1. Functionally graded materials in the modelling lightweight plates J AROSŁAW J ĘDRYSIAK B OHDAN M ICHALAK Department of Structural Mechanics Łódź University of Technology J OWITA R YCHLEWSKA C ZESŁAW W OŹNIAK Institute of Mathematics and Computer Sciences Technological University of Częstochowa

2. Outline Introduction Formulation of the problem Auxiliary concepts Modelling assumptions 2D-model equations Conclusions 1. 2. 3. 4. 5. 6.

3. 1. Introduction  Example of functionally graded materials Functionally graded materials (FGM) * are materials, which have properties continuously varying with position within the element to avoid interface problems, such as, stress concentrations, cf. Suresh and Mortensen (1998). S. Suresh, A. Mortensen,Fundamentals of functionally graded materials. The University Press, Cambridge 1998. Fig. 1. Etched cross- section of a Japanese sword blade (cf. Suresh and Mortensen (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)

4.  Examples of composite 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.  The dividing of the plate 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 as the functionally graded laminated elastic plate or the FGL plate. Fig. 4.

8. 3. Auxiliary concepts Averaging operation on the segment I n Averaging operation on ( ‑ l/2,l/2) Slowly varying sequence:{f n }  SV Slowly varying function:f(x,·)  SV Difference quotients: The shape function:g=g(z), z  [-H,H] Fig. 5. Example of the shape function

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

10.  Assumption 3 Terms  (  ) are assumed to be neglected in the course of the modelling, i. e.:  The plate-bending assumptions (for 2D model)

11. 5. 2D-model equations Denotations: The macroscopic 2D ‑ plate model without the segment thickness l. The microstructural 2D ‑ plate model where:underlined terms are dependent on the segment thickness l; where: u n =0 if n=0, if n=m.

12. 6. Conclusions  The new modelling approach proposed to describe FGL elastic plates is 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.