On tolerance averaged model of the propagation of boundary displacement fluctuationsin FGM-type laminated composite

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Scientific Researchof the Instituteof Mathematicsand Computer Science

ON TOLERANCE AVERAGED MODEL OF THE PROPAGATION OF BOUNDARY DISPLACEMENT FLUCTUATIONS

IN FGM-TYPE LAMINATED COMPOSITE

Lena Łacińska, Ewaryst Wierzbicki

Institute of Mathematics and Computer Science, Czestochowa University of Technology, Poland email: lena@imi.pcz.pl

Abstract. The object of the contribution is the analysis of a certain initial-boundary problems in elastodynamics of FGM-type multiphased laminated composites. The aim of the paper is to formulate an answer to the question what kind of assumptions are enough to be taken into account in the framework of the simplified tolerance model for the elastodynamics of the considered laminates the boundary effect will be observed.

Introduction

The aim of the contribution is to discuss of the tolerance description of be- haviours observed in the linear elastodynamics of multiphased FGM-type laminates being a certain effect of displacement perturbations situated on the boundary of the region occupied by the laminated composite material. To this end the second order tolerance model which is also named a simplified tolerance model is taken into account [1, 2]. It is a certain adaptation of the classical tolerance averaged model to the analysis of problems like boundary effect problems in elastodynamics of a laminated medium.

In the framework of the contribution the latin indices i, j, k, l, p, q, r, s run over 1, 2, 3, greek indices α, β, γ,... r over 1, 2, latin indices A,B,C,... over 1, ..., N, where N a number of shape function which should be postulated in every problem analysed in the framework of the tolerance averaging technique approach.

1. Model equations

The starting point of the considerations is the tolerance averaging model equ-

ation system of the FGM-type laminated composite. It will be assumed that the

Carthesian coordinate system have been introduced and the physical 3-dimensional

space and Ox

1

-axis direction is perpendicular to the laminae interfaces planes. The

microstructure of the FGM-type laminated composite determined by the length λ

(2)

of the repeated layer and thickness l

^A

( z ) of the A-th laminae, A = 1,...,N. We shall introduce averaging operator

/ 2 / 2

, ( / 2, / 2)

( ) 1 ( )

z l z l

z L l L l

f z f y dy

l

+

−

∈ − + −

〈 〉 = ∫

where f(·) is an integrable function defined in the interval (z-l/2, z+l/2). The basic unknowns of this system is the averaged displacement field u together with the fluctuation amplitudes v

^A

, A = 1,...,N. The displacement field U of the laminated composite can be determined under the formula

( , ) t = ( , ) t + h

^A

( ) x

1 ^A

( , ) t

U x u x v x (1)

in which x = (x

1

, x ), x = ( x

₁

, x

₂

) , and h

^A

= lg

^A

, A = 1,...,N, are locally-periodic functions named shape functions which form and a number are strictly connected with the microstructure of the FGM-type laminated composite. For particulars the reader is referred to [3]. Denoting by ρ ( ) ⋅ and C

_ijkl

( ) ⋅ the mass density field and elasticity modulus tensor fields, respectively, the tolerance model equations of the elastodynamics of FGM-type laminated composite can be written in the form

1 1

2 2

1 1 1 1

1 11 1 1 1

1 1

, )

( , ,

) , , , , )

, (

( 0

(

A A A A

i ijkl l k ij l l j ij l l

A B A A B B A B A B

i ij l l j i l i l

A B B A B A B B

i l l i l i l l

A

i kl l k

, g , l g ,

g g g g , g g g g

g g , g g g g ,

g , l

u C u C C

l l C l C C

l C C C

C u

γ γ

γ γ γ γ

βγ β γ β β

ρ

ρ υ υ

υ υ

〉 − =

〉 〉 +

−

〉 − <

〈 〉 − 〈 〉 +〈 〈 〉

〈 − 〈 + 〈 〉 − 〈 〉

− 〈 〉 + 〈 〉 〈 〉 +

+〈

&&

, ), 0

A

ijkl l k j

g C > u =

(2)

Considerations will be restricted to the case in which material gradation of the laminate is situated along the Ox

1

-axis direction. It is mean that coefficients in (2) depend on variable x

1

as on a certain parameter. Hence, model equations system (2) can be reformulated to the form

