Classical and Computational Solid Mechanics

by Fung, Y. C.

• ISBN13: 9789810241247
• ISBN10: 9810241240
• Edition: 1st
• Format: Paperback
• Copyright: 2001-10-01
• Publisher: World Scientific Pub Co Inc

Summary

This invaluable book has been written for engineers and engineering scientists in a style that is readable, precise, concise, and practical. It gives first priority to the formulation of problems, presenting the classical results as the gold standard, and the numerical approach as a tool for obtaining solutions. The classical part is a revision of the well-known text Foundations of Solid Mechanics, with a much-expanded discussion on the theories of plasticity and large elastic deformation with finite strains. The computational part is all new and is aimed at solving many major linear and nonlinear boundary-value problems.

Author Biography

Y. C. Fung works on solid mechanics, and has contributed to expand its frontiers bordering aerodynamics and biology. He helped to establish the fields of aeroelasticity and biomechanics. He received his BS and MS from the National Central University in China and PhD from the California Institute of Technology. He is the recipient of the National Medal of Science from the President of USA, the Founders Award of the National Academy of Engineering of USA, the von Karman Medal from ASCE. Timoshenko Medal from ASME. Poiseuille Medal from ISB. Borelli Medal from ASB. Landis Award from Microcirculatory Society, and Alza Award from BMES. He is a member of the US National Academy of Science. National Academy of Engineering. Institute of Medicine of NAS. Academia Sinica, and Chinese Academy of Science. He is Honorary Professor of 16 universities in China, and Distinguished Alumnus of Caltech Pin Tong is a preeminent developer of computational solid mechanics. He uses more and more general variational principles that admit less and less restrictive hypotheses. He received his BS degree from Taiwan University, and his MS and PhD from the California Institute of Technology. He was Research Fellow at Caltech. Assistant and Associate Professor of Aeronautics and Astronautics at MIT. Chief of Structures and Dynamics Division of US Department of Transportation. Transportation Systems Center. He is Professor of Mechanical Engineering and Founding Head of the Department at the Hong Kong University of Science and Technology. Author of over 100 papers and a book on Finite Element Method. Dr Tong is regional editor of the Int J of Fracture, and Int J Comp Mech, and President of Far East Oceanic Fracture Soc, HK Soc Theoretical and Applied Mechanics, and HK Sec ASME. Dr Tong received the von Karman Award for his outstanding contribution to structural materials, the Engineer of the Year Award, and the Award for Meritorious Achievement from the US Department of Transportation

