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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The testing velocity was 1 mm/minute. During the course of the experimental tests, a photographing equipment was used (Canon EOS 70D with 20 MPixel; Canon from Tokyo, Japan) to perform the a measurement and enable extracting other required parameters to calculate the J -integral. 2.3. J-integral formulation Rice (Rice 1968) developed the J -integral method in 1968, aiming to characterize the strain concentration near cracks and notches. The J -integral method can be applied to several tests, such as mode I, mode II and mixed mode, such as the SLB test. Ji et al. (Ji et al. 2012) proposed a J -integral formulation, which is the one used in this work, for the J I and J II assessment by the SLB specimen. J I can be expressed as

4 P P

n t 

( ) n 

( )



J

,

(1)

d    =

=

I

n n

n

0

where t n is the tensile stress,  n is the normal separation at the crack tip and  P is the relative rotation between the two adherends at the loading line. The J II expression is defined as

2

hQ

2 2 h Q a D T

1 2

     

+

0 

T

D

2



( ) s 

( ) s t



J

d  

=

=

,

(2)

II

s

s

s

2 h A D + 2 2

0

where t s is the shear stress, δ s is the shear separation at the crack tip, D is the bending stiffness of a beam (assuming equal beams), A is the axial stiffness of a beam and Q T is the shear forces resultant on the bonded joint. The direct method enables the assessment of the tensile and shear cohesive laws, respectively t n (  n ) or t s (  s ) laws, through differentiation of the J I -  n or J II -  s curves. This J -integral formulation relies on three important parameters:  p ,  n and  s . In this work, an optical technique was applied, in which these parameters were assessed by using a vector and geometric analysis of the images captured during the tests. Relative rotation between two beams at the load line (  p ) The parameter  P is required to determine the J I and t n . Concerning the vector analytical approach to determine  P , two vectors were defined, u and t . Both vectors are collinear with the top and bottom adherends’ leftmost edge. The u vector spans between the points E and F, and the t vector between the points G and H, as expressed by ( ) , , x x y y z z u F E F E F E = − − − , ( ) , , x x y y z z t H G H G H G = − − − (3)

Furthermore, in order to relate both vectors, the vector product between the vectors u and t results in the vector s

( ) P 

2 s s s u t = + + =   2

2 y

s

sin

s u t =  and

.

(4)

x

z

Therefore, rearranging this equation as a function of  P , results in

s u t        

.

(5)

arcsin = 



P