Micromechanical analysis of coupling between anisotropic damage and friction in quasi brittle materials: Role of the homogenization scheme

This paper deals with micromechanical analysis of anisotropic damage and its coupling with friction in quasi brittle materials. The anisotropic model is formulated in the framework of Eshelby-based homogenization methods. The emphasis is put on the study of effects of spatial distribution of microcracks and their interactions. Microcracks closure effects as well as coupling between damage evolution and frictional sliding on closed cracks lips are taken into account. The interaction of sliding and damage evolution is addressed by performing a global thermodynamic analysis on two macroscopic criteria established in the paper. The role of the homogenization scheme is discussed in detail through various applications.     

Micromechanical analysis of damage in saturated quasi brittle materials

In this paper, we propose a micromechanical analysis of damage and related inelastic deformation in saturated porous quasi brittle materials. The materials are weakened by randomly distributed microcracks and saturated by interstitial fluid with drained and undrained conditions. The emphasis is put on the closed cracks under compression-dominated stresses. The material damage is related to the frictional sliding on crack surface and described by a local scalar variable. The effective properties of the materials are determined using a linear homogenization approach, based on the extension of Eshelby’s inclusion solution to penny shaped cracks. The inelastic behavior induced by microcracks is described in the framework of the irreversible thermodynamics. As an original contribution, the potential energy of the saturated materials weakened by closed frictional microcracks is determined and formulated as a sum of an elastic part and a plastic part, the latter entirely induced by frictional sliding of microcracks. The influence of fluid pressure is accounted for in the friction criterion through the concept of local effective stress at microcracks. We show that the Biot’s effective stress controls the evolution of total strain while the local Terzaghi’s effective stress controls the evolution of plastic strain. Further, the frictional sliding between crack lips generates volumetric dilatancy and reduction in fluid pressure. Applications of the proposed model to typical brittle rocks are presented with comparisons between numerical results and experimental data in both drained and undrained triaxial tests.     

A new discontinuous FE formulation for crack path prediction in brittle solids

In this paper a new finite element (FE) formulation to simulate embedded strong discontinuity for the study of the fracture process in brittle or quasi-brittle solids is presented. A homogeneous discontinuity is considered to be present in a cracked finite element with the possibility to take into account the opening and the sliding phenomena which can occur across the crack faces. In such a context a new simple stress-based implementation of the discontinuous displacement field is proposed by an appropriate stress field correction introduced at the Gauss points level in order to simulate, in a fashion typical of an elastic–plastic classical FE formulation, the mechanical effects of the bridging and friction stresses due to crack faces opening and sliding which can occur during the loading–unloading process structural component or solid being analysed. The proposed formulation does not need to introduce special or modified shape functions to reproduce discontinuous displacement field but simply relaxes the stress field in an appropriate fashion. Both linear elastic and elastic–plastic behaviour of the non-cracked material can be considered. Several 2D problems are presented and solved by the proposed procedure in order to predict load–displacement curves of brittle structures as well as crack patterns that develop during the loading process.The proposed discontinuous new FE formulation gives the advantages to be simple, computationally economic and to keep internal continuity of the numerical FE model; furthermore the developed algorithm can be easily implemented in standard FE programs as a standard plasticity model.     

A boundary element application for mixed modeloading idealized sawtooth fracture surface

In this paper, a 2-D elastic-plastic BEM formulation predicting the reduced mode IIand the enhanced mode I stress intensity factors are presented. The dilatant boundary conditions (DBC) are assumed to be idealized uniform sawtooth crack surfaces and an effective Coulombsliding law. Three types of crack face boundary conditions, i.e. (1) BEM sawtooth model-elasticcenter crack tip; (2) BEM sawtooth model-plastic center crack tip; and (3) BEM sawtoothmodel-edge crack with asperity wear are presented. The model is developed to attempt todescribe experimentally observed non-monotonic, non-linear dependence of shear crack behavioron applied shear stress, superimposed tensile stress, and crack length. The asperity sliding isgoverned by Coulombs law of friction applied on the inclined asperity surface which hascoefficient of friction μ. The traction and displacement Greens functions which derive fromNaviers equations are obtained as well as the governing boundary integral equations for an infiniteelastic medium. Accuracy test is performed by comparison stress intensity factors of the BEMmodel with analytical solutions of the elastic center crack tip. The numerical results show thepotential application of the BEM model to investigate the effect of mixed mode loading problemswith various boundary conditions and physical interactions.     

