Rasmus ÿstergaard forside 100%.indd - Solid Mechanics

310 R.C. Østergaard, B.F. Sørensen
Fig. 6 The
load-independent phase
angle ω as function of the
elastic mismatch parameter
for a number of different
thickness ratios. The arrow
indicate the value
corresponding to a
symmetric, homogeneous
DCB specimen loaded by an
axial force
Fig. 7 The shear stress
contours
σ12 H/P = 0.1, 0.05, 0.01
for a sandwich specimen
with η = 1, β =−0.2 and
= 0.5, 0, 05 and 0.005,
respectively
ω [ ◦ ]
from a FEM solution with the CSDE method. First, the
method determines the load combination that results in
the largest deviation between Ganaand Gtrue and then
it calculates an relative error ξ = Gana−Gtrue
for this
Gtrue
combination of the loads. Details on the procedure is
found in Appendix C. Figure 8a–d shows curves for the
intact length of the specimen, (L − a)/H, that gives an
error ξ = 0.05.
The curves show that for >0.01, the length of
the intact part of the sandwich specimen must be larger
than 10H to ensure that the error, ξ, stays below 5%.
This holds for all analyzed values of η ≤ 8. For an
elastic mismatch = 0.001, (L − a) must be larger
to ensure low errors on G. For instance, a sandwich
specified by η = 1, = 0.001 and β = 0anerror
ξ = 5% is found for (L − a)/H = 27.
It was found that the error, ξ, is strongly dependent
on (L − a)/H and a small change of (L − a) changes
123
90
80
70
60
50
40
30
20
10
η = 10 η = 8
η = 4
β = 0
β = −0.3
η = 1
η = 0.2
0
0.001 0.01 0.1 1
Σ
η = 0, Σ= 1
ω = 49.1◦ ξ significantly. Increasing (L − a) with 2H, typically
results in an error that is practically zero; vice versa,
reducing the length with 2H results in large deviations
on G.
As seen in Fig. 7 the stress field also elongates slightly
towards the cracked end of the specimen (x1 =−a).
However, an investigation showed that errors above 1%
does not appear for a/H > 4.
5 Example: a method for measuring interfacial
fracture toughness of sandwich specimens under
mixed mode loadings
It is well known from experiments and modeling works
that the fracture toughness of weak interfaces can depend
on the mode mixity (Cao and Evans 1989;
Wang and Suo 1990; Liechti and Chai 1992;