Algorithms for non-linear contact constraints with application to stability problems of rods and shells

In this paper a class of non-linear problems is discussed where stability as well as post-buckling behaviour is coupled with contact constraints. The contact conditions are introduced via a perturbed Lagrangian formulation. From this formulation the penalty and Lagrangian multiplier method are derived. Both algorithms are investigated together with an algorithm based on an augmented Lagrangian method. The resulting finite element formulation is applied to structural problems of beams and shells undergoing finite elastic deflections and rotations. For the examination of the post-buckling behaviour the arc-length method is used. The performance of the element formulation and a comparison of the different contact algorithms are demonstrated by numerical examples.     

A numerical modelling of non linear 2D-frictional multicontact problems: application to post-buckling in cellular media

In this paper a numerical modelling of non linear problems involving large deformations and frictional contact conditions is proposed. The motivation of this work comes from the study of the cellular materials (such as wood or foams) undergoing strong deformations. We restrict our study to a regular cellular network of hexagonal cells with thin walls. Strong loadings can generate at first buckling phenomena, then self-contact in the cell. Renouncing homogenization procedures, not always pertinent in this case, we have developed direct simulations. After giving the mechanical and mathematical formulations of the problem, we present two advanced numerical tools to solve large non linear frictional multicontact problems. This numerical modelling is based on an arc-length continuation method which permits to snap through singular points due to buckling phenomena and on an optimal domain decomposition method adapted to frictional contact problems. Finally, mechanical investigations of the contactless buckling and the post-buckling provide some pertinent parameters controlling the deformation process.This work was supported in part by CINES, Centre Informatique National de lEnseignement Supérieur, (Montpellier-FRANCE), Project mgc2547. The different simulations run on the SGI3800 computer based at the CINES.     

Finite deformation frictional mortar contact using a semi‐smooth Newton method with consistent linearization

A two‐dimensional, finite deformation frictional contact formulation with Coulomb's law is presented. The approach considers multibody contact and is based on a mortar formulation. The enforcement of contact constraints is realized with dual Lagrange multipliers. These alternative multiplier spaces are constructed in a way that the multipliers can easily be eliminated from the global system of equations by static condensation such that the system size does not increase. Friction kinematic variables are formulated in an objective way and enter non‐smooth complementarity functions for expressing the contact constraints. An active set strategy is derived by applying a semi‐smooth Newton method, which treats contact nonlinearities, material and geometrical nonlinearities in one single iterative scheme. By further carrying out a consistent linearization for both normal and frictional contact forces and constraints, a robust and highly efficient algorithm for linear and higher‐order (quadratic) interpolation is achieved. Efficiency of the proposed method and quality of results are demonstrated in several examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Post-buckling analysis of shell and membrane structures by dynamic relaxation method

The family of arc-length method has been used for the post-buckling analysis of structures. However, it often becomes intractable for complex nonlinear problems. For instance, the incremental solution hangs about and never reaches the desired deformation when wrinkling appears after the buckling of membrane structures. In order to cope with this problem, alternative methods have been proposed, e.g., the dynamic relaxation (DR) method and the DR arc-length method. The applicability of these methods to the post-buckling analysis of shell and membrane structure is investigated in the present research, where the mixed interpolation of tensorial components (MITC) shell element is used for the FE modeling. It is shown that the DR and the DR arc-length methods are much more robust than the arc-length method, but for complicated nonlinear problems, a continuous deformation path can not be obtained by any of the three methods.     

Optimisation of geometrically non-linear composite structures based on load–displacement control

A design optimisation methodology for beam reinforced composite structures with non-linear geometric behaviour is proposed. The formulation involves displacement, stresses, buckling and size constraints. The Newton–Raphson iterative procedure and the arc-length method are used for tracing equilibrium path and later updating the buckling load and the first ply failure load. The proposed sensitivity analysis model is based on an approach of the adjoint variable method for structures with non-linear geometric behaviour. The optimal design performs on a multilevel scheme based on structural efficiency maximisation exploring the anisotropic properties of the composites and weight minimisation using the ply thickness and the cross-section variables of the stiffeners. To demonstrate the applicability of the proposed developments, optimisation problems considering first ply failure and buckling conditions are presented.     

A computational method for frictional contact problem using finite element method

Using the finite element method a numerical procedure is developed for the solution of the two-dimensional frictional contact problems with Coulomb's law of friction. The formulation for this procedure is reduced to a complementarity problem. The contact region is separated into stick and slip regions and the contact stress can be solved systematically by applying the solution technique of the complementarity problem. Several examples are given to demonstrate the validity of the present formulation.     

Three‐dimensional quasi‐static frictional contact by using second‐order cone linear complementarity problem

A new formulation is presented for the three‐dimensional incremental quasi‐static problems with unilateral frictional contact. Under the assumptions of small rotations and small strains, a second‐order cone linear complementarity problem is formulated, which consists of complementarity conditions defined by bilinear functions and second‐order cone constraints. The equilibrium configurations are obtained by using a combined smoothing and regularization method for the second‐order cone complementarity problem. Copyright © 2005 John Wiley & Sons, Ltd.     

