Spatial form

ˉu(x,0)=u0(X),˙ˉu(x,0)=˙u0(X).
When all data are independent of time, the BVP is referred to as static, so the BVP of elastostatics is written:
{divσ+ρf=0u=udonSut=σ.n=tdonSt
The equilibrium equations have already been obtained from the VWP, which constitutes a variational principle.
The weak form of previous initial BVP is obtained by multiplying the dynamical equilibrium equation by a test function, a virtual admissible velocity field written in the form of a variation of the displacement δu, and integrated over the domain Ω_t, so that we obtain the weak form of the equations of motion in the actual configuration
∫Ωt(−divσ.δu−ρf.δu+ρü.δu)dx=0ü
using the identity
divσ.δu=div(σ.δu)−σ:gradδu=div(σ.δu)−σ:δ(gradu)
and integrating by part (divergence th.) using the kinematic boundary condition for δu = 0 on S_u leads to
∫Ωt(σ:gradδu−(ρf−ρü).δu)dx−∫Sttd.δuds=0∫Ωtu(x,0).δudx=∫Ωtu0(X).δudx∫Ωt˙u(x,0).δudx=∫Ωt˙u0(X).δudxü
We have written the two initial conditions for the displacement and velocity fields in weak form. The first equation is called a variational equation.
We recognize the VWP when we identify the previous integrals
Pa=∫Ωtρü.δudx=0,Pe=∫Sttd.δuds+∫Ωtρfdx,Pi=−∫Ωt(σ:gradδu)dxü
as the virtual power of acceleration, external and internal forces respectively.
To be complete, we will show that the term gradδu is identified with the variation of the strain measure conjugated to Cauchy stress, namely the Almansi strain e.
The Gateaux derivative of the Eulerian strain, the spatial tensor e, is evaluated based on the concept of the Lie derivative: we first pull-back e to Ω₀, take the derivative there, and push-forward the result in Ω_t; this gives
δe=F−T.(Dδu(FT.e.F)).F−1=F−T.(DδuE).F−1
Thus, recalling that
δE={δ(FT).F+FT.δF}/2=sym(FT.Gradδu)↔δEAB=12(FaB∂δua∂XA+FaA∂δua∂XB)
we finally arrive at
δe=12(F−T.GradTδu+Gradδu.F−1)=12(gradTδu+gradδu)=sym(gradδu)
Inserting this variation back into P_i and considering the symmetry of Cauchy stress delivers
Pi=−∫Ωt(σ:δe)dx
in which we recognize the (already written) virtual power of internal forces. Thus, the previous BVP is equivalent to the VWP principle
Pa=Pe+Pi
It is important to emphasize that the VWP does not require the existence of a potential, and that the constitutive law is not incorporated at this stage. Note that a minimization principle (a stronger requirement) does not in general exist due to loads being non-conservative or to the non-existence of a strain energy function (case of Hypoelastic materials).
For hyperelastic materials, the stress is related to the strain through a strain energy function expressed in the reference configuration, thus it is more convenient to express the VWP in the reference configuration.
Lagrangian formulation: the strong form of the initial BVP is written in terms of the nominal stress
The dynamical equilibrium can alternatively be written based on the second Piola-Kirchhoff stress, which is a symmetrical tensor, by substituting P = F.S in the first equation. This BVP is now expressed over a fixed domain, involving the Lagrangian coordinate X; note the occurrence here of the material divergence. This BVP has to be complemented by a constitutive law expressing the stress versus a conjugated strain measure.
The weak form of the previous BVP is obtained using the same steps as for the spatial form, leading to the (already written) VWP involving Lagrangian stress measures.