An Extension of the Hu-Washizu Variational Principle in Linear Elasticity for Dynamic Problems

Abstract

The Hu-Washizu variational principle in linear elasticity in which displacement, strain, and stress can be independently varied is extended, for application to dynamic problems, to include independent variation of the velocities associated with the displacements. Such independent variation is of value in the construction of approximate theories for beams and plates.

The Linear Theory of Elasticity

“…1 We denote these subgroups by ~1 , ..• , ~1 1 • Table 1, which is due to CoLEMAN and NoLL, 2 lists the generators of each of these groups; in this table {k, l, m} denotes a right-handed orthonormal basis, q=Vf(k+l+m), and R: is the orthogonal tensor corresponding to a right- 1 The thirty-two crystal classes are discussed by ScHOENFLIESS [1891,1], VOIGT [1910,1], FLINT [1948, 4), DANA and HURLBUT [1959,2), SMITH and RIVLIN [1958,16], COLEMAN and NOLL [1964,5], TRUESDELL and NOLL [1965,22), BURCKHARDT [1966,5]. 1 We denote these subgroups by ~1 , ..• , ~1 1 • Table 1, which is due to CoLEMAN and NoLL, 2 lists the generators of each of these groups; in this table {k, l, m} denotes a right-handed orthonormal basis, q=Vf(k+l+m), and R: is the orthogonal tensor corresponding to a right- 1 The thirty-two crystal classes are discussed by ScHOENFLIESS [1891,1], VOIGT [1910,1], FLINT [1948, 4), DANA and HURLBUT [1959,2), SMITH and RIVLIN [1958,16], COLEMAN and NOLL [1964,5], TRUESDELL and NOLL [1965,22), BURCKHARDT [1966,5].…”

“…However, since the necessary restriction that S be self-equilibrated is absent in all of these investigations, they can at most be valid for a region whose boundary consists of a single closed surface. The elegant proof given here is due to CARLSON [1967,4] {see also CARLSON [1966,6]). The elegant proof given here is due to CARLSON [1967,4] {see also CARLSON [1966,6]).…”

Comments on the Free Vibrations of Beams With a Single-Edge Crack

“…The parameter m is the slope of the linear stress distribution at the cracked section and is calculated by assuming continuity of the bending moment at the crack site, as discussed later. The parameter is the rate of stress decay in the x direction and has to be estimated from experimental results [7] or from a detailed "nite element model [6]. Figure 2 shows a plot of the crack function in the vicinity of the crack tip.…”

“…The equations of motion of a cracked beam-like structure are derived through the Hu}Washizu}Barr [7] variational method, which can be viewed as an extension of the Hellinger}Reissner stationary principle [8] and allows independent kinematic assumptions on the displacement, velocities, strain and stress "elds in elastodynamic problems. The variational method states that for independent variations of displacements u G , strains GH , stresses GH , and momentum p G , one must have…”

On the modelling of dynamic structures with discontinuities

“…In this, they found that the bilinear frequency formula is a good approximation for the effective natural frequency of cracked beam. Based on Hu-Washizu-Barr variational formulation [13], considering cracked bar as a one-dimensional continuum equation of motion and boundary condition were developed in [14]. They found that breathing cracks result in a smaller drop in the dominant system frequency as compared to the natural frequencies of linear system with open cracks.…”