Foundations and applications of mechanics: 1 Continuum mechanics
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Delhi
Cambridge University Press
2015
|
| Edition: | Third edition |
| Series: | Cambridge IISc series
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| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| Physical Description: | xxiii, 852 Seiten Illustrationen, Diagramme |
| ISBN: | 9781107091351 |
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Record in the Search Index
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| adam_text | Cambridge ֊ IISc Series
Continuum mechanics studies the foundations of deformable body mechanics from a
mathematical perspective. It also acts as a base upon which other applied areas such
as solid mechanics and fluid mechanics are developed. After providing the necessary
mathematical background in the form of a detailed introduction to tensors, the book
discusses kinematics, balance laws and constitutive laws for simple materials. Major
emphasis is placed on topics that have come into prominence in the latter half of
the twentieth century such as material symmetry, frame-indifference and
thermomechanics. Besides a thorough discussion of the theory, the book also
discusses applications in the areas of linear and nonlinear elasticity. The chapter
on linear elasticity is detailed enough to be used as a textbook /reference for an
introductory graduate-level course on linear elasticity.
The book presents several advanced topics including fourth-order tensors,
differentiation of tensors, exponential and logarithmic tensors, and their application
to nonlinear elasticity. The implications of the Clausius-Duhem inequality, material
symmetry, frame-indifference, and material and elastic stability of nonlinear
hyperelastic materials are also discussed in detail.
C. S. Jog is Professor at the Department of Mechanical Engineering, Indian Institute
of Science, Bangalore. Having more than 20 years of experience in teaching and
research, Jog has published more than 40 papers in various international journals.
He teaches courses on solid mechanics, continuum mechanics, and the finite element
method, primarily at postgraduate level. His research interests include continuum
mechanics, elasticity, fluid-structure interaction problems and development of new
finite element strategies.
Cover image: Various stages of deformation in an elastica
problem obtained using a hybrid finite element method
Image source: Author
Cambridge
UNIVERSITY PRESS
www.cambridge.org
ISBN 978-1-107-09135-1
Contents
f}
List of Figures x
List of Tables xv
Preface xvii
Notation xxi
1 Introduction to Tensors 1
1.1 Vector Spaces 1
1.2 Vectors in 5ft3 1 8
1.3 Second-Order Tensors 12
1.3.1 The tensor product 15
1.3.2 Principal invariants of a second-order tensor 17
1.3.3 Inverse of a tensor 20
1.3.4 Eigenvalues and eigenvectors of tensors 27
1.4 Skew-Symmetric Tensors 29
1.5 Orthogonal Tensors 31
1.6 Symmetric Tensors 40
1.6.1 Principal values and principal directions 40
1.6.2 Positive definite tensors and the polar decomposition 48
1.6.3 Isotropic functions 53
1.7 Higher-Order Tensors 60
1.8 Isotropic Tensors 77
1.9 Differentiation of Tensors 80
1.9.1 The directional derivative 81
1.9.2 Product rule 83
1.9.3 Chain rule 84
1.9.4 Gradient, divergence and curl 85
1.9.5 Examples 97
1.10 The Exponential and Logarithmic Functions 99
1.11 Divergence and Stokes Theorems 117
1.12 Groups 119
2 Kinematics 137
2.1 Lagrangian and Eulerian Descriptions 137
2.2 Length, Area and Volume Elements in the Deformed Configuration 139
vi Contents
2.2.1 Length element in the deformed configuration: Strain tensors 140
2.2.2 Area element in the deformed configuration: The Piola transform 146
