problemi di frattura e danneggiamento in materiali e

PROBLEMI DI FRATTURA E DANNEGGIAMENTO IN MATERIALI E SISTEMI COMPOSITI

Parte 1:MECCANICA DELLA FRATTURA LINEARE E NON LINEARE NELLO

STUDIO DEI MATERIALI COMPOSITI

Roberta MassabòUniversità degli Studi di Genova

Università degli Studi di Pisa15 Febbraio 2010

DICAT - Department of Civil, Environmental and Architectural Engineering

INDICE DELLA PRESENTAZIONE

- Cenni di Meccanica delle Frattura Elastica LineareConcentrazione degli sforziIntensificazione degli sforzi - fattori di Intensificazione degli sforziCriterio di propagazione energeticoForza generalizzata di propagazione della fessuraFrattura in condizioni di modo mistoMeccanica della frattura non lineare – il modello di Dugdale

- Frattura in modo misto nei materiali compositi laminati

- I modelli coesivo e bridged-crack nello studio della frattura nei compositiIntroduzioneMetodo delle weight functionsLunghezze di scala e gruppi adimensionaliTransizioni nella risposta meccanica dei materiali compositiConfronto tra I due metodi

RICHIAMI DI MECCANICA DELLA FRATTURA ELASTICA LINEARE

FIBER-REINFORCED COMPOSITE LAMINATES

CHARACTERISTICS:- High degree of tailoring of elastic

properties- High strength to weight ratios- High stiffness to weight ratios(⇒ aeronautics/aerospace/defence)

DRAWBACKS:- Lack of strength in through-thickness direction- Low damage / impact tolerance and resistance- High sensitivity to interlaminar flaws- Catastrophic failures

(inter-ply layers: low-toughness fracture paths )

Lamina:unidirectionally reinforced composite

Laminate:stacking sequences: [0]n, [±α]n, [0/90]n, [0/±45/90]n, ….

Manufacturing procedures:- lay-up- resin impregnation (in PMC): RTM, RFI

REMEDIES:- Tougher matrices

- Inter-ply films, addition of particles,…

- Through-thickness reinforcement (trans-laminar reinforcement)

Cheeseman et al., Amptiac, (8), 2004.

STRUCTURAL COMPOSITE

SINGLE AND MULTIPLE DELAMINATION

Protective layer

Structural skin incomposite laminate

impactor

x zy

q(t)

(c)

(b)

Interfacial fracture

Delaminations

(Kim and Swanson, Comp. Struct., 2001)

COMPOSITE SANDWICH

p(t)Core crushing

Blast

Impactor

Rigid support

DelaminationsInterfacialfracture

(M.G. Andrews & R. Massabò, 2007, Engineering Fracture Mechanics)

h1

h2

a c

x z

esV2

esV1

esM 2

esN2

esN1

esM1esV0

esN 0

esM 0

0,1ϕΔ

0,2ϕΔ

h0

h2

h1

Reference solutions from the literature:

Suo, JAM, (1990): - axial loading only (bending moments and normal forces)- analytical expression for energy release rate, ≡ elementary beam theory- analytical mode decomposition based on dimensional considerations, linearity, relationship between ERR and SIF (Irwin, Sih)(derivation is analytical except for a single parameter extracted from numerical solution of one loading case

Li, Wang and Thouless, - accounts for effects of shear in bimaterial isotropic beams JMPS, (2004): - numerical FE solution for the energy release rate

- expressions for SIFs depending on 5 numerically determined constants

THE EFFECTS OF SHEAR AND NEAR TIP DEFORMATIONS ON ENERGY RELEASE RATE AND MODE MIXITY

OF EDGE-CRACKED ORTHOTROPIC LAYERS

Edge cracked layer subject to generalized end forces

- Plane problem- x and z = principal material axes

- Non dimensional orthotropy ratios (plane stress):

-so that crack tip fields independent of distance from load points


OF EDGE-CRACKED ORTHOTROPIC LAYERS (M.G. Andrews & R. Massabò, 2007, Engineering Fracture Mechanics)

xzzxxz

x

x

z

GEE

EE ννρλ −==

2, z

0 < ρ < 5 and 0 < λ < 1 (typical values for composites)

:λ −≥ 1/4min i a,c c = h (i = 0,1,2)

h1

h2

a c

x z

esV2

esV1

esM 2

esN2

esN1

esM1esV0

esN 0

esM 0

0,1ϕΔ

0,2ϕΔ

h0

h2

h1

Edge cracked homogeneous and orthotropic linear elastic layer subject to generalized end forces


OF EDGE-CRACKED ORTHOTROPIC LAYERS (M.G. Andrews & R. Massabò, 2007, Engineering Fracture Mechanics)

- Plane problem- x and z = principal material axes

- Non dimensional orthotropy ratios (plane stress):

-so that crack tip fields independent of distance from load points

xzzxxz

x

x

z

GEE

EE ννρλ −==

2, z

0 < ρ < 5 and 0 < λ < 1 (typical values for composites)

:λ −≥ 1/4min i a,c c = h (i = 0,1,2)

Aim of our work:

-Derive semi-analytical expressions for the energy release rate and the stress intensity factors in homogeneous orthotropic layers that depend on the crack tip stress resultants;

- The expressions have physical significance and allow separation of the different contributions

h1

h2

a c

x z

esV2

esV1

esM 2

esN2

esN1

esM1esV0

esN 0

esM 0

0,1ϕΔ

0,2ϕΔ

h0

h2

h1

Edge cracked homogeneous and orthotropic linear elastic layer subject to generalized end forces

ENERGY RELEASE RATE AND STRESS INTENSITY FACTORS

= h1

h2

N1 M1 V1

N2 M2 V2

N0 V0

M0

(a) Δx→0

+

(c1)

M0

N0

N M

M*

N

h1

h2

VS h1

h2

VS

VD

h1

h2

VD + +

(c2) (c3)

1 2* 2

h hM M N += +

= h1

h2

N1 M1 V1

N2 M2 V2

N0 V0

M0

(a) Δx→0

+

(c1) (b)

N0

M0 M0

N0

h1

h2

N M

M*

N

h1

h2

VS h1

h2

VS

VD

h1

h2

VD + +

(c2) (c3)

1 2* 2

h hM M N += +

Self equilibrated crack tip loading systems:

h1

h2

a c

x z

esV2

esV1

esM 2

esN2

esN1

esM1esV0

esN 0

esM 0

ENERGY RELEASE RATE AND STRESS INTENSITY FACTORSRelationship between energy release rate and stress intensity factors in orthotropic body (Sih, 1965):

Complex stress intensity factor:

