Performance of time‐stepping schemes for discrete models in fracture dynamic analysis

In this work we present a critical evaluation of different time integration schemes within the setting of non‐linear dynamic analysis of brittle fracture problem represented by a discrete model. The discrete model of this kind consists of Voronoi cells representing the grains of a heterogeneous structure, which are interconnected by cohesive forces modelled by beam‐like links capable of taking properly into account both brittle dynamic fracture and large displacement of a still connected pack of grains that might split from the structure. The brittle behaviour of cohesive links requires that the dynamic analysis of such a model be treated with care, and the best possible integration scheme be selected. Four different schemes are explored and compared in application to a dynamic traction test, including Newmark explicit and implicit schemes, HHT‐α scheme and energy‐decaying scheme. Copyright © 2005 John Wiley & Sons, Ltd.     

用三维离散元实现混凝土Ⅰ型断裂模拟

为了进行从连续介质到非连续介质转化的数值模拟,实现结构破坏全过程的仿真分析,本文基于三维变形体离散元法,分别采用弥散的旋转裂缝模型和分离裂缝模型分析了混凝土、岩石等准脆性材料的受拉开裂行为。通过单轴拉伸算例和对比三点弯梁试验的数值计算与试验结果,说明在确定的断裂能情况下,两种断裂模型均能够合理预测结构的荷载位移响应;并且基于离散单元法的分离裂缝模型不仅可以获得裂缝开裂过程荷载位移全曲线,而且可以实现系统部分或完全失效溃决过程的仿真分析。     

A fractional rate‐dependent cohesive‐zone model

This paper presents a novel formulation of a hereditary cohesive zone model able to effectively capture rate‐dependent crack propagation along a defined interface, over a wide range of applied loading rates and with a single set of seven input parameters only, as testified by the remarkable agreement with experimental results in the case of a double cantilever beam made of steel adherends bonded along a rubber interface. The formulation relies on the assumption that the measured fracture energy is the sum of a rate‐independent ‘rupture’ energy, related to the rupture of primary bonds at the atomic or molecular level, and of additional dissipation caused by other rate‐dependent dissipative mechanisms present in the material and occurring simultaneously to rupture. The first contribution is accounted for by introducing a damage‐type internal variable, whose evolution follows a rate‐independent law for consistency with the assumption of rate independence of the rupture energy. To account for the additional dissipation, a fractional‐calculus‐based linear viscoelastic model is used, because for many polymers, it is known to capture the material response within an extremely wide range of strain rates much more effectively than classic models based on an exponential kernel. To the authors' knowledge, this is the first application of fractional viscoelasticity to the simulation of fracture. © 2015 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.     

基于损伤演化的杆件断裂模拟方法及其应用

建立了钢材的弹塑性损伤本构模型,将其与有限单元法中分段纤维梁模型结合,实现了杆件不同位置损伤程度评估,根据评估结果实时修正杆件特性,建立了杆件断裂数值模拟方法。将所建立的杆件断裂模拟方法应用于单调递增荷载作用下的梁系结构和地震作用下的单层网壳结构,结果表明,所建立方法可有效地模拟杆件断裂;杆件断裂突然,与其相连的节点位移及杆件应变能突增,可能导致与其相连杆件破坏;强震作用下单层网壳结构可能发生杆件断裂,导致结构局部刚度降低,引起结构倒塌破坏。     

Comparison of a phase‐field model and of a thick level set model for brittle and quasi‐brittle fracture

This paper provides a comparison between one particular phase‐field damage model and a thick level set (TLS) damage model for the simulation of brittle and quasi‐brittle fractures. The TLS model is recasted in a variational framework, which allows comparison with the phase‐field model. Using this framework, both the equilibrium equations and the damage evolution laws are guided by the initial choice of the potential energy. The potentials of the phase‐field model and of the TLS model are quite different. TLS potential enforces a priori a bound on damage gradient whereas the phase‐field potential does not. The TLS damage model is defined such that the damage profile fits to the one of the phase‐field model for a beam of infinite length. The model parameters are calibrated to obtain the same surface fracture energy. Numerical results are provided for unidimensional and bidimensional tests for both models. Qualitatively, similar results are observed, although TLS model is observed to be less sensible to boundary conditions. Copyright © 2015 John Wiley & Sons, Ltd.     

