On critical assessment of the use of local and nonlocal damage models for prediction of ductile crack growth and crack path in various loading and boundary conditions

In this work, we have assessed the results of the local and nonlocal versions of Rousselier’s damage model, which have been used here for simulation of ductile crack growth. There are several issues regarding the accuracy of the results which has been addressed in this paper, e.g., accuracy in simulation of crack path, extent and width of the damaged region, fracture resistance behaviour in situations such as symmetric vs. non-symmetric boundary-value problems, mixed-mode loading vs. mode-I loading of the crack-tip, etc. It was also observed that the shape and orientation of the elements at the crack-tip, in addition to their size, influence the results of the local damage model. In this work, it was shown that the above issues can be resolved through the use of nonlocal damage models. The predictions of the nonlocal model are also consistent with the experimental observations unlike its local counterpart. Several examples were presented, where the results as obtained by both the local and nonlocal models were compared. From this experience, it is recommended that the local damage models should not be used blindly by the analysts for all kinds of mesh design, loading, boundary conditions, etc.     

Predicting the failure of ultrasonic spot welds by pull-out from sheet metal

A methodology for determining the cohesive fracture parameters associated with pull-out of spot welds is presented. Since failure of a spot weld by pull-out occurs by mixed-mode fracture of the base metal, the cohesive parameters for ductile fracture of an aluminum alloy were determined and then used to predict the failure of two very different spot-welded geometries. The fracture parameters (characteristic strength and toughness) associated with the shear and normal modes of ductile fracture in thin aluminum alloy coupons were determined by comparing experimental observations to numerical simulations in which a cohesive-fracture zone was embedded within a continuum representation of the sheet metal. These parameters were then used to predict the load–displacement curves for ultrasonically spot-welded joints in T-peel and lap-shear configurations. The predictions were in excellent agreement with the experimental data. The results of the present work indicate that cohesive-zone models may be very useful for design purposes, since both the strength and the energy absorbed by plastic deformation during weld pull-out can be predicted quite accurately.     

The influence of particle shape,volume fraction and distribution on post-necking deformation and fracture in uniaxial tension of AA5754 sheet materials

Finite element analysis was performed over a small particle field, edge constraint plane strain post-necking model. The aim is to understand the roles of particle shape, volume fraction and distribution over the post-necking deformation and fracture of AA5754-O sheet materials. For models containing one single particle, the post-necking deformation decreases when the particle varies from circular to elliptical. The inter-particle spacing, the major parameter of distribution to determine whether a pair of particles belongs to a stringer or not, was varied for models with two particles of circular or elliptical shape. The general trend is that the post-necking deformation and fracture strains decrease with decreasing spacing between particles. There is considerable difference in terms of both fracture topographies and strains for models containing 16 particles when distributions varied from random/uniform to stringer distributions. The post-necking deformation and fracture strains monotonically decrease with particle volume fractions for models with 4–64 particles of random or stringer distribution. This indicates that the post-necking behavior for AA5754-O alloys where the matrix material is rather ductile is not solely controlled by a single or pair of particles although they may become initiation places of damage. Multiple damaging sources such as stringers or large particles can act cooperatively and speed up the damaging propagation of the material, and therefore produce small post-necking deformation and early fracture. The center clustering of particles can be beneficial for post-necking behavior and bendability of sheet materials.     

Relations between a micro-mechanical model and a damage model for ductile failure in shear

Gurson type constitutive models that account for void growth to coalescence are not able to describe ductile fracture in simple shear, where there is no hydrostatic tension in the material. But recent micro-mechanical studies have shown that in shear the voids are flattened out to micro-cracks, which rotate and elongate until interaction with neighbouring micro-cracks gives coalescence. Thus, the failure mechanism is very different from that under tensile loading. Also, the Gurson model has recently been extended to describe failure in shear, by adding a damage term to the expression for the growth of the void volume fraction, and it has been shown that this extended model can represent experimental observations. Here, numerical studies are carried out to compare predictions of the shear-extended Gurson model with the shear failures predicted by the micro-mechanical cell model. Both models show a strong dependence on the level of hydrostatic tension. Even though the reason for this pressure dependence is different in the two models, as the shear-extended Gurson model does not describe voids flattening out and the associated failure mechanism by micro-cracks interacting with neighbouring micro-cracks, it is shown that the trends of the predictions are in good agreement.     

