Effects of material randomness on static and dynamic fracture

M. Ostoja-Starzewski and G. WangDept. 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

2

A

U

A

WG

Peeling a beam off a substrate

determine the critical crack length and stability

4

crack stability:

equilbrium stable 0

equilbrium neutral 0

equilbrium unstable 0

)(

2

2

A

5

Strain energy release rate: 2

A

U

A

WG

Peeling a random beam off a substrate

γ = random field

U = random functional

},);,({ Xxx

6

stiff inclusions in soft matrix

2)(

)(

10m

i

C

C

4)(

)(

10m

i

C

C

7

Dead-load conditions (for Euler-Bernoulli beam):

a

dxIE

MaU

0

2

2)(

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)

aB

UG

]},0[,);,({ axxE

8

U is a random integral

upon ensemble averaging:

In conventional formulation of deterministic fracture mechanics, random heterogeneities E′(x,ω) are disregarded ( )

? ?

a

dxxIE

MEaU

0

2

),(2))(,(

a

dxxEEI

MEaU

0

2

)],('[2),(

constEE

a

dxEI

MEaU

0

2

2),(

),(),( EaUEaU ),(),( EaGEaG

9

Note: random field E is positive-valued almost surely

by Jensen's inequality

EE

11

),(2

1

22),(

0

2

0

2

0

2

EaUdxIE

Mdx

EI

Mdx

EI

MEaU

aaa

10

Define:

G in hypothetical material:

G properly averaged in random material:

with side conditions

aB

EaUEaG

),(

),(

aB

EaUEaG

),(

),(

0),0( EU0),0( EU

),(),( EaGEaG

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:

aB

EaUEaG

),(

),(

aB

EaUEaG

),(

),(

0),0( EU0),0( EU

ExE ),(

),(),( EaGEaG

12

Define:

stress intensity factor in hypothetical material:

stress intensity factor properly averaged in random material:

),( EaK

),( EaK

),(),( EaKEaK

),(),( EaJEaJ

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:

0),(' xE

),,(),,(),,(1111* EaGEaGEaG

EE 11

14

Fixed-grip conditions:

G can be computed by direct ensemble averaging of E (and μ)

4

2

2

9

22 Ba

EIu

a

P

B

u

a

P

B

uG

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:

Pmixedu GGG

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.

Pmixedu GGG

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:

MPG

PG

MuG

uG

MPPu GGG MPMuu GGG

18

Stochastic crack stability:

equilbrium stable 0

equilbrium neutral 0

equilbrium unstable 0

)(

2

2

A

wide scatter of

random critical crack length!

[ASME J. Appl. Mech., 2004]

19

same result via random Legendre transformation

))(,())(,(* EaUMEaU

),(),(*11* EaGEaG

20

Observe

)()/1( EaEa cc

ca

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

21Vdd

δ

dδ

efftδ

t

δ

tRR

CCCC

CSSS)(SC

'

'

1

1111

1

'

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, 1981.

• D. Greenspan, Particle Modeling, 1997.

• R.W. Hockney and J.W. Eastwood, Computer Simulation Using Particles, 1999.

26

(a) potential energy (b) interaction force

qp r

H

r

G )1(, pq

27

• potential • interaction force

(cntd.)

28

Dynamic fracture simulations

[Comp. Mat. Sci., 2005]

29

Uniform stretching in y-direction at 0.5m/s

homogeneous material heterogeneous 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 material heterogeneous 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

2106.2 T(b) s

(c) s210007.3 T

(d) s210011.3 T

(p,q) = (3,5)

35

(Continued)

(a) T = 0.0 s.

210477.1 T(b) s

(c) s210485.1 T

(d) s210489.1 T

(p,q) = (5,10)

36

(Continued)

(a) T = 0.0 s.

210024.1 T(b) s

(c) s210026.1 T

(d) s210030.1 T

(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: ),('),( xCCxC

Is it isotropic?

Is it uniquely defined?

41

Basic model: ),('),( xCCxC

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– …