Supercritical as well as subcritical Hopf bifurcation in nonlinear flutter systems

The Hopf bifurcations of an airfoil flutter system with a cubic nonlinearity are investigated, with the flow speed as the bifurcation parameter. The center manifold theory and complex normal form method are Used to obtain the bifurcation equation. Interestingly, for a certain linear pitching stiffness the Hopf bifurcation is both supercritical and subcritical. It is found, mathematically, this is caused by the fact that one coefficient in the bifurcation equation does not contain the first power of the bifurcation parameter. The solutions of the bifurcation equation are validated by the equivalent linearization method and incremental harmonic balance method.     

Base force element method of complementary energy principle for large rotation problems

Using the concept of the base forces, a new finite element method （base force element method, BFEM） based on the complementary energy principle is presented for accurate modeling of structures with large displacements and large rotations. First, the complementary energy of an element is described by taking the base forces as state variables, and is then separated into deformation and rotation parts for the case of large deformation. Second, the control equations of the BFEM based on the complementary energy principle are derived using the Lagrange multiplier method. Nonlinear procedure of the BFEM is then developed. Finally, several examples are analyzed to illustrate the reliability and accuracy of the BFEM.     

New discrete element models for elastoplastic problems

The discrete element method （DEM） has attractive features for problems with severe damages, but lack of theoretical basis for continua behavior especially for nonlinear behavior has seriously restricted its application. The present study proposes a new approach to developing the DEM as a general and robust technique for modeling the elastoplastic behavior of solid materials. New types of connective links between elements are proposed, the interelement parameters are theoretically determined based on the principle of energy equivalence and a yield criterion and a flow rule for DEM are given for describing nonlinear behavior of materials. Moreover, a numerical scheme, which can be applied to modeling the behavior of a continuum as well as the transformation from a continuum to a discontinuum, is obtained by introducing a fracture criterion and a contact model into the DEM. The elastoplastic stress wave propagations and the tensile failure process of a steel plate are simulated, and the numerical results agree well with those obtained from the finite element method （FEM） and corresponding experiment, and thus the accuracy and efficiency of the DEM scheme are demonstrated.     

THE LARGE DEFLECTION ON PROBLEM OF CIRCULAR PLATE ON ELASTIC FOUNDATION AND IN CONJUNCTION WITH LINEAR ELASTIC STRUCTURE

This paper deals with the axisymmetrical deformation of the circular plate in largedeflection,which is on elastic foundation and in conjunction with a certain linear elasticstructure.The governing integral equations are established by the method of mixedboundary condition I and the simplified form is given.The pertrubation method is used toobtain the solutions and an example of the composite structure made up of a circular plateand a cylindrical shell is presented.     

An iterative parallel algorithm of finite element method

In this paper, a parallel algorithm with iterative form for solving finite element equation is presented. Based on the iterative solution of linear algebra equations, the parallel computational steps are introduced in this method. Also by using the weighted residual method and choosing the appropriate weighting functions, the finite element basic form of parallel algorithm is deduced. The program of this algorithm has been realized on the ELXSI-6400 parallel computer of Xi'an Jiaotong University. The computational results show the operational speed will be raised and the CPU time will be cut down effectively. So this method is one kind of effective parallel algorithm for solving the finite element equations of large-scale structures.     

Reduced projection augmented Lagrange bi-conjugate gradient method for contact and impact problems

Based on the numerical governing formulation and non-linear complementary conditions of contact and impact problems,a reduced projection augmented Lagrange bi- conjugate gradient method is proposed for contact and impact problems by translating non-linear complementary conditions into equivalent formulation of non-linear program- ming.For contact-impact problems,a larger time-step can be adopted arriving at numer- ical convergence compared with penalty method.By establishment of the impact-contact formulations which are equivalent with original non-linear complementary conditions,a reduced projection augmented Lagrange bi-conjugate gradient method is deduced to im- prove precision and efficiency of numerical solutions.A numerical example shows that the algorithm we suggested is valid and exact.     

Weight functions for interface cracks in dissimilar anisotropic materials

Bueckner‘s work conjugate integral customarily adopted for linear elastic materials is established for an interface crack in dissimilar anisotropic materials. The difficulties in separating Stroh‘s six complex arguments involved in the integral for the dissimilar materials are overcome and then the explicit function representations of the integral are given and studied in detail. It is found that the pseudo-orthogonal properties of the eigenfunction expansion form (EEF) for a crack presented previously in isotropic elastic cases, in isotopic bimaterial cases, and in orthotropic cases are also valid in the present dissimilar arbitrary anisotropic cases. The relation between Bueckner‘s work conjugate integral and the J-integral in these cases is obtained by introducing a complementary stressdisplacement state. Finally, some useful path-independent integrals and weight functions are proposed for calculating the crack tip parameters such as the stress intensity factors.     

