Building Materials IV:

Materials and Mechanics

ifb.ethz.ch/education

Major in Materials and Mechanics

1. S

emes

ter

(HS)

2. S

emes

ter

(FS)

3. S

emes

ter

(HS)

Wood Physics Fundamentals of Engn. Fracture Mechanics

NDT und Health Monitoring

Concrete Material Science

Building Materials and Sustainability Computational Statistical Physics

Durability and Maintenance of RC

Mechanics of Composite Materials

Science and Engineering of Glass and Natural Stone in Construction

Principles of Non-linear FEMBituminous Materials

Materials IV (recommendation) Herrmann; Flatt; Burgert 2V/3KP

Structural Analysis with FEM

Wood and Wood Composites

Kress; 3V/4KP

Frangi, etal; 2V/3KP

Concrete Technology Introduction to Computational Physics Herrmann; 2V/1U/4KP

Martinola, Bäumel; 2V/2KP

Building Physics: Moisture and Durability Carmeliet; Derome2G/3KP

Schindler; 2V/1U/4KP

Burgert,Niemz; 2G/3KP

Elsener; Angst2G/3KP

Kress; 2V/1U/4KP

Hora; 2V/2U/5KP

Herrmann; 2V/1U/4KP

Flatt etal.; 2G/3KP

Partl; 2G/3KP

Wittel; Wangler2G/3KP

Metallic Materials and Corrosion Elsener; 2G/3KP

Habert; 2G/3KP

Niemz, Elsener; 2G/3KP

Technology Research and Development Modelling und Simulation

Shrinkage and Cracking of ConcreteLura;2G/3KP

Wood Elaboration/Machining Niemz2G/2KP

Mechanics of Building Materials Wittel; 2V/3KP

Diverse expertise

Hans Herrmann Robert Flatt Falk WittelIngo Burgert

5

Building Materials IV

7

Fracture Mechanics

Materials IV

• H. Liebowitz (ed.): „Fracture: an advanced Treatise“, 7 Bände, (Academic Press, New York, 1968-72)

• T. L. Anderson, „Fracture Mechanics: Fundamentals and Applications“ (CRC Press,1995)

• Lawn, B.R., „Fracture of Brittle Solids“, (Cambridge Solid State Science Series, 2nd Edn., 1993)

8

Great Molasses Flood Boston, 1919

Tanks

9

Ships

Liberty Ships 1441 damaged ships 1942-46

Schenectady 1943

10

De Havilland Comet

Airplanes

BOAC Flight 781January 10,1954

11

Damage analysis of theDe Havilland Comet

Airplanes

12

Genova, Aug.20, 2018

13

Koror-Babelthuap bridgeSept.26, 1996

Palau Bridge

14

Minnessota I-35W, 2007

Bridges

15

In the US on average each year 25 bridges

collapse.

Bridges

17

brittle ↔ ductil

Rupture behaviour

“spröde” “zäh”

18

Crack Modes

19

Crack Modes• Mode I : symmetric opening

orthogonal to crack surface

• Mode II : shear of crack surface in the direction of crack propagation (plane shear)

• Mode III : shear of crack surface in the direction orthogonal to crack propagation (no plane shear)

20

for tension crack E/10, for shear failure G/10

Theoretical Strength

21

Infinite plate with elliptic hole under uni-axial load

Linear elastic solution

Guri Kolosov1909

circular hole a = b : σt max = 3 σ

maximal tangential stress:

Mode I

22

Deforming elliptic hole into a narrow crack: b → 0 gives a crack of length 2a

Linear elastic solution

Alan Arnold

Griffith1921

crack tip

23

Diverges at crack tip x = a.Scaling law with crack length a !

Griffith‘s solution

24

Polar coordinates and r/a << 1 :

Stress field close to tip

(Sneddon, 1946)

All components diverge as 1/r!

Mode I

25

Mode I

Stress field close to tip

26

The stress intensity factor KI describes for mode I the intensity of the stress field at the crack tip.

