An incremental procedure for deformation analysis of elastic-plastic frames

In the present paper an incremental procedure is formulated for first-order elastoplastic analysis of plane frame structures discretized into a finite number of beam elements and described by piecewise linear elastic-perfectly plastic constitutive laws, under the assumption of both reversible and irreversible behaviour of material and using piecewise linear yield conditions at any desired degree of discretization in the space of the active stress resultants (axial force, shear force and bending moment). The proposed method, by using the independent elastic-plastic kinematical compatibility equations, restrains the problem sizes within not more than twice the number of the redundant unknowns in the complete elastic frame, regardless of the degree of discretization of the piecewise linear yield conditions, still maintaining the advantage exhibited by the Mathematical Programming methods of requiring only one factorization of the matrix governing the problem when no local unloading occurs. Furthermore, the outlined algorithm allows the additional computational effort to be restrained in the case of local unloading, inasmuch as it requires a new factorization to be performed of a part of the matrix governing the problem, whose size is small with respect to the total size of the matrix.     

The incremental perturbation method for large displacement analysis of elastic-plastic structures

A procedure of applying the perturbation method is presented for the incremental numerical analysis of materially and combined non-linear problems of discrete and discretized structral systems. Small but finite strain and stress increments are strictly distinguished from the strain rates and stress rates, respectively. It is shown that, by applying the perturbation procedure not only to the non-linear strain-displacement relations and equilibrium equations, but also to the constitutive equations in terms of rate quantities, all the governing equations can be satisfied to any desired accuracy at every instantaneous configuration in between the starting and terminal points of an incremental step. The proposed method provides also means of finding, to a desired accuracy, every point on an equilibrium path of a discrete system at which a new element will start yielding or unloading and possible critical points on the path. The significance of the proposed method is expected to be appreciated particularly in numerical investigations of critical behaviours and post-buckling behaviours.     

Formal convergence characteristics of elliptically constrained incremental Newton-Raphson algorithms

Numerous situations in continuum mechanics, structural stability, optimization and related fields generate problems requiring the solution of nonlinear algebraic equations. To solve such problems, a large assortment of schemes has evolved over the years. This paper will consider the formal numerical properties of the newly developed constrained type incremental Newton-Raphson operator schemes. Specifically the evaluation of the formal behavior of the elliptically constrained version is treated in detail. Note this procedure has the versatility to efficiently handle a wide range of nonlinearities including the possibilities of positive, negative, semi and indefinite tangent properties in an inherently stable manner. The formalism includes such items as determining from both a global as well as local point of view the existence, uniqueness and convergence characteristics. Also included in the developments will be the determination of the occurrence of global safety zones wherein convergence is assured. The approach taken is general enough to provide a framework to enable applications to a wide variety of constrained schemes involving continuous, piecewise continuous, closed or open constraint conditions.     

Convergence analysis of the tabu-based real-coded small-world optimization algorithm

A novel, tabu-based real-coded small-world optimization algorithm (TR-SWA) is proposed. Tabu search is adopted to avoid duplicate searches of the real-coded small-world optimization algorithm (R-SWA). A crossover operator is introduced to construct search operators. The convergence behaviour of this TR-SWA scheme is shown by establishing the Markov model, and it is proved that TR-SWA meets the convergence theorem of a general random search algorithm proposed by Solis and Wets. Simultaneously, martingale convergence theorems are used to prove the nearly universal strong convergence of TR-SWA. Finally, five benchmark functions are introduced to evaluate the performance of TR-SWA: comparisons are made between TR-SWA, particle swarm optimization, binary-coded small-world optimization algorithm and R-SWA. Numerical experiments demonstrate that the addition of the tabu search improves the performance of R-SWA for most of the investigated optimization problems, and the global convergence of TR-SWA is guaranteed if the feasible set is bounded.     

Accelerating the convergence of elastic-plastic stress analysis

Some results are given in which a modified Aitken acceleration is applied to elastic-plastic stress analysis.     

