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.     

Post-yield behavior of fixed steel beams encased in concrete with plastic rotation capability under monotonic loads

Flexural strength calculated based on only curvature at sections shows some inconsistencies at large strains compared with nonlinear finite element analyses which considered damaged concrete plasticity. The prediction of the post-yield behavior of steel–concrete composite structures has been a complex issue, rendering analytically inaccurate prediction of the post-yield deflection of composite beams when plastic flows were not considered. The accurate prediction of the post-yield deformation of composite steel beams encased in structural concrete should account for the strain based on plastic rotation. The plastic rotation is influenced by the inelastic energy dissipation. The post-yield behavior of steel–concrete composite structures is affected by the inclination of diagonal cracks of concrete, and the stiffening effect of concrete tension between cracks. Plastic strain occurring in the steel section also contributes to in-elastic behavior of composite structures. The aim of this study was to idealize plastic flow of steel beams encased in structural concrete at fixed foundation. Plastic flows were calculated by non-linear finite element analysis including the consideration of concrete plasticity. The post-yield deflection was, then, predicted in a manner reflecting plastic deformation. The proposed idealization agreed well with numerical data obtained by means of nonlinear finite element analysis, providing a simplified but reliable procedure for practicing engineers designing composite structures in the inelastic region.     

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.     

基于微平面弹塑性应力-应变关系的混凝土本构模型

在B.P. Bazant等人提出的混凝土微平面本构模型M2的基础上,将微平面上的应力分解为体、偏、剪三个分量,根据各个分量的物理意义定义了相应的应力-应变关系函数,即理想弹塑性函数。引入了破断应变的概念,当微平面应变达到破断应变后应力减为零。介绍了模型参数的确定方法。最后通过三个算例初步验证了本文所建议模型的合理性和正确性。     

Formulation and numerical treatment of incompressibility constraints in large strain elastic–plastic analysis

The present paper is concerned with an efficient framework for a nonlinear finite element procedure for the rate‐independent finite strain analysis of solids undergoing large elastic‐isochoric plastic deformations. The formulation relies on the introduction of a mixed‐variant metric deformation tensor which will be multiplicatively decomposed into a plastic and an elastic part. This leads to the definition of an appropriate logarithmic strain measure which can be additively decomposed into the exact isochoric (deviatoric) and volumetric (spheric) strain measures. This fact may be seen as the basic idea in the formulation of appropriate mixed finite elements which guarantee the accurate computation of isochoric strains. The mixed‐variant logarithmic elastic strain tensor provides a basis for the definition of a local isotropic hyperelastic stress response whereas the plastic material behavior is assumed to be governed by a generalized J₂ yield criterion and rate‐independent isochoric plastic strain rates are computed using an associated flow rule. On the numerical side, the computation of the logarithmic strain tensors is based on higher‐order Padé approximations. To be able to take into account the plastic incompressibility constraint a modified mixed variational principle is considered which leads to a quasi‐displacement finite element procedure. Finally, the numerical solution of finite strain elastic‐plastic problems is presented to demonstrate the efficiency and the accuracy of the algorithm. Copyright © 1999 John Wiley & Sons, Ltd.     

Estimation of elastic‐plastic strain response at two‐dimensional notches

It is widely recognized that the accuracy of notch fatigue calculations can be improved significantly when those calculations are based on the elastic‐plastic response strain at the notch root, as opposed to the remotely applied loads or stresses. Two of the most widely used approximations for this response are Neuber's rule and Glinka's equivalent strain energy density method. In the present work, a survey of some of the many published evaluations of these methods was first conducted, and then, additional detailed comparisons with elastic‐plastic finite element analyses for a series of semicircular and V‐shaped notch configurations were performed. Based on the observed limitations of both the Neuber and Glinka approaches, and with the guidance of the elastic‐plastic finite element results, a new (and more robust) approach for the estimation of notch response strains is proposed. This approach calls for the definition of a generalized notch response curve (GNRC), which is dependent on both the material stress–strain curve and the notch geometry. Once defined, the GNRC allows the determination of the response strain for any applied stress.     

