Finite element analysis of elastic-plastic layered soil using discrete Fourier series expansion

A method is developed which basically concerns how to reduce the necessary storage and cost of solving a set of algebraic equations arising from applying the initial stress technique to the analysis of a uniformly layered elastic-plastic soil. Advantage is taken of the geometrical symmetry of the problem and the consequent symmetry of the stiffness matrix. The nodal values are expanded in a discrete Fourier series so that a two-dimensional problem is reduced to a number of one-dimensional problems.     

Finite element modelling of crack propagation in elastic-plastic media

Materials which are cyclically stressed by sliding indenters often undergo fatigue wear, as surface breaking vertical cracks and subsurface horizontal cracks propagate causing eventual loss of material. In this study, the authors model crack propagation in an elastic-plastic material using finite element techniques, and consider the influence of friction, elasticity, plasticity and degree of penetration on the J-integral at the tip of a vertical crack. Crack propagation directions are estimated using J-integral maxima as the determining variable. It is found that the J-integral values, as a measure of strain energy release rate, can be used to estimate the crack propagation angle. Its main advantage lies in the fact that it considers both modes (I, II) of crack propagation. Using the J-integral values, one finds that, in the absence of friction between the indenter and the material, the vertical crack is equally prone to propagation at both 45 and 135° angles. However, one notices that the vertical crack favours the direction opposite to the direction of rolling for non-zero values of friction, i.e. 135°. The effects of both the crack depth and the crack tip plasticity are also investigated. It is found that any experimental findings suggestive of crack orientations closer to the horizontal in the direction opposite to the sliding direction are probably a result of shallow vertical asperities or higher crack tip plasticity.     

Crack growth initiation in elastic-plastic materials

Abtract The problem of initiation of crack growth in an elastic-plastic material under idealized conditions of plane stress or plane strain was studied. The stress analysis of the cracked plate was performed by using an elastic-plastic finite element computer program based on the J ₂ flow theory of plasticity and a singular crack tip element. A method for determining the critical quantities at the moment of crack growth was presented. The strain energy density failure criterion was used according to which the crack starts to grow when material elements ahead of the crack absorb a critical amount of stored strain energy density which remains after subtracting the amount of irreversible energy dissipated by permanent deformation. The critical stress and the plastic zone at crack initiation were determined. No restrictions regarding the amount of plastic deformation at the crack tip and the form of the stress-strain diagram of the material in tension were imposed in the analysis. Results were given for a center cracked plate specimen for various crack lengths and ultimate stresses of the material in tension. The dependence of the critical stress on these factors and on the idealized conditions of plane stress or plane strain was established.

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Résumé On a étudié le problème de l'amorçage d'une fissure dans un matériau élastoplastique sous des conditions idéales d'état plan de tension ou d'état plan de déformation.L'analyse des contraintes d'une plaque fissurée a été effectuée à l'aide d'un programme de calcul élasto-plastique par éléments finis basé sur une théorie de l'écoulement plastique. On présente une méthode pour déterminer les paramètres critiques au moment de la croissance d'une fissure. On utilise comme critère de rupture la densité d'énergie de déformation. Selon ce critère, la fissure démarre lorsque le volume de matière en avant de celle-ci absorbe une quantité de densité d'énergie de déformation accumulée, sous déduction de la quantité d'énergie irréversiblement dissipée par la déformation permanente. On détermine la contrainte critique et la zone plastique lors de l'amorçage de la fissure. Dans l'analyse, on n'a pas imposé de restriction sur l'étendue de la déformation plastique à l'extrémité de la fissure ou sur le forme du diagramme tension-dilatation de matériau en traction. Les résultats sont fournis dans le cas d'une éprouvette plate à fissure centrale, pour diverse longueurs de fissure et résistances à la rupture du matériau. On établit une dépendance entre la contrainte critique et ces divers facteurs, ainsi qu'avec les conditions idéalisées d'état plan de tension ou de déformation.

     

Finite element analysis of deformation of strain-softening materials

A finite element analysis of strain-softening materials is presented in which the shear band of prescribed thickness is assumed to exist within elements where maximal stress intensity is reached. The incremental stiffness matrix of the element is derived including shear band deformation. Examples presented in the Paper demonstrate that the load-displacement curve and the displacement field are not sensitive to the mesh size used in the solution.     

