CNLD. Rupture of Rubber

Transcription

1 Rupture of Rubber Experiment: Paul Petersan, Robert Deegan, Harry Swinney Theory: Michael Marder Center for Nonlinear Dynamics and Department of Physics The University of Texas at Austin PRL (2004) PRL (2002) PRL (2005) cond-mat/ Supported by the National Science Foundation c 2005, Michael Marder

2 Outline Background: Limiting Speed of Cracks Experimental Observations Oscillations Sound and Rupture Speeds Numerical Investigations Multi-Particle Modeling Explorations Theory Elementary Shock Theory Continuum Theory Discrete Theory and Scaling c 2005, Michael Marder

3 Background Limiting Speed of Cracks Rayleigh wave speed limits cracks in tension (Yoffe, 1951; Stroh, 1957; Freund, 1972) Shear waves can also travel at 2c s (Andrews, 1976; Burridge, Conn and Freund, 1979; Rosakis, Samudrala, Coker, 1999; Gao, Huang, Abraham, 2001) In a discrete medium, there is no limit to the speed of tensile cracks (Slepyan, 1982; MM, 1995; Buehler, Gao, and Abraham, 2003) Integrity of crack surface behind tip is key to whether supersonic solutions survive (Ravi-Chandar and Knauss, 1984; Fineberg et al, 1991; MM) c 2005, Michael P Marder c 2005, Michael Marder

4 c 2005, Michael Marder Experimental Observations Balloons and First Experiments Balloons Controlled Experiments

5 c 2005, Michael Marder Experimental Observations Apparatus 60 cm y x 33cm 120 cm 13 cm 10 cm 66 cm

6 c 2005, Michael PMarder Experimental Observations Oscillations

7 c 2005, Michael PMarder Experimental Observations Oscillations 0.5 ( a ) y (cm) ( b ) 0.5 y (cm) x(cm)

8 c 2005, Michael PMarder Experimental Observations Oscillations y extension, λy Straight Oscillating x extension λ x

9 c 2005, Michael PMarder Experimental Observations Oscillations y extension, λy Straight Oscillating x extension λ x

10 c 2005, Michael PMarder Experimental Observations Oscillations No detailed theory yet for oscillations... but see Yang and Chen, Phys. Rev. Lett. 95, (2005) Wavelength (cm) Extension ratio λ x 0.1 Amplitude (cm)

11 c 2005, Michael Marder Experimental Observations Sound Speeds and Rupture Velocities y Tip has wedge-like shape like Mach cone x 1 cm

12 c 2005, Michael PMarder Experimental Observations Sound Speeds and Rupture Velocities Longitudinal speed measured by time of flight d + + pulse Fixed clamps

13 c 2005, Michael PMarder Experimental Observations Sound Speeds and Rupture Velocities Longitudinal speed measured by time of flight Longitudinal speed measured from force extension curves. W d + + pulse L F sin(ωt) Fixed clamps δx sin(ωt)

14 c 2005, Michael PMarder Experimental Observations Sound Speeds and Rupture Velocities longitudinal wave speed measured calculated Two methods agree Extension ratio λ x

15 c 2005, Michael PMarder Experimental Observations Sound Speeds and Rupture Velocities Shear wave speed measured from force extension curves. W L F sin(ωt) (25 Hz) δx sin(ωt)

16 Experimental Observations Sound Speeds and Rupture Velocities Speed (m/s) Longitudinal Rupture Shear Extension Ratio λ x c 2005, Michael PMarder

17 c 2005, Michael PMarder Experimental Observations Sound Speeds and Rupture Velocities Transform to reference frame 30 Speed (m/s)/λx Longitudinal Rupture Shear Extension Ratio λ x

18 Experimental Observations Sound Speeds and Rupture Velocities Over range of extensions covered by our experiments (λ x, λ y 200% 350%), the speed of sound waves is well described by the Mooney-Rivlin free energy: e(i 1, I 2 ) = A(I 1 + 2BI 2 ) = A [ (E xx + E yy + E zz ) + 2B ( E xx E yy E 2 xy ) + 2BEzz I 2 ] c 2005, MichaelPMarder

19 Experimental Observations Sound Speeds and Rupture Velocities Over range of extensions covered by our experiments (λ x, λ y 200% 350%), the speed of sound waves is well described by the Mooney-Rivlin free energy: e(i 1, I 2 ) = A(I 1 + 2BI 2 ) = A [ (E xx + E yy + E zz ) + 2B ( E xx E yy E 2 xy ) + 2BEzz I 2 ] c 2005, MichaelPMarder

20 Experimental Observations Sound Speeds and Rupture Velocities Over range of extensions covered by our experiments (λ x, λ y 200% 350%), the speed of sound waves is well described by the Mooney-Rivlin free energy: e(i 1, I 2 ) = A(I 1 + BI 2 ) = A [ (E xx + E yy ) + B ( E xx E yy E 2 xy )] Sound speeds (m/s) Longitudinal, c xl Shear, c xs λ x c 2005, MichaelPMarder

