Monte Carlo simulation of complex cohesive fracture in random heterogeneous quasi-brittle materials

A numerical method is developed to simulate complex two-dimensional crack propagation in quasi-brittle materials considering random heterogeneous fracture properties. Potential cracks are represented by pre-inserted cohesive elements with tension and shear softening constitutive laws modelled by spatially-varying Weibull random fields. Monte Carlo simulations of a concrete specimen under uni-axial tension were carried out with extensive investigation of the effects of important numerical algorithms and material properties on numerical efficiency and stability, crack propagation processes and load-carrying capacities. It was found that the homogeneous model led to incorrect crack patterns and load–displacement curves with strong mesh-dependence, whereas the heterogeneous model predicted realistic, complicated fracture processes and load-carrying capacity of little mesh-dependence. Increasing the variance of the tensile strength random fields with increased heterogeneity led to reduction in the mean peak load and increase in the standard deviation. The developed method provides a simple but effective tool for assessment of structural reliability and calculation of characteristic material strength for structural design.     

Numerical simulation of quasi-brittle fracture using damaging cohesive surfaces

The cohesive surface methodology is used in a numerical study of fracture of concrete. The traction vs. separation response is governed by an isotropic damage law in which damage evolves according to a prescribed one-dimensional linear or exponential softening law. Cohesive surfaces are immersed in the continuum to allow for a maximum freedom of crack path selection. The single edge notched four point shear beam and the double edge notched tensile bar are used to study: (i) the influence of the tangential cohesive response on the development of the fracture path and (ii) the mesh alignment sensitivity. It is shown that in the present formulation, the tangential cohesive response has a minor influence on both crack path and global characteristics. Mesh alignment does have a significant influence on the outcome of the numerical analysis.     

3-D numerical simulation of fracture processes in heterogeneous brittle materials

By using the lattice model combined with finite element methods and statistical techniques, a numerical approach is developed to establish mechanical models of three-dimensional heterogeneous brittle materials. A special numerical code is introduced, in which a lattice model and statistical approaches are used to simulate the initial heterogeneity of material properties. The size of displacement-load step is adaptively determined so that only few elements would fail in each load step. When the tensile principal strain in an element exceeds the ultimate strain of this element, the element is considered broken and its Young's modulus is set to be very low. Some important behaviors of heterogeneous brittle materials are indicated using this code. Load-displacement curves and figures of three-dimensional fracture patterns are also numerically obtained, which are similar to those observed in laboratory tests.     

Effective hydraulic conductivities in unsaturated heterogeneous media by Monte Carlo simulation

When modeling flow and transport through unsaturated heterogeneous geological deposits, it may be neither computationally nor technically feasible to account for the actual heterogeneity in the simulations. One would fall short in terms of technical feasibility because there is simply no way that the entire spatial domain could be characterized (e.g., you cannot measure hydraulic conductivity at every location at a site). With respect to computational feasibility, the non-linear nature of the Richards equation (which is used to model the flow process) makes simulation of most sites extremely computationally intensive. The computational roadblock is being dismantled as computer hardware advances, but our inability to precisely characterize geological heterogeneity is expected to remain with us for a very long time. To address this problem, the analyst typically uses average or effective properties to model flow and transport behavior through heterogeneous media. In this paper, a variety of approaches for developing effective unsaturated flow properties are assessed. Computational results have been obtained which give the hydraulic conductivity ratios (K _parallel/K _nomal) for highly nonisotropic layered materials. These results are compared with analytical models. Good agreement was obtained for all soil saturation levels except for extremely dry conditions.This work was performed at Sandia National Laboratories, which is operated for the U.S. Department of Energy under contract number DE-AC04-76DP00789.     

