PSI - Issue 39

Davide Leonetti et al. / Procedia Structural Integrity 39 (2022) 9–19 Author name / Structural Integrity Procedia 00 (2019) 000–000

12 4

d

Patch load

Rolling surface

RCF crack

Traffic direction

Elastic foundation

Fig. 2. Mechanical scheme of the cracked beam supported by the elastic foundation under a moving contact load.

2.1. The weight function for a inclined edge crack As previously mentioned, the weight function method allows the calculation of the SIF for a cracked body, independent on the load configuration. In general, the SIFs for mode I and mode II loading are: K I = � σ n ( x ) h 11 ( x,a ) a 0 dx+ � τ ( x ) h 12 ( x,a ) a 0 dx (3) K II = � σ n ( x ) h 21 ( x,a ) a 0 dx+ � τ ( x ) h 22 ( x,a ) a 0 dx (4) where h 11 , h 12 , h 21 , h 22 are the weight functions, σ n (x) and τ(x) are the istribution of the normal and tangential stresses acting along the perspective crack edge calculated considering the uncracked geometry. When Equations 3 and 4 are expressed in a matrix form, the matrix of the weight functions =h ij is a square matrix. Based on the crack edge displacements, Petrosky and Achenbach (1978) proposed the following formulation for the weight functions of an edge crack: h ji = � 2 π a � D n ji ( 1-x/a ) ( n-1 ) 2 ⁄ ∞ n=0 (5) where the index j refers to the mode of opening ( j =1 mode I, j =2 mode II), and the index i refers to the type of stress acting along the crack line ( i =1 normal, i =2 shear). To calculate weight functions effectively, it is often made use of the concept of Green’s function that is intimately related to the weight function, see Wu (1991). A Green’s function is the stress intensity factor for a pair of unit forces P localized along at the crack faces. It results that the stress distribution is ( ) = ( − 0 ) , where is the Kronecker delta function and 0 is the abscissa along the crack edge identifying the point of application of the unit forces. Wu (1991) reported Green’s functions for edge cracks with ω =0. Up to the best knowledge of the authors, there is no published reference reporting the complete set of Green’s functions, i.e. the contribution on both K I and K II for both a normal and tangential unit force, for edge cracks in finite width plates. In Fett (1997) some values are reported for a/W =0.6 and ω =22.5 ° retrieved from an unpublished work. By using this concept, it is possible to estimate the coefficients D n ji of the weight functions, if the Green’s function is available for modes I and II and for a number of reference cases in agreement with the finite expansion of the series in Equation 4. It is a common choice in Fett (1997) to select a structure for the weight functions in which the integer exponents of the term ( 1-x/a ) are neglected. By stopping the series at n=4, and considering 011 = 011 = 1 and 012 = 021 = 0 due to consistency with crack mouth deformations, see Fett (1997), eight conditions are needed to determine mode I and mode II SIFs for each geometrical configuration. With this scheme, four reference solutions are needed when the parameters are determined using the direct adjustment method.