Multi-field Dual-Mixed Variational Principles Using Non-symmetric Stress Field in Linear Elastodynamics

Abstract

Three types of multi-field dual-mixed variational principles using a priori non-symmetric stress field will be derived for dynamic problems of linearly elastic solids. Starting from the functional of complementary Hamilton’s principle it is shown how this variational formulation is modified with applying the Lagrange multiplier technique, to obtain a new four-field dual-mixed functional. The independent fields of this functional will be the displacement vector, the non-symmetric stress tensor, the skew-symmetric rotation tensor and the momentum vector. Then we present how to treat appropriately the six different types of initial conditions in a weak and/or strong sense by adding a new integral expression to the four-field functional. Modifying this functional through a Legendre transformation results in a new complementary energy-based five-field dual-mixed functional which allows the momentum- and velocity vectors to be independently approximated. A complementary energy-based three-field dual-mixed variational formulation is obtained by eliminating the momentum- and velocity fields through the strong enforcement of the kinematic equation and the impulse-velocity relation from the five-field principle.