21 Experimental Observations Sound Speeds and Rupture Velocities Over range of extensions covered by our experiments (λ x, λ y 200% 350%), the speed of sound waves is well described by the Mooney-Rivlin free energy: e(i 1, I 2 ) = A(I 1 + BI 2 ) = A [ (E xx + E yy ) + B ( E xx E yy E 2 xy 30 )] cyl (m/s) λ x c 2005, MichaelPMarder

22 c 2005, Michael Marder Numerical Studies Multi-Particle Modeling Tethered Membranes (Nelson, et al.) Virtual Bond Method (Gao and Klein) Peridynamics (Silling and Bobaru) Mesodynamics (Holian) Distance between near neighbors I 1i = 1 6a 2 u u ij2 ij3 j n(i) Failure extension Lattice spacing {( uij u ij a 2) if u ij < λ f λ 2 f a2 else u ij4 u ij1 u ij5 u ij6

23 c 2005, Michael Marder Numerical Studies Multi-Particle Modeling F i = 1 (` uij u ij a 2 6 λ 2 f a 2 G i = 1 9 H i = 1 27 j n(i) j n(i) j k n(i) (` uij u ij a 2 2 `λ2 f a 2 2 if u ij < λ f else if u ij < λ f else h(u ij )h(u ik ) ` u ij u ik + 2a 2 2, and h(u) = 1/(1 + e (u λ f )/u s ). or alternatively, I i 2 = 3 4 I i 2 = 9 8 I i 1 = F i a 2 1 a 4 `F 2 i G i, 1 a 4 (G i H i + 4).

24 Numerical Studies Multi-Particle Modeling Investigations Obtained ruptures resembling experiment. What was needed, what was not? c 2005, MichaelPMarder

25 Numerical Studies Multi-Particle Modeling Investigations Obtained ruptures resembling experiment. What was needed, what was not? Increase in sound speed near tip of rupture (hyperelasticity) c 2005, MichaelPMarder

26 Numerical Studies Multi-Particle Modeling Investigations Obtained ruptures resembling experiment. What was needed, what was not? Increase in sound speed near tip of rupture (hyperelasticity) c 2005, MichaelPMarder

27 Numerical Studies Multi-Particle Modeling Investigations Obtained ruptures resembling experiment. What was needed, what was not? Increase in sound speed near tip of rupture (hyperelasticity) Increase in sound speed as rubber retracts c 2005, MichaelPMarder

28 Numerical Studies Multi-Particle Modeling Investigations Obtained ruptures resembling experiment. What was needed, what was not? Increase in sound speed near tip of rupture (hyperelasticity) Increase in sound speed as rubber retracts c 2005, MichaelPMarder

29 Numerical Studies Multi-Particle Modeling Investigations Obtained ruptures resembling experiment. What was needed, what was not? Increase in sound speed near tip of rupture (hyperelasticity) Increase in sound speed as rubber retracts Second term in Mooney-Rivlin theory proportional to I 2 c 2005, MichaelPMarder

30 Numerical Studies Multi-Particle Modeling Investigations Obtained ruptures resembling experiment. What was needed, what was not? Increase in sound speed near tip of rupture (hyperelasticity) Increase in sound speed as rubber retracts Second term in Mooney-Rivlin theory proportional to I 2 c 2005, MichaelPMarder

31 Numerical Studies Multi-Particle Modeling Investigations Obtained ruptures resembling experiment. What was needed, what was not? Increase in sound speed near tip of rupture (hyperelasticity) Increase in sound speed as rubber retracts Second term in Mooney-Rivlin theory proportional to I 2 Dissipation (Kelvin, proportional to relative motion of neighbors) c 2005, MichaelPMarder

32 Numerical Studies Multi-Particle Modeling Investigations Obtained ruptures resembling experiment. What was needed, what was not? Increase in sound speed near tip of rupture (hyperelasticity) Increase in sound speed as rubber retracts Second term in Mooney-Rivlin theory proportional to I 2 Dissipation (Kelvin, proportional to relative motion of neighbors) Yes. βc 2 2 u t...(or more elaborate models) c 2005, MichaelPMarder

33 Numerical Studies Multi-Particle Modeling Investigations Obtained ruptures resembling experiment. What was needed, what was not? Increase in sound speed near tip of rupture (hyperelasticity) Increase in sound speed as rubber retracts Second term in Mooney-Rivlin theory proportional to I 2 Dissipation (Kelvin, proportional to relative motion of neighbors) Yes. βc 2 2 u t...(or more elaborate models) Toughening behind tip c 2005, MichaelPMarder

34 Numerical Studies Multi-Particle Modeling Obtained ruptures resembling experiment. What was needed, what was not? Increase in sound speed near tip of rupture (hyperelasticity) Increase in sound speed as rubber retracts Second term in Mooney-Rivlin theory proportional to I 2 Dissipation (Kelvin, proportional to relative motion of neighbors) Yes. βc 2 2 u t...(or more elaborate models) Toughening behind tip Yes Investigations c 2005, MichaelPMarder

35 c 2005, Michael Marder Theoretical Views Elementary Considerations Reference Frame Laboratory Frame (x, y) r u (λ x x, λ y y) θ c θ v c v = sin θ laboratory slope = λ y λ x v2 /c 2 1

36 c 2005, Michael Marder Theoretical Views Continuum Theory Following suggesion of Rice (following Shield) adopt Neo Hookean theory. Horizontal displacements are static, so obtain theory for vertical displacement u. ü = c 2 2 u + c 2 β 2 u u is not small. β describes Kelvin dissipation (E ) Boundary conditions: u y = β 2 u t y for x < 0; u = 0 for x > 0.