Imaginary component of complex SIF = 0

Rea

l com

pone

nt o

f co

mpl

ex S

IF =

0 │K │

Ψ

Dimensional considerations, linearity and Eq. (*) :

(*)


Imaginary component of complex SIF = 0

Rea

l com

pone

nt o

f co

mpl

ex S

IF =

0

│KM │

│KN ││K │

ΨM

ΨN

Components of complex stress intensity factor:

Assuming:

Stress Intensity Factors:

ROOT ROTATIONS

h1

h2

a c

x z

esV2

esV1

esM 2

esN2

esN1

esM1esV0

esN 0

esM 0

= h1

h2

N1 M1 V1

N2 M2 V2

N0 V0

M0

(a) Δx→0

+

(c1)

N M

M*

N

h1

h2

VS h1

h2

VS

VD

h1

h2

VD + +

(c2) (c3)

1 2* 2

h hM M N += +

= h1

h2

N1 M1 V1

N2 M2 V2

N0 V0

M0

(a) Δx→0

+

(c1) (b)

N0

M0 M0

N0

h1

h2

N M

M*

N

h1

h2

VS h1

h2

VS

VD

h1

h2

VD + +

(c2) (c3)

1 2* 2

h hM M N += +

0,1ϕΔ

0,2ϕΔ

h0

h2

h1

Root rotations:

10,1 1 1 1

1 1

1S D

MV VN

S Dx

a M a N a V a VE h h

ϕ⎛ ⎞

Δ = + + +⎜ ⎟⎝ ⎠

20,2 2 2 2

1 1

1S D

MV VN

S Dx

a M a N a V a VE h h

ϕ⎛ ⎞

Δ = + + +⎜ ⎟⎝ ⎠

1/ 21

DVxz Sa bλ ν κ−= −

1/ 22

DVxz Sa bλ ην κ−= +

1/ 21 1 (1 )SV

xz Sa bλ η ν κ−= − +1/ 2

2 (1 )SVxz Sa bλ η η ν κ−= + +

where:1/ 4

1 2 1 2, , ,M M N Na a a a bλ −=

1 2h hη =2

x zzx xz

xz

E EG

ρ ν ν= −z xE Eλ =and:

ROOT ROTATIONS

ROOT ROTATIONS

-Root rotations are generated by all crack tip stress resultants and strongly depend on the local two-dimensional fields.

- This result can explain limitations of models based on plate theory and an interface approach in accurately predicting fracture parameters for short and moderately long cracks


Energy release rate from 2D elasticity:

Energy release rate from first order shear deformation beam theory:

( ) ( )31 12 12Mf η η= +

( ) 2 31 1 4 6 32Nf η η η η= + + +

( ) 11 1, ,

1S

S

VV xz

S

f a ρη λ ρ νκ η λ

⎛ ⎞⎛ ⎞= + +⎜ ⎟⎜ ⎟+ ⎝ ⎠⎝ ⎠

( ) 1 21, , D D

D

V VV xz

S

f a a η ρη λ ρ νκ λ

⎛ ⎞+ ⎛ ⎞= − + +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

( ) ( ) 21 3 1

sinMM Nf f

η ηγ η − ⎛ ⎞+

= ⎜ ⎟⎝ ⎠

( ) 1 1, sin2S

S

N

VN V

af f

γ η ρ −⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠

( ) 1 1 2, sin2D

D

N N

VN V

a af f

γ η ρ −⎛ ⎞−

= ⎜ ⎟⎜ ⎟⎝ ⎠


Energy release rate from 2D elasticity:

Energy release rate from first order shear deformation beam theory:

( ) ( )31 12 12Mf η η= +

( ) 2 31 1 4 6 32Nf η η η η= + + +

( ) 11 1, ,

1S

S

VV xz

S


⎛ ⎞⎛ ⎞= + +⎜ ⎟⎜ ⎟+ ⎝ ⎠⎝ ⎠

( ) 1 21, , D D

D

V VV xz

S


⎛ ⎞+ ⎛ ⎞= − + +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

( ) ( ) 21 3 1

sinMM Nf f

η ηγ η − ⎛ ⎞+

= ⎜ ⎟⎝ ⎠

( ) 1 1, sin2S

S

N

VN V

af f

γ η ρ −⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠

( ) 1 1 2, sin2D

D

N N

VN V

a af f

γ η ρ −⎛ ⎞−

= ⎜ ⎟⎜ ⎟⎝ ⎠

-Expression for the energy release rate fully resolve all crack problems where conditions are either mode I or mode II (e.g. laboratory specimens)


( ) ( )31 12 12Mf η η= +

( ) 2 31 1 4 6 32Nf η η η η= + + +

( ) 11 1, ,

1S

S

VV xz

S


⎛ ⎞⎛ ⎞= + +⎜ ⎟⎜ ⎟+ ⎝ ⎠⎝ ⎠

( ) 1 21, , D D

D

V VV xz

S


⎛ ⎞+ ⎛ ⎞= − + +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

( ) ( ) 21 3 1

sinMM Nf f

η ηγ η − ⎛ ⎞+

= ⎜ ⎟⎝ ⎠

( ) 1 1, sin2S

S

N

VN V

af f

γ η ρ −⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠

( ) 1 1 2, sin2D

D

N N

VN V

a af f

γ η ρ −⎛ ⎞−

= ⎜ ⎟⎜ ⎟⎝ ⎠


Semi-analytically derived constants:


( ) ( )31 12 12Mf η η= +

( ) 2 31 1 4 6 32Nf η η η η= + + +

( ) 11 1, ,

1S

S

VV xz

S


⎛ ⎞⎛ ⎞= + +⎜ ⎟⎜ ⎟+ ⎝ ⎠⎝ ⎠

( ) 1 21, , D D

D

V VV xz

S


⎛ ⎞+ ⎛ ⎞= − + +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

( ) ( ) 21 3 1

sinMM Nf f

η ηγ η − ⎛ ⎞+

= ⎜ ⎟⎝ ⎠

( ) 1 1, sin2S

S

N

VN V

af f

γ η ρ −⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠

( ) 1 1 2, sin2D

D

N N

VN V

a af f

γ η ρ −⎛ ⎞−

= ⎜ ⎟⎜ ⎟⎝ ⎠


Semi-analytically derived constants:

-Expressions for the stress intensity factors fully resolve mixed mode fracture problems in statically determined systems. In statically undeterminatesystems calculation of the crack tip stress resultants is first required (using 2D analyses or beam theory models accounting for the root rotations.

