Prediction of fragment size and ejection distance of masonry wall under blast load using homogenized masonry material properties

The fragment hazard resulting from a nearby explosion is a major concern in the design of structures which may be subjected to blast loads. This paper presents a predictive method based on the theories of continuum damage mechanics and mechanics of micro-crack development, and numerical simulation to determine the probabilistic fragment size distribution and the launch distances. Theoretical derivations are presented to calculate fragment distribution. The fragmentation process is modeled according to the crack initiation and propagation, which depend on the material damage levels and are estimated using continuum damage mechanics theory. The proposed method involves two steps. First a finite element model is developed to estimate the material damage, fragment distribution and the ejection velocity. Then a simple algorithm is used to predict the fragment trajectory and the launch distance based on the fragment size and the ejection velocity. A masonry wall is used as an example in this study. The wall is modeled with both the distinctive consideration of the brick and mortar material properties and the homogenized masonry material properties. The reliability and efficiency of using the homogenized masonry material model in predicting the masonry wall damage and fragmentation are proven. The program AUTODYN is used in this study to conduct the numerical simulations with the proposed models linked to it as user subroutines. The numerical results indicate that the masonry fragments approximately follow the Weibull distribution, which is consistent with some empirical fragment distributions. The proposed method avoids using erosion technique, which inevitably results in a loss of fragment mass, and avoids discretizing the structure into particles or predefining the failure planes, which may lead to unrealistic prediction of damage propagation and evolution and therefore inaccurate fragmentation process and fragment size distributions.     

Penalty function method for combined finite–discrete element systems comprising large number of separate bodies

Large‐scale discrete element simulations, the combined finite–discrete element method, DDA as well as a whole range of related methods, involve contact of a large number of separate bodies. In the context of the combined finite–discrete element method, each of these bodies is represented by a single discrete element which is then discretized into finite elements. The combined finite–discrete element method thus also involves algorithms dealing with fracture and fragmentation of individual discrete elements which result in ever changing topology and size of the problem. All these require complex algorithmic procedures and significant computational resources, especially in terms of CPU time. In this context, it is also necessary to have an efficient and robust algorithm for handling mechanical contact. In this work, a contact algorithm based on the penalty function method and incorporating contact kinematics preserving energy balance, is proposed and implemented into the combined finite element code. Copyright © 2000 John Wiley & Sons, Ltd.     

Biaxial fragmentation of thin silicon oxide coatings on poly(ethylene terephthalate)

Crack patterns of 53 nm and 103 nm thick silicon oxide coatings on poly(ethylene terephthalate) films are analyzed under equibiaxial stress loading, by means of a bulging cell mounted under an optical microscope with stepwise pressurization of film specimens. The biaxial stress and strain are modeled from classical elastic membrane equations, and an excellent agreement is obtained with a finite element method. In the large pressure range, the derivation of the biaxial strain from force equilibrium considerations are found to reproduce accurately the measured data up to 25% strain. The examination of the fragmentation process of the coating under increasing pressure levels reveals that the crack onset strain of the oxide coating is similar to that measured under uniaxial tension. The fragmentation of the coating under biaxial tension is also characterized by complex dynamic phenomena which image the peculiarities of the stress field, resulting in considerable broadening of the fragment size distribution. The evolution of the average fragment area as a function of biaxial stress in the early stages of the fragmentation process is analyzed using Weibull statistics to describe the coating strength.     