An effective boundary method for the analysis of elastoplastic problems

In this paper, a series of effective formulae of the boundary element method is presented. In these formulae, by using a new variable two kernels are only of the weaker singularity of Lnr (where r is the distance between a source point and a field point). Hence, the singularities in the conventional displacement formulation and stress formulation at internal points are reduced respectively so that the "boundary-layer" effect which strongly degenerates the accuracy of stress calculation by using original formulae is eliminated. Also the direct evaluation of coefficients C (boundary tensor), which are difficult to calculate, is avoided. This method is used in elastoplastic analysis. The results of the numerical investigation demonstrate the potential advantages of this method.     

MESHLESS METHOD FOR 2D MIXED-MODE CRACK PROPAGATION BASED ON VORONOI CELL

A meshless method integrated with linear elastic fracture mechanics (LEFM) is presented for 2D mixed-mode crack propagation analysis. The domain is divided automatically into sub-domains based on Voronoi cells, which are used for quadrature for the potential energy. The continuous crack propagation is simulated with an incremental crack-extension method which assumes a piecewise linear discretization of the unknown crack path. For each increment of the crack extension, the meshless method is applied to carry out a stress analysis of the cracked structure. The J-integral, which can be decomposed into mode I and mode II for mixed-mode crack, is used for the evaluation of the stress intensity factors (SIFs). The crack-propagation direction, predicted on an incremental basis, is computed by a criterion defined in terms of the SIFs. The flowchart of the proposed procedure is presented and two numerical problems are analyzed with this method. The meshless results agree well with the experimental ones, which validates the accuracy and efficiency of the method.     

THE LINEAR APPROXIAIATION OF THE LINE CONTINUOUS DISTRIBUTION METHOD OF SINGULARITIES IN CREEPING MOTION

The linear approximation of the line continuous distri-bution method of singularities is proposed to treat the creeping motion of the arbitrary prolate axisymmetrical body. The analytic expressions in closed form for the flow field are obtained. The numerical results for the proiate spheroid and Cassini oval demonstrate that the convergence and the accuracy of the proposed method are better than the constant density approximation, furthermore, it can be applied to greater slender ratio. In this paper the example is yielded to show that the linear approximation of the singularities for the density on the partitioned segments can be utilized to consider the creeping motion of the arbitrary pointed prolaue axisymmetrical body.     

A FINITE ELEMENT—MATHEMATICAL PROGRAMMING METHOD FOR ELASTOPLASTIC PROBLEMS BASED ON THE PRINCIPLE OF VIRTUAL WORK

By expanding the yielding function according to Taylor series and neglecting the high order terms, the elastoplastic constitutive equation is written in a linear complementary form. Based on this linear complementary form and the principle of virtual work, a finite element-complementary method is derived for elastoplastic problem. This method is available for materials which satisfy either associated or nonassociated flow rule. In addition, the existence and uniqueness of solution for the method are also discussed and some useful conclusions are given. The project is supported by the National Natural Science Foundation of China     

Axisymmetric Thermoelastoplastic Stress State of Isotropic Solids of Revolution Under Impulsive Loading

Consideration is given to the solution of a dynamic problem for a solid of revolution with an arbitrary meridional section under impulsive thermomechanical loading inducing elastoplastic strains. The theory of small elastoplastic deformations is used. The constitutive equations are linearized by the variable-parameter method. The unloading process is described by a linear law. The solution technique involves the finite-element approximation in spatial coordinates and the finite-difference representation of time derivatives. Based on the principle of linear summation, recurrent relations are derived for successive evaluation of nodal displacements by an explicit scheme in time. Solution for cylinders and disks are presented to illustrate the influence of elastoplastic deformations on wave processes     

弹塑性接触问题的非光滑非线性方程组方法

将求解三维弹性摩擦接触问题的非光滑非线性方程组方法推广到弹塑性(Mises材料)情形，提出了两种应用方法：一种是将非光滑非线性方程组方法和求解弹塑性问题常用的Newton—Raphson迭代方法结合起来；另一种是将问题写成统一的非光滑非线性方程组，直接求解。数值算例验证了两种方法的有效性，并进行了结果比较。     

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...     

Generalized Inverses in Elastic-Plastic Analysis of Structures∗

This paper deals with elastic-plastic analysis of skeletal structures that are subjected to proportionally increasing loading. It is assumed that no local unloading occurs (holonomic behavior) and that yield conditions are piecewise linearized. A quadratic programming problem, which arises from the application of the minimum complementary energy principle for such structures, is shown to have an explicit form of solution. The matrix expression of this solution involves certain modifications of the Bott-Duffin generalized inverse. This inverse can be effectively calculated for a given structure and allows one to obtain a unique stress distribution under agiven load. Moreover, if the load is prescribed up to an unknown scalar factor, the ultimate value of this multiplier (collapse load) and the elastoplastic stress state at collapse can be found by the same method     

A novel semi-inverse solution method for elastoplastic torsion of heat treated rods