Numerical study of mixed-mode fracture in concrete

In the present paper, a finite element code based on the microplane model for concrete is used for the analysis of typical mixed-mode geometries: a Single-Edge-Notched beam, a Double-Edge-Notched specimen and a Dowel-Disk specimen. A local smeared fracture finite element analysis is carried out. As a regularization procedure, the crack band method is used. The principal objective of the study was to investigate whether the smeared fracture finite element code is able to predict mixed-mode fracture of concrete with no optimisation of the material model parameters. Comparison between experimental and numerical results shows that the used code predicts structural response and crack patterns realistically for all cases investigated. Moreover, it is shown that for most of the studied geometries a mixed-mode fracture mechanism dominates at crack initiation, however, with increase of the crack length mode-I fracture becomes dominant and the specimens finally failed in mode-I fracture.     

Influence of loading rate on concrete cone failure

Three different effects control the influence of the loading rate on structural response: creep of bulk material, rate dependency of growing microcracks and structural inertia. The first effect is important only at extremely slow loading rates whereas the second and third effects dominate at higher loading rates. In the present paper, a rate sensitive model, which is based on the energy activation theory of bond rupture, and its implementation into the microplane model for concrete are discussed. It is first demonstrated that the model realistically predicts the influence of the loading rate on the uniaxial compressive behaviour of concrete. The rate sensitive microplane model is then applied in a 3D finite element analysis of the pull-out of headed stud anchors from a concrete block. In the study, the influence of the loading rate on the pull-out capacity and on the size effect is investigated. To investigate the importance of the rate of the growing microcracks and the influence of structural inertia, static and dynamic analyses were carried out. The results show that with an increase of the loading rate the pull-out resistance increases. For moderate loading rates, the rate of the microcrack growth controls the structural response and the results of static and dynamic analysis are similar. For very higher loading rates, however, the structural inertia dominates. The influence of structural inertia increases with the increase of the embedment depth. It is shown that for moderately high-loading rates the size effect becomes stronger when the loading rate increases. However, for very high-loading rate the size effect on the nominal pull-out strength vanishes and the nominal resistance increases with an increase of the embedment depth. This is due to the effect of structural inertia.     

钢管混凝土结构材料非线性的一种有限元分析方法

为了更简单地考虑梁单元的材料非线性受力性能,把断面广义力和广义应变的概念运用于单元分析中,将单元的弹塑性刚度矩阵分离为弹性刚度矩阵和塑性刚度矩阵。这样,梁单元的变形可以由弹性变形和塑性变形简单地迭加,结构内力可通过弹性应变能的斜率(弹性刚度矩阵)与位移的乘积求得,从而在增量-迭代计算时可较准确且较快地计算出结构变形后的不平衡力。应用这一计算方法,推导了基于纤维模型的三维梁单元的钢管混凝土结构的有限元基本公式,并将其植入能考虑几何非线性的三维梁单元非线性计算程序NL_Beam3D中以计算结构的双重非线性问题。算例分析表明该方法和程序能较准确地反映钢管混凝土结构的双重非线性特性。     

Development of a reference strain approach for assessment of fracture response of reeled pipelines

As a result of recent increase in exploitation of hydrocarbon resources in harsher environments and also installation techniques which utilize the materials plastic deformation capacity, accurate assessment of fracture response of pipelines subject to large plastic strains (e.g., typical of reeled pipes) has attracted particular interest nowadays. In this paper, an approach, based on the evaluation of the J-integral, is developed for assessing the integrity of such pipelines, manifested in a model of a pipeline with a circumferential part-through crack subjected to plastic bending. The proposed approach is an extension of the reference strain method developed earlier by other researchers, and takes advantage of the displacement controlled loading nature in such pipes (thus being suitable for Strain Based Design methodologies), and the resulting high strain levels, which often cause fracture response of the material in the plastic regime. The developed formulation relates the fracture response of the pipe (in terms of the non-dimensionalized J-integral) as a linear function of the axial strain in the pipe at its uncracked state. A series of 300 3D nonlinear finite element models using the ABAQUS software were analyzed in preparation of the equation that could assess the fracture response of such pipes with great accuracy. The resulting equation, calibrated by the finite element results, can predict the fracture response of pipes with a maximum error of 2% for a practical uncracked material strain range of 1.5% ? ε_unc ? 4%.     

