Spatial form

u and Δu in D_ΔUδW_int (u, δu). The first term in D_ΔUδW_int (u, δu) represents the geometrical stress leading to the geometrical stiffness matrix, while the second term represents the material stiffness matrix.
The material form of the linearization of the virtual work principle leads to the total Lagrangian formulation. It is also possible to write a spatial form of the linearization of the virtual work principle (which is fully equivalent but written over the actual configuration), leading to the updated Lagrangian formulation. Both formulations are used in subsequent numerical solution procedures of the field equations of nonlinear elasticity using the finite element method; the description of those methods is however outside the scope of the volume.