-Expressions are very accurate (uncertainties below 2%) for crack lengths λ −≥ 1/4

min i a,c c = h (i = 0,1,2)

APPLICATION – The exemplary case of the DCB specimen

h

h

a c

P

P2h

h

h

a c

P

P2h

Dimensionless energy release rate:

Dimensionless energy release rate for degenerate orthotropic beams:

Elementary beamtheory

Near tip deformations due to shear

Near tip deformations due to shear

Shear deformationsalong the arms

Near tip deformations due to bending

(and shear)

To describe fracture processes where nonlinear mechanisms arise in narrow, finite size bands, or process zones, ahead of traction free cracks

(Steiger, Sadouki & Wittmann)

z-pins

crack

(Zok & Hom, 1990)

extensive microcrackingand grain interlocking

BRIDGED- AND COHESIVE-CRACK MODELS

Cohesive crack

PMMA/Al composite

localized crack tip mechanisms

extensive fiber bridging

carbon-epoxy laminate/Ti z-pins(Rugg, Cox & Massabò, 2002)

Bridged crack

epoxy mortar

bridging rodscrack tip microcracks

COHESIVE CRACK MODEL

COHESIVE CRACK MODEL BRIDGED CRACK MODEL

continuous model discontinuous model

COHESIVE CRACK MODEL BRIDGED CRACK MODEL

continuous model discontinuous model

cohesive/bridging traction laws

ICIC or KG

(in bridged crack model)

Model parameters:

+

BRIDGED CRACK MODEL - SOLUTION

●

=+= PM www

,...),( PPMfw =

●Crack propagation: ICIPIMI KKKK =−=

(Castigliano’s theorem)

(e.g. weight functions for orthotropic double cantilever beams in Brandinelli, Massabò & Cox, 2003; Brandinelli & Massabò, 2006)

+∫a

c

IMIP d2M abPM

KKE

ICK

0d2P a

c2

2IP =∫ ab

PK

E

Mw Pw

Extension to other problems: weight function method

(Carpinteri, 1981, 1984; Bosco & Carpinteri, 1992; Carpinteri & Massabò, 1996,1997)

statically indeterminate problem

Kinematic compatibility condition → determination of forces P

●

DEVELOPMENT OF BRIDGED-CRACK MODELSMaterial systems

Barenblatt (1959) crystalsDugdale (1960) metalsBilby, Cottrell & Swinden (1963) metals…Romualdi & Batson (1963) reinforced concreteCarpinteri (1981, 1984) reinforced concreteMarshall, Cox & Evans (1985) fiber reinforced ceramicsBudiansky, Hutchinson & Evans (1986) fiber reinforced ceramicsJenq & Shah (1985,1986) fiber reinforced concreteFoote, Mai & Cotterell (1986) fiber reinforced cementitious composites Rose (1987a,b) crack reinforcement by springs and patchesSwanson et al. (1987) coarse grain ceramicsErdogan & Joseph (1987) particle reinforced ceramicsMc Meeking & Evans (1990) metal matrix composites, fatigueKendall, Clegg & Gregory (1991) polymer crazingBower & Ortiz (1991) particle reinforced brittle matrix compositesSuo, Ho & Gong (1993) ceramic matrix composites, notch sensitivityBallarini & Muju (1993) brittle matrix compositesCarpinteri & Massabò (1996) fiber reinforced cementitious compositesMassabò & Cox (1999) through-thickness reinforced laminates…

(KI π 0)

(KI = 0)

(For reviews: Bao & Suo, 1992; Cox & Marshall, 1994; Massabò, 1999)

LEFM

SSBl

∞→a

IC b( )+G G

∞→t∞→h

∞→a

ICG

cw)(0 wσ

≡

∞→h∞→t

(Budiansky, Hutchinson & Evans, 1986; Rose, 1987)

ASYMPTOTIC SOLUTIONS FOR BRIDGED CRACKSTHE SMALL SCALE BRIDGING LIMIT

0uσ

cw

c

b 00

( )w

w dwσ= ∫G0σ

bGw

0u

cSSB σ

π Ewl8

=

LEFM

SSBl

∞→a

IC b( )+G G

∞→t∞→h

∞→a

ICG

cw)(0 wσ

≡

∞→h∞→t


2

b

IC

b

IC

0u

cSSB 1

8 ⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

GG

GG

σπ Ewl

(Bao & Suo, 1992; Cox & Marshall, 1994)


Small scale bridging characteristic length scales:

)0≠IC(G

(Bilby, Cottrell, Swinden, 1963; Hillerborg, 1976)

(For rectangular bridging law)

Bridged crack

)0=IC(GCohesive crack

0uσ

cw

c

b 00

( )w

w dwσ= ∫G0σ

bGw

0u

cSSB σ

π Ewl8

=

LEFM

SSBl

∞→a

IC b( )+G G

∞→t∞→h

∞→a

ICG

cw)(0 wσ

≡

∞→h∞→t


2

b

IC

b

IC

0u

cSSB 1

8 ⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

GG

GG

σπ Ewl

(Bao & Suo, 1992; Cox & Marshall, 1994)


Small scale bridging characteristic length scales:

)0≠IC(G

(Bilby, Cottrell, Swinden, 1963; Hillerborg, 1976)

(For rectangular bridging law)

Bridged crack

)0=IC(GCohesive crack

0uσ

cw

c

b 00

( )w

w dwσ= ∫G0σ

bGw

Small scale bridging characteristic length scale defines material brittleness and varies over many order of magnitude in different material systems

SSBl

t2

∞→a

∞→h

ICG

cw)(0 wσ

(Suo, Bao, Fan, 1992; Massabo & Cox, 1999)

ASYMPTOTIC SOLUTIONS FOR BRIDGED CRACKSTHE SMALL SCALE BRIDGING LIMIT IN SLENDER BODIES

Mode I and mode II characteristic length scales:

I 1/ 4 3/ 4SSB SSB( )l l t≈

II 1/ 2SSB SSB( )l l t≈

where lSSB is the characteristic length scale in an infinite body

(for mode I fracture)

(for mode II fracture)

0uσ

cw

c

b 00

( )w

w dwσ= ∫G0σ

bGw

SSBl

t2

∞→a

∞→h

ICG

cw)(0 wσ

(Suo, Bao, Fan, 1992; Massabo & Cox, 1999)

ASYMPTOTIC SOLUTIONS FOR BRIDGED CRACKSTHE SMALL SCALE BRIDGING LIMIT IN SLENDER BODIES

Mode I and mode II characteristic length scales:

I 1/ 4 3/ 4SSB SSB( )l l t≈

II 1/ 2SSB SSB( )l l t≈

where lSSB is the characteristic length scale in an infinite body

(for mode I fracture)

(for mode II fracture)

Characteristic length scales in slender bodies are smaller than those in infinite bodies

