Untitled

92 Riccardo Rossi, Vitaliani Renato, and Eugenio Onate
where we introduced the symbols g = j T j and G = J −1 .Operator g takes,
after some calculations, the simple form
gξ • gξ gη • gξ
g =
gξ • gη gη • gη
(15)
From the definition of he Green Lagrange strain tensor E = 1
2 (C − I)
we obtain immediately δE = 1δC.
This allows to write the equation of vir-
2
tual works in compact form as (taking in consideration only body forces and
pressure forces)
h0
2
Ω
δWint W = δWext W + δWpress W (16)
δC : S = h0
Ω
δx • b +
ω
pδx • n (17)
Internal Work
The term h0
2 δC : S describes the work of internal forces during the defor-
Ω
mation process. Operator G = J−1 is referred to the reference configuration
and is therefore strictly constant, follows immediately that
δC = G T δgG (18)
The term δC : S becomes in Einstein’s notation
1 1
δC : S =
2 2 δCIJ C SIJ = 1
2 δgijGiIGjJSIJ
introducing the symbols
1
2 δgij → 1
⎛ ⎞
δg11 1
δ {g} = ⎝ δg22 ⎠
2 2
2δg12
; SIJ →{S} = ⎝
GiIGjJ → [Q] T ⎛
(G11)
= ⎝
2 (G12) 2 0 (G22)
2G11G12
2 0
0 G12G22 G11G22
it is possible to express the (19) in Voigt form as
1 1
δC : S =
2 2 {δg}T [Q] T {S} = 1
⎛
S11
S22
S12
⎞
⎠
⎞
⎠
(19)
(20)
(21)
2 {δg}T {s} ; {s} =[Q] T {S} (22)
considering the definition(15), introducing the symbol {δx} T =
and taking in account the isoparametric approximation one obtains
1
2 δg11 = ∂NI N
∂ξ δxI • gξ = {δx} T
⎛ ∂N1
∂ξ
⎝
{gξ}
⎞
... ⎠
∂Nk
∂ξ {gξ}
{δx1} T
T
. ..{δxk}
(23)