利用无网格方法分析钢筋混凝土梁开裂问题

裂缝处理一直是混凝土数值分析中的一个难点,而无网格方法由于不需要单元网格,非常适用于分析断裂问题,因此可以考虑将其引入混凝土断裂分析领域。首先将混凝土中的裂缝分为微观裂缝和宏观裂缝,对于微观裂缝仍然使用传统的弥散裂缝模型;而宏观裂缝则利用无网格方法可以非常方便地调整节点分布的特点,通过增加裂缝节点和裂面边界的方法加以模拟。并给出了宏观裂缝产生,扩展的具体模拟方法。通过算例表明,利用提出的方法,可以较准确地分析混凝土宏观裂缝的产生、扩展以及裂缝宽度变化,得到一些传统有限元方法难以得到的计算结果。     

Cracking node method for dynamic fracture with finite elements

A new method for modeling discrete cracks based on the extended finite element method is described. In the method, the growth of the actual crack is tracked and approximated with contiguous discrete crack segments that lie on finite element nodes and span only two adjacent elements. The method can deal with complicated fracture patterns because it needs no explicit representation of the topology of the actual crack path. A set of effective rules for injection of crack segments is presented so that fracture behavior beginning from arbitrary crack nucleations to macroscopic crack propagation is seamlessly modeled. The effectiveness of the method is demonstrated with several dynamic fracture problems that involve complicated crack patterns such as fragmentation and crack branching. Copyright © 2008 John Wiley & Sons, Ltd.     

Modelling cohesive crack growth using a two-step finite element-scaled boundary finite element coupled method

A two-step method, coupling the finite element method (FEM) and the scaled boundary finite element method (SBFEM), is developed in this paper for modelling cohesive crack growth in quasi-brittle normal-sized structures such as concrete beams. In the first step, the crack trajectory is fully automatically predicted by a recently-developed simple remeshing procedure using the SBFEM based on the linear elastic fracture mechanics theory. In the second step, interfacial finite elements with tension-softening constitutive laws are inserted into the crack path to model gradual energy dissipation in the fracture process zone, while the elastic bulk material is modelled by the SBFEM. The resultant nonlinear equation system is solved by a local arc-length controlled solver. Two concrete beams subjected to mode-I and mixed-mode fracture respectively are modelled to validate the proposed method. The numerical results demonstrate that this two-step SBFEM-FEM coupled method can predict both satisfactory crack trajectories and accurate load-displacement relations with a small number of degrees of freedom, even for crack growth problems with strong snap-back phenomenon. The effects of the tensile strength, the mode-I and mode-II fracture energies on the predicted load-displacement relations are also discussed.     

Anisotropic dynamic damage and fragmentation of rock materials under explosive loading

This paper describes the development of a constitutive model for predicting dynamic anisotropic damage and fragmentation of rock materials under blast loading. In order to take account of the anisotropy of damage, a second rank symmetric damage tensor is introduced in the present model. Based on the mechanics of microcrack nucleation, growth and coalescence, the evolution of damage is formulated. The model provides a quantitative method to estimate the fragment distribution and fragment size generated by crack coalescence in the dynamic fragmentation process. It takes account of the experimental facts that a brittle rock material does not fail if the applied stress is lower than its static strength and certain time duration is needed for fracture to take place when it is subjected to a stress higher than its static strength. Numerical results are compared with those from independent field tests.     

工程弹塑性断裂随机分析的多维插值方法

为解决用确定性有限元或已有无量纲断裂参数表值时进行弹塑性断裂随机分析的问题，本文研究了利用数值插值方法进行随机参数计算的可行性。以单边裂纹板为例，根据确定性有限元计算的无量纲断裂参数表值，通过多维插值计算获得断裂参数及其对基本随机变量变化率，应用“弹塑性断裂随机分析的工程方法”进行弹塑性断裂随机分析。算例表明在敏感区用插值方法计算的结果小于用随机有限元方法计算的结果。     

Extended finite element method and fast marching method for three-dimensional fatigue crack propagation

A numerical technique for planar three-dimensional fatigue crack growth simulations is proposed. The new technique couples the extended finite element method (X-FEM) to the fast marching method (FMM). In the X-FEM, a discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modeled by finite elements with no explicit meshing of the crack surfaces. The initial crack geometry is represented by level set functions, and subsequently signed distance functions are used to compute the enrichment functions that appear in the displacement-based finite element approximation. The FMM in conjunction with the Paris crack growth law is used to advance the crack front. Stress intensity factors for planar three-dimensional cracks are computed, and fatigue crack growth simulations for planar cracks are presented. Good agreement between the numerical results and theory is realized.     

