[...]... strong form The weak form can be used to approximate the strong form by finite elements; solutions obtained by finite elements are approximate solutions to the strong form Strong Form to Weak Form A weak form will now be developed for the momentum equation (2.2.23) and the traction boundary conditions For this purpose we define trial functions u( X,t ) which satisfy any displacement boundary conditions and. .. developed for large deformation problems The appropriate description depends on the characteristics of the problem to be solved In addition to describing the several types of finite element formulations for nonlinear problems, this Chapter reviews some of the concepts of finite element discretization and finite element procedures These include the weak and strong forms, the operations of assembly, gather and. .. deformations Their advantage in these problems is a consequence of the fact that Eulerian elements do not deform with the material Therefore, regardless of the magnitudes of the deformation in a process, Eulerian elements retain their original shape Eulerian elements are particularly useful in modeling many manufacturing processes, where very large deformations are often encountered For each of the formulations,... and tensor form of second order tensors is indicated by the number of subscripts and the letter used We use subscripts beginning with letters i to q for tensors, and subscripts a to g for Voigt matrix indices Thus σ ij is replaced by σ a in going from tensor to Voigt notation The correspondence between the subscripts (i,j) and the Voigt subscript a is given in Table 1 for two and three dimensions For. .. LAGRANGIAN AND EULERIAN FINITE ELEMENTS IN ONE DIMENSION by Ted Belytschko Northwestern University @ Copyright 1997 2 1 Introduction In this chapter, the equations for one-dimensional models of nonlinear continua are described and the corresponding finite element equations are developed The development is restricted to one dimension to simplify the mathematics so that the salient features of Lagrangian and. .. Analysis of Solids and Structure, Vol 1, Wiley, New York T.J.R Hughes (1987), The Finite Element Method, Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, New York T.J.R Hughes (1996), personal communication M Kleiber (1989), Incremental Finite element Modeling in Non-linear Solid Mechanics, Ellis Horwood Limited, John Wiley J.T Oden (1972), Finite elements of Nonlinear Continua, McGraw-Hill,... and scatter, and the imposition of essential boundary conditions and initial conditions Mappings between different coordinate systems are discussed along with the need for finite element mappings to be one-to-one and onto Continuity requirements of solutions and finite element approximations are also considered While much of this material is familiar to most who have studied linear finite elements, they... Nuclear Engineering and Design, 42 , 41-52 T Belytschko and T.J.R Hughes (1983), Computational Methods for Transient Analysis, North-Holland, Amsterdam K.-J Bathe (1996), Finite Element Procedures, Prentice Hall, Englewood Cliffs, New Jersey R.D Cook, D.S Malkus, and M.E Plesha (1989), Concepts and Applications of Finite Element Analysis, 3rd ed., John Wiley M.A Crisfield (1991), Non-linear Finite Element... to discrete form; the derivatives in time are not discretized For static problems with rate-independent materials, the discrete equations are independent of time, so the finite element discretization results in a set of nonlinear algebraic equations Examples of the total and updated Lagrangian formulations are given for the 2-node, linear displacement and 3-node, quadratic displacement elements Finally,... translate this to a matrix expression in terms of column matrices for ε ij and a rectangular matrix for Bija , the kinematic Voigt rule is used for ε ij and the first two indices of BijKk and the nodal component rule is used for the second pair of indices of BijKk and the indices of ukK Thus 1-19 T Belytschko, Introduction, December 16, 1998 elements of [B] are Bab where (i , j ) → a by the Voigt rule, . many interesting challenges and opportunities in nonlinear finite element analysis. 1.2. RELATED BOOKS AND HISTORY OF NONLINEAR FINITE ELEMENTS Several excellent texts and monographs devoted either. usefulness and potential of nonlinear finite element analyses are very sanguine. In many industries, nonlinear finite element analysis have shortened design cycles and dramatically reduced the need for. linear finite element analysis, nonlinear finite element analysis confronts the user with many choices and pitfalls. Without an understanding of the implication and meaning of these choices and