Georges Cailletaud Centre des Matériaux MINES...
Transcript of Georges Cailletaud Centre des Matériaux MINES...
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Rheology and multiaxial criteria
Georges Cailletaud
Centre des MatériauxMINES ParisTech/CNRS
Non Linear Computational MechanicsAthens MP06
Georges Cailletaud | Plasticity 1/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Contents
1 Mechanical testsStructuresRepresentative material elements
2 Rheological modelsBasic building bricksPlasticityViscoelasticityElastoviscoplasticity
3 Multiaxial plasticity criterionMechanical testsMechanisms
4 Pressure dependent modelsIsotropic : Tresca, von MisesAn anisotropic criterion : Hill
5 Pressure independent models
6 Synthesis on criteria
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Tests on a civil plane
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Vibration of a wing
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Biological structures (1/2)
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Biological structures (2/2)
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Food industry
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Testing machines
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Tension test on a metallic specimen
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Mechanical tests
Basic tests
Time independent plasticityTension test, or hardening testCyclic load, or fatigue test
Time dependent plasticityTest at constant stress, or creep testTest at constant strain, or relaxation test
Other tests
Multiaxial loadTension–torsionInternal pressure
Bending tests
Crack propagation tests
Georges Cailletaud | Plasticity 10/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Typical result on an aluminum alloy
For a stress σ0.2, it remains 0.2% residual strain after unloading
Stress to failure, σu
0.2% residual strainElastic slope
Tension curve
ε(mm/mm)
σ(M
Pa)
0.040.030.020.010
600
500
400
300
200
100
0
E=78000 MPa, σ0.2=430 MPa, σu=520 MPa Doc. Mines Paris-CDM, Evry
Georges Cailletaud | Plasticity 11/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Typical result on an austenitic steel
Material exhibiting an important hardening : the yield stress increasesduring plastic flow
0.2% residual strainElastic slope
Tension curve
ε(mm/mm)
σ(M
Pa)
0.080.070.060.050.040.030.020.010
600
500
400
300
200
100
0
E=210000 MPa, σ0.2=180 MPa, σu=660 MPa Doc. ONERA-DMSE, Châtillon
Georges Cailletaud | Plasticity 12/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Push–pull test on an aluminum alloy
Test under strain control ± 0.3%
Positive residual strain at zero stress
Negative stress at zero strain
ε(mm/mm)
σ(M
Pa)
0.0050.0030.001-0.001-0.003-0.005
300
200
100
0
-100
-200
-300
Doc. Mines Paris-CDM, Evry
Georges Cailletaud | Plasticity 13/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Schematic models for the preceding results
σ
σ y
E
0 εa. Elastic–perfectly plastic
ε0
E
TE
σ
σy
b. Elastic–plastic (linear)
Elastoplastic modulus, ET = dσ/dε.
ET = 0 : elastic-perfectly plastic material
ET constant : linear plastic hardening
Et strain dependent in the general case
Georges Cailletaud | Plasticity 14/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
How does a plasticity model work ?
0 0’
A
B
ε
σ
Elastic regimeOA, O’B
Plastic flowAB
Residual strainOO’
Strain decomposition, ε = εe + εp ;
Yield domain, defined by a load function f
Hardening, defined by means of hardening variables,AI .