Important consequences when length scale is used to size the mesh in numerical descriptions of fracture processes

0uσ

cw

c

b 00

( )w

w dwσ= ∫G0σ

bGw

ASYMPTOTIC SOLUTIONS FOR BRIDGED CRACKSTHE ACK LIMIT IN SLENDER AND NON SLENDER BODIES

ACK characteristic length scales:

(Massabò & Cox, 1999)

1-α-21+α

1+αACK Ic

14 2El π α β

α+⎛ ⎞= ⎜ ⎟

⎝ ⎠G

II 1/ 2ACK ACK( )l l t≈

0 ( ) ( / 2)w w ασ β=For power law bridging:

in an infinite body

in a slender body (mode II)

(Cox & Marshall, 1994)

Slender body loaded in mode II

ACK limit

(Avenston, Cooper & Kelly, 1971)

ASYMPTOTIC SOLUTIONS FOR BRIDGED CRACKSTHE ACK LIMIT IN SLENDER AND NON SLENDER BODIES

ACK characteristic length scales:


1-α-21+α

1+αACK Ic

14 2El π α β

α+⎛ ⎞= ⎜ ⎟

⎝ ⎠G

II 1/ 2ACK ACK( )l l t≈

0 ( ) ( / 2)w w ασ β=For power law bridging:

in an infinite body

in a slender body (mode II)

(Cox & Marshall, 1994)

Slender body loaded in mode II

ACK limit

Comparing the characteristic length scales in long unnotchedbodies determines whether crack will approach:

- ssb limit (if lssb << lACK) → catastrophic failure

- ACK limit (if lACK << lssb) → noncatastrophic failure

The presence of a notch favours catastrophic failure

If length scales are similar or body is finite, crack growth is in large scale bridging and detailed calculations are required

LARGE SCALE BRIDGING FRACTURE INTHROUGH-THICKNESS REINFORCED LAMINATES

DISCONTINUOUS TTR(Fibrous/metallic Z-pins)

CONTINUOUS TTR (stitching / weaving)

Titanium z-pins

(3TEX)

3D woven composites




Titanium z-pins

Load versus mid-span deflection curves in ENF specimens

(3TEX)

3D woven composites

stitchedunstitched




Titanium z-pins

Load versus mid-span deflection curves in ENF specimens

(3TEX)

3D woven composites

stitchedunstitched

A through thichkess reinforcement:

- controls and even suppresses crack growth

- changes failure mechanisms

- Improves damage/impact tolerance


b b( ) (0)w wτ τ β= +1/ 2

b IIC

b IIC

(0) 2( )/ 80

τ β=

=

G

G G

1/ 2ACK b IIC(0) ( )τ τ β= + GIIACK ( 4 )l E tβ=

TRANSITION FROM NON-CATASTROPHIC TO CATASTROPHIC FAILURE IN SLENDER BODIES

IIACK

al

0IIACK

0.0 al

=

2t

L >> h a>>h

Crack length

Crit

ical

load

for c

rack

gro

wth

b b( ) (0)w wτ τ β= +1/ 2

b IIC

b IIC

(0) 2( )/ 80

τ β=

=

G

G G


IIACK

al

0IIACK

0.0 al

=

2t

L >> h a>>h

EXPERIMENTAL DERIVATION OF BRIDGING TRACTION LAW FOR A STITCHED COMPOSITE

b ( ) 12.7 102 MPaw wτ = +2

IIC

c

0.37 kJ/m0.5 mmw

=

=

G

continuous stitching

(Massabò, Mumm & Cox, 1998)


b b( ) (0)w wτ τ β= +1/ 2

b IIC

b IIC

(0) 2( )/ 80

τ β=

=

G

G G

IIACK

al

0IIACK

0.0 al

=

2t

L >> h a>>h




Crack length

Crit

ical

load

for c

rack

gro

wth

b b( ) (0)w wτ τ β= +1/ 2

b IIC

b IIC

(0) 2( )/ 80

τ β=

=

G

G G

IIACK

al

0IIACK

0.0 al

=

2t

L >> h a>>h




Crit

ical

load

for c

rack

gro

wth

Design of and with advanced composites:

In through-thickness reinforced laminates overdesigning should be avoided since insertion process degrades inplane properties → bridging action must be understood and quantified using characteristic length scales and bridged-crack modeling

MMB 5%Step 12, 1000 N

020406080

100120140160180200

0 5 10 15 20 25 30 35

Position (mm)

Dis

plac

emen

t (µ

m)

OpeningSliding

Mixed-Mode Bending

0200400600800

10001200140016001800

0 2 4 6 8 10 12

Cross-He ad Displace me nt (mm)

Load

(N

)

GripFailure

1.5% Z-fiber

5% Z-fiber

Control

CrackInitiation


z-pins

crack

Through thickness reinforced laminates

(Rugg, Cox & Massabo, 2002)

OTHER CHARACTERISTIC LENGTH SCALES

MMB 5%Step 12, 1000 N

020406080

100120140160180200

0 5 10 15 20 25 30 35

Position (mm)

Dis

plac

emen

t (µ

m)

OpeningSliding

Mixed-Mode Bending

0200400600800

10001200140016001800

0 2 4 6 8 10 12

Cross-He ad Displace me nt (mm)

Load

(N

)

GripFailure

1.5% Z-fiber

5% Z-fiber

Control

CrackInitiation


z-pins

crack

Through thickness reinforced laminates

(Massabo & Cox, 2001)

4

3

4EI 2β

πλ =

THEORETICAL MODEL

OTHER CHARACTERISTIC LENGTH SCALES

a

h∞→t

)(0 wσ

ICG

hlSSB

FRACTURE IN FINITE SIZE BODIESDIMENSIONLESS GROUPS GOVERNING MECHANICAL BEHAVIOR

hlSSB

b

IC

GG

and

(one group)

(two groups))0≠IC(G

Bridged crack:

)0=IC(GCohesive crack:

0uσ

cw

c

b 00

( )w

w dwσ= ∫G0σ

bGw

ICK

IC

5.0u

P KhN ρσ

=

(Carpinteri 1981, 1984; Massabò, 1994)

bhP uP ρσ=

FLEXURAL RESPONSE OF BRITTLE MATRIX COMPOSITES WITH DISCRETE DUCTILE REINFORCEMENTS


ICK

(Carpinteri 1981, 1984; Massabò, 1994)

bhP uP ρσ=

IC

5.0u

P KhN ρσ

=

Localized rotation

Crit

ical

mom

ent f

or c

rack

pro

paga

tion

EXPERIMENTS

(Bosco, Carpinteri & Debernardi, 1990)

