Finite Elasticity Solutions Using Hybrid Finite Elements Based on a Complementary Energy Principle—Part 2: Incompressible Materials

1979 ◽  
Vol 46 (1) ◽  
pp. 71-77 ◽  
Author(s):  
H. Murakawa ◽  
S. N. Atluri
Keyword(s):  
Finite Elasticity ◽  
Strain Analysis ◽  
Companion Paper ◽  
Complementary Energy ◽  
Energy Principle ◽  
Complementary Energy Principle ◽  
Elasticity Solutions ◽  
Strip Test ◽  
Nonlinear Elastic ◽  
Incompressible Rubber

In a companion paper [1], the authors presented a total Lagrangean rate formulation for a hybrid stress finite-element method, based on a rate complementary energy principle which involves both the rates of Piola-Lagrange stress and rotation as variables, for finite strain analysis of nonlinear elastic compressible solids. In this paper the method is extended to the case of precisely incompressible rubber-like materials. Two plane stress problems, one corresponding to a biaxial strip test and, the other, a sheet with a circular hole, both involving strains in excess of 100 percent, are solved and the numerical results are discussed.

Finite Elasticity Solutions Using Hybrid Finite Elements Based on a Complementary Energy Principle

1978 ◽  
Vol 45 (3) ◽  
pp. 539-547 ◽  
Author(s):  
H. Murakawa ◽  
S. N. Atluri
Keyword(s):  
Finite Elasticity ◽  
Strain Analysis ◽  
Elastic Solid ◽  
Element Model ◽  
Equilibrium Equations ◽  
Complementary Energy ◽  
Rotation Tensor ◽  
Energy Principle ◽  
Complementary Energy Principle ◽  
Nonlinear Elastic

The possibility of deriving a complementary energy principle, for the incremental analysis of finite deformations of nonlinear-elastic solids, in terms of incremental Piola-Lagrange (unsymmetric) stress alone, is examined. A new incremental hybrid stress finite-element model, based on an incremental complementary energy principle involving both the incremental Piola-Lagrange stress, and an incremental rotation tensor which leads to discretization of rotational equilibrium equations, is presented. An application of this new method to the finite strain analysis of a compressible nonlinear-elastic solid is included, and the numerical results are discussed.

Some Problems in the Theory of Nonconservative Elasticity

1986 ◽  
Vol 10 (1) ◽  
pp. 28-33
Author(s):  
Qi-hao Zhang ◽  
Dian-kui Liu
Keyword(s):  
Finite Element Analysis ◽  
Variational Principle ◽  
Variational Principles ◽  
Theory Of Elasticity ◽  
Complementary Energy ◽  
Element Analysis ◽  
Energy Principle ◽  
Nonconservative Systems ◽  
Potential Energy Principle ◽  
Complementary Energy Principle

This study develops the general quasi-variational principles for nonconservative problems in the theory of elasticity such as the quasi-potential energy principle, the quasi-complementary energy principle, the generalized quasi-variational principle and quasi-Hamilton principle. The application of these quasi-variational principles to finite element analysis is also discussed and illustrated with some examples. The total variational principle for nonconservative systems of two variables is also studied.

The Triangular Equilibrium Element in the Solution of Plate Bending Problems

1968 ◽  
Vol 19 (2) ◽  
pp. 149-169 ◽  
Author(s):  
L. S. D. Morley
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Plate Bending ◽  
Complementary Energy ◽  
Element Analysis ◽  
Energy Principle ◽  
The Finite Element Analysis ◽  
Complementary Energy Principle ◽  
Bending Problems

SummaryFurther details are given of a recently developed triangular equilibrium element which is then applied, in conjunction with the complementary energy principle, to the finite element analysis of some plate bending problems. The element is demonstrated to have a straightforward and satisfactory application and to possess advantages over the conventional triangular displacement element.

Application of 2D base force element method with complementary energy principle for arbitrary meshes

2014 ◽  
Vol 31 (4) ◽  
pp. 691-708 ◽  
Author(s):  
Yijiang Peng ◽  
Nana Zong ◽  
Lijuan Zhang ◽  
Jiwei Pu
Keyword(s):  
Lagrange Multiplier ◽  
Lagrange Multiplier Method ◽  
Multiplier Method ◽  
Complementary Energy ◽  
Energy Principle ◽  
Content Type ◽  
Complementary Energy Principle ◽  
Force Element ◽  
2D Model ◽  
Element Method

Purpose – The purpose of this paper is to present a two-dimensional (2D) model of the base force element method (BFEM) based on the complementary energy principle. The study proposes a model of the BFEM for arbitrary mesh problems. Design/methodology/approach – The BFEM uses the base forces given by Gao (2003) as fundamental variables to describe the stress state of an elastic system. An explicit expression of element compliance matrix is derived using the concept of base forces. The detailed formulations of governing equations for the BFEM are given using the Lagrange multiplier method. The explicit displacement expression of nodes is given. To verify the 2D model, a program on the BFEM using MATLAB language is made and a number of examples on arbitrary polygonal meshes and aberrant meshes are provided to illustrate the BFEM. Findings – A good agreement is obtained between the numerical and theoretical results. Based on the studies, it is found that the 2D formulation of BFEM with complementary energy principle provides reliable predictions for arbitrary mesh problems. Research limitations/implications – Due to the use of Lagrange multiplier method, there are more basic unknowns in the control equation. The proposed method will be improved in the future. Practical implications – This paper presents a new idea and a new numerical method, and to explore new ways to solve the problem of arbitrary meshes. Originality/value – The paper presents a 2D model of the BFEM using the complementary energy principle for arbitrary mesh problems.

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