Cosserat Theories
Search in google:
This book presents a unified hierarchical formulation of theories for three-dimensional continua, two-dimensional shells, one-dimensional rods, and zero-dimensional points. Moreover, it is shown that a Cosserat point can be used like a finite element for the numerical solution of problems in continuum mechanics. One objective of this book is to allow readers with varying backgrounds easy access to fundamental understanding of these powerful Cosserat theories. Therefore, the mathematical presentation has been simplified and a number of example problems have been solved. Also, complete specific constitutive equations for nonlinear orthotropic elastic structures have been presented in terms of easily identifiable material constants. This book can be used as a graduate text in continuum mechanics or as an invaluable reference for researchers in nonlinear structural mechanics. Booknews Presents a unified hierarchical formulation of theories for three- dimensional continua, two-dimensional shells, one-dimensional rods, and zero-dimensional points, and shows that a Cosserat point can be used like a finite element for the numerical solution of problems in continuum mechanics. Mathematical presentation is simplified, with solved examples. Complete specific constitutive equations for nonlinear orthoptropic elastic structures are presented in terms of easily identifiable material constants. Can be used as a graduate text in continuum mechanics or as a reference for researchers in nonlinear structural mechanics. Rubin teaches mechanical engineering at TechnionIsrael Institute of Technology. Annotation c. Book News, Inc., Portland, OR (booknews.com)
PrefacexiiiChapter 1Introduction11.1The basic idea of a Cosserat model11.2A brief outline of the book31.3Notation9Chapter 2Basic Tensor Operations in Curvilinear Coordinates112.1Covariant and contravariant base vectors112.2Base tensors and components of tensors132.3Basic tensor operations152.4Covariant differentiation and Christoffel symbols17Chapter 3Three-Dimensional Continua193.1Configurations and motion193.2Balance laws213.3Invariance under superposed rigid body motions273.4Mechanical power343.5An alternative derivation of the balance laws353.6An averaged form of the balance of linear momentum373.7Anisotropic nonlinear elastic materials383.8Constraints403.9Initial and boundary conditions433.10Material Symmetry443.11Isotropic nonlinear elastic materials473.12A small strain theory513.13Small deformations superimposed on a large deformation543.14Pure bending of an orthotropic rectangular parallelepiped573.15Torsion of an orthotropic rectangular parallelepiped603.16Forced shearing vibrations of an orthotropic rectangular parallelepiped633.17Free isochoric vibrations of an isotropic cube653.18An orthotropic rectangular parallelepiped loaded by its own weight663.19An isotropic circular cylinder loaded by its own weight673.20Plane strain free vibrations of an isotropic solid circular cylinder683.21Dissipation inequality and material damping69Chapter 4Cosserat Shells734.1Description of a shell structure734.2The Cosserat model of a shell774.3Derivation of the balance laws from the three-dimensional theory804.4Balance laws by the direct approach874.5Invariance under superposed rigid body motions924.6Mechanical power934.7An alternative derivation of the balance laws954.8Anisotropic nonlinear elastic shells974.9Constraints1004.10Initial and boundary conditions1064.11Further restrictions on constitutive equations for shells constructed from homogeneous anisotropic nonlinear elastic materials1084.12A small strain theory1134.13Small deformations superimposed on a large deformation1174.14Pure bending of an orthotropic rectangular plate1214.15Torsion of an orthotropic rectangular plate1294.16Forced shearing vibrations of an orthotropic rectangular plate1344.17Free isochoric vibrations of an isotropic cube1364.18An orthotropic rectangular plate loaded by its own weight1374.19Elastic shells1404.20Plane strain expansion of an isotropic circular cylindrical shell1434.21Plane strain free vibrations of an isotropic solid circular cylinder1474.22Expansion of an isotropic spherical shell1494.23Free vibrations of an isotropic solid sphere1564.24An isotropic circular cylindrical shell loaded by its own weight1584.25Isotropic nonlinear elastic shells1614.26A simple derivation of the local equations