Initiation and propagation of complex 3D networks of cracks in heterogeneous quasi-brittle materials: Direct comparison between in situ testing-microCT experiments and phase field simulations

We provide the first direct comparisons, to our knowledge, of complex 3D micro cracking initiation and propagation in heterogeneous quasi-brittle materials modelled by the phase field numerical method and observed in X-ray microtomography images recorded during in situ mechanical testing. Some material parameters of the damage model, including the process zone (internal) length, are identified by an inverse approach combining experimental data and 3D simulations. A new technique is developed to study the micro cracking at a finer scale by prescribing the local displacements measured by digital volume correlation over the boundary of a small sub-volume inside the sample during the numerical simulations. The comparisons, performed on several samples of lightweight plaster and concrete, show a remarkable quantitative agreement between the 3D crack morphology obtained by the model and by the experiments, without any a priori knowledge about the location of the initiation of the cracks in the numerical model. The results indicate that the crack paths can be predicted in a fully deterministic way in spite of the highly random geometry of the microstructure and the brittle nature of its constituents.     

Coupled continuum and discrete analysis of random heterogeneous materials: Elasticity and fracture

Recent work has suggested that the heterogeneous distribution of mechanical properties in natural and synthetic materials induces a toughening mechanism that leads to a more robust structural response in the presence of cracks, defects or other types of flaws. Motivated by this, we model an elastic solid with a Young′s modulus distribution described by a Gaussian process. We study the pristine system using both a continuum and a discrete model to establish a link between the microscale and the macroscale in the presence of disorder. Furthermore, we analyze a flawed discrete particle system and investigate the influence of heterogeneity on the fracture mechanical properties of the solid. We vary the variability and correlation length of the Gaussian process, thereby gaining fundamental insights into the effect of heterogeneity and the essential length scales of heterogeneity critical to enhanced fracture properties. As previously shown for composites with complex hierarchical architectures, we find that materials with disordered elastic fields toughen by a ‘distribution-of-weakness’ mechanism inducing crack arrest and stress delocalization. In our systems, the toughness modulus can increase by up to 30% due to an increase in variability in the elastic field. Our work presents a foundation for stochastic modeling in a particle-based micromechanical environment that can find broad applications within natural and synthetic materials.     

On the dynamics of grafted branched polymers—a Monte Carlo simulation study

A simplified model of grafted branched polymers was designed and investigated. The model consisted of star-branched chains constructed on a simple cubic lattice. The star polymers were built of three arms of equal lengths. The chains were attached to an impenetrable flat surface with one arm’s end. The arm attached to the surface (a stem) was built of segments different from those in two remaining arms (branches). During the Monte Carlo simulation of the system, the conformation of each chain was modified according to the metropolis sampling algorithm with local changes of chain conformations. The simulations were performed for different chain lengths and the temperature of the system (solvent conditions). The structure of a polymer film formed on the grafting surfaces depended strongly on the temperature and the low temperature films consisted of two separate layers with the insoluble layer located near the grafting surface. The short-time relaxation of the branches and stems of chains was also investigated. The analysis of the dynamics of the model system shows the influence of the structure of the system on relaxation times of various parameters. Paper presented at the AERC 2005 held on April 21–23, 2005 in Grenoble, France.     

Monte Carlo simulation of radiative transfer in a refractive layered medium

In this work, we adapted the Monte Carlo method to simulate radiative transfer in a two-layer scattering slab with continuously varying refractive index in each of the two layers and a jump of refractive index at the interface between the two layers. The hemispherical reflectance (R _h ) and transmittance (T _h ) of the slab are obtained by tracing photon bundles propagating along curved trajectories. There is a very satisfying correspondence between the present results and those obtained by numerical solution of integral radiative transfer equation for the special cases with constant refractive index in each of the layers. The magnitude of numerical uncertainty decreases with the increase of bundles. The results show that the R _h decreases with the increase of the positive gradient of the refractive index considered. For the cases with constant total thickness, the R _h and the T _h increase with the increase of the ratio of upper-layer thickness to lower-layer thickness.     

Dynamics of branched chain solutions in adsorbing slit. A Monte Carlo study

A coarse-grained model of branched polymers confined in a slit formed by two parallel impenetrable surfaces that were attractive for polymer segments was developed and studied. The model chains were regular stars consisting of f?=?3 branches of equal length. The positions of chains were restricted to vertices of a simple cubic lattice. The chains were at good solvent conditions—the excluded volume was the only interaction between the segments of the chain. The properties of the model chains were determined by means of Monte Carlo simulations with a sampling algorithm based on chain’s local changes of conformation. The dependence of chain mobility on the width of the slit and the polymer concentration was studied. The conditions at which the strongly adsorbed chains could swap between both confining surfaces were studied. The mechanism of the chain motion was also discussed.     

