Article

# A new algorithm for numerical solution of dynamic elastic–plastic hardening and softening problems

Department of Civil and Environment Engineering, University of California, Los Angeles, Los Angeles, CA 90095-1593, USA
(Impact Factor: 2.13). 08/2003; 81(17):1739-1749. DOI: 10.1016/S0045-7949(03)00167-6

ABSTRACT

The objective of this paper is to develop a new algorithm for numerical solution of dynamic elastic–plastic strain hardening/softening problems, particularly for the implementation of the gradient dependent model used in solving strain softening problems. The new algorithm for the solution of dynamic elastic–plastic problems is derived based on the parametric variational principle. The gradient dependent model is employed in the numerical model to overcome the mesh-sensitivity difficulty in dynamic strain softening or strain localization analysis. The precise integration method, which has been used for the solution of linear problems, is adopted and improved for the solution of dynamic non-linear equations. The new algorithm is proposed by taking the advantages of the parametric quadratic programming method and the precise integration method. Results of numerical examples demonstrate the validity and the advantages of the proposed algorithm.

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Available from: Jiun-Shyan Chen
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• "The precise time integration (PTI) method was first proposed by Zhong and Williams for the linear initial value problem of structural dynamics [12] . The PTI method has attracted much interest, and its application has been broadened to initial value problems like heat conduct problems, random response problems [13] [14] [15] , and some two-point boundary-value problems (TPBVPs) [16] [17] because of its prominent numerical advantages, such as high precision and high efficiency. "
##### Article: Precise integration method for solving singular perturbation problems
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ABSTRACT: This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method. Key wordssingular perturbation problem-first-order ordinary differential equation-two-point boundary-value problem-precise integration method-reduction method Chinese Library ClassificationO175.8-O241.81 2000 Mathematics Subject Classification65L10-76M45
Preview · Article · Nov 2010 · Applied Mathematics and Mechanics
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• "Most common mathematical models for analysis or optimization of linear hardened material structures implicate so called hardening matrices (Чирас 1986; Hongwu et al. 2003). For instance in paper (Чирас 1986) such a static formulation of extreme energy principle is stated: of all statically admissible residual internal forces vectors , the one that sum of self-equilibrium internal forces and hardening elastic potentials is minimal is the true one. "
##### Article: Optimization-based elastic-plastic analysis of steel frames by volumetric plasticity concept
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ABSTRACT: Structures composed of physical nonlinear finite elements under bending and compression or tension are considered in this paper. Material nonlinearity is considered as linearly hardened. In case of material hardening, plastic strains do not concentrate in one point but distribute in the certain volumes of finite element. Volumes of plas-tic strains zones in frame structure elements impact on elasticity modules of these elements sections by decreasing them. Technique of such elasticity modules decrease in finite elements sections is suggested. To realize such structure analysis, a treatment of strains in mathematical model is changed. Now strains are treated not as rotation angles and elongations of finite elements, but as longitudinal strains. Mathematical model including above mentioned modifica-tions is presented. Solving algorithm based on a modified Newton-Raphson method is particularly explained and employed for numerical example.
Preview · Article · Jan 2010
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• "The parametric variational principle, which was developed by Zhong et al. [17] [18] based on optimal control theory to deal with the unspecified boundary value problems in continuum mechanics, is used in the construction of the new Voronoi cell element. In the parametric variational principle, the constitutive relations of the physical phenomenon for a control system are taken into account by selecting some proper state and control variables, so the variational method is suitable to deal with the problems where the classical variational principle can not be applied directly [19] [20] . For elastic-plastic problems, this method avoids the limitation of Drucker hypothesis and can be applied to the non-associated plastic constitutive model in strain softening problems [21] . "
##### Article: Parametric variational principle based elastic-plastic analysis of heterogeneous materials with Voronoi finite element method
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ABSTRACT: The Voronoi cell finite element method (VCFEM) is adopted to overcome the limitations of the classic displacement based finite element method in the numerical simulation of heterogeneous materials. The parametric variational principle and quadratic programming method are developed for elastic-plastic Voronoi finite element analysis of two-dimensional problems. Finite element formulations are derived and a standard quadratic programming model is deduced from the elastic-plastic equations. Influence of microscopic heterogeneities on the overall mechanical response of heterogeneous materials is studied in detail. The overall properties of heterogeneous materials depend mostly on the size, shape and distribution of the material phases of the microstructure. Numerical examples are presented to demonstrate the validity and effectiveness of the method developed.
Preview · Article · Jul 2006 · Applied Mathematics and Mechanics