georges cailletaud - mines paristech
TRANSCRIPT
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Rheology
Georges Cailletaud
Centre des MatériauxMINES ParisTech/CNRS
WEMESURF contact course, Paris 21-25 juin
Georges Cailletaud | Rheology 1/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Contents
1 Mechanical testsTests on everythingResults on material elements
2 Rheological modelsBuilding bricksPlasticity
3 Plasticity criterionMultiaxial stress statesPlasticity mechanismsModels insensitive to hydrostatic pressureModels sensitive to hydrostatic pressureSynthesis on the criteria
4 3D plasticityThe ingredients of a 3D plasticity modelPlastic flowEvaluation of the flow directionsPerfectly plastic model
Georges Cailletaud | Rheology 2/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Contents
1 Mechanical testsTests on everythingResults on material elements
2 Rheological modelsBuilding bricksPlasticity
3 Plasticity criterionMultiaxial stress statesPlasticity mechanismsModels insensitive to hydrostatic pressureModels sensitive to hydrostatic pressureSynthesis on the criteria
4 3D plasticityThe ingredients of a 3D plasticity modelPlastic flowEvaluation of the flow directionsPerfectly plastic model
Georges Cailletaud | Rheology 3/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Tests on a civil plane
www.mts.com
Georges Cailletaud | Rheology 4/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Biological structures (1/2)
www.mts.com
Georges Cailletaud | Rheology 5/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Biological structures (2/2)
www.mts.com
Georges Cailletaud | Rheology 6/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Food industry
www.mts.com
Georges Cailletaud | Rheology 7/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Testing machines
www.mts.com
Georges Cailletaud | Rheology 8/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Tension test on a metallic specimen
www.mts.com
Georges Cailletaud | Rheology 9/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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 | Rheology 10/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Contents
1 Mechanical testsTests on everythingResults on material elements
2 Rheological modelsBuilding bricksPlasticity
3 Plasticity criterionMultiaxial stress statesPlasticity mechanismsModels insensitive to hydrostatic pressureModels sensitive to hydrostatic pressureSynthesis on the criteria
4 3D plasticityThe ingredients of a 3D plasticity modelPlastic flowEvaluation of the flow directionsPerfectly plastic model
Georges Cailletaud | Rheology 11/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Typical result on an aluminium 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 | Rheology 12/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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, Chllon
Georges Cailletaud | Rheology 13/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Push–pull test on an aluminium 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 | Rheology 14/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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 | Rheology 15/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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 .
Georges Cailletaud | Rheology 16/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Contents
1 Mechanical testsTests on everythingResults on material elements
2 Rheological modelsBuilding bricksPlasticity
3 Plasticity criterionMultiaxial stress statesPlasticity mechanismsModels insensitive to hydrostatic pressureModels sensitive to hydrostatic pressureSynthesis on the criteria
4 3D plasticityThe ingredients of a 3D plasticity modelPlastic flowEvaluation of the flow directionsPerfectly plastic model
Georges Cailletaud | Rheology 17/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Contents
1 Mechanical testsTests on everythingResults on material elements
2 Rheological modelsBuilding bricksPlasticity
3 Plasticity criterionMultiaxial stress statesPlasticity mechanismsModels insensitive to hydrostatic pressureModels sensitive to hydrostatic pressureSynthesis on the criteria
4 3D plasticityThe ingredients of a 3D plasticity modelPlastic flowEvaluation of the flow directionsPerfectly plastic model
Georges Cailletaud | Rheology 18/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Building bricks for the material models
Georges Cailletaud | Rheology 19/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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
Georges Cailletaud | Rheology 20/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Contents
1 Mechanical testsTests on everythingResults on material elements
