Cell Mechanics: Indentation of Elastic Shells Felix Wong October - - PowerPoint PPT Presentation

Cell Mechanics: Indentation of Elastic Shells

Felix Wong October 24, 2014

Review of elasticity

I Objects:

• 1. displacement field ui
• 2. strain field ij(x, t)
• 3. stress field "ij(x, t)
• 4. strain tensor "ij
• 5. stress tensor ij

I Equations:

• 1. momentum balance

@jij + bi = ⇢@2

t ui

• 2. displacement-strain

"ij = 1 2 (@jui + @iuj)

• 3. Hooke’s law

"i = 1 E (i ⌫(j + k)) + ↵∆T, "ij = 1 + ⌫ E ij

I Equilibrium plane (2D) problems:

• 1. reduces to computing the Airy stress function

ij = (1)i+j@i@j, ∆∆ = 0

• D. Vella et al. Indentation of Ellipsoidal and Cylindrical Elastic Shells.

Motivation: geometry-induced rigidity in nonspherical pressurized shells

I Ebending ⇠ t3, Estretching ⇠ t

• A. Lazarus et al. Geometry-Induced Rigidity in Nonspherical Pressurized Elastic Shells. PRL 109, 144301 (2012).

• D. Vella et al. Indentation of Ellipsoidal and Cylindrical Elastic Shells.

I 2D Objects:

• 1. Gaussian curvature g > 0
• 2. bending stiffness or flexural rigidity B = Et3/(12(1 ⌫2))
• 3. internal pressure p
• 4. Airy stress function
• 5. displacement w = w(x, y) = w(r, ✓)
• 6. point force F

I Equations:

• 1. force balance

Br4w + r2

k [, w] = p F

2⇡ (r) r

• 2. compatibility

1 Et r4 r2

kw = 1

2 [w, w]

r2

k = (Ri)−1@2 i ,

[f, g] := @2

xf@2 yg 2@x@yf@x@yg + @2 yf@2 xg

• D. Vella et al. The indentation of pressurized elastic shells: from polymeric capsules to yeast cells. J. R. Soc. Interface (2012) 9, 448455.

• D. Vella et al. Indentation of Ellipsoidal and Cylindrical Elastic Shells.

I Spherical shell: Rx = Ry

= ) F = 8 < : 4⇡ B

`2

b

p

(⌧2−1) tanh−1 p (1−⌧−2)

⌧ t ⇡pR t `b = ✓BR2 Et ◆1/4 , ⌧ = pR2 4(EtB)1/2 , = vertical displacement

• D. Vella et al. Indentation of Ellipsoidal and Cylindrical Elastic Shells.

I Unpressurized, dedimensionalized non-spherical shell equations:

• 1. force balance

r4W + r2Φ + ∆ M (@2

XΦ @2 Y Φ) = F

2⇡ (R) R

• 2. compatibility

r4Φ r2W ∆ M (@2

XW @2 Y W) = 0,

M = 1 2 (x + y), ∆ = 1 2 (x y), " = ∆ M

• 3. solution

F = ✓ 1 "2 2 + · · · ◆ 4⇡ B `2

b

lim

⌧→0

p (⌧ 2 1) tanh−1 p (1 ⌧ −2) = 8(BEtG)1/2

• D. Vella et al. Indentation of Ellipsoidal and Cylindrical Elastic Shells.

F = ✓ 1 "2 2 + · · · ◆ 4⇡ B `2

b

lim

⌧→0

p (⌧ 2 1) tanh−1 p (1 ⌧ −2) = 8(BEtG)1/2

• D. Vella et al. Indentation of Ellipsoidal and Cylindrical Elastic Shells.

I Pressurized non-spherical shell: we can’t solve this analytically

• 1. anisotropic initial stress

xx0 = 1 2 pRy,

yy = 1

2 pRy ✓ 2 Ry Rx ◆

• 2. effective radius of curvature R = −1

M

• 3. effective isotropic initial stress M = (0

xx + 0 yy)/2

• 4. effective pressure

⌧ = 1 2 M (tEB2

M)1/2 =

p 4(tEB)1/22

M

2⇠ + p1 ⇠ 1 ⇠(p1 ⇠ + 1) ⇠ := G/2

M,

M(S ⇥ R) = 3p/8M, M(S2) = p/2M,

• D. Vella et al. Indentation of Ellipsoidal and Cylindrical Elastic Shells.

k1 ⇠ 4⇡B `2

b

⌧ log 2⌧ , ⌧ 1 Compared to previous work, gives correct prefactor of (log 2⌧)−1. Large indentation t, ⌧ 1: same as before

• D. Vella et al. Indentation of Ellipsoidal and Cylindrical Elastic Shells.

Summary of highly pressurized ⌧ l shells: ( k1 =

⇡ log 2⌧ p−1 M 2⇠+√1−⇠−1 ⇠(√1−⇠+1)

⌧ t k2 = ⇡p−1

M

t = ) geometry-induced stiffness is accounted for by using mean curvature and mean base stress in the spherical case