Showing papers in "Archive for Rational Mechanics and Analysis in 1961"
TL;DR: In this article, the authors follow Toupin & Bernstein's approach to the general theory yet try to achieve the elegant and explicit directness of Ericksen's earlier treatment of isotropic incompressible materials.
Abstract: After the classical researches of Christoffel, Hugoniot, Hadamard, and Duhem on waves in elastic materials, it might seem that little remains to be learned. Such is not the case. As for most parts of mechanics, it has been necessary in the last decade to go over the matter again, not only so as to free the conceptual structure from lingering linearizing and to fix it more solidly in the common foundation of modern mechanics, but also so as to derive from it specific predictions satisfying modern needs for contact between theory and rationally conceived experiment. After reading the recent papers by Toupin & Bernstein [1961, 1] and by Hayes & Rivlin [1961, 2], I have seen that more can be learned than is there proved. In the present paper I follow Toupin & Bernstein’s approach to the general theory yet try to achieve the elegant and explicit directness of Ericksen’s earlier treatment of isotropic incompressible materials [1953]. At the same time, all the results of Hayes & Rivlin are obtained in shorter but more general form as immediate corollaries.
200 citations
TL;DR: In this paper, the authors apply the theory of superposition of infinitesimal deformations on finite deformations in an isotropic elastic material to the study of the propagation of surface waves in a semi-infinite body which is subjected to a static, pure homogeneous deformation.
Abstract: In the present paper we apply the theory [1]⋆ of the superposition of infinitesimal deformations on finite deformations in an isotropic elastic material to the study of the propagation of surface waves in a semi-infinite body which is subjected to a static, pure homogeneous deformation.
160 citations
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146 citations
TL;DR: In this article, pre-publication prices are valid through the end of the third month following publication, and therefore are subject to change subject to the availability of pre-publishing data.
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124 citations
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84 citations
TL;DR: In this article, the authors apply the theory of the superposition of infinitesimal deformations on finite deformations in an isotropic elastic material to the study of the propagation of a plane wave of small amplitude in an infinite body of the material which is subjected to a static, pure homogeneous deformation.
Abstract: In the present paper, we apply the theory [1] of the superposition of infinitesimal deformations on finite deformations in an isotropic elastic material to the study of the propagation of a plane wave of small amplitude in an infinite body of the material which is subjected to a static, pure homogeneous deformation. It is seen that the secular equation for the determination of the square of the velocity of propagation in a given direction has three real eigen-values and correspondingly three real mutually perpendicular eigen-directions. Provided these three eigen-values are all positive, travelling waves may be propagated in the body in the direction considered with linear polarisations along each of these eigen-directions. If one or more of the eigen-values is negative for any direction of propagation, the body is inherently unstable in the state of pure homogeneous deformation considered.
82 citations
TL;DR: In this article, a theory for the finite deformation of a thin membrane composed of homogeneous elastic material which is isotropic in its undeformed state is formulated for the case of a small deformation superposed on a known finite deformations of the membrane, and the stability of cylindrically symmetric modes for the inflated and extended cylinder with fixed ends is determined.
Abstract: A theory is formulated for the finite deformation of a thin membrane composed of homogeneous elastic material which is isotropic in its undeformed state. The theory is then extended to the case of a small deformation superposed on a known finite deformation of the membrane. As an example, small deformations of a circular cylindrical tube which has been subjected to a finite homogeneous extension and inflation are considered and the equations governing these small deformations are obtained for an incompressible material. By means of a static analysis the stability of cylindrically symmetric modes for the inflated and extended cylinder with fixed ends is determined and the results are verified by a dynamic analysis. The stability is considered in detail for a Mooney material. Methods are developed to obtain the natural frequencies for axially symmetric free vibrations of the extended and inflated cylindrical membrane. Some of the lower natural frequencies are calculated for a Mooney material and the methods are compared.
69 citations
66 citations
TL;DR: In this paper, the authors generalize fundamental theorems in three-dimensional classical elastostatics such as the traditional proofs of the uniqueness and reciprocal theorem or of the minimum e ergy principles to unbounded domains exterior to a finite number of closed surfaces.
Abstract: : Conventional proofs of fundamental theorems in three-dimensional classical elastostatics such as the traditional proofs of the uniqueness and reciprocal theorems or of the minimum e ergy principles are generaliz d to unbounded domains exterior to a finite number of closed surfaces.
TL;DR: The definition of differentiability for functions of several variables is defined in this paper, where it is shown that f(x) − f(a) − (x − a)f ′ (a) tends to 0 faster than |x − p| does.
Abstract: One way to interpret this expression is that f(x) − f(a) − (x − a)f ′(a) tends to 0 faster than |x − a| does and consequently f(x) is approximately equal to f(a) + (x− a)f ′(a). The equation y = f(a) + (x− a)f ′(a) is the equation of the line tangent to the graph of f at the point (a, f(a)). So f(x) is approximated very well by its tangent line. This observation is the bases for linear approximation. Using this form of the definition as a model it is possible to construct a definition of differentiability for functions of several variables. What goes in the denominator is fairly easy to see; namely, |P − P0|. Similarly the first two term in the numerator would become f(P )−f(P0). But what should replace the term (x−a)f ′(a)? First we note that it must be a number. One of the factors will be (P0 − P ) or better yet ( −−→ P0P ) — a vector. Consequently the other must also be a vector and the product will be the dot product. With these observation the definition of differentiability for functions of several variable is as follows.
