E. V. Makeev

Optimal Shape Design of Axisymmetric Shells for Crack Initiation and Propagation Under Cyclic Loading

ABSTRACT This paper describes a problem of axisymmetric shell optimization under fracture mechanics and geometric constraints. The problem is formulated as the weight (volume of a shell material) minimization and the meridional contour of the shell is taken as the design variable. The shell is made from quasi-brittle materials and through cracks arising are admitted. It is supposed that the shell is loaded by cyclic forces. A crack propagation process related to the stress intensity factor is described by the Paris fatigue law. The problem of finding the meridian shape (geometric design variable), of the shell having the smallest mass subject to constraint on the cyclic number for fatigue cracks, is investigated using a minimax (guaranteed) approach worked out in the theory of optimization under incomplete information. Optimal designs of the shells are found numerically with the application of genetic algorithms.

On the penetration of nonaxisymmetric bodies into a deformable solid medium and their shape optimization

We consider the problem of penetration of rigid pyramidal bodies (impactors) into a strained medium in the case of large speeds of penetration and estimate the depth of the impactor penetration. To this end, we use the two-stage penetration model proposed by Forrestall. We state the shape optimization problem for the penetrating body, which is based on the consideration of a set of bodies of pyramidal external shape with given fixed mass. We study both solid and hollow (shell-shaped) bodies. For the optimization functional we take the penetration depth of the penetrating body, and for the projection variable we take the number of faces of the pyramidal body. We present the results of computations of the penetration depth for different shapes of the impactor and show that, both for shells and solid impactors, the bodies of the shape of a circular cone are optimal. The problems of high-speed penetration of rigid bodies into a deformable medium are nowadays very topical problems [1] which have been studied by Russian and foreign authors [2–8].

Finding of Rigid Punch Shape and Optimal Contact Pressure Distribution

The problem of contact pressure optimization is formulated for the case of rigid punch interacted with elastic medium. Coupling of the punch penetration and action of external loads at the outside regions is taken into account. The shape of the punch is considered as an unknown design variable. The minimized integral functional characterizes the discrepancy between the actual contact pressure and the required pressure distribution. The problem is studied under condition that the total forces and moments applied to the punch and the loads acted at the outside regions are given. It is shown that the considered optimization problem can be split and transformed to two successively solved problems. Optimal shapes are found analytically for the punches having rectangular contact domains.

Some problems of optimizing shell shape and thickness distribution on the basis of a genetic algorithm

We consider problems related to designing axisymmetric shells of minimal weight (mass) and the development of efficient nonlocal optimization methods. The optimization problems under study consist in simultaneous search for the optimal geometry and the shell thickness optimal distribution from the minimal weight condition under strength constraints and additional geometric constraints imposed on the thickness function, the transverse cross-section radii distribution, and the volume enclosed by the shell. Using the method of penalty functions, we reduce the above optimal design problem to a nonconvex minimization problem for the extended Lagrange functional. To find the global optimum, we apply an efficient genetic algorithm. We present the results of numerical solution of the optimal design problem for dome-like shells of revolution under the action of gravity forces. We present some data characterizing the convergence of the method developed here.

Some Problems of Multipurpose Optimization for Deformed Bodies and Structures

Some problems of multipurpose analysis and optimization of deformed structures and thin-walled structural elements are studied in this paper under some constraints including incomplete data. The first problem is the multipurpose optimization of layered plate made from given set of materials in context of optimization of ballistic limit velocity. Incomplete data concerning the thickness of layers of optimized multilayered shield structure are taken into account. The Pareto-approach and numerical evolutionary method (genetic algorithm) are used for solving of the considered multipurpose problem. The second problem studied in the paper is the shape optimization problem for rigid punch moving on the surface of elastic half-space, which is solved analytically in multipurpose formulation taking into account friction of contacted surfaces, wear of materials and arising pressure distributions. The relative movement is considered in frame of quasi-static formulation. Formulated optimization problem is studied analytically using the developed decomposition approach and exact solutions are obtained for the punch which has a rectangular contact region and moves translationally with a constant velocity.

SOME ANALYTICAL AND COMPUTATIONAL ESTIMATES OF PARAMETERS OF OPTIMAL PROTECTIVE PLATE STRUCTURE

Ðàññìîòðåíà çàäà÷à îïòèìèçàöèè ñòðóêòóðû çàùèòíîé ïëèòû çàäàííîé òîëùèíû, ñîñòîÿùåé èç íåñêîëüêèõ ñëîåâ. Ïðåäïîëàãàåòñÿ, ÷òî ñëîè èçãîòîâëåíû èç ðàçëè÷íûõ ìàòåðèàëîâ ñ çàäàííûìè ñâîéñòâàìè, à ïîñëåäîâàòåëüíîñòü óêëàäêè è òîëùèíû ñëîåâ îïðåäåëÿþòñÿ èç óñëîâèÿ ìàêñèìèçàöèè ïðåäåëüíîé áàëëèñòè÷åñêîé ñêîðîñòè âûñîêîñêîðîñòíîãî óäàðíèêà, ïðîíèêàþùåãî â çàùèòíóþ ïëèòó. Ïîëó÷åíû àíàëèòè÷åñêèå âûðàæåíèÿ äëÿ ïðåäåëüíîé áàëëèñòè÷åñêîé ñêîðîñòè è ñîîòíîøåíèÿ, îïèñûâàþùèå ñòðóêòóðó ïëèòû, ïîñòðîåíû îïòèìàëüíûå ðåøåíèÿ äëÿ çàäàííîãî íàáîðà ìàòåðèàëîâ è ðàçëè÷íûõ çíà÷åíèé ïàðàìåòðîâ çàäà÷è. Ïðîâåäåíà òåîðåòè÷åñêàÿ îöåíêà ïàðàìåòðà, õàðàêòåðèçóþùåãî ôîðìó ãîëîâíîé ÷àñòè óäàðíèêà.

Reliable Optimal Design in Contact Mechanics

The problem of contact pressure optimization is formulated for the case of rigid punch interacted with elastic medium. Coupling of the punch penetration and action of external loads at the outside regions is taken into account. The shape of the punch is considered as an unknown design variable. The minimized integral functional characterizes the discrepancy between the actual contact pressure and the required pressure distribution. The problem is studied under condition that the total forces and moments applied to the punch and the loads acted at the outside regions are given. It is shown that the considered optimization problem can be splitted and transformed to two successively solved problems. Optimal shapes are found analytically for the punches having rectangular contact domains.

Optimization of flexible beams

We consider the optimal design problem for cantilever beams of variable rigidity loaded at the free end by an arbitrary transverse force. The value of the cantilever free end vertical displacement serves as the optimality criterion, and the distribution of the cantilever thicknesses (cross-sections) is usually used as the design variable. We present results of an asymptotic analysis and a numerical solution of the optimization problem and discuss specific features of the formation of optimal solutions under nonlinear bending.

On the stress state of shells penetrating into a deformable solid

The motion of an axisymmetric shell in a deformable solid medium is considered. It is assumed that the medium resistance is described by a two-term expression containing a constant term (the rigidity characteristic) and an inertial term quadratic with respect to the penetration velocity. A model of the impactor penetration with the normal interactions with the resisting medium taken into account is proposed. The membrane forces and the arising stresses are determined for decelerated motions of the impactor.