A (3,2) reduced degree-of-freedom unified zigzag laminated beam theory

A (3,2) unified zigzag beam theory is developed with a reduced number of degree-of-freedom. Comparing to previous methods in the field of zigzag beam theory, the main novelty in this paper's method is that a more general non-vanishing top/bottom surface's shear stress boundary conditions are satisfied automatically in strong form. The bottom surface shear stress condition and the interface shear stress continuity conditions are used to uniquely determine the coefficients of zigzag functions. For the top surface shear stress condition, it is used to eliminate one degree-of-freedom, changing the 7°-of-freedom (3,2) zigzag beam to a 6°-of-freedom (3,2) zigzag beam. The zigzag coefficients are derived with an explicit formulation. Since the proposed method's formula is based on the unified beam theory, the formulation can be applied to any specific beam theory. The corresponding zigzag coefficients are also dependent on the specific beam theory's thickness basis function.In the numerical test section, several benchmark problems are solved to verify the accuracy. It is observed that the proposed beam has accurate solution for both thick and thin beams. The shear stress accuracy is also good for both vanishing and non-vanishing shear stress boundary conditions on top/bottom surfaces.     

High-order tetrahedral finite elements applied to large deformation analysis of functionally graded rubber-like materials

In this paper, high-order tetrahedral finite elements are employed to analyze structures and solids composed of functionally graded rubber-like materials under finite displacements, finite strains, statically applied forces and isothermal conditions. In order to do so, the following concepts are used: geometrically nonlinear analysis, Green–Lagrange strain tensor, second Piola–Kirchhoff stress tensor, hyperelastic constitutive relations, isoparametric solid tetrahedral finite elements of any order of approximation, and functionally graded materials. The equilibrium of the body is achieved via the Principle of the Stationary Total Potential Energy. The elements are fully integrated via Gaussian quadratures, and the resultant processing time is reduced by means of parallel techniques. To solve the nonlinear system of equations, the Newton–Raphson iterative procedure is employed.The proposed formulation is validated by benchmark problems such as: the Cook’s membrane and the thick cylinder. Other interesting simulation, the Cook’s block is proposed in order to evaluate high strain gradient situations. The results show that, in the context of the present study, locking-free behavior is obtained with simple mesh refinement.     

基于曲率插值的大变形梁单元

线性梁单元的形函数在单元大转动时会引起虚假应变,不适用于几何非线性分析．传统的几何非线性梁单元由于位移插值和转角插值的相干性,常常引起剪切闭锁等问题．该文 提出了一种平面大变形梁单元,通过单元域内的曲率插值以及曲率与节点位移之间的函数关系,将单元节点力和节点位移表示为节点曲率的函数．由于曲率插值本质上是对梁的应变进行插值,保证了单元任意刚体运动不会产生虚假的节点力;且将梁的截面形心位移表示为曲率的函数,避免了传统单元中的剪切闭锁问题．因而所提方法特别适用于梁的几何非线性分析．数值算例说明了所提方法的正确性和有效性.     

基于剪切效应纤维梁单元的结构非线性有限元数值模拟

基于Euler-Bernoulli梁理论的经典纤维模型忽略了剪切变形给截面带来的影响,为了得到更加精确的梁单元模型,该文基于考虑剪切效应的纤维梁单元,根据Timoshenko梁理论,推导了该纤维梁单元的刚度矩阵,并结合弹塑性增量理论,同时考虑了几何非线性和材料非线性的双重影响,建立了压弯剪复杂应力状态下结构非线性有限元...     

Nonlinear,finite deformation,finite element analysis

The roles of the consistent Jacobian matrix and the material tangent moduli, which are used in nonlinear incremental finite deformation mechanics problems solved using the finite element method, are emphasized in this paper, and demonstrated using the commercial software ABAQUS standard. In doing so, the necessity for correctly employing user material subroutines to solve nonlinear problems involving large deformation and/or large rotation is clarified. Starting with the rate form of the principle of virtual work, the derivations of the material tangent moduli, the consistent Jacobian matrix, the stress/strain measures, and the objective stress rates are discussed and clarified. The difference between the consistent Jacobian matrix (which, in the ABAQUS UMAT user material subroutine is referred to as DDSDDE) and the material tangent moduli (C^e) needed for the stress update is pointed out and emphasized in this paper. While the former is derived based on the Jaumann rate of the Kirchhoff stress, the latter is derived using the Jaumann rate of the Cauchy stress. Understanding the difference between these two objective stress rates is crucial for correctly implementing a constitutive model, especially a rate form constitutive relation, and for ensuring fast convergence. Specifically, the implementation requires the stresses to be updated correctly. For this, the strains must be computed directly from the deformation gradient and corresponding strain measure (for a total form model). Alternatively, the material tangent moduli derived from the corresponding Jaumann rate of the Cauchy stress of the constitutive relation (for a rate form model) should be used. Given that this requirement is satisfied, the consistent Jacobian matrix only influences the rate of convergence. Its derivation should be based on the Jaumann rate of the Kirchhoff stress to ensure fast convergence; however, the use of a different objective stress rate may also be possible. The error associated with energy conservation and work-conjugacy due to the use of the Jaumann objective stress rate in ABAQUS nonlinear incremental analysis is viewed as a consequence of the implementation of a constitutive model that violates these requirements.     