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Result of a tension on a steel at hightemperature
Viscosity effect : Strain rate dependent behaviour
ε = 1.6 10−5s−1ε = 8.0 10−5s−1ε = 2.4 10−4s−1
725◦C
ε
σ(M
Pa)
0.10.080.060.040.020
80
60
40
20
0
Doc. Ecole des Mines, Nancy
Georges Cailletaud | Plasticity 16/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Creep test on a tin–lead wire
Mines Paris-CDM, Evry
Georges Cailletaud | Plasticity 17/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Creep on a cast iron
σ=25MPaσ=20MPaσ=16MPaσ=12MPa
t (s)
εp
10008006004002000
0.03
0.025
0.02
0.015
0.01
0.005
0
Doc. Mines Paris-CDM, Evry
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Schematic representation of a creep curve
Primary creep , with hardening in the material
Secondary creep , steady state creep : εp is a power function ofthe applied stress
Tertiary creep , when damage mechanisms start
pε
III
III
t
Georges Cailletaud | Plasticity 19/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Creep on a cast iron (2)
T=800◦CT=700◦CT=600◦CT=500◦C
σ (MPa)
εp(s−1
)
100101
0.001
0.0001
1e-05
1e-06
1e-07
1e-08
Doc. Mines Paris-CDM, Evry
Georges Cailletaud | Plasticity 20/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Relaxation test
Constant strain during the test
During the test :ε = 0 = ε
p + σ/E
dεp =−dσ/E
The viscoplastic strain increases meanwhile stress decreases
The asymptotic stress may be zero (total relaxation) or not (partialrelaxation)
Partial relaxation if there is an internal stress or a threshold in thematerial
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Schematic representation of a relaxation curve
The current point in stress space is obtained as the sum of a thresholdstress σs and of a viscous stress σv
The threshold stress represents the plastic behaviour that is reached forzero strain rate
σv
pε
σs
t
σ σ
E
Georges Cailletaud | Plasticity 22/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Contents
1 Mechanical testsStructuresRepresentative material elements
2 Rheological modelsBasic building bricksPlasticityViscoelasticityElastoviscoplasticity
3 Multiaxial plasticity criterionMechanical testsMechanisms
4 Pressure dependent modelsIsotropic : Tresca, von MisesAn anisotropic criterion : Hill
5 Pressure independent models
6 Synthesis on criteria
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Building bricks for the material models
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Various types of rheologies
Time independent plasticity
ε = εe + ε
p dεp = f (...)dσ
Elasto-viscoplasticity
ε = εe + ε
p dεp = f (...)dt
ViscoelasticityF(σ, σ,ε, ε) = 0
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Time independent plasticity
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Elastic–perfectly plastic model
The elastic/plastic regime is defined by means ofa load function f (from stress space into R)
f (σ) = |σ|−σy
Elasticity domainif f < 0 ε = ε
e = σ/E
Elastic unloading
if f = 0 and f < 0 ε = εe = σ/E
Plastic flowif f = 0 and f = 0 ε = ε
p
The condition f = 0 is the consistency condition
Georges Cailletaud | Plasticity 27/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Prager model
Loading function with two variables, σ and X
f (σ,X) = |σ−X |−σy with X = Hεp
Plastic flow if both conditions are verified f = 0 and f = 0.
∂f∂σ
σ+∂f∂X
X = 0
sign(σ−X) σ− sign(σ−X) X = 0 thus : σ = X
Plastic strain rate as a function of the stress rate
εp = σ/H
Plastic strain rate as a function of the total strain rate (once an elasticstrain is added)
εp =
EE +H
ε
Georges Cailletaud | Plasticity 28/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Equation of onedimensional elastoplasticity
Elasticity domainif f (σ,Ai) < 0 ε = σ/E
Elastic unloading
if f (σ,Ai) = 0 and f (σ,Ai) < 0 ε = σ/E
Plastic flow
if f (σ,Ai) = 0 and f (σ,Ai) = 0 ε = σ/E + εp
The consistency condition writes :
f (σ,Ai) = 0