Abschätzung von Bruchzähigkeitskennwerten aus der Bruch‐ oder Kerbschlagarbeit

Estimation of Fracture Toughness from the Charpy Fracture Energy By means of a simplistic mechanical model a mathematical relation between the total fracture energy of an edge‐cracked beam under bending and the fracture toughness was derived. Inserting the parameters of the standard Charpy specimens and accounting for the finite root radius and the shallowness of the notch, an equation to calculate fracture toughness from the Charpy fracture energy was obtained. Unlike the commonly used empirical correlation formulas, the presented equation is applicable to any elastic‐plastic material. From its theoretical basis and the underlying assumptions it is expected to be conservative in any case. Although the derivation only holds for the upper‐shelf regime, it also is applicable to the brittle‐to‐ductile transition regime as a lower bound. In this range, the degree of conservatism can be reduced by accounting for the well‐known shift of the transition temperature. Comparison with experimental data confirm these features of the presented formula.     

A fracture framework for Euler–Bernoulli beams based on a full discontinuous Galerkin formulation/extrinsic cohesive law combination

A new full Discontinuous Galerkin discretization of Euler–Bernoulli beam is presented. The main interest of this framework is its ability to simulate fracture problems by inserting a cohesive zone model in the formulation. With a classical Continuous Galerkin method, the use of the cohesive zone model is difficult because inserting a cohesive element between bulk elements is not straightforward. On one hand if the cohesive element is inserted at the beginning of the simulation, there is a modification of the structure stiffness and on the other hand inserting the cohesive element during the simulation requires modification of the mesh during computation. These drawbacks are avoided with the presented formulation as the structure is discretized in a stable and consistent way with full discontinuous elements and inserting cohesive elements during the simulation becomes straightforward. A new cohesive law based on the resultant stresses (bending moment and membrane) of the thin structure discretization is also presented. This model allows propagating fracture while avoiding through‐the‐thickness integration of the cohesive law. Tests are performed to show that the proposed model releases, during the fracture process, an energy quantity equal to the fracture energy for any combination of tension‐bending loadings. Copyright © 2010 John Wiley & Sons, Ltd.     

Intralaminar fracture mechanism in unidirectional CFRP composites — part II: analysis

Three kinds of representative carbon fiber reinforced unidirectional composite materials are used, and their intralaminar fracture behavior is investigated by using the double-cantilever beam (DCB) specimen with a simultaneous acoustic emission measuring. In Part I, the experimental results on the crack propagation, the bridging fibers, the intralaminar fracture toughness acoustic emission characteristics and microscope observations were obtained. Here, we use a bridging fiber model to analyze the debonding force acting on a bridging fiber and try to estimate the number of bridging fibers during the crack propagating process. At the same time, the intralaminar fracture toughness is calculated by both the adhesive force model and the finite element analysis. As a result, it is found that the intralaminar fracture toughness without the bridging fibers will have a constant value during the crack propagation, but it increases greatly when bridging fibers exist. It is clear that the bridging fibers play an important role in the intralaminar fracture toughness. The debonding forces acting on the bridging fibers and the number of bridging fibers are obtained. Furthermore, the quantitative estimation of the increment of the intralaminar fracture toughness contributed by bridging fibers is made according to the adhesive force model and it is comparable with the results obtained by the finite element analysis.     