2 Rheological modelsBuilding bricksPlasticity
3 Plasticity criterionMultiaxial stress statesPlasticity mechanismsModels insensitive to hydrostatic pressureModels sensitive to hydrostatic pressureSynthesis on the criteria
4 3D plasticityThe ingredients of a 3D plasticity modelPlastic flowEvaluation of the flow directionsPerfectly plastic model
Georges Cailletaud | Rheology 21/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Time independent plasticity
Georges Cailletaud | Rheology 22/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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 | Rheology 23/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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 | Rheology 24/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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 | Rheology 25/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Illustration of the two hardening types
Georges Cailletaud | Rheology 26/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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 | Rheology 27/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Contents
1 Mechanical testsTests on everythingResults on material elements
2 Rheological modelsBuilding bricksPlasticity
3 Plasticity criterionMultiaxial stress statesPlasticity mechanismsModels insensitive to hydrostatic pressureModels sensitive to hydrostatic pressureSynthesis on the criteria
4 3D plasticityThe ingredients of a 3D plasticity modelPlastic flowEvaluation of the flow directionsPerfectly plastic model
Georges Cailletaud | Rheology 28/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Contents
1 Mechanical testsTests on everythingResults on material elements
2 Rheological modelsBuilding bricksPlasticity
3 Plasticity criterionMultiaxial stress statesPlasticity mechanismsModels insensitive to hydrostatic pressureModels sensitive to hydrostatic pressureSynthesis on the criteria
4 3D plasticityThe ingredients of a 3D plasticity modelPlastic flowEvaluation of the flow directionsPerfectly plastic model
Georges Cailletaud | Rheology 29/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Biaxial loading path
Georges Cailletaud | Rheology 30/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Shear
Georges Cailletaud | Rheology 31/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Tension–torsion tests on tubes
Tension–torsion specimen
For a tube of length L, diameter2R and width e :
Strain measured by a straingauge, or use of the relationbetween the angle (β) andthe 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 | Rheology 32/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Biaxial testsBiaxial test on a cruciform specimen vinylester–glas fiber
More on the websiteSciences de l’Ingénieur, ENS Cachan
Georges Cailletaud | Rheology 33/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Shear test
Double shear Arcan(rubber) specimen
Doc. Centre des Matériaux, MINES ParisTech
Georges Cailletaud | Rheology 34/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Search of the yield surface in tension–shear
PhD Rousset, ENS Cachan
Georges Cailletaud | Rheology 35/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Yield surfaceInitial and after the first compression
PhD Rousset, ENS Cachan
Georges Cailletaud | Rheology 36/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Shear on basalt
Georges Cailletaud | Rheology 37/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Contents
1 Mechanical testsTests on everythingResults on material elements
2 Rheological modelsBuilding bricksPlasticity
3 Plasticity criterionMultiaxial stress statesPlasticity mechanismsModels insensitive to hydrostatic pressureModels sensitive to hydrostatic pressureSynthesis on the criteria
4 3D plasticityThe ingredients of a 3D plasticity modelPlastic flowEvaluation of the flow directionsPerfectly plastic model
Georges Cailletaud | Rheology 38/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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 | Rheology 39/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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 | Rheology 39/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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 | Rheology 39/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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 | Rheology 39/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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
Georges Cailletaud | Rheology 39/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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
