Aerohydroelasticity problems for bodies with initial stresses

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Numerous aerohydroelasticity problems (for time-independent and time-dependent problems, for quiescent and moving liquid and gas, in linear and nonlinear formulations, etc.) are under investigation at present. The results of these investigations are reflected in a large number of publications (1-6, 18, 19, 21-27) and others, where two-dimensional applied theories have been adopted in the description of the motion of elastic bodies whose formulation in- corporates the hypotheses of Kirchoff--Love, Timoshenko, and others in individual cases with the initial stresses taken into account (within the framework of the indicated theories). The formulation of linearized aerohydroelasticity problems is expounded on in this article for three-dimensional bodies with initial stresses which interact with compressible and incom- pressible ideal and viscous fluids: methods of solution are discussed which are based on the introduction of different general solutions; results are given for the solution of individual problems. We will introduce a single assumption which consists of the fact that we will investigate those dynamical processes in a system consisting of an elastic body and a liquid or gas for which the additional stresses (perturbations) which arise are significantly less than the initial stresses. This makes it possible to apply the relationships of linearized elasticity theory (8, 9, 12, 14). It is possible with such a formulation to investigate the effect of initial stresses on the values of parameters characterizing dynamical processes in a system consisting o'f an elastic body with initial stresses and a liquid or gas (wave propagation velocities, damping coefficients, and so on). Below we will discuss the initial state deter- mined within the framework of the theory of finite deformations (9, 14); it is necessary to introduce the appropriate simplifications indicated in (8, 9, 12, 14) for switching to dif- ferent versions of the theories of small initial deformations. Quantities referring to the initial deformed state in an elastic body and the initial motion in a liquid or a gas will be denoted by the index "0"; when the symbols for quantities referring to the elastic body as well as to the liquid or gas coincide, we will use a prime to denote the latter. Let us restrict ourselves to the case of a homogeneous initial deformed state in an elastic body whose displacements are determined from the following expressions: u~ = 6~j (Li -- 1) x~ (X l = const). (0.1) It is possible to obtain effective solutions of individual classes of problems with the same approach as has been outlined in (13) for elastic bodies which do not interact with the liquid or gas, which will be partially realized in the present article. i. Basic Relationships. We will discuss the formulation of the basic relationships for compressible and incompressible elastic bodies with initial stresses and for an ideal and viscous liquid and gas, and we will also discuss the conditions at the media interface. The indicated relationships, along with the boundary and initial conditions specified separately for the elastic body and the liquid or gas, represent a closed system of aerohydroelasticity problems for elastic bodies with initial stresses. We note that the relationships of linearized elasticity theory in (8-12, 14) and in other monographs are presented in Lagrangian coordinates of the natural (undeformed) state. It is necessary for the formulation of aerohydroelasticity problems to switch to the coordi- nates of the initial deformed state, since the relationships for a liquid and a gas are for- mulated in the case of linearized problems in the coordinates of the initial deformed state (with respect to an elastic body, a liquid, or a gas these coordinates correspond to the natural state). We will assume thatthe elasticbody (in the initial deformed state) occupies a volume VI and the liquid or gas occupies a volume V2; we will denote the media interface by S.