PSI - Issue 25

F.J.C.F.B. Loureiro et al. / Procedia Structural Integrity 25 (2020) 63–70 Loureiro et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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Normal separation at the crack tip (  n ) The parameter  n is required to estimate J I and t n . Concerning the geometrical approach to determine  n , three virtual points were defined: A at the leftmost vertex of the adhesive/upper adherend interface, A’ to the right side of point A and B at the leftmost vertex of the adhesive/lower adherend interface. A line segment ( y = mx + b ) was established between point A :( A x , A y ) and A’ :( A x ’ , A y ’ ) as ' AA . The slope ( m ) and y -coordinate at x =0 (defined as b ) of the line segment ' AA were obtained though ( ) ( ) ( ) ' and ' ' . ' y y y x x x A A m b A m A A A − = = −  − (6)

The chosen formulation for the minimum distance from point B :( B x , B y ) to line segment ' AA is

1 2 ax by c d a b + + = + 1 2

, with the line segment defined as ax + by + c =0

(7)

' AA equation according with Eq. (7) results in

Rewriting the line segment

' : 0 AA y mx b mx y b = +  − + = ,

(8)

1

1

1 1

and considering the virtual point B :( B x , B y ) as B :( x 1 , y 1 ), d may be expressed as

( ) 1 mx y b m − + + − 1 1 2 2

A d t  = − .

, with n

(9)

d

=

Shear separation at the crack tip (  s ) The parameter  s directly influences J II and t s , and it was obtained by a geometrical approach, similarly to  n . Three points were established ( A , A’ and B ). A line segment ' AA was defined with m obtained through Eq. (6). A perpendicular line to segment ' AA was defined, expressed as

x

1

' x b = − + , with b ’ defined by y 1 when x 1 = 0 as

.

(10)

' : AA y

' b y

⊥

= +

1

1

1

1

m

m

Following Eq. (7) formulation for the minimum distance from a point to a line, rearranging Eq. (10) accordingly resulted into

1

1

' ' 0 = − +  + − = , x b x y b

(11)

' : AA y

⊥

1

1

1

1

m

m

and considering the point B :( B x , B y ) as B :( x 1 , y 1 ), d may be expressed as

1

1 1 1 x y b + −   +     1 2 2 m

'

.

(12)

d m =

' AA ⊥ and B :( B x , B y ) represents δ s .

Therefore, the distance between