A rate-dependent damage model for brittle materials based on the dominant crack

A rate-dependent, continuum damage model is developed for brittle materials under dynamic loading. This model improves on the approach (ISOSCM) of [Addessio, F.L., Johnson, J.N., 1990. A constitutive model for the dynamic response of brittle materials. Journal of Applied Physics 67, 3275–3286] in several respects. (1) A new damage surface is found by applying the generalized Griffith instability criterion to the dominant crack (having the most unstable orientation), rather than by averaging the instability condition over all crack orientations as done previously. The new surface removes a discontinuity in the damage surface in ISOSCM when the pressure changes sign. (2) The strain due to crack opening is more consistent with crack mechanics, with only the tensile principal stresses contributing to the crack opening strain. This is achieved by incorporating a projection operator in the equation for the crack opening strain. One consequence of incorporating the projection operator is a prediction of shear dilatancy, which is not accounted for in ISOSCM. (3) The evolution of damage, which is based on the energy-release rate for the dominant crack, has a physical basis, whereas in the previous approach the damage growth rate was assumed to be an exponential function of the distance from the stress state to the damage surface without specific physical justification.An implicit algorithm has been developed so that a larger time step can be used than with the explicit algorithm used in ISOSCM. The numerical results of a silicon carbide (SiC) ceramic under several loading paths (hydrostatic tension/compression, uniaxial strain, uniaxial stress, and shear) and strain rates are presented to illustrate the main features of the model.     

A novel model of delamination bridging via Z-pins in composite laminates

A new micro-mechanical model is proposed for describing the bridging actions exerted by through-thickness reinforcement on delaminations in prepreg based composite materials, subjected to a mixed-mode (I–II) loading regime. The model applies to micro-fasteners in the form of brittle fibrous rods (Z-pins) inserted in the through-thickness direction of composite laminates. These are described as Euler–Bernoulli beams inserted in an elastic foundation that represents the embedding composite laminate. Equilibrium equations that relate the delamination opening/sliding displacements to the bridging forces exerted by the Z-pins on the interlaminar crack edges are derived. The Z-pin failure meso-mechanics is explained in terms of the laminate architecture and the delamination mode. The apparent fracture toughness of Z-pinned laminates is obtained from as energy dissipated by the pull out of the through-thickness reinforcement, normalised with respect to a reference area. The model is validated by means of experimental data obtained for single carbon/BMI Z-pins inserted in a quasi-isotropic laminate.     

混凝土复合型裂纹扩展的非局部近场动力学建模分析

基于非局部近场动力学理论,构建了修正的能反映混凝土宏观拉压异性和断裂特征的近场动力学本构模型,开发了相应的离散、加载和时间积分算法,实现典型混凝土构件中复合型裂纹扩展过程模拟。在物质点对尺度上定义局部损伤并考虑物质点对的相对转动,通过求解时空微-积分方程实现裂纹的自然萌生与扩展,避免裂尖不连续带来的求解奇异性、网格依赖性和网格重构以及常规近场动力学本构模型的泊松比限制。通过含单边和双边初始裂纹四点剪切混凝土梁裂纹扩展破坏全过程模拟,得到破坏形态、破坏荷载以及完整的荷载-裂纹开口滑移曲线,并与试验和其他数值模拟结果对比,验证了模型的精确性和算法的稳定性。     

DEM与FEM动态耦合算法研究

离散单元法作为一种有效的数值分析方法,能够模拟脆性材料的裂纹扩展及碎片飞散等破坏特性,但是无法从根本上克服计算效率低下的诟病;传统有限单元法具有计算高效稳定的优点,却难以描述脆性材料冲击破坏过程中连续体向非连续体的转化。本文首先提出一种基于罚函数法的改进型离散单元和有限单元耦合方法,以提高耦合分析精度。在此基础上提出了动态耦合算法:即在初始阶段,模型全部为有限单元,当局部即将发生破坏时,仅使即将发生破坏的有限单元及相邻单元自动转化为离散单元,在离散单元区域研究破坏问题。这种算法充分利用有限单元法计算高效的优点,同时最大限度克服了离散单元法计算效率的不足。最后,通过两个简单算例验证了改进型耦合算法和动态耦合算法的有效性。     

Modelling of progressive interface failure under combined normal compression and shear stress

The present work is concerned with an analysis of progressive interface failure under normal compressive stress and varying shear stress using the cohesive crack model. The softening model is assumed and frictional linear stress at contact is accounted for. A monotonic loading in anti-plane shear of an elastic plate bonded to a rigid substrate is considered. An analytical solution is obtained by neglecting the effect of minor shear stress component in the plate. The elastic and plate interface compliances are included into the analysis. Three types of solutions are distinguished in the progressive delamination analysis, namely short, medium and long plate solutions. The analysis of quasi-static progressive delamination process clarifies the character of critical points and post-critical response of the plate. The analytical solution provides a reference benchmark for numerical algorithms of analysis of progressive interface delamination. The case of a rigid–softening interface was treated in a companion paper, where also cyclic loading was considered.     