for shells1634.27A brief summary of the equations for shells1654.28Generalized membranes and membrane-like shells1704.29Simple membranes1724.30Expansion of an incompressible isotropic spherical shell1754.31Bending of an orthotropic plate into a circular cylindrical surface1794.3Linear theory of an isotropic plate1834.33Dissipation inequality and material damping187Chapter 5Cosserat Rods1915.1Description of a rod structure1915.2The Cosserat model of a rod1945.3Derivation of the balance laws from the three-dimensional theory1975.4Balance laws by the direct approach2045.5Invariance under superposed rigid body motions2075.6Mechanical power2085.7An alternative derivation of the balance laws2105.8Anisotropic nonlinear elastic rods2125.9Constraints2165.10Initial and boundary conditions2225.11Further restrictions on constitutive equations for rods constructed from homogeneous anisotropic nonlinear elastic materials2245.12A small strain theory2295.13Small deformations superimposed on a large deformation2325.14Pure bending of an orthotropic beam with rectangular cross-section2355.15Torsion of an orthotropic beam with rectangular cross-section2435.16Inhomogeneous shear of an orthotropic beam with rectangular cross-section2455.17Forced shearing vibrations of an orthotropic beam with rectangular cross-section2475.18Free isochoric vibrations of an isotropic cube2505.19An orthotropic beam with rectangular cross-section loaded by its own weight2515.20Elastic rods2545.21Plane strain expansion of an isotropic circular cylindrical shell2565.22Plane strain free vibrations of an isotropic solid circular cylinder2605.23An isotropic circular cylindrical shell loaded by its own weight2625.24Isotropic nonlinear elastic rods2655.25A simple derivation of the local equations for rods with rectangular cross-sections2665.26A brief summary of the equations for rods2705.27Linearized equations for beams with rectangular cross-sections2755.28Bernoulli-Euler rods2775.29Timoshenko rods2835.30Generalized strings2875.31Simple strings2885.32Transverse loading of an isotropic beam with a rectangular cross-section2905.33Linearized buckling equations2935.34An intrinsic formulation of Bernoulli-Euler rods with symmetric cross-sections3035.35Dissipation inequality and material damping309Chapter 6Cosserat Points3116.1Description of a point-like structure3116.2The Cosserat point model3136.3Derivation of the balance laws from the three-dimensional theory3156.4Balance laws by the direct approach3196.5Invariance under superposed rigid body motions3216.6Mechanical power3226.7An alternative derivation of the balance laws3236.8Anisotropic nonlinear elastic Cosserat points3256.9Constraints3286.10Initial Conditions3336.11Further restrictions on constitutive equations for Cosserat points constructed from homogeneous anisotropic nonlinear elastic materials3346.12A small strain theory3366.13Small deformations superimposed on a large deformation3376.14Forced shearing vibrations of an orthotropic rectangular parallelepiped3406.15Free isochoric vibrations of an isotropic cube3456.16Isotropic nonlinear elastic Cosserat points3466.17A brief summary of the equations for Cosserat points3476.18Dissipation inequality and material damping351Chapter 7Numerical Solutions using Cosserat Theories3557.1The Cosserat approach to numerical solution procedures for problems in continuum mechanics3557.2Formulation of the numerical solution of spherically symmetric problems using the theory of a Cosserat shell3577.3Formulation of the numerical solution of string problems using the theory of a Cosserat point3787.4Formulation of the numerical solution of rod problems using the theory of a Cosserat point3947.5Formulation of the numerical solution of three-dimensional problems using the theory of a Cosserat point4107.6Formulation of the numerical solution of two-dimensional problems using the theory of a Cosserat point418Appendix ATensors, Tensor Products and Tensor Operations in Three Dimensions429A.1Vectors and vector operations429A.2Tensors as linear operators430A.3Tensor products (special case)430A.4Indicial notation435A.5Tensor products (general case)437A.6Tensor transformation relations440A.7Additional definitions and results442Appendix BSummary of Tensor Operations in Specific Coordinate Systems447B.1Cylindrical polar coordinates447B.2Spherical polar coordinates449Exercises451Acknowledgments467References467Index475
