dissertation global and local fracture properties of metal matrix ...

Section 3
Fig. 3.6. Schematic view of the HRR-theory.
21
(3.1)
In Eq. 3.1, E is the Young modulus, J is the fracture toughness of material, σ0 is the yield
strength, r and θ are the polar coordinates, IN and dN are dimensionless constants both
depending on the strain hardening coefficient, N, and on σ0 /E, ~ σ ( N,
θ ) is a dimensionless
function listed in [59].
The J-integral in Eq. 3.1 can be substituted by COD using the relation proposed by Shih in
[58]
, (3.2)
where dN is dimensionless constant depending on the strain hardening coefficient, N, [58].
The material is assumed to follow a standard power-law work hardening behavior
, (3.3)
where according to [59], the σ0 is set to the yield strength, σy, and α is determined from the
tensile true stress-strain curve.
⎡ E J ⎤
σ σ
~
ij = 0 ⎢
σ ,
2 ⎥
ij
⎢⎣
ασ 0 I N r ⎥⎦
J =
d
1
σ 0
N
To evaluate the stress tensor at the moment of void initiation, σ HRR vi, the measured r, θ, and
the CODvi-values are inserted into Eqs. (3.2) and (3.1). From the stress tensor σ HRR vi, the
maximum principal stress, σ HRR max, can be calculated.
1/(
N + 1)
COD
ε
⎛ σ ⎞
= α ⎜
⎟
ε 0 ⎝σ
0 ⎠
N
r
θ
( N θ )
ij
σij