Nonlinear Finite Elements for Continua and Structures Part 1 pps

[...]... completely consistent For example, the expression for σ 12 from (A .1. 6) is σ 12 = C1 211 11 + C1 212 12 + C12 21 21 + C1222ε 22 (A .1. 10) The above translates to the following expression in terms of the Voigt notation σ 3 = C 31 1 + C33ε 3 + C32ε 2 (A .1. 11) which can be shown to be equivalent to (A .1. 10) if we use ε 3 = 12 + ε 21 = 2 12 and the minor symmetry of C : C1 212 = C12 21 It is convenient to... b (A .1. 8) and as indicated on the right, when writing the Voigt expression in matrix indicial form, indices at the beginning of the alphabet are used The Voigt matrix form of the elastic constitutive matrix is C 11 C12 [C] = C 21 C22  C 31 C32  C13  C 111 1 C 112 2 C23  = C2 211 C2222   C33  C1 211 C1222   1- 18 C 111 2  C2 212   C1 212   (A .1. 9) T Belytschko, Introduction, December 16 , 19 98... can generally not be expressed in closed form for arbitrary motions, but for the simple motion given in Eq (1. 4.3) it is given by X= x−t − t +1 (1. 4.7) 1 t2 2 Substituting the above into (3) gives 1- 10 T Belytschko, Introduction, December 16 , 19 98 v( x , t ) = 1 + ( x − t )(t − 1) 1 − x + xt − 1 t 2 2 1 t2 2 − t +1 = 1 t2 2 − t +1 (1. 4.8) Equations (1. 4.4) and (1. 4.8) give the same physical velocity... dimensions: σ 11  σ 1  σ 11 σ 12      σ≡  → σ 22  = σ 2  ≡ {σ} σ 21 σ 22      σ 12  σ 3  (A .1. 1) in three dimensions: σ 11 σ 12 σ ≡ σ 21 σ 22  σ 31 σ 32  σ 11  σ 1  σ  σ  2 22 σ 13      σ 33  σ 3   σ 23  →   =   ≡ {σ}  σ4 σ 23 σ 33       σ  σ  5 13     σ 12  σ 6  (A .1. 2) We will call the correspondence between the square matrix form of... tensor indices and the row numbers are identical, but the shear strains, i.e those with indices that are not equal, are multiplied by 2 Thus the Voigt rule for the strains is tensor → Voigt two dimensions  11   1   11 12      ε≡  →  ε 22  = ε 2  ≡ {ε} ε 21 ε 22      2 12  ε 3  in three dimensions 1- 17 (A .1. 3) T Belytschko, Introduction, December 16 , 19 98  11 12 ε 22 ε≡... dimensional problems, the matrix counterpart of BijKk is then written as  B11xK B K =  B22 xK  2 B12 xK  B11yK  B22 yK   2 B12 xK   (A .1. 18) The full matrix for a 3-node triangle is  Bxx1x  [B] =  Byy1x 2 Bxy1x  Bxx1y Byy1y 2 Bxy1y Bxx 2 x Byy 2 x 2 Bxy 2 x Bxx 2 y Byy 2 y 2 Bxy 2 y Bxx 3 x Byy3 x 2 Bxy3 x Bxx 3 y   Byy3 y  2 Bxy3 y   (A .1. 19) where the the first two indices have... the bottom row, the square matrix form of the tensor is indicated by boldface, whereas brackets are used to distinguish the Voigt form The correspondence is also given in Table 1 TABLE 1 Two-Dimensional Voigt Rule 1- 16 T Belytschko, Introduction, December 16 , 19 98 σ ij i 1 2 3 σa ¨ a 1 2 3 j 1 2 3 Three-Dimensional Voigt Rule σ ij i j 1 1 2 2 3 3 2 3 1 3 1 2 σa a 1 2 3 4 5 6 When the tensors are written... (A .1. 13) ui ( x ) = N I ( x )uiI , (A .1. 14)  ∂u ∂u j  1  ∂N I  ∂N I ε ij = 1  i +  = 2  ∂x δ ik + ∂x δ jk  ukI ≡ BijIk ukI 2  ∂x j ∂xi   j  i (A .1. 15) where To 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. .. Methods for Transient Analysis, North-Holland, Amsterdam K.-J Bathe (19 96), Finite Element Procedures, Prentice Hall, Englewood Cliffs, New Jersey R.D Cook, D.S Malkus, and M.E Plesha (19 89), Concepts and Applications of Finite Element Analysis, 3rd ed., John Wiley M.A Crisfield (19 91) , Non-linear Finite Element Analysis of Solids and Structure, Vol 1, Wiley, New York T.J.R Hughes (19 87), The Finite. .. 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 . Belytschko, Introduction, December 16 , 19 98 1- 11 vxt xtt tt xxt t tt (,)=+ − () − () −+ = −+ − −+ 1 1 1 1 1 1 2 2 1 2 2 1 2 2 (1. 4.8) Equations (1. 4.4) and (1. 4.8) give the same physical velocity. Belytschko, Introduction, December 16 , 19 98 1- 1 CHAPTER 1 INTRODUCTION by Ted Belytschko Northwestern University Copyright 19 96 1. 1 NONLINEAR FINITE ELEMENTS IN DESIGN Nonlinear finite element analysis. analysis are Belytschko and Hughes (19 83), Zienkiewicz and Taylor (19 91) , Bathe (19 95) and Cook, Plesha and Malkus (19 89). These books provide useful introductions to nonlinear finite element analysis.