High-strength reinforced concrete


IC

5.0u

P KhN ρσ

=

Localized rotation

EXPERIMENTS

(Levi, Bosco & Debernardi, 1998Bosco, Carpinteri & Debernardi, 1990)


IC

5.0u

P KhN ρσ

=

Localized rotation

EXPERIMENTS


IC

5.0u

P KhN ρσ

=

(Levi, Bosco & Debernardi, 1998Bosco, Carpinteri & Debernardi, 1990)

Flexural response of brittle matrix composites with ductile reinforcements:

- controlled by a single brittleness number

- brittle to ductile transition on increasing the beam depth

- local snap-through instabilities can be explained using bridged-crack model

(Carpinteri & Massabò, 1997)


EXPERIMENTS

(Jenkins et al., 1987)


EXPERIMENTS

(Zhu & Bartos, 1993)

TRANSITIONS IN THE FLEXURAL BEHAVIOR OF FIBER REINFORCED COMPOSITES

hhs c

0u

ICbc0u

c

E σσGGG +

== 250IC

b =GG

Fiber reinforced brittle matrix composite(e.g. fiber reinforced high strength concrete; fiber reinforced ceramic)

Bridged-crack model Cohesive-crack model

Two dimensionless groups govern mechanical behavior

(brittleness number, Carpinteri 1985)

(Massabo, 1999; Carpinteri & Massabò,1997)


hhs c

0u

ICbc0u

c

E)(

σσGGG +

==

250IC

b =GG

increasingEs

Es increasing


hhs c

0u

ICbc0u

c

E σσGGG +

==

250IC

b =GG


increasingEs

Es increasing


hhs c

0u

ICbc0u

c

E σσGGG +

==

250IC

b =GG

EsEs


EXPERIMENTS

(Jamet, Gettu et al., 1995; Jenq and Shah, 1986)

Es

Es


hhs c

0u

ICbc0u

c

E σσGGG +

==

250IC

b =GG


increasingEs

Es increasing

Double scaling transition: brittle to ductile to brittle in the flexural response of brittle matrix fiber reinforced composites on increasing the beam depth


hhs c

0u

ICbc0u

c

E σσGGG +

==

250IC

b =GG


increasingEs

Es increasing



bridged crack model

cohesive crack model

progressive brittleness

progressive ductility

Es

Es

increasing

increasing



bridged crack model

cohesive crack model

progressive brittleness

progressive ductility

Es

Es

increasingEs

Es increasingDrawbacks:

- The cohesive crack model is computationally very expensive

- The bridged crack model has limited range of applicability when matrices are not perfectly brittle

- Cohesive crack model must be applied to study crack initiation

PROBLEMI DI FRATTURA E DANNEGGIAMENTO IN MATERIALI E SISTEMI COMPOSITI

Parte 2:INTERAZIONE DI MECCANISMI MULTIPLI DI DANNEGGIAMENTO

Roberta MassabòUniversità degli Studi di Genova

Università degli Studi di Pisa15 Febbraio 2010

DICAT - Department of Civil, Environmental and Architectural Engineering

PEOPLE:

B.N. Cox (Teledyne Scientific, CA, U.S.A.) A. Cavicchi (post doc, University of Genova, Italy) M.G. Andrews (former grad. student, Northwestern University, Boeing Co. U.S.A)F. Campi (grad. student, University of Genova, Italy)

FUNDING:

U.S. Office of Naval Research, N00014-05-1-0098U.S. Army Research Office, DAAD19-99-C-0042Italian MIUR

OUTLINE

-Introduction and motivation

- Theoretical models to study the interaction of multiple damage mechanisms in laminates and composite sandwiches;

- Relevant results

MULTIPLE DAMAGE MECHANISMS IN LAMINATES AND SANWICHES

Cheeseman et al., Amptiac, (8), 2004.

STRUCTURAL COMPOSITE

Protective layer

Structural skin incomposite laminate

impactor

x zy

q(t)

(c)

(b)

Interfacial fracture

Delaminations

(Kim and Swanson, Comp. Struct., 2001)

COMPOSITE SANDWICH

p(t)Core crushing

Blast

Impactor

Rigid support

DelaminationsInterfacialfracture

MECHANICAL MODELSCOMPOSITE LAMINATES AND MULTILAYERED SYSTEMS

Bridging/contactsurfaces

Laminate

p(t)

Timoshenkobeams

p(t) Cohesive/contactinterfaces

Convenient forhomogeneous beams and

static loading

Convenient formultilayered beams and

dynamic loading

(Andrews, Massabò & Cox, IJSS, 2006)

(Andrews & Massabò, EFM, 2007)

(Andrews, Massabò, Cavicchi & Cox, IJSS, 2009)

(Andrews & Massabò, Comp. A, 2008)

(a) (b)


Core and/or interface - nonlinear Winkler foundation

Laminate or

upperface sheet

p(t)

Timoshenkobeams


Convenient forhomogeneous beams and

static loading

Convenient formultilayered beams and

dynamic loading


(Andrews & Massabò, EFM, 2007)

(Cavicchi and Massabò, 2009, proc. ICCM17)


(Andrews & Massabò, Comp. A, 2008)

(Cavicchi and Massabò, 2009, proc. AIMETA)

MECHANICAL MODELSFULLY BACKED COMPOSITE SANDWICHES

(a) (b)


Core/interface – n.l. Winkler foundation

p(t)

Timoshenko beam


(a) (b)

- Homogeneous, orthotropic, linear elastic skin- Perfectly brittle matrix + crack bridging/contact- Accurate mode decomposition (Andrews & Massabò, EFM, 2007)

k

1,−N

k kT1,−

Sk kT

, 1+N

k kT, 1+S

k kTCore/interface – n.l. Winkler foundation

- Multi-layered, orthotropic, linear elastic skin- Cohesive interfaces: perfect adhesion, brittleor cohesive fracture, contact, bridging

core/interface contact/bridging cohesive interfaces

w

fT

w

fT

Nw

NT

Nw

NTNTST

Sw Nw

NTST

Sw Nw

k k

MECHANICAL MODELS

Laminate or

upperface sheet

MECHANICAL MODELS

SOLUTION METHODsemi-analytic (iterative procedure needed todefine contact/bridging regions and plasticallyadmissible fields)

SOLUTION METHODnumerical: time/space discretization, finite difference solution scheme

( ), 1 1,1 2 S Sk k k k k k k m k kM V h T T I+ −′ − − + = &&ρ ϕ

, 1 1,S S

k k k k k m k kN T T S u+ −′ − + = &&ρ

, 1 1,+ −′ + − = &&N Nk k k k k m k kV T T S wρ

′=k kuε′= − +k k kwϕ γ

′=k kχ ϕ

=k k kN A ε=k k kM D χ

=k k kV G γ

,w ,k k ku ϕGeneralized displacements

Dynamic equilibrium Compatibility


Core/interface – n.l. Winkler foundation

p(t)