2.2.3 Volume element in the deformed configuration 149
2.3 Velocity and Acceleration 149
2.4 Rate of Deformation 151
2.5 Examples of Simple Motions 160
2.5.1 Pure extension 160
2.5.2 Rigid body motion 162
2.5.3 Simple shear 162
3 Balance Laws 166
3.1 The First Transport Theorem 167
3.2 Conservation of Mass 168
3.2.1 Lagrangian version 168
3.2.2 Eulerian version 169
3.3 The Second Transport Theorem 170
3.4 Generalized Transport Theorems 170
3.5 Balance of Linear Momentum 172
3.6 Balance of Angular Momentum 178
3.7 Properties of the Cauchy Stress Tensor 181
3.8 The Equations of Motion in the Reference Configuration 185
3.9 Variational Formulations 188
3.10 Energy Equation 191
3.11 Control Volume Form of the Balance Laws 199
4 Constitutive Equations 203
4.1 Frame of Reference 203
4.2 Transformation of Kinematical Quantities 206
4.3 Principle of Frame-Indifference 208
4.4 Principle of Material Frame-Indifference 219
4.5 Constitutive Relations for Simple Materials 222
4.6 Material Symmetry 226
4.7 Classification of Materials 231
5 Nonlinear Elasticity 240
5.1 Isotropic Elastic Particles 240
5.2 The Constitutive Equation of an Isotropic Solid for Small Deformations 244
5.3 Bounds on the Lame Constants 246
5.4 Hyperelastic Solids 247
Contents vii
5.5 Isotropic Hyperelastic Solids 253
5.6 St Venant-Kirchhoff Material 257
5.7 Examples of Nonlinear Compressible Hyperelastic Models 259
5.8 The Elasticity Tensors 263
5.9 Elastic and Material Stability 270
5.10 Nonuniqueness of Solutions in Elasticity 282
5.11 Exact Solutions for Homogeneous, Compressible, Isotropic Elastic Materials 283
5.11.1 Uniaxial stretch 284
5.11.2 Pure shear 285
5.11.3 Pure bending of a prismatic beam made of a particular
St Venant-Kirchhoff material 287
5.11.4 Torsion of a circular shaft made of a St Venant-Kirchhoff material 289
5.12 Exact Solutions for Homogeneous, Incompressible,
Isotropic Elastic Materials 290
5.12.1 Bending and stretching of a rectangular block 292
5.12.2 Straightening, stretching and shearing a sector of a cylinder 294
5.12.3 Torsion, inflation, bending, etc. of an annular wedge 295
5.12.4 Inflation/eversion of a spherical shell 300
6 Linearized Elasticity 308
6.1 Kinematics 308
6.2 Governing Equations 315
6.3 Energy and Variational Formulations in Linearized Elasticity 322
6.3.1 Single-field variational formulation 323
6.3.2 Two-field and three-field variational formulations 329
6.4 Uniqueness of Solution 331
6.5 Exact Solutions of some Special Problems in Elasticity 333
6.5.1 Torsion of a circular cylinder 333
6.5.2 Torsion of non-circular bars-Saint-Venant theory of torsion 335
6.5.3 Generalization of the Saint-Venant torsion theory to an
anisotropic, inhomogeneous bar 357
6.5.4 Torsion of circular shafts of variable diameter 371
6.5.5 Pure bending of prismatic beams 387
6.5.6 Bending of prismatic beams by terminal loads 392
6.5.7 Hollow sphere subjected to uniform pressure/Gravitating sphere 425
6.6 General Solutions for Elastostatics using Potentials 427
6.6.1 Cylindrical/elliptical elastic inclusion in an infinite domain with
a uniform state of stress at infinity 463
6.6.2 Rectangular domain (e.g., cantilever beam) loaded by tractions
on its edges
479
viti Contents
6.6.3 Circular disc loaded by a traction distribution on its rim 506
6.6.4 Wedge loaded by traction distributions on its edges 510
6.6.5 Thermal stresses 537
6.6.6 Thick hollow cylinder subjected to a linearly varying pressure
on the inner and outer surfaces 540
6.6.7 Sphere/spherical segment spinning about its axis 542
6.6.8 Circular cylinder with loading on its end faces, and lateral surfaces
traction-free 546
6.6.9 Circular cylinder with loading on the lateral surface, and end
