Introduction to materails modelling Lecture 11 - Viscoelasticity, creep Reijo Kouhia Tampere University, Structural Mechanics November 20, 2019 R. Kouhia (Tampere University, Structural Mechanics) Introduction to materails modelling November 20, 2019 1 / 17
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: σ = η d ε elastic spring σ = Eε, viscous dashpot d t = η ˙ ε Study of flow of matter is called rheology . R. Kouhia (Tampere University, Structural Mechanics) Introduction to materails modelling November 20, 2019 2 / 17
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) Introduction to materails modelling November 20, 2019 3 / 17
Basic Maxwell and Kelvin elements Maxwell: spring and dashpot in series Kelvin: spring and dashpot in parallel R. Kouhia (Tampere University, Structural Mechanics) Introduction to materails modelling November 20, 2019 4 / 17
Behaviour of the Maxwell model in creep and relaxation tests τ = η/E is the relaxation time R. Kouhia (Tampere University, Structural Mechanics) Introduction to materails modelling November 20, 2019 5 / 17
Behaviour of the Maxwell model in a constant strain rate test Stress-strain curve with three strain-rates ˙ ε = σ r /η, 1 . 5 σ r /η and 2 σ r /η . σ r is an arbitrary reference stress and ε r = σ r /E . R. Kouhia (Tampere University, Structural Mechanics) Introduction to materails modelling November 20, 2019 6 / 17
Behaviour of the Kelvin model in creep and relaxation tests τ = η/E is the relaxation time R. Kouhia (Tampere University, Structural Mechanics) Introduction to materails modelling November 20, 2019 7 / 17
Behaviour of the linear viscoelastic standard solid τ = η/ ( E 1 + E 2 ) is the relaxation time R. Kouhia (Tampere University, Structural Mechanics) Introduction to materails modelling November 20, 2019 8 / 17
Generalizations R. Kouhia (Tampere University, Structural Mechanics) Introduction to materails modelling November 20, 2019 9 / 17
Creep Creep of metals under stress means time dependent permanent deformation. Creep is significant at high temperatures when T > 0 . 3 T m , where T m is the melting temperature in absolute scale. virumismurto ε minimivirumisnopeusvaihe ε e t rup t I II III I primary creep, II secondary creep = steady-state creep, III tertiary creep R. Kouhia (Tampere University, Structural Mechanics) Introduction to materails modelling November 20, 2019 10 / 17
Application areas Important in the analysis of engines, power plant boilers & superheaters etc. Figures by Valmet Technologies Oy R. Kouhia (Tampere University, Structural Mechanics) Introduction to materails modelling November 20, 2019 11 / 17
Deformation mechanism maps 10 − 1 Theoretical Strength ˙ ε 4 ε 4 > ˙ ˙ ε 3 > ˙ ε 2 > ˙ ε 1 ˙ Plasticity ε 3 ˙ ε 2 10 − 2 ˙ Yield Strength ε 1 Breakdown (Low-Temperature Creep) Solidus Temperature 10 − 3 Power-Law Creep σ eq / G (High-Temperature Creep) 10 − 4 Elasticity Diffusional Flow 10 − 5 (Grain Boundary) (Lattice) 10 − 6 0 0.2 0.4 0.6 0.8 1.0 T / T m http://engineering.dartmouth.edu/defmech/ R. Kouhia (Tampere University, Structural Mechanics) Introduction to materails modelling November 20, 2019 12 / 17
Constitutive model Decomposition of strain into elastic, creep and thermal parts: ε = ε e + ε c + ε th σ = Eε e = E ( ε − ε c − ε th ) Creep strain rate ε c = f 1 ( T ) f 2 ( σ ) ε c = f 1 ( T ) f 2 ( σ, ε c , D ) ˙ or ˙ Temperature function is of Arrhenius type f 1 ( T ) ∼ exp( − Q/RT ) where Q is the activation energy and R the gas constant. Two common choices for the stress dependency � σ p Norton-Bailey model f 2 ( σ ) ∼ sinh p σ Garofalo model R. Kouhia (Tampere University, Structural Mechanics) Introduction to materails modelling November 20, 2019 13 / 17
Norton-Bailey type creep Creep strain rate in the Norton-Bailey model is � σ � p ε c = 1 ˙ exp( − Q/RT ) t c σ 0 where t c is a time parameter, related to the relaxation time and σ 0 is the drag stress. NB. The exponent p depends on temperature. R. Kouhia (Tampere University, Structural Mechanics) Introduction to materails modelling November 20, 2019 14 / 17
Multiaxial case Creep strain rate tensor in the Norton-Bailey model is � ¯ � p ∂ ¯ ε c = 1 σ σ ˙ exp( − Q/RT ) t c σ 0 ∂ σ where ¯ σ is the “ effective stress ” (scalar). Different versions of ¯ σ σ eff = √ 3 J 2 von Mises stress σ = ¯ ασ eff + (1 − α ) σ 1 convex combination of vM stress and largest principal stress α � σ 1 � + βI 1 + γσ eff isochronous form Hayhurst 1972 In the isochronous case α + β + γ = 1 . R. Kouhia (Tampere University, Structural Mechanics) Introduction to materails modelling November 20, 2019 15 / 17
Primary and tertiary creep Primary creep can be modelled by setting the drag stress dependent on the effective creep strain � � 2 ε c : ˙ ε c ε c ε c eff = ˙ eff d t, ε eff = ˙ 3 ˙ Continuum damage mechanics can be used to model tertiary creep σ = (1 − D ) C e ε e , � 2 r D = 1 exp( − Q d /RT ) � ¯ σ ˙ , t d (1 − D ) k (1 − D ) σ 0 where t d is a time parameter, Q d ”damage activation energy”. R. Kouhia (Tampere University, Structural Mechanics) Introduction to materails modelling November 20, 2019 16 / 17
Some empirical rule of thumb relations Monkman-Grant (1956) relationship ε min ) m t f = C MC ( ˙ Larson-Miller (1952) parameter P : P LM = T ( C + ln( t f )) , where C ≈ 20 and fracture time t f is given in hours. A recommendable form would be � σ � t f � � �� = Q ˜ P LM = T p ln + ln σ 0 t d R R. Kouhia (Tampere University, Structural Mechanics) Introduction to materails modelling November 20, 2019 17 / 17
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