1 initiation of joint research projects on piezoelectric composites (m. chafra, n. chafra, z....

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• Initiation of joint research projects on Piezoelectric composites (M. Chafra, N. Chafra, Z. Ounaies)

Fracture mechanics of Functionally Graded MagnetoElectroElastic Composites, FGMEEM (M. Rekik, S. El-Borgi and Z. Ounaies)

Flexoelectric properties of ferroelectrics and the nanoindentation size-effect (P. Sharma, M. Gharbi, S. El-Borgi)

• Four journal manuscripts completed in the area of fracture mechanics of FGMEEM (1 accepted et 3 under review)

• Two research proposals funded the Moroccan and Tunisian Governments Multifunctional materials and adaptive structures (L. Azrar, F. Najar, W.

Gafsi, S. El-Borgi)

Natural fiber composites for windmill applications (H. Kadimi, M. Chafra)

Contribution of Applied Mechanics and Systems Research Contribution of Applied Mechanics and Systems Research Laboratory Tunisia Polytechnic School, University of Carthage, Laboratory Tunisia Polytechnic School, University of Carthage,

Tunisia to IIMEC during 2011Tunisia to IIMEC during 2011

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• Dr. M. Chafra obtained a three-month Fulbright scholarship to work with Dr Z. Ounaies at Penn State University on Natural Fiber Composites

• Dr Najar, Dr Chafra and Dr Z Ounaies are jointly supervising PhD student Ahmed Jemai in his research dealing with the development of a top down approach to increase the performance of AFC for energy harvesting applications based on continuum modeling.

• Student exchange between EPT and Texas A&M University and Penn State University (4 students)

• Organizing oral sessions at the ICAMEM2010 conference (International Conf. on Advances in Mech. Eng. & Mechanics)

• Presenting our IIMEC work at the NSF funded US-Tunisia Workshop on Research and Educational Advances in Smart Micro-sensing and Biomimetic sensors which was organized by Michigan State and our laboratory in December 2010 in Tunisia.

Contribution of Applied Mechanics and Systems Research Contribution of Applied Mechanics and Systems Research Laboratory Tunisia Polytechnic School, University of Carthage, Laboratory Tunisia Polytechnic School, University of Carthage,

Tunisia to IIMEC during 2011Tunisia to IIMEC during 2011

3

•Collaboration with Prof Pradeep Sharma on nanoindentation of Ferroelectric materials

•Published a paper in international journal.

Contribution of Applied Mechanics and Systems Research Contribution of Applied Mechanics and Systems Research Laboratory Tunisia Polytechnic School, University of Carthage, Laboratory Tunisia Polytechnic School, University of Carthage,

Tunisia to IIMEC during 2011Tunisia to IIMEC during 2011

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• PhD student Mongi REKIK who started his research in a topic related to IIMEC will defend his thesis in March 2012.

• He was jointly supervised by Prof Zoubeida Ounaies and Sami El-Borgi

• He worked on fracture mechanics of Functionally Graded Magneto Electro Elastic Materials under Thermo Electro Magneto Mechanical Loading.

• Four journal manuscripts completed in the area of fracture mechanics of FGMEEM (1 accepted and 3 under review).

Contribution of Applied Mechanics and Systems Research Contribution of Applied Mechanics and Systems Research Laboratory Tunisia Polytechnic School, University of Carthage, Laboratory Tunisia Polytechnic School, University of Carthage,

Tunisia to IIMEC during 2011Tunisia to IIMEC during 2011

International Institute for Multifunctional Materials for Energy Conversion

18-19 January 2012, Texas A&M University, College Station, Texas, USA

An Embedded Crack in a Functionally Graded MagnetoElectroElastic Medium

Mongi Rekik, Sami EL-BORGIApplied Mechanics and Systems Research Laboratory

Tunisia Polytechnic SchoolUniversity of Carthage, Tunisia

Zoubeida OUNAIESDepartment of Mechanical Engineering

Pennsylvania State University, USA

6

•Introduction and motivation•Plane Problem

Problem description and formulation Derivation of Singular Integral Equations Solution of the SIE

•Axisymmetric Problem Problem description and formulation Derivation of Singular Integral Equations Solution of the SIE

•Results and discussion

OutlineOutline

7

High residual and thermo-magneto-electric stresses

Mismatch in thermo-Magneto-Electro-Mechanical properties

The interface fails due to cracks

Introduction and motivation (1)Introduction and motivation (1)

PiezoElectric

PiezoMagneticMagnetoElectoElastic

8

no thermo-magneto-electro-mechanical properties mismatch

Functionally Graded MagnetoElectoElastic Material

Introduction and motivation (2)Introduction and motivation (2)

FGMEEM

0%

100%

100%

0%

9

Introduction and motivation (3)Introduction and motivation (3)

Delamination at the interfaces

10

Former studies considered only mode III crack problem•Feng et al 2006, 2007•Ma et al 2007 & 2009•Li et al 2008-1,2•Zhou et al 2008•Guo 2009

The mode I and II crack problem is not studied yet

Introduction and motivation (4)Introduction and motivation (4)

The purpose of this project is to study the influence of the material The purpose of this project is to study the influence of the material non-homogeneity on the stress (non-homogeneity on the stress (mechanicalmechanical), electric displacement ), electric displacement and magnetic induction intensity factors.and magnetic induction intensity factors.

