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132 Antonio J. Gil
where K matIJ
and K geoIJ
stand for the elemental material or constitutive stiffness
matrix and the elemental geometrical or initial stress stiffness matrix, respectively.
By assuming that the body forces b and external surface tractions ¯t not associated
to pressure forces remain constant and by taking into account that the pressure
component is dependent upon the geometry due to changing orientation and surface
area of the structure, the linearization of the global vector of external forces is given
through the following derivation:
f I f ext =
Γ
tN I dΓ = −p
Γ
nN I dΓ (46)
By applying the Nanson rule for the unit normal n -see for instance [17] or [19]
for details- and particularizing for an isoparametric three-node linear finite element:
f I
f ext = −p JF
pret
−T n pret N I pret
−pΓ
dΓ = (
3
∂x
∂ξ pret ×
1
∂x
∂ξ pret ) (47)
2
K pIJ
ij
Γ
= −pΓ pret ɛilm
3
∂N
(δlj
J
∂ξ pret
Fm
1
F 2 + Fl
∂N
F 1δmj
J
∂ξ pret
2
where p is the pressure scalar acting on the considered finite element, ξ pret
1
) (48)
and ξ pret
2
are the local plane coordinates and ɛ is the socalled alternating third order tensor.
3.4 Direct Core Congruential Formulation (DCCF)
From the computational viewpoint, a very elegant procedure termed the Direct Core
Congruential Formulation (DCCF) may be applied to perform the implementation
stage of the formulation developed above. This methodology, which is hardly used
in the existing literature is due to pioneer studies in [20] and [21]. The main ideas
behind this formulation can be discovered in the notable paper due to [22].
The scope of the DCCF is establishing the set of global equilibrium equations
whose unknowns are the components of the displacement gradient tensor G which
is given as:
Gij = ∂ui
∂X pret
(49)
Xj
Therefore, this new set of equations is completely independent on the geometry
of the structure and on the adopted discretization properties. Afterwards, every
single component of the displacement gradient tensor may be easily expressed in
terms of the nodal displacements of the Lagrangian mesh. Naturally, it is right then
when properties concerning geometry and discretization are brought to light. The
consideration of only traslational degrees of freedom for the nodes of the Lagrangian
mesh makes the DCCF specially simple and easy of being implemented. The Fig. 4
shows a summary of this formulation:
Gradient
equations
=⇒ =
Congruential
transformation
=⇒ =
Fig. 4. DCCF scheme.
Equations in
DOFs