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Karlsruhe Institute of Technology
KIT – University of the State of Baden-Württemberg and National Research Center of the Helmholtz Association www.kit.edu
Continuum Models in Scale Dependent PlasticityAnalytical and Numerical Aspects
Christian Wieners
Institut für Angewandte und Numerische Mathematik, Karlsruhe
Karlsruhe Institute of TechnologyOutline
1. A Remark on Multiscale Plasticity
2. Generalized Standard Materials in Strain-Gradient Plasticity
3. Approximation of Strain-Gradient Plasticity
4. Algorithms in Strain-Gradient Plasticity
5. Results for Strain-Gradient Plasticity
Cooperations with ...
P. Neff (Essen), K. Chełminski (Warschau)B. Wohlmuth (München), D. Reddy (Capetown)W. Müller, A. Sydow, M. Sauter, D. Maurer (Karlsruhe)
1
Karlsruhe Institute of TechnologyDefects and Dislocations in a Single Crystal
Dislocations are micro-structuralline defects in the atomic latticeof a crystalline material alongwhich the crystallographicstructure is disturbed.
en.wikipedia.org/wiki/Dislocation
www.tf.uni-kiel.de/matwis/amat Weertman and Weertman 19822
Karlsruhe Institute of TechnologyDislocation Densities in a Single Crystal
We consider a single crystal with a finite set of slip systems A, determined by theslip direction sssα ∈ R3 and the slip-plane normal mmmα ∈ R3 for all α ∈ A.
Classical continuum theory of dislocationsThe displacement gradient can be decomposed into elastic and plastic part, i.e.,∇uuu = hhhe +hhhp. The dislocation density GGG is determined by∫
SGGGmmmda =
∫∂S
hhhpds ,
where S is a surface with normal mmm and ∂S is a dislocation loop, i.e., GGG = curlhhhp.
Constitutive setting in Crystal PlasticityWithin single crystal plasticity, the plastic distortion is of the form
hhhp(γγγ) = ∑αγ
α sssα ⊗mmmα ,
where γα is the plastic slip in the slip system α. This gives the macroscopicBurgers tensor and the edge and screw dislocation densities
GGG = curlhhhp = ∑α(∇γ
α ×mmmα )⊗sssα = ∑αρedge(mmmα ×sssα )⊗sssα + ρsrewsssα ⊗sssα .
Here we follow the monograph of Gurtin, Leele, Anand 2009.3
Karlsruhe Institute of TechnologyMaterials with Memory
We consider displacements
uuu : [0,T ]×Ω−→ R3
of a material with internal variables
zzz : [0,T ]×Ω−→ RN .
We assume that the free energy is of the form
Ψ(uuu,zzz) =∫
Ωψ(x ,∇u(x),z(x),∇z(x))dx
and the total energy is given
E (t ,uuu,zzz) = Ψ(uuu,zzz)−〈`(t),uuu〉 ,with the load functional 〈`(t),vvv〉=
∫Ω bbb(t) ·vvv dx +
∫ΓN
tttN (t) ·vvv da.
The plastic evolution is determined by a dissipation distance R.
Let VVV ⊂ H1(Ω,R3) and ZZZ ⊂ H1(Ω,RN ) are suitable spaces for (infinitesimal)displacements and internal variables, respectively. For simplicity, we assume thathomogeneous boundary conditions are included in the spaces.
4
Karlsruhe Institute of TechnologySmall Strain Plasticity
Within the infinitesimal setting we consider the following linear constitutiverelations: depending on the deformation we define the strain
εεε = sym(∇uuu) : [0,T ]×Ω−→ Sym(3)
and the plastic strain depending on the internal variables
εεεp(zzz) : [0,T ]×Ω−→ Sym(3) .
We assume that the free energy is of the form
Ψ(uuu,zzz) = Ψelastic(εεε(uuu)−εεε
p(zzz))
+ Ψdefect(zzz) ,
where the elastic energy is given by
Ψelastic(εεεe) =12
∫Ω
εεεe : C : εεε
edx .
The plastic evolution is determined by a convex dissipation functional
R : ZZZ −→ R∪∞ .Equivalently, the plastic evolution is determined by the plastic potential
χ : ZZZ ∗ −→ R∪∞defined by duality χ = R∗, i.e., χ(yyy) = supzzz∈ZZZ 〈yyy ,zzz〉−R(zzz).