Stress intensity factor (SIF)

George Rankin

Irwin1957

27

Comparing with Sneddon eq. gives the SIF for the Griffith crack :

units of the SIF :

Stress intensity factor (SIF)

28

For other geometries and loads consider general form:

Examples for the SIF

where YI is a correction term, which depends on:• the geometry of the intact structural element

• position, shape and size of crack

• external load (e.g. tension, bending)

a is crack length (or half-length for internal cracks)w is characteristic size of structural element

For Griffith crack YI = 1.In general YI is determined experimentally or numerically.

29

Other geometries and loads

example for YI :

30

Mode I :

Mode II :

Mode III :

SIF in three dimensions

I yy I

II yx II

III yz III

aK a Y

w

aK a Y

w

aK a Y

w

31

Unstable brittle crack growth begins, when KI reaches a critical value, namely

the “toughness” KIc .KIc is an experimentally determined material

constant, which depends on temperature, loading velocity and stress state.

The smaller KIc , the larger ist the danger of brittle failure.

Bruchzähigkeit

Irvine‘s fracture criterion

32

Determining the toughness

ASTM E399-83

Cut with a diamond saw a notch of width 300-400 µm.

single-notched 3-point flexural test

33

material KIc [MPa m1/2 ]

concrete 0.2-1.4steel 50aluminium 24 polystyrene 0.7-11glass 0.8wood 0.4-2.5

Toughness values

34

σc is the load, at which, at a fixed crack length, crack growth sets in.

Critical nominal load σc

→

external load

crack length toughness

35

Critical crack length ac

ac is the crack length, at which, at fixed load, crack growth sets in.

→

36

SIF is only relevant within the yellow region. Outside the neglected terms get more important. Closer to the crack tip the stress becomes so large, that the linear elastic approximation is

not anymore valid and one finds as well non-linear elastic as also plastic effects. At the crack tip itself the continuum

description breaks down.

Areas around the crack tip

crack

elastic field determined by K

plastic zone

process zone

37

The plastic zone

= plane stress

= plane strain

Local stress reduction is achieved through plastic flow, but due to the small size of the plastic zone the elastic stress field always remains

present and the SIF continues to the be decisive criterion for failure.

dog-bone modelcrack tip

38

a crack, with the elastic energy U0 without a crack:

Griffith criterion

0 a fU U U U

where Ua is the elastic energy,

which is released by the crack and Uf is the energy, which is needed

to create the crack surface.

Compare the elastic energy U of a uni-axially stretched plate with

39

Griffith criterion0 a fU U U U

where γ is the surface energy density.

Failure occurs, when the gained elastic energy U0 - Ua is larger than the required surface energy Uf , i.e. when U-U0

changes its sign.

2

c

E

a

0dU

dawhere is the energy release rate.

This gives a critical nominal stress of:

Energiefreisetzungsrate

40

The J-integral

‐ for the calculation of the energy flux in crack tip (also for inelastic materials)

- for determining the crack opening- as part of a failure criterion for ductile materials- path independent, exceptions: curved cracks, dynamical crack propagation, cracks with load on crack surface, viscoelastic behavior.

Genady Cherepanov1967

Jim Rice1968

41

The J-integral

path Γ without crack energy = 0path Γ with crack finite energythe crack must be enclosed by the path.

21

:u

J Udx n dsx

2 dimensions

0

~ijijdU elastic stretching energy density

displacement along Γ

element of the arc Γ

elastic energy work by displacing forces

u

ds

42

cJJ

222

2

1)(

1IIIIII K

GKK

EJ

2 / (1 ); E E E E EVZ : ESZ :

021

straight, unloaded crack surface = 0

In the purely elastic case and when Γ is the border of the sample, then one has:J-integral = energy release rate

Closed paths:

criterion for crack growth:

2D

The J-integral

43

Cracks are not straight, because

disorder dominates the fracture process.

