基于Updated Lagrangian法的三维粘弹性大变形问题的有限元分析

本文基于Updated Lagrangian增量迭加方法,采用以现时Kirchhoff应力增量和现时Green应变增量表示的虚功方程和Kirchhoff应力张量-Green应变张量的积分本构关系,导出粘弹性大变形的虚功方程。依此采用二十节点三维立方等参数元编制相应的计算程序。三个算例结果与以往一维、二维的计算结果完全符合。     

各向同性硬化材料弹塑性本构关系的渐近积分及其计算精度

1.引言由于材料塑性性质依赖于变形的历史,弹塑性有限无分析一般都采用增量方法。其基本点是从已知的现时(时间t,对于静力分析、t可理解为加载水平)状态出发,利用虚功原理建立到达邻近(时间t+△t))状态的变形增量的运动方程。这样逐步求解,从而得到结构在整个时间历程的变形和应力状态。为得到足够精确的结果,一般情况下,在非线性分析的每个增量步中,还要进行平衡迭代。但是无论是采用哪种迭代(例如Newton-Raphson迭代,常刚度迭代等),每个增量步或每次迭代以后都要进行状态决定,即根据求得的位移增量决定新的变形状态的应力、应变等,作为下一增量步或下次迭代求解的出发点。因此状态决定在增量有限元分析中不仅在计算量上占很大比例,而且对解的精度有直接的影响。在状态决定的计算中,从位移增量计算应变增量仅涉及几何关系,而由应变增量计算应力增量则涉及材料的本构关系。关于后者,弹塑性增量理论中建立的是应力和应变的微分关系:     

以Kirchhoff应力张量-Green应变张量表示本构关系的粘弹性大变形平面问题的有限元法

本文基于Updated Lagrangian增量叠加方法,采用以现时Kirchhoff应力增量和现时Green应变增量表示的虚功方程和Kirchhoff应力-Green应变的积分本构关系,导出粘弹性大变形的变分方程.依此采用八节点二维等参数元编制相应的平面问题的计算程序,算例的计算结果与以往的工作完全符合.     

再生碎石混凝土力学性能试验研究

针对再生粗骨料为碎石的再生混凝土,设计了99个试件,分别进行立方体抗压强度、棱柱体抗压强度、抗折强度、弹性模量和泊松比等力学性能的试验。本文研究了再生混凝土的强度指标、变形性能、损伤过程、能量耗散等特征,分析了骨料取代率对其各性能指标的影响;基于试验数据,得出各强度性能指标的换算关系;分别采用分段式本构关系模型和损伤本构关系模型对再生混凝土的应力-应变本构关系进行研究,并提出相应的本构方程。研究结果表明:与天然骨料混凝土相比,再生混凝土的各项强度指标均略为增大,但变形性能和耗能性能降低,应力-应变全曲线下降段更为陡峭;随着取代率的增大,弹性模量呈先减小后增大的变化趋势,泊松比无明显变化,与天然混凝土接近;其损伤发展过程加快,应变在ε=(3~5)×10-3时最为迅速。分段式本构方程和损伤本构方程的计算结果与试验值均能较好地吻合。     

考虑晶体滑移面分解正应力的细观损伤模型

材料内部的解理、滑移面剥离等细观损伤是引起宏观失效的根源, 从细观尺度研究损伤的发生和发展有助于深入认识材料的变形和失效过程. 本文基于晶体塑性理论, 从滑移系的受力和变形出发研究材料的细观损伤, 建立了考虑滑移面分解正应力的细观损伤模型, 为晶体材料解理断裂的分析提供了新方法. 首先, 在晶体弹塑性变形构型的基础上引入损伤变形梯度张量的概念, 从变形运动学着手建立了考虑损伤能量耗散的本构方程, 并推导了塑性流动方程与损伤演化方程; 然后, 建立了相应的数值计算方法, 给出了应力与状态变量的更新算法, 推导了Jacobian矩阵的表达式; 接着, 以$[100]$取向的单晶铜材料为例, 通过有限元计算与试验结果的对比, 并采用粒子群优化算法标定了11个材料细观参数; 最后, 将所提细观损伤模型应用于RVE单轴拉伸过程的模拟, 得到了考虑损伤影响的应力应变曲线, 并分析了材料的塑性流动与损伤演化过程. 结果表明, 本文所提模型能够计算材料在受载过程中的损伤累积效应, 合理反映晶体材料的细观损伤机理.      

