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.     

浅曲梁大挠度分析的余能有限元方法

以高玉臣提出的弹性大变形余能原理为基础,利用Lagrange乘子,放松平衡方程和力边界条件对余能泛函的约束,推导出广义的余能原理.根据极分解定理,将变形分为刚性转动和纯变形两部分,则余能也包含相应的两部分,一部分与刚性转动有关,而另一部分与纯变形有关.使用线弹性本构关系,建立了可用于几何非线性计算的有限元模型.应用更新的Lagrange列式法,给出了增量形式的有限元公式.数值计算结果表明,该方法可用于浅曲粱的几何大变形计算.     

A finite element implementation of the stress gradient theory

In this contribution, a finite element implementation of the stress gradient theory is proposed. The implementation relies on a reformulation of the governing set of partial differential equations in terms of one primary tensor-valued field variable of third order, the so-called generalised displacement field. Whereas the volumetric part of the generalised displacement field is closely related to the classic displacement field, the deviatoric part can be interpreted in terms of micro-displacements. The associated weak formulation moreover stipulates boundary conditions in terms of the normal projection of the generalised displacement field or of the (complete) stress tensor. A detailed study of representative boundary value problems of stress gradient elasticity shows the applicability of the proposed formulation. In particular, the finite element implementation is validated based on the analytical solutions for a cylindrical bar under tension and torsion derived by means of Bessel functions. In both tension and torsion cases, a smaller is softer size effect is evidenced in striking contrast to the corresponding strain gradient elasticity solutions.

     

Continuum damage models with non-conventional finite element formulations

In recent years, some research effort has been devoted to the development of non-conventional finite element models for the analysis of concrete structures. These models use continuum damage mechanics to represent the physically non-linear behavior of this quasi-brittle material. Two alternative approaches proved to be robust and computationally competitive when compared with the classical displacement finite element implementations. The first corresponds to the hybrid-mixed stress model where both the effective stress and the displacement fields are independently modeled in the domain of each finite element and the displacements are approximated along the static boundary, which is considered to include the inter-element edges. The second approach corresponds to a hybrid-displacement model. In this case, the displacements in the domain of each element and the tractions along the kinematic boundary are independently approximated. Since it is a displacement model, the inter-element boundaries are now included in the kinematic boundary. In both models, complete sets of orthonormal Legendre polynomials are used to define all approximations required, so very effective p-refinement procedures can be implemented. This paper illustrates the numerical performance of these two alternative approaches and compares their efficiency and accuracy with the classical finite element models. For this purpose, a set of numerical tests is presented and discussed.     

Finite deformation theory for soft ferromagnetic elastic solids

A general theory of finite deformation of soft ferromagnetic elastic solids is formulated following the linear theory developed earlier by Pao and Yeh. The constitutive equations, field equations, and the boundary conditions of this theory are applied to analyse the buckling of a plate under the action of a uniform magnetic field. A nontrivial equilibrium configuration for the deformed plate is shown to exist, and the critical value of the externally applied magnetic induction at which the plate buckles is determined. It is demonstrated that the non-linear deformation affects the critical magnetic induction considerably.     

粘弹性固体的精细积分有限元算法

粘弹性固体本构方程的数学表达式分为微分型和积分型两种，其数值求解主要是时域上离散计算。文中从微分型表达式出发导出其状态空间方程的数学表达式，通过严格推导论证了它与微、积分型表达式的等价性；引入状态空间方程，从而利用精细积分格式来求解粘弹性固体本构方程；给出了粘弹性固体本构方程的精细积分有限元算法，为求解粘弹性固体本构方程的数值解提供了一个新的途径，具有计算简便，求解精度高等优点。     

Macroscopical non-linear material model for ferroelectric materials inside a hybrid finite element formulation

A new approach for modeling hysteretic non-linear ferroelectric ceramics is presented, based on a fully ferroelectric/ferroelastic coupled macroscopic material model. The material behavior is described by a set of yield functions and the history dependence is stored in internal state variables representing the remanent polarization and the remanent strain. For the solution of the electromechanical coupled boundary value problem, a hybrid finite element formulation is used. Inside this formulation the electric displacement is available as nodal quantity (i.e. degree of freedom) which is used instead of the electric field to determine the evolution of remanent polarization. This involves naturally the electromechanical coupling. A highly efficient integration technique of the constitutive equations, defining a system of ordinary differential equations, is obtained by a customized return mapping algorithm. Due to some simplifications of the algorithm, an analytical solution can be calculated. The automatic differentiation technique is used to obtain the consistent tangent operator. Altogether this has been implemented into the finite element code FEAP via a user element. Extensive verification tests are performed in this work to evaluate the behavior of the material model under pure electrical and mechanical as well as coupled and multi-axial loading conditions.     

