Abstract:
An inverse coe cient problem for the nonlinear biharmonic equation
Au := (g(»2(u))(ux1x1 + (1/2)ux2x2)x1x1 + (g(»2(u))ux1x2)x1x2 + (g(»2(u))(ux2x2 +
(1/2)ux1x1))x2x2 = F(x), in ½ R2, is considered. This problem arises in computational
material science as a problem of identi cation of unknown properties of
inelastic isotropic homogeneous incompressible bending plate using surface measured
data. Within J2-deformation theory of plasticity these properties are described by
the coe cient g(»2(u)) which depends on the e ective value of the plate curvature:
»2(u) = (ux1x1)2 + (ux2x2)2 + (ux1x2)2 + ux1x1ux2x2 . The surface measured output data
is assumed to be the de ections wi, i = 1,M, at some points of the surface of a plate
and obtained during the quasistatic process of bending. For a given coe cient g(»2(u))
mathematical modeling of the bending problem leads to a nonlinear boundary value
problem for the biharmonic equation with Dirichlet or mixed types of boundary conditions.
Existence of the weak solution in the Sobolev space H2() is proved by using the
theory of monotone potential operators. A monotone iteration scheme for the linearized
equation is proposed. Convergence in H2-norm of the sequence of solutions of the linearized
problem to the solution of the nonlinear problem is proved, and the rate of
convergence is estimated. The obtained continuity property of the solution u 2 H2()
of the direct problem, and compactness of the set of admissible coe cients G0 permit
one to prove the existence of a quasi-solution of the considered inverse problem.
