Summary of Fracture, Impact and Fatigue Behaviors
- Brittle fracture
- Corrosion fatigue
- Creep
- Ductile fracture
- Ductile-to-brittle transition
- Fatigue
- Fatigue life
- Fatigue limit
- Fatigue strength
- Impact energy
- Intergranular fracture
- Stress raiser
- Thermal fatigue
- Transgranular fracture
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Creep
Creep is a time-dependent and permanent deformation of materials when subjected to a
constant load at a high temperature (> 0.4 Tm). Examples: turbine blades, steam generators.
(Tm in Kelvins)
Creep Stages
1. Instantaneous deformation, mainly elastic.
2. Primary/transient creep. Slope of strain vs. time decreases with time: work-hardening
3. Secondary/steady-state creep. Rate of straining is constant: balance of work-hardening and recovery.
4. Tertiary. Rapidly accelerating strain rate up to failure:
Formation of internal cracks, voids, grain boundary separation, necking, etc.
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Parameters of creep behavior
∆𝜀
The stage of secondary/steady-state creep is of longest duration and the steady-state creep rate 𝜀 ∙ = ∆𝑡 is the
most important parameter of the creep behavior in long-life applications.
Another parameter, especially important in short-life creep situations, is time to rupture, or the rupture
life time, tr.
The stage of secondary/steady-state creep is of longest duration and the steady-state creep rate is the most
important parameter of the creep behavior in long-life applications.
Another parameter, especially important in short-life creep situations, is time to rupture, or the rupture
lifetime, tr.
Creep: stress and temperature effects
With increasing stress or temperature:
*The instantaneous strain increases
*The steady-state creep rate increases
*The time to rupture decreases
Creep: stress and temperature effects
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The stress/temperature dependence of the steady-state creep rate can be described by
𝑄
𝜀 ∙ = 𝐴𝑒𝑥𝑝 {− 𝑅𝑇}---------------------- (1)
Where Q = the activated energy for the rate-controlling process
A = the material structural constant
T = the absolute temperature
R = the universal gas constant
The activated energy Q can be calculated by assuming the temperature interval is small so
that the creep mechanisms is not expected to change.
𝐴 = 𝜀1 ∙ 𝑒 𝑄/𝑅𝑇 = 𝜀2 ∙ 𝑒 𝑄/𝑅𝑇
𝑄=
𝜀1 ∙
𝑅 ln( ∙ )
𝜀2
1
1 ---------------------------- (2)
−
𝑇2 𝑇1
Mechanisms of Creep
Different mechanisms are responsible for creep in different materials and under different loading and
temperature conditions. The mechanisms include:
* Stress-assisted vacancy diffusion
* Grain boundary diffusion
* Grain boundary sliding
* Dislocation motion
Alloys for high-temperature use
(Turbines in jet engines, hypersonic airplanes, nuclear reactors, etc.)
Creep is generally minimized in materials with:
* High melting temperature
* High elastic modulus
* Large grain sizes (inhibits grain boundary sliding)
Following alloys are especially resilient to creep:
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- Stainless steels
- Refractory metals (containing elements of high melting point, like Nb, Mo, W, Ta)
- “Superalloys” (Co, Ni based: solid solution hardening and secondary phases)
Presentation of engineering creep data
Creep strength is defined as the stress at a given temperature, which produces a
steady-state creep rate (10-11 to 10-8 s-1.)
Creep data
Creep data can also be presented as a plot of stress and time to produce different amounts of total
strain.
The upper most curve is the stress rupture curve. The percentage beside each data
point is the percentage reduction at failure.
Example:
Determine the working stress at 600oC and 800oC for type
316 stainless steel if the design criterion is a creep strength based on
1 percent extension in 1000 hr. Use a factor of safety of 3.
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Example:
For the stress-minimum creep rate curve, determine the
activation energy at 700 0C and 800 0C for creep at a stress of 100 MPa.
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Viscoelasticity
Introduction
In classic elasticity there is no time delay between the application of a force and the deformation that it causes.
For many materials, however, there is additional time-dependent deformation that is recoverable. This is called
viscoelastic or anelastic deformation. When a load is applied to a material, there is an instantaneous elastic
response, but the deformation also increases with time. This viscoelasticity should not be confused with creep,
which is time-dependent plastic deformation. Anelastic strains in metals and ceramics are usually so small
that they are ignored. In many polymers, however, viscoelastic strains can be very significant. Anelasticity is
responsible for the damping of vibrations. A high damping capacity is desirable where vibrations might
interfere with the precision of instruments or machinery and for controlling unwanted noise. A low damping
capacity is desirable in materials used for frequency standards, in bells, and in many musical instruments.
Viscoelastic strains are often undesirable. They cause sagging of wooden beams, denting of vinyl flooring by
heavy furniture, and loss of dimensional stability in gauging equipment. The energy associated with damping
is released as heat, which often causes an unwanted temperature increase. Study of damping peaks and how
they are affected by processing has been useful in identifying mechanisms. The mathematical descriptions of
viscoelasticity and damping will be developed in this part.
