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El. knyga: Geometric Continuum Mechanics and Induced Beam Theories

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This research monograph discusses novel approaches to geometric continuum mechanics and introduces beams as constraint continuous bodies. In the coordinate free and metric independent geometric formulation of continuum mechanics as well as for beam theories, the principle of virtual work serves as the fundamental principle of mechanics. Based on the perception of analytical mechanics that forces of a mechanical system are defined as dual quantities to the kinematical description, the virtual work approach is a systematic way to treat arbitrary mechanical systems. Whereas this methodology is very convenient to formulate induced beam theories, it is essential in geometric continuum mechanics when the assumptions on the physical space are relaxed and the space is modeled as a smooth manifold. The book addresses researcher and graduate students in engineering and mathematics interested in recent developments of a geometric formulation of continuum mechanics and a hierarchical development of induced beam theories.

Recenzijos

This book presents elements of Geometric continuum Mechanics with application to rod theories. the book may be used in courses to the advanced undergraduate students that already have knowledge about the classical beam theories. Also it will be useful to the graduate students of Mechanics and the researchers in Mechanics. (Teodor Atanackovi, zbMATH 1330.74002, 2016)

1 Introduction
1(16)
1.1 Motivation
1(1)
1.2 The Virtual Work
2(2)
1.3 Literature Survey
4(4)
1.3.1 Foundations of Continuum Mechanics
4(3)
1.3.2 Beam Theory
7(1)
1.4 Aim and Scope
8(1)
1.5 Outline
9(8)
References
10(7)
Part I Geometric Continuum Mechanics
2 Kinematics
17(16)
2.1 Body and Space
17(3)
2.2 Spatial Virtual Displacement Field
20(5)
2.3 Configuration Space
25(4)
2.4 Affine Connection
29(4)
References
32(1)
3 Force Representations
33(12)
3.1 Principle of Virtual Work
33(3)
3.2 Classical Nonlinear Continuum Mechanics
36(9)
References
41(4)
Part II Induced Beam Theories
4 Preliminaries
45(10)
4.1 Fundamental Principles of a Continuous Body
45(3)
4.2 Constrained Position Fields
48(1)
4.3 Intrinsic and Induced Beam Theories
49(6)
References
52(3)
5 Classical Nonlinear Beam Theories
55(20)
5.1 Kinematical Assumptions
55(4)
5.2 Virtual Work Contributions
59(6)
5.2.1 Virtual Work Contributions of Internal Forces
60(1)
5.2.2 Virtual Work Contributions of Inertia Forces
61(2)
5.2.3 Virtual Work Contributions of External Forces
63(1)
5.2.4 The Boundary Value Problem
64(1)
5.3 Nonlinear Timoshenko Beam Theory
65(3)
5.4 Nonlinear Euler--Bernoulli Beam Theory
68(1)
5.5 Nonlinear Kirchhoff Beam Theory
69(1)
5.6 Literature Survey of Numerical Implementations
70(5)
References
71(4)
6 Classical Linearized Beam Theories
75(8)
6.1 Linearized Beam Kinematics
75(3)
6.2 The Boundary Value Problem of the Classical Linearized Beam Theory
78(1)
6.3 Linearized Timoshenko Beam Theory
79(1)
6.4 Linearized Euler--Bernoulli Beam Theory
80(1)
6.5 Linearized Kirchhoff Beam Theory
80(3)
Reference
81(2)
7 Classical Plane Linearized Beam Theories
83(18)
7.1 Constrained Position Fields in Linear Elasticity
84(1)
7.2 The Plane Linearized Timoshenko Beam
85(9)
7.2.1 Kinematics. Virtual Work and the Boundary Value Problem
85(4)
7.2.2 Constraint Stresses of the Plane Timoshenko Beam
89(5)
7.3 The Plane Linearized Euler--Bernoulli Beam
94(4)
7.3.1 Kinematics, Virtual Work and the Boundary Value Problem
94(2)
7.3.2 Constraint Stresses of the Plane Euler-Bernoulli Beam
96(2)
7.4 The Plane Linearized Kirchhoff Beam
98(3)
References
99(2)
8 Augmented Nonlinear Beam Theories
101(16)
8.1 The Nonlinear Cosserat Beam
101(6)
8.1.1 Kinematical Assumptions
102(1)
8.1.2 Virtual Work Contribution of Internal Forces
103(1)
8.1.3 Virtual Work Contribution of Inertia Forces
104(1)
8.1.4 Virtual Work Contribution of External Forces
105(1)
8.1.5 The Boundary Value Problem
105(1)
8.1.6 Constitutive Law and Restrictions on Internal Forces
106(1)
8.2 The Nonlinear Saint--Venant Beam
107(10)
8.2.1 Kinematical Assumptions
108(1)
8.2.2 Virtual Work Contribution of Internal Forces
109(2)
8.2.3 Virtual Work Contribution of Inertia Forces
111(1)
8.2.4 Virtual Work of External Forces
112(1)
8.2.5 The Boundary Value Problem
113(1)
8.2.6 Constitutive Laws
114(1)
References
114(3)
9 Conclusions and Outlook
117(6)
9.1 Geometric Continuum Mechanics
117(2)
9.2 Induced Beam Theories
119(4)
References
121(2)
Appendix A Multilinear Algebra 123(18)
Appendix B Properties of the Cross Product 141(2)
Index 143
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