Rasmus ÿstergaard forside 100%.indd - Solid Mechanics

Interface crack in sandwich specimen 309
Table 7 The phase angle ω (in degrees) computed with different values of the mismatch parameters and β for a specimen with
thickness ratio η = 6
β : 0.5 0.1 0.05 0.01 0.005 0.001
α: −1/3 −1/2
−0.4 49.7 54.7 63.5 65.7 66.6
−0.3 42.4 45.1 53.7 58.3 65.8 67.4 67.0
−0.2 46.9 49.5 57.5 61.6 68.0 69.1 67.4
−0.1 51.2 53.7 61.2 64.9 70.1 70.6 67.6
0.0 55.5 57.7 64.7 68.1 72.2 72.1 67.4
Table 8 The phase angle ω (in degrees) computed with different values of the mismatch parameters and β for a specimen with
thickness ratio η = 8
β : 0.5 0.1 0.05 0.01 0.005 0.001
α: −1/3 −1/2
−0.4 49.9 55.1 64.7 67.5 70.3
−0.3 42.4 45.2 53.9 58.7 67.1 69.4 70.8
−0.2 47.0 49.5 57.8 62.1 69.5 71.2 71.3
−0.1 51.3 53.8 61.5 65.4 71.7 72.9 71.7
0 55.6 57.9 65.1 68.7 73.9 74.5 71.9
Table 9 The phase angle ω (in degrees) computed with different values of the mismatch parameters and β for a specimen with
thickness ratio η = 10
β : 0.5 0.1 0.05 0.01 0.005 0.001
α: −1/3 −1/2
−0.4 50.0 55.3 65.4 68.5 72.6
−0.3 42.4 45.2 54.0 58.9 67.9 70.5 73.2
−0.2 47.0 49.6 57.9 62.4 70.3 72.3 73.8
−0.1 51.3 53.8 61.6 65.7 72.6 74.2 74.3
0 55.6 57.9 65.3 69.0 74.8 75.9 74.8
cant, except for small . However, for large , the
effect of β on ω is complex, see Tables 2–9.
4.4 Lengths of sandwich specimens
As mentioned, the analytical expression for the energy
release rate (16) is derived by evaluating the J-integral
along the exterior boundaries of the specimen using the
stress field corresponding to loading by moments and
axial forces. However, this distribution of the stresses
is only true when the stress field from the crack tip is
far from the ends of the specimen. For short specimens,
the crack tip stress field reaches the boundaries along
which the J-integral is evaluated. Since this is not taken
into account in the analytical result (16), the analytical
G will deviate from the true J-integral value. For
material combinations with small elastic mismatch this
problem is not very pronounced since the stress field
at the crack tip is confined to a small region around the
crack tip. For materials with larger elastic mismatch,
the extension of this zone is significantly larger. Figure
7 shows how the material mismatch parameter
affects the stress field. From the figure it is clear that
the stress field elongates, in particular towards the uncracked
end of the specimen, as the elastic mismatch
between core and face sheet becomes larger.
Consequently, when choosing the dimensions for a
test specimen, it is crucial that the specimen is sufficiently
long, so that the stresses along the ends of the
sandwich are unaffected by the stresses from the crack
tip, so that the results in Sects. 3 and 4 apply. For
this purpose we have established a novel method that
estimates the maximum deviation between the analytical
energy release rate (16) (here denoted Gana) and
the true energy release rate Gtrue. We calculate Gtrue
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