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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
THE EFFECTS OF SHEAR AND NEAR TIP DEFORMATIONS ON ENERGY RELEASE RATE AND MODE MIXITY
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
THE EFFECTS OF SHEAR AND NEAR TIP DEFORMATIONS ON ENERGY RELEASE RATE AND MODE MIXITY
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. (*) :
(*)
ENERGY RELEASE RATE AND STRESS INTENSITY FACTORS
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 AND STRESS INTENSITY FACTORS
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 AND STRESS INTENSITY FACTORS
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 −⎛ ⎞−
= ⎜ ⎟⎜ ⎟⎝ ⎠
-Expression for the energy release rate fully resolve all crack problems where conditions are either mode I or mode II (e.g. laboratory specimens)
ENERGY RELEASE RATE AND STRESS INTENSITY FACTORS
( ) ( )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 −⎛ ⎞−
= ⎜ ⎟⎜ ⎟⎝ ⎠
Stress Intensity Factors:
Semi-analytically derived constants:
ENERGY RELEASE RATE AND STRESS INTENSITY FACTORS
( ) ( )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 −⎛ ⎞−
= ⎜ ⎟⎜ ⎟⎝ ⎠
Stress Intensity Factors:
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)
ENERGY RELEASE RATE AND STRESS INTENSITY FACTORS
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
(Budiansky, Hutchinson & Evans, 1986; Rose, 1987)
2
b
IC
b
IC
0u
cSSB 1
8 ⎟⎟⎠
⎞⎜⎜⎝
⎛−+=
GG
GG
σπ Ewl
(Bao & Suo, 1992; Cox & Marshall, 1994)
ASYMPTOTIC SOLUTIONS FOR BRIDGED CRACKSTHE SMALL SCALE BRIDGING LIMIT
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
(Budiansky, Hutchinson & Evans, 1986; Rose, 1987)
2
b
IC
b
IC
0u
cSSB 1
8 ⎟⎟⎠
⎞⎜⎜⎝
⎛−+=
GG
GG
σπ Ewl
(Bao & Suo, 1992; Cox & Marshall, 1994)
ASYMPTOTIC SOLUTIONS FOR BRIDGED CRACKSTHE SMALL SCALE BRIDGING LIMIT
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:
(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
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
LARGE SCALE BRIDGING FRACTURE INTHROUGH-THICKNESS REINFORCED LAMINATES
DISCONTINUOUS TTR(Fibrous/metallic Z-pins)
CONTINUOUS TTR (stitching / weaving)
Titanium z-pins
Load versus mid-span deflection curves in ENF specimens
(3TEX)
3D woven composites
stitchedunstitched
LARGE SCALE BRIDGING FRACTURE INTHROUGH-THICKNESS REINFORCED LAMINATES
DISCONTINUOUS TTR(Fibrous/metallic Z-pins)
CONTINUOUS TTR (stitching / weaving)
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
(Massabò & Cox, 1999)
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
1/ 2ACK b IIC(0) ( )τ τ β= + GIIACK ( 4 )l E tβ=
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)
TRANSITION FROM NON-CATASTROPHIC TO CATASTROPHIC FAILURE IN SLENDER BODIES
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
(Massabò & Cox, 1999)
TRANSITION FROM NON-CATASTROPHIC TO CATASTROPHIC FAILURE IN SLENDER BODIES
1/ 2ACK b IIC(0) ( )τ τ β= + GIIACK ( 4 )l E tβ=
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
(Massabò & Cox, 1999)
TRANSITION FROM NON-CATASTROPHIC TO CATASTROPHIC FAILURE IN SLENDER BODIES
1/ 2ACK b IIC(0) ( )τ τ β= + GIIACK ( 4 )l E tβ=
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
continuous stitching
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
continuous stitching
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
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
FLEXURAL RESPONSE OF BRITTLE MATRIX COMPOSITES WITH DISCRETE DUCTILE REINFORCEMENTS
IC
5.0u
P KhN ρσ
=
Localized rotation
EXPERIMENTS
(Levi, Bosco & Debernardi, 1998Bosco, Carpinteri & Debernardi, 1990)
FLEXURAL RESPONSE OF BRITTLE MATRIX COMPOSITES WITH DISCRETE DUCTILE REINFORCEMENTS
IC
5.0u
P KhN ρσ
=
Localized rotation
EXPERIMENTS
FLEXURAL RESPONSE OF BRITTLE MATRIX COMPOSITES WITH DISCRETE DUCTILE REINFORCEMENTS
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)
FLEXURAL RESPONSE OF BRITTLE MATRIX COMPOSITES WITH DISCRETE DUCTILE REINFORCEMENTS
EXPERIMENTS
(Jenkins et al., 1987)
FLEXURAL RESPONSE OF BRITTLE MATRIX COMPOSITES WITH DISCRETE DUCTILE REINFORCEMENTS
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)
TRANSITIONS IN THE FLEXURAL BEHAVIOR OF FIBER REINFORCED COMPOSITES
hhs c
0u
ICbc0u
c
E)(
σσGGG +
==
250IC
b =GG
increasingEs
Es increasing
TRANSITIONS IN THE FLEXURAL BEHAVIOR OF FIBER REINFORCED COMPOSITES
hhs c
0u
ICbc0u
c
E σσGGG +
==
250IC
b =GG
(Massabo, 1999; Carpinteri & Massabò,1997)
increasingEs
Es increasing
TRANSITIONS IN THE FLEXURAL BEHAVIOR OF FIBER REINFORCED COMPOSITES
hhs c
0u
ICbc0u
c
E σσGGG +
==
250IC
b =GG
EsEs
(Massabo, 1999; Carpinteri & Massabò,1997)
EXPERIMENTS
(Jamet, Gettu et al., 1995; Jenq and Shah, 1986)
Es
Es
TRANSITIONS IN THE FLEXURAL BEHAVIOR OF FIBER REINFORCED COMPOSITES
hhs c
0u
ICbc0u
c
E σσGGG +
==
250IC
b =GG
(Massabo, 1999; Carpinteri & Massabò,1997)
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
TRANSITIONS IN THE FLEXURAL BEHAVIOR OF FIBER REINFORCED COMPOSITES
hhs c
0u
ICbc0u
c
E σσGGG +
==
250IC
b =GG
(Massabo, 1999; Carpinteri & Massabò,1997)
increasingEs
Es increasing
TRANSITIONS IN THE FLEXURAL BEHAVIOR OF FIBER REINFORCED COMPOSITES
(Massabo, 1999; Carpinteri & Massabò,1997)
