Samuel Forest - KIT · Samuel Forest Mines ParisTech ... Plan 1 Introduction Mechanics of...

Strain gradient and micromorphic plasticity

Samuel Forest

Mines ParisTech / CNRSCentre des Materiaux/UMR 7633

BP 87, 91003 Evry, [email protected]

Plan

1 IntroductionMechanics of generalized continua

2 Strain gradient plasticity theoryMethod of virtual powerConstitutive equationsShearing of a laminate microstructure

3 Micromorphic plasticity theoryBalance and constitutive equationsRate–independent plasticityShearing of a laminate microstructure

4 Discussion

NotationsCartesian basis

A = Ai e i , A∼ = Aij e i ⊗ e j , A∼ = A∼

= A = Aijk e i ⊗ e j ⊗ e k

tensor products

a ⊗ b = aibj e i ⊗ e j , A∼ ⊗ B∼ = AijBkl e i ⊗ e j ⊗ e k ⊗ e l

A∼ B∼ = AikBjl e i ⊗ e j ⊗ e k ⊗ e l

contractions

A · B = AiBi , A∼ : B∼ = AijBij , A∼...B∼ = AijkBijk

nabla operators

∇x = ,i e i , ∇X = ,I E I

u ⊗∇ = ui ,j e i ⊗ e j , σ∼ .∇ = σij ,j e i

3/53

Plan




4 Discussion

Mechanics of generalized continua

Principle of local action: the stress state at a point X depends onvariables defined at this point only

[Truesdell, Toupin, 1960] [Truesdell, Noll, 1965]

Continuous

Medium

local

action

nonlocal

actionnonlocal theory: integral formulation [Eringen, 1972]

Introduction 6/53

Mechanics of generalized continua

Simple material: A material is simple at the particle X if and only if itsresponse to deformations homogeneous in a neighborhood of Xdetermines uniquely its response to every deformation at X .


Continuous

Medium

local

action

nonlocal


simplematerialF∼

non simplematerial

Cauchy medium (1823)(classical / Boltzmann)

Introduction 7/53

Mechanics of generalized continuaSimple material: A material is simple at the particle X if and only if itsresponse to deformations homogeneous in a neighborhood of Xdetermines uniquely its response to every deformation at X .


Continuous

Medium

local

action

nonlocal


simplematerialF∼

non simplematerial

Cauchy continuum (1823)(classical / Boltzmann)

mediumof order n

mediumof grade n

Cosserat (1909)u ,R

∼

micromorphic[Eringen, Mindlin 1964]u ,χ

∼

second gradient[Mindlin, 1965]F∼,F∼⊗∇

gradient of internalvariable [Maugin, 1990]u ,α

Introduction 8/53

Plan




4 Discussion

Power of internal forces

• Model variables according to a first gradient theory

MODEL = v , v ⊗∇, p, ∇p velocity v and cumulative plastic strain p are assumed to beindependent degrees of freedom

• Virtual power of internal forces of a subdomain D ⊂ B of thebody

P(i)(v ∗, p∗) =

∫D

p(i)(v ∗, p∗) dV

simple stress tensor σ∼, generalized stresses a (unit MPa), b (unit

MPa.mm), microforces according to [Gurtin, 2002]

• The virtual power density of internal forces is a linear form onthe fields of virtual modeling variables

p(i) = σ∼ : (v ∗ ⊗∇) + a p∗ + b ·∇p∗

• The virtual power density of internal forces is invariant with respect to

superimposed rigid body motion ⇒ σ∼ is symmetric [Germain, 1973a]

Strain gradient plasticity theory 11/53

Power of contact forces

• Application of Gauss theorem to the power of internal forces∫D

p(i) dV =

∫∂D

v ∗ · σ∼ · n dS +

∫∂D

p∗ b · n dS

−∫D

v ∗ · σ∼ ·∇ dV −∫D

p∗ (b ·∇− a) dV

The form of the previous boundary integral dictates the formof the

• power of contact forces acting on the boundary ∂D of thesubdomain D ⊂ B

P(c)(v ∗, p∗) =

∫∂D

p(c)(v ∗, p∗) dS

p(c)(v ∗, p∗) = t · v ∗ + ac p∗

simple traction t (unit MPa), double traction ac (unitMPa.mm)


