A Synergistic Multiscale Modeling Approach to Damage in Composites
Multiscale Modelling of Multifunctional Composites
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Multiscale Modelling of Multifunctional Composites
Yehia Bahei-El-Din & Amany Micheal
Center of Advanced Materials CAMThe British University in Egypt BUE
Third Annual Meeting of IIMEC College Station, Texas, January 18-19, 2012
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SPONSORS
• International Institute for Multifunctional Materials for Energy Conversion IIMEC
• Air Force Office of Scientific Research AFOSR
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Collaboration With IIMEC
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• Participation of IIMEC affiliates in International Workshop on Advanced Materials for Wind Turbine Blades organized by CAM • IIMEC offered summer internships to BUE students • Collaboration with Drs. Zoubeida Ounaies and Pradeep Sharma
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IIMEC Egypt
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Multiscale Modeling (1/2)
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Laminate Analysis
&Composite (Ply)
Model
Laminate Scale Phase Scale
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Averaging Models
•Mori-Tanaka
•Hill’s SCM
fiberMatrix
M-T Composite
SCM
Periodic Array•PHA
Idealized
RVEThe British University in Egypt
Multiscale Modeling (2/2)
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Nye 1957The British University in Egypt
Multifunctionality
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Constitutive Laws Of A Single PhaseElectro-Thermo-Mechanical Coupling (1/5)
Direct Mechanical Effect
Stress σ (N/m2 ) Strain ε
Stiffness L (N/m2 )
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Compliance M (m2/N)
Electrical Displacement Electrical Field Intensity D (C/m2) E (V/m)
Direct Electrical Effect
Permittivity κ (C/Vm)
Permittivity-1 κ-1 (Vm/C)
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Constitutive Laws Of A Single SolidElectro-Thermo-Mechanical Coupling (2/5)
Thermo-Mechanical Coupling
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Strain ε Temperature (oC)
Coeff.of thermal expansion α /oC
Electrical Displacement D Strain ε (C/m2)
Piezoelectric constant e (C/m2)
Electro-Mechanical Coupling
Strain ε Electrical Field E (V/m)
Piezoelectric constant d T (m/V)
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Constitutive Laws Of A Single SolidElectro-Thermo-Mechanical Coupling (3/5)
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Temperature Electrical (oC) Displacement D (C/m2)
Pyroelectric constant q (C/m2/oC)
Thermo-Electrical Coupling
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Constitutive Laws Of A Single SolidElectro-Thermo-Mechanical Coupling (4/5)
,( )LL
E
Te E
,)(D qd
E
,)(M
T
EEd
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Direct Effect Coupling Effect
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M L
Constitutive Laws of A Single SolidElectro-Thermo-Mechanical Coupling (5/5)
• Remain in the system following mechanical loading/unloading
• Lump up of induced thermal, electrical and damage effect
• Function of mechanical and/or physical properties of material
eigenstress eigenstrain
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L
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Microscopically HeterogeneousMulti Phase Materials
• Levin’s (1967) formula for thermal eigenstresses
• Generalized by Dvorak & Benveniste (1992)
• Stress concentration factor B depends on local elastic properties and geometry
1,
Tr r r
r Q
c
B
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Local FieldsMulti Phase Materials
, 1) Due to overall fields
r r B
• Hill (1967)
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Local FieldsMulti Phase Materials
, 1) Due to overall fields
2) Self-Induced by eigen fields
r rr r F
• Eshilby (1956)
• Hill (1967)
r
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Local FieldsMulti Phase Materials
,
r
1) Due to overall fields
2) Self-Induced by eigen fields
3) Transformed by eigen fields
r rs s F
• Dvorak (1992)
s
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Transformation Field Analysis
1,
r r rs ss Q
B F
• Local eigen fields caused by deformation mechanisms are known functions of stress, temperature, Electric Field, internal parameters
؟
• Damage ?
• Dvorak (1992)
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1,
B Fr r i rs ss Q
• Bahei-El-Din (2004)
؟
Transformation Field Analysis of Damage
1.. .. .. .. ..
.. ...... .. .. ..
F Bs rs r i r
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Damage CriteriaFailure Criterion
Description
I Local Ply In-plane Phase Failure due to Tension or Compression Strength
II Matrix Failure Due to Transverse Shear
III Sliding Failure due to Local Longitudinal Shear
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Examples
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(0/90)s Laminate
(0/±45/90)s Laminate
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Electrical Field Intensity E versus Electrical Displacement D for Different
Layup
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Temperature-Electric Displacement for Different Layup
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Stress-Electric Field Intensity Path for a (0/90)s Laminate
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Stress-Strain Response for a (0/90)s Laminate Under Overall Stress in X1
Direction
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Stress-Electric Displacement for the 0 Ply in a (0/90)s Laminate Under Overall
X1 Stress
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Stress-Electric Displacement for the 90 Ply in a (0/90)s Laminate Under
Overall X1 Stress
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Electric Field Intensity-Electric Displacement at Different X1 Tension Stress Levels on a (0/90)s Laminate
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Stress-Strain Response of a Sym. (0/±45/90)s Laminate Under Tension in
Overall X1 Direction
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Stress-Electric Displacement for a (0/±45/90)s Under Overall X1 Stress
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Stress-Strain Response for a (0/±45/90)s Laminate Under Overall Shear Stress
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Stress-Electric Displacement for a (0/±45/90)s Under Overall Shear Stress
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Conclusion
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• A Multiscale Study is conducted on a Laminate composite
• Constituents are multifunctional materials with electro-thermo- mechanical coupling
• All effects other than mechanical , including damage, are lumped up and treated as transformation or eigen effects
• Laminate layup affects the direct electric response of PZT fibers in a certain ply due to confinement caused by other plies
• It is concluded that local damage due to all effects in a certain ply changes the electric response of piezoelectric fibers in all plies with different aspects
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