Locally enhanced Voronoi cell finite element model (LE‐VCFEM) for simulating evolving fracture in ductile microstructures containing inclusions

Ductile heterogeneous materials such as cast aluminum alloys undergo catastrophic failure that initiates with particle fragmentation, which evolves with void growth and coalescence in localized bands of intense plastic deformation and strain softening. The Voronoi cell finite element model (VCFEM), based on the assumed stress hybrid formulation, is unable to account for plastic strain‐induced softening. To overcome this shortcoming of material softening due to plastic strain localization, this study introduces a locally enhanced VCFEM (LE‐VCFEM) for modeling the very complex phenomenon of ductile failure in heterogeneous metals and alloys. In LE‐VCFEM, finite deformation displacement elements are adaptively added to regions of localization in the otherwise assumed stress‐based hybrid Voronoi cell finite element to locally enhance modeling capabilities for ductile fracture. Adaptive h‐refinement is used for the displacement elements to improve accuracy. Damage initiation by particle cracking is triggered by a Weibull model. The nonlocal Gurson–Tvergaard–Needleman model of porous plasticity is implemented in LE‐VCFEM to model matrix cracking. An iterative strain update algorithm is used for the displacement elements. The LE‐VCFEM code is validated by comparing with results of conventional FE codes and experiments with real materials. The effect of various microstructural morphological characteristics is also investigated. Copyright © 2008 John Wiley & Sons, Ltd.     

Families of 4-node and 9-node finite elements for a finite deformation shell theory. An assesment of hybrid stress, hybrid strain and enhanced strain elements

In this paper we discuss and compare three types of 4-node and 9-node finite elements for a recently formulated finite deformation shell theory with seven degrees of freedom. The shell theory takes thickness change into account and circumvents the use of a rotation tensor. It allows for the applicability of three-dimensional constitutive laws and equipes the configuration space with the structure of a vector space. The finite elements themselves are based either on a hybrid stress functional, on a hybrid strain functional, or on a nonlinear version of the enhanced strain concept. As independent variables either the normal and shear resultants, the strain tensor related to the deformation of the midsurface, or the incompatible enhanced strain field are taken as independent variables. The fields of equivalence of these different formulations, their limitations as well as possible improvements are discussed using different numerical examples. Received 10 December 1998     

A refined finite element model for the analysis of elastic–plastic frames

A finite element model for the elastic–plastic analysis of plane frames is proposed. The formulation is based on the independent modelling of the displacement and plastic strain fields; the latter is modelled both over the cross-section and along the element length as function of a finite number of parameters, which are considered as an extra set of independent variables, in addition to nodal displacements. Stress redistribution is allowed for over the cross-section, but not over the element length, where the distribution of stress resultants (axial forces and bending moments) is imposed consistently with the assumed displacement model; stress redistribution in terms of stress resultants becomes possible only because of the finite number of redundancies introduced when assembling. It is shown that the model can be formulated in such a way that not only compatibility and elasticity, but also equilibrium (in the sense of beam theory), are fully complied with and only the plastic portion of the constitutive relationship is approximately fulfilled, even if, in principle, to any desired level of accuracy. The model produces accurate results, including a detailed representation of the spreading of plastic zones, with a fairly limited number of elements.     

Phenomenological modelling of rate-dependent plasticity for high strain rate problems

The results of two related theoretical investigations for large-scale computations of elasto-plastic deformations at ultrahigh strain rates are summarized in this paper. The first effort concerns the development of a phenomenological constitutive model for finite deformation elasto-plasticity which includes the effects of thermal softening, strain hardening, rate-dependence, as well as the noncoaxiality of the plastic strain rate and the stress deviator, and the incorporation of this model in a large-scale explicit finite-element code. The second effort involves the investigation of localized deformations and shear banding at high strain rates. It is shown that the constitutive model considered, together with standard quadrilateral finite elements with one point integration (piece-wise constant strain and stress fields), can nicely produce the observed intense localized deformations without recourse to any special elements. The results are illustrated in terms of the shear localization observed in uniaxial extension, and of void collapse under uniaxial compression. In addition, the effect of the noncoaxiality of the plastic strain rate and the stress tensor is included in the model, and its influence on strain localization at high strain rates is investigated.     