)()()(),( III

ij

IIIII

ij

III

ij

I

ij r

k

r

k

r

kr Crack

y r

x

11

Plane Problem description and formulation (1)Plane Problem description and formulation (1)

1 3 10 30, , e ,yk k k k

15 31 33 150 310 330, , , , e ,y

1 3 10 30, , e ,yp p p p

1 3 10 30, , e ,ym m m m

11 13 33 44 110 130 330 440, , , , , , e ,yc c c c c c c c

15 31 33 150 310 330, , , , e ,ye e e e e e

15 31 33 150 310 330, , , , e ,yf f f f f f

11 33 110 330, , e ,y

11 33 110 330, , e ,yg g g g

11 33 110 330, , e ,y

Plane Problem description and formulation (2)Plane Problem description and formulation (2)

•Crack surfaces are assumed to be magnetoelectrically impermeable

•Crack surfaces are subjected to Thermal loading mechanical tangential and normal tractions electric displacement magnetic field

Which are related to external loads

•Body forces and local electric charge are neglected

13

•Small excitations

Plane Problem description and formulation (3)Plane Problem description and formulation (3)

•Linear constitutive relations

E grad >>>>>>>>>>>>>>

•Neglecting body forces, local electric charge and current the mechanical equilibrium and Gauss’s laws for electricity and magnetism

1

2

Tgrad u grad u >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

H grad >>>>>>>>>>>>>>

c e f

D e g E p T

f g mB H

>>>>>>>>>>>>>>>>>>>>>>>>>>>>

>>>>>>>>>>>>>>>>>>>>>>>>>>>>

0Div 0Div D >>>>>>>>>>>>>> 0Div B

>>>>>>>>>>>>>>

q k grad T>>>>>>>>>>>>>>

0Div q

14

• The MagnetoElectroElasticity partial differential equations are given by

Plane Problem description and formulation (4)Plane Problem description and formulation (4)

11 13 31 31 44 44 15 15

44 44 15 15 13 33 33 33

15 15 11 11 31 33

15 15 11 11

c c e f c c e fx y y y y x x x

c c e f c c e fy x x x x y y y

x ye e g e e

y x x x x

f f gy x x x

11

33

3333 33

31 33 33 33 33

ux

v y

pgyy y y

f f g mx y y y y

1 3 0T T T T

k kx x y y

15

Plane Problem description and formulation (5)Plane Problem description and formulation (5)

• Boundary conditions

The crack surface loadings :

Continuity conditions along the interface :

Regularity conditions :

x a

13 13,0 ,extx x 33 33,0 ,extx x

3 3,0 ,extD x D x 3 3,0 ,extB x B x

,0 ,0 ,u x u x ,0 ,0 ,v x v x

,0 ,0 ,x x ,0 ,0 ,x x

, 0,u x y , 0,v x y

, 0,x y , 0,x y

y

x a

*30 0

,0T xk k Q

y

,0 ,0T x T x

, 0 0T x y

• Sollution cracked medium:Where

• Integral equation

Where

• Projecting the density function on Chebyshev polynomials,The integral equation becomes

• Collocation

16

Thermal ProblemThermal Problem

12 1, n y i xT x y A e d

* 01 2

2 1 30

1 ,2

ai x t

th

a

Qn nie d t tdt k

n n k

2 230 10 0k n n k

2 2,0 ,0 ,0 ,0th x T x T x T x T xx x

1

0 *1

1 1

,1

1 ²

Nth n

n th nn

k t x T ta k U x dt k

t

2 1cos , 1.. ,

2i

ix i N

N

17

Derivation of the Singular Integral Equations (2)Derivation of the Singular Integral Equations (2)

above crack :

• The displacement fields

under crack:

