Some C0-continuous mixed formulations for general dipolar linear gradient elasticity boundary value problems and the associated energy theorems

The goal of this work is a systematic presentation of some classes of mixed weak formulations, for general multi-dimensional dipolar gradient elasticity (fourth order) boundary value problems. The displacement field main variable is accompanied by the double stress tensor and the Cauchy stress tensor (case 1 or μ ? τ ? u formulation), the double stress tensor alone (case 2 or μ ? u formulation), the double stress, the Cauchy stress, the displacement second gradient and the standard strain field (case 3 or μ ? τ ? κ ? ε ? u formulation) and the displacement first gradient, along with the equilibrium stress (case 4 or u ? θ ? γ formulation). In all formulations, the respective essential conditions are built in the structure of the solution spaces. For cases 1, 2 and 4, one-dimensional analogues are presented for the purpose of numerical comparison. Moreover, the standard Galerkin formulation is depicted. It is noted that the standard Galerkin weak form demands C¹-continuous conforming basis functions. On the other hand, up to first order derivatives appear in the bilinear forms of the current mixed formulations. Hence, standard C⁰-continuous conforming basis functions may be employed in the finite element approximations. The main purpose of this work is to provide a reference base for future numerical applications of this type of mixed methods. In all cases, the associated quadratic energy functionals are formed for the purpose of completeness.     

Convergence analysis of a finite element method based on different moduli in tension and compression

When analyzing materials that exhibit different mechanical behaviors in tension and compression, an iterative approach is required due to material nonlinearities. Because of this iterative strategy, numerical instabilities may occur in the computational procedure. In this paper, we analyze the reason why iterative computation sometimes does not converge. We also present a method to accelerate convergence. This method is the introduction of a new pattern of shear modulus that was strictly derived according to the constitutive model based on the bimodular elasticity theory presented by Ambartsumyan. We test this procedure with a numerical example concerning a plane stress problem. Results obtained from this example show that the proposed method reduces the cost of computation and accelerates the convergence of the solution.     

Canonical dual finite element method for solving post-buckling problems of a large deformation elastic beam

This paper presents a canonical dual mixed finite element method for the post-buckling analysis of planar beams with large elastic deformations. The mathematical beam model employed in the present work was introduced by Gao in 1996, and is governed by a fourth-order non-linear differential equation. The total potential energy associated with this model is a non-convex functional and can be used to study both the pre- and the post-buckling responses of the beams. Using the so-called canonical duality theory, this non-convex primal variational problem is transformed into a dual problem. In a proper feasible space, the dual variational problem corresponds to a globally concave maximization problem. A mixed finite element method involving both the transverse displacement field and the stress field as approximate element functions is derived from the dual variational problem and used to compute global optimal solutions. Numerical applications are illustrated by several problems with different boundary conditions.     

板桁结构考虑局部屈曲空间计算的横向条带板段单元法

用有限个横向条带法构造了板桁组合结构板段考虑局部屈曲的空间位移模式。基于三维连续体的虚功增量方程,导出了横向条带板段单元的ＵＬ列式,并考虑了板段单元位形变化的影响。此计算方法自由度少,计算精度高,能用于大型板桁结构的几何非线性分析。文末计算了广东西江桥板桁组合结构模型梁,计算结果与实测结果吻合较好。     

Secondary buckling and tertiary states of a beam on a non-linear elastic foundation

An axially compressed beam resting on a non-linear foundation undergoes a loss of stability (buckling) via a supercritical pitchfork bifurcation. In the post-buckled regime, it has been shown that under certain circumstances the system may experience a secondary bifurcation. This second bifurcation destablizes the primary buckling mode and the system “jumps” to a higher mode; for this reason, this phenomenon is often referred to as mode jumping. This work investigates two new aspects related to the problem of mode jumping. First, a three mode analysis is conducted. This analysis shows the usual primary and secondary buckling events. But it also shows stable solutions involving the third mode. However, for the cases studied here, there is no natural loading path that leads to this solution branch, i.e. only a contrived loading history would result in this solution. Second, the effect of an initial geometric imperfection is considered. This breaks the symmetry of the system and significantly complicates the bifurcation diagram.     

