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F.E.M. for Prestressed Saint Venant-Kirchhoff Hyperelastic Membranes 133
By employing a classical vectorization of the displacement gradient tensor -see
for instance [17] and [19]- intoacolumnvector g and by expressing this in terms of
the nodal displacements, it turns:
g = Bu (50)
where B is the matrix of the gradients of the shape functions for the chosen finite
element and u is the column vector that gathers the nodal displacements for the N
nodes of a single finite element:
u T = u 1 1 u 1 2 u 1 3 . .. u N 1 u N 2 u N
3
(51)
Analogously, the 2x2 submatrix of in-plane components of the Green-Lagrange
strain tensor E may bevectorizedintoacolumn vector e by following the kinematic
Voigt ruleas:
(52)
e T = E11 E22 2E12
Every component of the new vector e may be expressed in terms of the vector
g as:
ei = h T i g + 1
2 gT Hig (53)
where hi and Hi are a vector and a symmetric matrix of numerical values comprising
(1, 0). Eventually, the in-plane components of the second Piola-Kirchhoff stress
tensor may be transformed into a vector by means of the kinetic Voigt ruleas:
s T = S11 S22 S12
(54)
By substituting equations (53) and (54) back into (37) the vector of global
internal forces may be rewritten in an easier way as:
fint f =
V pret
B T φ intdV φ int = sihi + siHig (55)
where summation is implied for repeated indices according to Einstein’s notation.
By proceeding in the same manner, the contributions to the generic total tangent
stiffness matrix K mat and K geo may beobtained as:
K mat =
V pret
B T S mat BdV K geo =
V pret
B T S geo BdV (56)
S mat = (hi + Hig)Cij C (h T j + g T Hj) S geo = siHi (57)
where the fourth order tensor of elastic moduli, Cij C , has been transformed into a
3x3 matrix by applying the Voigt rule vectorization procedure to equation (33) to
come outwith:
s = σ pret + Ce (58)
In the next sections, two numerical examples will be presented with the purpose
of demonstrating the capabilities of the described technique. The Finite Element
Method was applied to study both a prestressed cable network and a prestressed
membrane. As a consequence, the DCCF was particularized on two different finite
elements: a two-noded and a three-noded linear isoparametric finite elements. The
latter is described in some detail in the appendix located at the end of the chapter.