Buckling and post-buckling of piezoelectric plates using finite element method

The finite element equations considering the geometrical nonlinearity of piezoelectric smart structures are derived based on the total Lagrange method under the assumption of weak coupling between electricity and mechanics. Buckling and post-buckling of piezoelectric-plate with various boundary conditions are investigated. The calculated results show that piezoelectric effects and external voltage can hardly affect the buckling and post-buckling characteristics of piezoelectric-plate under uniaxial pressure while the buckling caused by displacement in-plane has much to do with the electric field. The present work is supported by the National Natural Science Foundation of China (No. 59635140).     

A method for solving the factorized vorticity-stream function equations by finite elements

A new finite element method for solving the time-dependent incompressible Navier-Stokes equations with general boundary conditions is presented. The two second-order partial differential equations for the vorticity and the stream function are factorized, apart from the non-linear advection term, by eliminating the coupling due to the double specification on the stream function at (a part of) the boundary. This is achieved by reducing the no-slip boundary conditions to projection integral conditions for the vorticity field and by evaluating the relevant quantities involved according to an extension of the method of Glowinski and Pironneau for the biharmonic problem. Time integration schemes and iterative algorithms are introduced which require the solution only of banded linear systems of symmetric type. The proposed finite element formulation is compared with its finite difference equivalent by means of a few numerical examples. The results obtained using 4-noded bilinear elements provide an illustration of the superiority of the finite element based spatial discretization.     

Post-buckling analysis of slender elastic vertical rods subjected to terminal forces and self-weight

This article presents the behavior of slender elastic rods subjected to axial terminal forces and self-weight. The mathematical formulation is presented, a solution is sought for a double-hinged boundary condition and the analysis is carried out for different values of non-dimensional weight. The formulation derives from geometrical compatibility, equilibrium of forces and moments and constitutive relations yielding a set of six first order non-linear ordinary differential equations with boundary conditions specified at both ends, which characterizes a complex two-point boundary value problem. Furthermore, a perturbation method is used to find the critical buckling loads and initial post-buckling solutions. A numerical integration scheme based on a three parameter shooting method is employed in the post-buckling solutions.     

On non-associative anisotropic finite plasticity fully coupled with isotropic ductile damage for metal forming

In this work, non-associative finite strain anisotropic elastoplasticity fully coupled with ductile damage is considered using a thermodynamically consistent framework. First, the kinematics of large strain based on multiplicative decomposition of the total transformation gradient using the rotating frame formulation, is recalled and different objective derivatives defined. By using different anisotropic equivalent stresses (quadratic and non-quadratic) in yield function and in plastic potential, the evolution equations for all the dissipative phenomena are deduced from the generalized normality rule applied to the plastic potential while the consistency condition is still applied to the yield function. The effect of the objective derivatives and the equivalent stresses (quadratic or non-quadratic) on the plastic flow anisotropy and the hardening evolution with damage is considered. Numerical aspects mainly related to the time integration of the fully coupled constitutive equations are discussed. Applications are made to the AISI 304 sheet metal by considering different loading paths as tensile, shear, plane tensile and bulge tests. For each loading path the effect of the rotating frame, the equivalent stress (quadratic or non-quadratic) and the normality rule (with respect to yield function or to the plastic potential) are discussed on the light of some available experimental results.     

Electroelastic field concentrations ahead of electrodes in multilayer piezoelectric actuators: experiment and finite element simulation

The effects of applied voltage on the electroelastic field concentrations ahead of electrodes in multilayer piezoelectric actuators were examined in a combined experimental and numerical investigation. Experiments were performed to measure the strain near internal and surface electrodes at various electrical loading conditions. The finite element method was also used to solve the coupled electro-elastic boundary value problem. The strain, stress and electric displacement concentrations were calculated and a non-linear behavior induced by localized polarization switching was discussed. A comparison of strain concentration was made between experiment and simulation.     

Drilling couples and refined constitutive equations in the resultant geometrically non-linear theory of elastic shells

It is well known that distribution of displacements through the shell thickness is non-linear, in general. We introduce a modified polar decomposition of shell deformation gradient and a vector of deviation from the linear displacement distribution. When strains are assumed to be small, this allows one to propose an explicit definition of the drilling couples which is proportional to tangential components of the deviation vector. The consistent second approximation to the complementary energy density of the geometrically non-linear theory of isotropic elastic shells is constructed. From differentiation of the density we obtain the consistently refined constitutive equations for 2D surface stretch and bending measures. These equations are then inverted for 2D stress resultants and stress couples. The second-order terms in these constitutive equations take consistent account of influence of undeformed midsurface curvatures. The drilling couples are explicitly expressed by the stress couples, undeformed midsurface curvatures, and amplitudes of quadratic part of displacement distribution through the thickness. The drilling couples are shown to be much smaller than the stress couples, and their influence on the stress and strain state of the shell is negligible. However, such very small drilling couples have to be admitted in non-linear analyses of irregular multi-shell structures, e.g. shells with branches, intersections, or technological junctions. In such shell problems six 2D couple resultants are required to preserve the structure of the resultant shell theory at the junctions during entire deformation process.     