Power of forces acting at a distance

P(e)(v ∗, p∗) =

∫D

p(e)(v ∗, p∗) dV

p(e)(v ∗, p∗) = f · v ∗ + c∼ : (v ∗ ⊗∇) + ae p∗ + b e ·∇p∗

simple body forces f (unit N.mm−3), double body forces c∼ and ae

(unit N.mm−2), triple body force b e (unit N.mm−1)

P(e)(v ∗, p∗) =

∫∂D

(v · c∼ · n + p∗b e · n

)dS

−∫D

(v ∗ · (c∼ ·∇− f ) dV + p∗(b e ·∇− ae)

)dV


Principle of virtual powerIn the static case, ∀v ∗,∀p∗,∀D ⊂ B,

P(i)(v ∗, p∗) = P(c)(v ∗, p∗) + P(e)(v ∗, p∗)

[Germain, 1973b]


Principle of virtual powerIn the static case, ∀v ∗,∀p∗,∀D ⊂ B,

P(i)(v ∗, p∗) = P(c)(v ∗, p∗) + P(e)(v ∗, p∗)

which leads to∫∂D

v ∗ · (t − (σ∼ − c ) · n ) + p∗(ac − (b − b e) · n ) dS

+

∫D

v ∗ · ((σ∼ − c∼) ·∇ + f ) + p∗((b − b e) ·∇− a + ae) dV = 0

cf. [Germain, 1973b, Forest and Sievert, 2003,Kirchner and Steinmann, 2005, Lazar and Maugin, 2007,Hirschberger et al., 2007]


Balance and boundary conditions

The application of the principle of virtual power leads to the

• balance of momentum equation (static case)

(σ∼ − c∼) ·∇ + f = 0, ∀x ∈ B

• balance of generalized moment of momentum equation (staticcase)

(b − b e) ·∇− a + ae = 0, ∀x ∈ B

• boundary conditions

(σ∼ − c∼) · n = t , ∀x ∈ ∂B

(b − b e) · n = ac , ∀x ∈ ∂B


Plan




4 Discussion

Continuum thermodynamics

• Enhance the local balance of energy and the entropy inequality

ρε = p(i) − div q + ρr , −ρ(ψ + ηT ) + p(i) −q

T.∇T ≥ 0

• Decomposition of total strain

v = u , ε∼ =1

2(u ⊗∇ + ∇⊗ u )

ε∼ = ε∼e + ε∼

p

• Consider the constitutive functionals:

ψ = ψ(ε∼e ,T , p, α,∇p), η = η(ε∼

e ,T , p, α,∇p)

σ∼ = σ∼(ε∼e ,T , p, α,∇p)

a = a(ε∼e ,T , p, α,∇p), b = b (ε∼

e ,T , p, α,∇p)


State laws

• Clausius–Duhem inequality (isothermal)

(σ∼−ρ∂ψ

∂ε∼e) : ε∼

p+(a−ρ∂ψ∂p

)p+(b−ρ ∂ψ

∂∇p)·∇p+σ∼ : ε∼

p−ρ∂ψ∂α

α ≥ 0

• Derive the state laws [Coleman and Noll, 1963]

σ∼ = ρ∂ψ

∂ε∼e, X = ρ

∂ψ

∂α, R =

∂ψ

∂p, b =

∂ψ

∂∇p

• Residual dissipation

Dres = σ∼ : ε∼p + (a− R)p − X α ≥ 0


Flow rule and evolution law

• Introduce an equivalent stress measure σeq such that

σ∼ : ε∼p = σeqp

which defines the cumulative plastic strain rate p

Dres = (σeq + a− R)p − X α ≥ 0

• Introduce the viscoplastic potential Ω(σeq + a− R,X ) suchthat

ε∼p =

∂Ω

∂(σeq + a− R), α = −∂Ω

∂X

Ω convex with respect to the first, concave with respect to the second

variable for positive dissipation

• Rate–independent case; introduce the yield function

f (σ∼ , a,R) = σeq − R0 − R + a

where R0 is the initial yield stress

ε∼p = p

∂f

∂σ∼, σeq = σ∼ :

∂f

∂σ∼Strain gradient plasticity theory 20/53

Specific constitutive equations

• Free energy function

ρψ(ε∼e , p,∇p) =

1

2ε∼

e : Λ≈

: ε∼e +

1

2Hp2 +

1

2∇p · A∼ ·∇p

• State laws

σ∼ = Λ≈

: ε∼e , R = Hp, b = A∼ ·∇p

• Balance of generalized momentum (homogeneous material)

a = div b = div (A∼ ·∇p) = A∼ : (∇⊗∇p)

• Yield function f (σ∼ ,R, a) = J2(σ∼)− R0 − Hp + a

• Consistency condition; p is solution of a p.d.e.