冻土的有限变形本构关系的实验研究

根据有限变形理论,给出了冻土三轴蠕变实验数据处理时所需Green应变和Kirchhoff应力的计算公式,并根据冻土三轴蠕变实验结果给出冻土的有限变形本构关系及其蠕变参数。通过对冻土的实验数据对比分析可知:在相同试验条件下,应变较小时,小应变ε1 和Green应变Ez的数值几乎相等,随着应变的增加,两种表征方式的计算结果差别越来越大,可见对于冻土如果不考虑试件变形前后长度的变化,计算得出的应变偏离实际变形情况较大,因此冻土材料的本构关系采用有限变形表征更确切。根据冻土的有限变形本构关系计算得到的冻结壁内侧最大径向位移更接近实测结果,因此冻结壁设计计算依据冻土的有限变形本构关系更为合理。对今后冻结工程的设计具有参考价值。     

有机玻璃在高应变率下的损伤型非线性粘弹性本构关系及破坏准则

对两种有机玻璃高应变率下的大变形和破坏行为进行了实验研究,通过改进文献[2]本构关系的非线性弹性项并引入损伤参量,建立了一个适用于更大变形范围、能描述应力平台及本构失稳的损伤型非线性粘弹性本构方程。相应地,从临界损伤量概念出发,提出以应变和应变率为控制变量的破坏准则。不论是本构关系还是破坏准则,理论计算均与试验结果吻合良好。     

有机玻璃在高应变率下的损伤型非线性粘弹性本构关系及破坏准则

对两种有机玻璃高应变率下的大变形和破坏行为进行了实验研究,通过改进文献[2]本构关系的非线性弹性项并引入损伤参量,建立了一个适用于更大变形范围、能描述“应力平台”及“本构失稳”的损伤型非线性粘弹性本构方程。相应地,从临界损伤量概念出发,提出以应变和应变率为控制变量的破坏准则。不论是本构关系还是破坏准则,理论计算均与试验结果吻合良好。     

用于板料成形反向模拟的特殊应力更新算法

提出了一种改进的反向模拟法，以最终构型为研究对象，采用Euler坐标系，基于虚功原理 获得有限元列式. 改进的反向模拟法采用了一种基于塑性流动理论的本构方程，可以充分考 虑应变历史对塑性变形的影响. 为了避免流动理论应力更新算法过程中关于未知量\Delta \lambda 的非线性方程的求解，引入等效应力思想，无需Newton-Raphson迭代直 接计算未知量\Delta \lambda . 盒形件的拉深实例中，传统的基于塑性形变本 构方程的反向模拟法和改进的基于塑性流 动本构方程的反向模拟法计算结果，分别与基于增量有限元法的正向数值模拟求解器 LS-DYNA计算结果进行对比. 通过获得的坯料轮廓、成形极限图、等效应变分布、计算效率 等的比较，验证了所提出的基于塑性流动理论本构模型的应力更新算法的有效性.     

半解析有限元法分析非线性粘弹体在板材模具中的流动

聚合物熔体在模具中的均匀性决定板材成型的质量。熔体具有非线性记忆性的粘弹效应,同时流道十分复杂。为准确、简便地分析熔体流动情况,针对板材机头流道的特点,提出了流动计算新的方法——有限柱半解析法,即在流道的厚度方向采用富氏级数描述流动分布,把三维有限元方程简化成为二维,得到以流量和压力为未知量的有限元方程。同时采用K-BKZ积分型本构模型描述熔体的非线性记忆性,给出求解非线性方程组的迭代方法,取得较好的收敛性。对两种典型的板材模具进行了分析计算,得出模具出口处的流量分布,通过把结果与三维有限元分析的结果相比较,表明该方法可以准确地分析板材模具中熔体的流动。     

棒材通过锥形模静液挤压成形的无网格法分析

建立了一种基于初始构形及有限变形的粘塑性弹性本构关系，并由空间描述的Galerkin能量弱变分原理，经一致转换得一整体拉格朗日方程描述下的动量平衡方程．同时经线性化处理给出了显示中心差分法求解格式，以棒材通过锥形模的静液挤压成形为例进行了全面的EFG法数值模拟，从而证明了有限变形粘塑性EFG法对实际成形工艺分析、优化及设计的有效性。     

An efficient algorithm for strain history tracking in finite element computations of non-Newtonian fluids with integral constitutive equations

A new finite element technique has been developed for employing integral-type constitutive equations in non-Newtonian flow simulations. The present method uses conventional quadrilateral elements for the interpolation of velocity components, so that it can conveniently handle viscoelastic flows with both open and closed streamlines (recirculating regions). A Picard iteration scheme with either flow rate or elasticity increment is used to treat the non-Newtonian stresses as pseudo-body forces, and an efficient and consistent predictor-corrector scheme is adopted for both the particle-tracking and strain tensor calculations. The new method has been used to simulate entry flows of polymer melts in circular abrupt contractions using the K-BKZ integral constitutive model. Results are in very good agreement with existing numerical data. The important question of mesh refinement and convergence for integral models in complex flow at high flow rate has also been addressed, and satisfactory convergence and mesh-independent results are obtained. In addition, the present method is relatively inexpensive and in the meantime can reach higher elasticity levels without numerical instability, compared with the best available similar calculations in the literature.     

Three-dimensional extrudate swell experimental and numerical study of a polyethylene melt obeying a memory-integral equation

The present work deals with the experimental and numerical features of the flow of a linear low-density polyethylene melt (LLDPE) at 160°C at the exit of a die of square cross-section. The rheological properties of the fluid are fitted by a Wagner's memory-integral constitutive equation. The characteristics of the extrudate jet are determined by optical means at different flow rates. The stream-tube analysis, already applied to two-dimensional extrudate swell problems involving rate and integral constitutive equations, is considered to simulate the flow field. The method avoids particle tracking problems related to integral models and allows computation of the unknown free surface by considering only a `peripheral stream tube' limited by the wall and the jet surface and an inner stream surface. Those boundary surfaces are determined by considering the conservation equations together with boundary condition equations, solved by the Levenberg–Marquardt optimization algorithm. The method leads to a considerable reduction in the number of degrees of freedom and the storage area. The numerical results are found to be generally consistent with the experimental data and highlight the growing importance of stress peaks due to the singularity at the exit when the flow rate increases.     

Prediction of extrudate swell in polymer melt extrusion using an Arbitrary Lagrangian Eulerian (ALE) based finite element method

Accurate prediction of extrudate (die) swell in polymer melt extrusion is important as this helps in appropriate die design for profile extrusion applications. Extrudate swell prediction has shown significant difficulties due to two key reasons. The first is the appropriate representation of the constitutive behavior of the polymer melt. The second is regarding the simulation of the free surface, which requires special techniques in the traditionally used Eulerian framework. In this paper we propose a method for simulation of extrudate swell using an Arbitrary Lagrangian Eulerian (ALE) technique based finite element formulation. The ALE technique provides advantages of both Lagrangian and Eulerian frameworks by allowing the computational mesh to move in an arbitrary manner, independent of the material motion. In the present method, a fractional-step ALE technique is employed in which the Lagrangian phase of material motion and convection arising out of mesh motion are decoupled. In the first step, the relevant flow and constitutive equations are solved in Lagrangian framework. The simpler representation of polymer constitutive equations in a Lagrangian framework avoids the difficulties associated with convective terms thereby resulting in a robust numerical formulation besides allowing for natural evolution of the free surface with the flow. In the second step, mesh is moved in ALE mode and the associated convection of the variables due to relative motion of the mesh is performed using a Godunov type scheme. While the mesh is fixed in space in the die region, the nodal points of the mesh on the extrudate free surface are allowed to move normal to flow direction with special rules to facilitate the simulation of swell. A differential exponential Phan Thien Tanner (PTT) model is used to represent the constitutive behavior of the melt. Using this method we simulate extrudate swell in planar and axisymmetric extrusion with abrupt contraction ahead of the die exit. This geometry allows the extrudate to have significant memory for shorter die lengths and acts as a good test for swell predictions. We demonstrate that our predictions of extrudate swell match well with reported experimental and numerical simulations.     

The constitutive equations of finite strain poroelasticity in the light of a micro-macro approach

After recalling the constitutive equations of finite strain poroelasticity formulated at the macroscopic level, we adopt a microscopic point of view which consists of describing the fluid-saturated porous medium at a space scale on which the fluid and solid phases are geometrically distinct. The constitutive equations of poroelasticity are recovered from the analysis conducted on a representative elementary volume of porous material open to fluid mass exchange. The procedure relies upon the solution of a boundary value problem defined on the solid domain of the representative volume undergoing large elastic strains. The macroscopic potential, computed as the integral of the free energy density over the solid domain, is shown to depend on the macroscopic deformation gradient and the porous space volume as relevant variables. The corresponding stress-type variables obtained through the differentiation of this potential turn out to be the macroscopic Boussinesq stress tensor and the pore pressure. Furthermore, such a procedure makes it possible to establish the necessary and sufficient conditions to ensure the validity of an ‘effective stress’ formulation of the constitutive equations of finite strain poroelasticity. Such conditions are notably satisfied in the important case of an incompressible solid matrix.     