Timoshenko beam


k

1,−N

k kT1,−

Sk kT

, 1+N

k kT, 1+S

k kTCore/interface – n.l. Winkler foundation

k kLaminate

orupper

face sheet

Constitutive laws

(a) (b)

Assumptions:- System of two cracks in a cantilever beam- Frictionless contact- Homogeneous, isotropic and perfectly

brittle material

aL h

P

aU

L fixed

varying

0 1 2 3 4 5 6 7 8 9 100

200

400

600

800

1000

2D FEMTwo crack system

Equally spaced cracksaL = 5hL = 10h

2U EhP

G

/Ua

Energy release rate of upper crack vs. crack length

amplification

discontinuity

single crack solution

h

P

aU

L

h

0 1 2 3 4 5 6 7 8 9 1050

90

110 FEMCurrent Model Single crack

aU>aL

aU = varied aL = 5h h3 = h/3 h5 = h/3 L = 10h

/Ua h

tan-1

(KII

U/K

IU)

70

Relative amount of mode II to mode I(lower crack) vs. crack length

INTERACTION EFFECTS – QUASI STATIC LOADING AMPLIFICATION AND SHIELDING OF FRACTURE PARAMETERS

(Andrews, Massabò and Cox, IJSS, 2006; Andrews and Massabò, Comp. A, 2008)

aL h

P

aU

L fixed

varying

0 1 2 3 4 5 6 7 8 9 100

200

400

600

800

1000

2D FEMTwo crack system

Equally spaced cracksaL = 5hL = 10h

2U EhP

G

/Ua

Energy release rate of upper crack vs. crack length

amplification

discontinuity

single crack solution

h

P

aU

L

h

0 1 2 3 4 5 6 7 8 9 1050

90

110 FEMCurrent Model Single crack

aU>aL

aU = varied aL = 5h h3 = h/3 h5 = h/3 L = 10h

/Ua h

tan-1

(KII

U/K

IU)

70

Relative amount of mode II to mode I(lower crack) vs. crack length

INTERACTION EFFECTS – QUASI STATIC LOADING AMPLIFICATION AND SHIELDING OF FRACTURE PARAMETERS

(Andrews, Massabò and Cox, IJSS, 2006; Andrews and Massabò, Comp. A, 2008)

Assumptions:- System of two cracks in a cantilever beam- Frictionless contact- Homogeneous, isotropic and perfectly

brittle material

Interaction of multiple delaminations (static loading):- phenomena of amplification and shielding of energy release rate G

- sudden changes in G when tips are close - mode ratio variations- phenomena controlled by through-thickness spacing → predictions based on solutions for single delaminations may be unconservative

0 5 10 0

10

20

30

40

15 20 25 30 35 40

50

60

End Displacement (mm)

Load

(N)

40 mm 100 mm

Width = 20 mmExperimental (Robinson et al. 1999)

Proposed Model

(c)

20mm

3.18 mm 1.86 mm 1.59 mm

0 20 40 60 80 100 120 140 160 180

0

100

200

300

400

20mm

Crack tip position (mm)

Dis

plac

emen

t Inc

rem

ent

A

B

C

D B-CD

A

Hexcel HTA913 carbon epoxy Ex = 115 GPa νxy = .29 Ey = 8.5 GPa GIc = 330 J/m2 Gxy = 4.5 GPa GIIc = 800 J/m2

unstable growth of both cracks

shielding

amplification

unstable growth of left crack

Snap-back and snap-through instabilities due to local amplification and shielding of the crack tip stress fields

INTERACTION EFFECTS – QUASI STATIC LOADINGMACRISTRUCTURAL RESPONSE

(Andrews and Massabò, Comp. A, 2008)

Assumptions:- Frictionless contact- Homogeneous, orthotropic, perfectly brittle material

INFLUENCE OF CRACK SPACING ON MULTIPLE DELAMINATION

(I)

(III) (II)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

3hh

5 /h h

U L>G G

L U>G GB U LG > G ,G

h

P

a

h3

h5

Behavioral maps, depending on crack spacing, define regions characterized by equal length growth

- In grey region equality of length is maintained and is stable with respect to length perturbations

→ Is controlled delamination fracture a feasible tool to improve mechanical performance (e.g. energy absorption)?


Assumptions:- frictionless contact- perfectly brittle material

2L

2P

h

2a

h

2L

2P

h

2a

h

DYNAMIC INTERACTION EFFECTS IN HOMOGENEOUS SYSTEMSSTATIONARY DELAMINATIONS

Equally spaced, equal length delaminations

2L

2P

h

2a

h

P

time

LoadP

time

Load

In phase vibrations of delaminated beams

Dynamic and interaction effects uncoupled ifloading pulse is long enough

Unequally spaced or unequal length delaminations:

2L

2P

hh

Out of phase vibrations and hammering of delaminated beams after load removal

P

time

LoadP

time

Load

x zy

q(t)

(c)

(b)


DYNAMIC INTERACTION EFFECTS - STATIONARY DELAMINATIONSequally spaced, equal length delaminations, short pulses

Assumptions:- One crack in clamped-clamped beam- Frictionless contact- Homogeneous, isotropic, perfectly brittle

Time history of normalized energy release rate in systems with one delamination

0 1 2 3 4 50

0.5

1

1.5

2

static

GG

1t t

1

/ 2/ 2

a Lh h

==

1

1

1

/ 4

2

p

p

p

t tt tt t

=

=

=

P

tp/2 tp

Load

t

P(t)

a

h1 h

L

In phase vibrations of delaminated beams

Dynamic and interaction effects uncoupled ifpulse is long enough

DYNAMIC INTERACTION EFFECTS - STATIONARY DELAMINATIONSunequally spaced, unequal length delaminations, short pulses

Time history of energy release rate components

2

EhP

G

Load

P

Tp/2 Tp Lcth

P(t)

a

h1 h

L

Lcth

II-staticG

I-staticG

IIG

IG

1

105

/ 3311P

L ha hh hT

====

0 200 400 600 800 1000 12000

20

40

60 50

30

10

wEP

Lcth

0 200 400 600 800-1500

-1000

-500

0

500

1000

lower arm

upper arm

Loading:in phase

Free vibration: out of phase

(b)

Time history of load point displacements

hammering

(1) (2) (3)

Assumptions:- One crack in clamped-clamped beam- Frictionless contact- Homogeneous, isotropic, perfectly brittle

