Field/boundary element approach to the large deflection of thin flat plates

The problem of large deflections of thin flat plates is rederived here using a novel integral equation approach. These plate deformations are governed by the von Karman plate theory. The numerical solution that is implemented combines both boundary and interior elements in the discretization of the continuum. The formulation also illustrates the adaptability of the boundary element technique to nonlinear problems. Included in the examples here are static, dynamic and buckling applications.     

作大范围运动弹塑性平面板的动力学分析

研究作大范围运动弹塑性平面板的动力学特性.考虑了几何非线性和材料非线性,基于平面应力假设、Mises屈服条件和流动法则,采用绝对节点坐标法,用虚功原理建立了作大范围运动弹塑性平面板的动力学方程.在数值计算时将各时刻的塑性应变储存在全局数组中,实现了塑性应变的迭代计算.通过对带集中质量、作大范围运动平面板的数值仿真研究塑性效应对系统的动力学特性的影响.     

A spline wavelet finite element formulation of thin plate bending

The wavelet scaling functions of spline wavelets are used to construct the displacement interpolation functions of triangular and rectangular thin plate elements. The displacement shape functions are then expressed by spline wavelet functions. A spline wavelet finite element formulation of thin plate bending is developed by using the virtual work principle. Two numerical examples have shown that the bending deflections and moments of thin plates agree well with those obtained by the differential equations and conventional elements. It is demonstrated that the current spline wavelet finite element method (FEM) can achieve a high numerical accuracy and converges fast. The proposed spline wavelet finite element formulation has a wide range of applicability since it is developed in the same way like conventional displacement-based FEM.     

Incremental analysis of axisymmetric shallow shells with varying strain-displacement equations

The large displacement analysis of thin, shallow, axisymmetric shells is presented by using three different formulations for the strain-displacement relations. In the most general formulation all second-order terms of strains, rotations and their derivatives are taken into account; in the intermediary formulation the second-order terms of rotations only are accounted for; and in the third formulation the shallowness assumption is introduced and the Marguerre strain-displacement relations are adopted. The effectiveness of these formulations is investigated in numerical examples of a clamped circular plate and a clamped shallow spherical shell. Finally the stability of a clamped shallow spherical shell is examined using the Marguerre formulation between strains and displacements.     

A Non-Incremental Finite Element Procedure for the Analysis of Large Deformation of Plates and Shells in Mechanical System Applications

In this investigation, a non-incremental solution procedure for the finite rotationand large deformation analysis of plates is presented. The method, whichis based on the absolute nodal coordinate formulation, leads to plateelements capable of representing exact rigid body motion. In thismethod, continuity conditions on all the displacement gradients areimposed. Therefore, non-smoothness of the plate mid-surface at the nodalpoints is avoided. Unlike other existing finite element formulationsthat lead to a highly nonlinear inertial forces for three-dimensionalelements, the proposed formulation leads to a constant mass matrix, andas a result, the centrifugal and Coriolis inertia forces are identicallyequal to zero. Furthermore, the method relaxes some of the assumptionsused in the classical and Mindlin plate models and automatically satisfiesthe objectivity requirements. By using a generalcontinuum mechanics approach, a relatively simple expression for theelastic forces is obtained. Generalization of the formulation to thecase of shell elements is discussed. As examples of the implementationof the proposed method, two different plate elements are presented; oneplate element does not guarantee the continuity of the displacementgradients between the nodal points, while the other plate elementguarantees this continuity. Numerical results are presented in order todemonstrate the use of the proposed method in the large rotation anddeformation analysis of plates and shells. The numerical results, whichare compared with the results obtained using existing incrementalprocedures, show that the solution obtained using the proposed methodsatisfies the principle of work and energy. These results are obtainedusing explicit numerical integration method. Potential applications ofthe proposed method include high-speed metal forming, vehiclecrashworthiness, rotor blades, and tires.     

Linear and non-linear deflection analysis of thick rectangular plates—II. Numerical applications