37 c 2005, Michael Marder Theoretical Views Continuum Theory This theory has supersonic solutions (v > c). Recover laboratory slope of back edge = λ y λ x v2 /c 2 1 At the origin, the slope of the rupture is u y = x=0,y=0 λ y 1 c2 /v 2. The theory can be closed with rupture criterion λ y (4λ 2 f λ2 x)/3 = 1 c 2 /v 2

38 c 2005, Michael Marder Theoretical Views Continuum Theory Kelvin dissipation Sound speed Lattice spacing Dimensionless measure of dissipation is βc/ : this theory applies when βc/ 1. Displacements and strains are finite near tip. Stress diverges as exp[ x/(βc)]/ x near tip There is no energy release.

39 c 2005, Michael Marder Theoretical Views Discrete Theory Discrete theory can be solved using Wiener-Hopf techniques (Slepyan, MPM); find rupture speeds of 800-million-particle systems in five minutes ü y i = 2c2 3a 2 j n(i) ( u y ij + β uy ij) θ(λf u ij ).

40 c 2005, Michael Marder Theoretical Views Discrete Theory Discrete theory can be solved using Wiener-Hopf techniques (Slepyan, MPM); find rupture speeds of 800-million-particle systems in five minutes ü y i = 2c2 3a 2 ṽ = v/c, j n(i) ( u y ij + β uy ij) θ(λf u ij ). β = βc/a; z = 3 cos(ω/ṽ) 3ω 2 /[4(1 i βω)] 2 cos(ω/2ṽ) y = z + q z 2 1with abs(y) > 1, ; F (ω) = 8 < : y [N 1] y [N 1] y N y N 9 = 2z ; cos(ω/2ṽ) + 1 Q(ω) = F F 1 cos(ω/2ṽ) r ; λy = λy / (4λ 2 f λ2 x )/3 λy = 2 1 Z dω exp 4 2N + 1 4π 8 < 1 : iω (1 + β 2 ω 2 ) h ln Q(ω ) ln Q(ω i ) + β ln Q(ω ) β 2 ω 2 93 = 5. ;

41 c 2005, Michael Marder Theoretical Views Discrete Theory 2 Neo-Hookean Discrete Theory Continuum Approximation v/c Experiment Direct integration, N = 200 Discrete Solution, N = 200 Discrete Solution, N = 2000 Discrete Solution, N = λ y λ y / (4λ 2 f λ2 x)/3

42 c 2005, Michael Marder Theoretical Views Discrete Theory 2 Full Numerical Solution v/cs Continuum Approximation Experiment, λ f = 4.5 Longitudinal wave speed c lx Direct integration, Mooney-Rivlin Discrete Neo-Hookean Solution, N= λ y λ y / (4λ 2 f λ2 x)/3

43 c 2005, Michael Marder Theoretical Views Scaling Theory Subsonic fractures, scaled by energy System height N N = 99 N = 999 N = = λ y 2N + 1)

44 c 2005, Michael Marder Theoretical Views Scaling Theory Subsonic fractures, scaled by energy System height N N = 99 N = 999 N = = λ y 2N + 1) = λ y / 2N + 1

45 c 2005, Michael Marder Theoretical Views Scaling Theory Subsonic fractures, scaled by energy System height N N = 99 N = 999 N = = λ y 2N + 1) λ y

46 Theoretical Views Scaling Theory Subsonic fractures, scaled by energy System height N N = 99 N = 999 N = = λ y 2N + 1) λ y Supersonic ruptures, scaled by extension λ y c 2005, Michael Marder

47 Theoretical Views Scaling Theory Lattice dynamics, constraint of crack along line, explains work of Buehler, Abraham, and Gao (2003), not hyperelasticity 0.8 Crack Velocity v Rayleigh wave speed N = 60 N = 100 N = 140 N = 180 N extrapolated to infinity K I /K Ic c 2005, MichaelPMarder

48 c 2005, Michael Marder Theoretical Views Opening Angles Opening angle comparisons are less successful (simulations involve all measured features of experiment, including all sound speed variations.) 150 Opening angle (degrees) Simulation Experiment λ y

49 Conclusions New Sort of Failure The rupture is supersonic There is no energy release; energy arrives from very near tip. Dissipation and toughening of front behind tip are key physical ingredients; variations in sounds speed are present in real system but play no role in theory. Problem can be solved exactly in continuum and discrete formulations Comparison of rupture speeds with experiment is good. Comparison of rupture angles with experiment less satisfactory. Oscillatory instability and application to other systems remain to be studied. c 2005, Michael Marder