This paper proposes a multivariable formulation with stress and displacement fields as primary unknowns to solve dynamical problems. Displacement is approximated by a time-dependent linear combination of C 0 continuous basis functions over the spatial domain of the body. Strain is derived by differentiating the displacement field, which exists over each element. The acceleration fields are obtained by differentiating displacement twice with respect to time via the backward difference scheme. The stress field, satisfying equilibrium and traction continuity exactly within and between elements, is constructed using Airy’s stress functions based on C 2 continuous basis functions and by integrating the inertial terms spatially. The boundary value problems are cast as a constrained nonlinear minimization problem over the spatial domain, with the optimization carried out at predetermined time instances. The objective function is the Frobenius norm of the error in the constitutive relation, integrated over the spatial domain, with traction and displacement boundary conditions enforced as constraints at all time instances. The formulation is validated against analytical solutions for finite plate problems under time-varying uniaxial traction and extended to nonlinear elastic models.
Ansari et al. (Tue,) studied this question.