PSI - Issue 18

Johannes Scheel et al. / Procedia Structural Integrity 18 (2019) 268–273 J. Scheel et al. / Structural Integrity Procedia 00 (2019) 000–000

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the crack tip, to a certain extent implies arbitrariness. Irwin’s crack closure integral [Irwin (1958)] gave rise to the crack closure method (CCM) and is based on the assumption that the energy released by a crack extension ∆ a is equal to the amount of energy required for crack closure. In the CCM two finite element (FE) analyses are required, physi cally extending or closing the crack, in order to use the associated near tip fields. The modified crack closure integral (MCCI) technique [Rybicki and Kanninen (1977), Buchholz (1984)] is based on the assumption, that a virtual crack extension does not a ff ect the near tip fields, thus enabling the use of displacement jumps on the faces and nodal forces on the ligament of just one crack model and therefore one FE analysis. Although being quite accurate, the need of nodal reaction forces complicates and limits the use of this technique. In this work, an enriched modified crack closure integral (EMCCI) is introduced, based on the ideas of CCM and MCCI, however just requiring nodal displacements on the crack faces. Interpolation functions for the displacements are derived, involving prescribed crack face nodes, depending on the number of eigenfunctions of the Williams series [Williams (1957)] used to describe the crack fields. Also exploiting the near tip stress fields enables the calcula tion of the crack closure integral in closed form. Numerical examples (e.g. the double cantilever beam and three point-bending specimen) are used to verify the approach. The results are compared to other approaches, revealing appropriate accurateness combined with high flexibility and low implementation e ff ort, computational cost and mesh requirements.

Nomenclature

G

energy release rate crack extension

∆ a σ i j

stress tensor

u i displacements K I / II / III stress intensity factors µ shear modulus κ Kolosov’s constant L element length r , φ polar coordinates x 1 , x 2 Cartesian coordinates a 1 , b 1

parameters of the Williams series

E

Young’s modulus

ν Poisson’s ratio FE finite element DIM displacement interpretation method SIF stress intensity factor(s) MCCI modified crack closure integral EMCCI enriched modified crack closure integral CCM crack closure method CTEM crack tip element method

2. The enriched modified crack closure integral technique (EMCCI)

Basis of the EMCCI, as for the MCCI, is Irwins crack closure integral [Irwin (1958)], see Fig. 1:

1 2 ∆ a ∆ a

σ i 2 ( r , φ = 0) ∆ u i ( ∆ a − r , π ) dr ,

(1)

G = lim ∆ a → 0