Effects of material randomness on static and dynamic fracture M. Ostoja-Starzewski and G. Wang Dept. Mechanical Engineering | McGill Institute for Advanced Materials McGill University Montreal, Canada

2 1.Quasi-static fracture mechanics of random micro-beams 2.Dynamic fracture of heterogeneous media

3 Strain energy release rate: = material constant U = elastic strain energy of a homogeneous material Peeling a beam off a substrate determine the critical crack length and stability

4 crack stability:

5 Strain energy release rate: Peeling a random beam off a substrate γ = random field U = random functional

6 stiff inclusions in soft matrix

7 Dead-load conditions (for Euler-Bernoulli beam): where a = A/B, B = constant beam (crack) width From Clapeyron’s theorem: Note: randomness of E arises when Representative Volume Element (RVE) of deterministic continuum mechanics cannot be applied to a micro-beam need Statistical Volume Element (SVE) micro-beam is random: (wide-sense stationary)

8 U is a random integral upon ensemble averaging: In conventional formulation of deterministic fracture mechanics, random heterogeneities E′(x,ω) are disregarded () ? ?

9 Note: random field E is positive-valued almost surely by Jensen's inequality

10 Define: G in hypothetical material: G properly averaged in random material: with side conditions

11 Define: G in hypothetical material: G properly averaged in random material: with side conditions G computed by replacing random micro-beam by a homogeneous one ( ) is lower than G computed with E taken honestly as a random field:

12 Define: stress intensity factor in hypothetical material: stress intensity factor properly averaged in random material:

13 Remark 1: With beam thickness L increasing, mesoscale L/d grows deterministic fracture mechanics is then recovered Remark 2: Results carry over to Timoshenko beams:

14 Fixed-grip conditions: G can be computed by direct ensemble averaging of E (and μ)

15 Mixed-loading conditions:... both load and displacement vary during crack growth no explicit relation between the crack driving force and the change in elastic strain energy. … can get bounds from G under dead-load and G under fixed-grip:

16 Mixed-loading conditions:... both load and displacement vary during crack growth no explicit relation between the crack driving force and the change in elastic strain energy. … can get bounds from G under dead-load and G under fixed-grip: Note: in mechanics of random media, when studying passage from SVE to RVE, energy-type inequalites are ordered in an inverse fashion: kinematic (resp. force) conditions provides upper (resp. lower) bound.

17 Mixed-loading conditions for Timoshenko beam... four possibilities: P and M fixed: P and θ fixed: u and M fixed: u and θ fixed:

18 Stochastic crack stability: wide scatter of random critical crack length! [ASME J. Appl. Mech., 2004]

19 same result via random Legendre transformation

20 Observe 1.Potential energy Π(ω) is sensitive to fluctuations in E, which die out as L/d → ∞ (L beam thickness, d grain size) 2.Surface energy Γ(ω) is sensitive to fluctuations in γ, but randomness in γ independent of L/d cracking of micro-beams is more sensitive to randomness of elastic moduli than cracking of large plates 3.Under dead-load conditions: and small random fluctuations in E and γ lead to relatively much stronger (!) fluctuations in

21

22 1.Fracture mechanics of micro-beams 2.Dynamic fracture of heterogeneous media

23 Particle modeling of fracture/crushing of ores in comminution

24 From molecular dynamics (MD) to particle modeling (PM) Need to model dynamic fragmentation of heterogeneous materials having partially known (or unknown) interatomic potentials, e.g. ores Model should have - reasonable execution time - without complexity of FE schemes - allow asymmetry in tensile vs. compressive response - grasp nonlinear response PM is a lattice of quasi-particles interacting via potentials derived from MD lattice - based on equivalence of mass, energy, elastic modulus, and strength - dynamics computed via leap-frog scheme

25 D. Greenspan, Computer-Oriented Mathematical Physics, D. Greenspan, Particle Modeling, R.W. Hockney and J.W. Eastwood, Computer Simulation Using Particles, 1999.

26 (a) potential energy (b) interaction force

27 potentialinteraction force (cntd.)

28 Dynamic fracture simulations [Comp. Mat. Sci., 2005]

29 Uniform stretching in y-direction at 0.5m/s homogeneous materialheterogeneous material (p,q) = (3,5)(p,q) = (3,5) blue phase is 1% stiffer

30

31 Uniform stretching in y-direction at 0.5m/s homogeneous materialheterogeneous material (p,q) = (7,14)(p,q) = (7,14) blue phase is 1% stiffer

32 Observations on plates of homogeneous and heterogeneous (two-phase) materials similar behavior for stiff (7,14) and soft (3,5) materials at the onset of crack propagation the larger is the (p,q), the faster is the crack propagation crack propagation speed increases in presence of material randomness for lower (p,q): crack trajectory is initially straight, and then zigzags; for higher (p,q): coalescence of many cracks

33 Crack patterns in 7 nominally identical epoxy specimens under quasi-static loading [Al-Ostaz & Jasiuk, Eng. Fract. Mech., 1997]

34 (Continued) (a) T = 0.0 s (b) s (c) s (d) s (p,q) = (3,5)

35 (Continued) (a) T = 0.0 s. (b) s (c) s (d) s (p,q) = (5,10)

36 (Continued) (a) T = 0.0 s. (b) s (c) s (d) s (p,q) = (7,14)

(Cnd.) (a) 3D (b) Fx (c) Fy (d) Fz

38 (Cnd.) (a) 3D (b) Fx (c) Fy (d) Fz

microscale mesoscale macroscale

40 Basic model: Is it isotropic? Is it uniquely defined?

41 Basic model: Is it isotropic? Is it uniquely defined? Applications: random field models stochastic finite elements waves in random media FGM …

42 paradigm: FGM mesoscale property is anisotropic, and non-unique, bounded by Dirichlet and Neumann b.c.’s [Acta Mater., 1996]

43 conclude spatial inhomogeneity (gradient) prevents isotropy of approximating continuum it implies anisotropy of C tensor

44 Conclusions Michell truss-like continuum cannot really be attained RVE may be bounded by mesoscale responses –hierarchies of bounds involve variational principles, but are qualitative –quantitative results follow from computational mechanics Applications: –linear elastic microstructures –inelastic microstructures Examples –fiber-reinforced composites –random mosaics –cracked solids –smoothly inhomogeneous materials –…