Georges Cailletaud | Plasticity 29/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Illustration of the two hardening types
Georges Cailletaud | Plasticity 30/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Isotropic hardening model
Loading function with two variables, σ and R
f (σ,R) = |σ|−R−σy
R depends on p, accumulated plastic strain : p = |εp|dR/dp = H thus R = Hp
Plastic flow iff f = 0 and f = 0
∂f∂σ
σ+∂f∂R
R = 0
sign(σ) σ− R = 0 thus sign(σ) σ−Hp
Plastic strain rate as a function of the stress rate
p = sign(σ) σ/H thus εp = σ/H
Classical modelsRamberg-Osgood : σ = σy +Kpm
Exponential rule : σ = σu +(σy −σu)exp(−bp)
Georges Cailletaud | Plasticity 31/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Viscoelasticity
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Elementary responses in viscoelasticity
Serie, Maxwell model : ε = σ/E0 +σ/η
Creep under a stress σ0 : ε = σ0/E0 +σ0 t /η
Relaxation for a strain ε0 : σ = E0ε0 exp[−t/τ]
Parallel, Voigt model : σ = Hε+ηε or ε = (σ−H ε)/η
Creep under a stress σ0 : ε = (σ0 /H)(1−exp[−t/τ′])
The constants τ = η/E0 and τ′ = η/H are in seconds, τ denoting the socalled le relaxation time of the Maxwell model
Georges Cailletaud | Plasticity 33/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
More complex models
a. Kelvin–Voigt
(E0)
(H)
(η)
b. Zener
(η)(E2)
(E1)
Creep and relaxation responses
ε(t) = C(t)σ0 =
(1
E0+
1H
(1−exp[−t/τf ])
)σ0
σ(t) = E(t)ε0 =
(H
H +E0+
E0
H +E0exp[−t/τr ]
)E0ε0
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Elasto-viscoplasticity
Scheme of the model Tensile response
X = Hεvp
σv = ηεvp |σp|6 σy
σ = X +σv +σp
Elasticity domain, whose boundary is |σp|= σy
Georges Cailletaud | Plasticity 35/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Model equations
Three regimes
(a) εvp =0 |σp|= |σ−Hε
vp| 6σy
(b) εvp >0 σp =σ−Hε
vp−ηεvp =σy
(c) εvp <0 σp =σ−Hε
vp−ηεvp = −σy
(a) interior or boundary of the elasticity domain (|σp| < σy )(b),(c) flow (|σp|= σy and |σp| = 0 )
One can summarize the three equations (with < x >= max(x ,0)) by
ηεvp = 〈|σ−X |−σy 〉 sign(σ−X)
or :
εvp =
< f >
ηsign(σ−X) , with f (σ,X) = |σ−X |−σy
Georges Cailletaud | Plasticity 36/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Creep with a Bingham model
t
σ σo y-
H
εvp
Viscoplastic strainversus time
σ
σ
Xo
y
σ
vpεEvolution in the planestress– vsicoplastic
strain
εvp =
σo−σy
H
(1−exp
(− t
τf
))with : τf = η/H
Georges Cailletaud | Plasticity 37/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Relaxation with a Bingham model
σ
H-E
vpε
σ
y
Relaxation
H
ε
Transitoire : OA = BC
Relaxation : AB
Effacementincomplet : CDO
A
B
DC
vp
Fading memory
σ = σyE
E +H
(1−exp
(− t
τr
))+
Eεo
E +H
(H +E exp
(− t
τr
))with : τr =
η
E +H
Georges Cailletaud | Plasticity 38/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Ingredients for classical viscoplastic models
Bingham model
εvp =
< f >
ηsign(σ−X)
More generallyε
vp = φ(f )
φ(0) = 0 and φ monotonically increasing
εvp is zero if the current point is in the elasticity domain or on theboundary
εvp is non zero if the current point is outside from the elasticity domain
There are models with/without threshold, with/without hardening
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Viscoplastic models without hardening
Models without threeshold : the elastic domain is reduced to the origin(σ = 0)
Norton model
εvp =
( |σ|K
)n
sign(σ)
Sellars–Tegart model
εvp = Ash
( |σ|K
)sign(σ)
Models with a thresholdPerzyna model
εvp =
⟨ |σ|−σy
K
⟩n
sign(σ) , εvp = ε0
⟨ |σ|σy−1
⟩n
sign(σ)
Georges Cailletaud | Plasticity 40/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Viscoplastic models with hardening
The concept of additive hardening : The hardening comes from thevariables that represent the threshold (X and R)
εvp =
⟨ |σ−X |−R−σy
K
⟩n
sign(σ−X)
X stands for the internal stress (kinematical hardening)R +σy stands for the friction stress (isotropic hardening)σv is the viscous stress or drag stress