Georges Cailletaud | Rheology 39/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Slip systems in a single crystal
PhD F. Hanriot (ENSMP-CDM, Evry)
Georges Cailletaud | Rheology 40/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Slip systems in a polycrystal
Clavel (ECP, Châtenay)
Georges Cailletaud | Rheology 41/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Rupture under dynamic loading
Georges Cailletaud | Rheology 42/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Schmid law
The deformation comes from slip on systems s defined by a plane ofnormal ns, and a shear direction ls, iff the resolved shear stress, τs
reaches a critical value τc
Projection of the stress vector on the slip direction. For a single 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
Georges Cailletaud | Rheology 43/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Polycrystalline aggregate yield surfacesAssuming uniform elasticity
Directionally Polycrystalsolidifiedmaterial
Compute yield surfaces
Georges Cailletaud | Rheology 44/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Yield surfaces in tension–shear
001
σ11
σ 12
2001000-100-200
200
100
0
-100
-200
One cubic grain oriented along (001) axes
Georges Cailletaud | Rheology 45/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Yield surfaces in tension–shear
234001
σ11
σ 12
2001000-100-200
200
100
0
-100
-200
One grain oriented along (234)
Georges Cailletaud | Rheology 45/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Yield surfaces in tension–shear
2g234001
σ11
σ 12
2001000-100-200
200
100
0
-100
-200
One grain (001) and one grain (234)
Georges Cailletaud | Rheology 45/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Yield surfaces in tension–shear
10g2g
234001
σ11
σ 12
2001000-100-200
200
100
0
-100
-200
Ten randomly oriented grains
Georges Cailletaud | Rheology 45/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Yield surfaces in tension–shear
100g10g2g
234001
σ11
σ 12
2001000-100-200
200
100
0
-100
-200
Hundred randomly oriented grains
Georges Cailletaud | Rheology 45/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Yield surfaces in tension–shear
Tresca100g
10g2g
234001
σ11
σ 12
2001000-100-200
200
100
0
-100
-200
σ211 +4σ2
12 = σ2y , Tresca criterion
Georges Cailletaud | Rheology 45/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Contents
1 Mechanical testsTests on everythingResults on material elements
2 Rheological modelsBuilding bricksPlasticity
3 Plasticity criterionMultiaxial stress statesPlasticity mechanismsModels insensitive to hydrostatic pressureModels sensitive to hydrostatic pressureSynthesis on the criteria
4 3D plasticityThe ingredients of a 3D plasticity modelPlastic flowEvaluation of the flow directionsPerfectly plastic model
Georges Cailletaud | Rheology 46/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Characterization of the maximum shear
Stress 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
Georges Cailletaud | Rheology 47/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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
Georges Cailletaud | Rheology 48/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Model ingredients for 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|
Georges Cailletaud | Rheology 49/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Physical meaning of J
Sphere in the space of the deviatoric stresses
Octahedral shear stress :on a facet of normal (1,1,1), the normal component of the stress vectoris σoct and the tangential component is 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)
Georges Cailletaud | Rheology 50/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Von Mises criterion 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
Georges Cailletaud | Rheology 51/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Tresca versus 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
In the eigenstress space, cylinders of axis (1,1,1)
Georges Cailletaud | Rheology 52/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Comparisons of Tresca and von Mises criteria
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
Georges Cailletaud | Rheology 53/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
An anisotropic criterion : Hill
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, one coefficient only
Georges Cailletaud | Rheology 54/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Contents
1 Mechanical testsTests on everythingResults on material elements
2 Rheological modelsBuilding bricksPlasticity
3 Plasticity criterionMultiaxial stress statesPlasticity mechanismsModels insensitive to hydrostatic pressureModels sensitive to hydrostatic pressureSynthesis on the criteria