Variational methods are widely used for the solution of complex differential equations in mechanics for which exact solutions are not possible. The finite difference method, although well known as an efficient numerical method, was applied in the past only for the analysis of linear and non-linear thin plates. In this paper the suitability of the method for the analysis of non-linear deflection of thick plates is studied for the first time. While there are major differences between small deflection and large deflection plate theories, the former can be treated as a particular case of the latter, when the centre deflection of the plate is less than or equal to 0.2–0.25 of the thickness of the plate. The finite difference method as applied here is a modified finite difference approach to the ordinary finite difference method generally used for the solution of thin plate problems. In this analysis thin plates are treated as a particular case of the corresponding thick plate when the boundary conditions of the plates are taken into account. The method is first applied to investigate the deflection behaviour of clamped and simply supported square isotropic thick plates. After the validity of the method is established, it is then extended to the solution of rectangular thick plates of various aspect ratios and thicknesses. Generally, beginning with the use of a limited number of mesh sizes for a given plate aspect ratio and boundary conditions, a general solution of the problem including the investigation of accuracy and convergence was extended to rectangular thick plates by providing more detailed functions satisfying the rectangular mesh sizes generated automatically by the program. Whenever possible results obtained by the present method are compared with existing solutions in the technical literature obtained by much more laborious methods and close agreements are found. The significant number of results presented here are not currently available in the technical literature. The submatrices involved in the formation of the finite difference equations from the governing differential equations are generated directly by the computer program. The subroutine SOLINV using the change of variable technique illustrated elsewhere takes care of the solution of the general system. Simplicity in formulation and quick convergence are the obvious advantages of the finite difference formulation developed to compute small and large deflection analysis of thick plates in comparison with other numerical methods requiring extensive computer facilities.     

A geometrically exact beam element based on the absolute nodal coordinate formulation

In this study, Reissner’s classical nonlinear rod formulation, as implemented by Simo and Vu-Quoc by means of the large rotation vector approach, is implemented into the framework of the absolute nodal coordinate formulation. The implementation is accomplished in the planar case accounting for coupled axial, bending, and shear deformation. By employing the virtual work of elastic forces similarly to Simo and Vu-Quoc in the absolute nodal coordinate formulation, the numerical results of the formulation are identical to those of the large rotation vector formulation. It is noteworthy, however, that the material definition in the absolute nodal coordinate formulation can differ from the material definition used in Reissner’s beam formulation. Based on an analytical eigenvalue analysis, it turns out that the high frequencies of cross section deformation modes in the absolute nodal coordinate formulation are only slightly higher than frequencies of common shear modes, which are present in the classical large rotation vector formulation of Simo and Vu-Quoc, as well. Thus, previous claims that the absolute nodal coordinate formulation is inefficient or would lead to ill-conditioned finite element matrices, as compared to classical approaches, could be refuted. In the introduced beam element, locking is prevented by means of reduced integration of certain parts of the elastic forces. Several classical large deformation static and dynamic examples as well as an eigenvalue analysis document the equivalence of classical nonlinear rod theories and the absolute nodal coordinate formulation for the case of appropriate material definitions. The results also agree highly with those computed in commercial finite element codes.     

Direct numerical simulation of transitional flow at high Mach number coupled with a thermal wall model

In transitional and turbulent high speed boundary-layer flows the wall thermal boundary conditions play an important role and in many cases an assumption of a constant temperature or a specified heat flux may not be appropriate for numerical simulations. In this paper we extend a formulation for direct numerical simulation of compressible flows to include a thin plate that is thermally fully coupled to the flow. Even without such thermal coupling compressible flows with shock waves and turbulence represent a challenge for numerical methods. In this paper we review the scaling properties of algorithms, based on explicit high-order finite differencing combined with shock capturing, that are suitable for dealing with such flows. An application is then considered in which an isolated roughness element is of sufficient height to trigger transition in the presence of acoustic forcing. With the thermal wall model included it is observed that the plate heats up sufficiently during the simulation for the transition process to be halted and the flow consequently re-laminarises.     

基于面积坐标和解析试函数的薄板元

为体现离散法与解析法的互补和渗透,构造基于第二类四边形面积坐标的广义协调薄板元AATF-BQ4;根据薄板理论的控制方程,采用Kirchhoff直法线假设求解基本解析解,并作为试函数构造单元AATF-BQ4.数值算例表明,单元AATF-BQ4具有较高的精度和较好的稳定性,适用于实际工程计算应用.     

薄板的几何非线性求积元分析

针对薄板非线性迭代计算量很大的问题,依据von Kárman薄板非线性理论构造能量泛函,并用数值积分和数值微分进行离散,得到非线性方程组,从而利用求积元法(Quadrature Element Method,QEM)求解薄板的中等挠度的弯曲和非线性屈曲问题,得到可信的结果.算例表明:在处理薄板几何非线性问题上,QEM计算效率很高,应用潜力很大.     