Finite element simulation of ring expansion and fragmentation: The capturing of length and time scales through cohesive models of fracture

The expanding ring test of Grady and Benson (1983) is taken as a convenient yet challenging validation problem for assessing the fidelity of cohesive models in situations involving ductile dynamical fracture. Attention has been restricted to 1100-0 aluminum samples. Fracture has been modelled by recourse to an irreversible cohesive law embedded into cohesive elements. The finite element model is three-dimensional and fully Lagrangian. In order to limit the extent of deformation-induced distortion, we resort to continuous adaptive remeshing. The cohesive behavior of the material is assumed to be rate independent and, consequently, all rate effects predicted by the calculations are due to inertia and the rate dependency in plastic deformation. The numerical simulations are revealed to be highly predictive of a number of observed features, including: the number of dominant and arrested necks; the fragmentation patterns; the dependence of the number of fragments and the fracture strain on the expansion speed; and the distribution of fragment sizes at fixed expansion speed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Fracture and fragmentation of ice: a fractal analysis of scale invariance

The fracture and fragmentation processes of ice are reviewed using fractal concepts. Numerous evidences for the scale invariance of fracture and fragmentation patterns in ice are given, including fracture networks at small (laboratory) and large (geophysical) scales, the distribution of fragment sizes in crushed ice or the distribution of sea ice floe sizes, or self-affine fracture surfaces. These observations strongly argue for the scale invariance of fracture and fragmentation processes in ice. This implies that the fracture mechanisms and the physical parameters revealed at the laboratory scale are still relevant at large scale. However, apparent scale effects can be observed for some parameters if the fractal geometry is ignored or neglected. Scale invariance also implies that the homogenization procedures used in the damage mechanics of ice have to be taken with caution.     

Length scales and size distributions in dynamic fragmentation

The shatter of a cherished wine glass on impact with the kitchen tile, the spallation in the high-energy collision of atomic nuclei, the fragmentation of the Shoemaker-Levi comet on passage of the Roche limit of the Jovian gravitational field, collectively span vast length scales, yet are each examples of dynamic fragmentation with a number of commonalities. In the above examples, as well as many other dynamic fragmentation events, the consequence is the breakage of the body into some number of fragments that are distributed over size. At the heart of a satisfactory theory is the prediction of the number of fragments and the statistical distribution of these fragments over size. A theory based on energy principles is found to provide length scales that govern both the characteristic fragment size and the distribution spread. Fundamental failure and fracture properties of the material are central in determining the nature of the fragment size distribution. Fragment size distributions can range from relatively tight exponential functions to power-law relations spanning a number of decades in fragment size. The fragment distribution and the dynamic fracture processes leading to power-law distributions bear striking similarities to hydrodynamic turbulence. Onset of fracture asymptotes to a range of length scales in which destruction is self-similar and fractal, requiring that consequences, including the fragment size distributions, exhibit a power-law dependence on the length scale. The theory is described and supporting experimental evidence is provided.     

A hybrid multiscale finite element/peridynamics method

A hybrid method is presented that uses a representative volume element-based multiscale finite element technique combined with a peridynamics method for modeling fracture surfaces. The hybrid method dynamically switches from finite element computations to peridynamics based on a damage criterion defined on the peridynamics grid, which is coincident with the nodes of the finite element mesh. Nodal forces are either computed by the finite element method or peridynamics, as appropriate. The multiscale finite element method used here is a representative volume element-based approach so that inhomogeneous local scale material properties can be derived using homogenization. In addition, automatic cohesive zone insertion is used at the local scale to model fracture initiation. Results demonstrate that local scale flaw distributions can alter fracture patterns and initiation times, and the use of cohesive zone insertion can improve accuracy of crack paths.     