5
Karlsruhe Institute of TechnologyClassical Models in Infinitesimal Plasticity
Perfect Single Crystal Plasticity
Consider zzz = (γα )α∈A in ZZZ = L2(Ω,RN ) with
εεεp(γγγ) = ∑α
γα sym
(sssα ⊗mmmα
),
Ψdefect(γγγ) = 0 ,
R(γγγ) = ∑α∈A
∫Ω
Y0|γα |dx ,
where Y0 is the initial yield stress. Note that traceεεεp(γγγ) = 0.
Perfect von Mises Plasticity
Consider zzz = εεεp in ZZZ = εεεp ∈ L2(Ω,Sym(3)) : traceεεεp = 0 with12
Ψdefect(εεεp) = 0 ,
R(εεεp) =∫
ΩY0|εεεp|dx .
The models are not well-posed in H1×L2, only measure valued solutions can beobtained, see Temam, Suquet, Johnson, Seregin, Repin, ...
6
Karlsruhe Institute of TechnologyClassical Models with Hardening
Single Crystal Plasticity with Isotropic Hardening
Consider zzz = (γα ,µα )α∈A in ZZZ = L2(Ω,RN )×L2(Ω,RN ) with
Ψdefect((γγγ,µµµ)
)=
12 ∑α∈A
∫Ω
H0 |µα |2 dx ,
where H0 > 0 is the hardening modulus.We set domR = (γα ,µα ) ∈ZZZ : |γα | ≤ µα.
Von Mises Plasticity with Kinematic Hardening
Consider zzz = εεεp in ZZZ = εεεp ∈ L2(Ω,Sym(3)) : traceεεεp = 0 with
Ψdefect(εεεp) =
12
∫Ω
H0 εεεp : εεε
p dx .
Numerical analysis by Johnson, Han-Reddy, Carstensen, ...7
Karlsruhe Institute of TechnologyRepresentative Models in Strain Gradient Plasticity
Energy in a Single Crystal including Dislocation Densities ∇γα ×mmmα
Consider zzz = (γα ,µα )α∈A in ZZZ ⊂ H1(Ω,RN )×L2(Ω,RN ) with
Ψdefect((γγγ,µµµ)
)=
12 ∑α∈A
∫Ω
H0|µα |2 dx +∫
ΩL0|∇γ
α ×mmmα |2 dx ,
where L0 > 0 is the length scale parameter.
Energy including the Burgers Tensor curlhhhp
Consider zzz = hhhp in ZZZ = H(curl,Ω)3 with
Ψdefect(hhhp) =
12
∫Ω
H0hhhp : hhhp dx +∫
ΩL0 curlhhhp : curlhhhp dx
and εεεp(zzz) = symdevhhhp.
Studied by Gurtin, Needleman, Geers, Reddy, Svendson, Steinmann, Forest, ...
8
Karlsruhe Institute of TechnologyGeneralized Standard Materials
Assume that the energy
E : [0,T ]×VVV ×ZZZ −→ R
is bounded and uniformly convex in VVV ×ZZZ , and that the dissipation functional
R : ZZZ −→ R∪∞
is convex, proper, l.s.c., and positively 1-homogeneous.
Theorem For given admissible initial state zzz(0) ∈ domR a unique solution
(uuu,zzz) : [0,T ]−→VVV ×ZZZ
exists satisfying
Equilibrium 0 = ∂uuuE(t ,uuu(t),zzz(t)
)Flow Rule 0 ∈ ∂zzzE (t ,uuu(t),zzz(t)) + ∂R(zzz)
Generalized Standard Materials of monotone gradient type are analyzed byH.-D. Alber. Here, also the framework of Han-Reddy applies.
9
Karlsruhe Institute of TechnologyConjugated Variables and Duality
The conjugated variables (TTT ,yyy) for (uuu,zzz) are given by
TTT = ∂εεεe Ψelastic(εεε
e)yyy = −∂zzzΨ(uuu,zzz) ∈ZZZ ∗
This gives TTT = C : (εεε(uuu)−εεεp(zzz)) ∈ L2(Ω,Sym(3)).
For generalized standard materials, we obtain:
Macroscopic Balance Equation∫Ω
TTT : εεε(vvv)dx = 〈`(t),vvv〉 for all vvv ∈VVV
Microscopic Balance Equation
〈yyy ,www〉 =∫
ΩTTT : εεε
p(www)dx−〈∂ Ψdefect(zzz),www〉 for all www ∈ZZZ
Flow Rule
zzz ∈ ∂ χ(yyy) ⇐⇒ yyy ∈ ∂R(zzz)
10
Karlsruhe Institute of TechnologyRegularity of the Conjugated Variable
Lemma If R is bounded in L2 and ZZZ is dense in L2, the functional yyy ∈ZZZ ∗ with
yyy ∈ ∂R(zzz) , i.e., R(www)≥R(zzz)−〈yyy ,www − zzz〉 for all www ∈ZZZ ,
can be identified with a function in L2. (W.-Wohlmuth, SINUM 2011)
This allows to evaluate the microscopic balance equation and the plastic potential.
