numero25

Yu. G. Matvienko et alii, Frattura ed Integrità Strutturale, 25 (2013) 20-26; DOI: 10.3221/IGF-ESIS.25.04 22 2 1 2 1 (1 ) ( 4) ( 1) cos cos 2 2 2 2 (1 ) ( 4) ( 1) sin sin 2 2 2 2 n n n n n n n n r n n n n u A k E r n n n n v A k E                                                   (1) where u and v are in-plane displacement component in direction of x and y axis, respectively; μ is the Poisson’s ratio; E is the elasticity modulus; k = (3– μ )/(1+ μ ) for plane stress and k =(3–4 μ ) for plane strain conditions; A n are constants to be determined; r and θ are radial and angular distance from the crack tip as it is shown in Fig. 1. Values of stress intensity factor (SIF) K I and T-stress T are connected with coefficients of infinite series (1) by the following way [7]: 1 2 I K A   , 2 4 T A  (2) Generally initial experimental information represents a difference in absolute values of in-plane displacement components ( , ) n U r  and ( , ) n V r  for two cracks of length n a and 1 n a  : 1 ( , ) ( , ) ( , ) n n n U r u r u r       , 1 ( , ) ( , ) n n n V v r v r      (3) where 1 1 ( , ), ( , ) n n u r v r     and ( , ), ( , ) n n u r v r   are absolute values of in-plane displacement components in a point with polar co-ordinates ( , ) r  for a crack of 1 n a  and n a length, respectively. Eq. (3) are valid for any point belonging to the proximity of crack tip located at point n . But right hand sides of Eq. (3) include relative values of displacement components, which can not be directly used for a determination of A n -values from decomposition (1). The key point of the developed approach resides in the fact that each interference fringe pattern of type shown in Fig. 2 contains a set of specific points located at a crack border immediately. Absolute values of in-plane displacement components and then coefficients A n from formulae (1) for a crack of n a length can be determined at these points. First, specific points are located along the crack line between point n –1 and point n where displacement component 1 ( , ) n v r   equals to zero before making a crack length increment. Thus, interference fringe pattern shown in Fig. 2b allows determining absolute values of ( , ) n v r  -component for each point with polar co-ordinates 0 ≤ r ≤ Δ a n and θ = π . Developed approach employs four first coefficients of series (1) for deriving required fracture mechanics parameters. A distribution of ( , ) n v r  -displacement component along the crack line ( θ = π , see Fig. 1) is expressed as: 1 3 4 4 ( , ) 0( ) n n n r r r v r A A r E E       (4) Relation (4) shows that deriving K I value from Eq. (3) demands a determination of ( , ) n v r  -values at two points belonging to the interval 0 ≤ r ≤ Δ a n , θ = π , as minimum. It is conveniently to use two points with polar coordinates ( r =Δ a n , θ = π ) and ( r =Δ a n /2, θ = π ). Substituting these co-ordinates into relation (4) forms a system of linear algebraic equations, a solution of which is:     1 0.5 1 3 1 0.5 2 2 8 2 4 n n n n n n n n n E A v v a E A v v a a                 (5) where A n 1 and A n 3 are coefficients of decomposition (1) for a crack of a n length; Δ v n-1 = 2 v n ( r =Δ a n , θ = π ) and Δ v n-0.5 = 2 v n ( r =Δ a n /2, θ = π ) are crack opening values from Eq. (4), which have to be experimentally determined. SIF value for a crack of a n length follows from combining the first relations from Eqs. (2) and (5):   0.5 1 2 2 2 8 n I n n n E K v v a         (6)