Delamination resistance of GFRP‐epoxy rods with nanoparticle‐ and microparticle‐modified matrix and its correlation with the fracture properties of epoxy nanocomposites

Pultruded glass fiber–reinforced plastic (GFRP) rods are industrially produced for various applications. The mechanical properties of GFRP rods can be tailored, eg, with modified epoxy matrix by adding microsized and nanosized fillers. In this work, we investigated the transferability of the fracture properties of epoxy nanocomposites to GFRP epoxy rods. For this purpose, nanosilica particles, micron‐sized rubber, and micron‐sized calcium carbonate were used as fillers. Firstly, epoxy plates with different combinations of nanosized and micron‐sized fillers were produced and their fracture toughness evaluated. Secondly, using binary and ternary epoxy filler systems as matrices, GFRP rods were produced, and their fracture toughness was determined using a modified testing procedure derived from standard composite double cantilever beam specimens. Lastly, the correlation between fracture properties of epoxy nanocomposites and GFRP rods was analysed. Besides that, the fractographic investigations of both epoxy nanocomposites and GFRP rods were carried out and discussed.     

单丝断裂能量法研究温度对碳纤维/环氧界面性能的影响

针对两种不同上浆剂碳纤维/高温固化环氧树脂体系, 采用基于WND(Wagner-Nairn-Detassis)能量模型的单丝断裂法, 测试分析了从室温到130 ℃范围内单丝复合体系界面断裂能的变化规律, 研究了碳纤维上浆剂对界面耐热性能的影响, 并结合复合材料层板的短梁剪切性能, 分析了微观和宏观界面性能的关联性。结果表明: 在测试温度范围内, 碳纤维/环氧体系的界面断裂能随温度升高呈先下降而后基本不变的趋势, 去除上浆剂后界面断裂能及其随温度的变化程度与未去除上浆剂的情况存在差异, 说明上浆剂对界面耐热性有重要作用。碳纤维/环氧树脂层板层间剪切强度随温度升高线性下降, 与界面断裂能的变化规律不一致, 这与两种测试方法的原理及界面破坏位置的不同有关。      

Polygon‐based contact resolution for superquadrics

The representation of discrete objects in the discrete element modelling is a fundamental issue, which has a direct impact on the efficiency of discrete element implementation and the dynamic behaviour of particulate systems. Disks and spheres are the most commonly used geometric shapes due to their geometric simplicity and computational efficiency, but they are unable to provide resistance to rolling motion. For this reason, some non‐circular/spherical objects, such as polygons/polyhedrons, superquadrics, or the clustering of disks/spheres to form irregular shapes, are introduced. When superquadrics are used as discrete elements, the bottleneck of contact resolution is associated with the searching for intersections of two non‐linear functions, which is a very expensive operation and may sometimes fail in finding the solution. In this work, an efficient and robust algorithm is proposed for contact resolution of 2D superquadrics, in which any superquadric is approximated with a convex polygon through adaptive sampling; then by clipping two polygons, an efficient linear algorithm is performed to search for intersections and overlap area of the polygons; the contact forces and directions are determined by employing a newly established corner/corner contact model. It is important to highlight that the proposed methodology can also be extended to general non‐circular discrete object cases. The performance of the algorithm is demonstrated via numerical examples. Copyright © 2005 John Wiley & Sons, Ltd.     

Experimental Investigations of Fracture Process in Concrete by Means of X‐ray Micro‐computed Tomography

The paper describes investigation results on fracture in notched concrete beams under quasi‐static three‐point bending by the X‐ray micro‐computed tomography. The two‐dimensional (2D) and three‐dimensional image procedures were used. Attention was paid to width, length, height and shape of cracks along beam depth. In addition, the displacements on the surface of concrete beams during the deformation process were measured with the 2D digital image correlation technique in order to detect strain localisation before a discrete crack occurred. The 2D fracture patterns in beams were numerically simulated with the finite‐element method using an isotropic damage constitutive model enhanced by a characteristic length of micro‐structure. Concrete was modelled as a random heterogeneous four‐phase material composed of aggregate, cement matrix, interfacial transitional zones and air voids. The advantages of the X‐ray micro‐computed tomography were outlined.     