bridged crack model
cohesive crack model
progressive brittleness
progressive ductility
Es
Es
increasing
increasing
TRANSITIONS IN THE FLEXURAL BEHAVIOR OF FIBER REINFORCED COMPOSITES
(Massabo, 1999; Carpinteri & Massabò,1997)
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)
Bridging/contactsurfaces
Core and/or interface - nonlinear Winkler foundation
Laminate or
upperface sheet
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)
(Cavicchi and Massabò, 2009, proc. ICCM17)
(Andrews, Massabò, Cavicchi & Cox, IJSS, 2009)
(Andrews & Massabò, Comp. A, 2008)
(Cavicchi and Massabò, 2009, proc. AIMETA)
MECHANICAL MODELSFULLY BACKED COMPOSITE SANDWICHES
(a) (b)
Bridging/contactsurfaces
Core/interface – n.l. Winkler foundation
p(t)
Timoshenko beam
p(t) Cohesive/contactinterfaces
(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
Bridging/contactsurfaces
Core/interface – n.l. Winkler foundation
p(t)
Timoshenko beam
p(t) Cohesive/contactinterfaces
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)?
(Andrews, Massabò & Cox, IJSS, 2006)
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)
(Andrews, Massabò, Cavicchi & Cox, IJSS, 2009)
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
Assumptions:- frictionless contact- perfectly brittle material
2L
P
hh
2L
P
hhP
time
LoadP
time
Load
MULTIPLE DYNAMIC DELAMINATION FRACTURE
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)
Time history of crack growth
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
Assumptions:- frictionless contact- perfectly brittle material
2L
P
hh
2L
P
hhP
time
LoadP
time
Load
MULTIPLE DYNAMIC DELAMINATION FRACTURE
(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)
Assumptions:- frictionless contact- perfectly brittle material
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
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
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 cracks
equally spaced cracks
unequally spaced, upper crack
unequally spaced, lower crack
equally spaced cracks
CONTROLLED DELAMINATION FRACTURE TO IMPROVE MECHANICAL PERFORMANCE AGAINST DYNAMIC LOADINGS
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
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 cracks
equally spaced cracks
unequally spaced, upper crack
unequally spaced, lower crack
equally spaced cracks
Energy absorption through multiple delamination fracture in laminated plates can be optimized via a material design that favour crack formation along predefined planes
CONTROLLED DELAMINATION FRACTURE TO IMPROVE MECHANICAL PERFORMANCE AGAINST DYNAMIC LOADINGS
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
Displacement controlled dynamic growthwith fixed initial strain energy L
Time history of crack growth
ah
105o
L ha h
==
/Ltc h
cr / 0.086=G L
time
w(t) quasi-static while preventing crack growth
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
Results for: leading to small / moderate crack speeds
ENERGY ABSORPTION THROUGH MULTIPLE DELAMINATIONINFLUENCE OF CRACK BRIDGING MECHANISMS
Displacement controlled dynamic growthwith fixed initial strain energy L
Time history of crack growth
ah
105o
L ha h
==
/Ltc h
cr / 0.086=G L
time
w(t) quasi-static while preventing crack growth
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=
(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)
a
h
L
w(t)
Results for: leading to small / moderate crack speeds
ENERGY ABSORPTION THROUGH MULTIPLE DELAMINATIONINFLUENCE OF CRACK BRIDGING MECHANISMS
Displacement controlled dynamic growthwith fixed initial strain energy L
time
w(t) quasi-static while preventing crack growth
wst
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=
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
Time history of crack growth
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
INTERACTION EFFECTS IN SANDWICH BEAMSSHIELDING OF FRACTURE PARAMETERS AND ENERGY BARRIERS
(quasi static loading, elastic foundation)
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
INTERACTION EFFECTS IN SANDWICH BEAMSSHIELDING OF FRACTURE PARAMETERS AND ENERGY BARRIERS
(quasi static loading, elastic foundation)
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)
Perfect core/skin interface
elastic response
La
Uh
2P
hLh
plane ofsymmetry Elastic-plastic core
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
(crack growth criterion)
Perfect core/skin interface
a 12.8h
=
a 12.5h
=
a 12.0h
=
fcr
0.01T h
=crG
INFLUENCE OF CORE PLASTICITY ON FRACTURE PARAMETERS AND MECHANICAL RESPONSE
(static loading, elastic-plastic foundation)
La
Uh
2P
hLh
plane ofsymmetry Elastic-plastic core
Uh / h 0.5=L / h 100= crG = G
K
w
fT
fcrT
3hK 10E
−=5cr 10Eh
−=G
(crack growth criterion)
Perfect core/skin interface
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
INFLUENCE OF CORE PLASTICITY ON FRACTURE PARAMETERS AND MECHANICAL RESPONSE
(static loading, elastic-plastic foundation)
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
(crack growth criterion)
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
(crack growth criterion)
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 ….