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 u i 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 @ j � ij + b i = ⇢@ 2 t u i 2. displacement-strain " ij = 1 2 ( @ j u i + @ i u j ) 3. Hooke’s law " i = 1 " ij = 1 + ⌫ E ( � i � ⌫ ( � j + � k )) + ↵ ∆ T, � ij E 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 E bending ⇠ t 3 , E stretching ⇠ 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 sti ff ness or flexural rigidity B = Et 3 / (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 k � � [ � , w ] = p � F � ( r ) B r 4 w + r 2 2 ⇡ r 2. compatibility 1 k w = � 1 Et r 4 � � r 2 2 [ w, w ] r 2 k = ( R i ) − 1 @ 2 [ f, g ] := @ 2 x f @ 2 y g � 2 @ x @ y f @ x @ y g + @ 2 y f @ 2 i , x g 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: R x = R y p 8 ( ⌧ 2 − 1) 4 ⇡ B tanh − 1 p (1 − ⌧ − 2 ) � � ⌧ t < ` 2 = ) F = b ⇡ pR � � � t : ◆ 1 / 4 ✓ BR 2 pR 2 ` b = , ⌧ = 4( EtB ) 1 / 2 , � = vertical displacement Et

D. Vella et al. Indentation of Ellipsoidal and Cylindrical Elastic Shells. I Unpressurized, dedimensionalized non-spherical shell equations: 1. force balance r 4 W + r 2 Φ + ∆  Y Φ ) = � F � ( R ) ( @ 2 X Φ � @ 2  M 2 ⇡ R 2. compatibility r 4 Φ � r 2 W � ∆  ( @ 2 X W � @ 2 Y W ) = 0 ,  M  M = 1 ∆  = 1 " = ∆  2 (  x +  y ) , 2 (  x �  y ) ,  M 3. solution 1 � " 2 p ( ⌧ 2 � 1) ✓ ◆ 4 ⇡ B � = 8( BEt  G ) 1 / 2 � F = 2 + · · · lim ` 2 tanh − 1 p (1 � ⌧ − 2 ) ⌧ → 0 b

D. Vella et al. Indentation of Ellipsoidal and Cylindrical Elastic Shells. ( ⌧ 2 � 1) 1 � " 2 p ✓ ◆ 4 ⇡ B � = 8( BEt  G ) 1 / 2 � F = 2 + · · · lim ` 2 tanh − 1 p ⌧ → 0 (1 � ⌧ − 2 ) b

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 � x x 0 = 1 yy = 1 ✓ 2 � R y ◆ � 0 2 pR y , 2 pR y R x 2. e ff ective radius of curvature R =  − 1 M 3. e ff ective isotropic initial stress � M = ( � 0 xx + � 0 yy ) / 2 4. e ff ective pressure 2 ⇠ + p 1 � ⇠ � 1 ⌧ = 1 � M p M ) 1 / 2 = ⇠ ( p 1 � ⇠ + 1) ( tEB  2 4( tEB ) 1 / 2  2 2 M ⇠ :=  G /  2 � M ( S 2 ) = p/ 2  M , M , � M ( S ⇥ R ) = 3 p/ 8  M ,

D. Vella et al. Indentation of Ellipsoidal and Cylindrical Elastic Shells. k 1 ⇠ 4 ⇡ B ⌧ log 2 ⌧ , ⌧ � 1 ` 2 b 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: 2 ⇠ + √ 1 − ⇠ − 1 ( log 2 ⌧ p  − 1 ⇡ k 1 = � ⌧ t ⇠ ( √ 1 − ⇠ +1) M k 2 = ⇡ p  − 1 � � t M = ) geometry-induced sti ff ness is accounted for by using mean curvature and mean base stress in the spherical case