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

Weibulldistribution

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

6

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.

7

Orientation of Fibres

8

Tissue

rowlings before impregnation with resin

9

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“

10

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 m f1f f

11

Short Fibres

shear lag model :

The transfer of force on the fibre by the matrix

happens through shear at the interface.

12

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

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

cohesive elements to model debonding and the failure

modeling

Stress Calculation with FEM

16

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

17

Short fibres

and for shear:

Fibres must be longer than lc .

i f sinh cosh2

n nxE ns

r

lc is the critical

length

• cohesion

• friction

18

Interfaces

• tearing of fibres under tension

• failure of the matrix (under tension or shear)

• debonding = detachment at the interface

19

Failure of Fibre Reinforced Composites

20

Types of failure

21

Tearing of a fibre

glas

kevlarcarbon

fibre

successive fibre failures

0° 10°

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

Photoelastic Image of Stress Field

shear effects

Single fibre pull out

stresses along fibre

debonding

phase jumps:Debonding and Rupture

25

Failure of the matrix

transverse crack through glas reinforced polyester resin

26

Transversal stresses

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

Extensions

FBM

range of intreractionsload 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

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

to regimes :

-only partial failure

-no stationary state

-macroscopic stationarystate

-infinite life time

-monotonically increasing deformation

-global failure at finte 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

life time as function of size

universal decrease of life time with the number of fibres

PgN

tNt ff0

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

Damage evolution in wood

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 behaviourfor a=0.8 and different

values of kmax ; quenched disorder

Continuous Damage Model

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Strain [%]

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1.0

Dam

age

Par

amet

era=

EE

FF/E

0

a=0.3a=0.5a=0.8

0.0 0.5 1.0 1.5 2.0 2.5

Strain [%]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

F/N

kmax=1kmax=4kmax=8kmax=500

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Strain [%]

0.0

0.1

0.2

0.3

0.4

0.5

F/N

NCR=128NCR=64NCR=32 • maximal number of

failure events per fibre kmax

• damage parameter a

• system size

broad spectrum of scenarios

Continuous Damage Model

Avalanches and Clustering

0.0

0.3 0.9

simulation of avalanche statistics

at = c

D(

)

2/5~

2.2

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

Laminates

58

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

Faserdegradation

Rissmuster vollständig

Debonding / Mikroriss- /Segmentrisse

Spannungsfrei /Beginn der Pyrolyse

Kompression der Matrix über Fasern

Spannungsfrei bei Tempertemperatur

pyrolysis

tension stress

Multiple and Transverse Failures

degradation of layers perpendicular to the fibre orientation

micro-delaminations

transverse cracks

Sch

erd

eh

nu

ng

(

° D

EG)

-0.5

0

-0.5

0-0

.44

-0.3

8-0

.31

-0.2

5-0

.19

-0.1

2-0

.06

0.0

00

.06

0.1

20

.19

0.2

50

.31

0.3

80

.44

0.5

0

0.5

0

De

hn

un

g X

(

Tec

hn

isc

h %

)-1

.00

-1.0

0-0

.62

-0.2

50

.12

0.5

00

.88

1.2

51

.62

2.0

02

.38

2.7

53

.12

3.5

03

.88

4.2

54

.62

5.0

0

5.0

0

Degradation perpendicular to the fibre orientation

observations in GFK [02,902]s

Damage Evolution of the Transverse Layer

structure of the damage

evolution and distribution of porosity is important for Si deposition

Pyrolysis of C/hydroxylbenzene Laminates

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

•modell of Chang, Lessard 1991

UD [0n,90m]s

fibre-matrix shear failure

Failure Criteria for Simultaneous Mechanisms

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

• complex stress configurations

• activate various mechanisms

• local differences in stiffness

Puck's Theory

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 3 modes at the delamination front

Benzeggagh und Kenane:

BK-law

Wu und Reuter:

Potenz-law

Reeder-law

Crack Growth along Specified Paths: VCCT

example of single leg bending of a laminate

15.11

3022

IIIIII

a

IIIC

III

a

IIC

II

a

IC

I

equivC

equiv GGG

G

G

G

G

G

G

G

G onm

Crack Growth along Specified Paths: VCCT