Statement and solution of optimal problems for independent stress and deformation control by eigenstrain

Abstract

AbstractThe method of unique decomposition of eigenstrain into two constituents, namely in impotent eigenstrain, that does not cause stress and in the complementary nilpotent eigenstrain that does not induce any deformation in the linear elastic solid is considered in detail. Decomposition is performed in the Hilbert (energy) function space. These two complete classes of eigenstrains render optimal solutions by keeping shape and stress‐control problems well separated. Such an optimization is found to be of crucial importance in the design of smart structures, which can adapt to altering conditions. Benchmark solutions of both, shape control and optimal redistribution of elastic load stresses in discretized smart structures illustrate the power of the novel method and its convenience of application. Further, the problem of residual stress control past hot rolling of I‐beams is solved: the theorem on decomposition provides the crucial tool to define the required control parameters for forced cooling‐down with the goal of producing a favourable residual stress state as close as possible to a predefined (optimal) stress state. The latter is exemplarily selected to increase the buckling strength of such columns with given slenderness ratio.