Real Materials

models generality

realistic description

damage historydependence on the microscopic parameters

detailed determination of the microstructure and of the stress distribution

guidelines for theevaluation of experimentaldata and simulations

independent of the specific materialproperties

Modeling Real Cracks

statistical informationabout fracture mechanics

Coarse Graining

c

c

homogenisation or „micro-macro transition“

constitutive law

RVElargest defect < size of RVE < gradients

disorder

Representative

Volume Element

Coarse Graining

47

Numerical Techniques

• Finite Element Method

• Lattice Models

• Discrete Element Method

• Fibre bundles

Modeling Fracture Mechanics

• H.J. Herrmann and S. Roux (eds.): „Statistical Models for the Fracture of Disordered Media“ (North-Holland, Amsterdam, 1990)

• B. Bertram Broberg: „Cracks and Fracture“ (Elsevier, 2009)

• M.J. Alava, P.K.V.V. Nukkala and S. Zapperi: „Statistical Models of Fracture“, Advances in Physics, 55, 349 (2006)

• B.R. Lawn and T.R. Wilshaw: „Fracture of brittle solids“ (Cambridge Univ.Press, 1975)

49

Lattice Models

50

Beam modeldiscretized (Cosserat)

description of elasticity

iii Θyx on each site i:

Θlz

cellular solid

53

Breaking a beam

2 max ,i jp F q M M

Consider external shear

Can break by traction only or by bending

56

Breaking thresholdsA beam breaks when:

1,max2

M

ji

F t

MM

t

F

With the two thresholds distributed for instance as

1

1 0 1

1 0

xF F F

x xM M M

P t x t t ,

P t x q t t ,q

In practice we work with fixed external unitary displacements.

All forces, displacements etc are multiplied by a proportionalityconstant λ such that just the weakest bond breaks, i.e. one has totake the smallest λ for which

1,max2

M

ji

F t

MM

t

F

57

Disorder distributions

Uniform distribution

Weibull distribution

realistic description

two parameters:

: scale of strength

: amount of disorder

probability density cumulative distribution

th

)( thp

th

)( thP 1 ( )thp ( )th thP

1

( )

mth

thnP e

nm

58

Experimental verification of Weibull distribution

59

Disorder in Material

m

20

10

5

3

Crack2

eff

m

K

vibration period:

crack speed:

effKsv

m

Crack

atomistic mechanism at crack tip:

Dynamic Tearing

Random lattice of beams

62

Poisson lattice

63

Moukarzel construction

64

Regularized random lattice

Moukarzel

65

Breaking a macroscopic beam

67

3-Point Bending Test

Eric Schlangen

experiment

square lattice 100 × 100

Poisson lattice

Moukarzel lattice 30 × 30

Moukarzel lattice 40 × 40

Hybrid ModelingProblem: detailed model of the entire sytem is computationally too large,

but elastic embedding is important.

Strategy: Represent regions of large activity by detailed modeling.

Coupling and embedding of detailed models into coarser continuum models.

Examples:MD / lattice model

lattice model/ continuum FEM

DEM / FEM

Hybrid ModelingOverlapping domain

Edge-to-edgeHandshake Methods:

Kinematic coupling: sharp interface

Macroscopic, Atomistic and Ab initio Dynamics (MAAD)

(Abraham et al. 1998)

CGMD (Coarse Grained Molecular Dynamics)

Coupling to stiffness matrix

Virtual nodes in each domain

Overlap region: continuous transition

Bridging Scale Method (Liu)

Use Green functions to predict positions of atoms.

Strong dependence on the crystal structure

Bridging Method (T. Belytschko & S. Xiao)

Identical displacements in both domains

Spatial weighting of the energy contributions (Lagrange multipliers).

No hypothesis about the structure of the region

70

Crack Growth

notched sample under constant tension

• Morphology, speed and elastic waves are functions of the load and the material properties.

71

• crack speed larger than critical velocity

instability branching surface roughness

• non-linear material behaviour in process zone (PZ)

• beam lattice model with dynamics intrinsic length scale in PZ

• stabilization of crack propagation by wandering simulation windows load controls crack speed

?influence of disorder

?influence of crack speed

•bla2

H. Andersson (1969)

Fast Crack Growth

F K

Dynamic Damage in PMMA

FINEBERG&MARDER 1999

crack instability finite crack speed

0.70.2y x

critical speed vc < cR

universal power law for shape of branches

len

gth

of

bra

nch

[m

]Dynamic Damage in PMMA

Wave Propagation after Impact

longitudinal waves(compression waves)speed cl

transverse waves(shear waves)speed ct

Rayleigh waves(surface waves)speed cR

Elastic Waves

Point of View of Continuum

MOTT 1948dW dU dT

GdA dA dA

( )e od U dU

da da

GRIFFITH-criterion:

kinetic energy

Damage mechanics does not explain… why the RAYLEIGH-wave speed cR is not attained,

why instabilities appear,

why cracks branch and have rough surfaces,

what really happens in the process zone und its influence on G.