新型空间薄壁梁单元

基于Timoshenko梁理论和Vlasov薄壁杆件约束扭转理论,建立了具有内部结点的新型空间薄壁截面梁单元．通过对弯曲转角和翘曲角采取独立插值的方法,考虑了横向剪切变形,扭转剪切变形及其耦合作用,弯曲变形和扭转变形的耦合以及二次剪应力等因素影响,由Hellinger-Reissner广义变分原理,推得单元刚度矩阵．算例表明所建模型具有良好的精度,可用于空间薄壁杆系结构的有限元分析．     

Uniqueness and stability result for Cauchy’s equation of motion for a certain class of hyperelastic materials

We consider Cauchy’s equation of motion for hyperelastic materials. The solution of this nonlinear initial-boundary value problem is the vector field which discribes the displacement which a particle of this material perceives when exposed to stress and external forces. This equation is of greatest relevance when investigating the behavior of elastic, anisotropic composites and for the detection of defects in such materials from boundary measurements. This is why results on unique solvability and continuous dependence from the initial values are of large interest in materials’ research and structural health monitoring. In this article we present such a result, provided that reasonable smoothness assumptions for the displacement field and the boundary of the domain are satisfied for a certain class of hyperelastic materials where the first Piola–Kirchhoff tensor is written as a conic combination of finitely many, given tensors.     

极性连续统的增率型运动方程和边界条件

推导出了各种偶应力张量间和它们的变率间的关系,并建立起增率型角动量方程及其相应的边界条件。于是,把这些结果和匡震邦在“非线性连续介质力学基础”中给出的经典连续统力学的相应结果组合起来即得Cauchy形式和Piola形式以及Kirchhoff形式的极性连续统的增率型运动方程和边界条件。     

Large-amplitude free vibrations of functionally graded beams by means of a finite element formulation

The large-amplitude free vibration analysis of functionally graded beams is investigated by means of a finite element formulation. The Von-Karman type nonlinear strain–displacement relationships are employed where the ends of the beam are constrained to move axially. The effects of the transverse shear deformation and rotary inertia are included based upon the Timoshenko beam theory. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. A statically exact beam element which devoid the shear locking effect with displacement fields based on the first order shear deformation theory is used to study the geometric nonlinear effects on the vibrational characteristics of functionally graded beams. The finite element method is employed to discretize the nonlinear governing equations, which are then solved by the direct numerical integration technique in order to obtain the nonlinear vibration frequencies of functionally graded beams with different boundary conditions. The influences of power-law exponent, vibration amplitude, beam geometrical parameters and end supports on the free vibration frequencies are studied. The present numerical results compare very well with the results available from the literature where possible. Some new results for the nonlinear natural frequencies are presented in both tabular and graphical forms which can be used for future references.     

Derivation of the consistent flexibility matrix for geometrically nonlinear Timoshenko frame finite element

A consistent flexibility matrix is presented for a large displacement equilibrium-based Timoshenko beam–column element. This development is an improvement and extension to Neuenhofer–Filippou [1] (1998. ASCE J. Struct. Eng. 124, 704–711) for geometrically nonlinear Euler–Bernoulli force-based beam element. In order to find weak form compatibility and strong form equilibrium equations of the beam, the Hellinger–Reissner potential is expressed. During the formulation process, an extended displacement interpolation technique named curvature/shearing based displacement interpolation (CSBDI) is proposed for the strain–displacement relationship. Finally, the extended CSBDI technique is validated for geometric nonlinear examples and accuracy of the method is investigated concluding improved convergence rates with respect to the general finite element formulation. Also it is seen that the use of force based formulation removes shear locking effects. The results demonstrate considerable accuracy even in presence of high axial loading in comparison with the displacement based approach.     