Theoretical analysis of a class of mixed,C0 continuity formulations for general dipolar Gradient Elasticity boundary value problems

Mixed weak formulations, with two or three main (tensor) variables, are stated and theoretically analyzed for general multi-dimensional dipolar Gradient Elasticity (biharmonic) boundary value problems. The general structure of constitutive equations is considered (with and without coupling terms). The mixed formulations are based on various generalizations of the so-called Ciarlet–Raviart technique. Hence, C⁰ continuity conforming basis functions may be employed in the finite element approximations (or even, C⁻¹ basis functions for the Cauchy stress variable). All the complicated boundary conditions, especially in the multi-dimensional scenario, are naturally considered. The main variables are the displacement vector, the double stress tensor and the Cauchy stress tensor. The latter variable may be eliminated in some of the formulations, depending on the structure of the constitutive equations. The standard continuous and discrete Babuška–Brezzi inf–sup conditions for the constraint equation, as well as, solution uniqueness for both the continuous statements and discrete approximations, are established in all cases. For the purpose of completeness, two one-dimensional mixed formulations are also analyzed. The respective constitutive equations possess general structure (with coupling terms). For the 1-D formulations, all the inf–sup conditions are satisfied, for both the continuous and discrete statements (assuming proper selection of the polynomial spaces for the main variables). Hence, the general Babuška–Brezzi theory results in quasi-optimality and stability. For multi-dimensional problems, the difficulty of deducing the inf–sup condition on the kernel is examined. Certain aspects of methodologies employed to theoretically by-pass this problem, are also discussed.     

APPROXIMATE FORMULASOF IMPACT FORCE FOR NORMAL IMPACT OF A FINITE ELASTIC BEAM OF ELASTIC FOUNDATION BY A FINITE ROD

ＲｅｎＷｅｎｍｉｎ（任文敏）；ＨｕａｎｇＪｉａｎｍｉｎ（黄剑敏）；ＣｈｅｎＷｅｎ（陈文）（ＲｅｃｅｉｖｅｄＤｅｃ．２５，１９９４；ＣｏｍｍｕｎｉｃａｔｅｄｂｙＹａｎｇＧｕｉｔｏｎｇ）ＡＰＰＲＯＸＩＭＡＴＥＦＯＲＭＵＬＡＳＯＦＩＭＰＡＣＴＦＯＲＣＥＦＯ...     

Three-dimensional finite element modeling of a vibrating beam with a breathing crack

This paper develops a full three-dimensional finite element model in order to study the vibrational behavior of a beam with a non-propagating surface crack. In this model, the breathing crack behavior is simulated as a full frictional contact problem between the crack surfaces, while the region around the crack is discretized into three-dimensional solid finite elements. The governing equations of this non-linear dynamic problem are solved by employing an incremental iterative procedure. The extracted response is analyzed utilizing either Fourier or continuous wavelet transforms to reveal the breathing crack effects. This study is applied to a cracked cantilever beam subjected to dynamic loading. The crack has an either uniform or non-uniform depth across the beam cross-section. For both crack cases, the vertical, horizontal, and axial beam vibrations are studied for various values of crack depth and position. Coupling between these beam vibration components is observed. Conclusions are extracted for the influence of crack characteristics such as geometry, depth, and position on the coupling of these beam vibration components. The accuracy of the results is verified through comparisons with results available from the literature.     

张拉结构非线性分析的五节点等参单元

本文针对张拉结构的特点，提出了一种五节点等参数单元有限元模型，采用四次多项式作为位移插值函数及单元初始形状函数，并假定索是理想柔性的且满足虎克定律，基于修正的Ｌａｇｒａｎｇｉａｎ坐标描述法，建立了非线性有限元基本方程和切线刚度矩阵，利用Ｎｅｗｔｏｎ－Ｒａｐｈｓｏｎ法进行了实例计算。结果表明：本文方法精度极高，可供大跨度索网，索穹顶，拉线塔等张拉结构分析，设计时采用。     

Least-squares mixed finite element method for a class of stokes equation

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A simple criterion for finite time stability with application to impacted buckling of elastic columns