Table of Contents

Introduction	1	(29)
Hooke's Law	2	(7)
Linear Solids with Memory	9	(3)
Sinusoidal Oscillations in Viscoelastic Material: Models of Viscoelasticity	12	(2)
Plasticity	14	(1)
Vibrations	15	(3)
Prototype of Wave Dynamics	18	(4)
Biomechanics	22	(3)
Historical Remarks	25	(5)
Tensor Analysis	30	(36)
Notation and Summation Convention	30	(3)
Coordinate Transformation	33	(1)
Euclidean Metric Tensor	34	(4)
Scalars, Contravariant Vectors, Covariant Vectors	38	(1)
Tensor Fields of Higher Rank	39	(1)
Some Important Special Tensors	40	(2)
The Significance of Tensor Characteristics	42	(1)
Rectangular Cartesian Tensors	43	(1)
Contraction	44	(1)
Quotient Rule	45	(1)
Partial Derivatives in Cartesian Coordinates	46	(2)
Covariant Differentiation of Vector Fields	48	(1)
Tensor Equations	49	(3)
Geometric Interpretation of Tensor Components	52	(6)
Geometric Interpretation of Covariant Derivatives	58	(2)
Physical Components of a Vector	60	(6)
Stress Tensor	66	(31)
Stresses	66	(3)
Laws of Motion	69	(2)
Cauchy's Formula	71	(2)
Equations of Equilibrium	73	(5)
Transformation of Coordinates	78	(1)
Plane State of Stress	79	(3)
Principal Stresses	82	(3)
Shearing Stresses	85	(1)
Mohr's Circles	86	(1)
Stress Deviations	87	(1)
Octahedral Shearing Stress	88	(2)
Stress Tensor in General Coordinates	90	(4)
Physical Components of a Stress Tensor in General Coordinates	94	(1)
Equations of Equilibrium in Curvilinear Coordinates	95	(2)
Analysis of Strain	97	(30)
Deformation	97	(3)
Strain Tensors in Rectangular Cartesian Coordinates	100	(3)
Geometric Interpretation of Infinitesimal Strain Components	103	(1)
Rotation	104	(2)
Finite Strain Components	106	(2)
Compatibility of Strain Components	108	(5)
Multiply Connected Regions	113	(4)
Multivalued Displacements	117	(1)
Properties of the Strain Tensor	118	(3)
Physical Components	121	(2)
Example --- Spherical Coordinates	123	(2)
Example --- Cylindrical Polar Coordinates	125	(2)
Conservation Laws	127	(11)
Gauss' Theorem	127	(1)
Material and Spatial Descriptions of Changing Configurations	128	(3)
Material Derivative of Volume Integral	131	(2)
The Equation of Continuity	133	(1)
Equations of Motion	134	(1)
Moment of Momentum	135	(1)
Other Field Equations	136	(2)
Elastic and Plastic Behavior of Materials	138	(65)
Generalized Hooke's Law	138	(2)
Stress-Strain Relationship for Isotropic Elastic Materials	140	(3)
Ideal Plastic Solids	143	(3)
Some Experimental Information	146	(4)
A Basic Assumption of the Mathematical Theory of Plasticity	150	(6)
Loading and Unloading Criteria	156	(1)
Isotropic Stress Theories of Yield Function	157	(2)
Further Examples of Yield Functions	159	(7)
Work Hardening --- Drucker's Hypothesis and Definition	166	(1)
Ideal Plasticity	167	(4)
Flow Rule for Work-Hardening Materials	171	(6)
Subsequent Loading Surfaces --- Isotropic and Kinematic Hardening Rules	177	(12)
Mroz's, Dafalias and Popov's, and Valanis' Plasticity Theories	189	(6)
Strain Space Formulations	195	(4)
Finite Deformation	199	(1)
Plastic Deformation of Crystals	200	(3)
Linearized Theory of Elasticity	203	(35)
Basic Equations of Elasticity for Homogeneous Isotropic Bodies	203	(3)
Equilibrium of an Elastic Body Under Zero Body Force	206	(1)
Boundary Value Problems	207	(3)
Equilibrium and Uniqueness of Solutions	210	(3)
Saint Venant's Theory of Torsion	213	(9)
Soap Film Analogy	222	(2)
Bending of Beams	224	(5)
Plane Elastic Waves	229	(2)
Rayleigh Surface Wave	231	(4)
Love Wave	235	(3)
Solutions of Problems in Linearized Theory of Elasticity by Potentials	238	(42)
Scalar and Vector Potentials for Displacement Vector Fields	238	(3)
Equations of Motion in Terms of Displacement Potentials	241	(2)
Strain Potential	243	(3)
Galerkin Vector	246	(3)