A numerical analysis of the elastic-plastic peel test

The adhesive fracture energy, G_c, is determined from two types of elastic-plastic peel tests (i.e. the single-arm 90° and T-peel methods) and a linear-elastic fracture-mechanics (LEFM) test method (i.e. the tapered double-cantilever beam, TDCB method). A rubber-toughened epoxy adhesive, with both aluminium-alloy and steel substrates, has been used in the present work to manufacture the bonded joints. The peel tests are then modelled using numerical methods. The overall approach to modelling the elastic-plastic peel tests is to employ a finite-element analysis (FEA) approach and to model the crack advance through the adhesive layer via a node-release technique, based upon attaining a critical plastic strain in the element immediately ahead of the crack tip. It is shown that this ‘critical plastic strain fracture model (CPSFM)’ results in predicted values of the steady-state peel loads which are in excellent agreement with the experimentally-measured values. Also, the resulting values of G_c, as determined using the FEA CPSFM approach, have been found to be in excellent agreement with values from previously-reported analytical and direct-measurement methods. Further, it has been found that the calculated values of G_c are independent of whether a standard LEFM test or an elastic-plastic peel test method is employed. Therefore, it has been demonstrated that the value of the adhesive fracture energy, G_c, is independent of the geometric parameters studied and the value of G_c is indeed a characteristic of the joint, in this case for cohesive fracture through the adhesive layer. Finally, it is noted that the FEA CPSFM approach promises considerable potential for the analysis of peel tests which involve very extensive plastic deformation of the peeling arm and for analysing, and predicting, the performance of more complex adhesively-bonded geometries which involve extensive plastic deformation of the substrates.     

A remark on the application of the Newton-Raphson method in non-linear finite element analysis

Usually the notion Newton-Raphson method is used in the context of non-linear finite element analysis based on quasi-static problems in solid mechanics. It is pointed out that this is only true in the case of non-linear elasticity. In the case of constitutive equations of evolutionary-type, like in viscoelasticity, viscoplasticity or elastoplasticity, the Multilevel-Newton algorithm is usually applied yielding the notions of global and local level (iteration), as well as the consistent tangent operator. In this paper, we investigate the effects of a consistent application of the classical Newton-Raphson method in connection with the finite element method, and compare it with the classical Multilevel-Newton algorithm. Furthermore, an improved version of the Multilevel-Newton method is applied.     

Formulations in rates and increments for elastic-plastic analysis

Minimum principles in velocities, stress rates and plastic strain rates are extended in order to derive formulations for finite increments of displacement, stress and plastic strain fields defining complete numerical methods. Kinematical, statical and mixed principles are developed from a new variational formulation of the elastic-plastic work-hardening constitutive relation. The consequences of this time discretization are discussed independently of any discretization of the continuum. In particular, the incremental formulations derived from extended rate principles account for local elastic unloading and produce stress field approximations complying with equilibrium and plastic admissibility without any additional procedure, at least for piecewise linear yield functions. These properties are not fulfilled when the incremental analysis is based on direct discrete versions of classical rate principles. Finally, FEM approximations are formally introduced and the solution of the resulting finite dimensional quadratic optimization problem is considered.     

Linear programming methods of displacement analysis in elastic-plastic frames

The analysis is aimed at determination of displacements in elastic-perfectly plastic frames, in the case of monotonically increasing loads. Spread of partial plasticity from hinges and change of geometry on the conditions of equilibrium are not included. Extremum energy principles and linear programming methods are employed to solve the problem. Both the kinematic and the static formulations of the problem are presented. The mathematical models of the formulations constitute a dual pair of linear programming problems.     

Finite element analysis of void growth in elastic-plastic materials

Three-dimensional finite element computations have been carried out for the growth of initially spherical voids in periodic cubic arrays and for initially spherical voids ahead of a blunting mode I plane strain crack tip. The numerical method is based on finite strain theory and the computations are three-dimensional. The void cubic arrays are subjected to macroscopically uniform fields of uniaxial tension, pure shear and high triaxial stress. The macroscopic stress-strain behavior and the change in void volume were obtained for two initial void volume fractions. The calculations show that void shape, void interaction and loss of load carrying capacity depend strongly on the triaxiality of the stress field. The results of the finite element computation were compared with several dilatant plasticity continuum models for porous materials. None of the models agrees completely with the finite element calculations. Agreement of the finite element results with any particular constitutive model depended on the level of macroscopic strain and the triaxiality of the remote uniform stress field. For the problem of the initial spherical voids directly ahead of a blunting mode I plane strain crack tip, conditions of small scale yielding were assumed. The near tip stress and deformation fields were obtained for different void-size-to-spacing ratios for perfectly plastic materials. The calculations show that the holes spread towards the crack tip and towards each other at a faster rate than they elongate in the tensile direction. The computed void growth rates are compared with previous models for void growth.

