numero25

Yu. G. Matvienko et alii, Frattura ed Integrità Strutturale, 25 (2013) 20-26; DOI: 10.3221/IGF-ESIS.25.04 23 Taking into account only the first term from Eq. (1) leads to well-known Westergaard relation: * 1 2 8 n n I n v E K a      (7) A characterisation of T-stress values T is based on a determination of u -displacement component directed along x -axis. A distribution of ( , ) n u r  -displacement component for points belonging to the crack line ( θ = π , see Fig. 1), which corresponds to the second and the fourth terms of infinite series (1), is expressed as: 2 3 4 4 4 ( , ) 0( ) n n n r r u r A A r E E        (8) Absolute value of u -component for a crack of a n length can be again obtained at point n –1 with polar co-ordinates r =Δ a n and θ = π because at this point 1 ( , ) 0 n u r    . A substitution of n r a   and 1 ( , ) n n n u r a u        in Eq. (8) and taking into account the first from relations (3) lead to the following relation: 2 1 2 4 4 4( ) n n n n n a a u A A E E       (9) Eq. (9) gives us the first equation essential for a determination of T-stress value T when displacement component value u n- 1 is experimentally obtained. It should be noted that all experimental parameters needed for relations (5)-(7) and Eq. (9) can be derived from two interferograms, which correspond to Δ a n crack length increment. A formulation of the second required equation demands involving interference fringe pattern, which corresponds to crack length increasing from point n to point n +1 by Δ a n+1 increment (see Fig. 1). For point n +1 with polar co-ordinates ( r =Δ a n+1 , θ = 0 ) the first Eq. (4) can be written as: 1 1 1 1 1 1 ( ; 0) ( ; 0) 0 ( ; 0) n n n n n n n n U u r a u r a u r a u                         (10) Combining relations (1) and (10) gives: 2 1 1 1 1 1 2 1 1 3 4 2 (1 ) 4 4( ) (1 ) 2 n n n n n n n n n n a a a u A A a a A A E E E E                     (11) where 1 n u  is the absolute value of ( , ) n u r  displacement component at point n +1 (see Fig. 1). Relation (11) represents the second equation essential for a determination of T-stress because the values of coefficients A 1 and A 3 are already known from formulae (5) Note that a value of 1 n u  has to be experimentally derived from interference fringe pattern of type shown in Fig. 2a, which are recorded for Δ a n+1 crack length increment. If an estimation of T-stress value is restricted by coefficient A n 2 only, the following simplified formula is valid: n n n u T E a    (12) The T-stress value in Eq. (12) can be determined by using interference fringe pattern of type shown in Fig. 2a recorded for crack length increment Δ a n only. Electronic speckle-pattern interferometry (ESPI) serves for a determination of in-plane displacement components [11]. Well-known optical system with normal illumination with respect to plane object surface and two symmetrical observation directions is used. When a projection of illumination directions onto plane surface of the investigated object coincides with ξ -direction, interference fringe pattern is described as: 2sin d N     (12) where d ξ is in-plane displacement component in ξ -direction; N =  1;  2;  3, … are the absolute fringe orders;  = 0.532 μm is the wavelength of laser illumination;  = 45 degrees is the angle between inclined illumination and normal observation directions. When ξ -direction coincides with x -axis and y -axis displacement component u and v can be derived accordingly to formula (13), respectively.