采用共旋应变的三维热弹塑性有限变形有限元法

本文采用线性化共旋应变张量和增率型虚功原理，建立了有限变形热力耦合弹塑性有限元法。在该方法中，材料的流动应力取为应变总量、应变速率和温度的函数，推导了包含这种函数关系的本构矩阵。另外在温度场分析中，考虑了塑性功和摩擦功转化的热量。文后给出的算例表明该方法可以很好地模拟热加工过程。     

A marker-and-cell approach to viscoelastic free surface flows using the PTT model

This work is concerned with the development of a numerical method capable of simulating two-dimensional viscoelastic free surface flows governed by the non-linear constitutive equation PTT (Phan-Thien–Tanner). In particular, we are interested in flows possessing moving free surfaces. The fluid is modelled by a marker-and-cell type method and employs an accurate representation of the fluid surface. Boundary conditions are described in detail and the full free surface stress conditions are considered. The PTT equation is solved by a high order method which requires the calculation of the extra-stress tensor on the mesh contour. The equations describing the numerical technique are solved by the finite difference method on a staggered grid. In order to validate the numerical method fully developed flow in a two-dimensional channel was simulated and the numerical solutions were compared with known analytic solutions. Convergence results were obtained throughout by using mesh refinement. To demonstrate that complex free surface flows using the PTT model can be computed, extrudate swell and a jet flowing onto a rigid plate were simulated.     

Differential constitutive equations of incompressible media with finite deformations

A method is proposed for constructing a system of constitutive equations of an incompressible medium with nonlinear dissipative properties with finite deformations. A scheme of the mechanical behavior of a material is used, in which the points are connected by horizontally aligned elastic, viscous, plastic, and transmission elements. The properties of each element of the scheme are described with the use of known equations of the nonlinear elasticity theory, the theory of nonlinear viscous fluids, and the theory of plastic flow of the material under conditions of finite deformations of the medium. The system of constitutive equations is closed by equations that express the relation between the deformation rate tensor of the material and the deformation rate tensor of the plastic element. Transmission elements are used to take into account a significant difference between macroscopic deformations of the material and deformations of elements of the medium at the structural level. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 158–170, May–June, 2009.     

A finite strain kinematic hardening constitutive model based on Hencky strain: General framework, solution algorithm and application to shape memory alloys

The logarithmic or Hencky strain measure is a favored measure of strain due to its remarkable properties in large deformation problems. Compared with other strain measures, e.g., the commonly used Green-Lagrange measure, logarithmic strain is a more physical measure of strain. In this paper, we present a Hencky-based phenomenological finite strain kinematic hardening, non-associated constitutive model, developed within the framework of irreversible thermodynamics with internal variables. The derivation is based on the multiplicative decomposition of the deformation gradient into elastic and inelastic parts, and on the use of the isotropic property of the Helmholtz strain energy function. We also use the fact that the corotational rate of the Eulerian Hencky strain associated with the so-called logarithmic spin is equal to the strain rate tensor (symmetric part of the velocity gradient tensor). Satisfying the second law of thermodynamics in the Clausius-Duhem inequality form, we derive a thermodynamically-consistent constitutive model in a Lagrangian form. In comparison with the available finite strain models in which the unsymmetric Mandel stress appears in the equations, the proposed constitutive model includes only symmetric variables. Introducing a logarithmic mapping, we also present an appropriate form of the proposed constitutive equations in the time-discrete frame. We then apply the developed constitutive model to shape memory alloys and propose a well-defined, non-singular definition for model variables. In addition, we present a nucleation-completion condition in constructing the solution algorithm. We finally solve several boundary value problems to demonstrate the proposed model features as well as the numerical counterpart capabilities.     

各向同性率无关材料本构关系的不变性表示

在内变量理论的框架下，针对各向同性率无关材料，使用张量函数表示理论建立 了塑性应变全量及增量本构关系的最一般的张量不变性表示. 它们均由3个完备不可约的基 张量组合构成，这3个基张量分别是应力的零次幂、一次幂和二次幂. 因此得出，塑性应变、 塑性应变增量与应力三者共主轴. 通过对基张量的正交化，给出了本构关系式在主应力空间 中的几何解释. 进一步，全量(或增量)本构关系中3个组合因子被表达为应力、塑性应 变(或塑性应变增量)的不变量的函数. 当塑性应变(或塑性应变增量)的3个不变量之间 满足一定关系时，所给出的本构关系将退化为经典的形变理论(或塑性势理论). 最后，还讨论它与奇异屈服面理论的关系，当满足一定条件时，两者是一致的.