Showing papers in "Journal of Applied Mechanics in 1965"

Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55)

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Mechanics of Incremental Deformation

Abstract: This book embodies an approach to non-linear elasticity which marks a fundamental departure from classical and current trends. The basic theory was first published between the years 1934 and 1940 in seven papers listed at the end of this Preface. In addition to a systematic treatment of the general theory and extensions to viscoelasticity, the book includes comprehensive new developments and applications, many of which are presented here for the first time. The work is characterized by the use of cartesian concepts and of elementary mathematical methods that do not require a knowledge of the tensor calculus or other more specialized techniques. The explicit introduction of a local rotation field in the three-dimensional equations leads to a theory which separates the physics from the geometry and is equally valid for elastic and non-elastic materials, using either rectangular or curvilinear coordinates. As this book demonstrates, the scope of problems solved by these new methods goes far beyond the results which it has been possible to obtain by the more elaborate and less general traditional approach. New insights, leading to many discoveries and a unified outlook have been brought into such widely diversified areas as rubber elasticity, internal gravity waves in a fluid and tectonic folding in geodynamics. The theory provides rigorous and completely general equations governing the dynamics and stability of solids and fluids under initial stress in the context of small perturbations. It does not require that the medium be elastic or isotropic but is applicable to anisotropic, viscoelastic, or plastic media. No assumptions are introduced regarding the physical process by which the initial stress has been generated. The treatment of viscoelasticity, which constitutes a substantial portion of the book, incorporates some of the results established in my previous work on non-equilibrium thermodynamics. Non-linear theories of deformation and applications to problems of finite strain are obtained by extension ofthe concept of incremental deformation in a medium under initial stress. In contrast to the presentation in the papers listed at the end of this Preface, the concepts and methods are developed primarily in the context of the linearized mechanics of continuous media under initial stress as an independent theory.

Viscoelastic Behavior of Heterogeneous Media

Abstract: : The study of the macroscopic mechanical behavior of heterogeneous viscoelastic media is of considerable theoretical and practical interest. Such heterogeneous media are defined as mechanical mixtures of several discrete linear viscoelastic phases whose stress-strain relations are assumed to be known. It is assumed that specimens of such materials which are very large in comparison to phase regions are statistically homogeneous and statistically isotropic. Macroscopical mechanical behavior is defined as average strain or stress response of such specimens to boundary loading or displacements which would produce uniform states of strain or stress in homogeneous media. The problem considered in the report is that of analytical prediction of such responses in terms of phase geometry and phase stress-strain relations. (Author)

On the destabilizing effect of damping in nonconservative elastic systems.

Abstract: Linear viscous damping effect on nonconservative elastic system destabilization, determining critical loads from characteristic equation roots and degree of instability

Buckling of Axially Compressed Circular Cylindrical Shells at Stresses Smaller Than the Classical Critical Value

Abstract: : Closed-form solutions are given of the linear Donnell equations defining the buckling of thin-walled circular cylindrical shells subjected to uniform axial compression. In addition to the classical simple support conditions requiring the vanishing of the radial displacement, the axial bending moment resultant, the axial additional stress and the circumferential displacement, three other, equally justifiable, simple support conditions are defined and studied in the case of the semi-infinite shell. Two of them yield buckling stresses amounting to about one-half the classical critical stress. (Author)

Large-Amplitude Free Vibrations of a Beam

Abstract: Free vibrations of an initially straight elastic bar having free-free or clamped-free ends are analyzed by assuming that the bar is undergoing inextensional motion and that shearing effects and rotatory inertia are negligible. If Hamilton’s principle is applied, an isoperimetric variational problem is reached. Upon elimination of the subsidiary condition, the remaining Euler equation (a highly nonlinear partial differential equation) can be reduced to an ordinary nonlinear differential equation by means of an extension of Galerkin’s procedure (Bubnov method). Because of the particular type of the reduced equation, use can be made of Atkinson’s superposition method for the frequency of the periodic solutions of the equation. Finally, a plot shows the approximate non-linear period versus the initial amplitude, both rendered nondimensional by introducing proper linear expressions.