(H +∂f

∂σ∼: Λ≈

:∂f

∂σ∼)p + A∼ : (∇⊗∇p) =

∂f

∂σ∼: Λ≈

: ε∼

• Isotropic case + von Mises : Aifantis model [Aifantis, 1987]

A∼ = c21∼, σeq = R0 + Hp − c2∆p


Plan




4 Discussion

Confined plasticity

[Ashby, 1970]

start animate end

periodic simple shear test:classical solution

conventional continuum plasticity predicts homogenenous plasticdeformation in the layers


Confined plasticity

[Ashby, 1970]

start animate end

periodic shear test

Additional interface conditions associated with strain gradientplasticity induce size–dependent non–homogenous deformation


Laminate microstructure under shearUnit cell of a periodic two–phase laminate l = s + h

1

2

O

s hAifantis material in the white (soft) phase, purely elastic gray(hard) phase

• Form of the solution for imposed mean shear γ

u1 = γ x2, u2(x1) = u(x1), u3 = 0

unknown periodic functions u(x1), p(x1)

• Deformation gradient and strain

[∇u ] =

0 γ 0u,1 0 00 0 0

, [ε∼]

=

0 12(γ + u,1) 0

12(γ + u,1) 0 0

0 0 0


Resolution of the b.v.p.Let us consider homogeneous isotropic elasticity and no hardeningin the plastic phase for simplicity

• Equilibrium: homogeneous shear stress σ12 throughout the laminate

• Displacement in the hard phase

σ12 = µ(γ + uh,1) =⇒ uh

,1 = C , uh = Cx1 + D

• Plastic strain in the soft phase

ε∼p =

3

2p

s∼J2(σ∼)

, ε∼p =

√3

2p(e 1 ⊗ e 2 + e 2 ⊗ e 1)

from the yield condition we get

σ12 = µ(γ + uh,1) =⇒ uh

,1 = C , uh = Cx1 + D

so that the plastic strain is parabolic

p = α(x21 −

s2

4)

• Continuity of plastic strain at the interface p(±s/2) = 0


Resolution of the b.v.p.

• Displacement in the soft phase

σ12 = µ(γ + us,1 −

√3p) =⇒ us

,1 = C +√

3p

us = (C − α√

3s2

4)x1 + α

√3

3x31


Interface conditions

• Displacement continuity at x1 = ±s/2

us(s

2) = uh(

s

2) =⇒ −

√3α

s3

12= D

• Displacement periodicity at x1 = −s/2 and x1 = s/2 + h

us(− s

2) = uh(

s

2+ h) =⇒

√3α

s3

12= Cl + D

• Continuity of the stress vector at x1 = ±s/2

R0 − 2cα = µ√

3(γ + C )

• The wanted constants are deduced from the previousequations

C =R0 −

√3µγ

√3µ+

12cl√3s3

, D = −Cl

2, α = − 12√

3

D

s3


Plastic strain profile in the channel

b1/lR0

u2/lγp/γ

x/l

0.60.40.20-0.2

1.2

1

0.8

0.6

0.4

0.2

0

-0.2

µ (MPa) R0 (MPa) c (MPa.mm2) f l (µm) γ

30000 20 0.005 0.7 10 0.01

• Characteristic length: lc =√

c/µ = 0.4 µm, leading to strongsize effects in the micron range and below


Plastic strain profile in the channel

b1/lR0

u2/lγp/γ

x/l

0.60.40.20-0.2

1.2

1

0.8

0.6

0.4

0.2

0

-0.2

µ (MPa) R0 (MPa) c (MPa.mm2) f l (µm) γ

30000 20 0.005 0.7 10 0.01

• The higher order stress b1 = 2cα experiences a jump at theinterface s = ±s/2:

