Chap.7 yielding & fracture under combined stresses

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Nov 29, 2013 (3 years and 11 months ago)

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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



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

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


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


, 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)