Mixing shear and dilation in marginal solids Brian Tighe with Ren - PowerPoint PPT Presentation

Mixing shear and dilation in marginal solids Brian Tighe with René Pecnik and Ana Martin Calvo

Mixing shear and dilation... I. Dilation induced by shear II. Tuning shear compliance with pre-tension

Mixing shear �������� and dilation... I. Dilation induced by shear II. Tuning shear compliance �������� with pre-tension

Counterintuitive dilatancy Janmey et al. Nature Materials 2006 O. Reynolds 1885 Hutzler Conti & MacKintosh PRL 2008 Weaire & Hutzler, Phil. Mag. 2003 Weaire Packings expand, networks contract: Why the difference? packings : networks: foam liquid djl = 0.07 system expands system contracts (BoIton (a) 6 or pressure increases or pressure decreases (o), averaged 60, ( - ) , cljl < data

Dilatancy enhanced near jamming 휙 ↗ RCP Ren, Dijksman & Behringer PRL 2013

A nonlinear effect normal stress p 0 γ L symmetry: ϵ L ✏ = 1 2 R p � 2 + . . . Reynolds dilatancy coefficient ✓ @ 2 ✏ ◆ R p = @� 2 γ Ren, Dijksman & Behringer, PRL 2013 Weaire & Hutzler, Phil. Mag. 2003

Reynolds coefficient γ L assume a hyperelastic solid: ϵ L energy d U = − p d V − σ V d γ “enthalpy” d H = V d p − σ V d γ ✓ ∂ V ◆ ✓ ∂ σ V ◆ Maxwell = − ∂γ ∂ p p γ expression for R p Weaire & Hutzler, Phil. Mag. 2003 BPT, Gran. Matt. 2013

Reynolds coefficient ✓ ∂ G ◆ − G shear modulus G > 0 Young’s modulus E > 0 R p = ∂ p E typically E > G γ Weaire & Hutzler, Phil. Mag. 2003 BPT, Gran. Matt. 2013

Reynolds coefficient ✓ ∂ G ◆ − G shear modulus G > 0 Young’s modulus E > 0 R p = ∂ p E typically E > G γ magnitude >> 1 in marginal solids does compression ✓ ∂ G ◆ stiffen or soften the ' ∂ p shear modulus? γ Weaire & Hutzler, Phil. Mag. 2003 BPT, Gran. Matt. 2013

Soft spheres ✓ ∂ G ◆ R p ' ∂ p γ �������� G ∼ p 1 / 2 (Hookean) O’Hern, Silbert, Liu & Nagel, PRE 2003 1 R p ∼ p 1 / 2 > 0 packings expand BPT, Gran. Matt. 2013

Packings expand: verified in model foams ' % . Y E , averaged pressure 6 data (BoIton Weaire cljl < 1. Physical intuition? 0 0.05 '0.1 0 . 1 5 0 . 2 0.25 0.3 0 . 3 5 0.4 0 . 4 5 G.Y Hencky strain Hutzler shear strain 2. What about networks? foam 60, osn~otic with foam softivare n/(y/R) undefortned (a) liquid = 0 . 0 8 , ~ l 1 , ~ 6 > q~,,. n,,,,,, I 7 j3(al). djl = ( - ) , 0.07 (o), Weaire & Hutzler, Phil. Mag. 2003

Mixing shear �������� and dilation... I. Dilation induced by shear II. Tuning shear compliance �������� with pre-tension

�������� �������� �������� �������� �������� �������� �������� �������� � � � � � � � � unloaded state = floppy = tunable! �������� �������� �������� �������� �������� �������� �������� �������� �������� rest length p > 0 k e ff ∼ p �������� �������� �������� �������� �������� �������� �������� �������� �������� Tuning with tension

: Networks tension p = 0 p = 10 -3 p = 10 -1 floppy z < z c coordination rigid z > z c f/f max 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

: Networks tension p = 0 p = 10 -3 p = 10 -1 OFF z < z c coordination ON z > z c f/f max 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Manipulating marginal matter Brown et al, PNAS 2010 unjammed = OFF jammed = ON jamming transition as a switch

Rigidity induced by tension p = 0 p = 10 -3 p = 10 -1 z < z c measure G tension p rigidity: jamming transition as a switch ...or a knob

Shear modulus Ú ‡ ı Á Á Ú Ú Ì ‡ ı Ì Ú ‡ Á Ú ı ‡ connectivity Ì Ì Ì Ê Á Á Á Ú Ú Ï ‡ ‡ Ù Ù Ú Ú Ú Ú · · Û Û ‡ ı ı Ì Ê Á Ï Ú Ù Ú Ú Ú ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ · ‡ ‡ ‡ Ê ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı Ï Ï Û Ù Ù Ê Ê Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì · · Ï Ï Û Û Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Ú Ú Ú Ê ‡ Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ï Ê Ù Ì Ì Ì Ï · Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ù Ù Û Û ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ · · Û Û Ê Ê Ê Ê Ï Ï ‡ Ê Ê Ê Ï Ï 0.100 Ù Ú Ú Ú Ù ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ı ‡ ‡ · · · Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Û Û Û Û Ê Ê Ê Ê Ï Ï Ï Ï Ï Ù Ù Ù Ù ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ · Ê Ê · Û Û Ï Ï Û Û Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ê Ê Ê Ê Ï Ï Ï Ï Ù Ù · · · Ê Ê Ê Ê Ê Ï Ï Û Ì 0.050 Û Ù Ù Ê Á Ê · · Ï Ï Ï Ï Û Û Ù Ù G Ï Ï Ï Ê Ê · · Û Û Ù Ù Ù Ï Ê · · Ù Ï Û Û ‡ Ê Ê Ê Ê · · Û Û Ï Ï Ï Ï · Ê Ê Ï Ï Ù Ù modulus Ù Ù Û Û Û ı Ú Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê · · · Ï Ï Ï Ï Ï Ï Ï Ï Û Ï Ï Ì Ê Ê Ù Ù G Ù Ù Ù Ï Ï · · Û Û · · Ï Û Û Ê Ù Ê Ê Ï critical scaling Á · · · Ï Ï · · · Û Û Ù Ù Ù Ù 0.010 Ê Ê Û Û Û Ï Ï Á Ê Ï Ï Ï Û Ù Ù Ù Ù Ê Ê Ù Ù Ù Ê Ï Ï Ï · · · · · · · · Û Û Ï Ï Ï Ï Ï Ê Ê Û Û ✓ ◆ G ( p, z ) Ê Ê Ê Ê p 0.005 · Ï Ï Û Ù Ù Ù Û Ê | z − z c | µ = G · · · Ï Ï Ï Ù Ù Ù Ù Û Û Û Û Û Ê Ê Ï | z − z c | λ Ú Ê Ê Ï Ï Ï Ï Ù Ù Û Û Û · Ê Ê Ï Ï Ï · · · Ê Ê Ù Ï Û 0.001 0.001 0.01 0.1 10 - 4 tension p p z 4.5 3.5 4.0