A polygonal finite element formulation for modeling nearly incompressible materials

The objective of the present paper is to develop a finite element formulation for modeling nearly incompressible materials at large strains using polygonal elements. The present finite element formulation is a simplified version of the three-field mixed formulation and, in particular, it reduces the functional of the internal potential energy by expressing the field of the average volume-change in terms of the displacement field, where the latter is discretized using the Wachspress shape functions. The reduced mixed formulation eliminates the volumetric locking in nearly incompressible materials and enhances the computational efficiency as the static condensation is circumvented. A detailed implementation of the finite element formulation is presented in this study. Also, different example problems, including eigenvalue analysis, nonlinear patch test and other benchmark problems are presented for demonstrating the accuracy and the reliability of the developed formulation for polygonal elements.

     

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.     

A finite element model for rate-dependent behavior of ferroelectric ceramics

In this article, materials within a crystallite are modeled by continuum particles consisting of various types of ferroelectric variants which are characterized by their mass fractions. The constitutive behavior of each type of variant is characterized by a proposed Helmholtz free energy potential. Polarization switching is modeled by continuous changes of mass fractions which are governed by a onset criterion and a kinetic relation. A finite element algorithm is developed using the virtual work principle. The simulated results on the rate dependence in the polarization and strain responses to applied alternating electric field of different frequencies are in qualitative consistence with experimental observations. The rate-dependent behavior is explained in terms of changes of mass fractions of the variants that polarization switching involves, in response to the loading programs of different loading rates.     

Dual variational formulation for Trefftz finite element method of elastic materials

A dual variational principle is presented for Trefftz finite element analysis. The proof of the stationary conditions of the variational functional and the theorem on the existence of extremum are provided in this paper. They are boundary displacement condition, surface traction condition and interelement continuity condition. Based on the assumed intraelement and frame fields, element stiffness matrix equation is obtained which can easily be implemented into computer programs for numerical analysis with Trefftz finite element method. Two numerical examples are considered to illustrate the effectiveness and applicability of the proposed element model.     

A nonlinear finite element formulation for axisymmetric torsion of biphasic materials

In this paper, we present a finite element formulation for describing the large deformation torsional response of biphasic materials, with specific application to prediction of nonlinear coupling between torsional deformation and fluid pressurization in articular cartilage. Due to the use of a cylindrical coordinate system, a particular challenge arises in the linearization of the weak form. The torsional axisymmetric case considered gives rise to additional geometric terms, which are important for the robustness of the numerical implementation and that would not be present in a Cartesian formulation. A detailed derivation of this linearization process is given, couched in the context of a variational formulation suitable for finite element implementation. A series of numerical parametric studies are presented and compared to experimental measurements of the time dependent response of cartilage.     

A finite element finite-strain formulation for modeling colliding blocks of Gent materials

The present paper is devoted to the analysis of the contact/impact problems with Coulomb friction and large deformation between two hyperelastic bodies of Gent model. The total Lagrangian formulation is adopted to describe the geometrically non-linear behavior. For the finite element implementation, the explicit expression of the incremental law of Gent model is derived. A first order algorithm is applied for the numerical integration of the time-discretized equation of motion. Efficiency and accuracy of the resulting method is illustrated on a two-dimensional static contact problem and a three-dimensional dynamic contact problem as compared with ANSYS simulations.     

Incremental analysis for nonlinear rubber-like materials by hybrid stress finite element

In this paper, on the basis of the incremental Reissner variational principle, a nonlinear finite element analysis has been accomplished and a formulation of hybrid stress element has been presented for incompressible Mooney rubber-like materials. The corrected terms of the non-equilibrium force and the incompressibility deviation are considered in the formulation. The computed values of numerical example agree very closely with the exact solution.     

Hahn-Tsai复合材料的非线性杂交应力有限元方法

成功建立了Hahn-Tsai复合材料模型的非线性杂交应力有限元方程,采用Newton-Raphson迭代法求解结构的非线性位移方程。在迭代过程中,为了提高计算效率可采用简单迭代法由节点位移求解单元应力场。但是,当载荷增加到一定程度以后,非线性应力场由于循环迭代而无法收敛,显然,一般的加速方法不能解决这种循环迭代的发散问题。因此,本文发展了一种确实有效的非线性应力场迭代新方法,在不增加计算工作量的情况下,不仅极大地提高了收敛速度,而且对于较大载荷也能够很好地收敛,从而解决了大载荷下非线性杂交元方法失败的关键问题。数值算例表明该方法是确实可行的。     

A variational formulation for plate buckling problems by the hybrid finite element method

A functional is derived for development of stress hybrid finite elements for plate buckling problems. The equilibrium equations inside the element are identically satisfied in terms of Southwell stress functionsand the transverse displacement. Along the boundary of the element further displacement and normal slope functions are employed. These functions are so chosen as to satisfy the interelement compatibility requirements when the elements are connected. The boundary and internal displacements are selected entirely independently and comments are made on the choice of interpolation functions for the internal displacement.The stationarity of the functional is shown to lead to satisfaction of the equilibrium conditions along interelement boundaries, and the compatibility conditions inside the elements. The paper includes the details of a simple rectangular element and the results of a number of plate buckling problems analysed by the developed element.     

