Introduction to materails modelling Lecture 11 - Viscoelasticity, - - PowerPoint PPT Presentation

Introduction to materails modelling

Lecture 11 - Viscoelasticity, creep Reijo Kouhia

Tampere University, Structural Mechanics

November 20, 2019

• R. Kouhia (Tampere University, Structural Mechanics)

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Viscoelasticity

Some materials show pronounced influence of the rate of loading. Metals at elevated temperatures, concrete, plastics. Simple models can be build by using elastic spring and viscous dashpot models: elastic spring σ = Eε, viscous dashpot σ = η dε dt = η ˙ ε Study of flow of matter is called rheology.

• R. Kouhia (Tampere University, Structural Mechanics)

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Creep and relaxation

Creep: increase of strain when the specimen is loaded by a constant stress. Relaxation: decrease of stress when the strain is kept constant.

• R. Kouhia (Tampere University, Structural Mechanics)

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Basic Maxwell and Kelvin elements

Maxwell: spring and dashpot in series Kelvin: spring and dashpot in parallel

• R. Kouhia (Tampere University, Structural Mechanics)

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Behaviour of the Maxwell model in creep and relaxation tests

τ = η/E is the relaxation time

• R. Kouhia (Tampere University, Structural Mechanics)

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Behaviour of the Maxwell model in a constant strain rate test

Stress-strain curve with three strain-rates ˙ ε = σr/η, 1.5σr/η and 2σr/η. σr is an arbitrary reference stress and εr = σr/E.

• R. Kouhia (Tampere University, Structural Mechanics)

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Behaviour of the Kelvin model in creep and relaxation tests

τ = η/E is the relaxation time

• R. Kouhia (Tampere University, Structural Mechanics)

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Behaviour of the linear viscoelastic standard solid

τ = η/(E1 + E2) is the relaxation time

• R. Kouhia (Tampere University, Structural Mechanics)

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Generalizations

• R. Kouhia (Tampere University, Structural Mechanics)

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Creep

Creep of metals under stress means time dependent permanent deformation. Creep is significant at high temperatures when T > 0.3Tm, where Tm is the melting temperature in absolute scale.

t ε εe I II minimivirumisnopeusvaihe III trup virumismurto

I primary creep, II secondary creep = steady-state creep, III tertiary creep

• R. Kouhia (Tampere University, Structural Mechanics)

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Application areas

Important in the analysis of engines, power plant boilers & superheaters etc.

Figures by Valmet Technologies Oy

• R. Kouhia (Tampere University, Structural Mechanics)

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Deformation mechanism maps

0.2 0.4 0.6 0.8 1.0 10−6 10−5 10−4 10−3 10−2 10−1 Plasticity Diffusional Flow (Grain Boundary) (Lattice) Elasticity Power-Law Creep T/Tm σeq/G (Low-Temperature Creep) (High-Temperature Creep) Theoretical Strength Solidus Temperature Yield Strength Breakdown ˙ ε4 ˙ ε3 ˙ ε2 ˙ ε1 ˙ ε4 > ˙ ε3 > ˙ ε2 > ˙ ε1

http://engineering.dartmouth.edu/defmech/

• R. Kouhia (Tampere University, Structural Mechanics)

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Constitutive model

Decomposition of strain into elastic, creep and thermal parts: ε = εe + εc + εth σ = Eεe = E(ε − εc − εth) Creep strain rate ˙ εc = f1(T)f2(σ)

• r

˙ εc = f1(T)f2(σ, εc, D) Temperature function is of Arrhenius type f1(T) ∼ exp(−Q/RT) where Q is the activation energy and R the gas constant. Two common choices for the stress dependency f2(σ) ∼

• σp

Norton-Bailey model sinhp σ Garofalo model

• R. Kouhia (Tampere University, Structural Mechanics)

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Norton-Bailey type creep

Creep strain rate in the Norton-Bailey model is ˙ εc = 1 tc exp(−Q/RT) σ σ0 p where tc is a time parameter, related to the relaxation time and σ0 is the drag stress.

• NB. The exponent p depends on temperature.
• R. Kouhia (Tampere University, Structural Mechanics)

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Multiaxial case

Creep strain rate tensor in the Norton-Bailey model is ˙ εc = 1 tc exp(−Q/RT) ¯ σ σ0 p ∂¯ σ ∂σ where ¯ σ is the “effective stress” (scalar). Different versions of ¯ σ ¯ σ =      σeff = √3J2 von Mises stress ασeff + (1 − α)σ1 convex combination of vM stress and largest principal stress ασ1 + βI1 + γσeff isochronous form Hayhurst 1972 In the isochronous case α + β + γ = 1.

• R. Kouhia (Tampere University, Structural Mechanics)

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Primary and tertiary creep

Primary creep can be modelled by setting the drag stress dependent on the effective creep strain εc

eff =

• ˙

εc

effdt,

˙ εeff =

• 2

3 ˙ εc : ˙ εc Continuum damage mechanics can be used to model tertiary creep σ = (1 − D)C eεe, ˙ D = 1 td exp(−Qd/RT) (1 − D)k

• ¯

σ (1 − D)σ0 2r , where td is a time parameter, Qd ”damage activation energy”.

• R. Kouhia (Tampere University, Structural Mechanics)

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Some empirical rule of thumb relations

Monkman-Grant (1956) relationship ( ˙ εmin)mtf = CMC Larson-Miller (1952) parameter P: PLM = T(C + ln(tf)), where C ≈ 20 and fracture time tf is given in hours. A recommendable form would be ˜ PLM = T

• p ln

σ σ0

• + ln

tf td

• = Q

R

• R. Kouhia (Tampere University, Structural Mechanics)

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