Determination of Strength and Fracture Toughness from Indentation Tests
47
σ r =
3F
2π c 2
1 − 2ν
3
c 2
r 2
1 −
z
√
u
3
+
z
√
u
3
c 2 u
u 2 + c 2 z 2
+
z
√
u
u
1 − ν
c 2 + u
+ (1 + ν)
√
u
c
tan −1
c
√
u
− 2
(5)
Herein the quantity u is given by the geometrical entities r, z and c as
u =
1
2
r
2
+ z
2
− c
2
+
r 2 + z 2 − c 2
2 + 4c 2 z 2
(6)
While r and z denote the cylindrical coordinates in radial and depth direction the quantity
c is the contact radius. According to Johnson [10] it can be calculated through
c =
3
3FR
4
1 − ν 2
E
+
1 − ν 2
ind
E ind
(7)
Herein E and ν are the elastic properties of the specimen and E ind and ν ind characterize the
elastic properties of the indenter. In this way the stress criterion is completely specified.
5 Energy Criterion
For the calculation of the energy release rate we can follow the same approach as it
has already been used by Strobl [4] and Mouginot/Maugis [7]. In doing so, we assume
that the stress field in the uncracked half space (that means in the material specimen) in
essence remains the same during crack generation. Then the stress intensity factor for a
crack with the length z under mode I loading in good approximation can be calculated
with the method of weighting functions according to
K I (z) = 2
z
π
z
∫
0
σ r (˜ z)
√
z 2 − ˜
z 2
k(˜ z)d˜ z
(8)
(compare Tada [12]). Herein the function k can be approximated as k(˜ z) ≈ 1 +
0.3(1 − ˜
z/z). This way it is taken into account that the crack starts from a free surface. For linear elastic material behavior there is a simple relationship between the stress
intensity factor K I and the energy release rate G:
G =
1 − ν 2
E
K I .
(9)
As a consequence the energy criterion can be formulated as:
G =
1 − ν 2
E
1
a
a
0
K
2
I (z)dz ≥ G c .
(10)
The two integrations for the use of the energy criterion can be performed numerically.
Furtheron, by means of dimension analysis it can be shown that
G =
F
R
φ
ξ =
r 0
c
, η =
a
c
, ν
.
(11)
47
σ r =
3F
2π c 2
1 − 2ν
3
c 2
r 2
1 −
z
√
u
3
+
z
√
u
3
c 2 u
u 2 + c 2 z 2
+
z
√
u
u
1 − ν
c 2 + u
+ (1 + ν)
√
u
c
tan −1
c
√
u
− 2
(5)
Herein the quantity u is given by the geometrical entities r, z and c as
u =
1
2
r
2
+ z
2
− c
2
+
r 2 + z 2 − c 2
2 + 4c 2 z 2
(6)
While r and z denote the cylindrical coordinates in radial and depth direction the quantity
c is the contact radius. According to Johnson [10] it can be calculated through
c =
3
3FR
4
1 − ν 2
E
+
1 − ν 2
ind
E ind
(7)
Herein E and ν are the elastic properties of the specimen and E ind and ν ind characterize the
elastic properties of the indenter. In this way the stress criterion is completely specified.
5 Energy Criterion
For the calculation of the energy release rate we can follow the same approach as it
has already been used by Strobl [4] and Mouginot/Maugis [7]. In doing so, we assume
that the stress field in the uncracked half space (that means in the material specimen) in
essence remains the same during crack generation. Then the stress intensity factor for a
crack with the length z under mode I loading in good approximation can be calculated
with the method of weighting functions according to
K I (z) = 2
z
π
z
∫
0
σ r (˜ z)
√
z 2 − ˜
z 2
k(˜ z)d˜ z
(8)
(compare Tada [12]). Herein the function k can be approximated as k(˜ z) ≈ 1 +
0.3(1 − ˜
z/z). This way it is taken into account that the crack starts from a free surface. For linear elastic material behavior there is a simple relationship between the stress
intensity factor K I and the energy release rate G:
G =
1 − ν 2
E
K I .
(9)
As a consequence the energy criterion can be formulated as:
G =
1 − ν 2
E
1
a
a
0
K
2
I (z)dz ≥ G c .
(10)
The two integrations for the use of the energy criterion can be performed numerically.
Furtheron, by means of dimension analysis it can be shown that
G =
F
R
φ
ξ =
r 0
c
, η =
a
c
, ν
.
(11)