拉压异性纤维增强树脂基复合材料弹塑性问题的有限元分析

为了对复杂的非线性问题进行便捷求解,首先提出了考虑拉压异性的纤维增强树脂基复合材料统一非线性本构模型;然后,在此基础上进一步导出了本构模型的三维表现形式,以适用于非线性有限元分析工具的开发;随后,利用有限元软件ABAQUS提供的用户自定义子程序UMAT,自编了在二维和三维情况下的弹塑性应力分析程序;最后,应用程序对复合材料单向板和复合材料斜交板在偏轴拉伸/压缩下应力-应变曲线的预测与测试结果进行了比较,探讨了复合材料悬臂梁的弹塑性问题,并分析和比较了有无考虑拉压异性情况下应力分布和挠度响应的差异。结果表明:运用所提出的本构模型对考虑拉压不对称问题的弹塑性变形分析十分有效,这一本构模型有望成为实用数值分析工具,进而指导工程实践。     

Finite element analysis of slow crack growth

A finite element analysis of the process of slow crack growth is made for center-cracked specimens subjected to monotonically increasing load until the point of fast fracture is reached. The formulation is based on an incremental theory of plasticity with isotropic hardening. Numerical results and experimental data are compared. The need and the source of the governing equation for crack growth are discussed.     

Finite element simulation of delamination growth in composite materials using LS-DYNA

In this paper, a modified adaptive cohesive element is presented. The new elements are developed and implemented in LS-DYNA, as a user defined material subroutine (UMAT), to stabilize the finite element simulations of delamination propagation in composite laminates under transverse loads. In this model, a pre-softening zone is proposed ahead of the existing softening zone. In this pre-softening zone, the initial stiffness and the interface strength are gradually decreased. The onset displacement corresponding to the onset damage is not changed in the proposed model. In addition, the critical energy release rate of the materials is kept constant. Moreover, the constitutive equation of the new cohesive model is developed to be dependent on the opening velocity of the displacement jump. The traction based model includes a cohesive zone viscosity parameter (η) to vary the degree of rate dependence and to adjust the maximum traction. The numerical simulation results of DCB in Mode-I is presented to illustrate the validity of the new model. It is shown that the proposed model brings stable simulations, overcoming the numerical instability and can be widely used in quasi-static, dynamic and impact problems.     

Finite element methodology for elastic-plastic fracture problems in three dimensions

The governing finite element system for elastic-plastic analysis of fracture specimens in three dimensions is formulated. The formulation accounts for mixed material hardening, finite strains, finite rotations and plastic incompressibility. The implementation of these aspects into a computational formula is presented and alternative formulations are compared. Small strain theory is recovered as a special case of the present formulation. Analysis is performed on a finite thickness centre-cracked specimen. The grid characteristics required for converged solutions are discussed. The effects of material hardening model and specimen thickness are studied. The local yield state is examined as a gauge of the local deformation processes. The implications for the fracture behaviour of the specimen are discussed. Local surface displacements are compared to experimentally measured yield surfaces. The formulation is shown to predict extremely accurate local deformation in the neighbourhood of the crack front. Contrary to the few three-dimensional fracture studies carried out to date, this analysis concentrates on the local deformation behaviour which ultimately controls fracture. Accurate resolution of this behaviour is essential before meaningful fracture criteria in three dimensions can be developed.     

Finite element analysis of damaged woven fabric composite materials

Since fiber reinforced composite materials have been used in main parts of structures, an accurate evaluation of their mechanical characteristics becomes very important. Due to their anisotropic nature and complicated architecture, it is very difficult to reveal the damage mechanisms of these materials from the results of mechanical tests. Therefore, there is a need to conduct reliable simulations and analytical evaluations. In this paper, the damage behavior of FRP is simulated by finite element analysis using an anisotropic damage model based on damage mechanics. The proposed procedure is applied to an example; the finite element analysis of microscopic damage propagation in woven fabric composites. Experimental tests have been conducted to evaluate the validity of the proposed method. It is recognized that there is a good agreement between the computational and experimental results, and that the proposed simulation method is very useful for the evaluation of damage mechanisms.     