Experimental and finite element simulation methods for rate-dependent metal forming processes

An experimental procedure and a finite element simulation method for rate-dependent metal forming processes are developed. The development includes the formulation of a tangential stiffness matrix for an axisymmetric solid finite element with four node, eight degree of freedom, quadrilateral cross-section. The formulation includes the effects of elasticity, viscoplasticity, temperature, strain rate and large strains. The solution procedure is based on a Newton-Raphson incremental-iterative method which solves the non-linear equilibrium equations and gives temperatures and incremental stresses and strains. Three examples are studied. In example 1, finite element simulation for the upsetting of a cylindrical workpiece between two perfectly rough dies is performed and the results are compared with alternative finite element solutions. In examples 2 and 3, both experimental and finite element studies are performed for the upsetting of a cylindrical billet and the forging of a ball, respectively. Annealed aluminium 1100 workpieces are used in both examples. For the finite element analysis, uniaxial compression tests are first performed to provide the material properties. The tests generate elastic moduli and two sets of stress-strain curves (quasi-static and constant strain rate), which are used to establish a rate-dependent material model for input. For both examples 2 and 3, comparisons between the experimental and finite element simulation results for the forming force vs. die displacement relations and also for the deformed configurations show good agreement. The versatility of finite element methods allows for displaying detailed knowledge of the metal forming process, such as the distributions of temperature rise, yield stress, effective stress, plastic strain, plastic strain rate, forming forces and deformed configurations, etc. at any instance during the forming process.     

Assessment of strain gradients affecting the performance of 3-D strain rosettes – relevant to the analysis of cement mantle strains in total hip replacements

A large scale model analysis, using embedded strain gauges, of the strain distribution in the cement mantle surrounding a femoral prosthesis is underway. In order to predict, and so avoid, positions of locally high strain gradients in this model, a finite element and experimental analysis of a similar problem was undertaken. For this purpose, a loose fitting rectangular steel insert inside a surrounding rectangular epoxy sheath was used to model an extreme case of the torsional and bending components of hip joint load. The axial component of joint load was modelled using an axisymmetric finite element model of a tapered shaft. The finite element results were used to determine suitable positions for embedding gauges in the experimental model. Results showed that the finite element analysis failed to adequately model the close sliding fit between the steel insert and epoxy. Altering the experimental model to artificially replicate the finite element contact conditions produced good correlation in bending, with experimental strains agreeing with simple bending theory to within 6%. Satisfactory correlation under torsional loading was not obtained, but strain magnitudes were low. Predicted positions for embedding gauges give conservative results, lessening the possibility of strain gradient induced error in the large scale model test of the cement mantle and prosthesis.     

Numerical integration of a pressure‐dependent,non‐linear kinematic hardening constitutive model for large strain cyclic plasticity of metals

A Gurson‐based constitutive model is presented, which includes non‐linear mixed isotropic–kinematic hardening and creep, and allows the analysis of problems involving arbitrarily large plastic strains. This model was developed with the main objective of allowing, on the basis of a single set of material parameters, the numerical simulation of all the main features of cold metal forming processes, which usually imply severe loading–unloading cycles with very large plastic strains, difficult to be correctly reproduced numerically. A suitable integration scheme of the rate equations is described and implemented into a finite element code. The results obtained are compared with some reference experimental ones; an application of the model for the simulation of wire drawing processes is also presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Ductile damage analysis of elasto‐plastic shells at large inelastic strains

The objective of this contribution is to model ductile damage phenomena under consideration of large inelastic strains, to couple the corresponding constitutive law with a multi‐layer shell kinematics and to give finally an adequate finite element formulation. An elastic–plastic constitutive law is formulated by using a spatial hyperelasto‐plastic formulation based on the multiplicative decomposition of the deformation gradient. To include isotropic ductile damage the continuum damage model of Rousselier is modified so as to consider large strains and additionally extended by various void nucleation and macro‐crack criteria. In order to achieve numerical efficiency, elastic strains are supposed to be sufficiently small providing a numerical effective integration based on the backward Euler rule. Finite element formulation is enriched by means of the enhanced strain concept. Thus the well‐known deficiencies due to incompressible deformations and the inclusion of transverse strains are avoided. Several examples are given to demonstrate the performance of the algorithms developed concerning large inelastic strains of shells and ductile damage phenomena. Copyright © 2000 John Wiley & Sons, Ltd.     