Yoffe-instability: The Moving Griffith Crack

The stress field of the crack interacts with stress waves that are emitted by the growing crack.

The stress maximum jumps from 0°to 60° when the crack accelerates.

• Cracks become unstable at 66% cR.

E. Yoffe: Phil. Mag. Lond. 42 (1951)

Point of View of Continuum

Discrepancies Theory-Experiment-Simulation

• Observed branching velocity is 30%cR, and thus much smaller than predicted by Yoffe (66%cR). Formation of microcracks in front of crack tip interaction of microcracks with the main crack (Ravi-Chandar, Knauss 1984)

• The observed branching angle is smaller than 60°.

kinetic energy

deformation energy

Energies

= 0.027

m = 10

a = 0.7

m = 10

a = 0.7

( )min( ( ))

( ) (1 )( min )

dm dndn

wdm

w x xg x e

x x

•300x200 mass points

•switch in vertical and

horizontal direction

•add new elements

•damping at system boundaries

Shifted Observation Windows

82

Detail

83

Disorder in Material

m

20

10

5

3

connection of microcracks

Speed of Main Crack

vertical branching>system size

3.6

<vcrit

85

Speed of Branching Cracks

spee

d [

su]

time [su]

main crack

side branch

a=0.9; pre=1.8

89

0.70.2y x

• universal form of branches

• cascading cracks at weak disorder

Branch Morphologiesm

4

8

12

1000

Sharon, Fineberg (1998)

(x, y, z) (αx, αy, αζz)

Rescaled Range methodHurst (1965)

0.5-0.75 Bouchaud (2006)

roughness – increases with speed

- decreases with disorder

Crack Roughness

Mechanics and Failure of Fibre

Reinforced Composites (FRC)

Fibrous Materials

glas fibre reinforced plastic steel fibre

reinforced concrete

silicon carbide reinforced glassceramics

Natural Fibres: Wood

parallel bundle of small pipes

Anisotropic Creep of Wood

distribution of sample strengths follows a

Weibull distribution withm ≈ 9

• scaling as function of size

• tension experiment and modeling of softwood

1 expm

P

Weibull distribution

viscoelastic material

Fibre Reinforced Composites

fibre reinforced composites: two components

fibres

- carry most of the load

matrix

- carries nearly no load- ensures interaction .. between fibres

excellent mechanical properties

C-SiC

96

Mechanics of FRC

Stiffness and strength of the fibres

is much higher than that of matrix.

Typical fibres are made of glas, steel,

carbon and polymers.

Typical matrix materials are resins, rubber, ceramics and concrete.

97

Orientation of Fibres

98

Tissue

rowlings before impregnation with resin

99

Long Fibres

When the fibres completely span the sample

matrix and fibres are subjected to the same deformation and thus the stiffer fibre carries a larger proportion of the load

. → „load transfer“

100

Long Fibres

For the total tensile stress σA one

has the following mixing rule :

where σm is the stress in the matrix,

σf the stress in the fibre and f

the „enhancement factor“.

A 1m ff f

101

Short Fibres

shear lag model :

The transfer of force on the fibre by the matrix

happens through shear at the interface.

102

Short Fibres

The tensile stress in the center of the fibre

decays proportionally with the shear force

at the interface, which has a distance

r = fibre radius from the center of the fibre.

f id

dx r

103

Short fibres

where Ef is the Young modulus of the fibre and Em and νm

the Young modulus and the Poisson ratio of the matrix.

with

The shear at the interface can be approximated

by a hyperbolic function and one obtains then

the tensile stress inside the fibre:

H.L.Cox, 1952

f f 1 cosh coshE nx r ns

s is the ratio between length and diameter of the fibre.

