Crack Paths 2012

Due to the symmetry only one-quarter of the specimen is meshed with twenty-node

isoparametric elements. Near the crack front up to a distance of r = 0,5 m mand within a

layer thickness of 0,5 m mnear the free surface a very fine mesh with an element size of

0,01 m mis used. Towards the middle plane of the specimen with z/t = 0 the element

size is gradually increased in through-thickness direction up to a size of 0,1 mm.The

FE-simulations are carried out under the assumption of a linear-elastic material law with

E = 210.000 M P aand = 0,3.

The displacement field near the crack front can be written as superposition of two

terms

(3)

XLM

$LM ] U %LM 5

based on the known cylindrical singularity for the stresses inside the body and an

unknownvertex singularity in spherical coordinates with the exponent independent of

z [14]. The FE-analyses are evaluated with regard to the crack face displacements in y

direction in a cylindrical coordinate system. For every plane (z/t = const.) these

displacements in the vicinity of the crack front ( = 180°) can be expressed as follows:

XU &ÂUX 'ÂUX .

(4)

Boundary

z/t

§

z/t = 0,39 u = 0,54 5

layer

Figure 6. Influence of the vertex singularity on the exponent

u for different layers

For the numerical evaluation of the exponent u

it is advantageous to consider an

additional second term of higher order in Eq. (4). In the free surface the exponent u is

equal the vertex singularity exponent and tends towards 0,5 in the middle plane of the

specimen where the second term of Eq. (3) becomes zero. The thickness of the

boundary layer influenced by the vertex singularity is defined as the region near the free

surface in which

u differs from the classical value 0,5. The exponent

u is determined in

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