1 Prolegomena to the Theory.- (i) Historical Perspectives.- (ii) Toward the Rational Mechanics of Pseudo-rigid Bodies.- (iii) Toward the Study of Cross-theory Comparison.- 2 Foundations of the Theory of Pseudo-rigid Bodies.- (i) Event Worlds and Body Manifolds.- (ii) Kinematics.- (iii) Equations of Balance.- (iv) Forces and Force-moments.- (v) Rigid-body Mechanics: The Principle of Balance of Angular Momentum.- (vi) Changes of Frame.- (vii) Constitutive Relations.- 3 Consistency with Other Continuum Theories.- (i) The General Notion of Consistency.- (ii) Elasticity Theory. Subtheories.- (iii) The Subtheory of Pseudo-rigid Motions.- (iv) Symmetry. Uniqueness.- (v) Approximation of the Coarsifier.- 4 Explicit Motions of Pseudo-rigid Bodies.- (i) Rigid Motions.- (ii) Roto-deformations.- (iii) Pure Stretch Motions.- (iv) Shearing Motions.- (v) Plane Motions.- 5 Variational Formulation of the Theory.- (i) Lagrangian and Hamiltonian Formulations.- (ii) Symmetry and Conserved Quantities.- (iii) The Stability of Steady Motions: A Special Case.- (iv) Ignorable Coordinates: Remarks on the General Analysis of Stability.- 6 Approximations for Almost-rigid Bodies.- (i) The Problem of Transition to Rigidity.- (ii) Derivation of the Asymptotic Expansion.- (iii) Proof of the Spectral Theorem.- (iv) Special Results for Axially Symmetric Bodies.- (v) Gyroscopic Motions.- References.
This monograph concerns the development, analysis, and application of the theory of pseudo-rigid bodies. It collects together our work on that subject over the last five years. While some results have appeared else where, much of the work is new. Our objective in writing this mono graph has been to present a new theory of the deformation of bodies, one that has not only a firm theoretical basis, but also the simplicity to serve as an effective tool in practical problems. Consequently, the main body of the treatise is a multifaceted development of the theory, from foundations to explicit solutions to linearizations to methods of approximation. The fact that this variety of aspects, each examined in considerable detail, can be collected together in a single, unified treat ment gives this theory an elegance that we feel sets it apart from many others. While our goal has always been to give a complete treatment of the theory as it now stands, the work here is not meant to be definitive. Theories are not entities that appear suddenly one day and thereafter stand as given. Rather, they must mature and grow with time and experience. Our development is more correctly a beginning, tempting others to explore, appraise, and modify its features so as to produce something better.
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