1 2

f

2

1 ln 1m

m

En

E f

104

Short fibres

and for shear:

Fibres must be longer than lc .

i f sinh cosh2

n nxE ns

r

lc is the critical

length

Realistic Shear Stress

stress induced fieldof optical phase shifts

calculated field ofphase shifts

FE-stress field

equations of photoelasticity

d

nSIIIsps

sps

132

3311 4)(

n phase jumps

d width

S photoleastic constant

photoelastically active component is thedifference between first and second invariant of the stress tensor

analogy with heat conduction:integration of the phase rotation over the width (ray tracing)

Photoelastic Measurements

107

Transversal stresses

108

Failure of the matrix

transverse crack through glas reinforced polyester resin

cohesive elements to model debonding and the failure

modeling

Stress Calculation with FEM

• cohesion

• friction

110

Interfaces

• tearing of fibres under tension

• failure of the matrix (under tension or shear)

• debonding = detachment at the interface

111

Failure of Fibre Reinforced Composites

112

Types of failure

113

Tearing of a fibre

glas

kevlarcarbon

fibre

shear effects

Single fibre pull out

successive fibre failures

0° 10°

With Finite Elements one can calculate the stress field and from it the images of phase shifts.

Photoelastic Image of Stress Field

stresses along fibre

debonding

phase jumps:Debonding and Rupture

phenomenological•empirical expressions•macroscale

PUCK

•statistical approach

probabilisticmicromechanical•physical basis•microscale

combined:

Fibre composites: Failure models

FBM FBMlattice models

Fibre Bundle Model (FBM)

discrete set of parallel fibreson a regular lattice

perfectly brittle behaviour

range of load redistributionE

th

two parameters: E and th

distribution of failure thresholds

( )thP

two extremes GLS LLS

load parallel to fibresF

Daniels 1941

Macroscopic Behaviour of GLS

constitutive equation: 0 [1 ( )]P E E fraction ofintact fibres

load upon asingle fibre

for a Weibull distribution

0

mEne E

c

c

cc

macroscopic strength

strain controlled loading

Breaking process

GLS = global

loadsharing

Microstructure of Damage

no spatial correlations growing cracks

Global Load Sharing (GLS) Local Load Sharing (LLS)

Microscopic Damage Process

avalanches of breaking fibres

load redistribution

stress controlled loading

minth 1

newN

N

2 5 .( ) ~D

avalanche size distribution

2.5

acoustic emission

Experiments with Packings of Spheres

inversion of the continuous damage model to model force chains in

granular packings

8 acoustic sensors

Acoustic Emissionsize distribution of acoustic signals

experimental data (circles) and exponential fit 1.15±0.05 (solid line)

simulation results (dots) and analytical expression

with exponent -1

Extensions

FBM

range of interactionsload transfer function with adjustable parameter

failure criteriongradual degradationof fibre strength

time dependencetime dependent deformationcreep rupture

cyclic loadingdamage accumulationhealing

Creep Rupture

creep experiment

- several possible . mechanisms- material dependence

deformation - time

deformation rate

acoustic emission

Viscoelastic Fibres: Kelvin Element

E 0

/0

/0 1)( EtEt eeE

t

two parameters: E and

E

time evolution

Damage Evolution in Wood

Rupture of Bundles

E

P(ε) breaking threshold

load distribution

strain controlled breaking of fibres

E

P

)(10

coupling between breaking and viscoelasticity

in a global load sharing framework

damage enhanced creep

Analytic Solution

two regimes :

-only partial failure

-no stationary state

-macroscopic stationarystate

-infinite life time

-monotonically increasingdeformation

-global failure at finite .. time

0 c

0 c

Approaching the Critical Point

: relaxation time

relaxation by decreasingbreaking activity

2/10 )( c

/te

0 c

universal power law divergence

time to failure

global failure at finite time

continuous transition

2/10 )( cft

0 c

ft

universal power law divergence

Approaching the Critical Point

Simulation Results

Diverges with power law.

