PSI - Issue 13

Paul Judt et al. / Procedia Structural Integrity 13 (2018) 155–160 Author name / Structural Integrity Procedia 00 (2018) 000–000

156

2

have been introduced for the numerical crack tip loading analysis, such es special crack tip elements (CTE) [1] path independent integrals [15], the modified crack closure integral (MCCI) [16] or material forces [13]. These methods can be classified into local and global approaches of crack loading analysis. A local approach mainly employs numerical data in the vicinity of the crack tip, which are often inaccurate due to the weak numerical reproduction of the crack tip stress singularity. Global approaches, e.g. path-independent integrals in connection with remote integration contours [6, 7] provide accurate loading quantities as reliable numerical data far from the crack tip are extensively employed. No special crack tip meshing is required, only a mesh refinement at the crack tip region and the crack faces supports the accuracy of the loading analysis. The remote calculation of J k -integrals has successfully been applied to the prediction of crack paths in various specimens of anisotropic materials [9, 10]. Nevertheless, the implementation of local approaches typically is less complex and the evaluation of loading quan tities is faster. This is particularly important for loading analyses at three-dimensional cracks and the simulation of crack paths, where the calculation of path-independent integrals is even more complex. The material forces are closely related to path-independent integrals providing all relevant parameters to predict crack initiation and growth and can easily be implemented in three-dimensional numerical calculations. In the following, a methodology based on local material forces is presented for accurately calculating the crack driving force as well as parameters of crack initiation, being capable of a straight-forward extension to three-dimensional crack problems.

2. E ffi cient and accurate calculation of material forces

Eshelby [5] introduced the energy-momentum tensor Q k j = u δ k j − σ i j u i , k which is related to fundamental balance laws in the material space. The tensor Q k j , containing the elastic potential density of a system u , the identity tensor δ k j , the stress tensor σ i j and the displacement gradient u i , k , can be derived by applying the gradient operator to the Lagrangian density L = t − u , with t being the kinetic energy density. In the case of quasi-static processes ( t = 0), the balance equation

Q k j , j = ( u δ k j − σ i j u i , k ) , j = 0

(1)

is obtained, where comma convention and summation of repeated indices imply the divergence operator. In a defect free volume, the divergence of the energy-momentum tensor vanishes in the material space, analog to the vanishing divergence of the symmetric Cauchy stress tensor σ i j , j = 0 in the physical space, if volume tractions are zero. With the internal material forces d F k acting at a surface d A , the material traction vector is defined as

d F k d A =

q k =

Q k j n j ,

(2)

which is di ff erent from Q k j n k due to the non-symmetry of Q k j . From the integration of Eq. (2) within a closed boundary A O , the resulting material force at the corresponding domain V follows as F k = A O q k d A = A O Q k j n j d A = V Q k j , j d V (3)

and F k vanishes if V does not contain defects. Applying Eq. (3) to crack tips, the material force F k is called J k -integral [15] and represents the crack driving force. Considering plane problems in LEFM, the J k -integral vector is related to