The concept of multiplicative hardening : one plays on viscous stress, forinstance :
εvp =
( |σ|K (εp)
)n
sign(σ) =
( |σ|K0|εp|m
)n
sign(σ)
strain hardening
Georges Cailletaud | Plasticity 41/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
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, f = 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
dεvp = g(σ, . . .)dt
Georges Cailletaud | Plasticity 42/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Identification of the material parametersNorton model on tin–lead wires
0
0.02
0.04
0.06
0.08
0.1
0 1000 2000 3000 4000 5000
cree
p st
rain
time (s)
1534 g1320 g1150 g997 g720 g
0
2
4
6
8
10
12
14
0 5000 10000 15000 20000 25000
stre
ss (
MP
a)
time (s)
expsim
Creep test Relaxation ε=20%
Curves obtained with a Norton model
εp =
(σ
800
)2.3
I try by myself on the site mms2.ensmp.fr O
Georges Cailletaud | Plasticity 43/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Identification of the creep on salt
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0 0.5 1 1.5 2 2.5 3 3.5 4
stra
in
time (Ms)
expsim
Specimen Three level test (3, 6, 9 MPa)
Curves obtained with a Lemaitre model (strain hardening)
εp =
(σ
K
)n(εp + v0)
m
I try by myself on the site mms2.ensmp.fr O
Georges Cailletaud | Plasticity 44/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Contents
1 Mechanical testsStructuresRepresentative material elements
2 Rheological modelsBasic building bricksPlasticityViscoelasticityElastoviscoplasticity
3 Multiaxial plasticity criterionMechanical testsMechanisms
4 Pressure dependent modelsIsotropic : Tresca, von MisesAn anisotropic criterion : Hill
5 Pressure independent models
6 Synthesis on criteria
Georges Cailletaud | Plasticity 45/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Biaxial loading path
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Shear
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
From the lab to real world (1)
0.2% residual strainElastic slope
Tension curve
ε(mm/mm)
σ(M
Pa)
0.080.070.060.050.040.030.020.010
600
500
400
300
200
100
0
Georges Cailletaud | Plasticity 48/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
From the lab to real world (1)
0.2% residual strainElastic slope
Tension curve
connueCourbe de traction
ε(mm/mm)
σ(M
Pa)
0.080.070.060.050.040.030.020.010
600
500
400
300
200
100
0
In most of the cases, the material is characterizedby a simple tension curve
Georges Cailletaud | Plasticity 48/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
From the lab to real world (2)
0.2% residual strainElastic slope
Tension curve
connueCourbe de traction
ε(mm/mm)
σ(M
Pa)
0.080.070.060.050.040.030.020.010
600
500
400
300
200
100
0
212 53 10753 88 32
107 32 316
How can wetranspose ?
312312332211
Chargement reel complexe
t (s)
σ(M
Pa
)
6050403020100
400
350
300
250
200
150
100
50
0
Georges Cailletaud | Plasticity 49/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
From the lab to real world (2)
0.2% residual strainElastic slope
Tension curve
connueCourbe de traction
ε(mm/mm)
σ(M
Pa)
0.080.070.060.050.040.030.020.010
600
500
400
300
200
100
0
212 53 10753 88 32
107 32 316
How can wetranspose ?
312312332211
Chargement reel complexe
t (s)
σ(M
Pa
)
6050403020100
400
350
300
250
200
150
100
50
0
Georges Cailletaud | Plasticity 49/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
From the lab to real world (2)
0.2% residual strainElastic slope
Tension curve
connueCourbe de traction
ε(mm/mm)
σ(M
Pa)
0.080.070.060.050.040.030.020.010
600
500
400
300
200
100
0
212 53 10753 88 32
107 32 316
How can wetranspose ?
312312332211
Chargement reel complexe
t (s)
σ(M
Pa
)
6050403020100
400
350
300
250
200
150
100
50
0
Georges Cailletaud | Plasticity 49/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
How can one characterize multiaxial behaviour ?