surfaces traction-free ( Filon s problem ) 553
6.6.10 Clamped and simply supported circular cylinders 560
6.6.11 Circular cylinder spinning about its axis 565
6.6.12 Circular cylinder on a frictionless surface loaded under its
own weight 568
6.6.13 Point load in an infinite elastic body (Kelvin problem) 572
6.6.14 Point load acting normal to the boundary of an infinite half-space
(Boussinesq problem) 574
6.6.15 Truncated cone 575
6.6.16 Contact problems on a finite domain 589
6.6.17 Spherical cavity in an infinite domain with a uniaxial state of
stress at infinity 594
6.6.18 Prolate or oblate spheroidal cavity in an infinite domain
with a uniform stress state at infinity 599
6.6.19 Point load acting tangential to the boundary of an infinite
half-space (Cerruti problem) 603
6.7 Elastodynamics 605
6.7.1 Progressive waves 605
6.7.2 Solution to special problems 607
7 Thermomechanics 686
7.1 Thermoelastic materials 686
7.2 Viscoplastic materials 695
7.3 Restrictions on the constitutive relations for fluids 699
7.3.1 Thermodynamic relations for a perfect gas 713
7.3.2 The Navier-Stokes and energy equations 715
7.3.3 Summary of the governing equations for a Newtonian fluid 715
8 Rigid-Body Dynamics 718
Example 1: Central-force motion 725
Example 2: Cylinder rolling down a plane 727
Example 3: Sliding rod 728
Contents ix
Example 4: Motion of a sleigh 730
Example 5: Spinning disc 732
Example 6: Spinning top 734
Example 7: Force on a bar 736
Example 8: Rotating bar 737
Example 9: Slider-crank mechanism 738
Appendices 742
A Orthogonal Curvilinear Coordinate Systems 742
B Cylindrical Coordinate System 751
C Spherical Coordinate System 756
D Elliptic Cylindrical Coordinate System 760
E Bipolar Cylindrical Coordinate System 764
F Prolate Spheroidal Coordinate System 768
G Oblate Spheroidal Coordinate System 772
H Arbitrary Curvilinear Coordinate Systems 776
I Matrix Representations of Tensors for Engineering Applications 789
J Some Results in n-Dimensional Euclidean Spaces 802
K A Note on Boundary Conditions 820
Bibliography 825
Index 847
|
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| isbn | 9781107091351 |
| language | English |
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| spelling | Jog, Chandrashekhar S. Verfasser aut Foundations and applications of mechanics 1 Continuum mechanics C.S. Jog Third edition Delhi Cambridge University Press 2015 xxiii, 852 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Cambridge IISc series Tensoralgebra (DE-588)4505278-5 gnd rswk-swf Kontinuumsmechanik (DE-588)4032296-8 gnd rswk-swf Kontinuumsmechanik (DE-588)4032296-8 s Tensoralgebra (DE-588)4505278-5 s DE-604 (DE-604)BV043448250 1 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028198308&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028198308&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| subject_GND | (DE-588)4505278-5 (DE-588)4032296-8 |
| title | Foundations and applications of mechanics |
| title_auth | Foundations and applications of mechanics |
| title_exact_search | Foundations and applications of mechanics |
| title_full | Foundations and applications of mechanics 1 Continuum mechanics C.S. Jog |
| title_fullStr | Foundations and applications of mechanics 1 Continuum mechanics C.S. Jog |
| title_full_unstemmed | Foundations and applications of mechanics 1 Continuum mechanics C.S. Jog |
| title_short | Foundations and applications of mechanics |
| title_sort | foundations and applications of mechanics continuum mechanics |
| topic | Tensoralgebra (DE-588)4505278-5 gnd Kontinuumsmechanik (DE-588)4032296-8 gnd |
| topic_facet | Tensoralgebra Kontinuumsmechanik |
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