Rheology. October 2013

Transcription

1 Rheology Georges Cailletaud Centre des Matériaux MINES ParisTech/CNRS October 2013 Georges Cailletaud Rheology 1/44

2 Contents 1 Mechanical tests Structures Representative material elements 2 Rheological models Basic building bricks Plasticity Viscoelasticity Elastoviscoplasticity Georges Cailletaud Rheology 2/44

3 Tests on a civil plane Georges Cailletaud Rheology 3/44

4 Vibration of a wing Georges Cailletaud Rheology 4/44

5 Biological structures (1/2) Georges Cailletaud Rheology 5/44

6 Biological structures (2/2) Georges Cailletaud Rheology 6/44

7 Food industry Georges Cailletaud Rheology 7/44

8 Testing machines Georges Cailletaud Rheology 8/44

9 Tension test on a metallic specimen Georges Cailletaud Rheology 9/44

10 Mechanical tests Basic tests Time independent plasticity Tension test, or hardening test Cyclic load, or fatigue test Time dependent plasticity Other tests Test at constant stress, or creep test Test at constant strain, or relaxation test Multiaxial load Tension torsion Internal pressure Bending tests Crack propagation tests Georges Cailletaud Rheology 10/44

11 Typical result on an aluminum alloy For a stress σ 0.2, it remains 0.2% residual strain after unloading Stress to failure, σ u σ (MPa) Tension curve Elastic slope 0.2% residual strain 0.02 ε(mm/mm) E=78000 MPa, σ 0.2 =430 MPa, σ u =520 MPa Doc. Mines Paris-CDM, Evry Georges Cailletaud Rheology 11/44

12 Typical result on an austenitic steel Material exhibiting an important hardening : the yield stress increases during plastic flow σ (MPa) Tension curve Elastic slope 0.2% residual strain ε(mm/mm) E= MPa, σ 0.2 =180 MPa, σ u =660 MPa Doc. ONERA-DMSE, Châtillon Georges Cailletaud Rheology 12/44

13 Push pull test on an aluminum alloy Test under strain control ± 0.3% Positive residual strain at zero stress Negative stress at zero strain σ (MPa) ε(mm/mm) Doc. Mines Paris-CDM, Evry Georges Cailletaud Rheology 13/44

14 Schematic models for the preceding results σ σ E T σ y σ y E E 0 ε a. Elastic perfectly plastic 0 b. Elastic plastic (linear) ε Elastoplastic modulus, E T = dσ/dε. E T = 0 : elastic-perfectly plastic material E T constant : linear plastic hardening E t strain dependent in the general case Georges Cailletaud Rheology 14/44

15 How does a plasticity model work? σ B A Elastic regime OA, O B Plastic flow AB Residual strain OO 0 0 ε Strain decomposition, ε = ε e + ε p ; Yield domain, defined by a load function f Hardening, defined by means of hardening variables, A I. Georges Cailletaud Rheology 15/44

16 Result of a tension on a steel at high temperature Viscosity effect : Strain rate dependent behaviour 725 C 80 σ(mpa) ε = s 1 ε = s 1 ε = s 1 ε Doc. Ecole des Mines, Nancy Georges Cailletaud Rheology 16/44

17 Creep test on a tin lead wire Mines Paris-CDM, Evry Georges Cailletaud Rheology 17/44

18 Creep on a cast iron ε p σ=25mpa σ=20mpa σ=16mpa σ=12mpa t (s) Doc. Mines Paris-CDM, Evry Georges Cailletaud Rheology 18/44

19 Schematic representation of a creep curve Primary creep, with hardening in the material Secondary creep, steady state creep : ε p is a power function of the applied stress Tertiary creep, when damage mechanisms start p ε III I II t Georges Cailletaud Rheology 19/44

20 Creep on a cast iron (2) T=500 C T=600 C T=700 C T=800 C ε p (s 1 ) 1e-05 1e-06 1e-07 1e σ (MPa) Doc. Mines Paris-CDM, Evry Georges Cailletaud Rheology 20/44

21 Relaxation test Constant strain during the test During the test : ε = 0 = ε p + σ/e dε p = dσ/e The viscoplastic strain increases meanwhile stress decreases The asymptotic stress may be zero (total relaxation) or not (partial relaxation) Partial relaxation if there is an internal stress or a threshold in the material Georges Cailletaud Rheology 21/44

