Mathematical Theory of Elastic Equilibrium: Recent Results(7 Ergebnisse der angewandten Mathematik)

It is not my intention to present a treatise of elasticity in the follow­ ing pages. The size of the volume would not permit it, and, on the other hand, there are already excellent treatises. Instead, my aim is to develop some subjects not considered in the best known treatises of elasticity but nevertheless basic, either from the physical or the analytical point of view, if one is to establish a complete theory of elasticity. The material presented here is taken from original papers, generally very recent, and concerning, often, open questions still being studied by mathematicians. Most of the problems are from the theory of finite deformations [non-linear theory], but a part of this book concerns the theory of small deformations [linear theory], partly for its interest in many practical questions and partly because the analytical study of the theory of finite strain may be based on the infinitesimal one.

Table of Contents:
I—Kinematic Introduction.- § 1. Preliminaries.- § 2. Fundamental quantities describing the deformation.- § 3. A minimum theorem in the kinematics of large deformations.- II—Basic Equations of the Statics of Continuous Media.- § 1. Eulerian expressions of fundamental equations.- § 2. Lagrangean expressions of fundamental equations.- § 3. Thermodynamic potential.- III—Isotropic Bodies — Thermodynamic Potential.- § 1. Isotropic bodies.- § 2. Principal properties of the thermodynamic potential.- § 3. A property of the thermodynamic potential.- § 4. Elasticity of second grade. Signorini’s thermodynamic potential.- § 5. On the most general elasticity of second grade.- § 6. A new type of thermodynamic potential proposed by Tolotti.- §7. A type of thermodynamic potential proposed by Bordoni.- IV—Transformations Depending on a Parameter. Successive Equations of Elasticity.- § 1. Displacements depending on a parameter.- § 2. Successive systems. Linear elasticity.- § 3. Necessary conditions for the existence of solutions of the successive systems of equations.- § 4. On the conditions of compatibility of the successive systems of equations when constraints are present.- § 5. An example of a solution of order one.- § 6. Successive systems of equations forXrs.- §7. The second-order theory of Rivlin.- V—Analytical Problems Regarding the Fundamental Equations of Isothermal Static Elasticity.- § 1. Preliminaries.- § 2. Fundamental hypotheses. Statement of the theorems of existence and uniqueness in the case of a principal orientation of infinite order.- § 3. Demonstration of Theorem A).- § 4. On the expansion of the solution of the basic set of equations in a power series in ? when there is no axis of equilibrium.- § 5. On the existence of solutions and the possibility of expanding them in power series in ? when there is an axis of equilibrium.- § 6. Theorems of existence and uniqueness for the linear equations of isothermal static elasticity.- VI—Inequalities for the Equilibrium ofSlightly Deformable Elastic Bodies.- § 1. Preliminaries.- § 2. Integral properties.- § 3. A basic inequality.- § 4. A few consequences of inequality (6.16).- § 5. A new general inequality.- § 6. Application of (6.28) to the equilibrium of cylindrical bodies.- § 7. On the best at in a case similar to that of uniform bending.- § 8. Rectangular prism. A geometrical interpretation of the results.- § 9. Inequalities concerning the equilibrium of an arc.- § 10. Inequalities regarding the deformation of slightly deformable bodies.- § 11. On the deformation of a homogeneous elastic shell under pressure.- § 12. New inequalities regarding the displacements.- § 13. On the variation of temperature in an adiabatic transformation.- § 14. Upper bounds for displacements and stresses.- § 15. Other upper bounds for the stresses.- VII—Integration of the Fundamental Problem of Static Elasticity.- § 1. Preliminaries.- § 2. Analytical preface to the integration of the static elasticity problem.- § 3. Polynomial approximations of stress components.- § 4. Integration method.- § 5. Validity of Menabrea’s theorem in non-isothermal cases.- § 6. Adaptation of the integration method of § 4 to non-isothermal cases.- § 7. On the integration of the basic problem of isothermal elastic statics in the case of finite deformation.- § 8. Integration method of M. Picone.- § 9. On integration of the static problem of the homogeneous plate of arbitrary thickness.- § 10. Saint-Venant’s problem.- VIII—Plane Elasticity.- § 1. Preliminaries.- § 2. Structure of Airy’s stress function in the problem of plane deformations.- § 3. Meaning of the coefficients of the singular part of Airy’s function in the case of plane deformations.- § 4. Structure of Airy’s function in the problem of plane stress independent of y3.- § 5. Structure of Airy’s function in the problem of plane stress depending on y3.- IX—Hypo-Elasticity.- § 1. Preliminaries.- § 2. Basic equations of hypo-elasticity.- § 3. Hypo-elastic equilibrium.- §4. Lagrangean form of the equations of hypo-elasticity.- § 5. On the solution of the equations of hypo-elasticity in the static case for the body of grade zero.- § 6. Some considerations concerning the solution of the general problem of hypo-elastic statics.- X—Asymmetric Elasticity.- § 1. Preliminaries.- § 2. Basic Eulerian equations.- § 3. Basic Lagrangean equations.- § 4. Use of the thermodynamic potential.- § 5. Equilibrium equations in the linear theory.- § 6. Isothermal elastic potential for the small deformations of isotropic bodies.- § 7. Recapitulation of the general equations valid in the case of infinitesimal isothermal transformations.- § 8. Discussion of an example.- Index of Authors.