b1(s+

2)− b1(

s−

2) = 0− cαs, [[b1]](

s

2) = −cαs


Overall size effect

• Macroscopic stress strain relation

σ12

µ=

1

µfs2 + 4c

(√3

3fs2R0 + 4c γ

)bilinear response depending explicitly on channel size s

• Macroscopic stress vs mean plastic strain;

p =1

l

∫ s/2

−s/2p(x1) dx1 =⇒

√3p = f γ − C (1− f )− f

σ12

µ

σ12 =R0√

3+

4√

3c

f 3l2p

microstructure–induced linear hardening depending on unitcell size l

• Limit cases? thick channels: size independent threshold σ12 = R0/

√3

? thin films: scaling law σ12/p ∼ 1/l2


Plan




4 Discussion

General scalar microstrain gradient plasticity

• Classical and generalized plasticity

DOF = u , pχ STATE = ε∼e , p, α, pχ, ∇pχ

scalar plastic microstrain variable pχ

• Enhanced power of internal forces and extra balance equation

p(i) = σ∼ : ε∼ + a pχ + b .∇ pχ, p(c) = t .u + ac pχ

div b − a = 0, ∀x ∈ Ω, b .n = ac , ∀x ∈ ∂Ω

Micromorphic plasticity theory 34/53

General scalar microstrain gradient plasticity

• State lawsε∼ = ε∼

e + ε∼p

σ∼ = ρ∂ψ

∂ε∼e, R = ρ

∂ψ

∂p, X = ρ

∂ψ

∂α

a = ρ∂ψ

∂pχ+ av , b = ρ

∂ψ

∂∇pχ

• Evolution laws Dres = σ∼ : ε∼p + (av − R)p − X α ≥ 0

ε∼p = λ

∂f

∂σ∼, p = −λ ∂f

∂R, α = −λ ∂f

∂X

[Forest, 2009]


Explicit constitutive equations

• Quadratic free energy potential

ρψ(ε∼e , p, pχ,∇pχ) =

1

2ε∼

e : Λ≈

: ε∼e+

1

2Hp2+

1

2Hχ(p−pχ)2+

1

2∇pχ.A∼.∇pχ

[Forest, 2009, Dimitrijevic and Hackl, 2011]

• Constitutive equations

σ∼ = Λ≈

: ε∼e , a = −Hχ(p−pχ), b = A∼ .∇pχ, R = (H+Hχ)p−Hχpχ

• Substitution of constitutive into extra balance equations

pχ −1

Hχdiv (A∼ .∇pχ) = p

• Homogeneous and isotropic materials A∼ = A1∼

pχ −A

Hχ∆pχ = p, b.c. b · n = A∇pχ.n = ac

same p.d.e. as in the implicit gradient–enhancedelastoplasticity with ac = 0 [Engelen et al., 2003]


Link to Aifantis strain gradient plasticity

• Yield functionf (σ∼ ,R) = σeq − R0 − R

• Hardening law

R =∂ψ

∂p= (H + Hχ)p − Hχpχ

• Under plastic loading

σeq = R0 + Hpχ − A(1 +H

Hχ)∆pχ

compare with Aifantis model [Aifantis, 1987]

σeq = R0 + R(p)− c2∆p

The equivalence is obtained for Hχ = ∞ (internal constraint):

pχ ' p, A = c2


Plan




4 Discussion

Consistency condition

• Consistency condition

f =∂f

∂σ∼: σ∼ + +

∂f

∂RR

=∂σeq

∂σ: Λ≈

: (ε∼− ε∼p)− ∂R

∂pp − ∂R

∂pχpχ = 0

• Plastic multiplier

p =

N∼ : Λ≈

: ε∼−∂R

∂pχpχ

N∼ : Λ≈

: N∼ +∂R

∂p

, with N∼ =∂σeq

∂σ∼

where ε∼ and pχ are controllable variable.

• Even though the yield condition can be written as a partial differentialequation, there is no need for a variational formulation of the consistencycondition contrary to [Muhlhaus and Aifantis, 1991, Liebe et al., 2001].There is no need for a plastic front tracking technique. The plasticmicrostrain pχ and the generalized traction b .n are continuous across theelastic/plastic domain.