Forced vibrations

free vibrations

Out of phase vibrations, amplifications and hammering of delaminated beams after load removal

2L

P

hh

2L

P

hhP

time

LoadP

time

Load

In homogeneous systems under arbitrary dynamic loading conditions:

Equally spaced delaminations propagate to equal length configuration and equality of length is then maintained and stable with respect to length perturbations

150 200 250 3005

6

7

8

9

ah

/Ltc h

2

105

100

o

IIcr Icr

L ha h

PEh

==

=G = 2G

P(t)

a

h

L P

time

Load

MULTIPLE DYNAMIC DELAMINATION FRACTURE

(Andrews, Massabò, Cavicchi & Cox, 2008)

Time history of crack growth


2L

P

hh

2L

P

hhP

time

LoadP

time

Load


P

Tp/2 TpLct

h

P

Tp/2 TpLct

h

P(t)

a

h

L

02468

10

0 200 400

ah

/Ltc h02468

10

0 200 400

ah

/Ltc h

02468

10

0 200 400

ah

/Ltc h

lower loads or higher fracture toughness

lower speeds

higher loads or lower fracture toughness

Higher speeds

(Andrews, Massabò, Cavicchi & Cox, 2008)


In homogeneous systems under arbitrary dynamic loading conditions:

Unequally spaced delaminations: propagation at higher speed of a single dominant crack can be observed; response controlled by crack spacing


2L

P

hh

2L

P

hhP

time

LoadP

time

Load


(I)

(III) (II)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

3hh

5 /h h

U L>G G

L U>G GB U LG > G ,G

h

P

a

h3

h5

Behavioral maps, depending on crack spacing, define regions characterized by equal length growth

(Andrews, Massabò & Cox, 2006)


Displacement controlled dynamic growthwith fixed initial strain energy L

a

h

L

time

w(t) quasi-static while preventing crack growth

wst

Assumptions:cr /hG L(b)

w(t)a

h

L

(a) equally spaced cracks

unequally spaced cracks

0.1h

0.3h

w(t)Frictionless contactHomogeneous, isotropic, perfectly brittle material

leading to small / moderate crack speeds

CONTROLLED DELAMINATION FRACTURE TO IMPROVE MECHANICAL PERFORMANCE AGAINST DYNAMIC LOADINGS


a

h

L

time


wst


w(t)a

h

L



0.1h

0.3h



0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0 20 40 60 80 100 120 140 1600

2

4

6

8

10

12

0 20 40 60 80 100 120 140 160

Time history of crack growth Time history of expended energy

ah

105o

L ha h

==

/Ltc h /Ltc h

cr / 0.086h =G L

expEnergyL


equally spaced cracks

unequally spaced, upper crack

unequally spaced, lower crack




a

h

L

time


wst


w(t)a

h

L



0.1h

0.3h



0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0 20 40 60 80 100 120 140 1600

2

4

6

8

10

12

0 20 40 60 80 100 120 140 160

Time history of crack growth Time history of expended energy

ah

105o

L ha h

==

/Ltc h /Ltc h

cr / 0.086h =G L

expEnergyL



unequally spaced, upper crack

unequally spaced, lower crack


Energy absorption through multiple delamination fracture in laminated plates can be optimized via a material design that favour crack formation along predefined planes


INFLUENCE OF CRACK BRIDGING MECHANISMS STATIONARY DELAMINATIONS

0

10

20

30

40

50

60

0 200 400 600 800 1000 1200

II-staticG

I-staticG

IIG

IG

1

105

/3311P

L ha hh hT

====

LT t c h=

2

EhP

G

P

TP/2 TP

Load

LT t c h=

P(t)/2

a

h1 h

L

Time history of energy release rate components

Bridging mechanisms:

- shield crack tip field from

applied loading

- Problem becomes mode II

- stabilize free vibration phase

TN , TS

wN , wS

kN ,kS

Linear proportional bridging

, 0.01 /N Sk k E h=

Bridging tractions

(values for a typical stitched laminate)

0

1

2

3

4

5

6

7

8

9

10

0,00 20,00 40,00 60,00 80,00 100,00 120,00 140,00 160,00 180,00

w(t)a

h

L

Results for: leading to small / moderate crack speeds

ENERGY ABSORPTION THROUGH MULTIPLE DELAMINATIONINFLUENCE OF CRACK BRIDGING MECHANISMS



ah

105o

L ha h

==

/Ltc h

cr / 0.086=G L

time


wst

equally spaced cracks, no bridging

equally spaced cracks,bridging

cr / 0.086h =G L

TN , TS

kN ,kS

wc wN , wS

Bridging tractions

, 0.01 /N Sk k E h=

(values for a typical stitched laminate)0.1cw h=

a

h

L

(b)

w(t)a

h

L

(a) Equally spaced cracks

unequally spaced cracks, h3=0.1h, h5=0.3h

h3

h5w(t)

0,00

2,00

4,00

6,00

8,00

10,00

12,00

0,00 20,00 40,00 60,00 80,00 100,00 120,00 140,00 160,00 180,00





ah

105o

L ha h

==

/Ltc h

cr / 0.086=G L

time


wst

unequally spaced lower crack, no bridging

unequally spaced lower crack, bridging

cr / 0.086h =G L

TN , TS

kN ,kS

wc wN , wS

Bridging tractions

, 0.01 /N Sk k E h=


a

h

L

(b)

w(t)a

h

L

(a) Equally spaced cracks

unequally spaced cracks, h3=0.1h, h5=0.3h

h3

h5w(t)

a

h

L

w(t)




time


wst

cr / 0.086h =G L

TN , TS

kN ,kS

wc wN , wS

Bridging tractions

, 0.01 /N Sk k E h=


0,00

2,00

4,00

6,00

8,00

10,00

12,00

0,00 20,00 40,00 60,00 80,00 100,00 120,00 140,00 160,00 180,00


ah

105o

L ha h

==

/Ltc h

cr / 0.086=G L

unequally spaced lower crack, no bridging

unequally spaced lower crack, bridging

a

h

L

w(t)

Bridging mechanisms in the regime of low to moderate crack speeds:

- reduce crack speed

- may lead to crack arrest

- minimize differences in the response of systems with equally and unequally spaced cracks

INTERACTION EFFECTS IN SANDWICH BEAMSSHIELDING OF FRACTURE PARAMETERS AND ENERGY BARRIERS

(quasi static loading, elastic foundation)