Most polymers exhibit both elastic and viscous response to external forces.
Models for such a viscoelastic behavior consist of viscous and elastic elements. The spring illustrates purely
elastic behavior and the dashpot purely viscous behavior. If the spring is Hookean and dashpot is Newtonian,
the connection of these two elements in series gives the Maxwell model (see Fig. below); when they arranged
in parallel, the Kelven or Voigt model results (see Fig. below).
1- Maxwell model
Let G be the shear modulus of the spring and η the viscosity dashpot. When a load F is applied, causing the
𝜏
shear stress τ, the spring instantaneously stretches by the amount 𝐺 and the dashpot begins to elongate
𝜏
steadily at the rate 𝜂 .The total deformation in the Maxwell model is therefore distributed between two
elements(spring +dashpot).The two elements are, however, subjected to the entire stress τ. Thus we can
write
and
𝜏 = 𝜏𝑒 = 𝜏 𝑣
(1)
𝛾𝑡 = 𝛾𝑒 + 𝛾𝑣
(2)
Here
τ
is
the
total
stress
causing
the
total
deformation
𝜏𝑒 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑡𝑟𝑒𝑠𝑠 𝑐𝑎𝑢𝑠𝑖𝑛𝑔 𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝑠𝑡𝑟𝑎𝑖𝑛 𝛾𝑒 𝑎𝑛𝑑 𝜏𝑣 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑡𝑟𝑒𝑠𝑠 𝑐𝑎𝑢𝑠𝑖𝑛𝑔 𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑠𝑡𝑟𝑎𝑖𝑛 𝛾𝑣
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𝛾𝑡 ,
1
𝛾𝑒 = 𝜏
𝐺
𝑑𝛾𝑣
And
𝑑𝑡
The rate of the total strain with time is
𝑑𝛾𝑡
𝑑𝑡
1
=
𝑑𝛾𝑒
=
𝜏
𝜂
𝑑𝛾𝑣
+
𝑑𝑡
𝑑𝑡
(3)
(4)
(5)
Substituting Equations 3 and 4 into 5, we obtain
𝑑𝛾𝑡
𝑑𝑡
=
1 𝑑𝜏
𝐺 𝑑𝑡
1
+ 𝜏
(6)
𝜂
Equation 5 states that the total rate change of 𝛾𝑡 with time is determined by the rate of flow plus the rate of
change of the elastic deformation. If 𝛾𝑡 is constant,
1 𝑑𝜏
𝐺 𝑑𝑡
+
1
𝜂
𝑑𝛾𝑡
𝑑𝑡
=0 and Eq.6 reduces to
𝜏=0
(7)
To solution of this simple differential equation is obtained by transferring the terms on both sides and
integrating from time t=0 to time t, and from 𝜏0 to 𝜏 .
𝜏 = 𝜏0 𝑒
𝐺
𝜂
−( )𝑡
(8)
Eq.8 indicates that the stress 𝜏 gradually disappears exponentially, so that if the body is left to itself it will
𝜂
lose any internal stress. After a time t equal to 𝐺 , the stress will have decayed to 1/e of its original value. This
time is the relaxation time Trel for the material. Then
𝜏 = 𝜏0 𝑒 (−𝑡/𝑇𝑟𝑒𝑙 )
(9)
2- Kelvin or Voigt Model
For the Kelvin or Voigt model the situation is exactly reversed. The shear stress will be distributed
between the two elements, but any deformation will take place equally and simultaneously in both
elements. Mathematically this is represented by the following relationships.
𝛾 = 𝛾𝑒 = 𝛾𝑣
(10)
𝜏 = 𝜏𝑒 + 𝜏𝑣
(11)
Substituting 𝜏𝑒 = 𝐺𝛾 and 𝜏𝑣 = 𝜂(𝑑𝛾/dt) into Equation 11, we obtain
𝜂
𝑑𝛾
𝑑𝑡
+ 𝐺𝛾 = 𝜏
(12)
which, on solution, gives
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𝜏
𝛾 = [1 − 𝑒
𝐺
𝐺
𝜂
−( )𝑡
]
(13)
When the Kelvin element is stressed, part of the energy is stored in the spring, while the reminder of the
energy is dissipated in the dashpot and results in the retarded deformation. On releasing the stress, recovery
will occur because of the elasticity of the spring, but it is again retarded by the dashpot. The time needed to
achieve the equilibrium is called the retardation time, which equal to the ratio 𝜂/𝐺, as in the Maxwell
model. Equation 13 can then be written
𝜏
𝛾𝑡 = (1 − 𝑒 −𝑡/𝑇𝑟𝑒𝑡 )
(14)
𝐺
Where 𝛾𝑡 is called delayed or retarded elastic deformation.
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