# Design of Composite Plates of Extremal Rigidity

## Abstract

We consider design problems for plates possessing an extremal rigidity. The plate is assumed to be assembled from two isotropic materials characterized by different values of their elastic moduli; the amount of each material is given. We look for a distribution of the materials which renders the plate’s rigidity for either its maximal or minimal value. The rigidity is defined here as work produced by an extremal load on deflection of the points of the plate. The optimal distribution of the materials is characterized by some infinitely often alternating sequences of domains occupied by each of the materials (see [1], [2]). This leads to the appearance of anisotropic composites; their structures are to be determined at each point of the plate.

The exact bounds on the elastic energy density are obtained; the micro-structures of the optimal composites are found. The effective properties of such composites (we call them matrix laminate composites) are explicitly calculated. These composites extend the set of available materials. The initial problems are formulated and solved for an extended set of design parameters.

We use the results to solve the problem of optimal design of the plate with variable thickness, given mass, and with additional restrictions on the range of the thickness values. A rule is found that allows us to distinguish the cases in which the optimal composite is assembled of elements having only maximal or minimal thickness.

The number of optimal design problems for the clamped square plate of maximal and minimal rigidity are solved numerically.

## Keywords

Optimal Design Matrix Composite Composite Plate Stiffness Tensor Optimal Design Problem## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Lurie, K.A., Cherkaev, A.V., About the application, of the Prager’s theorem to the optimal design Problem of thin plates,
*Izvestia AN SSSR, Mechanics of Solids***6**(1976), (in Russian).Google Scholar - [2]Lurie, K.A., Cherkaev, A.V., Fedorov, A.V., Regularization of optimal design problems for bars and plates, I, II,
*J. Opt. Th. Appl.***37**(1982), 499–543.MathSciNetzbMATHCrossRefGoogle Scholar - [3]Lurie, K.A., Some problems of optimal bending and stretching of elastic plates,
*Izvestia AN SSSR, Mech. Solids***6**(1979), 86–93, (in Russian).Google Scholar - [4]Rikhlevsky, J.K., About the Hook’s law,
*Applied Mathematics and Mechanics***3**(1984), 420–436, (in Russian).Google Scholar - [5]Zhikov, V.V., Kozlov, S.M., Oleinik, O.A., Kha Tien Ngoan, Averaging and
*G*-convergence of differential operators,*Uspehi Mat Nauk***34**(5) (1979), 209–219, (in Russian).MathSciNetGoogle Scholar - [6]K.A. Lurie and A.V. Cherkaev, Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion,
*Proc. Roy. Soc. Edinburgh***99 A**(1984), 71–87 (first version: preprint 783, Ioffe Physico-technical Institute, Leningrad, 1982)MathSciNetCrossRefGoogle Scholar - [7]Lurie, K.A., Cherkaev, A.V., Fedorov, A.V., On the existence of solutions to some problems of optimal design for bars and plates,
*J. Opt Th. Appl.***43**(1984), 247–281.MathSciNetCrossRefGoogle Scholar - [8]Filippov, A.F., Some questions of the theory of optimal control,
*Vestnik MGU, Mathematics***2**(1959), 25–32, (in Russian).Google Scholar - [9]Morrey, C.B., Integrals in the calculus of variations, Springer-Verlag, Berlin, 1966.zbMATHGoogle Scholar
- [10]Dacorogna, B., Weak continuity and weak lower semicontinuity of nonlinear functionals,
*Lecture Notes in Math*.**922**, Springer-Verlag, New York, 1982.Google Scholar - [11]Ball, J.M., Currie, J.C., Olver, RJ. Null, Lagrangians, weak continuity, and variational problems of arbitrary order,
*Funct. Anal.***41**(2) (1981), 135–174.MathSciNetzbMATHCrossRefGoogle Scholar - [12]Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity,
*Arch. Rat. Mech. Anal.***63**(4) (1977), 337–403.zbMATHCrossRefGoogle Scholar - [13a]Kohn, R.V., Strang, G., Optimal design and relaxation of variational problems, I, II, III,
*Comm. Pure & Appl. Math.***39**(1986), 113–137,MathSciNetzbMATHCrossRefGoogle Scholar - [13b]Kohn, R.V., Strang, G., Optimal design and relaxation of variational problems, I, II, III,
*Comm. Pure & Appl. Math.***39**(1986), 139–182,MathSciNetzbMATHCrossRefGoogle Scholar - [13c]Kohn, R.V., Strang, G., Optimal design and relaxation of variational problems, I, II, III,
*Comm. Pure & Appl. Math.***39**(1986), 353–377.MathSciNetzbMATHGoogle Scholar - [14]K.A. Lurie and A.V. Cherkaev, Exact estimates of the conductivity of a binary mixture of isotropic materials,”
*Proc. Roy. Soc. Edinburgh***104 A**(1986), 21–38. (first version: preprint 894, Ioffe Physico-Technical Institute, Leningrad, 1984).MathSciNetCrossRefGoogle Scholar - [15]Raitum, U.E., Extension of the extremal problems for linear elliptic operators,
*Dokladi AN SSSR***243**(2), 281–283, (in Russian).Google Scholar - [16]Lurie, K.A., Fedorov, A.V., Cherkaev, A.V., Regularization of optimal design problems of bars and plates and correction of the contradictions in the system of necessary conditions of optimality, preprint,
*Phys. Tech. Inst. AN SSSR***667**(1980), (in Russian).Google Scholar - [17]Fedorov, A.V., Cherkaev, A.V., Optimal choice of the orientation of the axis for elastic symmetric orthotropic plate,
*Izvestia AN SSSR, Mech. Solids***3**(1983), 135–142, (in Russian).Google Scholar - [18]Lurie, K.A., Cherkaev, A.V.,
*G*-Closure of some particular sets of admissible material characteristic for the problem of bending of thin plates,*J. Opt. Th. Appl.***42**(1984), 305–315.MathSciNetzbMATHCrossRefGoogle Scholar - [19]Kartvellishvilli, V.M., Mironov, A.A., Samsonov, A.M., Numerical method of solution for the problems of optimal design of reinforced constructions,
*Izvestia AN SSSR, Mech. Solids***2**(1981), 93–103, (in Russian).Google Scholar - [20]Olhoff, N., Optimal Design of Constructions, Mir, Moscow, 198, (in Russian).Google Scholar
- [21]Banichuk, N.V., Shape Optimization of Elastic Bodies, Nauka, Moscow, 1980, (in Russian).Google Scholar
- [22]Olhoff, N., Lurie, K.A., Cherkaev, A.V., Fedorov, A.V., Sliding regimes and anisotropy in optimal design of vibrating axisymmetric plates,
*Denish Center Appl. Math. Mech.*, Report 192, 1980.Google Scholar - [23]Olhoff, N., Cheng, K.-T., An investigation concerning optimal design of solid elastic plates,
*Int. J. Solid Struc.***17**(1981), 305–321.MathSciNetzbMATHCrossRefGoogle Scholar - [24]Arman, J.-L. R, An Application of Optimal Control Theory for the System with Distributed Parameters to the Problems of Optimal Design of Constructions, Mir, Moscow, 1977, (in Russian).Google Scholar
- [25]Kohn, R.V., Vogelius, M., A new model for thin plates with rapidly varying thickness I,
*Proc. Un. of Maryland*, BN-988 (1982).Google Scholar - [26]Cherkaev, A.V., Some Problems of Optimal Design of for Elastic Elements of Constructions, PhD Thesis, Polytecnicheskij Institute, 1979, (in Russian).Google Scholar