Variational Principles of Elastodynamics
In this chapter both the classical Hamilton-Kirchhoff Principle and a convolutional variational principle of Gurtin’s type that describes completely a solution to an initial-boundary value problem of elastodynamics are used to solve a number of typical pr
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Variational Principles of Elastodynamics
In this chapter both the classical Hamilton-Kirchhoff Principle and a convolutional variational principle of Gurtin’s type that describes completely a solution to an initialboundary value problem of elastodynamics are used to solve a number of typical problems of elastodynamics.
5.1 The Hamilton-Kirchhoff Principle To formulate H-K principle we introduce a notion of kinematically admissible process, and by this we mean an admissible process that satisfies the straindisplacement relation, the stress-strain relation, and the displacement boundary condition. (H-K) The Hamilton-Kirchhoff Principle. Let P denote the set of all kinematically admissible processes p = [u, E, S] on B × [0, ∞) satisfying the conditions u(x, t1 ) = u1 (x), u(x, t2 ) = u2 (x) on B
(5.1)
where t1 and t2 are two arbitrary points on the t-axis such that 0 ≤ t1 < t2 < ∞, and u1 (x) and u2 (x) are prescribed fields on B. Let K = K{ p} be the functional on P defined by t2 K{ p} = [F(t) − K(t)] dt (5.2) t1
where F(t) = UC {E} −
b · u dv −
B
sˆ · u da
(5.3)
∂B2
M. Reza Eslami et al., Theory of Elasticity and Thermal Stresses, Solid Mechanics and Its Applications 197, DOI: 10.1007/978-94-007-6356-2_5, © Springer Science+Business Media Dordrecht 2013
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5 Variational Principles of Elastodynamics
and
1 K (t) = 2
ρ u˙ 2 dv
(5.4)
B
for every p = [u, E, S] ∈ P. Then δ K{ p} = 0
(5.5)
if and only if p satisfies the equation of motion and the traction boundary condition. Clearly, in the (H-K) principle a displacement vector u = u(x, t) needs to be prescribed at two points t1 and t2 of the time axis. If t1 = 0, then u(x, 0) may be identified with the initial value of the displacement vector in the formulation of an initial-boundary value problem, however, the value u(x, t2 ) is not available in this formulation. This is the reason why the (H-K) principle can not be used to describe the initial-boundary value problem. A full variational characterization of an initialboundary value problem of elastodynamics is due to Gurtin, and it has the form of a convolutional variational principle. The idea of a convolutional variational principle of elastodynamics is now explained using a traction initial-boundary value problem of incompatible elastodynamics. In such a problem we are to find a symmetric secondorder tensor field S = S(x, t) on B × [0, ∞) that satisfies the field equation ¨ = −B on B × [0, ∞) ˆ −1 (div S)] − K[S] ∇[ρ
(5.6)
subject to the initial conditions ˙ 0) = S˙ 0 (x) for x ∈ B S(x, 0) = S0 (x), S(x,
(5.7)
and the boundary condition s = Sn = sˆ on ∂B × [0, ∞)
(5.8)
Here S0 and S˙ 0 are arbitrary symmetric tensor fields on B, and B is a prescribed symmetric second-order tensor field on B × [0, ∞). Moreover, ρ, K, and sˆ have the same meaning as in classical elastodynamics. First, we note that the problem is equivalent to the following one. Find a symmetric second-order tensor field on B×[0, ∞) that satisfies the integro-differential equation ˆ −1 t ∗ (div S)] − K[S] = −Bˆ on
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