It has been considered that ductile fracture of metal in tension occurs by void coalescence. It has been reported in the preceding paper that (1) the volume fraction of voids,
vV, necessary to cause void coalescence by internal necking is given as a function of radial stress, σ
r, and (2) the area,
SV, of the V region in which the volume fraction of voids exceeds
vV can be expressed as a function of
S⁄πρ
2,
v and σ
r, where
S is the cross-sectional area of a specimen, ρ is a radius of void and
v is the mean volume fraction of voids of a specimen.
This paper describes the area of the strain hardening region H, which is considered to surround the V region, and the equation for the condition of ductile fracture. A criteria of fracture was considered to be
ΔPV+
ΔPH≤0, where
ΔPV(=
SV\left{\dfrac∂
Y∂ε\left[\dfrac34(1−
vV2)+\dfrac16ln\dfrac27256
vV3\
ight}−
Y\left(\dfrac32
vV+\dfrac12
vV\
ight)\
ight]\dfracδ
h, refer to the preceding paper) and
ΔPH are the increments of the load on the V and H regions after small deformation. (
Y: yield stress, δ: displacement,
h: a half height of the V regon equal to ρ⁄\sqrt
vV)
To calculate
ΔPH, it is assumed that (1) the strain in the H region varies in inverse proportion to the distance from the V region (i.e. strain concentration is relieved with distance) and (2) the distance from the center of the V region to the outer border of the H region in a certain direction is equal to the one which would move from the center of the V region before impinging on a void in the H region. Under these assumptions
ΔPH was expressed as follows:
ΔP_H = (1+σ_r/Y) \frac∂Y∂ε\left{\fracπ^2ρv+((S_V/π)^1/2-h) ∫_0^2 π ln\left(1+\fracπρ2vh ln\frac2 π2 π-θ\
ight)d θ\
ight}δ.
From the above equations, the condition of ductile fracture could be expressed as a function of
v, σ
r,
S⁄πρ
2 and the relative strain hardening ratio (∂
Y⁄∂ε)⁄
Y.
The theoretical results were in approximate agreement with experiment reported previously.
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