梯度塑性的有限元分析及应变局部化模拟

对梯度塑性连续体提出了一个有限元方法．内状态变量的Ｌａｐｌａｃｉａｎ的确定基于它在求积点邻域的最小二乘方多项式近似．具体地考虑了具有一点求积和Ｈｏｕｒｇｌａｓｓ控制特点的基于胡海昌－Ｗａｓｈｉｚｕ变分原理的混合应变元和单元平均意义下的ｖｏｎ－Ｍｉｓｅｓ屈服准则．解析地导出了梯度塑性下一致性单元切线刚度矩阵和速率本构方程的一致性积分算法．在所建议的非局部化途径中求积点的一致性条件在非局部化意义下逐点精确满足．数值例题表明所提出的非经典连续体的有限元方法求解应变局部化问题的有效性     

A combined parametric quadratic programming and precise integration method based dynamic analysis of elastic-plastic hardening/softening problems

The objective of the paper is to develop a new algorithm for numerical solution of dynamic elastic-plastic strain hardening/softening problems. The gradient dependent model is adopted in the numerical model to overcome the result mesh-sensitivity problem in the dynamic strain softening or strain localization analysis. The equations for the dynamic elastic-plastic problems are derived in terms of the parametric variational principle, which is valid for associated, non-associated and strain softening plastic constitutive models in the finite element analysis. The precise integration method, which has been widely used for discretization in time domain of the linear problems, is introduced for the solution of dynamic nonlinear equations. The new algorithm proposed is based on the combination of the parametric quadratic programming method and the precise integration method and has all the advantages in both of the algorithms. Results of numerical examples demonstrate not only the validity, but also the advantages of the algorithm proposed for the numerical solution of nonlinear dynamic problems. The project supported by the National Key Basic Research Special Foundation (G1999032805), the National Natural Science Foundation of China (19872016, 50178016, 19832010) and the Foundation for University Key Teacher by the Ministry of Education of China     

One plasticity model for problems of plastic metal working

Scalar and tensor models of plastic flow of metals extending plasticity theory are considered over a wide range of temperatures and strain rates. Equations are derived using the physico-phenomenological approach based on modern concepts and methods of the physics and mechanics of plastic deformation. For hardening and viscoplastic solids, a new mathematical formulation of the boundary-value plasticity problem taking into account loading history is obtained. Results of testing of the model are given. A numerical finite-element algorithm for the solution of applied problems is described. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 159–169, November–December, 2008.     

基于线性互补模型的梯度塑性连续体无网格方法

对梯度塑性连续体提出了一个归结为线性互补问题的数值分析方法. 塑性乘子与位移均为主要未知变量,并采用基于移动最小二乘的无网格方法分别在积分点与节点上插值. 联立弱形式下的平衡方程与积分点上逐点满足的非局部本构方程和屈服准则可以导出一个线性互补问题,并通过Lexico-Lemke算法求解. 构造了一个基于N-R方法的迭代方案,使得不需要形成一致性切线刚度矩阵而仍保持二阶收敛性. 一维和二维的数值算例证明了所提出的方法处理由应变软化引起的应变局部化问题的有效性.      

基于能量等效原理的应变局部化分析:Ⅱ.有限元解法

以非局部塑性理论为基础,应用状态空间理论,通过局部和非局部两个状态空间的塑性能量耗散率等效原理,提出了一种求解应变局部化问题的新方法,以得到与网格无关的数值解.针对二维问题的屈服函数和流动法则导出了求解非局部内变量的一般方程,并提出了在有限元环境中求解应变局部化问题的应力更新算法.为了验证所提出的方法,对1个一维拉杆和3个二维平面应变加载试件进行了有限元分析.数值结果表明,塑性应变的分布和载荷-位移曲线都随着网格的变小而稳定地收敛,应变局部化区域的尺寸只与材料内尺度有关,而对有限元网格的大小不敏感.对于一维问题,当有限元网格尺寸减小时,数值解收敛于解析解.对于二维剪切带局部化问题,数值解随着网格尺寸的减小而稳定地向唯一解收敛.当网格尺寸减小时,剪切带的宽度和方向基本上没有变化.而且得到的塑性应变分布和网格变形是平滑的.这说明,所提方法可以克服经典连续介质力学模型导致的网格相关性问题,从而获得具有物理意义的客观解.此模型只需要单元之间的位移插值函数具有C~0连续性,因而容易在现有的有限元程序中实现而无需对程序作大的修改.     

Shakedown analysis of engineering structures with limited kinematical hardening

We present a numerical method for the computation of shakedown loads of engineering structures with limited kinematical hardening under thermo-mechanical loading. The method is based on Melan’s statical shakedown theorem, which results in a nonlinear convex optimization problem. This is solved by an interior-point algorithm recently developed by the authors, specially designed for lower bound shakedown analysis of large-scale problems. Limited kinematical hardening is taken into account by use of a two-surface model, such that both alternating plasticity and incremental collapse can be captured. For the yield surface as well as for the bounding surface the von Mises criterion is used. The proposed method is validated by two examples, where numerical results are compared to those of literature where available.     

