On the quasi-static approximation in the initial boundary value problem of linearised elastodynamics

Abstract

Continuous data dependence estimates are employed to rigorously derive conditions that validate the quasi-static approximation for the initial homogeneous boundary value problem in the theory of small elastic deformations superposed upon large elastic deformations. This theory imposes no sign-definite assumptions on the linearised elastic moduli and in consequence the requisite estimates are established using methods principally motivated by known Lagrange identity arguments.

Appendix A

For ease reference, we list without proof two key results previously derived in [19]. Notation has been altered to that adopted here, and a body-force vector f(x, t) per unit mass is now included in the equation of motion (2.1) . We have for $0\le 2t\le T$ [19, eqn.(2.17)]:

$$\begin{aligned} 2\int _{\Omega (t)}\rho u_i(t) u_{i,t} (t)\, \mathrm{d}x= & {} \int _0^t \int _{\Omega (\eta )} \rho [u_i(2t-\eta )f_i(\eta )-u_i(\eta )f_i(2t-\eta )]\, \mathrm{d}x\, \mathrm{d}\eta \nonumber \\&+\int _{\Omega (t)} \rho u_i(2t) u_i^{(1)}\mathrm{d}x+\int _{\Omega (t)} \rho u_{i,t}(2t) u_i^{(0)}\mathrm{d}x. \end{aligned}$$

(A.1)

Secondly, when $f_i=0$, the following bound holds [19, eqn.(3.22)] for $0\le 2t\le T$:

$$\begin{aligned} \int _{\Omega (t)} \rho u_i(t) u_{i,t} (t)\, \mathrm{d}x\le \frac{ T}{4} \int _{\Omega } \rho u_i^{(0)} u_i^{(0)}\mathrm{d}x + \frac{M_1 T^{1/2}}{4} \left[ \int _{\Omega } \rho u_i^{(0)} u_i^{(0)}\mathrm{d}x +\frac{T}{\sqrt{2}} \int _{\Omega } \rho u_i^{(1)} u_i^{(1)}\mathrm{d}x\right] , \end{aligned}$$

(A.2)

where $M_1$ is given by (3.22).