Finite element analysis of rubber-like materials by a mixed model

A Reissner type variational principle is utilized to formulate a mixed finite element model for a finite-strain analysis of Mooney-Rivlin rubber-like materials. An incremental and stationary Lagrangian formulation is adopted. The functional consists of incremental displacements and incremental hydrostatic and distortional stresses as variables. In the finite element formulation the displacements are interpolated in terms of nodal displacements while the two different strss components are approximated independently. The stress parameters for the distortional stresses are eliminated at the element level and the resulting matrix equations for each incremental solution involve the incremental nodal displacements and the average hydrostatic pressure in each element as unknowns. Four-node quadrilateral plane stain elements were used to analyze the inflation of an infinitely long thick-walled cylinder subjected to internal pressure. Both resulting displacements and stresses are found to converge to exact values as the magnitude of the loading increments is decreasing.     

基于参数变分原理的含夹杂Voronoi单元法及非均质材料弹塑性计算

含夹杂Voronoi单元通过在基体单元中引入一任意夹杂, 可以更好地反映非均质材料中微结构特性。基于参数势能和余能原理, 推导了无夹杂和含夹杂Voronoi单元有限元列式, 并在此基础上形成二次规划求解模型。将含夹杂Voronoi单元应用于非均质材料宏观弹塑性性能预测计算中, 分析了非均质材料中夹杂对其宏观等效弹塑性力学性能的影响。数值结果与其它方法所得结果的比较证明了本文中所给出模型的正确性和工程可适用性。      

Finite deformation finite element analyses of tensile growing crack fields in elastic-plastic material

Finite deformation finite element analyses of plane strain stationary and quasi-statically growing crack fields in fully incompressible elastic-ideally plastic material are reported for small-scale yielding conditions. A principal goal is to determine the differences between solutions of rigorous finite deformation formulation and those of the usual small-displacement-gradient formulation, and thereby assess the validity of the (nearly all) extant studies of ductile crack growth that are based on a small-displacement-gradient formulation. The stationary crack case with a significantly blunted tip is studied first; excellent agreement in stress characteristics at all angles about the crack tip and up to a radius of about three times the crack tip opening displacement is shown between Rice and Johnson's [1] approximate analytical solution and our numerical solution. Outside this radius, the numerical results agree very well with Drugan and Chen's [2] small-displacement-gradient analytical characteristics solution in the region of principal plastic deformation. Thus we identify accurate analytical representations for the stress field throughout the plastic zone of a blunted stationary crack. For the growing crack case, the macroscopic difference in crack tip opening profiles between previous small-displacement-gradient solutions and the present results is shown to be negligible, as is the difference in the stress fields in plastic regions. The stress characteristics again agree very well with analytical results of [2]. The numerical results suggest—in agreement with a recent analytical finite deformation study by Reid and Drugan [3]—that it is the finite geometry changes rather than the additional spin terms in the objective constitutive equation that cause any differences between the small-displacement-gradient and the finite deformation solutions, and that such differences are nearly indistinguishable for growing cracks.     

Three-dimensional cell model analyses of void growth in ductile materials

Three-dimensional micromechanical models were developed to study the damage by void growth in ductile materials. Special emphasis is given to the influence of the spatial arrangement of the voids. Therefore, periodical void arrays of cubic primitive, body centered cubic and hexagonal structure are investigated by analyzing representative unit cells. The isotropic behaviour of the matrix material is modelled using either v. Mises plasticity or the modified Gurson-Tvergaard constitutive law. The cell models are analyzed by the large strain finite element method under monotonic loading while keeping the stress triaxiality constant. The obtained mesoscopic deformation response and the void growth of the unit cells show a high dependence on the value of triaxiality. The spatial arrangement has only a weak influence on the deformation behaviour, whereas the type and onset of the plastic collapse behaviour are strongly affected. The parameters of the Gurson-Tvergaard model can be calibrated to the cell model results even for large porosity, emphasizing its usefulness and justifying its broad applicability.     

Statistically informed dynamics of void growth in rate dependent materials

An inherently rate dependent material model is used to model the nucleation and growth of voids in metals, leading to spall fracture. Intrinsic material rate dependence introduces a third time scale in addition to those introduced by rates of nucleation and growth. Material rate dependence does increase the spall strength of metals, but it is not nearly as important as local inertia in doing so.     