The existing theories of finite-time stability depend on a prescribed bound on initial disturbances and a prescribed threshold for allowable responses. It remains a challenge to identify the critical value of loading parameter for finite time instability observed in experiments without the need of specifying any prescribed threshold for allowable responses. Based on an energy balance analysis of a simple dynamic system, this paper proposes a general criterion for finite time stability which indicates that finite time stability of a linear dynamic system with constant coefficients during a given time interval [0, t _f ] is guaranteed provided the product of its maximum growth rate (determined by the maximum eigen-root p₁ >0) and the duration t _f does not exceed 2, i.e., p₁t _f <2. The proposed criterion (p₁t _f =2) is applied to several problems of impacted buckling of elastic columns: (i) an elastic column impacted by a striking mass, (ii) longitudinal impact of an elastic column on a rigid wall, and (iii) an elastic column compressed at a constant speed (“Hoff problem”), in which the time-varying axial force is replaced approximately by its average value over the time duration. Comparison of critical parameters predicted by the proposed criterion with available experimental and simulation data shows that the proposed criterion is in robust reasonable agreement with the known data, which suggests that the proposed simple criterion (p₁t _f =2) can be used to estimate critical parameters for finite time stability of dynamic systems governed by linear equations with constant coefficients.     

Buckling of soft-core sandwich plates with angle-ply face sheets by means of a C0 finite element formulation

In order to avoid using C¹ interpolation functions in finite element implementation of the previous zig–zag theories, artificial constraints, in which the first derivatives of transverse displacement will be replaced by the assumed variables, are usually employed. However, such assumption will violate continuity conditions of transverse shear stresses at interfaces. Differing from previous work, this paper will propose a C⁰-type zig–zag theory for buckling analysis of laminated composite and sandwich plates with general configurations. The first derivatives of transverse displacement have been taken out from a displacement field of the proposed zig–zag theory. Thus, the C⁰ interpolation functions are only required in finite element implementations of the proposed model. Without use of any artificial constraints, an eight-node quadrilateral element based on the proposed model is presented by incorporating the terms associated with the geometric stiffness matrix. In order to verify performance of the proposed model, several buckling problems of sandwich plates with soft core have been analyzed. Numerical results show that the proposed model is able to predict accurately buckling loads of the soft-core sandwich plates with varying fiber orientations of face sheets.     

A mixed finite volume formulation for the solution of gradient elasticity problems

The present works aims at solving the equations of dipolar gradient elasticity via the finite volume method. Initially, a general, mixed, finite volume formulation is stated. As a first approach, the equations of 1D and 2D gradient elasticity are solved via the proposed method. Numerical implementation shows that the suggested model is in excellent agreement with analytic solutions. The approach seems to be promising for extension to 3D problems also.     

A novel beam finite element with singularities for the dynamic analysis of discontinuous frames

In this work, a model of the stepped Timoshenko beam in presence of deflection and rotation discontinuities along the span is presented. The proposed model relies on the adoption of Heaviside’s and Dirac’s delta distributions to model abrupt and concentrated, both flexural and shear, stiffness discontinuities of the beam that lead to exact closed-form solutions of the elastic response in presence of static loads. Based on the latter solutions, a novel beam element for the analysis of frame structures with an arbitrary distribution of singularities is here proposed. In particular, the presented closed-form solutions are exploited to formulate the displacement shape functions of the beam element and the relevant explicit form of the stiffness matrix. The proposed beam element is adopted for a finite element discretization of discontinuous framed structures. In particular, by means of the introduction of a mass matrix consistent with the adopted shape functions, the presented model allows also the dynamic analysis of framed structures in presence of deflection and rotation discontinuities and abrupt variations of the cross-section. The presented formulation can also be easily employed to conduct a dynamic analysis of damaged frame structures in which the distributed and concentrated damage distributions are modelled by means of equivalent discontinuities. As an example, a simple portal frame, under different damage scenarios, has been analysed and the results in terms of frequency and vibration modes have been compared with exact results to show the accuracy of the presented discontinuous beam element.     

Simultaneous solution of the Navier-Stokes and elastic membrane equations by a finite element method