Implicit stress-update algorithm for isotropic Cam-clay model based on the subloading surface concept at finite strains

A stress-update algorithm for the subloading surface Cam-clay model is developed. The model is re-formulated to be compatible with the multiplicative hyperelasto-plasticity for finite strains. The algorithm aims at synthesizing the phenomenological capability of the subloading surface model and the numerical advantage of the return mapping. A closed-form representation of the algorithmic tangent is also derived. The proposed algorithm is implemented into a finite-element code and is shown to be capable of producing accurate results even for large incremental steps. Numerical examples of boundary-value problems demonstrate the robustness of the algorithm.     

非经典大变形弹塑性本构方程及其算法研究

本文介绍了作者提出的非经典大变形弹塑性木构方程的增量形式，发展了相应的有限元分析方法。由于不采用屈服面的概念及相应的流动规律与加载准则，与传统的算法相比，本文提出的算法不仅简化了计算过程，而且改善了收敛速度和计算精度。作为验证和应用的例子，分析了圆柱试件的大变形拉伸颈缩过程和短圆柱的镦粗过程。与实验结果的比较证明了所发展方法的有效性。     

Calibrating a J2 plasticity material model using a 2D inverse finite element procedure

Material models are the key ingredients to accurately capture the global mechanical response of structural systems. The use of finite element analysis has proven to be effective in simulating nonlinear engineering applications. However, the choice of the appropriate material model plays a big role in the value of the numerical predictions. Such models are not expected to exactly reproduce global experimental response in all cases. Alternatively, the measured global response at specific domain or surface points can be used to guide the nonlinear analysis to successively extract a representative material model. By selecting an initial set of stress–strain data points, the load–displacement response at the monitoring points is computed in a forward incremental analysis without iterations. This analysis retains the stresses at the integration points. The corresponding strains are not accurate since the computed displacements are not anticipated to match the measured displacements at the monitoring points. Therefore, a corrective incremental displacement analysis is performed at the same load steps to adjust for displacements and strains everywhere by matching the measured displacements at the monitoring points. The stress–strain vectors at the most highly stressed integration point are found to establish an improved material model. This model is used within a multi-pass incremental nonlinear finite element analysis until the discrepancy between the measured and the predicted structural response at the monitoring points vanishes. The J2 flow theory of plasticity is used as a constitutive framework to build the tangent elastic–plastic matrices. The applicability of the proposed approach is demonstrated by solving 2D inverse continuum problems. The comparisons presented support the effectiveness of the proposed approach in accurately calibrating the J2 plasticity material model for such problems.     

Reduced augmented Lagrangian bi-conjugate gradient method for impact-contact problems

To avoid the numerical oscillation of the penalty method and non-compatibility with explicit operators of conventional Lagrange multiplier methods used in transient contact problems to enforce surface contact conditions, a new approach to enforcing surface contact constraints for the transient nonlinear finite element problems, referred to as “the reduced augmented Lagrangian bi-conjugate gradient method (ALCG)”, is developed in this paper. Based on the nonlinear constrained optimization theory and is compatible with the explicit time integration scheme, this approach can also be used in implicit scheme naturally. The new surface contact constraint method presented has significant advantages over the widely adopted penalty function methods and the conventional Lagrangian multiplier methods. The surface contact constraints are satisfied more accurately for each step by the algorithm, so the oscillation of numerical solution for the explicit scheme is depressed. Through the development of new iteration strategy for solving nonlinear equations, ALCG method improves the computational efficiency greatly. Project supported by State Education Commission Doctoral Foundation and Natural Science Foundation of Liaoning Province.     

Numerical and Experimental Study of Bubble Dynamics in Contact with a Solid Surface

A complex numerical and experimental method is proposed for studying 3D dynamics of a bubble contacting with a surface in the presence of an acoustic field. The numerical approach is based on the boundary element method for potential flows, which is most efficient for solving the problems in a 3D formulation. The use of heterogeneous computer architectures consisting of central graphic processors and becoming more and more popular makes it possible to increase the scale of the problem and sufficiently reduce the calculation time. The mesh destabilization problems are solved using a spherical filter. To describe the contact line dynamics, a semi-empirical law of motion is used. The experimental method is based on high-speed recording and optical microscopy. An air bubble contacts with the inner surface of an experimental cell made from acrylic glass and filled with distilled water. The acoustic field in the cell generated by a disk-shaped acoustic radiator is measured using a hydrophone. The behavior of the bubble contacting with a hydrophillic surface is considered for the cases of a fixed or moving contact line. The shape and volume oscillations of the bubble are investigated. The results of numerical simulations agree qualitatively with the experimental data.     

Enhanced measurement of strain distributions

A theoretical approach is presented that uses multiple strain gages to accurately measure complicated strain distributions. The technique is based on the method of weighted residuals in conjunction with measured strain data and is applicable for arbitrary in-plane strain distributions. Conventional measurements using strain gages are shown to represent a particular case of the approach presented. The experimental characterization of unidimensional strain fields is discussed in detail. Two approaches are presented; these are based on linear and quadratic approximations of the strain field. The strain distribution for two important practical problems is evaluated assuming ideal conditions to assess the performance of the proposed approach. In both cases, the simulated results demonstrate that measurement error resulting from the finite size of a strain gage may be reduced. That is, a larger strain gage may be used for a given maximum admissible error. The method also allows a minimal error of measured nonlinear strains.     

