On the existence of solutions to some problems of optimal design for bars and plates

Exact Relations for Effective Tensors of Polycrystals. II. Applications to Elasticity and Piezoelectricity

“…When it does not, we say that there is an exact relation for effective moduli. This can indeed happen as evidenced by recent results [4,10,12] on polycrystals. In this paper we describe a general method for finding such relations for effective moduli of laminates.…”

“…The results were almost always inequalities that the effective tensors had to satisfy. On rare occasions, however, researchers were discovering exact relations, or equalities for effective tensors [1,4,6,7,10,12]. These results were in large respect byproducts of the inequalities derived using one of the two methods mentioned above.…”

“…One of them was discovered by Hill [6,7]. Two were found in [10,12] and two more were described in [4]. One more is obtained as an intersection of the two manifolds from [4].…”

“…The term "translation method" has been coined by Milton [13] and has become a universally accepted term since. The two methods mentioned above have brought about complete solutions to many G-closure problems [3,4,8,9,10,11,12,16] as well as many optimal bounds on important functionals such as energy or complementary energy. The accumulated literature on the subject is too vast to allow a complete list (see, however, [13] for a review and comparison of the two methods).…”

A problem in the optimal design of networks under transverse loading

“…It is then a functional of the pretractions and these, in their turn, are the coefficients of an equilibrium operator. Thus, the problem reduces to studying the minimization of a cost functional defined over a certain class of operators, as is customary in many other optimization problems [1][2][3][4][5][6][7][8][9][10][11][12]. The theory of (/-convergence (see, e.g., [13,14]) proves to be the natural setting to discuss the existence of solutions for these problems.…”