Thermal effects

• For temperature dependent parameters

a = div b = div (A∇pχ) = A∆pχ +∂A

∂T∇T .∇pχ

pχ −A

Hχ∆pχ −

1

Hχ

∂A

∂T∇T .∇pχ = p

• Consistency condition

p =

N∼ : Λ≈

: (ε∼− ε∼th)− ∂R

∂pχpχ −

∂R

∂TT

N∼ : Λ≈

: N∼ +∂R

∂p


Plan




4 Discussion

Laminate microstructure under shearUnit cell of a periodic two–phase laminate l = s + h

1

2

O

s hMicromorphic material in the white (soft) phase, purely elasticmicromorphic gray (hard) phase

• Form of the solution for impose mean shear γ

u1 = γ x2, u2(x1) = u(x1), u3 = 0

unknown periodic functions u(x1), p(x1), pχ(x1)

• Deformation gradient and strain

[∇u ] =

0 γ 0u,1 0 00 0 0

, [ε∼]

=

0 12(γ + u,1) 0

12(γ + u,1) 0 0

0 0 0


Resolution of the b.v.p.Let us consider homogeneous isotropic elasticity, homogeneous Hχ

and no hardening in the plastic phase for simplicity

• The shear stress is uniform throughout the laminate and takesthe value

√3σ12 = R0 + R = R0 + Hχ(p − pχ) = R0 − Apχ,11

• Derivation of the previous equations with respect to x1 showsthat pχ,111 = 0 which leads to the parabolic profile of themicro–plastic deformation in the soft phase

pχ(x) = αx2 + β, ∀|x | ≤ s

2

Note that √3σ12 = R0 − 2Aα

• The parabolic plastic strain profile follows

p = αx2 + β − 2A

Hχ


Resolution of the b.v.p.

A new feature of the model is that the microplastic strain pχ doesnot vanish in general in the hard phase, whereas p does:

pχ −Ah

Hχ∆pχ = 0

phχ = αh coshωh(x −

l

2),

s

2≤ x ≤ s

2+ h, with ω2

h =Hχ

Ah

the phχ profile is of hyperbolic nature


Interface conditions

• Continuity of micro–plastic deformation at x = s/2:

αs2

4+ β = αh coshωh

h

2

• Continuity of the generalized stress component b1:

Aαs = −Ahαhωh sinhωhh

2


Interface conditionsThe displacement in the plastic and elastic phases can be expressed as

us = αx3

√3

+

„√3β − γ +

R0√3µ− 2Aα(

1√3µ

+

√3

Hχ)

«x

uh =

„1√3µ

(R0 − 2Aα)− γ«

x + C

They are used to exploit two additional interface conditions

• Continuity of the displacement at x = s/2:us(s/2) = uh(s/2)

αs3

8√

3µ+√

3(β − 2Aα

Hχ)s

2= C

• Periodicity of the displacement componentus(−s/2) = uh(s/2 + h)

−(σ12

µ− γ)l +

√3(β +

2Aα

Hχ)s

2− α

s3

8√

3= C


Plastic strain profiles in the channel

b1/lR0

u2/lγp/γ

pχ/γ

x/l

0.60.40.20-0.2

1.2

1

0.8

0.6

0.4

0.2

0

-0.2

µ (MPa) R0 (MPa) Hχ (MPa) A (MPa.mm2) f l (µm) γ

30000 20 50000 0.005 0.7 10 0.01


Overall size effect

The scaling law results from the expression of the overall stress σ12

as a function of the mean plastic strain over the unit cell:

p =1

l

∫ s2

− s2

(αx2 + β − 2Aα

Hχ) dx = βf

(1− 1

L2

(s2

12− 2A

Hχ

))

with L2 =s2

4+

A

Ah

s

ωhcotanh(ωh

h

2) = −β

α. The uniform stress

component can now be expressed as a function of the volumefraction f of the soft phase and of the unit cell size l :

√3σ12 = R0 +

2A

f

p

f 2l2

6+

2A

Hχ+

A

Ah

fl

ωhcotanh (ωh

h

2)

displaying a size–dependent overall linear hardening


Scaling lawsTwo limit cases naturally arise

• Internal constraint Hχ →∞ for which the strain gradientplasticity model is retrieved

