Algorithm of the method of characteristics for the analysis of nonlinear one-dimensional wave processes of conical and cylindrical shell deformation: PMM vol. 35, no. 4, 1971, pp.690–700

Abstract

An algorithm is proposed for realizing the method of characteristics for the analysis of one-dimensional wave processes excited by the edge effect and described by a quasilinear system of differential equations having several pairs of families of characteristics. The algorithm is written in Lagranglan coordinates for conical and cylindrical shells, on the basis of a quasilinear system of sixth order equations of a geometrically nonlinear theory of Timoshenko type [1], The algorithm presumes the absence of strong discontinuities i.e., of discontinuities in the first derivatives of the shell displacement, which will limit the class of admissible edge effects and permit carrying out the analysis up to the appearance of the first shock in problems where the shocks originate during wave propagation. Despite this, the proposed algorithm permits elucidation of specific properties of the wave solution in nonlinear theory. An illustrative example is given for a conical shell. In speaking of the one-dimensional transients of conical and cylindrical shell deformation, we shall have in mind the axisymmetric processes of these objects and the plane deformations of an infinite cylindrical shell (ring). One-dimensional transients of conical and cylindrical shell deformations have been investigated in a wave formulation in the past decade by using various models and methods within the scope of linear shell theory [2–4]. Fast-moving conical and cylindrical shell deformations have also been investigated within the scope of nonlinear shell theory [5–10,11], but methods to analyze the specific properties of traveling waves are still almost lacking. Indeed, mainly the application of 1. a) Methods of reducing the shell to a system with a finite number of degrees of freedom [12–16]; 2. b) Finite-difference methods of integration by using a mesh determined in advance without clarification and taking account of discontinuities in the solution [17–19]; 3. c) The method of lines [rays] with subsequent integration of a system of Runge-Kutta ordinary differential equations [20]; 4. d) A method of approximating the tangential displacements by the linear “column” solution with the subsequent calculation of the normal displacements from the nonlinear equations [21–23], is found in existing papers which are devoted primarily to the analysis of dynamic stability problems. These methods turned out to provide results in stability analysis, but do not allow complete characterization of the solution as a nonlinear transient of wave propagation. Hence, there is little information on how a geometrically nonlinear wave process, which is known to be small in some initial stage of the motion will differ from a linear wave process, will deviate more and more from the linear with the growth of time and become qualitatively different for high enough values of the time. Not explained are the limits of the well-founded applicability of linear theory, as well as the role of the shocks for a change in the character of the wave transient, including the appearance of large normal displacements (loss of stability). It is shown that the mentioned questions can be clarified by using the method of characteristics within the scope of nonlinear theory. However, an algorithm is needed for this, which is adapted to the case of the presence of several pairs of families of characteristics (there are three such pairs in a nonlinear theory of Timoshenko type in the case of one-dimensional wave processes). The algorithm proposed below consists of the following elements: 1. a) The representation of the initial equations [1] as a quasilinear system of first-order equations in the first derivatives of the displacement and additional formulas to calculate the displacements which are also in the coefficients of the system of equations; 2. b) The construction of equations governing the direction of the characteristics and the differential relations on the characteristics; 3. c) An algorithm of the iterative product of a computation by the second method of Masseau in the case of the presence of several pairs of families of characteristics. Let us note that the method of characteristics has been applied in [24–26] within the scope of a linear shell theory of Timoshenko type. The directions of the characteristics are constant in linear theory and independent of the solution, but in a theory of Timoshenko type two out of the three pairs of families of characteristics coincide. During the calculation of the linear solution (for comparison with the nonlinear solution), it was noted in passing that the difference between linear solutions obtained by the method of characteristics in [25,26] is a result of an error in one coefficient of the equations in [25].