A generalization of the equi-stress principle in optimizing the mechanical performance of two-dimensional grained composites

Abstract

A new optimization criterion of minimizing the variations of the boundary tractions in 2D bi-material elastostatic problems is proposed as a relaxation of the well-known equi-stress principle far beyond its primary application. This integral-type assessment of the local stresses offers significant numerical advantages over their direct minimization. In particular, it allows us to obtain reliable results at moderate computational cost through the same flexible scheme as in the author’s previous research on optimizing perforated plates. The scheme combines a genetic algorithm optimization with an efficient direct solver and with an economic shape parametrization, both formulated in the complex variable terms. After an extended analysis of the criterion, its effectiveness is illustrated, as before, by applying to a checkerboard grained plate where the equi-stress inclusions cease to exist at not-too-small volume fractions. The results obtained permit us to make some empirical conclusions which may stimulate further studies in both theoretical and practical directions.

Plane elastostatic stress analysis in complex variables: A wavelet processing perspective

“…The same situation also takes place for the contact stresses s nn (t), t nu (t) along a two-phase interface [7]:…”

“…In contrast, for shear-dominating loads, the equistressness makes no sense since any stress distribution should have a number of sign-changing points in locations predefined by the shear rotational antisymmetry. Instead, the less restrictive piecewise constant distribution of the hoop stresses (M-equistressness) js uu (t)j = Const, t 2 L ð3:11Þ was adopted and numerically shown to be attained in some non-trivial instances [7,10,15,17]. As each Walsh function is also piecewise constant, the WT of (3.11) involve only one or a few items thus providing an efficient stress optimization tool with respect to the M-equistressness.…”

Planar grained structures with multiple inclusions in a periodic cell: Elastostatic solution and its potential applications

“…The corresponding KM potentials are proven [11] to take the form which is readily extended to the multi-inclusion case…”

Inverse Problem of Elasticity for a Plate Weakened by Hole and Cracks

“…Mathematical Problems in Engineering Satisfying the boundary conditions on the crack faces by functions (20)- (21), after some transformations we get a singular integral equation with respect to the unknown function 1 1 ( 1 ):…”

“…where the complex potentials Φ (1) 0 ( ) and Ψ (1) 0 ( ) are determined by the relations similar to (36), in which 0 ( ) should be replaced by 1 ( ), and the analytic functions Φ (1) 1 ( ) and Ψ (1) 1 ( ) are sought in the form of power series (21). The coefficients and are found from formulas of type (22).…”