The effects of geometry and material properties on the fracture of single lap-shear joints

A review of the mechanics of lap-shear joints is followed by a detailed analysis of the problem using a cohesive-zone approach. The cohesive-zone model allows not only the influence of geometry to be considered, but also allows the cohesive properties of the interface and plastic deformation of the adherends to be included in the analysis. The first part of the paper examines the strength of elastic joints, with an emphasis on the effects of geometry, the cohesive strength of the adhesive, and mode-mixedness. The cohesive-zone models show a transition to the predictions of linear-elastic fracture mechanics under conditions where these are expected to apply. The second part of the paper examines the effect of plasticity in the adherends, and looks at the transition between the elastic and plastic regimes. The final part of the paper makes comparisons between the predictions of the numerical calculations and experimental observations for a model system consisting of a commercial adhesive used to bond an aluminum alloy. Using cohesive-zone parameters previously determined for this particular combination of materials, the numerical predictions show excellent agreement with the experimental observations.     

Diffusion Mass Transfer in Miscible Oil Recovery: Visual Experiments and Simulation

Co-and counter-current type transfers due to diffusion and -free- convection caused by the buoyant forces between fracture and matrix were studied experimentally using 2-D glass-bead models. Mineral oil and kerosene were used as the displaced phase. The model saturated with oil was exposed to solvent phase (pentane) under static conditions (no flow in fracture) to mimic matrix-fracture interaction during gas or liquid solvent injection in naturally fractured reservoirs. Displacement fronts and patterns were analyzed and quantified using fractal techniques to obtain correlations between the fractal properties and displacement type. Displacements resulted in a mixture of bulk diffusion and -free- convection mainly depending on the interaction type (co- or counter-current), oil type, and displacement direction (horizontal and vertical). Conditions yielding different types of displacement patterns were identified. Finally, a stochastic model that was inspired from invasion percolation and diffusion limited aggregation algorithms was developed for the horizontal displacement cases. The experimental observations were matched to the displacement patterns obtained through the stochastic modeling.     

A numerical study on the deformation and fracture modes of steel projectiles during Taylor bar impact tests

A heterogeneous material model based on macro-mechanical observations is proposed for simulation of fracture in steel projectiles during impact. A previous experimental study on the deformation and fracture of steel projectiles during Taylor bar impact tests resulted in a variety of failure modes. The accompanying material investigation showed that the materials used in the impact tests were heterogeneous on scales ranging from microstructure as investigated with SEM to variation in fracture strains from tensile tests. A normal distribution is employed to achieve a heterogeneous numerical model with respect to the fracture properties. The proposed material model is calibrated based on the tensile tests, and then used to independently simulate the Taylor bar impact tests. A preliminary investigation showed that the simulations are sensitive to assumptions regarding the anvil behaviour and friction properties. A flexible anvil and a yield-limited friction law are shown to be necessary to correctly reproduce the experimental behaviour. The proposed model is then shown to be capable of correctly reproducing all fracture modes but one, and also predict critical impact velocities for projectile fracture with reasonable accuracy. Fragmentation at velocities above the critical velocity is not well reproduced due to excessive element erosion. Measures to make the element erosion process more physical are proposed and discussed with their respective drawbacks. The use of a simple fracture criterion in combination with an element erosion technique accentuates the effect of distributing the fracture parameter.     

Mesofracture mechanics: a necessary link

Classical fracture mechanics is based on the premise that small scale features could be averaged to give a larger scale property such that the assumption of material homogeneity would hold. Involvement of the material microstructure, however, necessitates different characteristic lengths for describing different geometric features. Macroscopic parameters could not be freely exchanged with those at the microscopic scale level. Such a practice could cause misinterpretation of test data. Ambiguities arising from the lack of a more precise range of limitations for the definitions of physical parameters are discussed in connection with material length scales. Physical events overlooked between the macroscopic and microscopic scale could be the link that is needed to bridge the gap. The classical models for the creation of free surface for a liquid and solid are oversimplified. They consider only the translational motion of individual atoms. Movements of groups or clusters of molecules deserve attention. Multiscale cracking behavior also requires the distinction of material damage involving at least two different scales in a single simulation. In this connection, special attention should be given to the use of asymptotic solution in contrast to the full field solution when applying fracture criteria. The former may leave out detail features that would have otherwise been included by the latter. Illustrations are provided for predicting the crack initiation sites of piezoceramics. No definite conclusions can be drawn from the atomistic simulation models such as those used in molecular dynamics until the non-equilibrium boundary conditions can be better understood. The specification of strain rates and temperatures should be synchronized as the specimen size is reduced to microns. Many of the results obtained at the atomic scale should be first identified with those at the mesoscale before they are assumed to be connected with macroscopic observations. Hopefully, “mesofracture mechanics” could serve as the link to bring macrofracture mechanics closer to microfracture mechanics.     

