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E-raamat: Virtual Work Approach to Mechanical Modeling

  • Formaat: EPUB+DRM
  • Ilmumisaeg: 21-Feb-2018
  • Kirjastus: ISTE Ltd and John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781119510598
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Virtual Work Approach to Mechanical ModelingSuurem pilt
  • Formaat: EPUB+DRM
  • Ilmumisaeg: 21-Feb-2018
  • Kirjastus: ISTE Ltd and John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781119510598
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This book is centred about the Principle of virtual work and the related method for mechanical modelling. It aims at showing and enhancing the polyvalence and versatility of the virtual work approach in the mechanical modelling process.

This book is centred about the Principle of virtual work and the related method for mechanical modelling. It aims at showing and enhancing the polyvalence and versatility of the virtual work approach in the mechanical modelling process. The virtual work statement is set as the principle at the root of a force modelling method that can be implemented on any geometrical description. After experimentally induced hypotheses have been made on the geometrical parameters that describe the concerned system and subsystems, the method provides a unifying framework for building up consistently associated force models where external and internal forces are introduced through their virtual rates of work. Systems described as three-dimensional, curvilinear or planar continua are considered: force models are established with the corresponding equations of motion; the validation process points out that enlarging the domain of relevance of the model for practical applications calls for an enrichment of the geometrical description that takes into account the underlying microstructure.

Preface xiii
Chapter 1 The Emergence of the Principle of Virtual Velocities
1(18)
1.1 In brief
1(1)
1.2 Setting the principle as a cornerstone
1(1)
1.3 The "simple machines"
2(3)
1.4 Leonardo, Stevin, Galileo
5(4)
1.5 Descartes and Bernoulli
9(3)
1.5.1 Rene Descartes (1596--1650)
9(2)
1.5.2 Johann Bernoulli (1667--1748)
11(1)
1.6 Lagrange (1736--1813)
12(7)
1.6.1 Lagrange's statement of the principle
12(1)
1.6.2 Lagrange's proof of the principle
13(2)
1.6.3 Lagrange's multipliers
15(4)
Chapter 2 Dualization of Newton's Laws
19(24)
2.1 In brief
19(1)
2.2 Newton's statements
19(2)
2.2.1 First law
19(1)
2.2.2 Second law
20(1)
2.2.3 Third law
20(1)
2.2.4 Material points
21(1)
2.3 System of material points
21(4)
2.3.1 System of material points
21(2)
2.3.2 Subsystem
23(1)
2.3.3 Law of mutual actions
24(1)
2.3.4 Summing up
25(1)
2.4 Dualization and virtual work for a system of material points
25(8)
2.4.1 System comprising a single material point
25(1)
2.4.2 System comprising several material points
26(3)
2.4.3 Virtual velocity, virtual motion and virtual work
29(2)
2.4.4 Statement of the principle of virtual work (P.V.W.)
31(1)
2.4.5 Virtual motions in relation to the modeling of forces
32(1)
2.5 Virtual work method for a system of material points
33(8)
2.5.1 Presentation of the virtual work method
33(1)
2.5.2 Example of an application
34(6)
2.5.3 Comments
40(1)
2.6 Practicing
41(2)
2.6.1 Law of mutual actions
41(2)
Chapter 3 Principle and Method of Virtual Work
43(18)
3.1 Why and what for?
43(1)
3.2 General presentation of the virtual work method
44(6)
3.2.1 Geometrical modeling
44(1)
3.2.2 Virtual motions
45(1)
3.2.3 Virtual (rates of) work
45(1)
3.2.4 Principle of virtual (rates of) work
46(2)
3.2.5 Implementing the principle of virtual work
48(1)
3.2.6 Comments
49(1)
3.3 General results
50(3)
3.3.1 System, subsystems, actual and virtual motions
50(1)
3.3.2 Virtual (rates of) work in rigid body virtual motions
50(1)
3.3.3 Fundamental law of dynamics
51(1)
3.3.4 Wrench of internal forces
51(1)
3.3.5 Law of mutual actions
52(1)
3.3.6 Remarks
52(1)
3.4 Particular results
53(5)
3.4.1 System, subsystems, actual and virtual motions
53(1)
3.4.2 Momentum theorem
54(1)
3.4.3 Center of mass theorem
55(2)
3.4.4 Kinetic energy theorem
57(1)
3.4.5 Comments
58(1)
3.5 About equilibrium
58(3)
3.5.1 Equilibrium
58(2)
3.5.2 Self-equilibrating fields of internal forces
60(1)
Chapter 4 Geometrical Modeling of the Three-dimensional Continuum
61(18)
