Large elastic deformations of isotropic materials X. Reinforcement by inextensible cords

Abstract The theory of large elastic deformations of incompressible, isotropic materials developed in previous papers of this series is employed to examine some simple deformations of elastic bodies reinforced with cords. The cords are assumed to be thin, flexible and inextensible, and to lie parallel and close together in smooth surfaces in the undeformed body, which is thus divided into sections by boundary surfaces which are inextensible in certain directions. In the simple problems considered, the cords impose relationships upon the parameters which specify the deformation. The following examples are examined from this point of view: (i) the pure homogeneous strain of a thin uniform sheet containing a double layer of cords lying in a plane midway between its major surfaces; (ii) the combined pure homogeneous strain and flexure of a cuboid containing a double layer of cords lying in a plane parallel to a pair of opposite faces, the two sets of cords being unsymmetrically disposed in this plane with respect to the remaining faces of the cuboid, and the symmetrical case being obtained from this by a suitable choice of constants; (iii) the combined extension and flexure of a thin rectangular sheet with two sets of cords placed symmetrically in a plane parallel to its major surfaces, the problem being considered as a limiting case of (ii); (iv) the simultaneous extension, inflation and torsion of a cylindrical tube containing one or two sets of cords lying in helical paths concentric with the axis of the cylinder. Ifr cases, relations are obtained for the determination of the tensions in the cords in terms of the applied forces and the parameters which define the deformation.