Untitled

F.E.M. for Prestressed Saint Venant-Kirchhoff Hyperelastic Membranes 141
The geometrical and material components of the stiffness matrix S can be developed
as follows:
S geo
=
S mat
mat
S11 S
=
mat
12
(A.7)
geo
S11 Sgeo 12
S geo
21 Sgeo
22
S geo
11 = s1I3 S mat
11
S geo
12 = s3I3 S mat
12
S geo
21 = s3I3 S mat
21
S geo
22 = s2I3 S mat
22
= C11f1 f f T 1
f + C33
C f2 f f T 2
S mat
21
S mat
22
f + C13f1f
+ C31
f f T 2
C f2 f f T 1
f (A.8)
= C13f1 f f T
f + C32 C f2 f f T
f + C12f1 f f T
f + C33 C f2 f f T
f (A.9)
= C31
1
C f1 f f T 1
= C33
C f1 f f T 1
f + C23f2f
+ C33
f f T 2
2
2
C f1 f f T 2
f + C22f2f
+ C32
f f T 2
C f1 f f T 2
1
f + C21f2f
(A.10)
f f T 1
f + C23f2f
(A.11)
where si is the ith-component of the second Piola-Kirchfoff stress tensor in Voigt
notation, I3 is the identity 3x3 matrix, Cij C is a component of the fourth order tensor
of elastic moduli in Voigt notation and the vectors f1 f and f2 f constitute the first and
second column of the deformation gradient tensor which can be expressed in their
transpose forms as:
f T f 1 =
1+g1 g2 g3 f T f 2 =
g4 1 + g5 g6
(A.12)
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f f T 1