Torsion rods are a primary component of many power transmission and other mechanical systems. The behavior of these rods under elastoplastic torsion is of major concern for designers. Different methods have so far been proposed which deal with the elastoplastic torsion of rods, most of which assume constant yield stress. This assumption produces rough and inaccurate results when the rods are heat treated, since in the process of heat treatment the form of yield stress distribution across the rod cross section changes. We propose a new method for calculating elastoplastic torsion of rods of simply connected cross section which is based on heat treatment observations. In our method the full plastic stress function is obtained by using the semi-inverse method. Elastoplastic stress function is obtained by generalizing the idea of the membrane analogy and using a piecewise continuous stress function. Since the proposed form of yield stress distribution can not be handled by the current Finite Element packages, we produce a computer package with a 3D graphical interface capable of calculating and displaying the 3D elastoplastic stress function, shear stress contours, and torque-angle of rotation per unit length. We show that our method produces excellent agreement for several known cross sections in comparison to methods which assume constant yield stress.     

The General Mathematical Theory of Plasticity and the Il’yushin Postulates of Macroscopic Definability and Isotropy

The physical laws characterizing the relation between stresses and strains are considered and analyzed in the general modern theory of elastoplastic deformations and in its postulates of macroscopic definability and isotropy for initially isotropic continuous media. The fundamentals of this theory in continuum mechanics were developed by A.A. Il’yushin in the mid-twentieth century. His theory of small elastoplastic deformations under simple loading became a generalization of Hencky’s deformation theory of flow, whereas his theory of elastoplastic processes which are close to simple loading became a generalization of the Saint-Venant–Mises flow theory to the case of hardening media. In these theories, the concepts of simple arid complex loading processes arid the concept of directing form change tensors are introduced; the Bridgman law of volume elastic change and the universal Roche–Eichinger laws of a single hardening curve under simple loading are adopted; and the Odquist hardening for plastic deformations is generalized to the case of elastoplastic hardening media for the processes of almost simple loading without consideration of a specific history of deformations for the trajectories with small arid mean curvatures. In this paper we discuss the possibility of using the isotropy postulate to estimate the effect of forming parameters in the stress-strain state appeared due to the strain-induced anisotropy during the change of the internal structures of materials. We also discuss the possibility of representing the second-rank symmetric stress and strain tensors in the form of vectors in the linear coordinate six-dimensional Euclidean space. An identity principle is proposed for tensors and vectors.     

Bending of an elastoplastic Hencky bar-chain: from discrete to nonlocal continuous beam models

The static behavior of an elastoplastic one-dimensional lattice system in bending, also called a microstructured elastoplastic beam or elastoplastic Hencky bar-chain (HBC) system, is investigated. The lattice beam is loaded by concentrated or distributed transverse monotonic forces up to the complete collapse. The phenomenon of softening localization is also included. The lattice system is composed of piecewise linear hardening–softening elastoplastic hinges connected via rigid elements. This physical system can be viewed as the generalization of the elastic HBC model to the nonlinear elastoplasticity range. This lattice problem is demonstrated to be equivalent to the finite difference formulation of a continuous elastoplastic beam in bending. Solutions to the lattice problem may be obtained from the resolution of piecewise linear difference equations. A continuous nonlocal elastoplastic theory is then built from the lattice difference equations using a continualization process. The new nonlocal elastoplastic theory associated with both a distributed nonlocal elastoplastic law coupled to a cohesive elastoplastic model depends on length scales calibrated from the spacing of the lattice model. Differential equations of the nonlocal engineering model are solved for the structural configurations investigated in the lattice problem. It is shown that the new micromechanics-based nonlocal elastoplastic beam model efficiently captures the scale effects of the elastoplastic lattice model, used as the reference. The hardening–softening localization process of the nonlocal continuous model strongly depends on the lattice spacing which controls the size of the nonlocal length scales.     

两向地震波的输入方向对平—扭耦联系统动力反应的影响

本文按有限输入方向计算并采用线性予测的方法讨论了两向地震对平-扭耦联系统动力反应的影响.通过实例计算说明:弹塑性结构受地震波输入方向的影响显著大于相应弹性结构.对弹塑性结构而言,在本质上不存在确定不变的最不利地震波输入方向.     

Nonaxisymmetric Thermoelastoplastic Analysis of Branched Shells of Revolution by the Semianalytic Finite-Element Method

A technique is proposed to solve elastoplastic deformation problems for branched shells of revolution under the action of asymmetric forces and a temperature field. The kinematic equations are derived within the framework of the linear Kirchhoff–Love theory of shells and the thermoelastic relations within the framework of the theory of small elastoplastic strains. The problem is given a variational formulation based on the virtual-displacement principle and the Fourier-series expansion of the unknown functions and loads with respect to the circumferential coordinate. The additional-load method is used to solve a nonlinear problem and the finite-elements method is used to carry out a numerical analysis. As an example, an asymmetric stress–strain analysis is performed for a cylindrical shell reinforced by a ring plate.