TL;DR: In this article, the authors derived the restricted form applicable to small dynamic deformations superposed on large static deformations and applied it to problems of wave propagation in finitely deformed viscoelastic bodies.
Abstract: The non-linear stress-deformation relation for materials with memory has been discussed by Green & Rivlin [1] and Green,Rivlin & Spencer [2]. In the present paper we derive the restricted form applicable to small dynamic deformations superposed on large static deformations. Although this can be done by making an appropriate linearization of the final result of the non-linear theory, we avoid some of the complexities of that theory by introducing the linearization at an earlier stage. The results obtained will apply in particular to problems of wave propagation in finitely deformed viscoelastic bodies.
TL;DR: In this paper, the geometrical relationship between a closed surface and a solution surface is studied, and the mapping of a solution space into a closed closed surface in the case of regular variational problems gives rise to a canonical mapping of S onto the solution of a non-parametric, uniformly elliptic variational problem.
Abstract: : The geometrical relationships between a closed surface and a solution surface are studied. The mapping of a solution surface S, into a closed surface in the case of regular variational problems gives rise to a canonical mapping of S onto the solution of a non-parametric, uniformly elliptic variational problem. The principal application is to show that solutions of some variational problems behave qualitatively like minimal surfaces, in the same sense that solutions of uniformly elliptic equations behave like solutions of the Laplace equation. (Author)
TL;DR: In this article, the motion of the orbital plane as a rigid body is introduced and a non-elliptical orbit motion in this plane is defined, and the basic nonlinear features of the apsidal motion are incorporated in the analytical development so as to produce a theory valid at all angles of inclination of the orbit.
Abstract: A novel approach to the study of the orbits of artificial satellites is presented. Emphasis is placed upon the basic geometry and other aspects of satellite motion which are of first importance to satellite engineering. The motion of the orbital plane as a rigid body is introduced and a non-elliptical orbit motion in this plane is defined. The plane orbit so defined possesses the very desirable feature of representing a succession of satellite positions and hence reveals the true motion of the satellite. An analytical treatment yields a completely general second order theory of earth satellite motion which is suitable for engineering purposes. In the latter development, particular attention is paid to the apsidal motion of the orbit and the concomitant resonance effects at the critical orbit inclination. The basic nonlinear features of the apsidal motion, which have not been recognized in earlier theories, are incorporated in the analytical development so as to produce a theory valid at all angles of inclination of the orbit.
TL;DR: In this paper, an analysis of electro-magneto-optical effects, though based on nonlinear equations for which all but the simplest exact solutions are difficult to exhibit, may be simplified and made tractable by assuming that the dynamical part of the solution (the light wave) has such weak intensity that the vectors which describe it may be treated as infinitesimals.
Abstract: Electro-magneto-optical effects in stationary materials are observed when the materials are placed in strong, static electric or magnetic fields and an electromagnetic wave (light) traverses the medium. The Faraday effect is an example of the class of phenomena we have in mind. Clearly, the existence of electro-magneto-optical effects in material media is direct evidence that the equations for the electromagnetic field in a material medium, unlike the equations for it in vacuum, are non-linear since the sum of two solutions generally fails to be a solution. An analysis of electro-magneto-optical effects, though based on nonlinear equations for which all but the simplest exact solutions are difficult to exhibit, may be simplified and made tractable by assuming that the dynamical part of the solution (the light wave) has such weak intensity that the vectors which describe it may be treated as infinitesimals.
TL;DR: In this article, the Runge-Kutta Procedure for 2-HP is described as follows: 1. Definitions of the Matrix Quantit ies to be Used 2.
Abstract: Conten t s Pas~ w 1. Introduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 w Taylor Expansions . . . . . . . . . . . . . . . . . . . . . . . . . 39 w 3. Definitions of the Matrix Quantit ies to be Used . . . . . . . . . . . . . 40 w The Basic Equat ions for the Runge-Kut ta Procedure for 2-HP . . . . . . 43 w 5. Computations Prepara tory to Matching the Taylor Expansions . . . . . . 44 w 6. Requirements on the Parameters in the Case for u . . . . . . . . . . . . 48 w 7. Requirements on the Parameters in the Case gor p . . . . . . . . . . . . 5t w Requirements on the Parameters in the Case for q . . . . . . . . . . . . 53 w 9. Fur ther Requirements . . . . . . . . . . . . . . . . . . . . . . . . 54 w 10. Values for the Parameters . . . . . . . . . . . . . . . . . . . . . . 55 w t t . Determinat ion of U 0 and B~a . . . . . . . . . . . . . . . . . . . . . 57 w 12. Convergence of the Numerical Approximations Obtained by R K 2 . . . . 59