Chap.7 yielding & fracture under
combined stresses
7.1
yield
criteria ↔
fracture
criteria
└── Failure ─┘
◦
need for failure criteria
Yield strengths for a ductile metal under various states of stress: (a) uniaxial tension, (b)
tension with transverse compression, (c) biaxial tension, and (b) hydrostatic
compression.
응력상태에
따라
항복점
달라짐
.
◦
crack
존재
고려
Ⅹ
7.2 general form of failure criteria
◦
for isotropic materiasl
c
f
3
2
1
,
,
(at failure)
7.3 maximum normal stress fracture criterion
3
2
1
,
,
may
u
◦
graphical representation of
the normal stress criterion
Figure 7.2
Failure locus for the maximum normal stress fracture
criterion for plane stress.
◦
safety factor
1
u
N
D
D
X
Figure 7.3
Three

dimensional failure surface for the
maximum normal stress fracture criterion.
7.4 maximum shear stress yield criterion
s
msx
msx
msx
0
1
3
3
2
2
1
0
1
3
3
2
2
1
0
0
0
1
3
3
2
2
1
0
max
,
,
2
2
,
2
,
2
2
2
,
2
,
2
인장실험
◦
graphical representation of the maximum
shear stress criterion
Figure 7.5
Failure locus for the maximum shear stress yield
criterion for plane stress.
Figure 7.6
Three

dimensional failure surface for
the maximum shear stress yield criterion.
◦
hydrostatic stresses & the maximum shear
Stress criterion
7.5 octahedral shear stress yield criterion
◦
development of the octahedral shear stress
criterion
◦
graphical representation of the octahedral
shear stress criterion
◦
energy of distortion
7.6 discussion & comparison of the basic failure
criteria
◦
comparison of failure criteria
Figure 7.11
Plane stress
failure loci for three criteria.
These are compared with
biaxial yield data for ductile
steels and aluminum alloys,
and also with biaxial
fracture data for gray cast
iron. (The steel data are
from [Lessells 40] and
[Davis 45], the aluminum
data from [Naghdi 58] and
[Marin 40], and the cast
iron data from [Coffin 50]
and [Grassi 49].)
◦
hydrostatic & deviatoric stresses
◦
yield criteria for anisotropic & uneven materials
ex) polymer
Biaxial yield data for various polymers compared to a modified octahedral
shear stress theory. (After Raghava, Caddell, and Yeh [Raghava 73]; used
with permission.)
◦
fracture in brittle materials
“normal stress criterion good as long as the normal
stress having the largest absolute value is tensile.”
Biaxial fracture data
of gray cast iron
compared to
various frac

ture
criteria.
(
Data from
[Grassi 49.] )
Dominant stress compressive,
→ fracture occurs in planes of max shear stress
7.7 coulomb

mohr fracture criterion
i
Figure 7.15 Coulomb

Mohr fracture criterion as
related to Mohr
’
s circle
and predicted fracture
planes.
◦
development of the coulomb

mohr criterion
sin
2
2
cos
2
2
sin
cos
tan
1
,
3
1
3
1
3
1
i
i
일직선상에있으면
이
u
2
3
1
3
1
◦
graphical representation of
the coulomb

mohr criterion
)
,
(
1
2
,
1
2
b
a
m
m
u
uc
u
ut
Fracture planes predicted by the Coulomb

Mohr criterion for
uniaxial tests tension and compression.
0
,
2
3
1
Pure torsion and the fracture planes predicted by
the Coulomb

Mohr criterion.
◦
discussion of the coulomb

mohr criterion
Biaxial yield data for various polymers compared to a modified octahedral
shear stress theory. (After Raghava, Caddell, and Yeh [Raghava 73]; used
with permission.)
문제점
설명
p. 267
설명
7.8 modified mohr fracture criterion
tension
이
작으나마
존재할
경우
→
Some variations of the modified Mohr fracture criterion.
Three

dimensional failure surface for the modified Mohr fracture criterion for
the particular case of . The Coulomb

Mohr surface is truncated by
three faces of the maximum normal stress cube.
ut
i
7.9 additional comments on failure criteria
◦
brittle versus ductile behavior

brittle materials may exhibit considerable ductility
under highly comp. hydrostatic stress
Stress

strain data for limestone cylinders tested under axial com

pression with
various hydrostatic pressures ranging from one to 10,000 atmo

spheres, The
applied compressive stress plotted is the stress in the pressurized laboratory, that
is, the compression in excess of pressure. (Adapted from[Griggs 36]; used with
permission; coptright
ⓒ
The University of Chicago press.)
Effect of pressures ranging from one to 26,500 atmospheres on the tensile
behavior of a steel, specifically AISI 1045 whit stress in the
pressurized laboratory is plotted. (data from[Bridman 52] pp.47

61)
.
40
HRC
↔
즉
, ductile
–
brittle under testing hydro stress

fracture
&
yielding
are separate event
Modified Mohr Tresca or Mises
Relationships of the limiting surfaces for yielding and fracture
for materials that usually in a brittle manner.
◦
stress raiser effect
◦
effects of cracks &
their time

dependent
growth
(s.c.c)
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