The Role of Fredholm Conditions in Signorini's Perturbation Method G. CAPRIZ & P. PODIO GUIDUGLI Dedicated to Clifford Truesdell on his sixtieth birthday Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 2. The Perturbation Technique . . . . . . . . . . . . . . . . . . . . . . 263 3. Algebraic Consequences of Orthogonality ................. 266 4. Elastic Equilibrium in a Neighbourhood of Any Placement ......... 268 5. Successive Approximations . . . . . . . . . . . . . . . . . . . . . . . 270 6. Fredholm Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 272 7. Signorini's Case: Initial Placement at Ease ................ 274 8. Initial Placement under Stress: Some Properties of ker L ......... 278 9. Initial Placement under Stress: Integral Properties ............. 281 10. Significance of Static Incompatibilities .................. 283 11. A Critical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 1. Introduction We pursue here our study of SIGNORINI's analytic perturbation method in finite elasticity (rid. [1], [2] and [3]), considering the case where the starting equilibrium placement is under stress. This generalization is not at all trivial: on the one hand, the indeterminacy of the solution of the approximating linear problem of first order can here be large; on the other hand, the technical conditions for existence of solutions of the approximating linear problems of successive orders do not always reflect simply the requirement of balance of angular momentum. As is customary in the theory of perturbation methods, we introduce a parameter and study local properties of the mapping (parameter) ~ (solution). We find it convenient to present the formal developments in a general context, substantially adapting an analysis which is common to studies of bifurcation phenomena. We derive a series expansion that could be used to construct approximations (of increasing order with respect to the parameter) of the solution for the static balance equations of elastic bodies under dead loads. Archive for Rational Mechanics and Analysis, Volume 70, 9 by Springer-Verlag 1979