Deformable-body dynamics is a subject to investigate the states of strains and internal relative motions in deformable solids subject to the action of external forces．This is all old and interesting topic．and many problems still are unsolved or solved incompletely．Rethinking such problems in this topic may bring new vital to the modern science and technology．The first consideration of the nature of the resistance of deformable-bodies to rupture was given by Galileo in 1638．The theory of deformable．bodies．started from Galileos problem．is based on the discovery of Hookes Law in 1660 and the general differential equations of elasticity by Navier in 1821．The HookeS law is an experimental discovery about the stress and strain relation．This law provides the basis to develop the mathematical theory 0f deformable bodies．In 1821．Navier was the first to investigate the general equations of equilibrium and vibration of elastic solids．In 1850．Kirchhoff proposed two assumptions：(i)that linear filaments Of the plate initially normal to the middle．surface remain straight and normal to the middle．surface after deformed．and (ii) that all fibers in middle surface remain unstretched．Based on the Kirchhoff assumptions，the approximate theories for beams，rods，plates and shells have been developed for recent 1 50 years．From the theory of 3．dimensional de．formable body．with certain assumptions．this book will present a mathematical treatise of such approximate theories for thin deformable．bodies including cables．beams，rods，webs，membranes，plates and shells．The nonlinear theory for de．formable body based on the Kirchhoff assumptions is a special case to be discussed．This book consists of eight chapters．Chapter 1 discusses the history Of the deformable body dynamics．Chapter 2 presents the mathematical tool for the de．formation and kinematics of deformable．bodies．Chapter 3 addresses the deformation geometry．kinematics and dynamics of deformable body．Chapter 4 discusses constitutive laws and damage theory for deformable-bodies．In Chapter 5．nonlinear dynamics of cables is addressed．Chapter 6 discusses nonlinear plates and waves，and the nonlinear theories for webs，membranes and shells are presented in Chapter 7．Finally，Chapter 8 presents the nonlinear theory for thin beams and rods．

