Spatial form

Exercise
Write the VWP in Lagrangian format.
For conservative loads and hyperelastic materials, the virtual power of external forces is the Gateaux derivative of a functional, thus the WVP can be rewritten as
ü∫Ω0(S:δE)dX−∫S0tt0d.δudS−∫Ω0ρ0f0.δudX+∫Ω0ρ0ü.δudX=0→∫Ω0δW(E)dX−δ{∫S0tt0d.udS−∫Ω0ρ0f0dX}+δ(12∫Ω0ρ0u˙2dX)=0→δ{∫Ω0W(E)dX−∫S0tt0d.udS−∫Ω0ρ0f0dX+12∫Ω0ρ0u˙2dX}=0
Thus, the introduced functional
≔V[u]≔∫Ω0W(E)dX+12∫Ω0ρ0u˙2dX−∫S0tt0d.udS−∫Ω0ρ0f0dX
called the total potential energy enjoys an extremum principle; V[u] is the sum of the mechanical and kinetic energy (first and second terms) minus the work of external forces.

3.13.1 Linearization of the principle of virtual work in material description

We recall the VWP written in Lagrangian format
ü∫Ω0(S:δE)dX−∫S0tt0d.δudS−∫Ω0ρ0f0.δudX+∫Ω0ρ0ü.δudX=0
with the virtual work of internal forces
Pi=−∫Ω0(S:δE)dX≡−δWint(u,δu)
The Lagrangian strain is obviously a function of the displacement field u. We linearized the term δW_int (u, δu) based on the directional derivative
DΔUδWint(u,δu)=ddε[∫Ω0(S(E(u+εΔu)):δE(u+εΔu))dX]ε=0
Since differentiation and integration commute, we may write the previous derivative
DΔUδWint(u,δu)=[∫Ω0(S(E(u)):DΔuδE(u)+DΔuS(E(u)):δE(u))dX]ε=0
Using the chain rule, we obtain
DΔuS(E(u))=∂ES(E(u)):DΔuE(u)=C(u):DΔuE(u)
with ℂ(u) the fourth order tensor or elastic moduli, leading to the expression of the linear increment
DΔUδWint(u,δu)=[∫Ω0(S(E(u)):DΔuδE(u)+δE(u):C(u):DΔuE(u))dX]ε=0
in which we have used the minor symmetries of the elasticity tensor. Finally, inserting
≔δE(u)≔DδuE(u)=sym(FT.Gradδu)→DΔuδE(u)=ddε[sym(F(u+εΔu)T.Gradδu)]ε=0
using the relations
≔ΔF=DΔuF=GradΔu,ΔE(u)≔DΔuE(u)=sym(FT.GradΔu),
delivers the relation
DΔuδE(u)=sym(GradTΔuGradδu)
Thus the linearization of the virtual work of internal forces leads to the increments in both tensor and indicial notations
DΔuδWint(u,δu)=∫Ω0(Gradδu:GradΔu.S+FT.Gradδuδ:C(u):FT.GradΔu)dX=∫Ω0∂δua∂XB(δabSBD+FaAFbCCABCD)∂Δub∂XDdX
The terms δ_abS_BD and F_aAF_bCC_ABCD therein represent the effective elasticity tensor, or the tangent stiffness, a symmetrical tensor, due to the fact that we can interchange the independent variations δ