Untitled

130 Antonio J. Gil
both prestressed and final loaded states, are related by means of the incremental
displacement field u as follows:
x = X pret + u, xi = X pret
i + ui (29)
According to this nomenclature, the strong formulation of the problem in a
Lagrangian description with respect to the prestressed configuration is summarized
in Fig. 3.Theconfigurations ℜ pret represents a material body of domain V pret with
frontier Γ pret . As can be observed, the super index ∗relat has been suppressed for
the sake of simplicity.
1. Balance of linear momentum.
∂Pji P
∂X pret
Xj
2. Transformation of stress tensors.
Jσij = ∂xi
∂X pret
Psj
s
P = ∂xi
∂X pret
s
3. Green-Lagrange straintensor.
4. Constitutive law.
Eij = 1
2
+ ρ pret bi = 0 in V pret
∂xj
∂X pret
t
( ∂xk
∂X pret
i
Sij = σ pret
ij
Sst with J = det( ∂xi
X
∂xk
∂X pret
Xj
∂X pret
j
(30)
) (31)
− δij) (32)
+ Cijkl C Ekl (33)
5. Internal strain energy functional per unit volume of prestressed configuration.
wint = σ pret
ij Eij + 1
2 Cijkl C EijEkl (34)
6. Boundary conditions.
ti = Pji P n pret
j = ¯ti on Γ pret
Γt ui =ūi on Γ i pret
Γui Fig. 3. Strong formulation for a Lagrangian description.
Thus, the weak form may be developed in a Total Lagrangian Format (TLF) by
means of the so called Principle of Virtual Work. By neglecting inertia forces:
δWint W =
δWext W =
V pret
V pret
δWint
δFij F Pji P dV =
δuibidV +
(35)
W (δui,ui) = δWext W (δui,ui) (36)
V pret
Γ pret
δF T : PdV =
δui ¯tidΓ =
V pret
V pret
δEijSijdV =
δu T · bdV +
Γ pret
V pret
δE : SdV
(37)
δu T · ¯tdΓ (38)