Phase‐field modeling of ductile fracture at finite strains: A robust variational‐based numerical implementation of a gradient‐extended theory by micromorphic regularization

This work provides a robust variational‐based numerical implementation of a phase field model of ductile fracture in elastic–plastic solids undergoing large strains. This covers a computationally efficient micromorphic regularization of the coupled gradient plasticity‐damage formulation. The phase field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modeling with geometric features rooted in fracture mechanics. It has proven immensely successful with regard to the analysis of complex crack topologies without the need for fracture‐specific computational structures such as finite element design of crack discontinuities or intricate crack‐tracking algorithms. The proposed gradient‐extended plasticity‐damage formulation includes two independent length scales that regularize both the plastic response as well as the crack discontinuities. This ensures that the damage zones of ductile fracture are inside of plastic zones or vice versa and guarantees on the computational side a mesh objectivity in post‐critical ranges. The proposed setting is rooted in a canonical variational principle. The coupling of gradient plasticity to gradient damage is realized by a constitutive work density function that includes the stored elastic energy and the dissipated work due to plasticity and fracture. The latter represents a coupled resistance to plasticity and damage, depending on the gradient‐extended internal variables that enter plastic yield functions and fracture threshold functions. With this viewpoint on the generalized internal variables at hand, the thermodynamic formulation is outlined for gradient‐extended dissipative solids with generalized internal variables that are passive in nature. It is specified for a conceptual model of von Mises‐type elasto‐plasticity at finite strains coupled with fracture. The canonical theory proposed is shown to be governed by a rate‐type minimization principle, which fully determines the coupled multi‐field evolution problem. This is exploited on the numerical side by a fully symmetric monolithic finite element implementation. An important aspect of this work is the regularization towards a micromorphic gradient plasticity‐damage setting by taking into account additional internal variable fields linked to the original ones by penalty terms. This enhances the robustness of the finite element implementation, in particular, on the side of gradient plasticity. The performance of the formulation is demonstrated by means of some representative examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Combined single and smeared crack model in combined finite‐discrete element analysis

Large‐scale discrete element simulations, combined finite‐discrete element simulations as well as a whole range of related problems, involve a large number of separate bodies that interact with each other and in general deform and fracture. In this context there is a need for a robust fracture algorithm applicable to simultaneous multiple fracturing of large numbers of bodies. In this work a fracture model for both initiation and propagation of mode I loaded cracks in concrete in the context of the combined finite‐discrete element method is reported. The algorithm is based on accurate approximation of experimental stress–strain curves for concrete in tension. Copyright © 1999 John Wiley & Sons, Ltd.     

Fracture mechanics‐based progressive damage modelling of adhesively bonded fibre‐reinforced polymer joints

A quasi‐static progressive damage model for prediction of the fracture behaviour and strength of adhesively bonded fibre‐reinforced polymer joints is introduced in this paper. The model is based on the development of a mixed‐mode failure criterion as a function of a master R‐curve derived from the experimental results obtained from standard fracture mechanics joints. Consequently, the developed failure criterion is crack‐length and mode‐mixity dependent, and it takes into account the contribution of the fibre‐bridging effect. Energy release rate values for adhesively bonded double‐lap joints are obtained by using the virtual crack closure technique method in a finite element model, and the numerically obtained strain energy release rate is compared to the critical strain energy release rate given by the mixed‐mode failure criterion. The entire procedure is implemented in a numerical algorithm, which was successfully used for predicting the strength and R‐curve response of adhesively bonded double‐lap structural joints made of pultruded glass fibre‐reinforced polymers and epoxy adhesives.     