Incremental analysis of elastic-plastic frames at finite spread of yielding zones

In elastic-perfectly plastic frames finite zones of yielding develop under monotonically increasing loads. The classical deformation analysis assumes that the elastic rigidity reduces locally to zero at the generalized yield hinges only. Such an approach underestimates the deformations during the loading history and at collapse.A method is proposed in order to evaluate the displacements of elastic-plastic frames when the actual spread of plastic zones is included in the analysis. In comparison with the classical method the developed technique accounts for the axial and flexural stiffness variation along each beam member due to the partial yielding of cross-sections. The actual extent of plastic zones depends on the stress resultant distribution in the structure.The proposed method is based on an incremental analysis procedure formulated by means of the independent elastic-plastic kinematical compatibility equations, and casts a computeroriented technique for evaluating the influence of the finite extension of plastic zones, accounting for the interaction between axial force and bending moment, and for variable crosssections and loads along structural members.The evaluation of the stiffness reduction in the partially plastic elements is specified for Ishape cross-section and is performed according to two different concepts: (a) through an analytical evaluation of the elastic core size, by means of appropriate formulae developed for the cross-section shape under investigation; (b) through a step-wise linear rigidity variation, which requires a preliminary generation of interaction domains of equal stiffness in the space of the active stress resultants for standard profiles.The essential features of the method and those of the program application are illustrated in simple examples, showing the order of magnitude of the actual displacements in comparison with the results of the classical deformation analysis of elastic-plastic frames.     

A two-dimensional co-rotational Timoshenko beam element with XFEM formulation

Pin connections and plastic hinges produce non-smooth displacement fields in beam structures. By using an appropriate extended finite element method (XFEM), non-smooth solutions can be obtained by regular coarse meshes, which do not necessarily conform to pins or plastic hinges. In this paper, a two-dimensional co-rotational beam element with XFEM formulation to simulate pin connections and plastic hinges is presented. Enrichments for the rotation and the deflection approximations are embedded in a co-rotational frame to capture the non-smoothness in both small and large deformations. Numerical examples on pin connections and plastic hinges demonstrate the accuracy and robustness of the present formulation.     

Enrichment of enhanced assumed strain approximations for representing strong discontinuities: addressing volumetric incompressibility and the discontinuous patch test

We present a geometrically non‐linear assumed strain method that allows for the presence of arbitrary, intra‐finite element discontinuities in the deformation map. Special attention is placed on the coarse‐mesh accuracy of these methods and their ability to avoid mesh locking in the incompressible limit. Given an underlying mesh and an arbitrary failure surface, we first construct an enriched approximation for the deformation map with the non‐linear analogue of the extended finite element method (X‐FEM). With regard to the richer space of functions spanned by the gradient of the enriched approximation, we then adopt a broader interpretation of variational consistency for the construction of the enhanced strain. In particular, in those elements intersected by the failure surface, we construct enhanced strain approximations which are orthogonal to piecewise‐constant stress fields. Contrast is drawn with existing strong discontinuity approaches where the enhanced strain variations in localized elements were constructed to be orthogonal to constant nominal stress fields. Importantly, the present formulation gives rise to a symmetric tangent stiffness matrix, even in localized elements. The present modification also allows for the satisfaction of a discontinuous patch test, wherein two different constant stress fields (on each side of the failure surface) lie in the solution space. We demonstrate how the proposed modifications eliminate spurious stress oscillations along the failure surface, particularly for nearly incompressible material response. Additional numerical examples are provided to illustrate the efficacy of the modified method for problems in hyperelastic fracture mechanics. Copyright © 2003 John Wiley & Sons, Ltd.     

A robust non‐linear mixed hybrid quadrilateral shell element

In the paper a non‐linear quadrilateral shell element for the analysis of thin structures is presented. The variational formulation is based on a Hu–Washizu functional with independent displacement, stress and strain fields. The interpolation matrices for the mid‐surface displacements and rotations as well as for the stress resultants and strains are specified. Restrictions on the interpolation functions concerning fulfillment of the patch test and stability are derived. The developed mixed hybrid shell element possesses the correct rank and fulfills the in‐plane and bending patch test. Using Newton's method the finite element approximation of the stationary condition is iteratively solved. Our formulation can accommodate arbitrary non‐linear material models for finite deformations. In the examples we present results for isotropic plasticity at finite rotations and small strains as well as bifurcation problems and post‐buckling response. The essential feature of the new element is the robustness in the equilibrium iterations. It allows very large load steps in comparison to other element formulations. Copyright © 2005 John Wiley & Sons, Ltd.     