2 2

) , ,

( [ ] 0

{ } [ ]

0

A A A A

i ijkl l k ijl l j ij l l

AB A AB B AB B AB B A

i i l l i l l il l ikl l k

A

i kl l k

, , l g ,

r , ,

l g C u

u C u C C

l l c ls C C u

γ γ

βγ γβ β β

β β

ρ υ υ

υ υ υ υ

− =

−

− < > =

〈 〉 − 〈 〉 + 〈 〉

− + + +

&&

&& (3)

in which coefficients are given by

1 1 1 1 1 1

1 11 1

[ ] ,

, { } ,

A A AB A B A B

ijl ij l i l i l i l

AB A B AB A B AB A B

ijkl ijkl il i l

g g

C C g , s C g , C g ,

c C g g C g , C g , r g g

β β β

ρ

= =

= = =

〈 〉 〈 − 〉

〈 〉 〈 〉 〈 〉 (4)

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On tolerance averaged model of the propagation of boundary displacement fluctuations ... 135

It must be emphasized that

{ } C

_li^BA

= { } C

^AB_il

(5)

Now we shall to pass to the simplified model. To this end fluctuation amplitude

A

υ

i

will be replaced by the alternative variable w

_i^A

named as internal fluctuation amplitude and interrelated with υ

_i^A

by the formula

hom

A A A

i i

w

i

υ = υ + (6)

in which gradient part of the fluctuation amplitude is determined by

2 hom

1 11 1 1

( g

^A

, ) C

_i _j

υ

^A_j

g

^A

, C

_{i kl}

u

_{k l}

, 0

〈 〉 + 〈 〉 =

Hence, second equations from (3) takes the form

2 2 2 *

), ,

( { }

AB B AB B AB B AB B A

i i l l i l l il l i

r ,

l w && − l c

_βγ

w

_γ _β

+ ls w

_β _β

+ C w = − l res (7)

where

*^A_i

O l ( )

res ∈ (8)

Since res

^*^A_i

∈ O l ( ) the right hand side of (7) is of an order O(l

³

) and hence can be omitted. Denoting by

K_pr^AD

the inverse matrix to { } C

_rq^DB

, i.e.

AD

{ }

DB AB

pr rq pq

K C = δ δ

and introducing denotation

[ ] [ ]

eff A AB A

ijkl ijkl ijp

K

pq qkl

C = 〈 C 〉 − C C (9)

model equations system can be approximately rewritten as

2

,

), ,

[ ] ,

( [ ]

{ } 0

eff A A

i ijkl k l j ikl k l

AB A AB B AB B AB B

i ij l l i l l il l

, w

r w w w w

u C u C

l c

_βγ _γβ

ls

_β _β

C

ρ =

+ =

〈 〉 −

− +

&&

(10)

At the same time formula (1) for total displacements U

i

takes the form

3

1 1

( )

A AB A A A

i

u

i

lg K

ip

C

p rs s r

, lg w

i

O

U = − 〈 g , 〉 u + + λ (11)

It must be emphasized that in the special case of two-phased periodic linear elastic

laminated composite with exclusively shape function h = h

¹

takes the form

(4)

2

), ,

[ ]

( [ ]

, { } 0

eff

i ijkl k l j ikl k l

i ij l l i l l il l

, w

rw w , ls w

u C u C

l c

_βγ _γβ _β

w

_β

C

ρ

=

+ =

〈 〉 −

−

&&

(12) where

1 1

11 11 11

, ,

[ ] [ ]

{ } { }

i i ikl ikl

ij l ij l il il

w w

r r

C C

c

_βγ

c

_βγ

C C

= =

= = =

and s

_{i l}_β

= s

¹¹_{i l}_β

= 0 . Moreover, (11) reduces to the form

3

1 1

( )

i

u

i

lgK

ip

C

p rs s r

, lgw

i

O

U = − 〈 g, 〉 u + + λ

for g = g

¹

, K

_ip

= K

¹¹_ip

. Equations (10) represent simplified tolerance model of the linear elastodynamics of FGM-type laminated medium.