用3D离散元实现Ⅰ/Ⅱ型拉剪混合断裂的模拟

将变形体离散元分别与弥散式旋转裂缝模型和分离式裂缝模型耦合起来,采用上述两种非线性断裂力学模型分析了混凝土、岩石等准脆性材料的Ⅰ/Ⅱ型拉剪混合模式的开裂行为,以实现从连续介质到非连续介质转化的数值模拟。基于变形体离散元方法的界面接触本构关系提出了混凝土Ⅰ/Ⅱ混合型开裂的拉剪分区开裂准则;基于缝面法向开度增大对刚度强度的逐渐折减实现了界面裂缝扩展的模拟。作为数值验证,通过单边切口非对称三点弯梁试验的数值计算与试验结果对比,表明两种断裂力学模型均能够合理预测裂缝的起裂和扩展,在混合形式的荷载条件下,裂缝通常以Ⅰ/Ⅱ型拉剪混合模式起裂,而后以Ⅰ型为控制方式稳定扩展。文末指出,基于离散单元法的分离裂缝模型能够实现系统在外载作用下破坏全过程仿真。     

Geometrically non‐linear damage interface based on regularized strong discontinuity

The contribution of this paper concerns the fracture modelling of an interface with a fixed internal material surface in the context of geometrically non‐linear kinematics. Typical applications are composite laminates and adhesive/frictional joints in general. In the model development, a key feature is the concept of regularized strong discontinuity, which provides a regular deformation gradient within the interface. The deformation gradient within the interface is formulated in a multiplicative structure with a continuous part and a discontinuous part, whereby the interface deformation is interpreted as a transformation between the material damaged configuration and the actual spatial configuration. In analogy with the continuum formulation of hyper‐inelasticity, a constitutive framework is defined for the relation between the induced material traction and the displacement jump vector, which are defined on the material damaged interface configuration. Within this framework, a simple, but yet still representative, model for the delamination problem is proposed on the basis of a damage–plasticity coupling for the interface. The model is calibrated analytically in the large deformation context with respect to energy dissipation in mode I so that a predefined amount of fracture energy is dissipated. The paper is concluded with a couple of numerical examples that display the properties of the interface. Copyright © 2002 John Wiley & Sons, Ltd.     

Finite element analysis of the end notched flexure specimen for measuring mode II fracture toughness

This paper presents a finite element analysis of the End Notched Flexure (ENF) test specimen for Mode II interlaminar fracture testing of composite materials. Virtual crack closure and compliance techniques employed to calculate strain energy release rates from linear elastic two-dimensional analysis indicate that the ENF specimen is a pure Mode II fracture test within the constraints of small deflection theory. Furthermore, the ENF fracture specimen is shown to be relatively insensitive to process induced cracks offset from the laminate midplane. Frictional effects are investigated by including the contact problem in the finite element model. A parametric study investigating the influence of delamination length, span, thickness and material properties is presented to assess the accuracy of beam theory expressions for compliance and strain energy release rate, G_II. Finite element results indicate that data reduction schemes based upon beam theory underestimate G_II by approximately 20–40% for typical unidirectional graphite fiber composite test specimen geometries. Consequently, an improved data reduction scheme which retains the simplicity of beam theory but also includes the accuracy of the finite element method is proposed.     

Use of fine aggregate matrix for computational modeling of low temperature fracture of asphalt concrete

In this study, a discrete element computational model is applied to simulate the fracture behavior of asphalt mixtures at low temperatures. In this model, coarse aggregates are explicitly represented by rigid spherical particles. The bonds that connect these particles represent the fine aggregate matrix (FAM), which is defined as the combination of asphalt binder and fine aggregates. The bending beam rheometer (BBR) tests are performed to determine the strength and Young’s modulus of FAM at low temperatures. The model is then used to simulate the semi-circular bend (SCB) tests on the mixtures. The model is verified by a series of BBR and SCB tests on both conventional and graphite nano-platelet modified asphalt materials. The comparison between the experimental and simulated results indicates that the peak load capacity of the SCB specimens is primarily governed by the tensile strength of the FAM. However, in order to capture the entire load–displacement curve of the SCB specimens, one needs to employ a softening constitutive model of the FAM, which requires the information on its fracture energy. Several experimental methods for measuring the fracture energy of FAM are discussed for future prediction of the complete load–displacement response of asphalt mixtures at low temperatures.