PSI - Issue 5

J. V. Araújo dos Santos et al. / Procedia Structural Integrity 5 (2017) 1198–1204 J. V. Araújo dos Santos et al. / Structural Integrity Procedia 00 (2017) 000 – 000

1200

3

x x w x y  y y w x y  ( , ) ( , )



 

x y ( , ) 4 





(1)

xx





 

x y ( , ) 4 





(2)

yy



where ( , ) x y yy   are, respectively, the phase maps if we consider the shearing direction as x or y . In Equations (1) and (2), x  and y  are the shearing amounts in the x and y directions, respectively, and  is the wavelength of the laser light. The partial spatial derivatives w x y x   ( , ) and w x y y   ( , ) can be viewed as the rotations in the x and y directions. Since we are interested in the analysis of beams, the y direction can be discarded, such that we just take into account a simplified form of Equation (1): ( , ) x y xx   and

dx x dw x

( )



 

x ( ) 4

xx   

(3)

In view of Equation (3) and due to its own nature, shearography can be simulated by taking the out-of-plane displacement of the beam w ( x ), obtained for instance with a finite element model, and differentiate it with a backward or forward finite difference formulas. These formulas as given, respectively, by:

x w x w x x     ( ) (

x w x x w x     ) ( ) (

( )

)

( )

dx x dw x 

dx x dw x 

( ) 

( ) 





or

(4)

Therefore, we obtain an approximation of the rotation field as a function of the displacement field w ( x ) and the shearing amount x  .

2.2. Computation of modal curvatures

In the present work, we consider the finite element method as a source of data used as reference, namely the modal displacement field needed in Equation (4). The modal curvatures ( ) x  are obtained directly from the relation between the bending moment M ( x ) and the bending stiffness EI : ( ) ( ) EI x M x   (5) and the relation among bending moment, stiffness matrix K , circular natural frequency  , mass matrix M , and rotation field ( ) x  :

) ( ) ( ) ( 2 M x K M x    

(6)

( ) x r  :

Thus, from Equations (5) and (6) we obtain the reference modal curvature

2

) ( )

( ) ( EI x K M r r     

x

(7)

where ( ) x r  are obtained directly from the finite element degrees of freedom. We can also define the modal curvature central finite difference to the rotations defined in Equation (4):

( ) x s  , coming from the simulation of shearography, by applying the