Bài giảng Strength of materials - Chương 5: Failure theories - Nguyễn Sỹ Lâm
CHAPTER 5: FAILURE THEORIES
Failure – A part is permanently distorted (bị bóp méo) and will not function
properly.
A part has been separated into two or more pieces.
Material Strength
Sy = Yield strength in tension, Syt = Syc
Sys = Yield strength in shear
Su = Ultimate strength in tension, Sut
Suc = Ultimate strength in compression
Sus = Ultimate strength in shear = 0.67 Su
CHAPTER 5: FAILURE THEORIES
A ductile material deforms significantly before fracturing. Ductility is measured by %
elongation at the fracture point. Materials with 5% or more elongation are considered
ductile.
Brittle material yields very little before fracturing, the
yield strength is approximately the same as the ultimate
strength in tension. The ultimate strength in compression
is much larger than the ultimate strength in tension.
FAILURE THEORIES – DUCTILE MATERIALS
Yield strength of a material is used to design components made of
ductile material
• Maximum shear stress theory (Tresca 1886)
(Thuyết bền: ứng suất tiếp lớn nhất – TB3)
(max )component > ( )obtained from a tension test at the yield point
Failure
Sy
= Sy
=
2
To avoid failure
Sy
2
(max )component
<
= Sy
n = Safety factor
Design equation
=Sy
FAILURE THEORIES – DUCTILE MATERIALS
SPECIAL CASES
A special planar state of stress
τxy= τ
Sy
TB3 2 4 2
σx= σ
2 2n
Purely shear state of stress
τxy= τ
Sy
2 2n
FAILURE THEORIES – DUCTILE MATERIALS
• Distortion energy theory (von Mises-Hencky)
(Thuyết bền: Thế năng biến đổi hình dáng lớn nhất – TB4)
Simple tension test → (Sy)t
Hydrostatic state of stress → (Sy)h
t
h
(Sy)h
(Sy)t
>>
Distortion contributes to
failure much more than
change in volume.
h
h
t
(total strain energy) – (strain energy due to hydrostatic stress) = strain energy
due to angular distortion > strain energy obtained from a tension test at the
yield point → failure
FAILURE THEORIES – DUCTILE MATERIALS
The area under the curve in the elastic region is called the Elastic Strain Energy.
U = ½ ε
3D case
UT = ½ 1ε1 + ½ 2ε2 + ½ 3ε3
Stress-strain relationship
2
E
1
3
E
3
1
E
2
v
v
v
ε1
ε2
ε3
v
v
v
=
=
=
E
1
E
E
2
E
E
3
E
1
2E
2
2
2
UT =
(1 + 2 + 3 ) - 2v (12 + 13 + 23)
FAILURE THEORIES – DUCTILE MATERIALS
Distortion strain energy = total strain energy – hydrostatic strain energy
Ud = UT – Uh
1
2E
2
2
2
UT =
(1 + 2 + 3 ) - 2v (12 + 13 + 23)
(1)
Substitute 1 = 2 = 3 = h
1
2E
2
2
2
Uh =
(h + h + h ) - 2v (hh + hh+ hh)
Simplify and substitute 1 + 2 + 3 = 3h into the above equation
(1 + 2 + 3)2
2
3h
(1 – 2v)
Uh =
(1 – 2v) =
2E
6E
Subtract the hydrostatic strain energy from the total energy to
obtain the distortion energy
1 + v
(2 – 3)2
(1 – 2)2 + (1 – 3)2 +
(2)
Ud = UT – Uh =
6E
FAILURE THEORIES – DUCTILE MATERIALS
Strain energy from a tension test at the yield point
Substitute in equation (2)
1= Sy and 2 = 3 = 0
1 + v
(2 – 3)2
(1 – 2)2 + (1 – 3)2 +
Ud = UT – Uh =
6E
1 + v
(S )2
Utest =
y
3E
To avoid failure, Ud < Utest
(2 – 3)2
(1 – 2)2 + (1 – 3)2 +
½
< Sy
2
FAILURE THEORIES – DUCTILE MATERIALS
(2 – 3)2
(1 – 2)2 + (1 – 3)2 +
½
< Sy
2
2D case, 3 = 0
½
2
2
(1 – 12 + 2 ) < Sy =
Where is von Mises stress
Design equation
FAILURE THEORIES – DUCTILE MATERIALS
τxy= τ
Pure torsion, = 1 = – 2
2
2
2
(1 – 2 1 + 2 ) = Sy
32 = Sy2
Sy
Sys = Sy / √ 3 →
or
3 3n
Relationship between yield strength in
tension and shear
1, 2 = x/2 ± [(x)/2]2 + (xy)2
If y = 0, then
the design equation can be written in terms of the dominant component
stresses (due to bending and torsion)
τxy= τ
Sy
2
2
TB4 1 31
σx= σ
3 3n
FAILURE THEORIES – DUCTILE MATERIALS
Distortion energy theory
Sy
Maximum shear stress theory
Sy
max =
2n
′ =
n
Sy , Su
• Select material: consider environment, density, availability →
• Choose a safety factor
n
Size Weight Cost
The selection of an appropriate safety factor should be based
on the following:
Degree of uncertainty about loading (type, magnitude and direction)
Degree of uncertainty about material strength
Uncertainties related to stress analysis
Consequence of failure; human safety and economics
Type of manufacturing process
Codes and standards
FAILURE THEORIES – DUCTILE MATERIALS
Design Process
Use n = 1.2 to 1.5 for reliable materials subjected to
loads that can be determined with certainty.
Use n = 1.5 to 2.5 for average materials subjected to
loads that can be determined. Also, human safety and
economics are not an issue.
Use n = 3.0 to 4.0 for well known materials subjected to
uncertain loads.
FAILURE THEORIES – DUCTILE MATERIALS
Design Process
Sy , Su
• Select material, consider environment, density, availability →
• Choose a safety factor
• Formulate the von Mises or maximum shear stress in terms of size.
• Use appropriate failure theory to calculate the size.
Sy
Sy
max =
′ =
2n
n
• Optimize for weight, size, or cost.
FAILURE THEORIES – BRITTLE MATERIALS
One of the characteristics of a brittle material is that the ultimate strength
in compression is much larger than ultimate strength in tension.
Suc
>> Sut
Mohr’s circles for compression and tension tests.
Suc
3
1
Stress
state
Sut
Tension test
Compression test
Failure envelope
The component is safe if the state of stress falls inside the failure envelope.
1 > 3 and 2 = 0
FAILURE THEORIES – BRITTLE MATERIALS
Modified Coulomb-Mohr theory
(Thuyết Bền Morh – TB5)
3 or 2
3 or 2
Sut
Sut
I
Safe
Safe
Safe
Sut
Sut
1
Suc
II
-Sut
Safe
III
Suc
Suc
Cast iron data
Three design zones
FAILURE THEORIES – BRITTLE MATERIALS
Modified Coulomb-Mohr theory
2
Sut
Zone I
2 > 0
and 1 > 2
1 > 0 ,
I
Sut
Design equation
1
II
-Sut
Zone II
III
2 < 0
and 2 < Sut
1 > 0 ,
Suc
Design equation
Zone III
2 < 0
and 2 > Sut
1 > 0 ,
Design equation
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