Probabilistic and dynamic contact and fracture models for

quasi-brittle fracture

Reza Abedi

Mechanical, Aerospace & Biomedical Engineering

University of Tennessee Knoxville / Space Institute

1

Integrated Fracture/Contact model

Mesoscopic interface subdivisions

2

Riemann problem set-up

3

Riemann solutions

4

Numerical verification:

Identical bars

Unlike other solutions,

SDG results are not overly

damped and are free of

numerical oscillations and

overshoot / undershoot5

Brake simulation

Contact mode transitions (high slip velocity)

Color: strain energy; Height: velocity

click to play movie

YouTube link

Depending on sliding velocity there are different mode transitions:

6

A rate-dependent interfacial damage modelStructural

Health Monitoring

7

Interfacial rate-dependent damage

model

Fineberg & Marder 1999

Conical (parabolic) marking on the crack surface

Ravi-Chandar, Knauss 1984

Motivated by mesoscale features:

8

Rate-dependent damage model:

RegularizationD interfacial damage parameter

Relaxation time is the maximum damage rate:

Recently a second term (kinematic-based term) is added

to damage evolution equation.9

Rate-dependent damage model:

Static damage

New effective stresses are used

recently that beside being realistic for

tensile dominant fracture problems

work for fracture in compressive

mode.

10

Rate effect:

influence of loading rate and k

𝑘 = 1

Both terms are equally important

𝑡/ ǁ𝜏

ሶ𝜔

ǁ𝜏

Increases loading rate:

1. Increases maximum stress

2. Increase or keep almost constant displacement jump at full damage

3. Increase or keep almost constant time at full damage

11

Influence of loading rate and k on

fracture energy

Regardless of how values are normalized

(w.r.t. stress scale 𝜏𝑠or velocity scale 𝜏𝑣):

1. At higher loading rates fracture energy increases.

2. At loading rates fracture energy is insensitive to

loading rate.12

Combination of contact and fracture models Health

Monitoring

13

Contact/damage examples

Color: log(strain energy); Height: velocity

click to play movieYouTube link

cyclic loading for a stiff circular inclusion

with an initial defect

Mode I: cyclic loading

YouTube link14

Computational Fracture Mechanics:SDG methods powerful adaptive operations in spacetime align element

boundaries with any crack propagation direction.

15

Solution-dependent crack path

Element boundaries are aligned with arbitrary propagation direction by

spacetime meshing operations.

Unlike XFEM, no need to introduce discontinuous features within elements.

Maximum effective stress governs direction.

Arbitrary crack extension:

Example: Mid-crack

propagation under

dynamic loading.

16

Probabilistic models for dynamic fracture:

Structural Health Monitoring• Spatial inhomogeneity in material property is important in capturing

realistic fracture patterns for quasi-brittle materials.

• Modeling statistical variations of material properties is essential in

capturing variations in macroscopic response (e.g. ultimate load,

absorbed energy, etc.)

17

1. Why randomness is important?

Al-Ostaz 1997: Epoxy sheets with holess

A. Different fracture patterns for the

same loading / geometry Quasi-brittle fracture of cellular ceramic

structures (Genet, Ritchie 2014)

Specimen size

Mean strength

Variation in strength (2x

standard deviation)

B. Mean strength and variation in

fracture strength decrease as specimen

size increase

J Kozicki and J Tejchman (2007)

(in the context of concrete response)

C. High variability in ultimate strength /

fracture energy

Area under the

curve

18

2. Why spatial inhomogeneity is

important?

Uniform fracture strength:

All points fail at the same point!

Quasi-static loading

Uniform

tensile

loading

19

Our models for stochastic fracture nucleation:

1. Phenomenological model based on Weibull fracture model

20

Implicit Representation of Material Randomness

Implicit approach: A random crack nucleation model represents

rock inhomogeneities and randomness

Weibull model is used for probabilistic crack

nucleation

• Fracture strength is

given by a Weibull

distribution.

• Strength at each

vertex is sampled by

an area modulated

Weibull model.

Area of the triangles

around vertex: A

21

Comparison of deterministic and random

(Weibull-based) fracture models

And the fracture pattern from the two models deviates even further at higher loadings:

Loading rate: tr = 10 ms Propellant rock fracture (high rate loading)

Deterministic model Weibull model

More microcracks are observed with deterministic model!

22

Comparison of details of fracture

patterns between the two models

Uniform strength

Uniform strength causes very

close microcracking as all points

experience high stresses

Nonuniform strength

(using Weibull model)

More realistic response23

Our models for stochastic fracture nucleation:

2. Stochastic volume elements (characterization) and Karhunen–Loève

(random field realization)

24

Current work (Abedi):

Stochastic fracture modeling

Statistical

Volume Element

(SVE)1

2

4 Random field realizations

5 Fracture Simulations

Random field statistics

3

1. By using SVEs material inhomogeneities & sample to

sample variations (randomness) are preserved.

2. Still no need to resolve all microscale details!

A very efficient and

accurate model for

fracture modeling25

Sample stochastic fracture results

SVE1x1 SVE8x8

Realizations for fracture strength based on

SVE2x2

Window sizes uses for statistical volume elements

Uniform fracture strength

Fracture

patterns

under

uniform load

in horizontal

direction

Very unrealistic fracture pattern with uniform fracture strength model

26

Acknowledgments

• University of Illinois at Urbana-Champaign (past and present):

Robert Haber, Jeff Erickson, Michael Garland,

Scott Miller, Boris Petrakovici, Alex Mont, Shuo-Heng Chung,

Shripad Thite, Yong Fan, Morgan Hawker, Jayandran

Palaniappan, Yuan Zhou.

• University of Tennessee:

Philip Clarke, Omid Omidi (post-doc), Bahador Bahmani

• University of Saint Thomas (Minnesota):

Katherine Acton, Sarah Baxter

• Others:

Olivier Allix, Karel Matous.

27