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Mechanical Engineering

MECHENG 382

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Winter

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Lecture 29 - Introduction to creep
C REEP
• Instantaneous response to a load is elastic or plastic deformation
ε = f σ)
• In general, all materials exhibit additional deformation with time
ε = f σ,t)
This is known as creep
• Importance of instantaneous response vs. creep response depends on
i) Time scale
Ice is elastic, but glaciers creep over a period of years
ii) Temperature
Al2O3is elastic at room temperature, but creeps at 1300 °C
• Creep deformation occurs in response to shear (constant volume) - like yield
Therefore, loading parameter for creep is von Mises effective normal stress
˜ 1 2 2 2
σH= [σ 1 σ 2) + σ 1σ 3) + σ −2σ 3)]
2
• Constitutive equations for creep are of the form:
dε˜H
dt = f ( H
where, the effective strain is given by
€ 2 2 2 2
εH= [ε1−ε 2) ( ε1−ε 3) (+ ε2−ε 3) ]
9
• Remember, that the von Mises effective stress was used so that the multiaxial yield
criterio€ was the same as the criterion for uniaxial tension.
• The same reasoning applies to this definition of effective strain:
(i) Creep is deformation by shear
(ii) Therefore, volume is conserved
(iii) In uniaxial creep:1σ =oσ ;2σ = 0;3σ = 0
If 1 = o , then conservation of volume (at small strains) implies
4/iv/14 1 Lecture 29 - Introduction to creep
ε = - ε /2; ε = - ε /2
2 0 3 o
˜ ˜
Therefore, σ H =σ o ε =H o
Linear creep
Strain rates proport€onal to stress
Multi-axial constitutive law gives results for any stress state:
dε˜
H = ε = Bσ ˜
dt H H
where B has units of (Pa.s) .
• Linear creep occurs by diffusion of atoms or molecules
(grain boundary or lattice diffusion in solids)
∴ B is of the form B = Boexp(−Q/RT)
dεxx
Uniaxial tension: =ε xxBσ xx (But, see below)
dt
€
Pure shear:
Mohr’s circle of stress ⇒ σ 1 τ xyσ 2 -τ ;xy =30
Mohr’s circle of strain ⇒ ε = γ /2; ε = -γ /2; ε3= 0
1 xy 2 xy
˜
∴ σ H 3τ xy & ε H γ ˙xy/ 3
∴ γxy = 3Bτ xy
dγ xy τxy
But we know from definition of viscosity: = γ˙xy=
dt η
where η is viscosity (Newtonian fluids) (Pa.s)
dε˜H σ˜H
• Linear creep law can also be expressed as: = εH=
dt 3η
where η =η oxp(+Q/RT)
dεxx σ xx
Uniaxial tension: =ε xx
€ dt 3η
Non-linear creep (power-law creep)
• Strai€ rates are not linearly related to stress
4/iv/14 2 Lecture 29 - Introduction to creep
Multi-axial law gives results for any stress state:
˜ n n
εH= Aσ ˜H = εo( Hσ o)
n and A

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