Functionally graded materials

in the modelling lightweight plates

JAROSŁAW JĘDRYSIAK

BOHDAN MICHALAK

Department of Structural MechanicsŁódź University of Technology

JOWITA RYCHLEWSKA

CZESŁAW WOŹNIAK

Institute of Mathematics and Computer Sciences Technological University of Częstochowa

Outline

Introduction

Formulation of the problem

Auxiliary concepts

Modelling assumptions

2D-model equations

Conclusions

1.

2.

3.

4.

5.

6.

1. Introduction

Example of functionally graded materials

Functionally graded materials (FGM)* are materials, which have properties continuously varying with position within the elementto 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)

Examples of composite plates

x

phase 1z

H

H

phase 2

x

phase 1z

H

H

phase 2

Fig. 2. Fragment of a sandwich-type plate

Fig. 3. Fragment of a FGM-type plate

2. Formulation of the problem

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

x

phase 1z

H

H

phase 2

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.

The dividing of the plate

where: H=ml, l is a segment thickness, m is a natural number, m‑1<<1.

RemarkThe plate is made of the very thin laminae and that is way it will be referred to asthe functionally graded laminated elastic plate or the FGL plate.

Fig. 4.

x

z

z=-ml

z=ml nth segment (n=±1,…,±m)

nth interface (n=0,±1,…,±(m–1))

n=±m – lower and upper boundary

3. Auxiliary concepts Averaging operation on the segment In

Averaging operation on (‑l/2,l/2)

Slowly varying sequence: {fn}SV

Slowly varying function: f(x,·)SV Difference quotients: The shape function: g=g(z), z[-H,H]

),( ),,( zfzf

Fig. 5. Example of the shape function

z

g 3l

3l

1 1 1 1 1 1

2 2 2 2 2 2

1 – phase 1; 2 – phase 2

4. Modelling assumptionsw=w(x,z,t) - the displacement field of the plate, .RtHHz ],,[ ,x

)],(),([))1((),(

),,(

11 ttllnzt

tz

nnn

n

xuxuxu

xw

Assumption 1The averaged part of the displacementis linearly approximated.

)]},(),([))1((

),(){(),,(~

11 ttllnz

tzgtz

nn

nnn

xvxv

xvxw

Assumption 2The residual part of the displacementis linearly approximated.

Fig. 6.

uin

ui(n+1) z

„n”

„n–1”

„n+1”

1

2

inw

in

i(n+1) z

„n”

„n–1”

„n+1”

1

2

inw~

).(

),(

),(

O

O

O

nnnn

nnnn

nnnn

g

g

g

vuw

vuw

vuw

Assumption 3Terms O() are assumed to be neglected in the course of the modelling, i. e.:

The plate-bending assumptions (for 2D model)

0),( ),,(),(

),,(),( ),,(),(

3

3

ttt

tutututu

nnn

nnn

xxx

xxxx

5. 2D-model equationsDenotations:

.,,)(

,,,),,(),,(

11

2121

m

n n

m

n nnnn

nnnnnnn

nnnnnn

g

gCuu

FFeeH

eeGeeFeeeevu

The macroscopic 2D‑plate model

without the segment thickness l.

0):()]([

,0)(1

nnnnnn

m

n nn

u

uu

uuEu

uEE

The microstructural 2D‑plate model

where: underlined terms are dependent on the segment thickness l; where: un=0 if n=0, if n=m.

1,,1,0)():(

,0):(])([

,0)(

22

11

mnull

u

uu

nnnnnnnn

nnnnnnnn

m

n nn

m

n nn

uGvvHv

uvGuFu

vGuFF

0 unu

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.