Axisymmetric deformation of shells of revolution made of fiber-reinforced elastomeric layers

Abstract

The study of the deformation of shells of fiber-reinforced elastomers is based on a structural approach in which the elastomeric matrix and reinforcement are considered separately, without reduction of the composite to a medium having continuously changing characteristics. The concept underlying this approach was first explained in [1], where a momentless model of deformation was used as the basis for describing a method of designing a rubber shell with a reticular reinforcement and no binder. In [2], calculations were performed for certain shells of revolution made of an elastomer and reinforced along the middle surface using fibers with little or no tensility. The effect of the reinforcement was accounted for by a method employed to analyze momentless shells. Line~-r relations from the theory of elasticity were used in [3] to construct a momentless model of the deformation of a shell with a reticular reinforcement with allowance for the effect of the binder. In [4], a method was presented for solving momentless problems of the deformation of multilayered shells of revolution. Here, the mechanical properties of the reinforcing fibers and binders were described with the use of nonlinear elastic relations. The well-known models of deformation do not consider the moment character of the stress state of composite shells made of elastomeric layers. They are suitable for the design of very thin axisymmetric shells, in which the stress state is of a membrane (momentless) character. Here, we construct a structural model that describes the axisymmetric deformation of moment shells of revolution composed of elastomeric layers unidirectionally reinforced with fibers. The method is based on an iterative procedure for the design of shells of this type and employs the hypothesis of a compressible normal element. In accordance with this hypothesis, a normal element of the shell remains straight and normal to the plane of reference during deformation, undergoing elongations that vary through the thickness of each layer. The volume occupied by the reinforcing fibers is assumed to be small compared to the volume occupied by the binder. In accordance with the smallness of the relative volume of the reinforcement, the cross section of the fibers is assumed to be infinitely small. We will use an iterative method to determine the parameters of the deformed shell at each point of discretization of the given boundary-value problem. 1. We will examine a thin composite shell of N elastomeric layers unldirectionally reinforced with fibers. We will assume that the fibers are crossed (arranged antisymmetrically) in any two adjacent layers to form a reinforcing grid with rhomboid cells. The fibers may be arranged in the meridional direction in one layer and in the circumferential direction in the next layer. Some of the layers have no reinforcing material and are purely elastomeric. The thicknesses of the layers is assumed to be constant or to change smoothly in the meridional direction. We take the internal bounding surface of the shell as the plane of reference (the base coordinate plane). We refer this plane to an orthogonal coordinate system established by the lines of principal curvature. The transverse coordinate z is reckoned along an outer normal to the initial reference plane. As the running meridional coordinate, we take the length t of the generatrix of the coordinate surface from a point on the bounding contour to the current point. We will use the following notation: h is the total thickness of the shell; h k is the thickness of the k-th layer; z k and Zk+ 1 are values of the coordinate z for the lower and upper surfaces bounding the k-th layer; k 1 and k 2 are the principal curvatures of the coordinate surface of the shell; r is the distance from a point on the coordinate surface to the x axis (symmetry axis) of the shell; 0 is the angle between the normal to the coordinate surface and the symmetry axis; vkf is the frequency of fibers in the k-th layer; c~kf is the angle of inclination of the k-th layer fibers to the meridian. Quantities pertaining to the reinforcing