4

1

, ,km y i x th i xk

k

u x y C e e d U e d

4

1

, ,km y i x th i xk k

k

v x y C r e e d V e d

4

1

, ,km y i x th i xk k

k

x y C s e e d e d

4

1

, ,km y i x th i xk k

k

x y C t e e d e d

8

5

, ,km y i x th i xk

k

u x y C e e d U e d

8

5

, ,km y i x th i xk k

k

v x y C r e e d V e d

8

5

, ,km y i x th i xk k

k

x y C s e e d e d

8

5

, ,km y i x th i xk k

k

x y C t e e d e d

where are the roots of the characteristic polynomial of the magnetoelectroelasticity equations system and are the UNKNOWNS.

km

, 1..8kC k

18

Derivation of the Singular Integral Equations (1)Derivation of the Singular Integral Equations (1)

• Injecting Fields Fourier transforms in the system of PDE leads to a system of ODE with 8th order characteristic polynomial 4 2

2 1 02

4

0X a X a X a

X m m a

• Roots extraction

2 2 11 2 4 2

1, 4 2 2 2 2 ,

2

am m a b b a

b

2 2 13 4 4 2

1, 4 2 2 2 2 ,

2

am m a b b a

b

2 2 15 6 4 2

1, 4 2 2 2 2 ,

2

am m a b b a

b

2 2 17 8 4 2

1, 4 2 2 2 2 ,

2

am m a b b a

b

19

Derivation of the Singular Integral Equations (3)Derivation of the Singular Integral Equations (3)

• Applying the continuity along the interface

A system of 8 linear equations relating the

to the density functions

• Introducing the density functions:

, 1..8kC k

0y

,0 ,0 ,u x u x u xx

,0 ,0 ,v x v x v x

x

,0 ,0 ,x x xx ,0 ,0 ,x x x

x

20

Derivation of the Singular Integral Equations (4)Derivation of the Singular Integral Equations (4)

become the only UNKNOWNS of the problem.

, ,u v and

•Injecting the Fourrier transforms into the constitutive equations and applying the crack surface loading conditions yields:

11 12 13 14 13 13, , , , ,a a a a

ext thu v

a a a a

K t x t dt K t x t dt K t x t dt K t x t dt x x

21 22 23 24 33 33, , , , ,a a a a

ext thu v

a a a a

K t x t dt K t x t dt K t x t dt K t x t dt x x

31 32 33 34 3 3, , , , ,a a a a

ext thu v

a a a a

K t x t dt K t x t dt K t x t dt K t x t dt D x D x

41 42 43 44 3 3, , , , ,a a a a

ext thu v

a a a a

K t x t dt K t x t dt K t x t dt K t x t dt B x B x

21

• The system of coupled singular integral equations:

Derivation of the Singular Integral Equations (5)Derivation of the Singular Integral Equations (5)

*

*

0 0 0

0 0

, sin sin sin ,D

ij ij ij ij ij ij

D

K t x d k t x d d k t x d k t x d

*

* *

1 1

0

cos, cos cos ,

D

ij ij ij ij ij

D D

t xK t x d t x d d k t x d k d

, 1,1 2..4, 2..4i j or

, 1, 2..4 2..4,1i j or

0

1sin ,t x d

t x

*

*

*

0

cos cos 1ln ln ,

D

D

t x t xd d D t x

22

• The dominant kernels singularities are of Cauchy type;

Solution of the plane SIE (1)Solution of the plane SIE (1)

• Truncated series:

• From the physics of the problem:

0ic 1ib

1 1

2 21 1 ,i ib c

i it t t t , , , ,i u v

1

,1 ²

Nn

u nn

at T t

t

1

.1 ²

Nn

v nn

bt T t

t

1

,1 ²

Nn

nn

ct T t

t

1

.1 ²

Nn

nn

dt T t

t

23

• Analytically integrating the singular terms:

Solution of the plane SIE (2)Solution of the plane SIE (2)

1 111 120 1

11 1 121 1 1

1 113 141 1

13 14 13 13

1 1

, ,1 1

1 ² 1 ²

, ,1 1,

1 ² 1 ²

Nn n n

n n nn

n n n n ext thn n

k t x T t T x k t x T ta k U x dt b k dt

nt t

T x k t x T t T x k t x T tc k dt d k dt

n nt t

1 11 21 0

1 2 11 1 1

33 331 13 40 0

3 1 4 1 3 3

1 1

, ,1 1

1 ² 1 ²

, 2, ,1 1

, 31 ² 1 ²

Nn i n i n

n i n i nn

ext th

i n i n ext thn i n n i n

T x k t x T t k t x T ta k dt b k U x dt

n t t

ik t x T t k t x T t

c k U x dt d k U x dt D D it t

3 3 , 4ext thB B i

24

• writing the system in N collocation points a system of 4N equation with 4N unknowns.

