Some general theorems and generalized and piecewise generalized variational principles for linear elastodynamics (Q1206676)

The general theorems and various types of variational principles of linear elastodynamics are investigated and summed up systematically. The presented principles and theorems are as follows: the principle of admissible work action, the principle of virtual displacements, the principle of virtual stress-momenta, the theorem of action of strain and kinematic energy, the reciprocity theorem for two different states of dynamic equilibrium, the reciprocity theorem of work action, the reciprocity theorem of displacements, theorems of the conditions at time limits \(t_ 1\) and \(t_ 2\), the principle of potential energy action, the principle of complementary energy action, the dynamic version of the Hu-Washizu and Hellinger-Reissner principles, the constitutive variational principles, the three types of piecewise variational principle (piecewise space-time domain, piecewise space domain, piecewise time domain). The concept of the action has a central role in the formulation of principles and theorems. The action of the time dependent variable \(f(t)\) in the time interval \([t_ 1,t_ 2]\) is given by the integral \(I=\int^{t_ 2}_{t_ 1}f(t)dt\). If the time dimension is not considered, the formulations obtained in the paper will lead to the corresponding ones for elastostatics.