4 3D plasticityThe ingredients of a 3D plasticity modelPlastic flowEvaluation of the flow directionsPerfectly plastic model
Georges Cailletaud | Rheology 55/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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
Georges Cailletaud | Rheology 56/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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
Georges Cailletaud | Rheology 57/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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
Georges Cailletaud | Rheology 58/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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
Georges Cailletaud | Rheology 59/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Contents
1 Mechanical testsTests on everythingResults on material elements
2 Rheological modelsBuilding bricksPlasticity
3 Plasticity criterionMultiaxial stress statesPlasticity mechanismsModels insensitive to hydrostatic pressureModels sensitive to hydrostatic pressureSynthesis on the criteria
4 3D plasticityThe ingredients of a 3D plasticity modelPlastic flowEvaluation of the flow directionsPerfectly plastic model
Georges Cailletaud | Rheology 60/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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 convexThe criter can depend or not from the hydrostatic pressure
Georges Cailletaud | Rheology 61/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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 convexThe criter can depend or not from the hydrostatic pressure
Georges Cailletaud | Rheology 61/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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 convexThe criter can depend or not from the hydrostatic pressure
Georges Cailletaud | Rheology 61/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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 convexThe criter can depend or not from the hydrostatic pressure
Georges Cailletaud | Rheology 61/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
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 convexThe criter can depend or not from the hydrostatic pressure
Georges Cailletaud | Rheology 61/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Contents
1 Mechanical testsTests on everythingResults on material elements
2 Rheological modelsBuilding bricksPlasticity
3 Plasticity criterionMultiaxial stress statesPlasticity mechanismsModels insensitive to hydrostatic pressureModels sensitive to hydrostatic pressureSynthesis on the criteria
4 3D plasticityThe ingredients of a 3D plasticity modelPlastic flowEvaluation of the flow directionsPerfectly plastic model
Georges Cailletaud | Rheology 62/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Contents
1 Mechanical testsTests on everythingResults on material elements
2 Rheological modelsBuilding bricksPlasticity
3 Plasticity criterionMultiaxial stress statesPlasticity mechanismsModels insensitive to hydrostatic pressureModels sensitive to hydrostatic pressureSynthesis on the criteria
4 3D plasticityThe ingredients of a 3D plasticity modelPlastic flowEvaluation of the flow directionsPerfectly plastic model
Georges Cailletaud | Rheology 63/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
From elastic to plastic behaviour
σ11
σ12
σ11
σ22
Tresca von Mises
Georges Cailletaud | Rheology 64/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
From elastic to plastic behaviour
σ11
σ12
σ11
σ22
Tresca von Mises
Georges Cailletaud | Rheology 64/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
From elastic to plastic behaviour
σ11
σ12
σ11
σ22
von MisesElastic domain
Von Mises criterion
Georges Cailletaud | Rheology 64/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
From elastic to plastic behaviour
σ11
σ12
σ11
σ22
Elastic regimevon Mises
Elastic domainVon Mises criterion
Georges Cailletaud | Rheology 64/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
From elastic to plastic behaviour
σ11
σ12
σ11
σ22
Elastic regimevon Mises
Elastic domainVon Mises criterion
Georges Cailletaud | Rheology 64/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
From elastic to plastic behaviour
σ11
σ12
σ11
σ22
Elastic regimevon Mises
Elastic domainVon Mises criterion
Georges Cailletaud | Rheology 64/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
From elastic to plastic behaviour
σ11
σ12
σ11
σ22
Plasticityvon Mises
Elastic domainVon Mises criterion
Georges Cailletaud | Rheology 64/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
3D plasticity
Strain decomposition
ε∼ = ε∼e + ε∼
th + ε∼p + ε∼
vp
Criteriondefined by the function f (σ∼ ,AI)
Flow rulesε∼
p = ...