Fracturing damage process in dynamic split experiments of a brittle glass

In this study, the 3-dimensional discrete element method is firstly introduced to explain the fracturing damage process of the dynamic split experiment of a special brittle glass ZnS. The corresponding dynamic split experiment is also performed by using the split Hopkinson pressure bar. Then the numerical results correspond closely to those obtained by experiments, and the fracturing damage mode shows that the sample under high strain rate loading would crack along vertical diameter in the band region between two loading edges, which differs from the static damage mode. Furthermore, by comparing a group of contrast numerical tests, the numerical results prove that loading area upon the top side of samples would influence the fracture mode of dynamic split experiments, which indicates that the narrow loading plane is better.     

Modelling of quasi-brittle behaviour: a discrete approach coupling anisotropic damage growth and frictional sliding

The paper provides development of the model of anisotropic damage by microcracking proposed by Bargellini et al. 2006. This model is based on a discrete approach, which introduces a finite set of microcrack densities associated with fixed directions. This approach avoids inconveniences encountered when using a single second order tensor damage variable D (non uniqueness of the free energy) and strain decomposition into positive and negative parts (spurious dissipation at crack closure). Frictional sliding on closed microcracks is introduced as an additional dissipative mechanism; it is represented by a second order sliding variable in each damage direction. Corresponding sliding criteria and non-associated sliding evolution laws, formulated in the strain space for the model coherence, permit to account for hysteretic phenomena. Unilateral effect is taken into account; Young's and shear moduli are correctly restored at microcrack closure. The crucial requirements of continuity of the energy and of stress–strain response are ensured through relevant conditions on parameters and sliding variables values at opening-closure. The discrete approach, associated with some hypotheses concerning damage evolution, permits to couple damage and dissipative sliding. The pertinence of the proposed theory is illustrated by simulating first elastic properties at constant damage, then by considering a specific loading path involving both damage and friction evolutions.     

基于承载接触特性的直齿面齿轮滑动摩擦啮合效率研究

为了解决直齿面齿轮滑动摩擦啮合效率的问题,基于弹性流体动力润滑理论,提出了一种计算直齿面齿轮啮合效率的方法.首先,运用轮齿接触分析(TCA)和轮齿承载接触分析技术(LTCA)对直齿面齿轮承载啮合过程进行数值仿真;其次,运用非牛顿准稳态热弹流理论建立滑动摩擦系数的计算模型,从而建立直齿面齿轮啮合效率的计算模型,最后分析了输入扭矩、转速等对啮合效率的影响.结果表明:滑动摩擦系数是影响齿轮啮合效率的重要因素;齿面不同位置滑动摩擦系数也不相同;滑动摩擦系数受输入转速、输入扭矩的影响.该方法为直齿面齿轮的进一步优化计算提供一定的理论依据.     

An enhanced beam-theory model of the mixed-mode bending (MMB) test—Part II: Applications and results

The paper presents an enhanced beam-theory (EBT) model of the mixed-mode bending (MMB) test, whereby the specimen is considered as an assemblage of two sublaminates partly connected by an elastic–brittle interface. Analytical expressions for the compliance, energy release rate, and mode mixity are deduced. A compliance calibration strategy enabling numerical or experimental evaluation of the interface elastic constants is also presented. Furthermore, analytical expressions for the crack length correction parameters—analogous to those given by the corrected beam-theory (CBT) model for unidirectional laminated specimens—are furnished for multidirectional laminated specimens, as well. Lastly, an example application to experimental data reduction is presented.     

Computational crack path prediction

A computer program has been developed for the numerical prediction of curved crack growth paths under proportional loading conditions. The numerical prediction is performed by the step-by-step method in cooperation with the stress analysis ahead of the crack tip and the determination of the curved increment of the crack growth. The stress analysis is performed by the method of superposition of analytical and finite-element solutions, and the results are then utilized to determine the coefficients of the analytical expression of the curved crack path obtained by the first order perturbation method.The first numerical example is given for the crack path prediction in DCB-type specimen, where we often observe abrupt crack curving. Computational prediction is performed by introducing slight and small initial branching at the original crack tip. Within few steps of numerical calculations unstable crack curving is obtained, and the predicted path shows extremely good agreement with the experimentally measured path. The second numerical prediction is made for an edge crack approaching a circular hole, which may be considered as a crack arrester. In the present case the effect of the initially introduced slight kink diminishes with increasing crack length. The crack turns back to the original direction, resulting arrest at the hole.     

