Final answer:
To develop a finite element model, the displacement and virtual displacement fields are expressed as a weighted sum of nodal shape functions. The resultant equations are derived by substituting these expressions into the weak form of the system's governing equations, and linearizing strain-displacement relationships for small strains.
Step-by-step explanation:
To develop a finite element model associated with a weak form, the given approximations for displacement u(x,y) and virtual displacement w(x,y) are expanded using shape functions. The shape functions Ni and Nj are a set of basis functions corresponding to each node in the finite element mesh. The displacement field is thus represented as a weighted sum of these shape functions with uj being the nodal displacements, and the virtual work w(x,y) is similarly expressed as a weighted sum with Ci as the coefficients for virtual displacements.
To generate the finite element equations, one substitutes these expressions into the weak form of the governing differential equations of the system - typically the balance of linear momentum, also incorporating boundary conditions when necessary. For instance, if the system involves elastic deformation, this would result in a system of linear equations where the stiffness matrix (derived by integrating the product of the derivatives of Ni and Nj times the material's elasticity tensor) is multiplied by the vector of nodal displacements, equaling the external force vector.
In the presence of small strains and assuming linear elasticity, non-linear terms can often be neglected. By doing so, the change in local length elements and the associated volume change due to strain can be incorporated using a linearized strain-displacement relationship, typically through the strain tensor components Uij.