4.1 The concept of a continuum
61(3)
4.1.1 Geometrical modeling
61(1)
4.1.2 Experiments
61(3)
4.1.3 Comments
64(1)
4.2 System and subsystems
64(3)
4.2.1 Particles and system
64(1)
4.2.2 Actual motions. Eulerian and Lagrangian descriptions
64(3)
4.3 Continuity hypotheses
67(3)
4.3.1 Lagrangian description
67(2)
4.3.2 Eulerian description
69(1)
4.3.3 Conservation of mass. Equation of continuity
70(1)
4.4 Validation of the model
70(4)
4.4.1 Weakening of continuity hypotheses
70(3)
4.4.2 Physical validation
73(1)
4.5 Practicing
74(5)
4.5.1 Homogeneous transformation
74(1)
4.5.2 Simple shear
75(1)
4.5.3 "Lagrangian" double shear
76(1)
4.5.4 Rigid body motion
76(3)
Chapter 5 Kinematics of the Three-dimensional Continuum
79(36)
5.1 Kinematics
79(11)
5.1.1 The issues
79(1)
5.1.2 Material time derivative of a vector
79(2)
5.1.3 Strain rate
81(1)
5.1.4 Extension rate
82(1)
5.1.5 Distortion rate
82(1)
5.1.6 Principal axes of the strain rate tensor
83(1)
5.1.7 Volume dilatation rate
84(1)
5.1.8 Spin tensor
85(2)
5.1.9 Rigid body motion
87(1)
5.1.10 Geometrical compatibility of a strain rate field
87(3)
5.2 Convective derivatives
90(8)
5.2.1 General comments
90(1)
5.2.2 Convective derivative of a "point function"
91(1)
5.2.3 Equation of continuity
91(1)
5.2.4 Acceleration field
92(1)
5.2.5 Convective derivative of a volume integral
93(2)
5.2.6 Euler's theorem
95(2)
5.2.7 Kinetic energy theorem
97(1)
5.3 Piecewise continuity and continuous differentiability
98(6)
5.3.1 Convective derivative of a volume integral
98(1)
5.3.2 Equation of continuity
99(2)
5.3.3 Momentum theorem
101(1)
5.3.4 Euler's theorem
102(1)
5.3.5 Kinetic energy theorem
103(1)
5.4 Comments
104(1)
5.5 Explicit formulas in standard coordinate systems
104(2)
5.5.1 Orthonormal Cartesian coordinates
104(1)
5.5.2 Cylindrical coordinates
105(1)
5.5.3 Spherical coordinates
105(1)
5.6 Practicing
106(9)
5.6.1 Rigid body motion
106(1)
5.6.2 Simple shear
107(1)
5.6.3 "Lagrangian" Double shear
108(1)
5.6.4 "Eulerian" Double shear
109(1)
5.6.5 Irrotational and isochoric motions
109(1)
5.6.6 Point vortex
110(1)
5.6.7 Fluid sink (or source)
111(1)
5.6.8 Geometrical compatibility of a thermal strain rate field
112(3)
Chapter 6 Classical Force Modeling for the Three-dimensional Continuum
115(54)
6.1 Virtual motions
115(1)
6.2 Virtual rates of work
116(4)
6.2.1 Virtual rate of work by quantities of acceleration
116(1)
6.2.2 Virtual rate of work by external forces for the system
117(1)
6.2.3 Virtual rate of work by external forces for subsystems
118(2)
6.2.4 Virtual rate of work by internal forces
120(1)
6.3 Implementation of the principle of virtual work
120(7)
6.3.1 Specifying the virtual rate of work by internal forces
120(2)
6.3.2 Equations of motion for the system
122(2)
6.3.3 Equations of motion for a subsystem
124(1)
6.3.4 The model
125(2)
6.3.5 Consistency
127(1)
6.4 Piecewise continuous and continuously differentiable fields
127(8)
6.4.1 The issues
127(1)
6.4.2 Piecewise continuous and continuously differentiable U(x, t) and σ(x, t)
128(3)
6.4.3 Piecewise continuous and continuously differentiable virtual velocity fields
131(4)
6.5 The stress vector approach
135(6)
6.5.1 The stress vector
135(1)
6.5.2 The stress vector as the historical fundamental concept
135(6)
6.6 Local analysis
141(4)
6.6.1 Components of the stress vector and stress tensor
141(1)
6.6.2 Normal stress, shear or tangential stress
142(1)
6.6.3 Principal axes of the stress tensor. Principal stresses
143(2)
6.6.4 Isotropic Cauchy stress tensor
145(1)
6.7 The hydrostatic pressure force modeling
145(2)
6.8 Validation and implementation
147(2)
6.8.1 Relevance of the model
147(1)
6.8.2 Implementation
148(1)
6.9 Explicit formulas for the equation of motion in standard coordinate systems
149(1)
6.9.1 Orthonormal Cartesian coordinates
149(1)
6.9.2 Cylindrical coordinates
149(1)
6.9.3 Spherical coordinates
150(1)
6.10 Practicing
150(19)
6.10.1 Spherical and deviatoric parts of the stress tensor
150(1)
6.10.2 Extremal values of the normal stress
151(1)
6.10.3 Stress vector acting on the "octahedral" facet
152(1)
6.10.4 Mohr circles
153(2)
6.10.5 Self-equilibrating stress fields
155(2)
6.10.6 Tresca's strength condition
157(1)
6.10.7 Maximum resisting rate of work for Tresca's strength condition
158(3)
6.10.8 Spherical shell
161(2)
6.10.9 Rotating circular ring
163(2)
6.10.10 Loading parameters, load vector