Hardening rulesAI = ... later
? Today, model without hardening, e.g. :
f (σ∼) = J(σ∼)−σy
Georges Cailletaud | Rheology 65/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Contents
1 Mechanical testsTests on everythingResults on material elements
2 Rheological modelsBuilding bricksPlasticity
3 Plasticity criterionMultiaxial stress statesPlasticity mechanismsModels insensitive to hydrostatic pressureModels sensitive to hydrostatic pressureSynthesis on the criteria
4 3D plasticityThe ingredients of a 3D plasticity modelPlastic flowEvaluation of the flow directionsPerfectly plastic model
Georges Cailletaud | Rheology 66/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Principle of maximal power
Formulation due to Hill (1951) :The power of the real stress tensor σ∼ associated to realviscoplastic strain rate ε∼
p is larger than the power computed withany other admissible stress tensor σ∼
∗ (id est that does notovercome f = 0) associated with ε∼
p
(σ∼−σ∼∗) : ε∼
p > 0
The form F is generated to combine the function that must bemaximized and the constraint by means of a plastic multiplyier(non linear optimization problem)
F(σ∼) = σ∼ : ε∼p− λ f (σ∼)
Its derivative with respect to σ∼ must be zero to reach anextremum. The extremum is a max iff the function f is convex
ε∼p = λ
∂f∂σ∼
Georges Cailletaud | Rheology 67/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Principle of maximal power
Formulation due to Hill (1951) :The power of the real stress tensor σ∼ associated to realviscoplastic strain rate ε∼
p is larger than the power computed withany other admissible stress tensor σ∼
∗ (id est that does notovercome f = 0) associated with ε∼
p
(σ∼−σ∼∗) : ε∼
p > 0
The solution of the plastic flow problem maximizes the plasticpower σ∼ : ε∼
p
The form F is generated to combine the function that must bemaximized and the constraint by means of a plastic multiplyier(non linear optimization problem)
F(σ∼) = σ∼ : ε∼p− λ f (σ∼)
Its derivative with respect to σ∼ must be zero to reach anextremum. The extremum is a max iff the function f is convex
ε∼p = λ
∂f∂σ∼
Georges Cailletaud | Rheology 67/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Principle of maximal power
Formulation due to Hill (1951) :The power of the real stress tensor σ∼ associated to realviscoplastic strain rate ε∼
p is larger than the power computed withany other admissible stress tensor σ∼
∗ (id est that does notovercome f = 0) associated with ε∼
p
(σ∼−σ∼∗) : ε∼
p > 0
The solution of the plastic flow problem maximizes the plasticpower σ∼ : ε∼
p
One has to develop a maximization under constraint, since onemust have also f (σ∼) 6 0
The form F is generated to combine the function that must bemaximized and the constraint by means of a plastic multiplyier(non linear optimization problem)
F(σ∼) = σ∼ : ε∼p− λ f (σ∼)
Its derivative with respect to σ∼ must be zero to reach anextremum. The extremum is a max iff the function f is convex
ε∼p = λ
∂f∂σ∼
Georges Cailletaud | Rheology 67/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Principle of maximal power
Formulation due to Hill (1951) :The power of the real stress tensor σ∼ associated to realviscoplastic strain rate ε∼
p is larger than the power computed withany other admissible stress tensor σ∼
∗ (id est that does notovercome f = 0) associated with ε∼
p
(σ∼−σ∼∗) : ε∼
p > 0
The form F is generated to combine the function that must bemaximized and the constraint by means of a plastic multiplyier(non linear optimization problem)
F(σ∼) = σ∼ : ε∼p− λ f (σ∼)
Its derivative with respect to σ∼ must be zero to reach anextremum. The extremum is a max iff the function f is convex
ε∼p = λ
∂f∂σ∼
Georges Cailletaud | Rheology 67/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Principle of maximal power
