MECHANICS OF 2D MATERIALS

Transcription

1 MECHANICS OF 2D MATERIALS Nicola Pugno Cambridge February 23 rd, 2015

2 2 Outline Stretching Stress Strain Stress-Strain curve Mechanical Properties Young s modulus Strength Ultimate strain Toughness modulus Size effects on energy dissipated

3 3 Linear Elastic Fracture Mechanics (LEFM) Stress-intensity factor Energy release rate Fracture toughness Size-effect on fracture strength Quantized Fracture Mechanics (QFM) Strength of graphene (and related materials) Bending Flexibility Bending stiffness Elastic line equation Elastic plate equation

4 4 Exercises by me 1. Apply QFM for deriving the strength of realistic thus defective graphene (and related 2D materials); 2. Apply LEFM for deriving the peeling force of graphene (and related 2D materials). Exercises by you 1. Apply LEFM for measuring the fracture toughness of a sheet of paper; 2. Apply the Bending theory for calculating the maximal curvature before fracture.

5 5 Stretching Fiber under tension F l 2 Stress: σ = F A l A Strain: ε = l l F l 2

6 6 Stress-strain curve: Signature of the material and main tool for deriving its mechanical properties x = failure σ x If the curve is monotonic, a force-control is sufficient. C A You can derive AB in displacement control, CD in crack-opening. The dashed area is the kinetic energy released per unit volume under displacement control. D x B x ε

7 7 Mechanical properties σ σ max dσ dε 0 E Young s modulus 1 E E V x σ max = maximum stress Strength ε u = ultimate strain ε u ε ε u 0 σdε = E V = Energy dissipated per unit volume or Toughness modulus. E = Fdl V = Al E V = F dl A l = σdε The toughness modulus is a material property only for ductile materials.

8 8 Size-effects The post-critical behaviour is size-dependent especially for brittle materials. σ 0 l l 0 ductile l brittle ε

9 9 For brittle materials E has no meaning: instead of the toughness modulus we V have to use the fracture energy. E = G c A Fracture G c fracture energy or energy dissipated per unit area σdε E V = G ca la = G c l The reality is in between: energy dissipated on a fractal domain: E V ld 3 3 l = V D is the fractal exponent, 2 D 3

10 10 Fracture Mechanics σ K I Stress intensity factor 2a σ tip r σ tip ~ K I 2πr constant Elastic solution is singular: We cannot say σ f stress at fracture σ f : σ tip = σ max

11 11 Linear Elastic Fracture Mechanics (LEFM) Griffith s approach G = dw da W = E L External work Energy release rate Crack surface area Total potential energy Stored elastic energy Crack propagation criterion: G = G C G c fracture energy G = K I E Elastic solution K I only for a function of external load and geometry K I = K IC K IC = G C E Fracture toughness

12 12 K I values reported in stress-intensity factor Handbooks E.g., infinite plate (i.e. width crack length ~ graphene) K I = σ πa Strength of graphene with a crack of length 2a K I = σ πa = K IC σ f = K IC πa Assuming statistically a l structural size: σ f l 1 2 Size effect on a fracture strength larger is weaker problem of the scaling up.

13 13 Paradox σ f a 0 With graphene pioneer Rod Ruoff we invented Quantized Fracture Mechanics (2004). The hypothesis of the continuous crack growth is removed: existence of fracture quanta due to the discrete nature of matter.

14 14 Related papers: N. M. Pugno, Dynamic quantized fracture mechanics, Int. J. Fract, 2006 N. M. Pugno, New quantized failure criteria: application to nanotubes and nanowires, Int. J. Fract, 2006

15 15 Quantized Fracture Mechanics (QFM) G = W A A = fracture quantum of surface area = qt Quantized energy release rate G = G C Fracture quantum of length Plate thickness G = W A = G = dw da K I 2 E da A W = G = G c GdA = 1 A K I 2 E da K I 2 E da = K IC E 2 Quantized stress-intensity factor (generalized): K I = 1 A K I 2 da = K IC

16 16 Monolayer graphene and examples C. Lee, X. Wei, J.W. Kysar, J. Hone, Measurement of the elastic properties and intrinsic strength of monolayer graphene, Science, 2008 : Young s modulus E = 1 TPa Intrinsic strength σ int = 130 GPa Ultimate strain ε u = 25 % Monolayer graphene hanging on a silicon substrate (scale bar: 50µm) Tensile test on macro samples of graphene composites Tennis racket made of graphene (Head )

17 17 Carbon nanotubes Min-Feng Yu, Oleg Lourie, Mark J. Dyer, Katerina Moloni, Thomas F. Kelly, Rodney S. Ruoff, Strength and Breaking Mechanism of Multiwalled Carbon Nanotubes Under Tensile Load, Science, 2000: The singlewalled carbon nanotubes (SWCNTs) presents: Young s modulus E =1000 GPa Tensile strength σ = 300 GPa Ultimate strain ε u = 30 % The MWCNTs breaks in the outermost layer ( sword-in-sheath failure), Young s modulus E =250 to 950 GPa Tensile strength σ = 11 to 63 GPa Ultimate strain ε u = 12 % Multiwalled carbon nanotubes (MWCNTs)

18 18 Exercise 1 E.g., infinite plate K I = σ πa K I = 1 qt a+q σ 2 πat da a = σ π 2q a + q 2 a 2 = σ π a + q 2 Strength of graphene K I = K IC σ f = K IC a σ f Unstable crack growth π a + q 2

19 19 q 0 Correspondence Principle QFM LEFM a 0 σ f = σ ideal σ ideal = K IC π q 2 q = 2 π K IC 2 σ ideal 2 q σ f = K IC π a + q 2 Very close to predictions by MD and DFT σ ideal ~ 100 GPa σ ideal ~ E 10 E~1 TPa K IC ~ 3MPa (QFM 2004) m Very close to experimental measurement (2014) P. Zhang, L. Ma, F. Fan, Z. Zeng, C. Peng, P. E. Loya, Z. Liu, Y. Gong, J. Zhang, X. Zhang, P. M. Ajayan, T. Zhu & J. Lou, Fracture toughness of graphene, Nature, 2014

20 20 Peeling of graphene Exercise 2 l θ δ F E E 0 δ = l l cos θ G = dw da = dl da G = d Fl 1 cos θ bdl L = Fd da = bdl F 1 cos θ = b G = G C F C = bg C 1 cos θ F C strongly dependent on θ N. M. Pugno, The theory of multiple peeling, Int. J. Fract, 2011

21 21 Exercise 3 Apply LEFM for measuring the fracture toughness of a sheet of paper with a crack of a certain length in the middle:

22 22 Bending of beams dφ X Bending theory b h R X Bending Moment M x M x dz dφ X 2 y εdz 2 y dφ X 2 z dz σ z = M x y Ky M I x = σ z ybdy = KI x x h 2 h 2 Moment of inertia I x = bh3 12 χ x = 1 R x = dφ x dz = ε z y = σ z Ey = M x I x E χ max = ε z,max h 2

23 23 Exercise 4 Derive the maximal graphene curvature. For graphene h = t 0.34 nm χ max huge flexibility is more a structural than a material property φ x = dv dz d2 v dz 2 = M x I x E Load per unit length M T N dz q N T M dt dz = q dm dz = T d4 v dz 4 = q I x E Elastic line equation Load per unit area For plates: 4 W = q D Et 3 Elastic plane equation Bending stiffness D = 12 1 ν 2