Effect of Initial Stresses on the Small Deformations of a Composite Rod

Abstract

A system of linear equations governing the small deformations of an initially stressed, curved, twisted composite rod is obtained through the use of the principle of virtual work. Apart from the extensions! and bending deformations, transverse shear as well as warping of the rod cross section are incorporated into the assumed displacement field. The resulting equations are applicable for transversely isotropic rods for a wide range of values of the ratio of the shear modulus to Young's modulus. Special cases of the governing equations are also presented for plane-curved symmetric rods. N this paper, we consider the effect of initial stresses on the small deformations of a curved, twisted rod. The rod is assumed to be transversely isotropic with a heterogeneous cross section. The effect of initial axial stress on the torsional rigidity of straight uniform rods is well-known.1'2 An examination of the formula in Ref. 1 for the effective torsional rigidity reveals that the effect of axial stress is particularly significant whenever the shear modulus of the material is small compared to Young's modulus. Corresponding to this range of values of the shear modulus, the shear deformation of the rod may be significant. In the following treatment, we obtain the effective torsional rigidity for space curved rods in the presence of shear deformation as part of our results. The analysis of the initial stress problem is based on the linearized three-dimensional initial stress problem, and the associated variational principles discussed in Ref. 2. In conjunction with the principle of virtual work, we assume a displacement field incorporating transverse shear deformation and warping of the cross section. In the subsequent considerations, approximations are introduced on the basis of the thinness of the rod to simplify the resulting relations. Inertia forces, associated with the small deformations, are included through the use of D'Alembert's principle. A system of equations for the treatment of symmetric, plane-curved rods is obtained by specializing the general relations. These equations readily uncouple into two sets— one describing the in-plane deformations, the other describing the out-of-plane deformations of the rod. Explicit forms of these relations are also stated for the case of negligible transverse shear deformation. An analysis of superposed small displacements on finite deformations of rods has been previously given by Green et al.3 The constitutive relations obtained in Ref. 3, on the basis of thermodynamically consistent strain energy functions, introduce unnecessary complexity in an approximate treatment of practical rod problems, such as the dynamic stability