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Résumé On a effectué des calculs par éléments finis à trois dimensions pour l'étude de la croissance de lacunes, initialement sphériques, selon des arrangements en cubes périodiques, et pour des lacunes initialement sphériques se formant en avant de l'extrémité arrondie d'une fissure sollicitée en état de déformation plane et selon un Mode I. La méthode numérique est basée sur la théorie des dilatations finies et les calculs sont effectués en trois dimensions.La disposition en cubes des lacunes est sujette à des champs macroscopiquement uniformes de tractions uniaxiales, de cisaillement pur et de contraintes à haute triaxialité. Le comportement macroscopique en contraintes-dilatations et le changement de volume des lacunes ont été obtenus pour deux fractions du volume initial de cavités.Les calculs montrent que la forme des cavités, leurs interactions et la perte résultante de capacité de change dépendent fortement de la triaxialité du champ de contraintes. On compare les résultats des calculs par éléments finis avec divers modèles de continuum à plasticité dilatante applicable à des matériaux poreux. Aucun de ces modèles n'est en accord complet avec les calculs par éléments finis. L'accord des résultats par éléments finis avec un modèle constitutif particulier dépend du niveau de déformation macroscopique et de la triaxialité du champ des contraintes uniformes agissant à distance. On a supposé des conditions d'écoulement plastique à petite échelle, pour résoudre le problème des lacunes initialement sphériques se trouvant directement en avant de l'extrémité d'une fissure arrondie sollicitée en Mode I et en déformation plane.Les champs de contrainte et de déformation près de la pointe de la fissure sont obtenus pour des matériaux parfaitement plastiques et pour différents rapports dimensions/espacements.Les calculs montrent que les cavités se développent vers l'extrémité de la fissure et l'une vers l'autre à une vitesse plus grande qu'elles ne s'allongent dans la direction de la contrainte de traction.On compare les vitesses calculées de croissance des lacunes avec celles fournies par de précédents modèles.

     

Some computational aspects of elastic-plastic large strain analysis

The governing equations for large strain analysis of elastic-plastic problems are reconsidered. An improved form of these equations is derived, which is valid for small increments of strain and large increments of rotation. Special attention is paid to the integration procedures for these equations in the deformation history. It is shown that the tangent modulus procedure for integration of the constitutive equations is conditionally stable, and that implicit methods, such as the ‘mean normal’ method, are to be preferred. A novel procedure is introduced for the treatment of nonlinear geometric effects. The performance of various element types is examined, with specific attention to effects of ‘locking’ and distortion. Several applications are discussed to illustrate the various aspects of the formulation developed ein this paper.     

Static non-linear three-dimensional analysis of a riser bundle by a substructuring and incremental finite element algorithm

The problem of static, non-linear, large three-dimensional deformation of riser bundles used in offshore oil and gas production is studied within the limits of small strain theory. The mathematical model consists of the models of component-risers and connectors which hold risers together. Each riser is modelled as a thin walled, slender, extensible or inextensible tubular beam-column. It is subject to non-linear three-dimensional deformation dependent hydrodynamic loads, torsion and distributed moments, varying axial tension, and internal and external fluid forces. The problem is solved numerically by developing an algorithm which features substructuring, condensation and non-linear incremental finite elements. Substructuring is used to decompose the riser bundle problem into those of individual component-risers and equilibria of connectors. Condensation is used along with the connector equilibrium equations to produce connector forces and moments. Strong non-linearities present in the model are handled by an incremental finite element approach. Accuracy of the computer code is verified by solving simple three-dimensional cases. Two three-dimensional applications are solved for a bundle with seven component-risers and up to a total of 1267 degrees of freedom. Finally, a comparison is made with numerical results of a two-dimensional analysis code. The influence of problem size on total CPU time is discussed.     

多点复合渐进成形与单点渐进成形的对比分析

为提高金属板材渐进成形的成形质量、成形精度、成形效率和成形极限,了解不同渐进成形工艺对制件成形性能的影响,本文以典型方锥台制件为研究对象,利用有限元软件MSC.Marc对2种渐进成形工艺进行了三维建模,对比分析了单点渐进成形和多点复合渐进成形对制件等效塑性应变、厚度分布和成形精度的影响.数值模拟结果表明:单点渐进成形的等效塑性应变和厚度减薄主要集中在制件相邻侧壁间的拐角处,而多点复合渐进成形的等效塑性应变和厚度减薄均匀地分布在制件成形区;相同成形工艺参数下,相比单点渐进成形,多点复合渐进成形更有利于制件的成形效率、成形质量、成形精度和成形极限的提高,更有利于抑制破裂等失稳现象的产生.2种渐进成形工艺的成形试验表明,数值模拟结果与试验相符.     