Energy including Dislocation Densities ∇γα
For the micro-stress holds L0∇γα ∈ H(div,Ω), for yyy = (πππ,ggg) = (πα ,gα )α∈A holds
πα = sssα ·TTTmmmα + L0∆γ
α , gα =−H0µα
and dom χ = (πππ,ggg) ∈ L2(Ω,R2N ) : |πα |+ gα ≤ Y0 and gα ≤ 0 for all α ∈ A.
Energy including the Burgers Tensor curlhhhp
For the Burgers tensor holds curlhhhp ∈ H(curl,Ω)3, for yyy holds
yyy = devTTT −H0hhhp−L0 curlcurlhhhp ,
and dom χ = yyy ∈ L2(Ω,R3×3) : |symdevyyy | ≤ Y0.11
Karlsruhe Institute of TechnologyPoint-wise Complementarity
The regularity of yyy =−∂zzzΨ(uuu,zzz) allows for a point-wise evaluation of the flow rule
zzz ∈ ∂ χ(yyy) ⇐⇒ yyy ∈ ∂R(zzz)
Energy including Dislocation Densities ∇γα
For yyy = (πα ,gα ) = (τα −ζ α ,−H0µα ) with τα = sssα ·TTTmmmα and ζ α =−L0∆γα holds
(γα , µ
α ) = λα (sgnπ
α ,1)
and the complementarity condition for the consistency parameters λ α
λα ≥ 0 , |πα |+ gα −Y0 ≤ 0 ,
(|πα |+ gα −Y0
)λ
α = 0 .
Energy including the Burgers Tensor curlhhhp
For yyy = devTTT −βββ with βββ = H0hhhp−L0 curlcurlhhhp holds
εεεεεεεεεp = γ
devTTT −βββ
|devTTT −βββ |and the complementarity condition for the consistency parameters λ
|devTTT −βββ | ≤ Y0 , λ ≥ 0 , λ (|devTTT −βββ |−Y0) = 0 .
12
Karlsruhe Institute of TechnologyIncremental Infinitesimal Plasticity
Let 0 = t0 < t1 < t2 < · · · be a time series. In step n, find (uuun,zzzn) ∈VVV ×ZZZ with
0 = ∂uuuE(tn,uuun,zzzn)
0 ∈ ∂zzzE (tn,uuun,zzzn) + ∂R(4zzzn) ⇐⇒ 4zzzn ∈ ∂ χ(−∂zzzE (tn,uuun,zzzn)
)for given (uuun−1,zzzn−1), where 4zzzn = zzzn−zzzn−1.
The incremental solution (uuun,zzzn) ∈VVV ×ZZZ is the unique minimizer of
4J n(uuu,zzz) = Ψ(uuu,zzz) +R(zzz−zzzn−1)−〈`n,uuu〉 .
Let VVV h×ZZZ h ⊂VVV ×ZZZ and let Πh be the L2 projection onto piecewise constants.
The fully discrete solution (uuun,h,zzzn,h) ∈VVV h×ZZZ h is the unique minimizer of
4J n,h(uuuh,zzzh) = Ψh(uuuh,zzzh) +Rh(zzzh−zzzn−1,h)−〈`n,uuun,h〉 ,
where Ψh(uuu,zzz) = Ψelastic(εεε(uuu)−εεεp(Πhzzz)) + Ψdefect(zzz) and Rh(zzz) = R(Πhzzz).
13
Karlsruhe Institute of TechnologyPrimal-Dual Constraint Convex Minimization
Let (uuun,h,zzzn,h) ∈VVV h×ZZZ h be the discrete solution. Define
TTT n,h = C : (εεε(uuun,h)−εεεp(Πhzzzn,h)) ,
τττn,h = −Πh∂ Ψ∗elastic(TTT n,h) .
Lemma If ΠhZZZ h = BBBh ⊂ L2(Ω,RN ), a discrete ’back stress’ βββ n,h ∈BBBh existssatisfying
〈βββ n,h,wwwh〉= 〈∂ Ψdefect(zzzn,h),wwwh〉 , wwwh ∈ZZZ h .