Equivalent Galerkin Vectors	249	(1)
Example --- Vertical Load on the Horizontal Surface of a Semi-Infinite Solid	250	(2)
Love's Strain Function	252	(2)
Kelvin's Problem --- A Single Force Acting in the Interior of an Infinite Solid	254	(5)
Perturbation of Elasticity Solutions by a Change of Poisson's Ratio	259	(3)
Boussinesq's Problem	262	(1)
On Biharmonic Functions	263	(5)
Neuber-Papkovich Representation	268	(2)
Other Methods of Solution of Elastostatic Problems	270	(1)
Reflection and Refraction of Plane P and S Waves	270	(3)
Lamb's Problem --- Line Load Suddenly Applied on Elastic Half-Space	273	(7)
Two-Dimensional Problems in Linearized Theory of Elasticity	280	(33)
Plane State of Stress or Strain	280	(2)
Airy Stress Functions for Two-Dimensional Problems	282	(6)
Airy Stress Function in Polar Coordinates	288	(7)
General Case	295	(4)
Representation of Two-Dimensional Biharmonic Functions by Analytic Functions of a Complex Variable	299	(2)
Kolosoff-Muskhelishvili Method	301	(12)
Variational Calculus, Energy Theorems, Saint-Venant's Principle	313	(66)
Minimization of Functionals	313	(6)
Functional Involving Higher Derivatives of the Dependent Variable	319	(1)
Several Unknown Functions	320	(3)
Several Independent Variables	323	(2)
Subsidiary Conditions --- Largrangian Multipliers	325	(3)
Natural Boundary Conditions	328	(2)
Theorem of Minimum Potential Energy Under Small Variations of Displacements	330	(5)
Example of Application: Static Loading on a Beam --- Natural and Rigid End Conditions	335	(4)
The Complementary Energy Theorem Under Small Variations of Stresses	339	(7)
Variational Functionals Frequently Used in Computational Mechanics	346	(9)
Saint-Venant's Principle	355	(4)
Saint-Venant's Principle---Boussinesq-Von Mises-Sternberg Formulation	359	(3)
Practical Applications of Saint-Venant's Principle	362	(3)
Extremum Principles for Plasticity	365	(4)
Limit Analysis	369	(10)
Hamilton's Principle, Wave Propagation, Applications of Generalized Coordinates	379	(28)
Hamilton's Principle	379	(4)
Example of Application --- Equation of Vibration of a Beam	383	(10)
Group Velocity	393	(3)
Hopkinson's Experiment	396	(2)
Generalized Coordinates	398	(1)
Approximate Representation of Functions	399	(3)
Approximate Solution of Differential Equations	402	(1)
Direct Methods of Variational Calculus	402	(5)
Elasticity and Thermodynamics	407	(21)
The Laws of Thermodynamics	407	(5)
The Energy Equation	412	(2)
The Strain Energy Function	414	(2)
The Conditions of Thermodynamic Equilibrium	416	(2)
The Positive Definiteness of the Strain Energy Function	418	(1)
Thermodynamic Restrictions on the Stress-Strain Law of an Isotropic Elastic Material	419	(2)
Generalized Hooke's Law, Including the Effect of Thermal Expansion	421	(2)
Thermodynamic Functions for Isotropic Hookean Materials	423	(2)
Equations Connecting Thermal and Mechanical Properties of a Solid	425	(3)
Irreversible Thermodynamics and Viscoelasticity	428	(28)
Basic Assumptions	428	(3)
One-Dimensional Heat Conduction	431	(1)
Phenomenological Relations---Onsager Principle	432	(4)
Basic Equations of Thermomechanics	436	(4)
Equations of Evolution for a Linear Hereditary Material	440	(4)
Relaxation Modes	444	(3)
Normal Coordinates	447	(3)
Hidden Variables and the Force-Displacement Relationship	450	(4)
Anisotropic Linear Viscoelastic Materials	454	(2)
Thermoelasticity	456	(31)
Basic Equations	456	(3)
Thermal Effects Due to a Change of Strain; Kelvin's Formula	459	(1)
Ratio of Adiabatic to Isothermal Elastic Moduli	459	(2)
Uncoupled, Quasi-Static Thermoelastic Theory	461	(1)
Temperature Distribution	462	(2)
Thermal Stresses	464	(2)
Particular Integral: Goodier's Method	466	(1)
Plane Strain	467	(3)
An Example --- Stresses in a Turbine Disk	470	(3)
