Mechanics of elastic rods
Khalid Jawed www.khalidjawed.com
Postdoctoral Fellow Integrated Soft Materials Laboratory Carnegie Mellon University PhD (Mechanics) 2016 Massachusetts Institute of Technology
Rods across length-scales
Definition of rod:
Rods and geometric nonlinearity
• Equations of equilibrium not updated with the deformation
Geometrically linear
• e.g. Euler-Bernoulli beam • Deviation from original geometry is
negligible
Rods and geometric nonlinearity
Geometrically nonlinear
Wikimedia.org
Tendrils
Wikihow.com
Telephone cord
R. Stocker (ETHZ)
Flagella
Cookingwithoutborders.com
Food
Colourbox.com
Garden hose
Gray et al, Adv. Mater. (2004)
Flexible electronics
Jay Miller (MIT)
Human hair
[Lyubchenko et al, Proc. Natl. Acad. Sci (1997)]
DNA
Theory and simulation of rods
Elastic energies I) Bending II) Twisting III) Stretching Image: Basile Audoly
Assume, Initially straight rod with circular cross-section
Kirchhoff 1824-1887
Undeformed Bending Twisting Stretching
Theory and simulation of rods
Kirchhoff 1824-1887
Elastic energies I) Bending II) Twisting III) Stretching
Bending energy per length,
Undeformed Bending Twisting Stretching
Theory and simulation of rods
Kirchhoff 1824-1887
Elastic energies I) Bending II) Twisting III) Stretching
Twisting energy per length, Twist,
Undeformed Bending Twisting Stretching
Theory and simulation of rods
Kirchhoff 1824-1887
Elastic energies I) Bending II) Twisting III) Stretching
Stretching energy per length,
Stretch,
Undeformed
Deformed
Undeformed Bending Twisting Stretching
Discrete Elastic Rods (DER) [Bergou et al., SIGGRAPH 2010]
Continuous
Discrete
1. Sum of elastic (bending + twisting + stretching) energies over the entire rod:
2. Forces (or moments) from energies:
3. Balance of forces:
m=massI=momentofiner.a
DER in animation industry
The Hobbit
Weta Digital
England
With Fang Da, J. Joo, Eitan Grinspun Columbia Computer Graphics Group
Columbia University
France
PipeLine Under The Ocean (PLUTO), 1944
England
France
PipeLine Under The Ocean (PLUTO), 1944
Submarine Cable Map submarinecable.com
Tangles in cables [Goyal et al, Int. J. Non Linear Mech. (2008)]
https://youtu.be/Nv9lBqPVuoE
Experiments and Simulations
Experiments: Deposit a thin elastic rod onto a moving rigid substrate
Discrete Elastic Rods (DER) Simulations: Implements discrete geometric model of thin flexible rods
[M. K. Jawed et al, Proc. Natl. Acad. Sci. 2014 M. K. Jawed et al, Extreme Mechanics Letters 2014 M. K. Jawed et al, J. Appl. Mech. 2015]
Camera Injector
Conveyor belt
Vbelt
Vinj H
Coiling of rods on static substrate [Jawed et al., Proc. Natl. Acad. Sci. 2014]
Coiling radius, Ro=?
Dimensional analysis Parameters
1. Flexural rigidity, EI 2. Torsional rigidity, GJ 3. Stretching rigidity, EA 4. Radius, r0 5. Height, H 6. Specific weight, 𝜌𝑔 7. Radius of coil, Ro
Gravito-bending length:
Buckingham Pi theorem Physical variables, n = 6 Physical dimensions, k = 3 Dimensionless groups, p = n - k = 3
Inextensible
We only consider incompressible material and circular cross-section (GJ/EI = 2/3)
Normalized height,
Normalized coiling radius,
Ratio of twisting and bending rigidity,
Dimensionless groups
Prediction of coiling radius
Normalized height,
Nor
mal
ized
coi
ling
radi
us,
Fit
Simula.ons
We can predict the coiling radius of a naturally straight rod
Inversion in coiling
Natural curvature can lead to inversion in coiling direction
Pattern formation [Jawed et al, Proc. Natl. Acad. Sci. 2014 Jawed & Reis, Extreme Mechanics Letters 2014]
Mea
nder
ing
Alte
rnat
ing
loop
s Tr
ansl
atin
g co
ils
Dimensionless speed mismatch
Prediction of pattern type
Gravito-bending length:
Governing length-scale
[Jawed & Reis, Extreme Mechanics Letters 2014]
We can predict the coiling pattern of a naturally straight rod
Robotic surgery and 3D printing
Interactive simulation of surgical needle insertion and steering
Chentanez et al, IEEE Trans. Biomed. Eng. 2005
Molten glass sewing machine [Brun et al (MIT)]
Instability-assisted fused deposition modeling [Passieux et al., Adv. Mater. 2015]
Questions? Thank you Prof. Pedro Reis (MIT) Fang Da (Columbia, now at Google) Jungseock Joo (UCLA) P. T. Brun (MIT)
References 1. Discrete viscous threads M Bergou, B Audoly, E Vouga, M Wardetzky, E Grinspun ACM Transactions on Graphics 29 (4), 116 2. Coiling of elastic rods on rigid substrates MK Jawed, F Da, J Joo, E Grinspun, PM Reis Proceedings of the National Academy of Sciences 111 (41), 14663-14668 3. Pattern morphology in the elastic sewing machine MK Jawed, PM Reis Extreme Mechanics Letters 1, 76-82 Project page and source code
www.cs.columbia.edu/cg/elastic_coiling/ Email: [email protected]