Issue 47

V. Rizov, Frattura ed Integrità Strutturale, (2047) 468-481; DOI: 10.3221/IGF-ESIS.47.37

material properties which describe the hardening behavior of the material (actually, the second term of the right-hand side of (6) models the material non-linearity). Each layer exhibits smooth material inhomogeneity along the width and length of the layer. Thus, it is assumed that the modulus of elasticity in the i -th layer varies continuously along the width according to following cosine law:

  

 

1 y y 



i

1

E E E  

cos

(7)

 

i

d

f

y

y

2

i

i

1 1 i 

i

1

where

 

y

1 y y  1 1 i

. (8)

i

1

In (7), E is a material property which governs the material gradient along the width. Apparently, the value of the modulus of elasticity at the left-hand lateral surface of the layer is i i d f E E  . The continuous variation of i d E in the length direction of the i -th layer is written as i d E is the value of the modulus of elasticity at the right-hand lateral surface of the layer, i f

  

  

l  



l

x

1 2

3

E E E  

(9)

cos

d

g

r



l

l

2

i

i

i

1 2

where





3 1 2 2 x l l   

(10)

0





0 x  and

1 2 l   , of the

x -axis is shown in Fig. 1. In (9),

g E is the value of

d E at the two end sections, 3

2 x l

The 3

3

i

i

r E is a material property which governs the material gradient in the length direction. It is obvious that the value of

beam,

i

i r E E  . For the Ramberg-Osgood stress-strain relation, * 0 i L u d E in the mid-span is i i g

* L U by (5) can be written as [17,

which is needed in order to calculate

18]



m

1

i

m

2

i 

i i 

m

i

* 0

 

u

(11)

L

1

E

2

i

i





m



m H

1

i

i

i

By substituting of (7) in (11), one arrives at



m

1

i

m

2

i 

i i 

m

i

* 0





u

(12)

L

1

  

  

  

 

1 y y 



i





i

E E 

1

2

cos

m



m H

1  

i

d

f

i

i

y

y

2

i

i

1 1 i 

i

1

x . For this purpose, (9) is re

E in (12) should be expressed as a function of 1

In order to perform the integration in (5), i d

written as

 

 

x



(

)

1 1 2 l

E E E  

(13)

cos

   

d

g

r

l

2

i

i

i

472