A new method for modelling cohesive cracks using finite elements

A model which allows the introduction of displacements jumps to conventional finite elements is developed. The path of the discontinuity is completely independent of the mesh structure. Unlike so‐called ‘embedded discontinuity’ models, which are based on incompatible strain modes, there is no restriction on the type of underlying solid finite element that can be used and displacement jumps are continuous across element boundaries. Using finite element shape functions as partitions of unity, the displacement jump across a crack is represented by extra degrees of freedom at existing nodes. To model fracture in quasi‐brittle heterogeneous materials, a cohesive crack model is used. Numerical simulations illustrate the ability of the method to objectively simulate fracture with unstructured meshes. Copyright © 2001 John Wiley & Sons, Ltd.     

Micromechanical modelling of ductile crack growth in the binder phase of WC–Co

This study focuses on numerical simulation of ductile failure in the Co binder phase of WC–Co hardmetal. The growth of edge cracks under mode I loading is considered. A computational micromechanics approach is taken where the Co binder ligaments are explicitly represented in finite element models. An embedding technique is employed. Crystal plasticity theory is used to represent plastic deformation in the Co ligaments. Crack propagation in the binder is simulated using an element removal technique based on a modified Rice and Tracey model for ductile void growth, and fracture resistance curves are generated. Parameter studies are performed for variations in microstructrual parameters such as numbers of Co ligaments ahead of the crack tip and local Co volume fraction. The importance of thermal residual stresses and finite element mesh density are also investigated.     

Plane stress finite element prediction of mixed-mode rubber fracture and experimental verification

This study demonstrates through experimental validation, that one can predict critical loads of arbitrarily shaped cracked rubber specimens of the mixed-mode type (mode I and II) using a plane stress finite element method and utilizing material constants that characterize the mechanical and fracture properties of SBR (Styrene Butadiene Rubber) material determined from experimental tests on a mode I specimen. Conversely, the finite element method can be used to extract useful critical tearing energy information from complicated, arbitrarily shaped cracked rubber specimens. The predicted critical loads or critical tearing energies for crack growth initiation and final fracture, as well as the crack growth initiation direction are compared to the experimental data with good agreement.     

Finite element method of thermal shock problem in a non-homogeneous isotropic hollow cylinder with two relaxation times

A solution of thermal shock problem of generalized thermoelasticity of a non-homogeneous isotropic hollow cylinder using finite element method is developed. Lord–Shulman and Green–Lindsay for the generalized thermoelasticity model are selected for that purpose which reduces to the classical model by appropriate choice of the parameters. The problem has been solved numerically using a finite element method (FEM). Numerical results for the temperature distribution, displacement, radial stress and hoop stress are represented graphically. The results indicate that the effects of nonhomogeneity are very pronounced. The effects of nonhomogeneity is presented with the three theories.     

Investigation of crack tunneling in ductile materials

Crack tunneling has been commonly observed in crack growth experiments on specimens made of ductile materials such as steel and aluminum alloys. The objective of this study is to investigate the crack tunneling phenomenon and study the effects of crack tunneling on the distribution of several mechanics parameters controlling ductile fracture. Three-dimensional (3D) elastic-plastic finite element analyses of stable tearing experiments involving tunneling fracture are carried out. Two model problems based on stable tearing experiments are considered. The first model problem involves a plate specimen containing a stationary, single-edge crack with a straight or tunneled crack front, under remote mode I loading. In the numerical analyses, the crack tip opening displacement, the von Mises effective stress, the mean stress, the stress constraint and the effective plastic strain around straight and tunneled crack fronts are obtained and compared. It is found that crack tunneling produces significant changes in the stress and deformation fields around the crack front. The second model problem involves a specimen containing a stably growing single-edge crack with a straight or tunneled crack front, under remote mode I loading. Crack growth events with a straight or tunneled crack front are simulated using the finite element method, and the effect of crack tunneling on the prediction of the load-crack-extension response based on a CTOD fracture criterion is investigated.