Formulation due to Hill (1951) :The power of the real stress tensor σ∼ associated to realviscoplastic strain rate ε∼
p is larger than the power computed withany other admissible stress tensor σ∼
∗ (id est that does notovercome f = 0) associated with ε∼
p
(σ∼−σ∼∗) : ε∼
p > 0
The form F is generated to combine the function that must bemaximized and the constraint by means of a plastic multiplyier(non linear optimization problem)
F(σ∼) = σ∼ : ε∼p− λ f (σ∼)
Its derivative with respect to σ∼ must be zero to reach anextremum. The extremum is a max iff the function f is convex
ε∼p = λ
∂f∂σ∼
Georges Cailletaud | Rheology 67/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Geometrical aspects of Hill’s principle
(σ∼−σ∼∗) : ε∼
p > 0
σ∼∗ on the yield surface, σ∼ in the domain, ε∼
p = 0∼Normality rule, by working in the tangent plane, (σ∼ on the surface),
k t∼∗ : ε∼
p > 0 and − k t∼∗ : ε∼
p > 0
so that t∼∗ : ε∼
p = 0
Sign of the multiplyier, by working on the interior normal, (σ∼ on thesurface), (σ∼−σ∼
∗) = k n∼ colinear to n∼ (k>0), and :
k n∼ : λn∼ > 0 thus λ > 0
Georges Cailletaud | Rheology 68/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Convexity of the yield surface
.ε. p
f<0σ
n
~
~
~
a. Illustration of the normality rule
*
~
~~σ n
σ
b. Convexity of f
Hill’s principle≡ (f convex and normality of the flow)
Georges Cailletaud | Rheology 69/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Contents
1 Mechanical testsTests on everythingResults on material elements
2 Rheological modelsBuilding bricksPlasticity
3 Plasticity criterionMultiaxial stress statesPlasticity mechanismsModels insensitive to hydrostatic pressureModels sensitive to hydrostatic pressureSynthesis on the criteria
4 3D plasticityThe ingredients of a 3D plasticity modelPlastic flowEvaluation of the flow directionsPerfectly plastic model
Georges Cailletaud | Rheology 70/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
NOTE : computation of ∂J/∂σ∼
One will have to express∂J∂σ∼
to compute n∼The derivative s∼ with respect to σ∼ , writes with index notation :
Jijkl =12
(δik δjl + δilδjk )− 13
δijδkl
since :s∼ = J
≈: σ∼
Derivative of J with respect to σ∼ :
∂J∂σ∼
=∂J∂s∼
:∂s∼∂σ∼
=∂((3/2)s∼ : s∼)1/2
∂s∼: J≈
=32
s∼J
: J≈
=32
s∼J
Other solution :
J2 =32
sijsij thus 2J dJ = 3sij dsij = 3sij dσij
∂J∂σij
=32
sij
J∂J∂σ∼
=3s∼2J
Georges Cailletaud | Rheology 71/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
NOTE : computation of ∂J/∂σ∼
One will have to express∂J∂σ∼
to compute n∼The derivative s∼ with respect to σ∼ , writes with index notation :
Jijkl =12
(δik δjl + δilδjk )− 13
δijδkl
since :s∼ = J
≈: σ∼
Derivative of J with respect to σ∼ :
∂J∂σ∼
=∂J∂s∼
:∂s∼∂σ∼
=∂((3/2)s∼ : s∼)1/2
∂s∼: J≈
=32
s∼J
: J≈
=32
s∼J
Other solution :
J2 =32
sijsij thus 2J dJ = 3sij dsij = 3sij dσij
∂J∂σij
=32
sij
J∂J∂σ∼
=3s∼2J
Georges Cailletaud | Rheology 71/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
NOTE : computation of ∂J/∂σ∼
One will have to express∂J∂σ∼
to compute n∼The derivative s∼ with respect to σ∼ , writes with index notation :
Jijkl =12
(δik δjl + δilδjk )− 13
δijδkl
since :s∼ = J
≈: σ∼
Derivative of J with respect to σ∼ :
∂J∂σ∼
=∂J∂s∼
:∂s∼∂σ∼
=∂((3/2)s∼ : s∼)1/2
∂s∼: J≈
=32
s∼J
: J≈
=32
s∼J
Other solution :
J2 =32
sijsij thus 2J dJ = 3sij dsij = 3sij dσij
∂J∂σij
=32
sij
J∂J∂σ∼
=3s∼2J
Georges Cailletaud | Rheology 71/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
NOTE : computation of ∂J/∂σ∼
One will have to express∂J∂σ∼
to compute n∼The derivative s∼ with respect to σ∼ , writes with index notation :
Jijkl =12
(δik δjl + δilδjk )− 13
δijδkl
since :s∼ = J
≈: σ∼
Derivative of J with respect to σ∼ :
∂J∂σ∼
=∂J∂s∼
:∂s∼∂σ∼