Homogenisation of random composites via the multiscale finite‐element method

The transition between the chosen microstructure and microvariables and the material properties on the macrolevel is always a sensitive point in the theory of homogenisation. In this talk we will observe the transfer of data between the scales based on the multiscale finite element method where in each Gauss point of the macromesh a micromesh is attached. For a given deformation gradient provided from the macroscale one calculates microfluctuations satisfying periodic boundary conditions and from those the effective first Piola‐Kirchhoff stress tensor for each Gauss point. The latter provides a possibility to calculate the elasticity tensor on the macrolevel. We study a microstructure containing elliptical cracks of random aspect ratio and orientation. The results based on such procedure show the dependence of the macrovariables on the crack ellipticity. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A second order theory for large deflections of slender beams

The classical theory of finite deformations for slender beams was mainly developed by Euler and Kirchhoff and assumes linear-elastic constitutive behaviour. But finite deformations lead a priori to a problem of nonlinear elasticity. Introducing dimensionless quantities, one realizes the importance of two small parameters which characterize the slenderness of the beam and the relation between the loading and the stiffness of the material. The asymptotic approximation of first order with respect to these two parameters confirm in the case of the “elastica” the Kirchhoff equations. For the second order approximation we restrict the discussion to the case, where the centerline of the beam bends as a plane curve in first order. Under this assumption, we derive general equations of second order and discuss some special cases. We show that local stress and strain fields of second order are always influenced by physical nonlinearities. For the second order global deformation of the centerline, the role of the physical nonlinearities depends on the shape of the cross section and its symmetry. The importance of the nonlinear effects is illustrated in some numerical examples. These applications show that physical nonlinearities have a greater influence than the geometrical nonlinearities of the problem. Thus, in higher approximations than the first, restriction to Hooke's law is not permissible in general.     

Finite element analysis of arches undergoing large rotations - I: Theoretical comparison

In this two part paper, the first part deals with five different nonlinear theories applicable to the analysis of arches in the context of solving the large displacement and the large rotation problem. These theories include, classical theory, first-order shear deformation theory, third-order shear deformation theory, modified classical theory and the Donnell-type theory. All the theories are developed using the Total Lagrangian approach. Simplifications and assumptions used in each of the theory are discussed. Explicit strain displacement gradient relations and element independent equilibrium equations in terms of displacement gradients are given for all the theories. Limitations of each of theory are discussed. In the second part of this paper, application of these theories for the classification of arch geometries is considered.     

基面力单元法在空间几何非线性问题中的应用

基于基面力的概念,并结合Euler角的位移描述方法,提出了适用于几何非线性计算的空间6结点余能基面力单元.使用MATLAB语言编程并对典型梁、板结构进行弹性大变形数值模拟.由计算结果可以看出,基于余能原理的基面力元法(BFEM)在计算构件的空间大变形时有较好的计算精度,对比传统有限元计算方法具有网格尺寸影响小和抗畸变能力强的特点,有良好的计算性能.     

Complete system of two-dimensional invariants for formulation of triangular finite element of composite shells

The paper describes a system of invariants of symmetric two-dimensional tensors defined on a plane or a surface. The system comprises the well-known first and second invariants and a new quantity called the combined invariant of two tensors. The focus is on the expression for the invariants in terms of normal components of the tensors determined in three different directions on the surface. The system of invariants is used to construct a triangular finite element for geometrically nonlinear analysis of shear deformable anisotropic shells subject to the Reissner–Mindlin assumptions. The relations obtained allow one to readily determine the strain energy of the element for the normal components of the stress and strain tensors in the direction of the element edges. Numerical examples are given to demonstrate some nonlinear capabilities of the element.     

The representation of the displacement gradient of isotropic elastic body in terms of the piola stress tensor : PMM vol. 40, n 6, 1976, pp. 1070–1077

The representation of the displacement gradient of an isotropic elastic body is analyzed. It is shown on the basis of a single controlling inequality and a polar expansion of the Piola tensor that such representation has generally four branches. The mechanical meaning and the nature of that ambiguity is explained. It is established that when the angles of turn of material fibers are not excessively large, only one of the four branches is obtained. Particular cases in which the nature of ambiguity is more complex are investigated. It is noted that in many practical problems the representation of the displacement gradient by the Piola stress tensor is unambiguous.The considered problem is associated with the variational principle of complementary energy in the nonlinear theory of elasticity, where the statistically feasible fields of the asymmetric Piola stress tensor is varied [1], A method was proposed there for expressing the displacement gradient in terms of the Piola stress tensor for an isotropic elastic body. Later the concept of complementary energy and the representation of the strain gradient in terms of the Piola stress tensor were considered in [2, 3]. Examples of the use of the complementary energy concept are given in [2] and the case of an anisotropic body is considered in [3], These investigations disclosed that the considered representation of the strain tensor leads to ambiguity, but the character and nature of the ambiguity were not fully investigated.     

