The document summarizes research on modeling the shapes of suspended curved rods, like hair. It describes developing a theoretical model using the rod's mechanical properties and running simulations and experiments with rods of varying curvature. The results show the model can accurately predict the rod transitioning from planar to helical shapes as curvature increases. An inner layer approximation was also developed to predict the local-to-global helical transition point. The work demonstrates how physics can explain the shapes of curved elastic rods.

1. Unveiling the physics of hair shapes
Journal Club
Krissia de Zawadzki
Instituto de F´ısica de S˜ao Carlos - Universidade de S˜ao Paulo
April 3, 2014
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2. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Outline
1 Introduction
2 Theoretical Background
3 Results
4 Theoretical Analysis
5 Conclusions
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3. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Shapes of suspended curly hair
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4. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Shape of a Ponytail and the Statistical Physics of
Hair Fiber Bundles
Raymond E. Goldstein, Patrick B. Warren, and Robin C. Ball
PRL 108, 078101 (2012)
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5. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Motivation
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6. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Shapes of a suspended curly hair
Is it possible to predict the shape of a naturally curved rod?
Model for an elastic rod
suspended under its own weight
Metaphor for a curly hair!
precision
experiments
+
computational
simulations
+
theoretical
analysis
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7. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Mechanical properties of rods
Curvature, Young's modulus, Poisson Number
휅(푠) = ||T′(푠)|| = ||훾′′(푠)||
휈 = −
푑휀푡푟
푑휀푎푥
휀푡푟 transverse strain
휀푎푥 axial strain
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8. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Mechanical properties and coordinate system
¯푟(푠) = position of the centerline
orthonormal director basis
(d1(¯푠), d3(¯푠), d3(¯푠))
r′
3 = d3 tangent vector
cartesian basis e푖
¯푟(퐿) = 0
(d1, d2, d3)푠=퐿 = (e푦,−e푥, e푧)
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9. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Mechanical equilibrium: minimization problem
퐸 = Young's modulus
퐼 = moments of inertia
퐸3 = 퐺 = 퐸/2(1+휈) =
shear modulus
퐼3 = 퐽 = the moment
of twist
휈 = Poisson's number
ℰ푒 =
Σ︁3
k=1
퐸푘퐽퐾
2
∫︁ 퐿
0
(휅푘(푠) − ^휅푘(푠))2푑푠
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10. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Reescalings
Rationalizing → reescalings:
Arc length: ¯푠 = 푠휅푛
Length: ¯퐿
= 휅푛퐿
Energies: ¯퐸
= 퐵휅푛,
with 퐵 = 퐸퐼 is the bending stiness and
퐼 = 휋푟4/4 moment of inertia
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11. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Total energy of the rod
¯ ℰ =
∫︁ ¯퐿
0
(︂
1
2
[︀
(¯휅1 − 1)2 + ¯휅22
+ ¯ 퐶¯휅3
]︀
− ¯ 푤¯푠 cos 훽
)︂
푑¯푠
strain
energy
gravitational
potential
Control
parameters
퐿, 휌, 푟, 퐸, 휈 and
휅푛 (휅1, 휅2)
¯ 퐶 = the radio between the twisting and bending
moduli
¯ 푤 = 푤/퐵휅3
푛 = dimensionless weight
푤 = 휌휋푟2푔 = weight per unit of length
¯ 퐶(휈) = (1 + 휈)−1 = 2/3
3 Stationary points of ℰ
1 퐿[푐푚] 20
휌 = 1200 kg/m3
푟 = 1.55 mm
퐸 = 1290 ± 12 kPa
휈 ≈ 0.5
0 휅푛[푚−1] 62
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12. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Experiments
Physical experiments:
3 custom fabrication
of rods (hair)
3