考虑应变梯度效应的三点弯梁模型解析研究--第二部分:局部化带的位移及转角

基于梯度塑性理论,分析了应变软化及真实裂纹扩展阶段的局部化带的张拉位移和转角。在弹性阶段,可以由弹性理论来确定二者的关系。真实裂纹出现后,利用平衡条件、几何条件及梯度以来的应变软化本构关系,得到了真实裂纹长度与局部化带长度的关系。当真实裂纹刚出现时,局部化带长度达到最大值。在任何阶段,局部化带到中性轴的距离单调降低,局部化带的张拉位移和转角受梁深、带宽、弹模及下降模量等的影响。弹模及下降模量越大,带宽越小,则局部化带的张拉位移和转角都增加。而且,在前两个阶段,张拉位移都线性增加,但在后两个阶段,转角都非线性增加。     

Equivalent localization element for crack band approach to mesh‐sensitivity in microplane model

The paper presents in detail a novel method for finite element analysis of materials undergoing strain‐softening damage based on the crack band concept. The method allows applying complex material models, such as the microplane model for concrete or rock, in finite element calculations with variable finite element sizes not smaller than the localized crack band width (corresponding to the material characteristic length). The method uses special localization elements in which a zone of characteristic size, undergoing strain softening, is coupled with layers (called ‘springs’) which undergo elastic unloading and are normal to the principal stress directions. Because of the coupling of strain‐softening zone with elastic layers, the computations of the microplane model need to be iterated in each finite element in each load step, which increases the computer time. Insensitivity of the proposed method to mesh size is demonstrated by numerical examples. Simulation of various experimental results is shown to give good agreement. Copyright © 2004 John Wiley & Sons, Ltd.     

A stabilized one‐point integrated quadrilateral Reissner–Mindlin plate element

A new quadrilateral Reissner–Mindlin plate element with 12 element degrees of freedom is presented. For linear isotropic elasticity a Hellinger–Reissner functional with independent displacements, rotations and stress resultants is used. Within the mixed formulation the stress resultants are interpolated using five parameters for the bending moments and four parameters for the shear forces. The hybrid element stiffness matrix resulting from the stationary condition can be integrated analytically. This leads to a part obtained by one‐point integration and a stabilization matrix. The element possesses a correct rank, does not show shear locking and is applicable for the evaluation of displacements and stress resultants within the whole range of thin and thick plates. The bending patch test is fulfilled and the computed numerical examples show that the convergence behaviour is better than comparable quadrilateral assumed strain elements. Copyright © 2004 John Wiley & Sons, Ltd.     

Efficient mixed Timoshenko–Mindlin shell elements

The precise representation of rigid body motions in the displacement patterns of curved Timoshenko–Mindlin (TM) shell elements is considered. This consideration requires the development of the strain–displacement relationships of the TM shell theory with regard to their consistency with the rigid body motions. For this purpose a refined TM theory of multilayered anisotropic shells is elaborated. The effects of transverse shear deformation and bending‐extension coupling are included. The fundamental unknowns consist of five displacements and eight strains of the face surfaces of the shell, and eight stress resultants. On the basis of this theory the simple and efficient mixed models are developed. The elemental arrays are derived using the Hu–Washizu mixed variational principle. Numerical results are presented to demonstrate the high accuracy and effectiveness of the developed 4‐node shell elements and to compare their performance with other finite elements reported in the literature. Copyright © 2002 John Wiley & Sons, Ltd.     

形状记忆聚氨酯热力耦合变形行为实验和有限元模拟

在室温下对形状记忆聚氨酯进行不同应变率下的单调拉伸实验,结合红外测温仪对试样表面温度进行同步监测,研究拉伸过程中的热力耦合效应。结果表明:当应力达到屈服峰后,分子链解缠导致了屈服软化,同时分子链之间的摩擦诱发了局部化温升;随着载荷继续增加,分子链在拉伸方向优先取向导致应变硬化发生,响应的应力和温度不断升高。同时发现,屈服峰和局部化温升均随着应变率的增加而显著增加,然而材料耗散生热诱导的应变软化和应变硬化之间存在竞争机制,使得局部化塑性流动过程对应变率的敏感性降低。基于有限元软件ABAQUS建立板状试样拉伸的有限元模型,对形状记忆聚氨酯的拉伸变形进行热力耦合分析。通过比较不同时刻的塑性应变场和温度场云图发现,局部化的塑性流动和温升均从初始缺陷处萌生,并逐渐向中间移动直至扩展到整个试样。进而提取不同加载速率下的平均温升曲线与实验结果进行了对比,发现二者吻合度较高。     