Multiaxial mechanical tests
Research on the physical deformationmechanisms
Georges Cailletaud | Plasticity 50/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Tension–torsion tests on tubes
Tension–torsion specimen
For a tube of length L, diameter2R and width e :
Strain measured by a gauge,or use of the relation betweenthe angle (β) and the strain(γ) :
β = γLR
Relation between themoment (M) and the shear(τ) :
M = 2πeR2τ0 0 0
0 0 σθz
0 σθz σzz
(rθz)
Georges Cailletaud | Plasticity 51/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Biaxial testsBiaxial test on a cruciform specimen vinylester–glas fiber
More on the websiteSciences de l’Ingénieur, ENS Cachan
Georges Cailletaud | Plasticity 52/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Shear test
Double Arcanshear specimen
(rubber)
Doc. Centre des Matériaux, MINES ParisTech
Georges Cailletaud | Plasticity 53/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Shear setup
Georges Cailletaud | Plasticity 54/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Search of the yield surface in tension–shear
PhD Rousset, ENS Cachan
Georges Cailletaud | Plasticity 55/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Initial surf. and after the first compression
PhD Rousset, ENS Cachan
Georges Cailletaud | Plasticity 56/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Initial surface and square-shape loading
PhD Rousset, ENS Cachan
Georges Cailletaud | Plasticity 57/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Shear on basalt
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Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Summary of the experimental observations
Crystalline material, where deformation comes from shear (alloys, rocks)
Crystal network No volume change
Powders, geomaterials, damaged materials
Georges Cailletaud | Plasticity 59/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Summary of the experimental observations
Crystalline material, where deformation comes from shear (alloys, rocks)
Crystal network No volume change
Powders, geomaterials, damaged materials
Georges Cailletaud | Plasticity 59/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Summary of the experimental observations
Crystalline material, where deformation comes from shear (alloys, rocks)
Crystal network No volume change
Powders, geomaterials, damaged materials
Georges Cailletaud | Plasticity 59/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Summary of the experimental observations
Critical variable ?
Crystalline material, where deformation comes from shear (alloys, rocks)
Crystal network No volume change
Powders, geomaterials, damaged materials
Georges Cailletaud | Plasticity 59/82
Plan Mechanical tests Rheological models Summary on rheology Multiaxial plasticity criterion Pressure dependent models Pressure independent models Synthesis on criteria
Summary of the experimental observations
Critical variable ?
Crystalline material, where deformation comes from shear (alloys, rocks)
Crystal network No volume change
ShearDeviator
Powders, geomaterials, damaged materials
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Summary of the experimental observations
Critical variable ?
Crystalline material, where deformation comes from shear (alloys, rocks)
Crystal network No volume change
ShearDeviator
Powders, geomaterials, damaged materials
Deviator+ spherical
part
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Slip systems in a monocrystal
PhD F. Hanriot (ENSMP-CDM, Evry)
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Slip systems in a polycrystal
Clavel (ECP, Châtenay)
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Rupture under dynamic loading
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Schmid law
The deformation comes from slip on systems s defined by a plane ofnormal ns, and a shear direction ls, iif the resolved shear stress, τs