复合材料加筋结构后屈曲的自开发程序

针对复合材料层合加筋结构的后屈曲问题,开发出自编Fortran程序,并嵌入Patran进行前后处理。采用该程序计算出线性屈曲模态,将其作为后屈曲非线性计算的初始挠度;采用增量迭代技术、弧长控制及能量收敛准则求解非线性方程;最后引入复合材料Tsai-Hill失效准则,对单元的损伤进行判断及刚度折减。采用复合材料加筋板的压缩试验对该系统进行初步验证。针对不同加载方式、选材及结构型式的机翼盒段,利用该系统进行后屈曲计算和分析,数值结果与试验数据吻合较好。     

Complementarity methods for multibody friction contact problems in finite deformations

This paper deals with the frictional contact occurring between deformable elastoplastic bodies subjected to large displacements and finite deformations. Starting from a standard slave/master formulation we have developed a symmetrical formulation with which the unilateral contact conditions and the friction law are satisfied for each body. From the continuum equations, the discretized frictional contact problem is set as a complementarity problem and solved using Lemke's mathematical programming method. The efficiency of the method is illustrated in the case of several examples. Copyright © 2001 John Wiley & Sons, Ltd.     

Thermal post-buckling of slender composite and FGM columns through a simple and novel FE formulation

A simple and novel finite element (FE) formulation is proposed to study the thermal post-buckling of composite and FGM columns with axially immovable ends and operating in severe thermal environment. A linear eigenvalue analysis gives the critical buckling temperature but practically the buckled columns can withstand additional thermal load beyond critical temperature, which can be obtained using von-Karman geometric nonlinearity, applicable for moderately large deflections. In the present study, the solution of the non-linear post-buckling problem is obtained by treating it as a linear eigenvalue problem using the concept of effective stiffness. Here, the total degrees of freedom (dof) of the discretized column are reduced and the post-buckling load is obtained without the need for iterative analysis. Comparison of the numerical results obtained from this FE formulation is in very good agreement with those obtained from the earlier FE formulations.     

DAE for Frictional Contact Modeling of Constrained Multi-Flexible Body Systems

A general formulation for modeling frictional contact interactions in a constrained multi-flexible body system is outlined in this paper. The governing differential-algebraic equations (DAE) for the constrained motion contains not only a frictional term but also, the unknown contact conditions. These contact conditions are characterized by a set of nonlinear complementarity equations. To demonstrate the model, a falling-spinning beam impacting a rough elastic ground with damping is solved and comparison with Stewart-Trinkles' results provided.     

薄壳屈曲问题的有限质点法求解

有限质点法是一种新型的数值方法,它从牛顿力学的角度出发,以质点为研究对象,将求解域离散为质点系统,同时着眼于该系统内各个质点所受的力,进而追踪到整个质点系统的运动状态,在处理结构或机构的大变位、大变形等非线性问题时具有独特的优势。该文将有限质点法应用于薄壳的屈曲问题研究,为追踪其完整的屈曲路径,将显式弧长法的加载策略与其相结合;针对屈曲或大变形后出现的接触问题,改进了一种适用于显式求解的接触算法。最后,通过自编程序,分别选取薄壳屈曲问题的静力、动力等经典算例,并将该方法的计算结果和相关文献及试验结果进行了对比。结果表明,该文的方法用于薄壳的屈曲问题求解是可行的,能有效捕捉薄壳屈曲的完整过程。     

Contact Analysis for Solids Based on Linearly Conforming Radial Point Interpolation Method

To simulate the contact nonlinearity in 2D solid problems, a contact analysis approach is formulated using incremental form of the subdomain parametric variational principle (SPVP). The formulation is based on a linearly conforming radial point interpolation method (LC-RPIM) using nodal integration technique. Contact interface equations are also presented using a modified Coulomb frictional contact model and discretized by contact point-pairs. In the present approach, the global discretized system equations are transformed into a standard linear complementarity problem (LCP) that can be solved readily using the Lemke method. The present approach can simulate various contact behaviors including bonding/debonding, contacting/departing, and sticking/slipping. An intensive numerical study is performed to validate the proposed method via comparison with the ABAQUS^® and to investigate the effects of the various parameters used in computations. These parameters include normal and tangential adhesions, frictional coefficient, nodal density, the dimension of local nodal support domain, nodal irregularity, shape parameters used in the radial basis function and the external load. The numerical results have demonstrated that the present approach is accurate and stable for contact analysis of 2D solids.     

On a model of elastic-plastic structure optimized for post-buckling behaviour

The post-critical analysis performed for a structure optimized only for buckling constraints shows in many cases that the behaviour of the designed element after buckling is unstable. That should be considered as a drawback of the standard design since the applicability of such an unstable optimal structure may be questionable. Recently the new concept of structural optimization against instability has been proposed. The post-buckling analysis is included directly in the optimization problem and the modified design is formulated. As a result, the modified optimal structure with stable behaviour after buckling is created. Many modified design problems for structures exposed to elastic instability have been already formulated and solved. This paper gives some proposals of extending that approach for the design of elastic-plastic structures.     

