The torsion problem of a circular nonlinear elastic rod loaded by end moments is considered. The solution constructed by the method of successive approximations taking into account second-order effects is compared with the solution obtained by a semi-inverse method. It is shown that the deadloading assumption breaks the symmetry of the Cauchy stress tensor in a certain region. A refined formulation of Saint Venant's principle is proposed for the problem of determining integral strain characteristics.Introduction. In designing many current high-precision devices, it is necessary to take into account the effects of physical and geometrical nonlinearities. (For example, in designing and calibrating a rod dynamometer, one should take into account the Pointing effect -the elongation of the rod due to torsion.)Analysis of this problem reduces to determining second-order effects in the torsion problem of a nonlinear elastic circular rod loaded by end moments. This classical problem has been the subject of many studies, among which we mention the work of Lur'e. In [1], he proposed a method of successive approximations for determining second-order effects in the deformation of bodies of various shapes. In [2], it is shown that the axial elongation of a cylinder obtained by the method of successive approximations [1] differs from that obtained by the semiinverse method. In the present paper, it is shown that the reason for this difference is that although the integral characteristics of the external load (axial force and torque) are equal in these problems, the solutions constructed correspond to different force distributions over the end surfaces of the cylinder. Moreover, the effect of this difference on the integral strain characteristic -cylinder elongation -is discussed.Method of Successive Approximations. The essence of the method described in [1] is that the problem of the equilibrium of a nonlinear elastic body of the form
The paper presents an exact solution for the problem of large deformations of torsion, axial tension–compression, and radial expansion or shrinkage of an elastic hollow circular cylinder equipped with pre-stressed elastic coatings. Surface coatings are modeled using the six-parameter nonlinear shell theory. The constitutive material of the cylinder is described by a three-dimensional nonlinear model of the isotropic incompressible body of the general form. Special boundary conditions describe the interaction of this material with thin coatings on the inner and outer surface of the pipe. Based on the solution obtained, numerical calculations were performed on the effect of preliminary stresses in coatings on the stress–strain state of a cylindrical pipe.
The torsion problem of a cylinder with a circular transverse cross section twisted by end moments that are equal in magnitude and opposite in direction isTile phenomenon of variation of tile length of an elastic cylinder in torsion was discovered experio mentally and described by Poynting in the early twentieth century. Quantitatively, this effect is manifested weakly: when the torsion angles are approximately 15-20 ~ per unit length, the relative elongation of tile sample does not exceed 0.01. However, in manufacturing precision measuring devices and in determining experimentally the elastic constants of materials, the influence of the Poynting effect should be taken into account.The phenomenon discovered by Poynting can be explained by means of the nonlinear torsion problem. The torsion problem of a compressible (changing its volume upon deformation) cylinder with allowance for axial elongation was treated both in known books dealing with continuum mechanics [1,2] and in recent studies. In particular, M. Chen and Z. Chen [3] analyzed this problem with tim use of asymptotic methods, and Koczyk and Weese [4] solved it by the finite-element method; the torsion problem of circular cylinders w~:~s analyzed numerically by Zubov [5] and Gavrilyachenko [6]. However, most of these studies considered concrete models of an elastic material.The goal of the present study is to estimate quantitatively and qualitatively the Poynting effect with tim use of various governing relations for isotropic compressible materials and compare the behavior of their models.Governing Relations. We introduce the reference (unstrained) and real (strained) configurations of the medium. The radius vector of a material point in the real configuration is denoted by R. The second-rank tensor C, which is called a strain gradient, is specified by the relation C --grad R, where grad is the gradient operator in the basis of the reference configuration.The Piola stress tensor defined in the reference configuration is expressed in terms of the "true" stress tensor T as follows [1]: D = (ct)-lTdet C.Rostov State Construction University, Rostov-on-Don 344022. Rostov State University, Rostov-onDon 344090.
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