La

Uh

2P

hLh

plane ofsymmetry

Elastic core

Uh / h 0.5=L / h 10=

Kw

fT

-Face-core interactions shieldcrack tips from applied loads

- Transition in energy release rate on increasing the core-skinstiffness ratio

- Minimum in G defines energybarrier to crack propagation

Perfect core/skin interface

0

5

10

15

20

25

30

35

0 2 4 6 8 10

EhP

G

ah

( )a( )b

( )c

( )d

( )a 0hKE

=

( ) 4b 10hKE

−=

( ) 3c 10hKE

−=

( ) 2d 10hKE

−=

L 10=h

0

5

10

15

20

25

30

35

0 2 4 6 8 10

EhP

G

ah

( )a( )b

( )c

( )d

( )a 0hKE

=

( ) 4b 10hKE

−=

( ) 3c 10hKE

−=

( ) 2d 10hKE

−=

L 10=h

Normalized crack length

Dim

ensi

onle

ssen

ergy

rele

ase

rate

no foundation

Rigid Support

p(t)

-Energy barrier for a characteristic lengthof the crack depending on the core-skinstiffness ratio

0

5

10

15

20

25

30

0 5 10 15 20 25 30

La

Uh

2P

hLh

plane ofsymmetry

Elastic core

Crack length

Crit

ical

load

forc

rack

pro

paga

tion

Uh / h 0.5=L / h 100=

Kw

fT

ah

4hK 10E

−=

3hK 10E

−=

2hK 10E

−=

cr

cr

PEhG hK

E

crG = GPerfect core/skin interface

(crack growth criterion)

Brittle skin, crG



0

5

10

15

20

25

30

0 5 10 15 20 25 30

La

Uh

2P

hLh

plane ofsymmetry

Elastic core

Crack length

Crit

ical

load

forc

rack

pro

paga

tion

Kw

fT

ah

4hK 10E

−=

3hK 10E

−=

2hK 10E

−=

cr

cr

PEhG

0

15

30

45

60

75

90

0 5 10 15 20 25 30

II

I

arctan⎛ ⎞

φ = ⎜ ⎟⎝ ⎠

GG

Am

ount

of m

ode

II to

mod

e I

ahCrack length

φ

3hK 10E

−=

-Energy barrier for a characteristic lengthof the crack depending on the core-skinstiffness ratio

- Sudden transition from mode II to mixedmode fracture at the barrier



0

5

10

15

20

25

0 5 10 15 20

(a)

(b)

(c) (a)

(b)

(c)

0.003P Eh ≤

0.005P Eh =

0.006P Eh =

2

EhP

G

ah

(elastic case)

La

Uh

2P

hLh

plane ofsymmetry Elastic-plastic core

Crack length

Dim

ensi

onle

ssen

ergy

rele

ase

rate

Uh / h 0.5=L / h 100=

- Plasticity of the core reduceselastic shielding and modifiesdependence of energy release rate on applied load

K

w

fT

fcrT

3hK 10E

−=f

3crT 10E

−=

INFLUENCE OF CORE PLASTICITY ON FRACTURE PARAMETERS AND MECHANICAL RESPONSE

(static loading, elastic-plastic foundation)


elastic response

La

Uh

2P

hLh


Uh / h 0.5=L / h 100= crG = G

K

w

fT

fcrT

3hK 10E

−=5cr 10Eh

−=G

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.008 0.010 0.012 0.014 0.016

cr

GG

PEh

Nor

mal

ized

ener

gyre

leas

era

te

Applied load

- The problem solution is not unique

- The response depends on the loadingand propagation histories



a 12.8h

=

a 12.5h

=

a 12.0h

=

fcr

0.01T h

=crG



La

Uh

2P

hLh


Uh / h 0.5=L / h 100= crG = G

K

w

fT

fcrT

3hK 10E

−=5cr 10Eh

−=G



0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

0.020

0 5 10 15 20 25 30ah

crPEh f

cr

0.001T h

=crG

Crit

ical

load

fori

nitia

lpro

paga

tion

fcr

0.01T h

=crG

fcr

0.02T h

=crG

fcrT h / crG

Notch length

- The problem solution is not unique

- The response depends on the loadingand propagation histories



Uh / h 1/ 3=L / h 10=

≥ crG G

MULTIPLE DYNAMIC DELAMINATION FRACTURE OF THE SKIN(dynamic loading, elastic foundation)

3hK 10E

−=

Elastic core

Kw

fT

L

Ua

h

La

1h2h3h

2P

plane of symmetry

Load

Time tp

P180p Lt c

h=

Lh / h 1/3=

cr

1=maxPEhG

0

2

4

6

8

10

0 50 100 150 200 250 300 350

lower crackupper crack

Ltc h

ah

-400

-200

0

200

400

600

800

0 50 100 150 200 250 300 350Ltch

Upper-middlearms

Lower arm

cr /w

G h E

Nor

mal

ized

crac

k le

ngth

s

Dimensionless time

Dimensionless timeD

ispl

acem

ents

at lo

adpo

int


Elastic core

Kw

fT

L

Ua

h

La

1h2h3h

2P

plane of symmetry

Load

Time tp

P180p Lt c

h=

0

2

4

6

8

10

0 50 100 150 200 250 300 350

lower crackupper crack

Ltc h

ah

-400

-200

0

200

400

600

800

0 50 100 150 200 250 300 350Ltch

Upper-middlearms

Lower arm

cr /w

G h E

Nor

mal

ized

crac

k le

ngth

s

Dimensionless time

Dimensionless timeD

ispl

acem

ents

at lo

adpo

int

Dynamic loading and inertial effects trigger mechanisms of multiple crack propagationthat are absent in quasi-static cases

Uh / h 1/ 3=L / h 10=

≥ crG G

3hK 10E

−=

Lh / h 1/3=

cr

1=maxPEhG


MULTIPLE DYNAMIC DELAMINATION FRACTURE OF THE SKIN(dynamic loading, elastic foundation)

CONCLUSIONS

- Static and dynamic interaction effects of multiple damage mechanisms in composite laminates, multilayered systems and composite sandwiches have been investigated

- Interaction effects on fracture parameters include: amplification and shielding of the energy release rate of one crack due to the presence of other cracks, modification of mode ratios, crack shielding and energy barriers due to the presence of elastic cores

- Interaction effects on macrostructural response include: snap-back and snap-through instabilities, hyper-strength, crack pull along

- Controlled delamination fracture can be a viable tool to improve damage/impact tolerance and energy absorption

- Crack bridging mechanisms stabilize the response of multiply delaminated beams in the free vibration phase after the removal of the load; they reduce crack speed and may lead to crack arrest

- Plasticity of the core reduces shielding of the fracture parameters; the problem solution becomes non unique and the response dependent on loading and propagation history; work is in progress ….

problemi di frattura e danneggiamento in materiali e

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