ABSTRACT
The Hooke tensors can be determined by their eigenvalues λi, called Kelvin moduli, and by eigentensors ωi, called proper states, satisfying the orthogonality conditions: ωi ·ωj = δij , where i, j = 1, 2, 3 in 3D case and i, j = 1, 2 in 2D case, see Rychlewski (1984). In the remainder we consider thin, transversely homogeneous plates and shells of uniform thickness h and we refer all quantities to appropriate mid-surfaces. Structures of this kind are characterized by two stiffness tensorsA= hC andD= h312C, whereC represents the tensor of reduced elastic moduli corresponding to the so-called generalized plane stress assumptions. TensorsA andD are characterized by three Kelvin moduli of the former one, i.e. λ1 ≥ λ2 ≥ λ3 and corresponding proper states ω1, ω2, ω3 defined on the middle surface and, consequently, within the whole body of the shell. In the sequel we deal with the problem posed as follows:
For fixed values λi of Kelvin moduli, characterize optimal directions of proper states ωi subject to the following conditions
to be satisfied pointwise. Proper state fields ω1 and ω2 defined on the middle
surface and subject to constraints in (1) have to be chosen optimally, i.e giving the total compliance smallest possible value.