Woven fabric composite material model with material nonlinearity for nonlinear finite element simulation

The objective of the current investigation is to develop a simple, yet generalized, model which considers the two-dimensional extent of woven fabric, and to have an interface with nonlinear finite element codes. A micromechanical composite material model for woven fabric with nonlinear stress-strain relations is developed and implemented in ABAQUS for nonlinear finite element structural analysis. Within the model a representative volume cell is assumed. Using the iso-stress and iso-strain assumptions the constitutive equations are averaged along the thickness direction. The cell is then divided into many subcells and an averaging is performed again by assuming uniform stress distribution in each subcell to obtain the effective stress–strain relations of the subcell. The stresses and strains within the subcells are combined to yield the effective stresses and strains in the representative cell. Then this information is passed to the finite element code at each material point of the shell element. In this manner structural analysis of woven composites can be performed. Also, at each load increment global stresses and strains are communicated to the representative cell and subsequently distributed to each subcell. Once stresses and strains are associated to a subcell they can be distributed to each constituent of the subcell i.e. fill, warp, and resin. Consequently micro-failure criteria (MFC) can be defined for each constituent of a subcell and the proper stiffness degradation can be modeled if desired. This material model is suitable for implicit and could be modified for explicit finite element codes to deal with problems such as crashworthiness, impact, and failure analysis under static loads.     

无额外自由度广义有限元非线性分析

将无额外自由度的广义有限元法由线弹性分析扩展到弹塑性大变形分析.局部强化函数的构建依赖于已有节点,不引入额外自由度,避免了线性相关性问题.在更新拉格朗日框架下,通过控制方程弱形式的线性化推导得到了节点内力的率形式,并分为材料和几何两部分.考虑超弹性和亚弹-塑性两种材料模型,采用Newton-Raphson迭代求解,给出...     

A hybrid finite element formulation for polycrystal plasticity with consideration of macrostructural and microstructural linking

A hybrid finite element formulation for the plastic deformation of FCC metals with anisotropy is outlined. Polycrystal plasticity theory is used to develop the constitutive response. The hybrid approach facilitates introduction of the microscale stress in the macroscopic statement of equilibrium. Convergence of the hybrid formulation is contrasted with that of a velocity-pressure formulation. It is demonstrated that the hybrid formulation is well suited for studies where significant spatial variations in constitutive response result from having only one, or a very few, crystal orientations represented in each finite element. A simulation of channel die compression is made with one crystal per finite element. The resulting texture evolution is compared with other texture evolution models and experimental data for cold rolled aluminum. It is demonstrated that the brass texture component, observed in the experimental data, is developed through shear deformations arising from grain-to-grain interactions.     

Non-linear static-dynamic finite element formulation for composite shells

Three non-linear finite element formulations for a composite shell are discussed. They are the simplified large rotation (SLR), the large displacement large rotation (LDLR), and the Jaumann analysis of general shells (JAGS). The SLR and the LDLR theories are based on total Lagrangian approach, and the JAGS is based on a co-rotational approach. Both the SLR and LDLR theories represent the in-plane strains exactly the same as Green's strain-displacement relations, whereas, only linear displacement terms are used to represent the transverse shear strain. However, a higher order kinematic through the thickness assumption is used in the SLR theory, which leads to parabolic transverse shear stress distribution compared to a first order kinematic through the thickness relationship used in the LDLR theory that leads to linear transverse shear stress distribution. Furthermore, the LDLR theory uses an Euler-like angle in the kinematics to account for the large displacement and rotation. The JAGS theory decomposes the deformation into stretches and rigid body rotations, where an orthogonal coordinate system translates and rotates with the deformed infinitesimal volume element. The Jaumann stresses and strains are used. Layer-wise stretching and shear warping through the thickness functions are used to model the three-dimensional behavior of the shell, where displacement and stress continuities are enforced along the ply interfaces. The kinematic behavior is related to the original undeformed coordinate system using the global displacements and their derivatives. Numerical analyses of composite shells are performed to compare the three theories. The commercial code ABAQUS is also used in this investigation as a comparison.     

