Functionally graded materials in the modelling lightweight plates

1. Functionally graded materialsin the modelling lightweight plates JAROSŁAWJĘDRYSIAK BOHDANMICHALAK Department of Structural MechanicsŁódź University of Technology JOWITARYCHLEWSKA CZESŁAWWOŹNIAK Institute of Mathematics and ComputerSciences Technological University of Częstochowa

2. Outline Introduction Formulation of the problem Auxiliary concepts Modelling assumptions 2D-model equations Conclusions

3. 1. Introduction Functionally graded materials (FGM)* are materials, which have properties continuously varying with position within the elementto avoidinterface problems, such as, stress concentrations,cf. Suresh and Mortensen (1998). • S. Suresh, A. Mortensen, Fundamentals of functionally graded materials.The University Press, Cambridge 1998. (*) The term ”FGM” was introduced in the mid-1980s in Japan (the Spaceplane project): M. Niino, S. Maeda (1990); M. Koizumi (1992); T. Hirai (1996) ØExample offunctionally graded materials Fig. 1. Etched cross-section of a Japanese sword blade(cf. Suresh and Mortensen (1998))

4. ØExamples ofcomposite plates Fig. 2. Fragment of a sandwich-type plate Fig. 3. Fragment of a FGM-type plate

5. 2. Formulation of the problem Object under consideration:linear-elastic plates made of functionally graded material along the axis perpendicular to the plate midplane.

6. The aim of the contribution: (i)to propose new 2D-model of elastic plates, made of a functionally graded material, to analyse dynamical problems, employing concepts introduced in the tolerance averaging technique for periodic structures, cf. Woźniak and Wierzbicki (2000). • Cz. Woźniak, E. Wierzbicki, Averaging Techniques in Thermomechanics of Composite Solids,Wydawnictwo Politechniki Częstochowskiej, Częstochowa, 2000.

7. ØThe dividing of the plate Fig. 4. where: H=ml, l is a segment thickness, m is a natural number, m‑1<<1. Remark The plate is made of the very thin laminae and that is way it will be referred to asthefunctionally graded laminated elastic plate orthe FGL plate.

8. üAveraging operation on the segmentIn üAveraging operation on (‑l/2,l/2) üSlowly varying sequence: {fn}ÎSV üSlowly varying function: f(x,·)ÎSV üDifference quotients: üThe shape function: g=g(z), zÎ[-H,H] Fig. 5. Example of the shape function 3. Auxiliary concepts

9. w=w(x,z,t) - the displacement field of the plate,. ØAssumption 1 The averaged part ofthe displacement is linearly approximated. Fig. 6. ØAssumption 2 The residual part of thedisplacement is linearly approximated. 4. Modelling assumptions

10. ØAssumption 3 Terms O(e) are assumed to be neglected in the course of the modelling, i. e.: ØThe plate-bending assumptions (for 2D model)

11. Denotations: The microstructural 2D‑platemodel where:underlined terms are dependent onthe segment thicknessl; where: un=0 if n=0, if n=m. The macroscopic 2D‑platemodel without thesegment thickness l. 5. 2D-model equations

12. 6. Conclusions • The new modelling approach proposed to describe FGL elastic platesis based on the concepts formulated for periodic composites and structures within the tolerance averaging technique, cf. Woźniak and Wierzbicki (2000). • The governing equations of 2D-model of functionally graded elastic plates are derived. • Using the proposed model, lightweight FGL plates can be designed and analysed, avoiding stress concentrations, typical for sandwich plates.