If inf‖βββ h‖0=1
sup‖zzzh‖ZZZ =1
〈βββ h,zzzh〉 ≥ c0 > 0, βββ n,h is unique. (W.-Wohlmuth, SINUM 2011)
The primal-dual solution (uuun,h,zzzn,h,βββ n,h) is characterized bythe linear Macroscopic Balance Relation
〈TTT n,h,εεε(vvvh)〉= 〈`n,vvvh〉 , vvvh ∈VVV h ,
the linear Microscopic Balance Relation〈βββ n,h,wwwh〉= 〈∂ Ψdefect(zzz
n,h),wwwh〉 , wwwh ∈ZZZ h
the convex Plastic Admissibilityτττ
n,h ∈ dom χ +βββn,h .
14
Karlsruhe Institute of TechnologyThe Radial Return
The radial return allows to compute TTT n,h and Πh4zzzn,h from
TTT trial = C : (εεε(uuun,h)−εεεp(Πhzzzn−1,h)) ,
yyy trial = yyyn−1,h−βββn,h
locally. This defines also the plastic response Πhzzzn,h = Rn(uuun,h,βββ n,h).
Theorem The primal-dual solution (uuun,h,zzzn,h,βββ n,h) is the unique solution of thenonlinear equation⟨
C :(εεε(uuun,h)−εεε
p(Rn(uuun,h,βββ n,h))),vvvh⟩−〈`n,vvvh〉 = 0 , vvvh ∈VVV h ,
−〈βββ n,h,wwwh〉+ 〈∂ Ψdefect(zzzn,h),wwwh〉 = 0 , wwwh ∈ZZZ h ,
〈Rn(uuun,h,βββ n,h),ηηηh〉−〈zzzn,h,ηηηh〉 = 0 , ηηηh ∈BBBh .
This nonlinear system is the critical point of an incremental saddle pointfunctional. Thus, the linearization is symmetric but indefinite.The system is strongly semi-smooth and a generalized Newton method issuperlinear convergent.For bubble-enhanced finite elements optimal order convergence estimates canbe provided.
15
Karlsruhe Institute of TechnologyThe Radial Return for Single-Crystal Plasticity
For given ζζζ ∈ L2(Ω,RN ), let P(·, ·;ζζζ ) be the orthogonal projection onto
CCC(ζζζ ) =
(TTT ,ggg) ∈ L2(Ω,Sym(3))×L2(Ω,RN ) : ϕα (TTT ,ggg,ζζζ )≤ 0 , α = 1, ...,N
with respect to the metric induced by
‖(TTT ,ggg)‖2 =∫
Ω
(TTT : C−1 : TTT + H−1
0 ∑α|gα |2
)dx ,
where ϕα (TTT ,ggg,ζζζ ) = |TTT : NNNα −ζ α |+ gα −Y0 and NNNα = sym(sssα ⊗mmmα ).
For given (TTT trial,gggtrial), the projection is uniquely determined by the KKT system
0 = C−1(TTT −TTT trial) +∑αλ
α sgn(TTT : NNNα −ζα )NNNα ,
0 = H−10 (ggg−gggtrial) +λλλ ,
0≤ λλλ , ϕα (TTT ,ggg,ζζζ )≤ 0 , ∑α
λα
ϕα (TTT ,ggg,ζζζ ) = 0 .
The solution of the KKT system defines the radial return
(TTT ,ggg) = P(TTT trial,gggtrial;ζζζ )
and thus also the response function for the plastic slips.16
Karlsruhe Institute of TechnologyEnergy including Dislocation Densities: Results
Distribution of the plastic slips γα for a shear test with Ω = (0, lΩ)2× (0,3lΩ) andlΩ = 20 [µ]. The results for the slip planes 111, 111, 111, 1 11 coincide inthe slip directions 〈011〉, 〈110〉, 〈101〉 up to rotation and sign changing.(joint work with B. D. Reddy and B. I. Wohlmuth)
17
Karlsruhe Institute of TechnologyClassical Single-Crystal Plasticity: Results
Distribution of the plastic slips γα for a shear test with Ω = (0, lΩ)2× (0,3lΩ) andlΩ = 20 [µ]. The results for the slip planes 111, 111, 111, 1 11 coincide inthe slip directions 〈011〉, 〈110〉, 〈101〉 up to rotation and sign changing.(joint work with B. D. Reddy and B. I. Wohlmuth)
18
Karlsruhe Institute of TechnologyClassical Single-Crystal Plasticity: Results
192 cells1536 cells
12288 cells98304 cells
t0.080.060.040.020
t ‖TTT‖SSS‖εεε(uuu)‖EEE
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
Convergence in space of the stress-strain relation for the shear test.