Materially and geometrically non-linear woven composite micro-mechanical model with failure for finite element simulations

A computational micro-mechanical material model of woven fabric composite material is developed to simulate failure. The material model is based on repeated unit cell approach. The fiber reorientation is accounted for in the effective stiffness calculation. Material non-linearity due to the shear stresses in the impregnated yarns and the matrix material is included in the model. Micro-mechanical failure criteria determine the stiffness degradation for the constituent materials. The developed material model with failure is programmed as user-defined sub-routine in the LS-DYNA finite element code with explicit time integration. The code is used to simulate the failure behavior of woven composite structures. The results of finite element simulations are compared with available test results. The model shows good agreement with the experimental results and good computational efficiency required for finite element simulations of woven composite structures.     

Analysis of the electromechanical behavior of ferroelectric ceramics based on a nonlinear finite element model

A nonlinear finite element (FE) model based on domain switching was proposed to study the electromechanical behavior of ferroelectric ceramics. The incremental FE formulation was improved to avoid any calculation instability. The problems of mesh sensitivity and convergence, and the efficiency of the proposed nonlinear FE technique have been assessed to illustrate the versatility and potential accuracy of the said technique. The nonlinear electromechanical behavior, such as the hysteresis loops and butterfly curves, of ferroelectric ceramics subjected to both a uniform electric field and a point electric potential has been studied numerically. The results obtained are in good agreement with those of the corresponding theoretical and experimental analyses. Furthermore, the electromechanical coupling fields near (a) the boundary of a circular hole, (b) the boundary of an elliptic hole and (c) the tip of a crack, have been analyzed using the proposed nonlinear finite element method (FEM). The proposed nonlinear electromechanically coupled FEM is useful for the analysis of domain switching, deformation and fracture of ferroelectric ceramics.The project supported by the National Natural Science Foundation of China (10025209, 10132010 and 90208002), the Research Grants of the Council of the Hong Kong Special Administrative Region, China (HKU7086/02E) and the Key Grant Project of the Chinese Ministry of Education (0306)     

Variational formulation of a three-dimensional surface-related solid-shell finite element

The solution of structural analysis problems, especially of shell structures, demands an efficient numerical solution strategy. Since unilateral contact problems are investigated, the shell model is formulated with respect to one of the outer surfaces, i.e., the shell formulation is surface-related. In particular, the investigation of textile reinforced strengthening layers (Brameshuber (ed.) in State-of-the-Art Report of RILEM Technical Commitee 201—TRC, 2006) will be carried out by this approach. Since shells are three-dimensional structures, i.e., bodies, the field equations of continuum mechanics are the starting point. This set of partial differential equations with pertinent boundary conditions has to be solved. An efficient numerical solution of this problem becomes easier, if the problem is reformulated using variational formalism. A corresponding mathematically abstract formulation of the underlying variational principle of the three-dimensional surface-related solid-shell finite element is stated. The discretization of the mathematically abstract principle is, among others, the source of several locking phenomena. The presented shell formulation assumes linear shell kinematics with six displacement parameters, circumventing a rotation formulation. This low-order shell kinematics produces parasitical strains and stresses, leading to poor approximations of the solution or even useless results. Therewith, extensions and/or adjustments of well-known techniques to prevent or at least reduce locking like the assumed natural strain method (Simo and Hughes in J Appl Mech 53:52–54, 1986) and the enhanced assumed strain method (Simo and Rifai in Int J Numer Methods Eng 29:1595–1638, 1990) have to be carried out. Using these adapted methods, a reliable and efficient solid-shell element with tremendously reduced locking properties is obtained. This concept comprises the utilization of unmodified three-dimensional constitutive relations by a minimal number of kinematical parameters. Finally, two nonlinear examples illustrate the reliability and the efficiency of the new solid-shell element.     

More generalized hybrid variational principle and corresponding finite element model

According to recent studies of the generalized variational principle by Professor Chien Weizang, the more generalized hybrid variational principle for finite element method is given, from which a new kind of the generalized hybrid element model is etablished.Using the thin plate bending element with varying thickness as an example, we compare various hybrid elements based on different generalized variational principles.     

基于新型裂尖杂交元的压电材料断裂力学研究

提出了一种裂尖邻域杂交元模型,将其与标准杂交应力元结合来求解压电材料裂纹尖 端的奇性电弹场和断裂参数的数值解．裂纹尖端杂交元的建立步骤为：1) 利用高次内插有限元特征法求解特征问题,得到反映裂尖奇异性电弹场状况的特 征值和特征角分布函数;2) 利用广义Hellinger-Reissner变分泛函以及特征问题的解来建立裂尖邻域杂交元模型．该 方法求解电弹场时,摒弃了传统有限元方法中裂尖奇异性场需要借助解析解的做法,也避免 了单纯有限元方法中需要在裂尖端部进行高密度单元划分．采用PZT5板中心裂纹问题 作为考核例,数值结果显示了良好的精确性．作为进一步应用,求解了含中心界面裂纹 的PZT4-PZT5两相压电材料的应力强度因子和电位移强度因子．所有的算例都考虑 了3种裂纹面电边界条件．