22 Schematic representation of a relaxation curve The current point in stress space is obtained as the sum of a threshold stress σ s and of a viscous stress σ v The threshold stress represents the plastic behaviour that is reached for zero strain rate σ σ σ v E σ s p ε t Georges Cailletaud Rheology 22/44

23 Contents 1 Mechanical tests Structures Representative material elements 2 Rheological models Basic building bricks Plasticity Viscoelasticity Elastoviscoplasticity Georges Cailletaud Rheology 23/44

24 Building bricks for the material models Georges Cailletaud Rheology 24/44

25 Various types of rheologies Time independent plasticity ε = ε e + ε p dε p = f(...)dσ Elasto-viscoplasticity ε = ε e + ε p dε p = f(...)dt Viscoelasticity F(σ, σ,ε, ε) = 0 Georges Cailletaud Rheology 25/44

26 Time independent plasticity Georges Cailletaud Rheology 26/44

27 Elastic perfectly plastic model The elastic/plastic regime is defined by means of a load function f (from stress space into R) f(σ) = σ σ y Elasticity domain Elastic unloading if f < 0 ε = ε e = σ/e Plastic flow if f = 0 and ḟ < 0 if f = 0 and ḟ = 0 ε = ε e = σ/e ε = ε p The condition ḟ = 0 is the consistency condition Georges Cailletaud Rheology 27/44

28 Prager model Loading function with two variables, σ and X f(σ,x) = σ X σ y with X = Hε p Plastic flow if both conditions are verified f = 0 and ḟ = 0. f f σ + σ X Ẋ = 0 sign(σ X) σ sign(σ X)Ẋ = 0 thus : σ = Ẋ Plastic strain rate as a function of the stress rate ε p = σ/h Plastic strain rate as a function of the total strain rate (once an elastic strain is added) ε p = E E + H ε Georges Cailletaud Rheology 28/44

29 Equation of onedimensional elastoplasticity Elasticity domain if f(σ,a i ) < 0 ε = σ/e Elastic unloading if f(σ,a i ) = 0 and ḟ(σ,a i ) < 0 ε = σ/e Plastic flow if f(σ,a i ) = 0 and ḟ(σ,a i ) = 0 ε = σ/e + ε p The consistency condition writes : ḟ(σ,a i ) = 0 Georges Cailletaud Rheology 29/44

30 Illustration of the two hardening types Georges Cailletaud Rheology 30/44

31 Isotropic hardening model Loading function with two variables, σ and R f(σ,r) = σ R σ y R depends on p, accumulated plastic strain : ṗ = ε p dr/dp = H thus Ṙ = Hṗ Plastic flow iff f = 0 and ḟ = 0 sign(σ) σ Ṙ = 0 f f σ + σ R Ṙ = 0 thus sign(σ) σ Hṗ Plastic strain rate as a function of the stress rate ṗ = sign(σ) σ/h thus ε p = σ/h Classical models Ramberg-Osgood : σ = σ y + Kp m Exponential rule : σ = σ u + (σ y σ u )exp( bp) Georges Cailletaud Rheology 31/44

32 Viscoelasticity Georges Cailletaud Rheology 32/44

33 Elementary responses in viscoelasticity Serie, Maxwell model : ε = σ/e 0 + σ/η Creep under a stress σ 0 : ε = σ 0 /E 0 + σ 0 t /η Relaxation for a strain ε 0 : σ = E 0 ε 0 exp[ t/τ] Parallel, Voigt model : σ = Hε + η ε or ε = (σ H ε)/η Creep under a stress σ 0 : ε = (σ 0 /H)(1 exp[ t/τ ]) The constants τ = η/e 0 and τ = η/h are in seconds, τ denoting the so called le relaxation time of the Maxwell model Georges Cailletaud Rheology 33/44

34 More complex models (H) (E 1 ) (E 0 ) (η) (E 2 ) (η) a. Kelvin Voigt b. Zener Creep and relaxation responses ( 1 ε(t) = C(t)σ 0 = + 1 ) E 0 H (1 exp[ t/τ f ]) σ 0 ( H σ(t) = E(t)ε 0 = + E ) 0 exp[ t/τ r ] E 0 ε 0 H + E 0 H + E 0 Georges Cailletaud Rheology 34/44

