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F.E.M. for Prestressed Saint Venant-Kirchhoff Hyperelastic Membranes 129
where λ and µ are known as the Lamé ´ constants. These two constants can be related
to the classical Young modulus E and Poisson’s ratioνas follows:
λ =
νE
(1 + ν)(1 − 2ν)
µ =
E
2(1+ν)
Another important feature which needs to be obtained is the incremental strain
energy accumulated into the structure along the deformation path from the primary
to the secondary states. By performing again a Taylor series expansion truncated
after the second order, the internal strain energy functional per unit of nominally
stressed volume may be developed as:
w ∗ int = wint + ∂wint
(21)
(E
∂EAB
∗ AB −EAB)+ 1 ∂
2
2 wint
(E
∂EAB∂ECD
∗ AB −EAB)(E ∗ CD −ECD) (22)
This above formula can be rewritten as:
where the terms Ω and Υ canbedepictedas:
w ∗ int = wint + Ω + Υ (23)
Ω = SABX pret
X E ∗relat
i,A Xpret
j,B
Υ = 1
pret
CABCDX
i,A 2 Xpret Xj,B E ∗relat
ij X pret
k,C Xpret kl
By substituting (24) and (25) back into (22):
w ∗ int − wint = J[σijE ∗relat
ij
ij
l,D E∗relat
+ 1
2 Cijkl
= JσijE ∗relat
ij
= 1
2 JCijkl C E ∗relat
ij E ∗relat
kl
(24)
(25)
C E ∗relat
ij E ∗relat
kl ]=JΨ (26)
By integrating over the initial undeformed volume V 0 corresponding to the configuration
ℜ0, and by applying the mass conservation principle from this volume V 0
to the prestressed volume V pret corresponding to the configuration ℜt, we obtain
the incremental Helmholtz’s free energy functional, which is given as:
∆Wint W =
V
(w
0
∗ int−wint)dV =
V
JΨdV =
0
V pret
ΨdV =
V pret
w ∗relat
int dV (27)
Therefore, the internal strain energy functional per unit of volume of the primary
state takes the final form:
w ∗relat
int
= σijE ∗relat
ij
+ 1
2 Cijkl
3 Finite Element Discretization
C E ∗relat
E ∗relat
3.1 From the Strong to the Weak Formulation
The above mentioned primary and secondary states can be understood as an initial
prestressed state ℜ pret and a final in service loading state ℜ due to the consideration
of live and dead load. Henceforth, the coordinates of any body’s particle, in
ij
kl
(28)