压电/纤维复合材料耦合旋转驱动器力学设计模型

本文在经典层合板理论和三维压电材料本构方程的基础上,依据压电材料与纤维复合材料之间的相互作用、位移连续条件,建立了压电—纤维复合材料旋转耦合驱动器力学模型,并推导出电耦合方程。计算和初步实验说明,这种新型结构的压电伸缩变形—离轴纤维复合层扭转变形耦合旋转驱动器比同类型驱动器能量密度高、转矩大,且这种旋转电机结构简单紧凑、工作稳定、寿命长、预计随着研究的进一步深入,可望成为一种新型的微致驱动器而广泛应用于微型机械、精密测量和自动控制等领域。     

Generalized variational principle on nonlinear theory of naturally curved and twisted closed thin-walled composite beams

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Periodic solutions of rigid body–viscous flow interaction

This paper describes the work on extending the finite element method to cover interactions between a viscous flow and a moving body. The problem configuration of interest is that of an arbitrarily shaped body undergoing a simple harmonic motion in an otherwise undisturbed incompressible fluid. The finite element modelling is based on a primitive variables representation of the Navier-Stokes equations using curved isoparametric elements. The non-linear boundary conditions on the moving body are obtained using Taylor series expansion to approximate the velocities at the fixed finite element grid points. The method of averaging is used to analyse the resulting periodic motion of the fluid. The stability of the periodic solutions is studied by introducing small perturbations and applying Floquet theory. Numerical results are obtained for several example body shapes and compared with published experimental results. Good agreement is obtained for the basic non-linear phenomenon of steady streaming.     

Modelling of finite piezoelectric patches: Comparing an approximate power series expansion theory with exact theory

Plate equations for a plate consisting of one elastic layer and one piezoelectric layer with an applied electric voltage have previously been derived by use of power series expansions of the field variables in the thickness coordinate. These plate equations are here evaluated by the consideration of a time harmonic 2D vibration problem with finite layers. The boundary conditions at the sides of the layers then have to be considered. Numerical comparisons of the displacement field are made with solutions from two other theories; a solution with equivalent boundary conditions for a thin piezoelectric layer applied on an elastic plate and an exact solution based on Fourier series expansions. The two approximate theories are shown to be equally good and they both yield accurate results for low frequencies and thin plates.     

Impact of a finite elastic-viscoplastic bar

A method for predicting the response of strain-rate sensitive structures under dynamic loading is developed. It is based on a finite difference method, the incremental theory of plasticity, and an elastic work-hardening viscoplastic material idealization. The strain-rate effect, loading and unloading conditions, and wave interactions are automatically accounted for, and adjusted if necessary, as the deformation proceeds. No iteration is required even if the field equations are nonlinear (e.g. non-linear constitutive equations, large deformation, or complicated geometry). We solve as an example the small deflection of a finite bar with a concentrated tip mass. The accuracy is comparable to that obtained by the well-known method of characteristics, a powerful tool for solving elastic-viscoplastic wave problems but which is restricted to small deflections and simple geometry. Because of the form of the constitutive relation selected (elastic work-hardening visco-plastic), several important new features of the dynamics response are brought out. These features are not revealed when simpler, computationally-convenient constitutive relations, such as rigid ideal-viscoplastic, rigid work-hardening viscoplastic and elastic ideal-viscoplastic are used.     

Post-buckling behaviour of slender structures with a bi-linear bending moment-curvature relationship

Certain classes of slender structures of complex cross-section or fabricated from specialised materials can exhibit a bi-linear bending moment-curvature relationship that has a strong influence on their global structural behaviour. This condition may be encountered, for instance, in (a) non-linear elastic or inelastic post-buckling problems if the cross-section stiffness may be well approximated by a bi-linear model; (b) multi-layered structures such as stranded cables, power transmission lines, umbilical cables and flexible pipes where the drop in the bending stiffness is associated with an internal friction mechanism. This paper presents a mathematical formulation and an analytical solution for such slender structures with a bi-linear bending moment versus curvature constitutive behaviour and subjected to axial terminal forces. A set of five first-order non-linear ordinary differential equations are derived from considering geometrical compatibility, equilibrium of forces and moments and constitutive equations, with hinged boundary conditions prescribed at both ends, resulting a complex two-point boundary value problem. The variables are non-dimensionalised and solutions are developed for monotonic and unloading conditions. The results are presented in non-dimensional graphs for a range of critical curvatures and reductions in bending stiffness, and it is shown how these parameters affect the structure's post-buckling behaviour.