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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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