time of last breaking

life time~

uniform distribution

N= 107 fibres

cft 0

uniform distribution

Weibull distributionwith m=2 and m=5

good agreementwith analytic predictions

2/10 )( cft

Simulation Results

Distribution of inter-event times

strain-time diagrame

P is uniform

c 0 c 0

1,i it t

Role of Load Distribution

E

load transfer function

completely global completely local

1add

ij

Zr

0

0.0 0.5 1.0 1.5 2.0

Strain [%]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

F/N

gamma=0gamma=3gamma=9

Role of Load Distribution

Size Scaling

.ln

)( constN

Nc

simulation results

Continuous Damage Model

• Multiple failures k up to a value kmax are allowed.

• After each failure event the stiffness of the failed fibre changes as Ei` = aEi .

• The new failure threshold can be the same as before (quenched disorder) or sampled again from the disorder distribution (annealed disorder).

PkaFP ;,,)1( max

quenched disorder

annealed disorder

if

if

id

3

id

1

id

2

id

Continuous Damage Model

constitutive equations: annealed disorder

•after one restructuring event

•after two restructuring events

•after kmax restructuring events

PaP 1

aPPa

aPPaP

12

111

1

0

1

0

1

0

max

max

max

1k

i

ii

ki

j

jj

ii

k

i

i aPaaPaPa

Continuous Damage Model

a) hardening,

b) macroscopic failure: set residual stiffness to zero after k* = kmax failure events.

constitutive behaviour for a=0.8 and different

values of kmax ; quenched disorder

Continuous Damage Model

Damage in Fibre Reinforced Compositeslaminate of crossed layers [0,90]n

micro-cracks (stress whitening)

fibre detachment (debonding)

fissures in transverse layers

microdelaminations

fibre failure

failure of fibre bundles

rupture

laminate of crossed layers [+45,-45]n

strong non-linearity

mechanisms interact

complex failure patterns

big dispersion in strength

ductile brittle failure, depending on which

mechanism is activated

Damage in Fibre Reinforced Composites

Failure Criteria for Simultaneous Mechanisms

failure modes:(1) matrix failure under tension

(3) fibre-matrix shear failure

Yt

Yc

Xc

Sc

transverse tensile strength of laminate

compressive strength of matrix

buckling strength of fibre

shear strength of individual layer

(2) fibre buckling

(4) matrix failure under compression

•spontaneous reduction in stiffness each time one criterion becomes effective•individual criteria are only coupled through the deformation

•model of Chang, Lessard 1991

pressure

tension

22 2( ) ( )2 1 2 2

22 ( ) ( )2 1 2 2

2 2

2 1 2( )

2

1 1

11

12 1

T

T

C

C

Yp p

S S Y S

p pS

Y

Yp S

2 11 2

1 1

2 11 2

1 1

11

11

FF

T F

FF

C F

mE

mE

mode A

mode B

mode C

fib

re f

ailu

refa

ilure

bet

wee

n f

ibre

s

Puck's Theory

Alfred Puck

phenomenological model based on the various mechanisms

crack planecrack between fibres

crack between fibres

smeared reduction in stiffness

mode A

mode B

mode C

Puck's Theory

• good agreement with experiments and other theories• winner of the WWFE 2D

Puck's Theory

Laminates

162

Structure of laminates

structure of a filter of

laminated polyester resin

with glas fibre fabricaluminium – GVK for

the hull of the A380

GF-reinforced damping PE floor covering

Pyrolysis of C/hydroxylbenzene Laminates

fibre degradation

crack pattern complete

debonding / microcracks /segmentation cracks

stress-free /beginning of pyrolysis

compression of the matrix through the fibres

stress-free at tempering temperature

Delamination

Delaminationsprobleme

delamination with different modes

simulation of delamination inglued compounds

considerations on the level of the structural element

Delamination

delamination zone

prediction about the advance in delamination by comparing the energy release rates

problem: combination of the three modes at the delamination front

Benzeggagh und Kenane:

BK-law

Wu und Reuter:

Power-law

Reeder-law

Crack Growth along Specified Paths: VCCT