=∂((3/2)s∼ : s∼)1/2
∂s∼: J≈
=32
s∼J
: J≈
=32
s∼J
Other solution :
J2 =32
sijsij thus 2J dJ = 3sij dsij = 3sij dσij
∂J∂σij
=32
sij
J∂J∂σ∼
=3s∼2J
Georges Cailletaud | Rheology 71/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Flow direction ssociated to von Mises’ criterion
Perfectly plastic model f (σ∼) = J(σ∼)−σy
εpij = λnij ε∼
p = λn∼
nij =∂J∂σij
=32
sij
Jn∼ =
∂J∂σ∼
=32
s∼J
One denotes by accumulated plastic strain, p, the length of the pathrepresenting plastic flow in the plastic strain space. At time t ,
p(t) =Z t
0p(τ)dτ p =
(23
εpij ε
pij
)1/2
=
(23
ε∼p : ε∼
p)1/2
With von Mises’s criterion
p =
(23
λ32
s∼J
: λ32
s∼J
)1/2
= λ
Georges Cailletaud | Rheology 72/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Von Mises’ model
Only one stress component is non zero
σ∼ =
σ 0 00 0 00 0 0
s∼ =2σ
3
1 0 00 −1/2 00 0 −1/2
Computation of the normal, with J = |σ|
n∼ =32
s∼J
=
1 0 00 −1/2 00 0 −1/2
sign(σ)
Components of the plastic strain rate
εp11 = λsign(σ) = p sign(σ) ε
p22 = ε
p33 =−ε
p11
Georges Cailletaud | Rheology 73/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Flow direction associated to Tresca’s criterion
Si σ1 > σ2 > σ3 : f (σ∼) = σ1−σ3−σy , there is a plastic flow in pureshear, with ε∼
p22 = 0
ε∼p = λ
1 0 00 0 00 0 −1
For simple tension , for example with σ1 > σ2 = σ3 = 0 :
f (σ∼) = σ1−σ2−σy or f (σ∼) = σ1−σ3−σy
ε∼p = λ
1 0 00 0 00 0 −1
+ µ
1 0 00 −1 00 0 0
Georges Cailletaud | Rheology 74/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Flow direction associated to Tresca’s criterion
Si σ1 > σ2 > σ3 : f (σ∼) = σ1−σ3−σy , there is a plastic flow in pureshear, with ε∼
p22 = 0
ε∼p = λ
1 0 00 0 00 0 −1
For simple tension , for example with σ1 > σ2 = σ3 = 0 :
f (σ∼) = σ1−σ2−σy or f (σ∼) = σ1−σ3−σy
ε∼p = λ
1 0 00 0 00 0 −1
+ µ
1 0 00 −1 00 0 0
Georges Cailletaud | Rheology 74/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Flow direction associated to Drucker–Pragercriterion
Expression of the criterion
f (σ∼) = (1−α)J(σ∼) + α trace(σ∼)−σy
The normal contains the identity tensor
n∼ =32
(1−α)s∼J
+ α I∼
Volume increase for any applied load :
trace(ε∼p) = λ trace n∼ = 3αλ
Georges Cailletaud | Rheology 75/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Flow direction associated to Drucker–Pragercriterion
Expression of the criterion
f (σ∼) = (1−α)J(σ∼) + α trace(σ∼)−σy
The normal contains the identity tensor
n∼ =32
(1−α)s∼J
+ α I∼
Volume increase for any applied load :
trace(ε∼p) = λ trace n∼ = 3αλ
Georges Cailletaud | Rheology 75/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Flow direction associated to Drucker–Pragercriterion
Expression of the criterion
f (σ∼) = (1−α)J(σ∼) + α trace(σ∼)−σy
The normal contains the identity tensor
n∼ =32
(1−α)s∼J
+ α I∼
Volume increase for any applied load :
trace(ε∼p) = λ trace n∼ = 3αλ
Georges Cailletaud | Rheology 75/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Contents
1 Mechanical testsTests on everythingResults on material elements
2 Rheological modelsBuilding bricksPlasticity
3 Plasticity criterionMultiaxial stress statesPlasticity mechanismsModels insensitive to hydrostatic pressureModels sensitive to hydrostatic pressureSynthesis on the criteria
4 3D plasticityThe ingredients of a 3D plasticity modelPlastic flowEvaluation of the flow directionsPerfectly plastic model
Georges Cailletaud | Rheology 76/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Perfectly plastic model
Flow to determine from f (σ∼) = 0 and f (σ∼) = 0
ε∼p = λ
∂f∂σ∼
= λn∼
Consistency condition written on f (σ∼) = J−σy
f =∂f∂σ∼
: σ∼ = n∼ : σ∼ = 0
During plastic flow, the current stress point can only stay around theyield surface. The plastic multiplyier cannot be determined in terms ofstress rate.
