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Failure Theories
2103320 Des Mach Elem Mech. Eng. Department
Chulalongkorn University
• Review stress transformation
• Failure theories for ductile
materials
•Maximum-Shear-Stress Theory
• Distortion-Energy Theory
• Coulomb-Mohr Theory
• Failure theories for brittle materials
•Maximum-Normal-Stress Theory
•Modifications of the Mohr Theory
Stress transformation
• At a point, there is only on stress state (σx, σy, τxy)
• Using different coordinate, identical stress state can be
written by different σx, σy, τxy (ex σ′x, σ′y, τ′xy)
• At a proper coordinate, τ′xy = 0 and only σ1, σ2 exist. This coordinate is called “Principal coordinate”
xσxyτ
yσ
Torque Bending moment
P x
y x′ y′
xσ ′xyτ ′yσ ′
Identical stress state, but
is displayed by different
coordinate system
1σ2σ
θτθσσσσ
σ 2sin2cos2
)(2
)(xy
yxyxx +
−+
+=′
θτθσσσσ
σ 2sin2cos2
)(2
)(xy
yxyxy −
−−
+=′
θτθσσ
τ 2cos2sin2
)(xy
xyxy +
−=′
θ
Principal stresses
2/1
22
21 22)(
,
+
−±
+= xy
yxyx τσσσσ
σσPrincipal stress
Maximum shear stress 2
31max
σστ −=
2 Dimension 3 Dimension
Principal stress 321 ,, σσσ 321 σσσ ≥≥
Maximum shear stress
231
maxσστ −
=
xσxyτ
yσ
Principal stresses are the solution of the
following equation
0322
13 =−+− III σσσ
zyxI σσσ ++=1
yyx
xyx
zzx
xzx
zzy
yzyIσττσ
σττσ
σττσ
++=2
zzyzx
yzyyx
xzxyx
Iστττστττσ
=3
),( xyx τσ
σ
),( xyy τσ
CCWτ
CWτ
1σ2σ
Failure Theories
• For simple load, failure can be known by simple test (tension test, compression test).
• For the combination of loading modes, failure theory is required to predict the failure.
• There is no universal theory of failure for the general case of material properties and stress state.
• Consideration are separated depended on metal behavior (ductile or brittle).
• Data used in the failure theories are based on the simple test (tension test, Compression test).
Ductile Materials
05.0≥fε (Elongation ≥ 5%)
• Maximum shear stress theory (MSS)
• Distortion energy theory (DE)
• Ductile Coulomb-Mohr (DCM)
Brittle Materials
05.0<fε (Elongation < 5%)
• Maximum normal stress theory (MNS)
• Brittle Coulomb-Mohr (BCM)
• Modifier Mohr (MM)
Maximum shear stress theory (1)
• The maximum shear stress (MSS) theory predicts that yielding begins whenever the maximum
shear stress in any element equals or exceeds the maximum shear stress in a tension-test
specimen of the same material when that specimen begins to yield.
• MSS theory is also referred to as the Tresca or Guest theory.
Stress state at a point Tension-test
xσxyτ
yσ
231
maxσστ −
=22max
yS==
στ
AP
=σ
yS== 1σσ
2max yS=τ
2231
maxyS
≥−
=σστYielding begins yS≥− 31 σσor and ysy SS 5.0=
nS y
2231
max ≥−
=σστ
Incorporate a
factor of safety nSy≥− 31 σσor
Maximum shear stress theory (2)
2 Dimension - Plane stress
σA σB σ1 σ3 Yield condition
+ + σA 0
+ - σA σB
- - 0 σB
• Consider at principal direction
• Assuming that principal stress
• No stress in the normal plane, hence the
other principal stress = 0
BA σσ ≥
yA S≥σ
yBA S≥−σσ
yB S−≤σ
Yield if a stress state is outside the
nonyield region
Distortion-Energy theory (1)
• The distortion energy theory predicts that yielding occurs when the distortion strain energy per
unit volume reaches or exceeds the distortion strain energy per unit volume for yield in simple
tension or compression of the same material.
• The distortion energy theory is also called the von Mises or von Mises-Hencky theory or the
octahedral-shear-stress theory
Angular distortion element
At principal direction
Pure volume change Pure angular distortion
3321 σσσσ ++
=av
Strain energy per
unit volume
Strain energy for
producing only
volume change
Distortion energy = +
Distortion-Energy theory (2)
Strain energy per unit volume
F
F
s
L
graph S-Funder Area work =⋅= ∫ dsF
graph -under Area work εσσ =⋅=⋅= ∫∫ dεLds
AF
V
F
s
Work
(σ)
(ε)
simple tension:
strain energy per unit volume εσσ21 =⋅= ∫ dεu
[ ]33221121 σεσεσε ++=u
[ ])(1zyxx E
σσνσε +−=
[ ])(1xzyy E
σσνσε +−=
[ ])(1yxzz E
σσνσε +−=
Hooke’s law
[ ])(221
13322123
22
21 σσσσσσνσσσ ++−++=
Eu
Distortion-Energy theory (3)
[ ])(221
13322123
22
21 σσσσσσνσσσ ++−++=
Eu
)21(2
3 2
νσ−=
Eu av
v
)222(2
21133221
23
22
21 σσσσσσσσσν
+++++−
=E
uv
strain energy per unit volume
strain energy per unit volume
Distortion energy
−+−+−+=−=
2)()()(
31 2
132
322
21 σσσσσσνE
uuu vd
Distortion-Energy theory (4)
• The distortion energy theory predicts that yielding occurs when the distortion strain energy per
unit volume reaches or exceeds the distortion strain energy per unit volume for yield in simple
tension or compression of the same material.