Lattice Boltzmann model for elastic thin plate with small deflection

In this paper, a lattice Boltzmann model for solving problems of elastic thin plate with small deflection is proposed. In order to recover the Sophie–Germain equation for elastic thin plate by lattice Boltzmann method, we transform the equation into a set of Poisson equations. Two sets of distribution functions are employed in the lattice Boltzmann equation to recover the Poisson equations. Based on this model, some problems on the rectangular elastic thin plate with small deflection are simulated. The comparisons between the numerical results and the analysis solutions are given in detail. The numerical examples show that the lattice Boltzmann model can be used to solve problems of the elastic thin plate with small deflection.     

Finite element formulation including interlaminar stress calculations

A quadrilateral plate element is developed on the basis of utilizing the compatibility equations to obtain the in-plane stresses, and the equilibrium equations to obtain both transverse shear and normal stresses. A plate as opposed to shell or solid formulation serves to provide efficient solutions for thin to moderately thick laminated composite configurations. The element formulation involves relaxation of the Kirchhoff hypothesis via superposition of a shear rotation upon a midplane rotation. The displacement field is carefully selected to obtain the desired transverse stress variation. Results are compared to both closed form and numerical solutions.     

On the use of shape functions in the cell centered finite volume formulation for plate bending analysis based on Mindlin–Reissner plate theory

This paper deals with a new formulation of the cell centered finite volume application for plate bending analysis based on Mindlin–Reissner plate theory. In this formulation shape functions are used to represent the variation of the unknown variables across the control volumes’ faces, which facilitates the calculation of stress resultants on the faces. The performance of the formulation for the computation of displacements and stress resultants for thin and thick plates is evaluated in a number of test problems. This testing reveals that the proposed approach enhances the predictive capability of the finite volume method in the analysis of thin to thick plates.     

A study of moderately thick quadrilateral plate elements based on the absolute nodal coordinate formulation

Finite element analysis using plate elements based on the absolute nodal coordinate formulation (ANCF) can predict the behaviors of moderately thick plates subject to large deformation. However, the formulation is subject to numerical locking, which compromises results. This study was designed to investigate and develop techniques to prevent or mitigate numerical locking phenomena. Three different ANCF plate element types were examined. The first is the original fully parameterized quadrilateral ANCF plate element. The second is an update to this element that linearly interpolates transverse shear strains to overcome slow convergence due to transverse shear locking. Finally, the third is based on a new higher order ANCF plate element that is being introduced here. The higher order plate element makes it possible to describe a higher than first-order transverse displacement field to prevent Poisson thickness locking. The term “higher order” is used, because some nodal coordinates of the new plate element are defined by higher order derivatives. The performance of each plate element type was tested by (1) solving a comprehensive set of small deformation static problems, (2) carrying out eigenfrequency analyses, and (3) analyzing a typical dynamic scenario. The numerical calculations were made using MATLAB. The results of the static and eigenfrequency analyses were benchmarked to reference solutions provided by the commercially available finite element software ANSYS. The results show that shear locking is strongly dependent on material thickness. Poisson thickness locking is independent of thickness, but strongly depends on the Poisson effect. Poisson thickness locking becomes a problem for both of the fully parameterized element types implemented with full 3-D elasticity. Their converged results differ by about 18 % from the ANSYS results. Corresponding results for the new higher order ANCF plate element agree with the benchmark. ANCF plate elements can describe the trapezoidal mode; therefore, they do not suffer from Poisson locking, a reported problem for fully parameterized ANCF beam elements. For cases with shear deformation loading, shear locking slows solution convergence for models based on either the original fully parameterized plate element or the newly introduced higher order plate element.     

Large displacement analysis of plates by a stress-based finite element approach

A mixed-hybrid incremental variational formulation, involving orthogonal rigid rotations and a symmetric stretch tensor, is proposed for finite deformation analysis of thin plates and shells. Isoparametric eight-noded elements are based upon the Kirchhoff-Love hypotheses, the assumption of plane stress, and the moderately large rotations of Von Karman plate theory. Semilinear elastic isotropic material properties are assumed, and the right polar decomposition of the deformation gradient is used. The symmetrized Biot-Luré (Jaumann) stress measure gives a unique complementary energy density and a set of variational principles with a priori satisfaction of linear momentum balance, a posteriori angular momentum balance, and interelement traction reciprocity by means of Lagrange multipliers. The incremental modified Newton-Raphson solution procedure is generated by a truncated Taylor series expansion of the functional in a total Lagrangian formulation. The theory is applied to laterally loaded and buckled thin plates, and numerical results are compared with truncated series solutions.     