Georges Cailletaud | Rheology 77/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Perfectly plastic model
Flow to determine from f (σ∼) = 0 and f (σ∼) = 0
ε∼p = λ
∂f∂σ∼
= λn∼
Consistency condition written on f (σ∼) = J−σy
f =∂f∂σ∼
: σ∼ = n∼ : σ∼ = 0
During plastic flow, the current stress point can only stay around theyield surface. The plastic multiplyier cannot be determined in terms ofstress rate.
Georges Cailletaud | Rheology 77/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Perfectly plastic model
Flow to determine from f (σ∼) = 0 and f (σ∼) = 0
ε∼p = λ
∂f∂σ∼
= λn∼
Consistency condition written on f (σ∼) = J−σy
f =∂f∂σ∼
: σ∼ = n∼ : σ∼ = 0
During plastic flow, the current stress point can only stay around theyield surface. The plastic multiplyier cannot be determined in terms ofstress rate.
Georges Cailletaud | Rheology 77/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Perfectly plastic behaviour, strain control
Elasticityσ∼ = Λ
≈: (ε∼− ε∼
p) and n∼ : σ∼ = 0
Projection on the normal to the surface
n∼ : σ∼ = n∼ : Λ≈
: (ε∼− ε∼p) = n∼ : Λ
≈: ε∼−n∼ : Λ
≈: λn∼
Expression of the plastic multiplyier as a function of the strain rate
λ =n∼ : Λ
≈: ε∼
n∼ : Λ≈
: n∼Expression of the plastic strain rate
ε∼p = λn∼ =
n∼ : Λ≈
: ε∼n∼ : Λ
≈: n∼
n∼
Georges Cailletaud | Rheology 78/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
The elastoplastic tensor L≈
ep
Stress rate σ∼ = Λ≈
: (ε∼− ε∼p)
By replacing the plastic strain rate
σ∼ = Λ≈
: ε∼−Λ≈
:
(n∼ : Λ
≈: ε∼
n∼ : Λ≈
: n∼n∼
)Since ΛijklnpqΛpqrs εrsnkl = ΛijklnklnpqΛpqrs εrs, it comes σ∼ = L
≈ep ε∼, with
L≈
ep = Λ≈−
(Λ≈
: n∼)⊗ (n∼ : Λ≈
)
n∼ : Λ≈
: n∼
Formally, this is an elastic type rule, but expressed in terms of rate
Georges Cailletaud | Rheology 79/80
Plan Mechanical tests Rheological models Plasticity criterion 3D plasticity
Isotropic elasticity and of von Mises’ criterion
Possible simplifications
Λijkl = λδijδkl + µ(δik δjl + δilδjk ) ; nij =32
sij
J
nij Λijkl = 2µnkl ; nij Λijkl nkl = 3µ ; nij Λijkl εkl = 2µnkl εkl
λ =23
n∼ : ε∼
Case of the onedimensional loading : during plastic flow, the plasticstrain rate is equal to the total strain rate, the normal is then the diagonal(1,−1/2,−1/2)sign(σ∼ ) . The total strain rate is the diagonal(ε
p11,−ε
p11/2,−ε
p11/2
), so that one get :
λ = εp11 sign(σ)
Georges Cailletaud | Rheology 80/80