移动接触下求解二维闭合裂纹的双重迭代边界元法

使用子域边界元法对受移动接触弹性体作用下的二维闭合裂纹问题进行了数值计算。由于两弹性体的接触界面和裂纹表面的接触范围的大小和接触状态事先是未知的 ,对此 ,在两个接触表面同时采用迭代的方法进行了求解。在裂纹的每个裂尖上都采用了四分之一的奇异单元以保证裂尖位移场和应力场奇异性的满足。用我们编制的二维裂纹问题程序对一些中心裂纹问题进行了计算 ,计算结果与经典断裂力学的理论值比较吻合。在无摩擦的条件下 ,对一些具有不同角度且受移动接触弹性体作用下的闭合裂纹问题进行了数值计算 ,得到了一些耦合作用下的应力强度因子的计算结果     

An interacting micro-crack damage model for failure of brittle materials under compression

A model is developed for brittle failure under compressive loading with an explicit accounting of micro-crack interactions. The model incorporates a pre-existing flaw distribution in the material. The macroscopic inelastic deformation is assumed to be due to the nucleation and growth of tensile “wing” micro-cracks associated with frictional sliding on these flaws. Interactions among the cracks are modeled by means of a crack-matrix-effective-medium approach in which each crack experiences a stress field different from that acting on isolated cracks. This yields an effective stress intensity factor at the crack tips which is utilized in the formulation of the crack growth dynamics. Load-induced damage in the material is defined in terms of a scalar crack density parameter, the evolution of which is a function of the existing flaw distribution and the crack growth dynamics. This methodology is applied for the case of uniaxial compression under constant strain rate loading. The model provides a natural prediction of a peak stress (defined as the compressive strength of the material) and also of a transition strain rate, beyond which the compressive strength increases dramatically with the imposed strain rate. The influences of the crack growth dynamics, the initial flaw distribution, and the imposed strain rate on the constitutive response and the damage evolution are studied. It is shown that different characteristics of the flaw distribution are dominant at different imposed strain rates: at low rates the spread of the distribution is critical, while at high strain rates the total flaw density is critical.     

接触面损伤演化过程的数值模型

结合接触面在细观尺度上的非均匀性及其损伤演化规律,提出了一种可以考虑接触面损伤演化过程的数值模型.数值模拟得到的剪切应力-剪切位移关系表明:随着剪切位移的增加,接触面上产生的损伤单元导致曲线斜率逐渐降低;当接触面被完全剪断后,模型中的剪切应力出现了一定程度的下降,此后的剪应力在摩擦力的作用下基本保持不变.模型所能承受的...     

Hydride-induced embrittlement and fracture in metals—effect of stress and temperature distribution

A mathematical model for the hydrogen embrittlement of hydride forming metals has been developed. The model takes into account the coupling of the operating physical processes, namely: (i) hydrogen diffusion, (ii) hydride precipitation, (iii) non-mechanical energy flow and (iv) hydride/solid-solution deformation. Material damage and crack growth are also simulated by using de-cohesion model, which takes into account the time variation of energy of de-cohesion, due to the time-dependent process of hydride precipitation. The bulk of the material, outside the de-cohesion layer, is assumed to behave elastically. The hydrogen embrittlement model has been implemented numerically into a finite element framework and tested successfully against experimental data and analytical solutions on hydrogen thermal transport (in: Wunderlich, W. (Ed.), Proceedings of the European Conference on Computational Mechanics, Munich, Germany, 1999, J. Nucl. Mater. (2000a) 279 (2-3) 273). The model has been used for the simulation of Zircaloy-2 hydrogen embrittlement and delayed hydride cracking initiation in (i) a boundary layer problem of a semi-infinite crack, under mode I loading and constant temperature, and (ii) a cracked plate, under tensile stress and temperature gradient. The initial and boundary conditions in case (ii) are those encountered in the fuel cladding of light water reactors, during operation. The effects of near-tip stress intensification as well as of temperature gradient on hydride precipitation and material damage have been studied. The numerical simulation predicts hydride precipitation at a small distance from the crack-tip. When the remote loading is sufficient, the near-tip hydrides fracture. Thus a microcrack is generated, which is separated from the main crack by a ductile ligament, in agreement with experimental observations.