PSI - Issue 13

Paul Judt et al. / Procedia Structural Integrity 13 (2018) 155–160

157

Author name / Structural Integrity Procedia 00 (2018) 000–000

3

the SIF [2]. Furthermore, J k is related to the ERR [12] by

d Π d a

J k z k = BG = −

(4)

,

where B represents the constant thickness of the specimen, Π the total potential energy and a is the crack length. To locally calculate the material force F α k at a node α , Mueller et al. [13] introduced an e ffi cient finite element based method, applying the weak form of the balance equation (1). In this method each finite element incorporating node α provides a contribution to the resulting material force at that node. Thus, to calculate the crack driving force J k = −F tip k , field values in elements adjacent to the crack tip node are employed and therefore this method is classified as local approach of crack tip loading analysis. Without special numerical treatment, local approaches in general are less accurate due to numerical deviations in the crack tip stress and strain fields. This leads to large deviations, especially in the second coordinate J 2 or F tip 2 , respectively. The remote calculation of the crack driving force according to Eq. (3) is beneficial, essentially employing nu merically reliable data. However, large integration contours in general require an integration along the crack faces, providing numerically inaccurate values approaching the crack tip, finally still leading to large deviations in J 2 . Im provements for an accurate calculation of J 2 have been presented by Eischen [4] and Judt and Ricoeur [6]. In the following, improvements applying the local approach of the calculation of material forces are presented and discussed. The inaccurate representation of the theoretical stress singularity in LEFM is related to large deviations of local fields in the vicinity of the crack tip. This induces large deviations of F tip k applying the local approach as local stress and displacement quantities are employed. In the crack tip region artificial material forces F α, art k thus arise, even though material defects are not present apart from the crack tip. These forces are related to the smeared crack tip stresses and an improvement of F tip k is obtained by adding up all material forces within a certain domain surrounding the crack tip [3] Nevertheless, this improvement does not apply in cases of curved cracks or mixed-mode loadings. In these cases real material forces act on the crack faces which cannot be distinguished from the artificial ones. For increasing domains the contribution of the real material forces along the crack face segment of length r leads to an increasing deviation in F sum k (r). Schu¨ tte [18] presented a method where F sum 2 ( r ) is linearly extrapolated to the crack tip ( r → 0) providing an improved value of F tip 2 . The same approach is feasible regarding F sum 1 ( r ) in connection with curved cracks. By summarizing forces within a certain domain the benefit of a local approach vanishes and other methods such as path independent integrals provide equal or even better results. Especially in cases where the crack tip approaches another boundary or is close to a second crack tip, the summation of material forces within a domain is doubtable as these forces are not uniquely assignable. To maintain the benefits of the local approach in connection with accurate loading analysis a new methodology is presented in the following. Assuming that the real crack driving force J k is composed of the material force acting at the crack tip, of artificial body forces appearing at nodes α and of artificial forces appearing at nodes β along the crack faces, i.e. J k = −    F tip k + α F α, art k + β F β, art k    , (6) J k = −F sum k = −   F tip k + α F α, art k   . (5)