Solution of the plane SIE (3)Solution of the plane SIE (3)

2 1cos , i=1..N

2i

ix

N

• The mechanical stresses, electric displacement and magnetic induction intensity factors:

1 220 230 2401

1 1 ,N

n

n n nn

k k b k c k d

2 1101

1 1 ,N

n

nn

k k a

320 330 3401

1 1 ,N

n

D n n nn

k k b k c k d

420 430 4401

1 1 ,N

n

B n n nn

k k b k c k d

25

Results and discussion - plane problem (1) Results and discussion - plane problem (1)

normalized Temperature under uniform thermal loading

10.5

0

-0.5

-1

-1.5

-2

-2.5-2 -1 0 1 2

T(x

,0- )/

T0 a

nd

T(x

,0+)/

T0

x/a

a

-1

0

1

2

-2 -1 0 1 2

a= 0a=1a=2a=3T

(x,0

- )/T

0 a

nd

T(x

,0+)/

T0

x/a

26

Results and discussion - plane problem (2) Results and discussion - plane problem (2)

normalized fields’ intensity factors under normal electric displacement

0

0.04

0.08

0.12

0 1 2 3

k1(1)k1(-1)

k 1/(D

0e 33/

33)

Nonhomogeneity parameter a

-0.1

-0.05

0

0.05

0.1

0 1 2 3

k2(1)k2(-1)

k 2/(D

0e 33/

33)

Nonhomogeneity parameter a

1

1.1

1.2

1.3

0 1 2 3

kD(1)kD(-1)

k D/D

0

Nonhomogeneity parameter a

0.2

0.6

1

0 1 2 3

kB(1)kB(-1)

k B/(

D0g 33

/33

)

Nonhomogeneity parameter a

27

Results and discussion - plane problem (3)Results and discussion - plane problem (3)

0

0.01

0.02

0.03

0 1 2 3

k1(1)k1(-1)

k 1/(B

0f 33/

33)

Nonhomogeneity parameter a

-0.04

-0.02

0

0.02

0.04

0 1 2 3

k2(1)k2(-1)

k 2/(B

0f 33/

33)

Nonhomogeneity parameter a

-0.3

-0.2

-0.1

0

0 1 2 3

kD(1)kD(-1)

k D/(

B0g 33

/33

)

Nonhomogeneity parameter a

1

1.05

1.1

1.15

0 1 2 3

kB(1)kB(-1)

k B/B

0

Nonhomogeneity parameter a

normalized fields’ intensity factors under normal magnetic induction

28

• The problem of an embedded crack in a FGMEEM was considered;

• The problem was formulated using the method of Singular Integral Equations (SIEs);

• The SIEs were solved numerically using orthogonal polynomial solutions (Chebyshev polynomials).

• Fields intensity factors (mechanical, electric and magnetic) increase with the nonhomogeneity parameter ;

• Mode I, electric displacement and magnetic induction intensity factors have the same parity opposite to that of mode II intensity factor.

ConclusionConclusion

29

• Completed solving the following problems:

Functionally Graded magneto-electro-elastic Axisymmetric Infinite Medium Subjected to Magneto-Electro-Mechanical Loading

Functionally Graded Pyro-magneto-electro-elastic Plane Infinite Medium Subjected to Arbitrary Loading including Thermal

Functionally Graded Pyro-magneto-electro-elastic Axisymmetric Infinite Medium Subjected to Arbitrary Loading including Thermal

Completed work (Fracture Mechanics of Completed work (Fracture Mechanics of FGMEEM)FGMEEM)

• Submitted three papers about these studies for journal publication (under review)

30

• Consideration of additional crack problems

more complicated geometries (half plane, layer, coating bonded to a homogeneous medium …) and

different crack configurations (embedded or surface crack)

in plane strain or axisymmetric conditions

to be solved analytically using the method of Singular

Integral Equations.

• Development of a finite element for the numerical solution of such problems

Future work (Fracture Mechanics of Future work (Fracture Mechanics of FGMEEM)FGMEEM)

31

Work in progress (Contact Mechanics of Work in progress (Contact Mechanics of FGMEEM)FGMEEM)

(y, v)

FGMEEM half-plane

aterial gradient

N, Q, J

F= ηNElectromagnetic conductor rigid

stamp

a b

(x, u)

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• Consideration of additional contact problems

More complicated geometries (half plane, layer, coating bonded to a homogeneous medium …)

Different punch profiles (flat, triangular, circular, parabolic)

Partial slip contact versus sliding contact

plane strain or axisymmetric conditions

to be solved analytically using the method of Singular

Integral Equations.

• Development of a boundary finite element based tool to solve more complicated contact mechanics problems.

Future work (Contact Mechanics of Future work (Contact Mechanics of FGMEEM)FGMEEM)

33

شكراشكراThank youThank you