PSI - Issue 37

T. Oliveira et al. / Procedia Structural Integrity 37 (2022) 698–705

700

T. Oliveira et al. / Structural Integrity Procedia 00 (2022) 000–000

3

the theory of damage mechanics employed. The first method used here is proposed by Ladeve`ze and Lubineau (2003) and it studies the microcracking in the matrix while the comparatively much stronger fibers are considered to stay intact; several sets of values are tested for this model, and the one selected for further analysis is denominated DC1, on which ˜ E 2 = 0 . 21 E 2 and ˜ G 12 = 0 . 42 G 12 and the remaining elastic constants are kept equal. The method used by Moreno-Garc´ıa et al. (2014) uses a much simpler approach of multiplying all elastic constants, except the Poisson’s ratio, by a damage parameter, such that [ ˜ D ( e ) ] = (1 − d ( e ) ) ∗ [ D ( e ) ] . As before, di ff erent values of d ( e ) are tested and the one which is found most suitable is denominated DC2 where d ( e ) = 0 . 7; through the analysis of the damage indices, it has been concluded that DC2 is about ten times more severe than DC1, which is a di ff erence suitable to test the response of the damage detection methods to di ff erent severity levels. The employed damage indices are based on the di ff erentiation of the transverse displacement of each node of the plate. The numerical di ff erentiation process chosen for this operation is the Finite Di ff erence method, which uses the values of the displacement and their spacing, uniform in each direction, to approximate the derivatives. The formula for the first derivative can be found on Equation 1, and the subsequent derivative orders are calculated by substituting the variable on the equation by the equation itself (i.e. f ′′ ( x ) = f ′ ( f ′ ( x ))) and simplifying the result. To obtain di ff erent values of accuracy in the computations, di ff erent orders of the finite di ff erences can be used. However, they require more points and, thus, disabling the possibility to compute the derivative on nodes close to the edges of the plate. In view of this, a choice was made to restrict the computations using only the second order central finite di ff erence. 2.3. Finite di ff erence method

1 2 w ( x − h , y ) +

1 2 w ( x + h , y )

= −

∂ w ∂ x

(1)

h

2.4. Damage indices

Three di ff erent damage indices are used, two of which were already applied in previous studies and a new one is proposed in this work. Moreno-Garc´ıa et al. (2014) proposed the use of the DFD (Di ff erence in Field Derivatives) damage index, which calculates the derivatives in the x direction up to the fourth order, as shown in Equation 2, for each mode.

∂ P w

∂ P ˜ w

q ( x , y )

q ( x , y ) ∂ x P

q ( x , y ) =

DFD ( P )

, P = 1 , 2 , 3 , 4

(2)

−

∂ x P

Arau´ jo dos Santos et al. (2006) proposed the use of an index which calculates the di ff erence between the averages of the displacements and first two derivatives of the first ten modes of the plates, as shown in Equation 3, 4 and 5, where in each derivative order the Euclidean norm of the vector’s components is used.

n q = 1 n q = 1 n q = 1

1 n

w kq − ˜ w kq

(3)

TD ( w k , ˜ w k ) =

1 n

˜ θ kq

θ kq 2 −

2

S D ( w k , ˜ w k ) =

(4)

1 n

˜ k kq

k kq 2 −

2

CD ( w k , ˜ w k ) =

(5)