Modeling of transient bending wave in an infinite plate and its coupling to arbitrary shaped piezoelements

Estimating the location and energy of impacts is of primary importance for assessing the condition of structures. Particularly, such estimation can be easily obtained from the energy flow in the structures, which is usually derived from the Poynting vector. In order to measure the Poynting vector in a thin plate using piezoelements bonded on the plate, an analytical formulation of the impulse response in thin infinite plates is presented. The knowledge of the impulse response of any linear time invariant (LTI) system is precious information for the determination of its behavior under arbitrary inputs. When dealing with propagation, and especially mechanical wave propagation, a common approach consists in using numerical methods that are often time-consuming, especially for multi-coupled systems. This paper proposes a new approach for modeling the impulse response of an infinite plate with surface-bonded piezoelectric elements. The proposed analytical formulation allows bypassing numerical analysis drawbacks, in particular instabilities occurring at high frequencies, case-dependent systems and computational requirements, while giving the response for any time and space domain values using a simple convolution. The proposed model relies on flexural wave decomposition over the spatial frequency domain and corresponds to a time generalization of the angular spectrum theory, thus introducing flexural wave propagation as a time-varying spatial filter. Once the impulse is know in the spatial frequency domain, the inverse Fourier transform is applied and leads to the impulse response in the physical domain. From this model, an analytical expression of the impulse voltage response of the piezoelectric transducers and the Poynting vector can be derived quite easily. The predicted impulse response is then compared to FEM simulation results and experimental measurements in order to assess the model.     

Optimization of the reinforcement of a 3D medium with thin composite plates

The reinforcement with a thin composite plate of a 3D linear elastic medium on its external boundary or inside is considered. A linear analysis of the 3D problem leads to a variational formulation in which the reinforcement is modelled by a Kirchhoff–Love plate. Considering the sum of the compliance and a cost as the design objective, a numerical example of the optimization of this reinforcement is performed taking into account the in-plane membrane rigidity only (i.e. the bending aspects are not treated numerically).     

Experimental validation of rigid-flexible coupling dynamic formulation for hub–beam system

The aim of this paper is to compare the accuracy of the absolute nodal coordinate formulation and the floating frame of reference formulation for the rigid-flexible coupling dynamics of a three-dimensional Euler–Bernoulli beam by numerical and experimental validation. In the absolute nodal coordinate formulation, based on geometrically exact beam theory and considering the torsion effect, the material curvature of the beam is derived, and then variational equations of motion of a three-dimensional beam are obtained, which consist of three position coordinates, two slope coordinates, and one rotational coordinate. In the floating frame of reference formulation, the displacement of an arbitrary point on the beam is described by the rigid-body motion and a small superimposed deformation displacement. Based on linear elastic theory, the quadratic terms of the axial strain are neglected, and the curvatures are simplified to the first order. Considering both the linear damping and the quadratic air resistance damping, the equations of motion of the multibody system composed of air-bearing test bed and a cantilevered three-dimensional beam are derived based on the principle of virtual work. In order to verify the results of the computer simulation, two experiments are carried out: an experiment of hub–beam system with large deformation and a dynamic stiffening experiment. The comparison of the simulation and experiment results shows that in case of large deformation, the frequency result obtained by the floating frame of reference formulation is lower than that obtained by the experiment. On the contrary, the result obtained by the absolute nodal coordinate formulation agrees well with that obtained by the experiment. It is also shown that the floating frame of reference formulation based on linear elastic theory cannot reveal the dynamic stiffening effect. Finally, the applicability of the floating frame of reference formulation is clarified.     

An extension of 3-D procedure to large strain analysis of shells

A numerical stress integration procedure for general 3-D large strain problems in inelasticity, based on the total formulation and the governing parameter method (GPM), is extended to shell analysis. The multiplicative decomposition of the deformation gradient is adopted with the evaluation of the deformation gradient practically in the same way as in a general 3-D material deformation. The calculated trial elastic logarithmic strains are transformed to the local shell Cartesian coordinate system and the stress integration is performed according to the GPM developed for small strain conditions. The consistent tangent matrix is calculated as in case of small strain deformation and then transformed to the global coordinate system.A specific step in the proposed procedure is the updating of the left elastic Green–Lagrangian deformation tensor. Namely, after the stresses are computed, the principal elastic strains and the principal vectors corresponding to the stresses at the end of time step are determined. In this way the shell conditions are taken into account appropriately for the next step.Some details are given for the stress integration in case of thermoplastic and creep material model.Numerical examples include bulging of plate (plastic, thermoplastic, and creep models for metal) and necking of a thin sheet. Comparison of solutions with those available in the literature, and with solutions using other type of finite elements, demonstrates applicability, efficiency and accuracy of the proposed procedure.