35 Elasto-viscoplasticity Scheme of the model Tensile response X = Hε vp σ v = η ε vp σ p σ y σ = X + σ v + σ p Elasticity domain, whose boundary is σ p = σ y Georges Cailletaud Rheology 35/44

36 Model equations Three regimes (a) ε vp =0 σ p = σ Hε vp σ y (b) ε vp >0 σ p =σ Hε vp η ε vp =σ y (c) ε vp <0 σ p =σ Hε vp η ε vp = σ y (a) interior or boundary of the elasticity domain ( σ p < σ y ) (b),(c) flow ( σ p = σ y and σ p = 0 ) One can summarize the three equations (with < x >= max(x,0)) by or : η ε vp = σ X σ y sign(σ X) ε vp = < f > η sign(σ X), with f(σ,x) = σ X σ y Georges Cailletaud Rheology 36/44

37 Creep with a Bingham model σ - o H ε vp σ y σ σ o X σ y Viscoplastic strain versus time ε vp = σ o σ y H ( t 1 exp Evolution in the plane stress vsicoplastic strain )) ( tτf with : τ f = η/h vp ε Georges Cailletaud Rheology 37/44

38 Relaxation with a Bingham model σ σy -E H vp ε A O D B C H Transitoire : OA = BC Relaxation : AB Effacement vp incomplet : CD ε Fading memory Relaxation ( ( E σ = σ y 1 exp t )) + Eε ( ( o H + E exp t )) E + H τ r E + H τ r with : τ r = η E + H Georges Cailletaud Rheology 38/44

39 Ingredients for classical viscoplastic models Bingham model More generally ε vp = < f > sign(σ X) η ε vp = φ(f) φ(0) = 0 and φ monotonically increasing ε vp is zero if the current point is in the elasticity domain or on the boundary ε vp is non zero if the current point is outside from the elasticity domain There are models with/without threshold, with/without hardening Georges Cailletaud Rheology 39/44

40 Viscoplastic models without hardening Models without threeshold : the elastic domain is reduced to the origin (σ = 0) Norton model Sellars Tegart model Models with a threshold Perzyna model σ ε vp σy = K ( ) σ n ε vp = sign(σ) K ( ) σ ε vp = Ash sign(σ) K n σ n sign(σ), ε vp = ε 0 1 sign(σ) σ y Georges Cailletaud Rheology 40/44

41 Viscoplastic models with hardening The concept of additive hardening : The hardening comes from the variables that represent the threshold (X and R) σ X R n ε vp σy = sign(σ X) K X stands for the internal stress (kinematical hardening) R + σ y stands for the friction stress (isotropic hardening) σ v is the viscous stress or drag stress The concept of multiplicative hardening : one plays on viscous stress, for instance : ( ) σ n ( ) σ n ε vp = sign(σ) = sign(σ) K (ε p ) K 0 ε p m strain hardening Georges Cailletaud Rheology 41/44

42 For plasticity and viscoplasticity... Elasticity defined by a loading function f < 0 Isotropic and kinematic variables For plasticity : Plastic flow defined by the consistency condition f = 0,ḟ = 0 Plastic flow : dε p = g(σ,...)dσ For viscoplasticity : Flow defined by the viscosity function if f > 0 Possible hardening on the viscous stress Delayed viscoplastic flow dε vp = g(σ,...)dt Georges Cailletaud Rheology 42/44

43 Identification of the material parameters Norton model on tin lead wires exp sim creep strain stress (MPa) g g 1150 g 997 g 720 g time (s) Creep test time (s) Relaxation ε=20% Curves obtained with a Norton model ( σ ) 2.3 ε p = 800 I try by myself on the site mms2.ensmp.fr O Georges Cailletaud Rheology 43/44

44 Identification of the creep on salt strain time (Ms) exp sim Specimen Three level test (3, 6, 9 MPa) Curves obtained with a Lemaitre model (strain hardening) ( σ ) n ε p = (ε p + v 0 ) m K I try by myself on the site mms2.ensmp.fr O Georges Cailletaud Rheology 44/44