Convergence and bias in the LSG algorithm for adaptive lattice filters

The lattice filter has several desirable characteristics that make it attractive in adaptive applications. The Lattice Stochastic Gradient (also called Gradient Adaptive Lattice algorithm) is popularly used to adapt the lattice filter. However, a theoretical study of the bias in the reflection coefficient and the convergence of the LSG algorithm have not been studied extensively yet. This paper presents some theoretical results on these issues. It is hoped that the results will also present some insights into the factors affecting the convergence of the filter.     

Convergence of the Schulz-Snyder phase retrieval algorithm to local minima

The Schulz-Snyder iterative algorithm for phase retrieval attempts to recover a nonnegative function from its autocorrelation by minimizing the I-divergence between a measured autocorrelation and the autocorrelation of the estimated image. We illustrate that the Schulz-Snyder algorithm can become trapped in a local minimum of the I-divergence surface. To show that the estimates found are indeed local minima, sufficient conditions involving the gradient and the Hessian matrix of the I-divergence are given. Then we build a brief proof showing how an estimate that satisfies these conditions is a local minimum. The conditions are used to perform numerical tests determining local minimality of estimates. Along with the tests, related numerical issues are examined, and some interesting phenomena are discussed.     

Three-dimensional elastic-plastic finite element analysis of small surface cracks

Three-dimensional, elastic and elastic-plastic finite element analysis of small surface cracks was performed. The elastic analysis is in good agreement with other solutions. For a round surface with a radius equal to six times the crack depth, the $\text{K}$ at the surface is about 4% higher than the $\text{K}$ for a flat surface. The results of the elastic-plastic analyses show a unique variation of the effective $\text{K}$ (J-integral) along the crack front with a decrease in $\text{K}$ at the surface due to a lack of plane strain constraint and an increase in the effective $\text{K}$ at the maximum depth point with increasing plasticity. Similar behavior was observed for a semielliptical crack. Increasing the strain hardening exponent from 10 to 20 produced similar results with slightly higher effective A's for high applied strains. These results are useful in understanding the fracture behavior of small surface cracks.     

An analytical-numerical alternating method for elastic-plastic analysis of cracks

A new algorithm based on the Schwartz-Neumann alternating technique is developed for the solution of elastic-plastic fracture mechanics problems. An analytical solution for an elastic crack, with arbitrary crack-face loading, is used inside an initial stress iterative procedure as an addition to the finite element solution for the uncracked body. Iteration processes of the alternating method and of the initial stress method are performed simultaneously. Numerical examples show that the proposed elastic-plastic alternating method in conjunction with the equivalent domain integral method provides reasonable values of the J-integral.This work was supported under the FAA Center of Excellence for Computational Modeling of Aircraft Structures at the Georgia Institute of Technology.     

Optimum/adaptive incremental sequence in nonlinear analysis

An approach to improve the accuracy of the incremental solutions to a nonlinear problem, through a strategy to control the size of the increment, based on stationary of an argumented energy functional, is presented. The problem of control of an optimum step size in the incremental theory is formulated for a fixed number of increments. The variables in this argumented functional are: (i) the incremental displacement vector, (ii) the scalar parameters _i which characterize the size of each of the increments, i = 1,..., N, and (iii) a Lagrange multiplier which enforces the constraint that the sum of all the normalized increments, i. e., _i is equal to 1. The optimality condition provides us a rigorous approach which gives rise to an iterative procedure because of nonlinearity of the stationary condition. If the number of increments is not prescribed, a noniterative procedure can be obtained, where the incremental sequence is controlled adaptively with less computational effort. The extension of the proposed method to non-selfadjoint problems, where a potential energy function does not exist, is also discussed. Numerical examples demonstrate the remarkable improvement in the accuracy of the solution by optimizing the incremental sequence, as well as the effectiveness of the adaptive control procedure proposed.Paper presented at The 16th International Congress of Theoretical and Applied Mechanics, Lyngby, Denmark, August 19–25, 1984