PSI - Issue 2_B

Maria Kashtalyan et al. / Procedia Structural Integrity 2 (2016) 3377–3384 Author name / Structural Integrity Procedia 00 (2016) 000–000

3379

3

Fig. 2. (a) Schematics showing a composite sample with multiple cracks in the ceramic layer;(b) representative segment bounded by two cracks without crack widening; (c) representative segment bounded by two cracks with crack widening l 2 . The equilibrium equations in terms of layer stresses, i.e. stresses averaged across the thickness of the layer and the depth of the sample, have the form

c ~ ~ ( ) ( )  m

ij   (1 ) , ,

1,2,

h h /

i j

(1a)

   







ij

ij

m c

~

~

h

w

1

2 ( ) 22   c c  2

1 ( ) 12   c c  2

d

d





c

~

 

c ( )

c ( )

x x x dx dx 1 2 3 1 3 ( , , )

,

,

(1b)







0

0

ij

ij

2

wh

dx h

dx h

c

  w

h

c

where 1 2 ,   are the interface shear stresses at the metal/ceramics interface. Assuming that out-of-plane shear stresses vary linearly with 3 x and in the metal layer this variation is restricted to the shear layer of thickness s h , i.e. , 1, 2 ), ( ; , 3 3 ( ) 3 3 3 ( ) 3          j h h x h x h h h x x h h s c c s c s j m j c c j c j     (2) the interface shear stresses 1 2 ,   can be expressed in terms of the in-plane displacements ~ , ~ , 1,2 ( ) ( )  j u u m j c j , and shear moduli c m G G , of the ceramic and metal phases as

c m (1 (1 ) / 2) 3 G G

(3)

c ), ( ~ ~ ( ) ( ) 

m K u u

,

h h /

K











j

j

j

j

j

s

m

h G

m c   h G

  

c m

The constitutive equations in terms of layer strains and stresses are                                ( ) 12 ( ) 22 ( ) 11 ( ) 66 ( ) 22 ( ) 12 ( ) 12 ( ) 11 ( ) 12 ( ) 22 ( ) 11 ~ ~ ~ 0 0 0 0 ~ ~ ~ c c c c c c c c c c c S S S S S       ,                 ( ) 22 ( ) 12 ( ) 12 ( ) 11 ( ) 12 ( ) 22 ( ) 11 0 0 ~ ~ ~ m m m m m m m S S S S S   

~ ~ ~

    

    

    

( ) m

0 0

  

11

( ) m

(4)

22

( ) m

( ) 66 m

12

( ) 11 ~ ~ m c    , and crack surfaces are stress-free, i.e. 0 11 ( )

In addition, it is also assumed that

~

0, ~  (5) Equations (1)-(4) can be reduced to two uncoupled second-order ordinary differential equations with respect to in-plane layer stresses in the ceramic layer. 0 ~ ~ 0, ~ ~ 12 ( ) 12 ( ) 12 ( ) 2 2 ( ) 12 2 22 ( ) 22 11 ( ) 11 ( ) 22 ( ) 1 2 ( ) 22 2                  c c c c c c c c c L dx d L dx d (6) Solutions of these equations satisfying specified boundary conditions (5) are found as ( ) s l ( ) c 12 ( ) s l ( ) c 22 2 2       x x 