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: mkjawed@andrew.cmu.edu