岩石、混凝土类材料断裂破坏有限元数值模拟中的有效方法

岩石、混凝土类材料断裂破坏有限元数值模拟中的网格重划,依据单元畸变和裂缝介质间的单元干涉作为网格重划判据,采用几何体重构技术把几何实体分解成能在ANSYS上实现六面体网格划分的几个部分,利用体积判断法确定新结点在旧单元的单元编号,在场量传递上采用基于解析性质的等参有限元逆变换,把旧网格场量信息传递到新网格中。本文对ANSYS进行二次开发,实现了三维网格重划,网格重划采用单元畸变和界面干涉两个判据,在网格再划分前进行几何体重构,提取变形后的点线面信息重新生成实体,充分利用AN-SYS的函数和体积判断法找到新结点在旧网格中的位置,在新旧网格间的场量传递中采用基于解析逆等参单元法。在平台上实现了三维有限元网格重划技术,最后利用方料的单轴压缩断裂模拟计算检验了传递前后等效塑性应变分布用载荷信息的变化,证明了所开发系统的正确性。     

Monte Carlo simulation of the heat flow in a dense hard sphere gas

The one-dimensional steady heat flow in a dense hard sphere gas is studied solving the Enskog equation numerically by a recently proposed DSMC-like particle scheme. The accuracy of the solutions is assessed through a comparison with solutions obtained from a semi-regular method which combines finite difference discretization with Monte Carlo quadrature techniques. It is shown that excellent agreement is found between the two numerical methods. The solutions obtained from the Enskog equation have also been found in good agreement with the results of molecular dynamics simulations.     

The advanced simulation of fatigue crack growth in complex 3D structures

An advanced incremental crack growth algorithm for the three-dimensional (3D) simulation of fatigue crack growth in complex 3D structures with linear elastic material behavior is presented. To perform the crack growth simulation as effectively as possible an accurate stress analysis is done by the boundary-element method (BEM) in terms of the 3D dual BEM. The question concerning a reliable 3D crack growth criterion is answered based on experimental observations. All criteria under consideration are numerically realized by a predictor–corrector procedure. The agreement between numerically determined and experimentally observed crack fronts will be shown on both fracture specimens and an industrial application.     

延性金属膨胀环的运动及断裂的三维数值模拟

利用LS-DYNA三维动力有限元软件对延性金属环的膨胀运动与断裂进行数值模拟。在膨胀环圆周加入泊松随机分布断裂成核点,利用J-C本构模型,研究诸如颈缩形成时间、颈缩区与均匀变形区的温度、应力、应变的对比等颈缩形成机理,以及讨论了环圆周上断裂成核点的泊松随机分布碎裂特性的影响。数值计算结果与实验结果、理论分析结果吻合较好,表明施加泊松随机断裂成核点的数值模拟方法是合理的。     

A novel 3D parallel mechanism for the passive motion simulation of the patella-femur-tibia complex

Parallel mechanisms have been exploited for the kinematic modelling of the passive motion, i.e. the motion under virtually unloaded conditions, of the patella-femur-tibia human joint. In particular, a new mechanism is devised in this paper: a 3D model of the patella-femur relative motion is presented which, combined with a previous simplified model of the femur-tibia relative motion, provides a suitable tool for the design of knee prostheses. Although less accurate than a previously presented model of the patella-femur-tibia joint, the new mechanism still replicates passive knee motion quite well and is simpler from a mechanical point of view. Experimental results validate the efficiency of the proposed model.     

How to compute plastic zones of heterogeneous materials: A simple approach using classical continuum and fracture mechanics

This is a short technical paper on how to use classical continuum and fracture mechanics to calculate the plastic zones caused by cracks on heterogeneous or composite materials. As an example, a sample consisting of an α-phase and β-phase is used. A crack is introduced to the sample, and stress is then applied. The plastic zone in front of the crack resulting from the applied stress is then calculated using commercial software. The concept uses two-level modeling: a global model using homogenized stiffness from a unit cell of heterogeneous material and a local model for the α-phase and β-phase. While this paper is written for general purposes, a concrete example using ferrite and martensite is also presented along with the experimental data. General agreement between the model and the experiment is observed. This method eliminates the need for a cumbersome analytical approach.     