It is the well known fact that that term ls w

_{i l}_β _l

,

_β

in the second of the above equation in the case of the saw-like function vanish [4], but in general case this pro- perty cannot be satisfied. For example for the shape function defined as the periodic solution related to the lower frequency ω to the problem

2

1 1 1111 1 1 1

1111 1 1 1

( ) ( ) ( ) ( ) 0 (0, ) \ { : 0,1,..., / }

[ , ]( ) 0 , 0,1,..., /

x h x C x h x for x L nl n L l

C h x for x nl n L l

ω ρ + = ∈ =

= = =

 



&&

(13)

we have s

_{i l}_β

≠ 0 for any admissible indices α ^, β ^{, l.}

2. Formulation of the problem

Let us consider the FGM-type laminated linear-elastic composite which occupy in the reference configuration the region Ω = ( − L,L) ×Π, where Π is a certain region in R

²

. For Π = (0,H) ×(0,H

1

) laminated composite under consideration is illustrated on Figure 1.

The aim of the paper is the analysis of the following initial-boundary problem for internal fluctuation amplitudes w

_i^A

Find amplitudes α

^A_i

= α

^A_i

( , ), x

₁

x ( , ) x

₁

x ∈Ω , such that vibrations ( , ) ( ) cos

A A

i i

w ⋅ = t α ⋅ ω t of internal fluctuation amplitude w

_i^A

( , ) ⋅ t satisfy in Ω the second from equations (10) and are a certain expansion into the region Ω of the boundary perturbations

1 1

( , ) cos , ( , ) , 0

A A

i i

w = α x x ω t x x ∈∂Ω t >

of internal fluctuation amplitude w

_i^A

( , ) ⋅ t .

(P

w

)

(5)

L

H 2 H 2

H 1

x 2

x 1

x 3

Fig. 1. Example of a laminated solid

In the special case mentioned at the end of Sec. 1 in which we deal with exclusively one saw-like shape function for two-phased linear-elastic the periodic laminated composite, the above problem was analyzed in [1, 2]. For two-phased FGM-laminated composite case the problem was discussed in [3].

The aim of this paper is to discuss this problem from mathematical viewpoint for multi-phased FGM-type laminated linear elastic composites. To this end an alternative form of the simplified model which will be more useful for the analysis will be obtained. It can be obtained by exponential transformation given by the formula

(

^Qx

)

A AB B

k ij j

w = e

^α

π

where

2 1 1

1 1 2

1 1 2 2 1 1

0

1 1 1 1

, ( ) ... ,

!

,..., 1, 2,3, ,..., 1,...,

n n n

n

C C C B

AC C C

Q n AB

ij ip p p p p p j

n

n n

e Q Q Q Q Q Q

n

p p C C N

− − −

∞

=

− −

≡ =

= =

∑

for

AB AB

ij ij

Q = s

_{α α}

x under which equations (10) yield

2

( ), [ ] ( ) ,

[ ] 0

eff A Qx AB B

i ijkl k l j ikl kp p l

AB B AB B AB B

i i l l il l

u C u , C e

l r c , M

α

βγ γβ

ρ π

π π π

〈 〉 − =

− + =

&&

&& ⁽¹⁴⁾

(6)

where

1 1

{ } 0.25

( ) , ( )

AB AB AC CD DB CD DE

ij ij i p p s j s p s s q

CE

M = C + s_α c⁻ _αβ s_β c⁻ _αβ c_βγ

= δ δ δ

pq _αγ

(15) The basic unknowns of this new form of simplified tolerance system are averaged displacements u

i

and introducing above knew unknown π

^B_j

which will be called generalized internal fluctuation amplitude. Now, we can reformulate problem (P

w

) to the form

Find amplitudes β

^A_i

= β

^A_i

( , ), x

₁

x ( , ) x

₁

x ∈Ω , such that vibrations ( , ) ( ) cos

A A

i

t

i

t

π ⋅ = β ⋅ ω of generalized internal fluctuation amplitude ( , )

A

i

t

π ⋅ satisfy in Ω the second from equations (13) and are a certain expansion into the region Ω of the boundary perturbations

1 1

( , ) cos , ( , ) , 0

A A

i i

x t x t

π = β x ω x ∈∂Ω >

of internal fluctuation amplitude π

_i^A

( , ) ⋅ t .