Variational Principle for Uncoupled Thermoelasticity	473	(1)
Variational Principle for Heat Conduction	474	(4)
Coupled Thermoelasticity	478	(3)
Lagrangian Equations for Heat Conduction and Thermoelasticity	481	(6)
Viscoelasticity	487	(27)
Viscoelastic Material	487	(4)
Stress-Strain Relations in Differential Equation Form	491	(6)
Boundary-Value Problems and Integral Transformations	497	(3)
Waves in an Infinite Medium	500	(3)
Quasi-Static Problems	503	(4)
Reciprocity Relations	507	(7)
Large Deformation	514	(73)
Coordinate Systems and Tensor Notation	514	(7)
Deformation Gradient	521	(4)
Strains	525	(1)
Right and Left Stretch Strain and Rotation Tensors	526	(2)
Strain Rates	528	(1)
Material Derivatives of Line, Area, and Volume Elements	529	(3)
Stresses	532	(7)
Example: Combined Tension and Torsion Loads	539	(4)
Objectivity	543	(5)
Equations of Motion	548	(2)
Constitutive Equations of Thermoelastic Bodies	550	(7)
More Examples	557	(5)
Variational Principles for Finite Elasticity: Compressible Materials	562	(6)
Variational Principles for Finite Elasticity: Nearly Incompressible or Incompressible Materials	568	(5)
Small Deflection of Thin Plates	573	(8)
Large Deflection of Plates	581	(6)
Incremental Approach to Solving Some Nonlinear Problems	587	(37)
Updated Lagrangian Description	587	(3)
Linearized Rates of Deformation	590	(3)
Linearized Rates of Stress Measures	593	(4)
Incremental Equations of Motion	597	(1)
Constitutive Laws	598	(6)
Incremental Variational Principles in Terms of T	604	(6)
Incremental Variational Principles in Terms of r*	610	(2)
Incompressible and Nearly Incompressible Materials	612	(5)
Updated Solution	617	(3)
Incremental Loads	620	(2)
Infinitesimal Strain Theory	622	(2)
Finite Element Methods	624	(132)
Basic Approach	626	(3)
One Dimensional Problems Governed by a Second Order Differential Equation	629	(9)
Shape Functions and Element Matrices for Higher Order Ordinary Differential Equations	638	(5)
Assembling and Constraining Global Matrices	643	(8)
Equation Solving	651	(4)
Two Dimensional Problems by One-Dimensional Elements	655	(2)
General Finite Element Formulation	657	(7)
Convergence	664	(1)
Two-Dimensional Shape Functions	665	(7)
Element Matrices for a Second-Order Elliptical Equation	672	(4)
Coordinate Transformation	676	(3)
Triangular Elements with Curved Sides	679	(3)
Quadrilateral Elements	682	(8)
Plane Elasticity	690	(12)
Three-Dimensional Shape Functions	702	(6)
Three Dimensional Elasticity	708	(6)
Dynamic Problems of Elastic Solids	714	(12)
Numerical Integration	726	(5)
Patch Tests	731	(4)
Locking---Free Elements	735	(15)
Spurious Modes in Reduced Integration	750	(4)
Perspective	754	(2)
Mixed and Hybrid Formulations	756	(39)
Mixed Formulations	756	(4)
Hybrid Formulations	760	(7)
Hybrid Singular Elements (Super-Elements)	767	(15)
Elements for Heterogeneous Materials	782	(1)
Elements for Infinite Domain	782	(6)
Incompressible or Nearly Incompressible Elasticity	788	(7)
Finite Element Methods for Plates and Shells	795	(53)
Linearized Bending Theory of Thin Plates	795	(10)
Reissner-Mindlin Plates	805	(8)
Mixed Functionals for Reissner Plate Theory	813	(6)
Hybrid Formulations for Plates	819	(3)
Shell as an Assembly of Plate Elements	822	(10)
General Shell Elements	832	(11)
Locking and Stabilization in Shell Applications	843	(5)
Finite Element Modeling of Nonlinear Elasticity, Viscoelasticity, Plasticity, Viscoplasticity and Creep	848	(25)
Updated Lagrangian Solution for Large Deformation	849	(3)
Incremental Solution	852	(2)
Dynamic Solution	854	(1)
Newton-Raphson Iteration Method	855	(2)
Viscoelasticity	857	(2)
Plasticity	859	(10)
Viscoplasticity	869	(1)
Creep	870	(3)
Bibliography	873	(36)
Author Index	909	(10)
Subject Index	919