A Numerical Method for Distributed Parameter Structural Optimization Problems with Repeated Eigenvalues

A numerical method for solving distributed parameter structural optimization problems in which repeated eigenvalues may occur is formulated. Recent results on directional differentiability of repeated eigenvalues are used to develop a generalized steepest descent method for structural optimization. The algorithm is shown to overcome technical difficulties associated with nondifferentiability of repeated eigenvalues. The method is used to optimize design of a clamped-clamped column in which a repeated eigenvalue occurs.     

饱和多孔介质中的混合有限元法和有限应变下应变局部化分析

对基于Biot理论的饱和多孔介质中动力-渗流耦合分析提出了一个耦合场混合元．固相位移、应变和有效应力以及流相压力、压力梯度和Darcy速度在单元内均处理为独立变量分别插值．基于胡海昌-Washizu三变量广义变分原理给出的饱和多孔介质动力-渗流耦合问题控制方程的单元弱形式，导出了单元公式．进一步导出了考虑压力相关非关联塑性的非线性单元公式和发展了相应的一致性算法．对几何非线性分析，采用了共旋公式途径．数值结果例题显示所发展耦合场混合元模拟大应变下由应变软化引起以应变局部化为特征的渐进破坏现象的性能．     

An adaptive multigrid finite-volume scheme for incompressible Navier–Stokes equations

An algorithm for the solutions of the two-dimensional incompressible Navier–Stokes equations is presented. The algorithm can be used to compute both steady-state and time-dependent flow problems. It is based on an artificial compressibility method and uses higher-order upwind finite-volume techniques for the convective terms and a second-order finite-volume technique for the viscous terms. Three upwind schemes for discretizing convective terms are proposed here. An interesting result is that the solutions computed by one of them is not sensitive to the value of the artificial compressibility parameter. A second-order, two-step Runge–Kutta integration coupling with an implicit residual smoothing and with a multigrid method is used for achieving fast convergence for both steady- and unsteady-state problems. The numerical results agree well with experimental and other numerical data. A comparison with an analytically exact solution is performed to verify the space and time accuracy of the algorithm.     

Plastic spin associated with a non-normality theory of plasticity

A plasticity model using a vertex-type plastic flow rule on a smooth yield surface for an anisotropic solid has been proposed recently. This model is here completed by incorporating the effect of plastic spin. Simple shear with a large shear strain is one of the hardest tests on finite strain anisotropic plasticity models, and it is here shown which plastic spin expression is needed to avoid unrealistic oscillatory behavior of the shear stress under large shear strains. The idea of using non-normality with a smooth yield surface originates from a recent proposal of using an abrupt strain path change to determine the subsequent yield surface shape. For this method both polycrystal plasticity calculations and experiments have shown a vertex-type response on the apparently smooth yield surface.     

An improved technique for solving two‐dimensional shallow water problems

A numerical model for solving the 2D shallow water equations is proposed herewith. This model is based on a finite volume technique in a generalized co‐ordinate system, coupled with a semi‐implicit splitting algorithm in which a Helmholtz equation is used for the surface elevation. Several benchmark problems have proven the good accuracy of this method in complex geometries. Nevertheless, several numerical perturbations were noted in the surface elevation. After finding the origin, a new numerical technique is suggested, to avoid these perturbations. Several severe tests are proposed to validate this technique. Copyright © 1999 John Wiley & Sons, Ltd.     

A new computational algorithm for contact friction modeling of large plastic deformation in powder compaction processes

In this paper, the large deformation frictional contact of powder forming process is modeled based on a new computational algorithm by imposing the contact constraints and modifying the contact properties of frictional slip. A simple and efficient numerical algorithm is presented for imposing the contact constraints and frictional contact properties based on the node-to-surface contact technique to simulate the large deformation contact problem in the compaction process of powder. The Coulomb friction law is used to simulate the friction between the rigid punch and the workpiece by the use of penalty approach. A double-surface cap plasticity model is employed together with the nonlinear contact friction algorithm within the framework of large FE deformation in order to predict the non-uniform relative density distribution during large deformation of powder die-pressing. Finally, the numerical schemes are examined for accuracy and efficiency in modeling of a set of powder components.     

Loosening of elastic inclusions

A numerical method is presented for simulating the occurrence of localized slip and separation along the interfaces of multiple, randomly distributed, circular elastic inclusions in an infinite elastic plane. The method is an extension of a direct boundary integral approach previously described elsewhere for solving problems involving perfectly bonded circular inclusions. Here, we allow displacement discontinuities to develop along the inclusion/matrix interfaces in accordance with a linear Mohr–Coulomb yield condition combined with a tensile strength cut-off. The displacements, tractions, and displacement discontinuities on the inclusion boundaries are all represented by truncated Fourier series, and an explicit iterative algorithm is adopted to determine zones of slip and separation under the prevailing loading conditions. Several examples are given to demonstrate the accuracy and generality of the approach.