• Unit cell size l → 0 leads to saturation stress√

3σ12 − R0 ∼ Hχ1− f

fp

micromorphic modelAifantis model

l (mm)

σ 12−

R0/√

3(M

Pa)

1010.10.010.0010.00011e-051e-06

1e+10

1e+08

1e+06

10000

100

1

0.01

0.0001

1e-06


Plan




4 Discussion

Main comments

• The choice of quadratic potentials with respect to straingradients leads to the existence of a size dependent overalllinear hardening modulus in a laminate microstructure

• The corresponding scaling law according to Aifantis straingradient plasticity (one new material parameter) is 1/l2

• In contrast the micromorphic model (two new materialparameters) leads to a saturation at nano–scales

• Physical metallurgy hints rather at scaling laws of the Orowan(1/l) or Hall-Petch (1/

√l) types

Discussion 51/53

Needed improvements

• Improve constitutive equations; for instance

ρψ(∇p) =√

∇p · A∼ ·∇p

[Conti and Ortiz, 2005, Okumura et al., 2007]

• Add higher order dissipative parts[Forest, 2009, Anand et al., 2012]

• Enhance interface conditions[Gurtin and Needleman, 2005, Acharya, 2007,Gurtin and Anand, 2008]

Discussion 52/53

Extension to crystal plasticity• In crystal plasticity, the relevant variable is not the gradient of the

cumulative plastic strain but rather the dislocation density tensor

Γ∼ = −curlH∼p, H∼ = u ⊗∇ = H∼

e + H∼p

[Cermelli and Gurtin, 2001, Svendsen, 2002]but no effects in laminates...

• A strain gradient and a micromorphic theory can be designed based onthe introduction of the dislocation density tensor in the free energyfunction [Aslan et al., 2011]

• Similar effects arise in laminate microstructures but the overall linearhardening is of kinematic nature

x = curl curlH∼p : m ⊗ n

[Cordero et al., 2010, Cottura et al., 2012]

• Hall–Petch related size effects can be predicted in polycrystals based onfull field simulations[Forest et al., 2000, Bayley et al., 2007, Neff et al., 2009,Bargmann et al., 2010, Cordero et al., 2012,Wulfinghoff and Boehlke, 2012]

• Constitutive equations are still unrealistic...

Discussion 53/53

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Anand L., Aslan O., and Chester S.A. (2012).A large–deformation gradient theory for elastic–plasticmaterials: Strain softening and regularization of shear bands.International Journal of Plasticity, vol. 30–31, pp 116–143.

Aslan O., Cordero N. M., Gaubert A., and Forest S. (2011).Micromorphic approach to single crystal plasticity and damage.International Journal of Engineering Science, vol. 49, pp1311–1325.

Bargmann S., Ekh M., Runesson K., and Svendsen B. (2010).

Discussion 53/53

Modeling of polycrystals with gradient crystal plasticity: Acomparison of strategies.Philosophical Magazine, vol. 90, pp 1263–1288.

Bayley C.J., Brekelmans W.A.M., and Geers M.G.D. (2007).A three–dimensional dislocation field crystal plasticityapproach applied to miniaturized structures.Philosophical Magazine, vol. 87, pp 1361–1378.

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Conti S. and Ortiz M. (2005).

Discussion 53/53

Dislocation Microstructures and the Effective Behavior ofSingle Crystals.Arch. Rational Mech. Anal., vol. 176, pp 103–147.

Cordero N.M., Gaubert A., Forest S., Busso E., Gallerneau F.,and Kruch S. (2010).Size effects in generalised continuum crystal plasticity fortwo–phase laminates.Journal of the Mechanics and Physics of Solids, vol. 58, pp1963–1994.

Cordero N. M., Forest S., and Busso E. P. (2012).Generalised continuum modelling of grain size effects inpolycrystals.Comptes Rendus Mecanique, vol. 340, pp 261–274.

Cottura M., Le Bouar Y., Finel A., Appolaire B., Ammar K.,and Forest S. (2012).A phase field model incorporating strain gradientviscoplasticity: application to rafting in Ni-base superalloys.

Discussion 53/53

Journal of the Mechanics and Physics of Solids, vol. 60, pp1243–1256.

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