An Analytical Model of Porosity–Permeability for Porous and Fractured Media

The classic Kozeny–Carman equation (KC) uses parameters that are empirically based or not readily measureable for predicting the permeability of unfractured porous media. Numerous published KC modifications share this disadvantage, which potentially limits the range of conditions under which the equations are applicable. It is not straightforward to formulate non-empirical general approaches due to the challenges of representing complex pore and fracture networks. Fractal-based expressions are increasingly popular in this regard, but have not yet been applied accurately and without empirical constants to estimating rock permeability. This study introduces a general non-empirical analytical KC-type expression for predicting matrix and fracture permeability during single-phase flow. It uses fractal methods to characterize geometric factors such as pore connectivity, non-uniform grain or crystal size distribution, pore arrangement, and fracture distribution in relation to pore distribution. Advances include (i) modification of the fractal approach used by Yu and coworkers for industrial applications to formulate KC-type expressions that are consistent with pore size observations on rocks. (ii) Consideration of cross-flow between pores that adhere to a fractal size distribution. (iii) Extension of the classic KC equation to fractured media absent empirical constants, a particular contribution of the study. Predictions based on the novel expression correspond well to measured matrix and fracture permeability data from natural sandstone and carbonate rocks, although the currently available dataset for fractures is sparse. The correspondence between model calculation results and matrix data is better than for existing models.     

New Correlative Models to Improve Prediction of Fracture Permeability and Inertial Resistance Coefficient

Presence of fracture roughness and occurrence of nonlinear flow complicate fluid flow through rock fractures. This paper presents a qualitative and quantitative study on the effects of fracture wall surface roughness on flow behavior using direct flow simulation on artificial fractures. Previous studies have highlighted the importance of roughness on linear and nonlinear flow through rock fractures. Therefore, considering fracture roughness to propose models for the linear and nonlinear flow parameters seems to be necessary. In the current report, lattice Boltzmann method is used to numerically simulate fluid flow through different fracture realizations. Flow simulations are conducted over a wide range of pressure gradients through each fracture. It is observed that creeping flow at lower pressure gradients can be described using Darcy’s law, while transition to inertial flow occurs at higher pressure gradients. By detecting the onset of inertial flow and regression analysis on the simulation results with Forchheimer equation, inertial resistance coefficients are determined for each fracture. Fracture permeability values are also determined from Darcy flow as well. According to simulation results through different fractures, two parametric expressions are proposed for permeability and inertial resistance coefficient. The proposed models are validated using 3D numerical simulations and experimental results. The results obtained from these two proposed models are further compared with those obtained from the conventional models. The calculated average absolute relative errors and correlation coefficients indicate that the proposed models, despite their simplicity, present acceptable outcomes; the models are also more accurate compared to the available methods in the literature.     

On the predictive capabilities of the shear modified Gurson and the modified Mohr-Coulomb fracture models over a wide range of stress triaxialities and Lode angles

The predictive capabilities of the shear-modified Gurson model [Nielsen and Tvergaard, Eng. Fract. Mech. 77, 2010] and the Modified Mohr-Coulomb (MMC) fracture model [Bai and Wierzbicki, Int. J. Fract. 161, 2010] are evaluated. Both phenomenological fracture models are physics-inspired and take the effect of the first and third stress tensor invariants into account in predicting the onset of ductile fracture. The MMC model is based on the assumption that the initiation of fracture is determined by a critical stress state, while the shear-modified Gurson model assumes void growth as the governing mechanism. Fracture experiments on TRIP-assisted steel sheets covering a wide range of stress states (from shear to equibiaxial tension) are used to calibrate and validate these models. The model accuracy is quantified based on the predictions of the displacement to fracture for experiments which have not been used for calibration. It is found that the MMC model predictions agree well with all experiments (less than 4% error), while less accurate predictions are observed for the shear-modified Gurson model. A comparison of plots of the strain to fracture as a function of the stress triaxiality and the normalized third invariant reveals significant differences between the two models except within the vicinity of stress states that have been used for calibration.     