基于接触协同作用结构非线性屈曲分析

工程结构中结构或构件之间存在接触摩擦现象是较为普遍的,接触协同作用使得结构非线性屈曲问题的数值求解变得困难。该文从这一实际问题出发,采用子结构凝聚自由度的有限元方法得到基于接触界面问题的非线性有限元平衡方程,通过增广拉格朗日乘子法对接触问题的约束条件进行处理,并与柱面弧长法结合,对非线性屈曲问题进行全过程跟踪求解。算例结果表明接触协同作用对结构的非线性屈曲过程影响明显,有必要在实际工程结构设计中考虑接触协同作用的影响。     

Mechanism of buckling development in elastic bars subjected to axial impact

By use of the finite difference method, the non-linear equations governing the elastic dynamic post-buckling deformations are solved for two types of impact buckling problems for straight bars. The initial dynamic buckling mode with a small amplitude parameter, given by the twin-characteristic-parameter solution, is used as the initial condition of the non-linear post-buckling solution. Particular attention is paid to the mechanism of growth and spread of buckling deformation in the bar and the interaction between the axial stress wave and the buckling deformation in the process of impact. It is found that the initial buckling deflection with one half-wave, occurring near the impacted end, spreads forward and develops into the higher mode as the axial stress wave propagates in the bar. The theoretical results are in good agreement with the experimental results reported in the literatures.     

Frictional contact of flexible and rigid bodies

 This paper is about planar frictional contact problems of both flexible and rigid bodies. For the flexible case a nonlinear finite element formulation is presented, which is based on a modified Coulomb friction law. Stick-slip motion is incorporated into the formulation through a radial return mapping scheme. Linearly interpolating four node elements and three node contact elements are utilized for the finite element discretization. The corresponding tangent stiffness matrices and residual vectors of the equations of motion are presented. In the rigid body case the contact problem is divided into impact and continual contact, which are mathematically described by linear complementarity problems. The impact in normal direction is modeled by a modified Poisson hypothesis, which is adapted to allow multiple impacts. The formulation of the tangential impact is grounded on Coulombs law of friction. The normal contact forces of the continual contact are such that colliding bodies are prevented from penetration and the corresponding tangential forces are expressed by Coulombs law of friction. Examples and comparisions between the different methods are presented. Received: 10 January 2001     

基于双特征参数解的直杆弹性动力后屈曲研究

利用差分方法求解动力后屈曲非线性方程解，研究了弹性直杆的2类轴向碰撞屈曲问题．将双特征参数解得出的含有小幅值参数的初始动力屈曲模态作为非线性后屈曲解的初始条件．理论计算的结果与文献中的实验数据达到了很好的一致，由此验证了双特征参数方法的正确性．研究结果还揭示了碰撞过程中屈曲变形扩展和发展的机理，以及轴向应力波和屈曲变形的相互作用规律．     

Boundary element analysis of two-dimensional elastoplastic contact problems

A general boundary element formulation for contact problems, capable of dealing with local elastoplastic effects and friction, is presented. Both conforming and non-conforming problems may be analysed. The contact problem is solved by means of a direct constraint technique, in which compatibility and equilibrium conditions are directly enforced in the general system of equations. The contact areas are modelled with linear interpolation functions, and quadratic interpolation functions are used everywhere else. Elastoplasticity is solved by a BEM initial strain approach The Von Mises yield criterion with its associated flow rule is adopted. Both perfectly plastic and work hardening materials are studied in the proposed formulation.

An incremental loading technique is proposed, which allows accurate development of the loading history of the problem. The non-linear nature of these problems demands the use of an iterative procedure, to determine the correct frictional conditions at every node of the contact area and the value of the plastic strains at selected points where local yielding may have occurred. Several numerical examples are presented to demonstrate the efficiency of the proposed formulation.     

复合材料翼面壁板轴压稳定性

基于T型长桁铺层数不同的两块复合材料翼面加筋壁板试件SC-1和SC-2,开展轴压稳定性试验研究,并提出一种预测屈曲载荷及最小后屈曲承载能力的工程分析方法,结合有限元特征值屈曲分析方法、有限元弧长法对试件的屈曲载荷、屈曲模态及后屈曲承载能力进行分析。试验结果表明,铺层数较多的试件SC-1的蒙皮局部屈曲应变较高,壁板也具有更高的屈曲载荷。在后屈曲阶段,SC-2加载到试验屈曲载荷的2.4倍未发生材料破坏和长桁蒙皮间脱粘损伤。工程分析方法和特征值屈曲分析能够准确预测壁板的屈曲载荷,最大误差分别为-9.3%和-2.8%,工程分析得到SC-2的最小后屈曲承载能力为试验屈曲载荷的2.09倍。有限元弧长法分析得到两件试件的屈曲载荷误差均小于1%,并具有壁板轴压屈曲模态预测和变形跟踪能力。