48
J. Hahn and W. Becker
Correspondingly it is sufficient to determine the dimensionless function φ in order to
determine the energy release rate for any arbitrary load F and any indenter radius R. In
this connection also the normalized crack radius ξ and the normalized crack length η
are introduced. Both quantities are normalized with respect to the contact radius c.
6 Numerical Determination of the Energy Release Rate
For the verification of the analytically determined energy release rate a finite element
model is implemented for the material specimen. The effect of the indenter is taken
into account by prescribed displacements u z in depth direction over the contact surface.
For a spherical indenter according to Fischer-Cripps [11] the displacements are of the
following form:
u z
u z,max
= 1 −
r 2
2a 2 ∀r ≤ c
(12)
Herein u z,max denotes the maximal indentation displacement. The simulation is performed for the uncracked configuration and for a series of cracked configurations. The
energy release rate then results from the difference of the elastic total potential. As the
indenter displacements are prescribed the crack initiation does not cause a change of the
Fig. 3. Results of finite fracture mechanics for an indenter radius R = 1 mm. In the upper half
the failure load F
f in dependence of the normalized crack radius ξ = r 0 /c. In the lower half the
normalized crack length η 0 = a/c again in dependence of the normalized crack radius.
J. Hahn and W. Becker
Correspondingly it is sufficient to determine the dimensionless function φ in order to
determine the energy release rate for any arbitrary load F and any indenter radius R. In
this connection also the normalized crack radius ξ and the normalized crack length η
are introduced. Both quantities are normalized with respect to the contact radius c.
6 Numerical Determination of the Energy Release Rate
For the verification of the analytically determined energy release rate a finite element
model is implemented for the material specimen. The effect of the indenter is taken
into account by prescribed displacements u z in depth direction over the contact surface.
For a spherical indenter according to Fischer-Cripps [11] the displacements are of the
following form:
u z
u z,max
= 1 −
r 2
2a 2 ∀r ≤ c
(12)
Herein u z,max denotes the maximal indentation displacement. The simulation is performed for the uncracked configuration and for a series of cracked configurations. The
energy release rate then results from the difference of the elastic total potential. As the
indenter displacements are prescribed the crack initiation does not cause a change of the
Fig. 3. Results of finite fracture mechanics for an indenter radius R = 1 mm. In the upper half
the failure load F
f in dependence of the normalized crack radius ξ = r 0 /c. In the lower half the
normalized crack length η 0 = a/c again in dependence of the normalized crack radius.