Chapter 1 Introduction

1.1.Deforfflable.body dynamics

1.1.1.Cable dynamics

1.1.2.Beams and rods

1.1.3.Plates and shells

1.1.4.Soft webs

1.2.Book layout

References

Chapter 2 Tensor Analysis

2.1.Vectors and tensors

2.1.1.Vector algebra

2.1.2.Base vectors and metric tensors

2.1.3.Local base vector transformation

2.1.4.Tensor algebra

2.2.Second.order tensors

2.2.1.Second.order tensor algebra

2.2.2.Basic properties

2.2.3.Tensor decompositions

2.2.4.Tensor functions

2.3.Tensor calculus

2.3.1.Differentiation

2.3.2.Invariant differential operators and integral theorems

2.3.3.Riemann-Christoffel curvature tensors

2.4.Two.point tensor fields

2.4.1.Two-point tensors

2.4.2.Independent coordinates

2.4.3.Correlated coordinates

2.4.4.Shifter tensor fields

References

Chapter 3 Deformation，Kinematics and Dynamics

3.1.Deformation geometry

3.1.1.Curvilinear coordinates

3.1.2.Deformation gradient and tensors

3.1.3.Green-Cauchy strain tensors and engineering strain

3.1.4.Principal strains and directions

3.2.Kinematics

3.2.1.Material derivatives

3.2.2.Strain rates

3.3.Dynamics

3.3.1.Forces and stresses

3.3.2.Transport theorem

3.3.3.Cauchy stress and couple-stress tensors

3.4.Energy conservation

References

Chapter 4 Constitutive Laws and Damage Theory

4.1.Constitutive equations

4.2.Material damage and effective stress

4.3.Equivalence principles

4.4.An anisotropic damage theory

4.5.Applications

4.5.1.Uniaxial tensional models

4.5.2.Pure torsion

4.5.3.Elastic perfectly-plastic materials

References Chapter 5 Nonlinear Cables

5.1.A nonlinear theory of cables

5.2.Traveling and rotating cables

5.3.Equilibrium of traveling elastic cables

5.3.1.Existence conditions

5.3.2.Displacements

5.3.3.Applications

5.4.Nonlinear dynamics of cables

5.4.1.Equations of motion

5.4.2.Motions of inextensible cables

5.4.3.Motions ofdef01Tnable cables

References

Chapter 6 Nonlinear Plates and Waves

6.1.A nonlinear theory of plates

6.1.1.Deformation of a 3-D body

6.1.2.Strains in thin plates

6.1.3.Equations of motion

6.1.4.Reduction to established theories

6.2.Waves in traveling plates

6.2.1.An approximate theory

6.2.2.Perturbation analysis

6.2.3.Static waves

6.2.4.Nonlinear waves

6.2.5 Chaotic waves

6.3.Waves in rotating disks

6.3.1.Equations of motions

6.3.2.Nonlinear waves

6.3.3.Resonant and stationary waves

6.4.Conclusions

References

Chapter 7 Nonlinear Webs.Membranes and Shells

7.1.Nonlinear webs

7.1.1.Cable-network webs

7.1.2.Cable-fabric webs

7.1.3.Continuum webs

7.2.Nonlinear membranes

7.2.1.A membrane theory based on the Cartesian coordinates-

7.2.2.A membrane theory based on the curvilinear coordinates

7.3.Nonlinear shells

7.3.1.A shell theory based on the Cartesian Coordinates

7.3.2.A shell theory based on the curvilinear coordinates

References

Chapter 8 Nonlinear Beams and Rods

8.1.Differential geometry of curves

8.2.A nonlinear theory of straight beams

8.3.Nonlinear curved beams

8.3.1.A nonlinear theory based on the Cartesian C00rdinates

8.3.2.A nonlinear theory based on the curvilinear coordinates

8.4.A nonlinear theory of straight rods

8.5.Nonlinear curved rods

8.5.1.A curved rod theory based on the Cartesian coordinates

8.5.2.A curved rod theory based on the curvilinear coordinates

References

Subject Index

　　l．2．Book layout

　　This book consists of eight chapters．Chapter 1 discusses the history of the deformable body dynamics．In Chapter 2．the mathematical tool for the deformation and kinematics 0f the deformable bodies will be presented．Chapter 3 will address the deformation geometry．kinematics and dynamics of deformable bodies．Chapter 4 will present constitutive laws and damage theory for deformable bodies．In Chapter 5．nonlinear cable dynamics will be presented．Chapter 6 will discuss the nonlinear theory and vibration waves of plates．In Chapter 7，the nonlinear theory for webs．membranes and shells will be presented．Finally．Chapter 8 will present the nonlinear theory for beams and rods．The main contents in this book are summarized as follows．

　　Chapter 2 will review the basic vector algebra first．The base vectors and metric tensors will be introduced．and the local base vectors in curvilinear coordinates and tensor algebra will be presented．The second-order tensors will be discussed in detail．The differentiation and derivatives of tensor fields will be presented．And the gradient．invariant differential operators and integral theorems for tensors are presented．The Riemann-Christoffel curvature tensor will also be discussed．Finally．two．point tensor fields will be presented．

　　Chapter 3 will present the defornlation geometry．kinematics and dynamics of Continuous media．To discuss deformation geometry．the deformation gradients will be introduced in the local curvilinear Coordinate systems．and the Green and Cauchy strain tensors will be presented．The stretch and angle changes for line elements will be discussed through Green and Cauchy strain tensors．The velocity gradient will be introduced for kinematics．and the material derivatives of deformation gradient．Infinitesimal line element．area and volume in the deformed con．figuration will be presented．The Cauchy stress and couple stress tensors will be defined to discuss the dynamics of continuous media．and the local balances for the Cauchy momentum and angular momentum will be discussed．The PiolaKirchhoff stress tensors will be introduced and the Boussinesq and Kirchhoff local balance of momentum will be discussed．The 10cal principles of the energy con．servation will be discussed by the virtual work principle．