reaches a critical value τc
Projection of the stress vector on the slip direction. For a sungle crystalsubmitted to σ∼
τs = (σ∼ .n
s).ls
There is as many criteria linear in stress as the number of slip systems
f (σ∼) = τs− τc
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Yield surfaces for polycrystals (Uniformelasticity)
Disrectionally Polycrystalsolidifiedmaterial
Compute yield surfaces
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Loading surfaces in tension–shear
001
σ11
σ 12
2001000-100-200
200
100
0
-100
-200
One cubic grain oriented along (001) axes
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Loading surfaces in tension–shear
234001
σ11
σ 12
2001000-100-200
200
100
0
-100
-200
One grain oriented along (234)
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Loading surfaces in tension–shear
2g234001
σ11
σ 12
2001000-100-200
200
100
0
-100
-200
One grain (001) and one grain (234)
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Loading surfaces in tension–shear
10g2g
234001
σ11
σ 12
2001000-100-200
200
100
0
-100
-200
Ten randomly oriented grains
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Loading surfaces in tension–shear
100g10g2g
234001
σ11
σ 12
2001000-100-200
200
100
0
-100
-200
Hundred randomly oriented grains
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Loading surfaces in tension–shear
Tresca100g
10g2g
234001
σ11
σ 12
2001000-100-200
200
100
0
-100
-200
σ211 +4σ2
12 = σ2y , Tresca criterion
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Contents
1 Mechanical testsStructuresRepresentative material elements
2 Rheological modelsBasic building bricksPlasticityViscoelasticityElastoviscoplasticity
3 Multiaxial plasticity criterionMechanical testsMechanisms
4 Pressure dependent modelsIsotropic : Tresca, von MisesAn anisotropic criterion : Hill
5 Pressure independent models
6 Synthesis on criteria
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Characterization of the maximum shear
Stres tensor in the eigendirections :=
σ1 0 00 σ2 00 0 σ3
Stress vector for a normal n in the plan (x1–x2) (with θ = angle(x1,n) :
Tn = σ1 cos2θ+σ2 sin2
θ =σ1 +σ2
2+
σ1−σ2
2cos2θ
|Tt |=(T 2−T 2
n
)1/2=|σ1−σ2|
2sin2θ
Mohr circles : (Tn− σ1 +σ2
2
)2
+T 2t =
(σ1−σ2
2
)2
Max shear
|T maxt |= |σ1−σ2|
2
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Tresca criterion
σ1σ2σ3Tn
Tt
Tmax
The maximum shear remains smaller than a critical value
Maxi,j |σi −σj |−σy = 0
σy is the elastic limit in tension
→WIKI
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Representation of an isotropic material
- Invariants of the stress tensor :
I1 = trace(σ∼) =σii
I2 =(1/2) trace(σ∼)2 =(1/2)σijσji
I3 =(1/3) trace(σ∼)3 =(1/3)σijσjk σki
- Invariants of the deviator (s∼ = σ∼− (I1/3) I∼) :
J1 = trace(s∼) =0
J2 =(1/2) trace(s∼)2 =(1/2)sijsji
J3 =(1/3) trace(s∼)3 =(1/3)sijsjk ski
- One notes :
J = ((3/2)sijsji)0,5 =
((1/2)
((σ1−σ2)
2 +(σ2−σ3)2 +(σ3−σ1)
2))0,5= |σ|
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Physical meaning of J
Sphere in the space of the deviatoric stresses
Octahedral shear stress :one a facet of normal (1,1,1), the stres vector has the followingcomponents : normal stress σoct and tangential stress τoct :
σoct = (1/3) I1 ; τoct = (√
2/3)J
The elastic distorsional energy (associated to the deviatoric part of σ∼and ε∼).
Wed =12
s∼ : e∼ =16µ
J2
Von Mises criterionf (σ∼) = J−σy
Note : formulated by Maxwell in 1865, and Huber in 1904 (→WIKI)
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Von Mises contour in the deviatoric plane
CS
CI
TS
σ1
CS
TSσ2
CS
TS
σ3
TS stands for the points that are equiva-lent to simple tension, CS those that areequivalent to simple compression (forinstance a biaxial load, since a stressstate like σ1 = σ2 = σ is equivalent toσ3 =−σ), CI corresponds to shear
f (σ∼) = J−σy
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Criteria without hydrostatic pressure
