PSI - Issue 13

Ramdane Boukellif et al. / Procedia Structural Integrity 13 (2018) 85–90

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Boukellif et al. / Structural Integrity Procedia 00 (2018) 000–000

dislocations. Within a continuum mechanics framework, these dislocations are no lattice defects but displacement discontinuities describing the local crack opening displacement. Thus, it is not necessary to discretize the domain around the crack, considerably saving computation time and data, which is crucial for an e ffi cient solution of the inverse problem. Solving the inverse problem, e.g. with a genetic algorithm [Holland (1992)] or an adaptive simulated annealing [Metropolis et al. (1953)], this allows the identification of external loading, crack position parameters, such as length, location or inclination angles and the calculation of stress intensity factors for straight and even for curved cracks in partly or fully bounded plane structures.

2. Theoretical Background

The stress field in a body with cracks can be calculated by superposition of the stresses σ D

i j due to dislocations,

which are distributed at the positions of the cracks. The total stress field σ ∗

i j ( ˆ x , ˆ y ) in local coordinates ( ˆ x , ˆ y ) is as

follows:

M � I = 1

σ D ; I

∞ i j ,

σ ∗

(1)

i j ( ˆ x , ˆ y ) =

i j ( ˆ x , ˆ y ) + σ

where M is the number of cracks and i j = ˆ x ˆ x , ˆ y ˆ y , ˆ x ˆ y . Using the principle of superposition, the induced stresses in particular at the points on the I -th crack line due to the distribution of dislocations of all the M cracks, can be calculated by:    σ D ; I ˆ x ˆ x ( ˆ x I ) σ D ; I ˆ y ˆ y ( ˆ x I ) σ D ; I ˆ x ˆ y ( ˆ x I )    = 2 µ π ( κ + 1) M � J = 1 a J � − a J    G IJ ˆ x ˆ x ˆ x ( ˆ x I ; ˆ ξ J ) G IJ ˆ y ˆ x ˆ x ( ˆ x I ; ˆ ξ J ) G IJ ˆ x ˆ y ˆ y ( ˆ x I ; ˆ ξ J ) G IJ ˆ y ˆ y ˆ y ( ˆ x I ; ˆ ξ J ) G IJ ˆ x ˆ x ˆ y ( ˆ x I ; ˆ ξ J ) G IJ ˆ y ˆ x ˆ y ( ˆ x I ; ˆ ξ J )    � B ˆ x ( ˆ ξ J ) B ˆ y ( ˆ ξ J ) � d ˆ ξ J , | ˆ x I | < a I , (2) where κ is Kolosov’s constant and µ is the shear modulus. Eq. (2) is applied to incorporate boundary conditions at crack faces. The dislocation influence functions G ki j are given in Hills et al. (1996); Dundurs et al. (1964). These describe stresses at a field point ( ˆ x , ˆ y ) due to a unit Burgers vector at the source point ˆ ξ J . The index k = ˆ x , ˆ y in the influence function indicates the direction of dislocations whereas i j = ˆ x ˆ x , ˆ y ˆ y , ˆ x ˆ y denote the components of induced stresses. The Eq. (2) gives a set of singular integral equations with Cauchy kernels, which can be solved using Gauss-Chebyshev numerical quadrature. The dislocation densities B k ( ˆ ξ J ) are determined accounting for boundary conditions. The first condition is that the crack surfaces are traction free. Secondly, the stresses on the external boundaries are equal to the subjected boundary loads. Finely, the displacement jumps at the crack tips are equal to zero and the gradient fields at this points are singular. Once having computed B k ( ˆ ξ J ), the strain at arbitrary points is calculated assuming plane stress conditions.

3. Crack detection and parameter identification

3.1. Finite and semi-infinite plate with inclined cracks at arbitrary locations

First verifications have been carried out numerically. The strain ε i j ( P m ) emerging from the distributed dislocation technique is ”measured” at points P m representing the positions of virtual strain gauges aligned along the edges of a rectangle with corner coordinates ( ¯ x , ¯ y ) and ( ˜ x , ˜ y ). First example is a finite plate (20 mm x 20 mm) with two inclined cracks and P m ( m = 1 , ..., 12) measuring points, ( ¯ x (a) ; ¯ y (a) ) = (1; 19) mm, ( ˜ x (a) ; ˜ y (a) ) = (19; 1) mm, as shown in Fig. 1 (a) . The second example is a semi-infinite plate with inclined interior crack and P m ( m = 1 , ..., 4) measuring points, ( ¯ x (b) ; ¯ y (b) ) = (2; 20) mm, ( ˜ x (b) ; ˜ y (b) ) = (18; − 20) mm, w = 20 mm, as shown in Fig. 1 (b) . The third example is a semi-infinite plane with inclined edge crack and P m ( m = 1 , ..., 4) measuring points, ( ¯ x (c) ; ¯ y (c) ) = (2; 20) mm,