HRB500/HRB600钢筋作纵筋的混凝土框架梁端弯矩调幅试验研究

为研究HRB500钢筋和HRB600钢筋作纵筋的混凝土框架梁端弯矩调幅规律,完成了12榀单层两跨混凝土框架静力加载试验。试验结果表明:由于受拉纵筋屈服强度提高,一方面梁端塑性铰出现推迟,塑性铰形成前会发生一定的弯矩调幅;另一方面锚固于节点内的梁端受拉纵筋应变渗透引起较大的梁端附加转角,加大了弯矩调幅能力。将试验框架梁端弯矩调幅分塑性铰形成前后两阶段进行考察,第一阶段弯矩调幅幅度为10.35%~33.42%,第二阶段弯矩调幅幅度为3.39%~30.5%。基于试验结果,建立了与梁端控制截面相对受压区高度呈幂函数减小趋势、与受拉纵筋屈服强度和受拉纵筋屈服时刻应变渗透引起的梁端附加转角呈线性增长趋势的第一阶段弯矩调幅系数计算公式;建立了与总塑性转角（塑性铰区范围内的塑性转角与应变渗透引起的梁端附加塑性转角之和）呈幂函数增长趋势、与受拉纵筋屈服强度呈线性减小趋势的第二阶段弯矩调幅系数计算公式。     

Shifted integration technique in one-dimensional plastic collapse analysis using linear and cubic finite elements

The present study is concerned with the physical explanations of the linear and the cubic finite elements for beams and axisymmetric shells through comparisons of their strain energy approximations with those of the Rigid Bodies-Spring Models which are discrete elements suitable for plastic collapse analysis using the concepts of plastic hinges and hinge lines. The established conditions for the equivalence between these two modellings, which are given as the relations between the locations of the numerical integration points and those of the occurrence of plastic hinges, can be conveniently used in the economical plastic collapse analysis of framed structures and axisymmetric shells where the locations of plastic hinge formations are controlled by the movement of numerical integration points. Some numerical results are shown in order to prove numerically the obtained relations and to verify the validity of the proposed shifting technique of numerical integration points, which is identified as ‘the shifted integration technique’ in the present paper.     

Finite rotation quadrilateral element for multi‐layer beams

The paper presents the finite rotations beam equations derived on use of the generalized Reissner hypothesis with a scalar parameter for the transverse extension. The beam strain and change of curvature measures are obtained from the right stretch strain, and the virtual work is given for Biot‐type stress and couple resultants. The strain energy for the first‐order isotropic elastic material is assumed in terms of the right stretch strain, and constitutive equations for the beam stress and couple resultants are derived. Two finite rotation elements are developed from the derived beam equations: a beam element with the transverse stretch and a quadrilateral element. First, the beam element with the uniformly under‐integrated tangent operator is developed. Next, the formula linking the middle‐line variables and the interface variables of the beam is introduced consistently with the generalized Reissner kinematics. Linearization of this formula is performed, and the derived tangent operator is used to convert the two‐node beam element to a four‐node quadrilateral. Both the finite elements have been tested on several numerical examples, some of highly non‐linear characteristics, and their accuracy is very good. It has been established that the quadrilateral element, which is intended for applications to multi‐layer beams, performs very well for high elemental aspect ratios, and can therefore be applied to modelling of very thin layers. Copyright © 1999 John Wiley & Sons, Ltd.     

晶须转动对铝基复合材料热压缩变形行为的影响

采用平面应变有限元方法研究了晶须转动对晶须增强铝基复合材料热压缩变形行为的影响.数值结果表明:晶须转动不仅引起晶须承载能力的下降,而且也影响基体的塑性变形行为,同时复合材料在热压缩变形过程中表现出明显的应变软化行为.数值分析研究进一步证明:晶须转动是复合材料出现应变软化的重要原因.     

Meshless analysis of shear deformable shells: the linear model

This work develops a kinematically linear shell model departing from a consistent nonlinear theory. The initial geometry is mapped from a flat reference configuration by a stress-free finite deformation, after which, the actual shell motion takes place. The model maintains the features of a complete stress-resultant theory with Reissner-Mindlin kinematics based on an inextensible director. A hybrid displacement variational formulation is presented, where the domain displacements and kinematic boundary reactions are independently approximated. The resort to a flat reference configuration allows the discretization using 2-D Multiple Fixed Least-Squares (MFLS) on the domain. The consistent definition of stress resultants and consequent plane stress assumption led to a neat formulation for the analysis of shells. The consistent linear approximation, combined with MFLS, made possible efficient computations with a desired continuity degree, leading to smooth results for the displacement, strain and stress fields, as shown by several numerical examples.