Mechanics of Curves and Surfaces
Introduction
Zhiping Xu
Email: xuzp@tsinghua.edu.cn
2022/02/21
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Why we are interested in curves and surfaces?
Principle of design: beauty, capacity, rigidity and cost
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The catenary problem
Hinges can transmit force but not moment, and structures are
usually ‘stronger’ under stretching/compression than bending!
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Why we are interested in curves and surfaces?
Geometry endows structural rigidity!
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What’s new in lower dimensions?
With the dimension reduced,
the problem of elasticity should be easier to solve?
3D
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2D
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1D
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What’s new in lower dimensions?
Frames are moving & there are both strain AND ROTATION of line elements
The rise of geometrical nonlinearity
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The beauty of geometry in elasticity
Geometry couples different modes of deformation
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and the complexity it leads to.
Geometry drives bifurcation of elastic deformation
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Ways to solve the problem.
• Setup fundamental equations (geometry, constitutive laws,
equilibrium) of 3D elasticity and reduce them to lower dimensions.
• Or propose kinematical assumptions, solve the equilibrium equations
with stress calculated from deformation using the constitutive laws.
• The key step is the kinematics - the modes of deformation should be
compatible and orthogonal to each other for the simplicity of
formulation.
• Simplified models could deduced following these procedures, named
usually to their original contributors – Euler, Kirchhoff, Love, Föppl, von
Kármán, and so on.
• Even within the scope of linear elasticity where material nonlinearity
can be ignored, the geometrical nonlinearity leads to technical
difficulties that can only be solved numerically or by scaling arguments.
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Scientific understandings from theory and data
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to be predicted
theory w/
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theory w/
approx. #1
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data from exp. or numerics
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How to learn?
• Understand popular models by deriving them and applying them.
• Encode the principles into a workable computer program that can
solve not-so-simple problems, and analyze the results with the theories.
• Read scientific papers, follow the logic of reasoning and technical
recipes in problem solving.
• This course will guide you to these steps.
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Syllabus
Part I
Preparation
Theory of Elasticity
Part II
Rods, Beams, Shells
Part III
Stability,
Dynamics, Fluid-Structure Coupling
Fluctuation, Defects and Failure
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Preparation
https://www.engineer4free.com/mechanics-of-materials.html
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Preparation
vector and tensor analysis
geometry of curves and surfaces
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Theory of elasticity
strain
compatibility
stress
equilibrium
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Rods
Gustav Robert Kirchhoff,
1824-1887
https://vfxserbia.com/2017/05/02/simulation-ready-hair-capture-by-disney/
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Plates
August Föppl,
1854-1924
Theodore von Kármán,
1881-1963
Measuring interlayer shear stress in bilayer graphene
Guorui Wang, …, Zhong Zhang
et al., Phys. Rev. Lett. 119, 036101, 2017
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Shells
shells of
revolution
shallow shell
Indentation of ellipsoidal and cylindrical elastic shells
Dominic Vella, Amin Ajdari, Ashkan Vaziri, and Arezki Boudaoud
Phys. Rev. Lett. 109, 144302 (2012)
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Elastic stability
Geometry and physics of wrinkling
E. Cerda and L. Mahadevan
Phys Rev Lett 90, 074302, 2003
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Dynamics
https://en.wikipedia.org/wiki/Bending#Kirchhoff%E2%80%93Love_theory_of_plates
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Fluid-structure interaction
The clapping book: Wind-driven oscillations
in a stack of elastic sheets
P. Buchak, C. Eloy, and P. M. Reis
Phys Rev Lett 105, 194301, 2010
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Thermal fluctuation
13 keV
Graphene kirigami
Melina K. Blees, … Paul L. McEuen
Nature 524, 204, 2015
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Defects and failure
detonation-driven fracture of Al 6061-T6 tubes
Unzipping of carbon nanotubes is geometry- dependent
Zhigong Song, Xin Mu, Tengfei Luo and Zhiping Xu
Nanotechnology 27, 015601, 2015
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Textbooks – Audoly & Pomeau (nonlinear, scaling arguments)
Part 0
Introduction
3D Elasticity
Part I Rods
Equations of Elastic Rods
mechanics of the human hair, rippled leaves, uncoiled springs
Part II Plates
The Equations for Elastic Plates
end effects in plate buckling, finite amplitude buckling of a
strip, crumpled paper, fractal buckling near edges
Part III Shells
Geometric Rigidity of Surfaces
Shells of Revolution
the elastic torus, spherical shell pushed by a wall
Appendix
variation, boundary and interior layers, geometry of helices,
plate equations from 3D elasticity
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Textbooks – Huang et al. (linear theories w/ solutions)
Part I Plates
Theories
Bending in Polar and Rectangular Coordinates
Energy Principles
Bending Theory with Shear
Large Deflection (Bending and Stretching)
Part II Shells
Theories
Membrane and Bending Theories
Bending of Shells of Revolution
Shallow Shells
Part III Stability
Stability of Plates
Stability of Shells
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Problem sets
• ~2 problems for each chapter (1-5).
• Solutions should be submitted within 2 weeks after the
corresponding chapter(s) is finished.
• 3 reading tasks for rods, plates, and shells.
• Questions and suggestions should be addressed to:
xuzp@tsinghua.edu.cn
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