1 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 Laboratory Tunisia Polytechnic School, University of Carthage, Tunisia to IIMEC during 2011 Contribution of Applied Mechanics and Systems Research Laboratory Tunisia Polytechnic School, University of Carthage, Tunisia to IIMEC during 2011

2 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 Laboratory Tunisia Polytechnic School, University of Carthage, Tunisia to IIMEC during 2011 Contribution of Applied Mechanics and Systems Research Laboratory Tunisia Polytechnic School, University of Carthage, Tunisia 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 Laboratory Tunisia Polytechnic School, University of Carthage, Tunisia to IIMEC during 2011 Contribution of Applied Mechanics and Systems Research Laboratory Tunisia Polytechnic School, University of Carthage, Tunisia to IIMEC during 2011

4 PhD student Mongi REKIK who started his research in a topic related to IIMEC will defend his thesis in March 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 Laboratory Tunisia Polytechnic School, University of Carthage, Tunisia to IIMEC during 2011 Contribution of Applied Mechanics and Systems Research Laboratory Tunisia Polytechnic School, University of Carthage, Tunisia to IIMEC during 2011

International Institute for Multifunctional Materials for Energy Conversion January 2012, Texas A&M University, College Station, Texas, USA An Embedded Crack in a Functionally Graded MagnetoElectroElastic Medium Mongi Rekik, Sami EL-BORGI Applied Mechanics and Systems Research Laboratory Tunisia Polytechnic School University of Carthage, Tunisia Zoubeida OUNAIES Department 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 Outline Outline

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 PiezoMagnetic MagnetoElectoElastic

8 no thermo-magneto- electro-mechanical properties mismatch Functionally Graded MagnetoElectoElastic Material Introduction and motivation (2) Introduction and motivation (2) FGMEEM 0% 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 ,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 non- homogeneity on the stress (mechanical), electric displacement and magnetic induction intensity factors. Crack y r  x

11 Plane Problem description and formulation (1)

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) Linear constitutive relations Neglecting body forces, local electric charge and current the mechanical equilibrium and Gauss’s laws for electricity and magnetism

14 The MagnetoElectroElasticity partial differential equations are given by Plane Problem description and formulation (4)

15 Plane Problem description and formulation (5) Boundary conditions The crack surface loadings : Continuity conditions along the interface : Regularity conditions :

Sollution cracked medium: Where Integral equation Where Projecting the density function on Chebyshev polynomials, The integral equation becomes Collocation 16 Thermal Problem

17 Derivation of the Singular Integral Equations (2) Derivation of the Singular Integral Equations (2) above crack : The displacement fields under crack: where are the roots of the characteristic polynomial of the magnetoelectroelasticity equations system and are the UNKNOWNS.

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 8 th order characteristic polynomial Roots extraction

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:

20 Derivation of the Singular Integral Equations (4) Derivation of the Singular Integral Equations (4) become the only UNKNOWNS of the problem. Injecting the Fourrier transforms into the constitutive equations and applying the crack surface loading conditions yields:

21 The system of coupled singular integral equations: Derivation of the Singular Integral Equations (5) Derivation of the Singular Integral Equations (5)

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:

23 Analytically integrating the singular terms: Solution of the plane SIE (2) Solution of the plane SIE (2)

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

25 Results and discussion - plane problem (1) Results and discussion - plane problem (1) normalized Temperature under uniform thermal loading

26 Results and discussion - plane problem (2) Results and discussion - plane problem (2) normalized fields’ intensity factors under normal electric displacement

27 Results and discussion - plane problem (3) Results and discussion - plane problem (3) 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. Conclusion Conclusion

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

31 Work in progress (Contact Mechanics of FGMEEM) Work in progress (Contact Mechanics of FGMEEM) (y, v) FGMEEM half-plane  aterial gradient N, Q, J F= ηN Electromagnetic conductor rigid stamp a b (x, u)

32 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 FGMEEM) Future work (Contact Mechanics of FGMEEM)

33 شكرا Thank you