1. 1 Experimental determination of K I by Annex IV Speckle interferometry

2. 2 crack x y r ux(r,)ux(r,) u y (r,  )  Asymptotic displacements at the crack tip (see eqs 4.40): with plane stress plane strain E : Young’s modulus  Poisson’s ratio Method proposed by Barker et al. (1985) ( see annex IV ) Mode-I loading

3. 3 General expressions for the displacement (Mode I): (plane stress conditions)

4. 4 Explicit form for v : For asymptotic expressions recovered (plane stress ) The 5 first terms in the series are expanded as:

5. 5 = 632nm Sensitive to in-plane displacement along y Optical set-up (interferometer) considered : CCD camera object surface (optically rough) Mirror   z y x Microscopic lens Experimental in-plane displacements

6. 6 Principle of the speckle interferometry (ESPI): Intensity of the speckle field: Initial state: Deformed state (after loading) : with v : y-component of the displacement Subtraction of the intensity images Interference fringes - Dark / bright fringes when - Fringe gap : - Achieved in real time by the commercial software Pisa z y CCD L Q’ f Q   a0a0 a1a1

7. 7 Loading cell: P[N] Specimen and loading grips CCD Stepper motor Lens Experimental set-up:

8. 8 Optical Arrangements n Isovalues of the horizontal displacement u Isovalues of the vertical displacement v  Light rays 

9. 9 Choice of the experimental points N=0 N integer On another dark fringe, the displacement components are Reference dark fringe: N = 0 (typically) N=1 N=2 N= -1 N=-2 ~ 0,7µm How to number the fringes?

10. 10 P i (r i,  i ;N i ) : current point on a dark fringe riri PiPi ii Algorithm : system formation S i,k (r,  ) known functions depending on the considered displacement: C k : unknown to be determined. for u for v N i : associated fringe order (integer) (typically K = 4) System:

11. 11 Solid rigid displacement taken into account with u rigid u rigid = R + P r cos  + Q r sin  (should be considered in practice) → 8 unknown Resolution of the system: M points P(r i,  i ) taken on the dark fringes (M > 40) : with

12. 12 Least square method : and also,

13. 13 Application : SEN and CT specimens used : Sensitivity: f = 0.73µm Fringes pictures (vertical displacement) : 6 mm E (GPa) a/bF (N) 0.38 2.4SEN CT 0.38 2.5 6/25 27.6/48 30 10 Treatment of the (fringes) images and implementation of the algorithm : Program speckle (Matlab):

14. 14 K I th =0.12 MPa m 1/2 Fringe of zero order 48 points selected E, f K I =0.12 MPa m 1/2 SEN specimen: Theoretical value:

15. 15 Fringe of zero order 47 points selected E, f K I =0.046 MPa m 1/2 K I th =0.047 MPa m 1/2 CT specimen: Theoretical value: