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IMA Journal of Applied Mathematics Advance Access originally published online on December 17, 2004
IMA Journal of Applied Mathematics 2005 70(1):129-146; doi:10.1093/imamat/hxh052
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IMA Journal of Applied Mathematics Vol. 70 No. 1 © Institute of Mathematics and its Applications 2005; all rights reserved.

Recovery of residual stress in a vertically heterogeneous elastic medium*

Sergei A. Ivanov1, Chi-Sing Man2 and Gen Nakamura3

1 Russian Center of Laser Physics, St Petersburg University, St Petersburg, 198904, Russia, 2 Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA, 3 Department of Mathematics, Graduate School of Sciences, Hokkaido University, Sapporo, 060-0810, Japan

We study the problem of identifying residual stress within a thin subsurface layer in an elastic medium occupying a region {Omega} = {(x1, x2, x3) , R3: 0 < x3 < L, where L ≤ {infty}} in space, where all parameters depend only on the depth x3. Under the theoretical framework of linear elasticity with initial stress, the incremental elasticity tensor of each material point is written as a sum of two terms, namely the elasticity tensor and the acoustoelastic tensor, both of which are taken here as isotropic functions of their arguments. By imposing impulsive loads and measuring the displacements at the boundary x3 = 0, we recover the residual stress and its gradient there. If the residual stress has a diagonal form, we can recover the residual stress inside the subsurface layer.

Keywords: elastic medium; residual stress; subsurface layer.

* Dedicated to Ray W. Ogden on the occasion of his 60th birthday


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