Pseudo-Spectral simulation of 1D nonlinear propagation in heterogeneous elastic media

In this work, a 1D Pseudo-Spectral Time Domain (PSTD) algorithm has been developed for solving elastic wave equation in nonlinear heterogeneous solids using FFTs for calculation of the spatial differential operator on staggered grid. The solver uses a staggered fourth order Adams–Bashforth method, by which stress and particle velocity are updated at alternating half time steps, to integrate forward in time. To circumvent wraparound inherent to FFT-based pseudo-spectral simulation, Convolution Perfectly Matched Layer (CPML) boundary condition has been used to eliminate implementation problems linked in classical PML to the introduction in nonlinear elasticity of a time dependent bulk modulus. Different kinds of nonlinear elastic models (quadratic and cubic nonlinearity, Nazarov hysteretic nonlinearity, bi-modular nonlinearity, PM-Space nonlinearity) have been implemented. The present study will focus on the comparison of nonlinear signature (harmonics generation, shock, frequency shift and attenuation) of these different kinds of nonlinearity for rod resonance, shock wave generation. These results are expected to be useful in helping to determine the predominant nonlinear mechanism in a specific experiment.     

Monte Carlo simulation of self-avoiding lattice chains subject to oscillatory shear flow in polymer solution

 This paper has introduced a pseudo-potential in bond-fluctuation model to simulate oscillatory shear flow of multiple self-avoiding chains in three dimensions following our previous work under simple shear flow. The oscillatory flow field was reasonably reproduced by lattice Monte Carlo simulation using this pseudo-potential neglecting hydrodynamic interaction. By sampling the configuration distribution functions, the macroscopic viscoelasticity of semi-concentrated polymer solution was determined. Both Newtonian and non-Newtonian regimes were studied. The complex modulus and dynamic viscosity exhibit a reasonable power relation with oscillatory frequency, which is consistent with present theories and experiments. Consequently, lattice Monte Carlo simulation has been extended to model free-draining self-avoiding multi chains subject to oscillatory shear flow and to investigate associated viscoelasticity on the molecular level. Received: 1 October 1999 Accepted: 19 October 1999     

脆性材料破坏模拟的网格生成算法

脆性材料的破坏过程具有随机性,当前的网格生成算法没有充分考虑脆性材料破坏时裂纹扩展和碎块生成的随机性。在Persson网格生成算法与Delaunay随机网格剖分理论基础上,提出了一种可根据模拟需要动态控制网格品质的网格生成算法。通过对随机分布点的Delauna三角化,生成初始网格,然后将网格体系比拟为桁架结构,网格节点即为桁架节点。桁架节点在虚拟力作用下可动态调整位置,并最终达到整个体系受力平衡。对Persson 算法中的尺寸分布函数和收敛条件进行了修正,从而提高了收敛速度,并适用于任意形状对象的网格剖分。 基于VC++平台开发了算法程序。通过实例对算法进行了验证,表明算法能够满足脆性材料破碎模拟的需要。     

Combined interval analysis - Monte Carlo simulation approach for the analysis of uncertainties in parallel manipulators

The quantification of the impact of uncertainties may increase the reliability and robustness of parallel manipulators. Monte Carlo simulation (MCS) and interval analysis are among the most common techniques used in uncertainty quantification. Interval analysis provides guaranteed performance since the interval evaluation of a function always contains the exact result. Nevertheless, interval analysis estimations are very conservative, frequently yielding overestimated results. Conversely, Monte Carlo Simulation avoids overestimation, but does not provide guaranteed performance. This paper proposes a novel hybrid algorithm combining the best features of interval analysis and Monte Carlo simulation for estimating probabilities of failure in the positioning error of parallel manipulators. A 3RRR manipulator is employed as case-study. The hybrid approach provides information on the bounds (minimum and maximum values) and estimated values of failure probabilities. The simulations herein compare the hybrid approach with pure interval analysis and pure Monte Carlo simulation. The results reveal that the hybrid technique may estimate not only the probability of failure in a continuous region, but also the worst and best case probabilities, much faster than interval analysis based approaches.