Fracture mechanics of composites in compression along cracks

Conclusions Thus, based on the concepts in [5–7], the beginning of fracture of a composite in compression along a plane containing cracks can be represented as follows. If the material has not fractured already, then local fracture necessarily occurs next to the cracks at a value of compressive load corresponding to surface instability. Here, the upper limit of the ultimate theoretical strength corresponds to the minimum value of the shear modulus. This fracture mechanism can be seen in composites reinforced with high-modulus fibers. It should be noted that fracture can also take place on free surfaces of the material at these load values as a result of surface instability [1]. If the area of the free surface (lateral surface) of the material is substantially less than the total area of the cracks present throughout the volume of the material, then obviously the cracks will be the deciding factor in the fracture mechanism. The above-noted conclusions and quantitative abalysis were made only for a linearly elastic, orthotropic material with a high shear stiffness (brittle fracture), while the general results obtained in the present article also pertain to nonlinearly elastopiastic models. The results can be refined for more complex models. It must also be noted that the theoretical ultimate strength may be reduced substantially if the interaction of cracks located in parallel planes during instability is considered. This feature of the fracture mechanism was noted in note 6 in the article [6]. Composite materials generally have a fairly large number of cracks in planes and surfaces along the reinforcing elements. In connection with this, for composites it is best to allow for interaction of cracks located in parallel planes, which in turn should lead to a substantial reduction in the theoretical ultimate strength value obtained in the present article.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 18, No. 6, pp. 3–9, June, 1982.     

Hydraulic Fracture Propagation with 3-D Leak-off

Hydraulic fracture models typically couple a fracture elasticity model with a geological reservoir model to forecast the rate of fluid leak-off from the propagating fracture. The most commonly used leak-off model is that originally specified by Carter, which involves the assumption that the fracture is embedded within an infinite homogenous porous medium where flow only occurs perpendicular to the fracture plane. The objectives of this paper are: (1) to show that assuming one-dimensional leak-off can lead to erroneous conclusions, (2) to present a robust numerical methodology for simulating three-dimensional leak-off from propagating hydraulic fractures, and (3) to present and compare a new analytical method based on assuming three-dimensional flow of an incompressible fluid through an incompressible porous formation from a circular planar fracture. Provided the fluid and formation compressibility can be ignored within the reservoir flow model, the three-dimensional leak-off from a circular planar fracture can be written in closed-form as a function, which depends linearly on fracture pressure and radial extent. This simple expression for leak-off can be easily coupled to a range of circular fracture elasticity models. As a comparison example, the Carter model, our new function and a three-dimensional numerical model of the full problem are coupled to the PK-radial fracture model. Comparison with the numerical model shows that our new function overestimates fracture growth during intermediate times but accurately predicts both the early and late-time asymptotic behavior. In contrast, the Carter model fails to replicate both the early and late-time asymptotic behavior. Our new function additionally improves on the Carter model by not requiring the evaluation of convolution integrals and allowing easy evaluation of both the spatial leakage flux distribution across the fracture face and the three-dimensional pressure distribution within the porous formation.     

裂隙岩体中非饱和渗流与运移的概念模型及数值模拟

探讨了裂隙岩体中非饱和地下水渗流与溶质运移的几种概念模型的构造及数值模拟问题 ,如裂隙网络模型、连续体模型、等效连续体模型、双孔隙度 (单渗透率 )模型、双渗透率模型、多组份连续体模型等。在裂隙岩体中 ,非饱和地下水的渗流可能只局限于岩体中的岩石组份、或裂隙网络 ,也可能在裂隙和岩石中同时发生 ;对前一种情形只需考虑单一连续体中的流动 ,而后一种情况则需要包括地下水在岩石和裂隙之间的交换。岩体中的裂隙网络往往是溶质运移的主要通道 ;但当溶质在裂隙与岩石之间的渗透和扩散是重要的运移机制时 ,就需要考虑岩石与裂隙界面处的溶质交换。为了模拟岩石与裂隙之间地下水和溶质的交换 ,就需要了解岩石与裂隙之间相互作用的模式和范围 ,使得这类问题的概念模型较单一连续体模型多了一层不确定性、其数值模拟也变得更为困难。因为在实际问题中不易、甚至根本不能判别非饱和渗流的实际形态 ,具体采用哪种模型主要取决于分析的目的和对现场数据的掌握程度。不论哪种模型都会受到模型及参数不确定性的影响 ,因此必须考虑与其他辅助模型的比较.     