(joint work with B. D. Reddy and B. I. Wohlmuth)
19
Karlsruhe Institute of TechnologyStrain-Gradient Crystal Plasticity: Results
1 593 d.o.f.10 059 d.o.f.71 847 d.o.f.
544 191 d.o.f.
t0.070.060.050.040.030.020.010
t ‖TTT‖SSS‖εεε(uuu)‖EEE
0.05
0.04
0.03
0.02
0.01
0
Convergence in space of the stress-strain relation for the shear test.
(joint work with B. D. Reddy and B. I. Wohlmuth)
20
Karlsruhe Institute of TechnologyClassical Single-Crystal Plasticity: Results
Distribution of the plastic slip γα for an indentation test.(joint work with B. D. Reddy and B. I. Wohlmuth)
21
Karlsruhe Institute of TechnologyStrain-Gradient Crystal Plasticity: Results
Distribution of the stress TTT , the plastic strain εεεp and the plastic slip γα withmmmα = 111 and sssα = 〈011〉 for the indentation test (Ω = (0, lΩ)3 with lΩ = 5) att = 0.1. Here we use 4 096 hexahedral cells and 184 287 degrees of freedom.
22
Karlsruhe Institute of TechnologyEnergy including Dislocation Densities: Results
lΩ = 5lΩ = 7.5lΩ = 10lΩ = 20lΩ = 30lΩ = ∞
t0.080.060.040.020
t ‖TTT‖SSS‖εεε(uuu)‖EEE
0.05
0.04
0.03
0.02
0.01
0
Stress-strain relation for the indentation test in dependence of the sample size lΩ.
(joint work with B. D. Reddy and B. I. Wohlmuth)
23
Karlsruhe Institute of TechnologyEnergy including Dislocation Densities: Results
lΩ = 5lΩ = 7.5lΩ = 10lΩ = 20lΩ = 30lΩ = ∞
0.10.080.060.040.020
‖ζζζ‖∞
6000
5000
4000
3000
2000
1000
0
Evolution of the back stress for the indentation test depending on lΩ.
(joint work with B. D. Reddy and B. I. Wohlmuth)
24
Karlsruhe Institute of TechnologyEnergy including Burgers Tensor: Macroscopic Limit
Dependence on the length scale parameter L0 = µS l20 .
l0 0.1 0.01 0.001 0.0001 0Example 1 ‖TTT h‖SSS 2.0543 2.0881 2.1109 2.1121 2.1122
‖hhhph‖QQQ 0.4864 0.3834 0.2973 0.2888 0.2884
‖εεε(uuuh)‖EEE 24.73 29.29 41.83 42.59 42.61‖curlhhhp
h‖0 0.0106 0.0751 0.300 0.364Example 2 ‖TTT h‖SSS 2.5231 2.5687 2.6317 2.6374 2.6382
‖hhhph‖QQQ 1.1145 0.9786 0.6899 0.6551 0.6493
‖εεε(uuuh)‖EEE 78.75 83.96 91.73 92.51 92.67‖curlhhhp
h‖0 0.00566 0.154 0.703 1.294 25
Karlsruhe Institute of TechnologyEnergy including Burgers Tensor: Convergence
Convergence history (up to ε ≤ 10−9) of the semi-smooth Newton method.
k ρ0 ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7 ρ8 ρ9
0 160.3 16.4 0.06 10−6 ε
1 89.6 46.2 17.24 0.971 0.002 ε
2 47.1 39.5 13.11 2.842 0.072 2 ·10−5 ε
3 24.1 12.4 11.68 8.649 7.727 1.572 0.19 0.0011 ε
4 12.2 6.7 6.13 4.661 4.585 1.235 0.15 0.0016 2 ·10−5 ε
Convergence for successive uniform refinements of the displacementat a test point x = (0,0,7) and for the stress maximum.
level k d.o.f. # cells # plastic cells |uuuh(zzz)| ‖TTT h‖∞
0 1 426 50 8 0.0167 581.641 8 903 400 77 0.0158 879.082 62 419 3 200 836 0.0192 1161.383 466 499 25 600 7 122 0.0214 1506.844 3 605 251 204 800 60 622 0.0228 1941.87
(joint work with B. I. Wohlmuth)
26
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