviour of rock formations. These elastoplastic models are then adopted to develop rigorous analytical solutions for the drained wellbore drilling problem subjected to in-plane isotropic stress field, and to further predict the borehole collapse failure. The illustration numerical examples show the distributions of stress components and the progressive development of the plastic deviatoric strain, in addition to the effective stress trajectory for a rock point at the borehole surface due to the wellbore drilling. The advantage of the desired wellbore drilling curve presented lies in the fact that it can track the deformation responses of the borehole surface from elastic to elastoplastic states and down to the peak and ultimate/residual conditions, thus offering much more flexibility and more appropriate mud pressure design over the conventional elastic-brittle models. The critical (minimum) mud pressures, predicted by the current elastoplastic analysis based on different wellbore instability criteria, are compared with the value corresponding to the elastic theory. The solution procedure proposed can also be utilized for the tunnelling analysis and extended to the design of lining required to stabilize a tunnel.

Theorems for masonry solids with brittle time-decaying tensile limit strength

In this paper, we introduce a phenomenological model approximating the behaviour of masonry structures, which is based on a low-tension elastic–brittle (EB) assumption with evolutionary tensile behaviour. The EB model is conceived by embedding a decaying tensile strength in the material behaviour, and it is able to achieve good agreement with the real behaviour of masonry. Since the model is quite sophisticated, non-holonomic, and the EB solution depends—amongst other things—on the loading path, it is worthwhile to investigate the relationships with more manageable and stable models rather than searching for unreliable solutions that depend on poorly predictable data. Namely, whereas it is quite clear and largely agreed upon that structural models widely applied in engineering (like perfectly plastic or no-tension models or other ones) are well-conditioned problems, the same does not apply to brittle structures. In this case, exact solutions are hard to be found and are scarcely attractive from the engineering point of view since they also depend on the load history and on unverifiable variables such as the local tensile strength. In view of these considerations, in this paper it is proved that stress fields in tensioned EB problems can be approached by highly stable solutions, on the upper and lower sides of the relevant complementary energy, and that the approximation gets closer as the limit tensile strength of the brittle material becomes lower.

Behavior of brittle anisotropic materials with different orientation of mechanical properties at the edge of piercing

The problem of normal interaction between a compact steel isotropic cylindrical projectile and an orthotropic plate at the edge of piercing in the range of impact velocity from 50 m/s to 400 m/s. The obstacle is made of an organoplastic material with some initial orientation of its mechanical properties or the same material whose properties are obtained by a 90° rotation of the initial material about the axis OY. The destruction of obstacles is studied and the efficiency of their protective properties is comparatively analyzed depending on the orientation of the elastic and strength properties of the anisotropic material. The problem is solved numerically by the finite element method in the three-dimensional statement. The behavior of the projectile material is described by an elastoplastic model, while the response of the obstacle anisotropic material is described in the framework of the elastic-brittle model with different tensile and compressive strengths.

Numerical modeling of development of fracture in anisotropic composite materials at low-velocity loading

The problem of normal interaction of the steel isotropic compact cylindrical projectile with the orthotropic plate on ballistic limit in range of velocities of impact from 50 to 400 m/s is considered. Target material is as organoplastic with initial orientation of mechanical properties, and the material, which properties received by turn on 90° relative to the axis OY of an initial material. Fracture of targets is investigated; the comparative analysis of efficiency of their protective properties depending on orientation of elastic and strength properties of an anisotropic material is carried out. The task is solved numerically, using the method of finite elements in three-dimensional statement. The behavior of a material of the projectile is described by elastic–plastic model; the behavior of anisotropic material of the target is described by elastic–brittle model with various ultimate strength limits on pressure and tension.

A simple J-integral governed bilinear constitutive relation for simulating fracture propagation in quasi-brittle material

The present paper develops a bilinear constitutive relation for material at fracture tip based on the cohesive zone model and the J-integral theory. In this bilinear constitutive relation, the stress linearly decreases rather than suddenly drops down to zero at the post-peak stage. The slope of stress degradation is related to the critical J-integral of material and the element size. By this bilinear constitutive relation, the tensile and mixed fracture propagation can be well simulated for the strain energy release rate can be preserved with element size decreasing. The simulation results are almost free of the element size sensitivity. The present model and its implementation are very simple. For the elements at fracture tip, the derived bilinear constitutive relation is adopted while for the rest elements, the ideal elastic-brittle model is adopted. For the fracture propagation is represented by the deterioration of stiffness through constitutive relation, no separate fracture criterion is needed, which makes the evaluation of the fracture propagation by separate fracture criterion and the mesh modification after each load step unnecessary. The simulation examples of tensile and mixed fracture suggest that the present method is valid and highly efficient.