It is easy to verify that if β

^A_i

= β

^A_i

( , ), x

₁

x ( , ) x

₁

x ∈Ω , is a certain solution to the problem (P

π

) then α

^A_k

= ( e

^Qx^α

)

_ij^AB

β

^B_j

is a certain solution to the problem (P

w

) and vice versa and hence we can restrict considerations to the analysis of the problem (P

_π

).

3. Special case

In the framework of this paper we are to discuss this problem in the special case of two-phased laminated composites. Under the above assumption we shall apply exclusively one shape function h = h

¹

and under denotations π π ⁼

¹

^, ^{[ ]} C

ikl

= ^{[ ]} C

¹ikl

, r = r

11

, c

_ij_βγ_l

= c

¹¹_ij_βγ_l

, M

_il

= M

_il¹¹

^, ⁽ e

^Qx^α

⁾

kp

= ⁽ e

^Qx^α

⁾

¹¹kp

, equations (14) are reduced to

2

), ( ) ,

( )

( [ ]

0

Qx eff

i ijkl k l j ikl kp p l

i i l l il l

, e

r ,

u C u C

c M

α

βγ γβ

π

π π π

ρ λ

=

− + =

〈 〉 −

&&

(16)

for

11 11

1

1 1

{ } 0.25

( ) ,

{ } { }

,

, ( ) ( )

ij ij ij i p p s j s ij ij

j s j s p s p s

M M C s c s C C

s s c c

α αβ β

β β αβ αβ

−

− −

= = +

=

= = (17)

(P

π

)

(7)

Moreover M is a certain symmetric matrix and hence problem (P

_ij _π

) is equiva- lent to the eigenvalue problem

2 2 2

( ) 0

ij l il l

c

_βγ

,

_γβ

r M

λ π + λ ω − π = (18)

4. Illustrative example

To illustrate considerations we shall assume that the planes x

α

= const. are symmetry planes in every constituent of the considered laminated composite.

In this case we shall observe that similarly as in the case of saw-like function the equation (18) separate into two independent scalar equations for π

1

and π

2

2 2 2

1221 1 22 11 1

2 2 2

2222 2 22 22 2

( )

0 0

, r

l c l M

π ω π

+ − =

+ − = (19)

every of which posses the same mathematical form. Hence we can rewrite (19)

1

and (19)

2

in the nondimensional form

2

2 2 2

2

(1 ) 0

d d

λ ζ γ ζ

ξ ⁺ ^{− Ω} ⁼ ⁽²⁰⁾

with respect to a certain length entity d, nondimensional variables

2

/ , /

x d l d

ξ = λ = where nondimensional unknown ζ ξ ( ) { ( ∈ π ξ π ξ

₁

d ),

₂

( d )} , 0 ≤ ≤ ξ H d / , and denotations

2 2 2 2

11 22 2

1212 2222 11 22

,

M M r r

c c M M

λ λ

ω ω

γ

^ ^ ^ ^

   

∈ Ω ∈ (21)

Conclusions

In the paper the problem (P

w

) of the propagation of the boundary displacement fluctuations into the interior of the region occupied by the multiphased FGM-type laminated composite materials has been analyzed. It was shown that in the special case of the two-phased laminated composite the problem (P

w

) can be reformulated to a certain problem (P

_π

) and separated on this way to the two independent one-dimensional eigenvalue problems (20). Problems of this type was discussed and solved for the periodic case in [2]. For saw-like function taken as the shape function these solutions has been adapted for a certain FGM-type laminates in [3].

An extended version of this paper will be published elsewhere.

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References

[1] Wierzbicki E., Woźniak C., ŁacińskaL., Boundary and initial fluctuation effect on dynamic behaviour of a laminated solid, Arch. Appl. Mech. 2005, 74, 618-628.

[2] Łacińska L., Wierzbicki E., Boundary and initial fluctuation effect on dynamic behaviour of a laminated solid, Scientific Research of the Institute of Mathematics and Computer Science 2005, 1(4), 110-122.

[3] Rychlewska J., Woźniak C., Boundary layer phenomena in elastodynamics of functionally graded laminates, Arch. Mech. 2006, 58 (4-5), 431-444.

[4] Woźniak C., Wierzbicki E., Averaging Techniques in Thermomechanics of Composite Solids, Wydawnictwo Politechniki Częstochowskiej, Częstochowa 2000.