Aspects of the application of a projective approach to the construction of an applied dynamical theory of multilayer shells

Abstract

UDC 539.3 The change from three-dimensional to two-dimensional equations in the theory of plates and shells is interpreted as a reduction in the dimensional of the initial equations of the spatial problem of elasticity theory. The form of the reduced equations that are thus obtained depends on the particular method used in the reduction of the dimension. The variational approach is the most well-founded approach to the reduction of dimension in plate theory. In [1] the basic principles and a sequence for the construction of two-dimensional equations of the refined theory of multilayer shells, including dynamical equations, were considered on the basis of the variational approach. Analysis of the different types of dynamical equations that have been obtained shows that the inertial terms corresponding to different unknowns of the two-dimensional equations are related to each other by means of the inertia matrix; note that this matrix is nondiagonal. This circumstance makes it far more difficult to apply different computational algorithms for the solution of nonsteady-state nonstationary problems that involve integration of the two-dimensional dynamical equations of generalized shell theory in view of the initial conditions (initial boundary value problem). By means of algebraic methods it is, in theory, possible to transform these types of equations in such a way as to render the inertia matrix diagonal. However, under ordinary conditions this is difficult to achieve. Moreover, the transformed equations will contain as unknowns linear combinations of the initial unknowns as new unknowns. These new unknowns lack any physical meaning. Thus arises the problem of constructing a variant of two-dimensional decision equations that will possess, in terms of the inertial terms, a normal form. At the same time, in such a construction the unknowns occurring in these equations will retain a physical meaning. Such a construction is possible if the underlying ideas of the projection method are used to reduce the dimension of the initial equations, bearing in mind certain features that are intrinsic to approximate theories of multilayer shells.