不同材料冠状动脉支架膨胀行为分析

冠状动脉支架作为经皮穿刺冠状动脉成形术中保持病变血管畅通的核心器件,其在手术过程中受球囊作用的扩张特性以及球囊撤出后的反弹行为对支架植入术的成功有着重要的影响.利用有限元的方法系统,建立专有支架单独膨胀和血管支架膨胀模型,分析了316L不锈钢和L605钴铬合金两种材料支架筋尺寸和支架扩张尺度的变化及血管对其膨胀行为的影响.结果显示,支架所选材料是决定支架膨胀行为的主要因素,L605材料支架所需的临界内压力及反弹行为明显大于316L不锈钢支架;材料一定时,增加支架筋的宽度或厚度提高支架迅速扩张临界内压力;支架轴向长度的变化只与结构和最终膨胀状态相关.有限元模拟对支架性能的评价和设计有一定指导意义.     

Cohesive fracture modeling of elastic-plastic crack growth in functionally graded materials

This work investigates elastic-plastic crack growth in ceramic/metal functionally graded materials (FGMs). The study employs a phenomenological, cohesive zone model proposed by the authors and simulates crack growth by the gradual degradation of cohesive surfaces ahead of the crack front. The cohesive zone model uses six material-dependent parameters (the cohesive energy densities and the peak cohesive tractions of the ceramic and metal phases, respectively, and two cohesive gradation parameters) to describe the constitutive response of the material in the cohesive zone. A volume fraction based, elastic-plastic model (extension of the original Tamura-Tomota-Ozawa model) describes the elastic-plastic response of the bulk background material. The numerical analyses are performed using WARP3D, a fracture mechanics research finite element code, which incorporates solid elements with graded elastic and plastic properties and interface-cohesive elements coupled with the functionally graded cohesive zone model. Numerical values of volume fractions for the constituents specified at nodes of the finite element model set the spatial gradation of material properties with isoparametric interpolations inside interface elements and background solid elements to define pointwise material property values. The paper describes applications of the cohesive zone model and the computational scheme to analyze crack growth in a single-edge notch bend, SE(B), specimen made of a TiB/Ti FGM. Cohesive parameters are calibrated using the experimentally measured load versus average crack extension (across the thickness) responses of both Ti metal and TiB/Ti FGM SE(B) specimens. The numerical results show that with the calibrated cohesive gradation parameters for the TiB/Ti system, the load to cause crack extension in the FGM is much smaller than that for the metal. However, the crack initiation load for the TiB/Ti FGM with reduced cohesive gradation parameters (which may be achieved under different manufacturing conditions) could compare to that for the metal. Crack growth responses vary strongly with values of the exponent describing the volume fraction profile for the metal. The investigation also shows significant crack tunneling in the Ti metal SE(B) specimen. For the TiB/Ti FGM system, however, crack tunneling is pronounced only for a metal-rich specimen with relatively smaller cohesive gradation parameter for the metal.     

Three-dimensional elastic-plastic finite element analysis of small surface cracks

Three-dimensional, elastic and elastic-plastic finite element analysis of small surface cracks was performed. The elastic analysis is in good agreement with other solutions. For a round surface with a radius equal to six times the crack depth, the $\text{K}$ at the surface is about 4% higher than the $\text{K}$ for a flat surface. The results of the elastic-plastic analyses show a unique variation of the effective $\text{K}$ (J-integral) along the crack front with a decrease in $\text{K}$ at the surface due to a lack of plane strain constraint and an increase in the effective $\text{K}$ at the maximum depth point with increasing plasticity. Similar behavior was observed for a semielliptical crack. Increasing the strain hardening exponent from 10 to 20 produced similar results with slightly higher effective A's for high applied strains. These results are useful in understanding the fracture behavior of small surface cracks.     

Finite element analysis of elastic–plastic materials displaying mixed hardening

The problem of non-radial loading (in the stress space) of structures of elastic–plastic metals is discussed. A concept of combined isotropic and kinematic hardening is proposed. For a material displaying such mixed hardening, constitutive equations are derived and incorporated in a finite element program for the analysis of elastic–plastic 2D-structures. When determining the numerical solution of the non-linear equations, the program enables the investigator to select an optimal combination of step by step and iteration procedures. The computed results are compared with experimental results and with results from calculations based on the overlay model, Reference 25.