M. Levinson

The instability of slightly compressible rectangular rubberlike solids under biaxial loadings

Abstract The instability of a rectangular solid loaded in plane strain by constant axial and transverse pressures is investigated using a variational method. Axial pressures are applied through rigid platens, and lateral pressures can be either hydrostatic or constant-directional. Two types of slightly compressible rubberlike material are studied in the problem, both of which reduce to Rivlins neo-Hookean material in the incompressible case. Previous studies using a “standard” material and the neo-Hookean material have yielded qualitatively different results. It is found that the “standard” material is unsuitable for studies of such finite strain problems and that the behaviour of the compressible materials studied tends smoothly towards that of the corresponding incompressible material as Poissons ratio approaches one half.

Indentation of an elastic layer(s) bonded to a rigid cylinder—I. Quasistatic case without friction

Abstract A quasistatic, frictionless indentation problem for an incompressible elastic layer(s) bonded to a rigid cylinder is solved by use of a stress function in the form of a series. The method of Yau is then applied to obtain the solution of the dual series equations resulting from the mixed-type boundary condition which categorizes the problem. Although the analysis is given in detail only for a single-layered cylinder, comments on the many-layered case are made and some numerical results are given for a two-layered covering. In addition, some observations are made concerning a specific numerical example typical of rubber-covered steel rolls of the sort used in paper mills.

The post-flutter oscillations of discrete symmetric structural systems with circulatory loading

Abstract The steady oscillatory behaviour of discrete symmetric structural systems with circulatory non-conservative loads is investigated in the region of the flutter point. The investigation is necessarily non-linear, and is accomplished by using a two-parameter perturbation approach. It is found that the non-vanishing steady oscillations are nearly harmonic. It is further shown that flutter can be classified as either “hard” or “soft”, and a method of classifying a particular system is obtained. The perturbation scheme can be carried on ad infinitum to give the complete non-linear steady oscillation behaviour of any given system in an extended neighbourhood of the critical point.

The long fluid storage bag: A contact problem for a closed membrane

Abstract A study of the mechanical behaviour of a reasonable model of the reinforced, rubber bags sometimes used as fuel storage tanks in remote locations is made. The non-linear differential equation governing the behaviour of the model is derived and it is found to have a solution which can be expressed in terms of elliptic integrals of the first and second kinds. The relations between cross-sectional shape, base contact length, volume, membrane forces and pressure are given algebraically and graphically. Some comments on practical considerations also are made.

Indentation of an elastic layer bonded to a rigid cylinder—II. Unidirectional slipping with Coulomb friction

Abstract The solution to the title problem, based on a stress function represented by a series, is presented in this paper. Slipping is assumed over the entire contact surface where Coulombs law of friction is assumed to apply. The series method of Yau is used to solve a set of dual series equations which result from the mixed type of boundary conditions defining the problem. The angular deviation of the center of the contact surface from the line joining the centers of the cylinders is determined by means of the principle of minimum potential energy. Some numerical results of interest for both compressible and incompressible elastic layers are presented graphically.

Pure Flexure of a Layered Orthotropic Shell-Elasticity Study of a Tire-Related Problem

Abstract This paper presents the analysis of the plane strain, pure bending of a portion of a three-layered cylindrical shell which is taken as a simple model of a part of a belted pneumatic tire. The two inner layers, representing the body and belts, respectively, are modeled as orthotropic, incompressible materials whose properties can be found from the appropriate cord and matrix material propties in a manner given in an Appendix. The outer layer, representing the tread rubber, is considered to be an isotropic, incompressible material. We concern ourselves with the study of the stresses tending to separate the various layers. These stresses, for two particular cases, are found to be strongly dependent on the crown angles of the cords in the body and belt layers. The results are presented in graphical form with the separation stress as a function of the body crown angle and with the belt crown angle entering as a parameter.