Distortion strain energy at a point Tension-test
−+−+−+=
2)()()(
31 2
132
322
21 σσσσσσνE
ud
At yield yS=1σ 032 ==σσ
2
31
yd SE
u ν+=
Yielding begins when
yS≥
−+−+−2/12
132
322
21
2)()()( σσσσσσ
22
132
322
21
31
2)()()(
31
ySEEνσσσσσσν +
≥
−+−+−+
Von Mises stress
Distortion-Energy theory (5)
2/1213
232
221
2)()()(
−+−+−=′
σσσσσσσVon Mises stress
In xyz coordinate, the von Mises stress can be calculated from
[ ] 2/1222222 )(6)()()(2
1zxyzxyxzzyyx τττσσσσσσσ +++−+−+−=′
Von Mises stress σ′ ≥ Yield strength Sy
nS y≥′σ
Yielding begins when
Incorporate a factor of safety
Yield if a stress state is outside the nonyield region
Distortion-Energy theory (6)
2/122 )( BBAA σσσσσ +−=′
2D - Plane stress
BA σσ ≥
Von Mises stress
• The nonyield region of the distortion energy theory is wider than the
region of the Maximum shear stress theory.
• The prediction from the distortion energy agrees well with all data for
ductile behavior. Hence, it is the most widely used theory for ductile
materials and is recommended for design problems.
• Consider at principal direction
• Assuming that principal stress
• No stress in the normal plane, hence the
other principal stress = 0
Distortion-Energy theory (7)
2 Dimension - Plane stress + pure shear
0== yx σσxyτ
[ ] 2/1222222 )(6)()()(2
1zxyzxyxzzyyx τττσσσσσσσ +++−+−+−=′von Mises stress equation
2/12 )3( xyτσ =′
yxy S≥=′ 2/12 )3( τσ
yy
xy SS
577.03=≥τ ysy SS 577.0=
Shear yield strength Yielding begins when
Example
A material has the yield strength Syc = Syt = 100 MPa, and εf = 0.55. Determine the factor of safety of the following cases.
(Mpa) σx σy τxy
a 70 70 0
b 60 40 -15
c 0 40 45
d -40 -60 15
e 30 30 30
Coulomb-Mohr Theory (Ductile Materials) (1)
• Can be used for materials whose strengths in
tension and compression are not equal.
• Use data from tension test and compression
test to draw Mohr’s circles
• Draw failure enveloped tangent to the circles
• Yield if a stress state is outside the envelope
31
1133
21
1122
CCCBCB
CCCBCB −
=−
Triangles OBiCi are similar, therefore
211 tSCB =
2)( 3122 σσ −=CB
233 cSCB =
2)( -origin 312 σσ +=C
2 -origin 1 tSC =
2 -origin 3 cSC =
131 =−ct SS
σσ
131 ≥−ct SS
σσ
nSS ct
131 ≥−σσ
Yielding begins
Incorporate a
factor of safety
Coulomb-Mohr Theory (Ductile Materials) (2)
σA σB σ1 σ3 Yield condition
+ + σA 0
+ - σA σB
- - 0 σB
tA S≥σ
1≥−c
B
t
A
SSσσ
cB S−≤σ
BA σσ ≥• Consider at principal direction
• Assuming that principal stress
• No stress in the normal plane, hence the
other principal stress = 0
2D - Plane stress
Yield if a stress state is outside the
nonyield region
Maximum-Normal-Stress Theory (Brittle)
• The maximum normal stress (MNS) theory states that failure occurs whenever one of the three
principal stresses equals or exceeds the strength.
321 σσσ ≥≥Principal stress
utS≥1σ ucS−≤3σor
nSut≥1σ n
Suc−≤3σor
Note
Yield strength of the brittle materials can not be
observed, hence the ultimate tensile strength or ultimate
compressive strength are used instead
Yielding begins
Incorporate a
factor of safety
Modifications of the Mohr Theory (Brittle)
Brittle-Coulomb-Mohr
Plane stress + factor of safety
σA σB σ1 σ3 Yield condition
+ + σA 0
+ - σA σB
- - 0 σB
nSutA ≥σ
nSS uc
B
ut
A 1≥−
σσ
nSucB −≤σ
Modified Mohr
σA σB σ1 σ3 Yield condition
+ + σA 0
+ -
σA σB
+ -
σA σB
- - 0 σB
nSutA ≥σ
nSSSSS
uc
B
utuc
Autuc 1)(≥−
− σσ
nSucB −≤σ
1≤AB σσ
1>AB σσ
nSutA ≥σ
Plane stress + factor of safety
Modifications of the Mohr Theory (Brittle)
Selection of Failure Criteria