Von Mises criterion
f (σ∼) = J−σy
Tresca criterion
f (σ∼) = Maxi,j |σi −σj |−σy
Use of the second and third invariant
f (σ∼) = fct(J2,J3)
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Comparaison Tresca–von Mises
In the tension–shear plane
− von Mises : f (σ,τ) =(σ
2 +3τ2)0,5−σy
−Tresca : f (σ,τ) =(σ
2 +4τ2)0,5−σy
In the plane of eigenstresses (σ1,σ2)
− von Mises : f (σ1,σ2) =(σ
21 +σ
22−σ1σ2
)0,5− σy
− Tresca : f (σ1,σ2) = σ2−σy if 0 6 σ1 6 σ2
f (σ1,σ2) = σ1−σy if 0 6 σ2 6 σ1
f (σ1,σ2) = σ1−σ2−σy if σ2 6 0 6 σ1
(symmetry with respect to axis σ1 = σ2)
In the deviatoric plane, von Mises = circle, Tresca = hexagon
Int the eigenstress space, cylindres of axis (1,1,1)
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Comparisons Tresca–von Mises
σ12
σ11
τt
τm
σyσy
τm
τt-
-
-
a. In tension–shear (von Mises :τm = σy/
√3, Tresca : τt = σy/2)
σ1
σ2
σy
σy
σy
σy-
-
b. In biaxial tension
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Anisotropic criteria
f (σ∼) = ((3/2)Hijkl sij skl)0,5−σy (or Hijkl σij σkl)
Hill’s criterion
In the orthotropy axes :
f (σ∼) =(F(σ11−σ22)2 +G(σ22−σ33)
2 +H(σ33−σ11)2
+2Lσ212 +2Mσ
223 +2Nσ
213)
0,5−σy
Transverse, 3 independent coefficients
Cubic symmetry cubique, one coefficient only
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Contents
1 Mechanical testsStructuresRepresentative material elements
2 Rheological modelsBasic building bricksPlasticityViscoelasticityElastoviscoplasticity
3 Multiaxial plasticity criterionMechanical testsMechanisms
4 Pressure dependent modelsIsotropic : Tresca, von MisesAn anisotropic criterion : Hill
5 Pressure independent models
6 Synthesis on criteria
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Drucker–Prager criterion
Linear combination of the first and second invariant (with 0 < α < 0.5)
f (σ∼) = (1−α)J +α I1−σy
Elastic yield in tension (σt ) and in compression (σc)
σt = σy σc =−σy/(1−2α)
σ1
2
3
σ
σ
In the eigenstress space
I1
J
σy
1−α
σy/α
In the plane I1− J
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Mohr–Coulomb criterion
Combination of the tangential and normal stresses in the Mohr plane
|Tt |<− tan(φ)Tn +C
Could also be expressed as the combination of the sum and thedifference of the extremal stresses (σ3 6 σ2 6 σ1)
f (σ∼) = σ1−σ3 +(σ1 +σ3)sinφ−2C cosφ
f<0
σ 3 σ1
T
Tn
t
C cohesion, φ internal friction ofthe material
If C is zero and φ non zero,powder material
If φ is zero and C non zero,coherent material
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Representation of Mohr-Coulomb’s criterion
σ
σσ
1
2
3
In the deviatoric plane, one et a regularhexagon
TS = 2√
6(C cosφ−p sinφ)/(3+ sinφ)
CS = 2√
6(−C cosφ+p sinφ)/(3−sinφ)
As a function of Kp and of the elasticitylimit in compression, Rp :
f (σ∼) = Kp σ1−σ3−Rp
Kp =1+ sinφ
1− sinφ= tan2
(π
4+
φ
2
)Rp =−2 cosφC
1− sinφ
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Closed criteriaThe material cannot be infinitely strong in compression
Cap model, closes by one ellipse Drucker–Prager’s criterion
Cam–clay model has its limit curve defined by two ellipses in the plane(I1− J)
−I1
J Criticalline
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Contents
1 Mechanical testsStructuresRepresentative material elements
2 Rheological modelsBasic building bricksPlasticityViscoelasticityElastoviscoplasticity
3 Multiaxial plasticity criterionMechanical testsMechanisms
4 Pressure dependent modelsIsotropic : Tresca, von MisesAn anisotropic criterion : Hill
5 Pressure independent models
6 Synthesis on criteria
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Criteria, synthesis
The boundary of the initial elasticity domain is defined by a function fromthe stress space in R, that can be
Piecewise linear (Schmid, Tresca)Quadratic, or more
The elastic domain is convex
The criter can depend or not from the hydrostatic pressure
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