13. ne control of 휅푛
PVC
exible tubes
Vinylpolyxiloxane
Comparison with simulations:
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14. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Simulations
Finite-dierence method
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15. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Equilibrium shapes of suspended rods
퐿, 휌, 푟, 퐸, 휈 ctes
휅푛 varying
3 휅푛 = 0 → straight
3 휅푛 small → planar
3 increasing 휅푛 → nonplanar 3D helical shape
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16. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Vertical elevation versus arc length
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17. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Phase diagram in the parameter space (¯퐿, ¯ 푤)
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18. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Comparison between experiments and simulations
3 11 110 equilibrium
shapes
0 휅푛[m−1] 100
Curvature 휅푛 in
uence in shapes
Vertical elevation versus length
Phase diagram in the parameter space
(¯퐿, ¯ 푤)
3 quantitative agreement between
experiments and simulations
3 small 휅푛 → planar shapes for all
lengths
high value of 휅푛 → planar for
퐿 . 0.1m, non-planar 퐿 0.1m
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19. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Shape classification
(i) planar
2D curves
recov. 3D model
¯휅1 = 훽′ and
¯휅2 = ¯휅3 = 0
(ii) nonplanar localized
helical
localized helices near
free end
helical portion 95 %
of the total length
(iii) nonplanar global
helical
typical 3D helix helical
portion 95 % of the
total length
3 2D - 3D transition: it's predictable!
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20. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Local helical configuration quantified by 훽(¯푠)
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21. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Classification of helical configurations
¯퐿
≪ 1
d2.e푧 ≈ 0
Parametrization
Euler angles 훽(¯푠) and 훾(¯푠)
d1 = cos 훽(−sin 훾e푥 + cos 훾ey) + sin 훽e푧
d2 = −cos 훽e푥(−sin 훾e푦
d3 = −sin 훽(−sin 훾e푥 + cos 훾ey)
¯휅1 = 훾′
¯휅2 = −훽′
¯휅3 = 훾′ cos 훽
훾′ = 휅푛 sin 훽/ sin2 훽 + ¯ 퐶 cos2 훽
¯ ℰ3퐷 =
∫︁ ¯퐿
0
(︂
푓( ¯ 푤¯푠, 훽(¯푠)) +
1
2
훽′(¯푠)2
)︂
푑¯푠
2 (1 + tan2 훽/퐶)−1 − 푢 cos 훽
푓(푢, 훽) = 1
Equilibrium con

22. gurations: stationary points of ℰ3퐷 with respect to 훽(¯푠)
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23. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Local helix approximation failure
퐿퐻 = ( ¯ 푤 ¯ 퐶)−1: 훽 varies quickly
7 Instability at ¯푠*
7 ¯푠 ≤ ¯푠*
퐿퐻 → upper part of the rod remains vertical
7 0 ≤ ¯푠 ≤ ¯푠*
퐿퐻 → 훽̸= 0 → helical con

24. guration at the end
Inner layer approximation
restoring 훽′(¯푠)
푓(¯푠 ≈ ¯푠퐿퐻) ≈ 푓0 +
1
2
휕2푓
휕훽2 훽2 +
1
24
휕4푓
휕훽4 훽4
휕2푓
휕훽2 = ¯ 푤(¯푠 − ¯푠*)
휕4푓
휕훽4 = 3(4 − 3 ¯ 퐶/ ¯ 퐶2)
∫︁ We have to minimize the functional
(( ¯ 푤(¯푠 − ¯푠*)/2)훽2 + 푓4/24훽4 + 1/2훽′2)푑¯푠
Changing variables:
¯ 푆 = ¯푠 − ¯푠*/ ¯ 푤−1/3
퐵(푆) =
√︀
푓4/12 ¯ 푤−1/3훽(¯푠)
∫︁New functional
(푆퐵2 + 퐵4 + 퐵′2)푑푆
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25. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Inner layer approximation
3 New functional
∫︁
(푆퐵2 + 퐵4 + 퐵′2)푑푆
3 Euler-Lagrange condition → second Painleve equation
3 퐵′′(푆) = 푆퐵(푆) + 2퐵3
3 It has a unique solution connecting 퐵 → 0 for 푆 → ∞ (symmetric)
to the 퐵
√︀
−푆/2 for 푆 → ∞ (bifurcated)
Hastings-McLeon solution
훽퐼퐿(¯푠) =
2 ¯ 퐶 ¯ 푤1/3
√
4 − 3 ¯ 퐶
퐵퐻푀퐿
(︂
¯푠 − ¯푠*
퐿퐻
¯ 푤−1/3
)︂
3 IL solution successfully
describes the smooth
transition between the
helical and straight portions
of rod near ¯푠*
퐿퐻
3 We now are able to predict the transition local → global !
3 훽퐼퐿(0.95¯ 퐿) = 1.5표
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26. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Local helical configuration quantified by 훽(¯푠)
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27. Introduction Theoretical Background Results Theoretical Analysis Conclusions
Conclusions
3 Agreement between experiments and simulations
3 Model rcovers both 2D and 3D shapes and allow to explain the
transition planar → helical con

28. gurations
3 Model applicable of a variety of engineering, naturally curved
systems, wires, cables, pipes
3 Helpful to the inverse problem of manufacture rodlike structures
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