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94 Riccardo Rossi, Vitaliani Renato, and Eugenio Onate
Linearization
Equation (16) is nonlinear, its practical use needs therefore its linearization.
The best rate of convergence is theoretically given by Newton-Raphson
technique which guarantees quadratical convergence to the solution. Defining
Ψ = δWint W − δWext W − δWpr W each Newton–Raphson step takes the form
dΨ + Ψ =0 (35)
The term Ψ can be explicitated using expression (30)(34) we therefore miss
only the differential dΨ that can be evaluated from the linearization of the
different contributions
Linearization of internal work
The term connected to the internal works can be linearized as follows
h0
d (Wint W ) = d
2
δC : S == h0
2
d (δC) : S + h0
2
δC : d (S) (36)
the first terms gives, by using (22)
h0
d (δC) :S =
2
h0
2
Ω
Ω
= h0
2
Ω
Ω
Ω
d {δg} {s} =
∂ {δg}
d
T
d {x} {s} (37)
∂ {x}
now it canbe seen that
1
d
2 {δg}T
=
δgξ • dgξ δgη • dgη δgη • dgξ + δgξ • dgη {s} =
= s11δgξ • dgξ s22δgη • dgη s12 (δgη • dgξ + δgξ • dgη)
(38)
substitution of the shape functions gives immediately a set of equalities in the
form
T
∂NI N ∂NJ N
s11δgξ • dgξ = s11
∂ξ ∂ξ δijδxI
∂NI N
• dxjJ x = s11
∂ξ
which makes possible towrite
1
d
2 {δg}T
=
∂NI N ∂NJ N
s12
∂η ∂ξ
s11
∂NI N
∂ξ
∂NJ N
∂ξ
∂NI N
+
∂ξ
∂NJ N
∂η
∂NI N
+ s22
∂η
Ω
∂NJ N
∂ξ δijδxiIdxjJ (39)
∂NJ N
∂η +
δijδxiIdxjJ
(40)