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

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

IIMEC Egypt

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Multiscale Modeling (1/2)

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Laminate Analysis

&Composite (Ply)

Model

Laminate Scale Phase Scale

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)

Nye 1957The British University in Egypt

Multifunctionality

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)

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)

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

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

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

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

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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