基于Hangmann骨折的植骨融合钢板内固定术有限元稳定性分析

Hangmann骨折是近年来发病率逐渐上升的上部颈椎损伤之一。植骨融合钢板内固定术作为一种较新的手术方案刚刚开始在国内采用。本文采用基于医学影像的快速自动三维有限元模型的构建方法,建立了上颈椎C2～C4两个运动节段的三维有限元模型,对上颈椎正常状态、植骨融合钢板系统内固定术方案在模拟生理荷载下的三维六自由度的稳定性、内固定器植入后的应力情况进行了计算和比较分析,并通过实验验证了内固定手术对上颈椎的生物力学稳定性和内固定器的力学疗效特征。本文结果显示：对Hangmann骨折治疗,植骨融合钢板系统内固定技术是一个较好的可供选择的措施之一。     

Experimental Investigation of Fracture Process Zone in Rocks Damaged Under Cyclic Loadings

Compared with other materials, most rocks generally fail in a brittle fashion rather than exhibiting yielding or purely plastic deformation. However, the initiation and coalescence of micro-cracks in the nonlinear region, known as the ‘fracture process zone’ (FPZ), are the primary reason for fracture propagation in rocks. Different elasticity-related models proposed for determining the features of the FPZ have not achieved an adequate understanding of its various fracture patterns. Based on previous experiments and numerical models, micro-crack density has been shown to be a function of loading history and to vary depending on whether the loading is monotonic or cyclic. The aim of the study reported here was to examine the different patterns of the FPZ under various types of cyclic loading and to quantitatively define damage and fracture patterns through the grains or rock matrix. Considerable laboratory testing was conducted, and fractured samples were investigated by computerised tomography scanning, supported by thin-section analysis. In the study, two different types of cyclic loading were tested: stepped and continuous. A diametral compressive loading was applied at predetermined amplitude and frequency with the continuous cyclic loading. The applied cyclic diametral compressive load was returned to the original level after each step, and at the next step, the amplitude started from zero, with stepped cyclic loading (SCL). An average 30 % strength reduction was found due to the SCL and emergence of high micro-fracture density in the FPZ. We presume that hard rock breakage techniques will be improved, especially for rock-cutting technologies, such as drag bits and oscillating disc cutting, by understanding the effects of cyclic loading on rock strength.     

Meshfree modelling of fracture—a comparative study of different methods

Different fracture methods in meshfree methods are studied and compared. Our studies focuses on the elementfree Galerkin (EFG) method though similar results were obtained with SPH and MPM. Three major fracture approaches are tested: Natural fracture, smeared crack method and discrete crack method. In the latter method, the crack is represented as continuous line and as set of discrete crack segment. Natural fracture is a key feature of meshfree methods. In some numerical examples, we will show that natural fracture criterion cannot handle even simple fracture adequately. Moreover, we will show in our numerical examples that smeared crack models can capture global behavior appropriately for simple examples but not for complex examples involving branching cracks. The most accurate methods are discrete fracture methods.     

The damage process zone characteristics at crack tip in concrete

1.IntroductionBynowfracturemechanicsisuniversallyacknowledgedasthetoolofanalysingtheinveshgationofconcretecracking.Andaftermanyyearsofdevelopmentonnumerousconstitutivemodelsforuniaxial,biaxialandtriaxialmodelsofconcreteundercompressivestresses,andtheirvariousrefinements,researchershavefinallyturnedtheirattentiontothecornerofthestressstraincurveandaregivingmoreattentiontotheeffectofcrackingonthenonlinearresponse.ofconcretestructures.Havingexperimentallyobservedthatthemacroscopicresponseofconcre…     

Phase-field modeling of crack propagation in piezoelectric and ferroelectric materials with different electromechanical crack conditions

We present a family of phase-field models for fracture in piezoelectric and ferroelectric materials. These models couple a variational formulation of brittle fracture with, respectively, (1) the linear theory of piezoelectricity, and (2) a Ginzburg–Landau model of the ferroelectric microstructure to address the full complexity of the fracture phenomenon in these materials. In these models, both the cracks and the ferroelectric domain walls are represented in a diffuse way by phase-fields. The main challenge addressed here is encoding various electromechanical crack models (introduced as crack-face boundary conditions in sharp models) into the phase-field framework. The proposed models are verified through comparisons with the corresponding sharp-crack models. We also perform two dimensional finite element simulations to demonstrate the effect of the different crack-face conditions, the electromechanical loading and the media filling the crack gap on the crack propagation and the microstructure evolution. Salient features of the results are compared with experiments.