功能梯度材料Timoshenko梁的热过屈曲分析

研究了功能梯度材料Timoshenko梁在横向非均匀升温下的热过屈曲．在精确考虑轴线伸长和一阶横向剪切变形的基础上,建立了功能梯度Timoshenko梁在热-机械载荷作用下的几何非线性控制方程,将问题归结为含有7个基本未知函数的非线性常微分方程边值问题A·D2其中,假设功能梯度梁的材料性质为沿厚度方向按照幂函数连续变化的形式．然后采用打靶法数值求解所得强非线性边值问题,获得了横向非均匀升温场内两端固定Timoshenko梁的静态非线性热屈曲和热过屈曲数值解．绘出了梁的变形随温度载荷及材料梯度参数变化的特性曲线,分析和讨论了温度载荷及材料的梯度性质参数对梁变形的影响．结果表明,由于材料在横向的非均匀性,均匀升温时的梁中存在拉-弯耦合变形．     

Nonlinear vibration of moderately thick laminated beams using finite element method

A finite element model is developed to study the large-amplitude free vibrations of generally-layered laminated composite beams. The Poisson effect, which is often neglected, is included in the laminated beam constitutive equation. The large deformation is accounted for by using von Karman strains and the transverse shear deformation is incorporated using a higher order theory. The beam element has eight degrees of freedom with the inplane displacement, transverse displacement, bending slope and bending rotation as the variables at each node. The direct iteration method is used to solve the nonlinear equations which are evaluated at the point of reversal of motion. The influence of boundary conditions, beam geometries, Poisson effect, and ply orientations on the nonlinear frequencies and mode shapes are demonstrated.     

An efficient mixed interpolated curved beam element for geometrically nonlinear analysis

In this study, a curved beam element is developed for geometrically nonlinear analysis of planar structures. The main contribution of this research is to use high-performance formulation to alleviate locking phenomena and consider finite rotation. This scheme is based on the mixed interpolation of the strain fields. In this study, special tying points are found and utilized. One of the interesting advantages of the proposed element is the ability to model tapered structures. Moreover, the First-order Shear Deformation Theory (FSDT) and the Green-Lagrange strain are included. Several complicated and applicable nonlinear problems are solved to depict the efficiency and high accuracy of the proposed element, especially by fewer numbers of elements.     

A homogeneous limit methodology and refinements of computationally efficient zigzag theory for homogeneous,laminated composite,and sandwich plates

The Refined Zigzag Theory (RZT) for homogeneous, laminated composite, and sandwich plates is revisited to offer a fresh insight into its fundamental assumptions and practical possibilities. The theory is introduced from a multiscale formalism starting with the inplane displacement field expressed as a superposition of coarse and fine contributions. The coarse displacement field is that of first‐order shear‐deformation theory, whereas the fine displacement field has a piecewise‐linear zigzag distribution through the thickness. The resulting kinematic field provides a more realistic representation of the deformation states of transverse‐shear‐flexible plates than other similar theories. The condition of limiting homogeneity of transverse‐shear properties is proposed and yields four distinct variants of zigzag functions. Analytic solutions for highly heterogeneous sandwich plates undergoing elastostatic deformations are used to identify the best‐performing zigzag functions. Unlike previously used methods, which often result in anomalous conditions and nonphysical solutions, the present theory does not rely on transverse‐shear‐stress equilibrium constraints. For all material systems, there are no requirements for use of transverse‐shear correction factors to yield accurate results. To model homogeneous plates with the full power of zigzag kinematics, infinitesimally small perturbations in the transverse shear properties are derived, thus enabling highly accurate predictions of homogeneous‐plate behavior without the use of shear correction factors. The RZT predictive capabilities to model highly heterogeneous sandwich plates are critically assessed, demonstrating its superior efficiency, accuracy, and a wide range of applicability. This theory, which is derived from the virtual work principle, is well‐suited for developing computationally efficient, C⁰ a continuous function of (x₁,x₂) coordinates whose first‐order derivatives are discontinuous along finite element interfaces and is thus appropriate for the analysis and design of high‐performance load‐bearing aerospace structures. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010