Fluid flow through a significantly compressed elastic tube occurs in a variety of physiological situations. Laboratory experiments investigating such flows through finite lengths of tube mounted between rigid supports have demonstrated that the system is one of great dynamical complexity, displaying a rich variety of self-excited oscillations. The physical mechanisms responsible for the onset of such oscillations are not yet fully understood, but simplified models indicate that energy loss by flow separation, variation in longitudinal wall tension and propagation of fluid elastic pressure waves may all be important. Direct numerical solution of the highly non-linear equations governing even the most simplified two-dimensional models aimed at capturing these basic features requires that both the flow field and the domain shape be determined as part of the solution, since neither is known a priori. To accomplish this, previous algorithms have decoupled the solid and fluid mechanics, solving for each separately and converging iteratively on a solution which satisfies both. This paper describes a finite element technique which solves the incompressible Navier-Stokes equatikons simultaneously with the elastic membrane equations on the flexible boundary. The elastic boundary position is parametized in terms of distances along spines in a manner similar to that which has been used successfully in studies of viscous free surface flows, but here the membrane curvature equation rather than the kinematic boundary condition of vanishing normal velocity is used to determine these diatances and the membrane tension varies with the shear stresses exerted on it by the fluid motions. Bothy the grid and the spine positions adjust in response to membrane deformation, and the coupled fluid and elastic equations are solved by a Newton-Raphson scheme which displays quadratic convergence down to low membrane tensions and extreme states of collapse. Solutions to the steady problem are discussed, along with an indication of how the time-dependent problem might be approached.     

On variational formulations with rigid-body constraints for finite elasticity: Applications to 2D and 3D finite element simulations

We investigate a recently proposed variational principle with rigid-body constraints and present an extension of its implementation in three dimensional finite elasticity problems. Through numerical examples, we illustrate that the proposed variational principle with rigid-body constraints is applicable to both single field and mixed finite elements of arbitrary order and geometry, e.g. triangular/tetrahedral and quadrilateral/hexagonal elements, in two and three dimensions. Moreover, we demonstrate that, as compared to the commonly adopted approach of discretizing the rigid domains with standard finite elements, the proposed formulation requires neither discretization nor numerical integration in the interior of each rigid domain. As a comparative result, the variational formulation may reduce the total number of degrees of freedom of the resulting finite element system and provide improved accuracy.     

On a pure complementary energy principle and a force-based finite element formulation for non-linear elastic cables

A complementary-dual force-based finite element formulation is proposed for the geometrically exact quasi-static analysis of one-dimensional hyperelastic perfectly flexible cables lying in the two-dimensional space. This formulation employs as approximate functions the exact statically admissible force fields, i.e., those that satisfy the equilibrium differential equations in strong form, as well as the equilibrium boundary conditions. The formulation relies on a principle of total complementary energy only expressed in terms of force fields, being therefore called a pure principle. Under the assumption of stress-unilateral behavior, this principle can be regarded as being dual to the principle of minimum total potential energy, corresponding therefore to a maximum principle. Some numerical applications, including cables suspended from two and three points at the same level or at different levels, with both Hookean and Neo-Hookean material behaviors, are presented. As it will be shown, in contrast to the standard two-node displacement-based formulation derived from the principle of minimum total potential energy, the proposed dual force-based formulation is capable of providing the exact solution of a given problem only using a single finite element per cable. Both the proposed principle of pure complementary energy and its corresponding force-based finite element formulation can be easily extended to the case of cables lying in the three-dimensional space.     

Variational principles of perforated thin plates and finite element method for buckling and post-buckling analysis

On the basis of the general theory of perforated thin plates under large deflections^{[1, 2]}, variational principles with deflectionw and stress functionF as variables are stated in detail. Based on these principles, finite element method is established for analysing the buckling and post-buckling of perforated thin plates. It is found that the property of element is very complicated, owing to the multiple connexity of the region. Project supported by National Natural Science Foundation of China.     

结构有限元分析中阶跃型弹性约束的一种有效处理方法

本文通过引入弹性约束刚度矩阵和结构位移约束列阵，提出了结构有限元分析中处理阶跃型弹性约束的一种有效方法。该法通过改变弹性约束系数及位移非约束量大小，可方便有效地处理结点常弹性约束，刚性约束，阶跃型弹性的约束及阶跃型刚性约束等问题。     

On the stability of bubble functions and a stabilized mixed finite element formulation for the Stokes problem

In this paper we investigate the relationship between stabilized and enriched finite element formulations for the Stokes problem. We also present a new stabilized mixed formulation for which the stability parameter is derived purely by the method of weighted residuals. This new formulation allows equal‐order interpolation for the velocity and pressure fields. Finally, we show by counterexample that a direct equivalence between subgrid‐based stabilized finite element methods and Galerkin methods enriched by bubble functions cannot be constructed for quadrilateral and hexahedral elements using standard bubble functions. Copyright © 2008 John Wiley & Sons, Ltd.