Modelling of reinforced‐concrete structures providing crack‐spacing based on X‐FEM,ED‐FEM and novel operator split solution procedure

In this work, we present a novel approach to the finite element modelling of reinforced‐concrete (RC) structures that provides the details of the constitutive behavior of each constituent (concrete, steel and bond‐slip), while keeping formally the same appearance as the classical finite element model. Each component constitutive behavior can be brought to fully non‐linear range, where we can consider cracking (or localized failure) of concrete, the plastic yielding and failure of steel bars and bond‐slip at concrete steel interface accounting for confining pressure effects. The standard finite element code architecture is preserved by using embedded discontinuity (ED‐FEM) and extended (X‐FEM) finite element strain representation for concrete and slip, respectively, along with the operator split solution method that separates the problem into computing the deformations of RC (with frozen slip) and the current value of the bond‐slip. Several numerical examples are presented in order to illustrate very satisfying performance of the proposed methodology. Copyright © 2010 John Wiley & Sons, Ltd.     

High rate material behaviour at hot forming conditions

Modern processes of hot forming use very high strain rates and large, mostly incrementally applied strains. For the simulation of such forming processes relevant material data are needed, which have to be recorded under accordant forming conditions. This places extraordinary demands on the experimental technique, because high temperatures, high strain rates and large strains have to be implemented simultaneously. In the following contribution such an aligned experimental technique is introduced. In order to apply a very large range of strain rate, compression tests are performed using different technical equipment up to a drop tower and Hopkinson pressure bar. To reach large plastic strains, a hot torsion test was developed, which allowed true plastic shear strains up to 10 at strain rates in the order of 100 1/s at hot forming temperatures.     

An iterative procedure for determining effective stress–strain curves of sheet metals

An iterative correction procedure using 3D finite element analysis (FEA) was carried out to determine more accurately the effective true stress–true strain curves of aluminum, copper, steel, and titanium sheet metals with various gage section geometries up to very large strains just prior to the final tearing fracture. Based on the local surface strain mapping measurements within the diffuse and localized necking region of a rectangular cross-section tension coupon in uniaxial tension using digital image correlation (DIC), both average axial true strain and the average axial stress without correction of the triaxiality of the stress state within the neck have been obtained experimentally. The measured stress–strain curve was then used as an initial guess of the effective true stress–strain curve in the finite element analysis. The input effective true stress–strain curve was corrected iteratively after each analysis session until the difference between the experimentally measured and FE-computed average axial true stress–true strain curves inside a neck becomes acceptably small. As each test coupon was analyzed by a full-scale finite element model and no specific analytical model of strain-hardening was assumed, the method used in this study is shown to be rather general and can be applied to sheet metals with various strain hardening behaviors and tension coupon geometries.     

Notch effect on creep–fatigue life for Sn–3.5Ag solder

This paper studies the creep–fatigue crack initiation and failure lives of Sn–3.5Ag solder notched specimens focused on the multiaxial strain at the notch root. Push–pull creep–fatigue tests were performed using three circumferential notched specimens using four kinds of creep–fatigue strain waveforms. Multiaxial strains at the notched section were calculated by finite element (FE) analysis under four kinds of creep–fatigue loading. Creep–fatigue damage laws were applied for evaluating the crack initiation and failure lives using the multiaxial strains obtained by the FE analysis. von Mises equivalent strain at the notch root estimated the crack initiation lives with a large scatter as well as the failure lives. Instead, the mean value of von Mises equivalent strain over the cross section of the notch root estimated the crack initiation and failure lives with a small scatter.