165(4)
Chapter 7 The Curvilinear One-dimensional Continuum
169(46)
7.1 The problem of one-dimensional modeling
169(1)
7.2 One-dimensional modeling without an oriented microstructure
170(17)
7.2.1 Geometrical modeling
170(1)
7.2.2 Kinematics: actual and virtual motions
171(1)
7.2.3 Virtual rate of work by quantities of acceleration
172(1)
7.2.4 Virtual rate of work by external forces for the system
172(1)
7.2.5 Virtual rate of work by external forces for subsystems
173(1)
7.2.6 Virtual rate of work by internal forces
174(1)
7.2.7 Implementation of the principle of virtual work
175(3)
7.2.8 Piecewise continuous fields
178(2)
7.2.9 Consistency and validation of the model
180(3)
7.2.10 An example
183(4)
7.3 One-dimensional model with an oriented microstructure
187(14)
7.3.1 Guiding ideas
187(1)
7.3.2 Geometrical modeling
188(1)
7.3.3 Kinematics: actual and virtual motions
189(1)
7.3.4 Virtual rate of work by quantities of acceleration
190(1)
7.3.5 Virtual rate of work by external forces for the system
191(2)
7.3.6 Virtual rate of work by external forces for subsystems
193(1)
7.3.7 Virtual rate of work by internal forces
193(1)
7.3.8 Implementation of the principle of virtual work
194(4)
7.3.9 Comments
198(1)
7.3.10 Piecewise continuous fields
198(3)
7.3.11 Integration of the field equations of motion
201(1)
7.4 Relevance of the model
201(7)
7.4.1 Physical interpretation
201(1)
7.4.2 Matching the one-dimensional model with the Cauchy stress model
202(4)
7.4.3 Terminology and notations
206(2)
7.5 The Navier-Bernoulli condition
208(4)
7.5.1 Virtual rate of angular distortion
208(1)
7.5.2 The Navier-Bernoulli condition
209(1)
7.5.3 Discontinuity equations
210(1)
7.5.4 Virtual rate of work by internal forces
211(1)
7.5.5 Comments
211(1)
7.6 Analysis of systems
212(3)
7.6.1 Static determinacy
212(1)
7.6.2 Systems made of one-dimensional members
212(3)
Chapter 8 Two-dimensional Modeling of Plates and Thin Slabs
215(46)
8.1 Modeling plates as two-dimensional continua
215(6)
8.1.1 Geometrical modeling
215(1)
8.1.2 Kinematics: actual and virtual motions
216(5)
8.2 Virtual rates of work
221(6)
8.2.1 Virtual rate of work by quantities of acceleration
221(1)
8.2.2 Virtual rate of work by external forces for the system
221(3)
8.2.3 Virtual rate of work by external forces for subsystems
224(1)
8.2.4 Virtual rate of work by internal forces
225(1)
8.2.5 Tensorial wrench field of internal forces
226(1)
8.3 Equations of motion
227(8)
8.3.1 Statement of the principle of virtual work
227(1)
8.3.2 Boundary equations
228(1)
8.3.3 Specifying the tensorial wrench of internal forces
229(1)
8.3.4 Field equations of motion
230(1)
8.3.5 Field equations of motion in terms of reduced elements
231(1)
8.3.6 Entailment of the equations of motion
232(3)
8.4 Physical interpretation and classical presentation
235(6)
8.4.1 Internal forces
235(1)
8.4.2 Field equations of motion
236(4)
8.4.3 Boundary equations
240(1)
8.5 Piecewise continuous fields
241(3)
8.5.1 Piecewise continuous field of internal forces
241(1)
8.5.2 Piecewise continuous virtual motions
242(2)
8.6 Matching the model with the three-dimensional continuum
244(6)
8.6.1 The matching procedure
244(1)
8.6.2 Three-dimensional virtual velocity field
245(1)
8.6.3 Virtual rate of work by quantities of acceleration
246(1)
8.6.4 Virtual rate of work by external forces
247(1)
8.6.5 Virtual rate of work by internal forces
247(1)
8.6.6 Identifying the reduced elements of the internal force wrench field
248(2)
8.7 The Kirchhoff-Love condition
250(3)
8.7.1 Virtual rate of angular distortion: Kirchhoff-Love condition
250(1)
8.7.2 Discontinuity equations
251(1)
8.7.3 Virtual rate of work by internal forces
252(1)
8.8 An illustrative example: circular plate under a distributed load
253(8)
8.8.1 Load carrying capacity
253(4)
8.8.2 Resistance of the two-dimensional plate element
257(2)
8.8.3 An equilibrated internal force wrench distribution
259(2)
Appendix 1 Introduction to Tensor Calculus 261(26)
Appendix 2 Differential Operators 287(22)
Appendix 3 Distributors and Wrenches 309(14)
Bibliography 323(8)
Index 331
Jean Salenc'on, Member of the Académie des sciences (France); Member (Emeritus) of the Académie des technologies (France); Senior Fellow of the Institute for Advanced Study, City University of Hong Kong.
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