On the Wellbore Stress Change Caused by Drawdown and Depletion: An Analytical Model for a Vertical Well in a Thin Reservoir

Summary An analytical model is presented to describe the stress change at the wall of a vertical wellbore because of drawdown and reservoir depletion. The model predicts a lower effective tangential stress and higher effective axial stress (at a given drawdown) than the popular model of Risnes, Bratli, and Horsrud (RBH) model (Risnes et al. 1982; Bratli et al. 1983; Fjær et al. 2008). The reason for this difference is that our model is valid for a finite reservoir thickness ("thin reservoir"), whereas the RBH model is constrained by the condition of an infinite reservoir thickness. We modeled the production-induced stress change in two reservoirs with different rock properties: Reservoir A has a porosity of approximately 20% and a permeability of up to 30 md; Reservoir B has a porosity of approximately 30% and a permeability of a few darcies. Calculated wellbore stress paths were combined with mechanical properties from core deformation experiments to evaluate the risk of drawdown-induced/depletion-induced shear failure. If we consider only the experimental shear-failure data collected on samples from Reservoir A for the shear-failure limit, then, according to the present model, the planned drawdown of 34 MPa will not lead to wellbore shear failure. Even after 34 MPa of depletion, a drawdown of 34 MPa can be applied safely in Reservoir A. For high-permeability Reservoir B, the model predicts that shear failure of the borehole wall will not occur in the first phase of production, when there is only a drawdown of up to 2 MPa and no depletion. Depending on the shear-failure criterion chosen, the model predicts shear failure after 25 to 42 MPa of depletion. Massive sand production was observed only after some 40 MPa of depletion, confirming that elastic-brittle models are conservative in predicting drawdown-induced or depletion-induced shear failure in a borehole, notably in high-porosity rocks. Our examples show their value in qualitative comparative analysis of shear failure and sand-production risk.

Connection between the deformation properties of rock joints and their fractal dimension

The mechanism of interaction between the edges of discontinuities under the normal loading is proposed based on the elastic-brittle model of a rock mass. The respective constitutive relationships are synthesized by numerical experiments, and the functional dependence of an average normal rigidity of a joint on the fractal dimension of its edge is established.

Geodetic evidence for rifting in Afar a brittle-elastic model of the behaviour of the lithosphere

An important episode of rifting occurred in November 1978 in southwest Afar, in the first subaerial section of the accreting plate boundary between the African and Arabian plates. Horizontal rifting of more than 2 m took place, with vertical displacements of about 1 m, earthquakes of magnitude up to 5.3, and a fissural volcanic eruption of basaltic lavas. Very precise geodetic measurements were carried out in order to study this crisis and strains of the order of 3 × 10 −4 were measured, both tensile and compressive. This paper presents an analysis of the mechanical behaviour of the lithosphere. It is shown that an elastic-brittle model with a rebound mechanism fits very well the data, and it is suggested that such a model, with magma injection in the resulting open fissures, should be used to describe accretion at plate boundaries.

A continuum model for plastic-brittle behaviour of rock and concrete

The present work deals with a continuum model for time-independent behaviour of progressively fracturing solids. The rate theory with fracture parameter is discussed and applied to rock-like materials. In particular, elastic-brittle and plastic-brittle idealizations are considered with the fracturing affecting elastic and plastic strains, respectively. The elastic-brittle model incorporates, however, the plasticity-like concept of varying fracture surface. Also the idea of a limit (failure) locus is introduced as the stress-path dependent condition, instead of being prescribed a priori. The plasticity model attempts to cover both stable and unstable portions of the stress-strain relation while the fracture parameter is coupled with irreversible volumetric dilatancy. The general model is applicable to wider class of engineering materials, ƒ. ex. to metals in advanced yielding accompanied with microfracturing.