On stress-gradient materials: formulation and homogenization

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On stress-gradient materials: formulation and homogenization

Sébastien Brisard, Vinh Phuc Tran, Karam Sab, Johann Guilleminot

To cite this version:

Sébastien Brisard, Vinh Phuc Tran, Karam Sab, Johann Guilleminot. On stress-gradient materials:

formulation and homogenization. Encounter of the third kind on generalized continua and microstructures, Apr 2018, Arpino, Italy. �hal-01758985�

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S. Brisard¹, V.P. Tran^1,2, K. Sab¹, J. Guilleminot³

On stress-gradient materials:

formulation and homogenization

4 April 2018

Encounter of the Third Kind — Generalized continua and microstructures

1Université Paris-Est, Laboratoire Navier, UMR 8205, CNRS, ENPC, IFSTTAR, F-77455 Marne-la-Vallée, France

2Université Paris-Est, Laboratoire Modélisation et Simulation Multi Échelle (MSME), UMR 8208, CNRS, F-77454 Marne-la-Vallée, France

3Department of Civil and Environmental Engineering, Duke University, Durham, NC 27708, USA

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Stress-gradient elasticity in a nutshell

The Stress Principle of Cauchy

σ⋅∇+b=0 and ⟦σ⟧⋅n=0 (SA) Cauchy elasticity

W^c[σ]=∫ ^w^c⁽^σ⁾^d ^V ^W ^c^[^σ^]=∫ ^w^c⁽^σ ^, ^σ^⊗∇)^d ^V

Stress-gradient elasticity

Forest and Sab (2012), Mechanics Research Communications 40 (first derivation) Sab, Legoll and Forest (2016), Journal of Elasticity 123(2) (mathematical justification) Forest and Sab (2017), Mathematics and Mechanics of Solids (extension to finite strain)

Polizzotto (2014, 2016), International Journal of Solids and Structures

Alternative model (not discussed here) A mature model

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In the present talk

A brief overview of the stress-gradient model

►Decomposition of the stress-gradient

►Stress boundary conditions

Homogenization of stress-gradient materials

►Hill–Mandel lemma, boundary conditions

►Softening size effect

Application to composites with spherical inclusions

►Eshelby’s inhomogeneity problem

►Mori–Tanaka estimates of effective compliance Some open questions

Tran, Brisard, Guilleminot and Sab (2018), International Journal of Solids and Structures

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Derivation of the stress-gradient model for elasticity

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Roadmap

Minimize complementary stress energy under SA constraint

►Cauchy elasticity: elastic equilibrium of fixed solid is retrieved

►Stress-gradient elasticity: elastic equilibrium of fixed solid is defined

►At this point, “fixed” not really meaningful, since dofs not (yet) defined

►“Trace” of the stress-gradient is prescribed

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Decomposition of stress-gradient (1/2)

W^c[σ]=∫ ^w^c⁽^σ ^, ^σ^⊗∇)^d ^V

Classical equilibrium equation (σ⊗∇):I₂=σ⋅∇=−b

Orthogonal decomposition of stress gradient σ⊗∇=Q+R with R :I₂=0 and Q∴R=0 R=I'₆∴(σ⊗∇)

Complementary stress energy of stress-gradient materials

W^c[σ]=∫ ^w^c⁽^σ ^, ^Q^,^R⁾^d^V

Equivalent expression of complementary stress energy

(projection onto space of trace-free, third-rank tensors)

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Decomposition of stress-gradient (2/2)

Modeling assumption Q=−¹₂ I₄⋅b

W^c[σ]=∫ ^w^c⁽^σ ^, ^Q^,^R⁾^d^V ⁼∫ ^w^c⁽^σ ^,^R⁾^d ^V

No strain measure attached to Q (generalized prestress) Orthogonal decomposition of stress gradient

σ⊗∇=Q+R with R :I₂=0 and Q∴R=0 Q is in fact fully prescribed

(straightforward linear algebra)

with R=I' ∴(σ⊗∇)

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Constrained minimization of energy

Initial problem

Stress-gradient as independent variable

Minimize: W^c[σ]=∫ ^w^c⁽^σ ^,^I^'⁶^∴⁽^σ^⊗∇))^d ^V

subject to: σ⋅∇+b=0

Minimize: W^c[σ ,R]=∫ ^w^c⁽^σ ^,^R⁾^d ^V

subject to: {^R⁼^σ^I^{⋅∇ +}^'⁶^∴⁽^b^σ⁼^⊗∇)⁰ ^∴^⋅^u^Φ

Lagrange multipliers

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Elastic equilibrium of fixed, SG bodies

Field equations

R=I'₆∴(σ⊗∇) Φ=∂_R w^c

e=∂_σ w^c σ⋅∇+b=0

e=Φ⋅∇ +ε[u]

Generalized stress-strain relations Generalized strain-displacement relations Generalized equilibrium equations

Boundary conditions

Φ⋅n+sym(u⊗n)=0 6 scalar boundary conditions

Continuity conditions

⟦Φ⋅n+sym(u⊗n)⟧=0

⟦σ⟧=0

Weaker continuity of “displacements”

Stronger continuity of “stresses”

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Linear elasticity

Complementary stress energy density w^c(σ ,R)=¹₂ σ :S:σ+¹₂ R ∴M∴R

No σ–R coupling for centrosymmetric materials!

Stress-strain relationships e=S∴σ and Φ=M∴R

S and M have major and minor symmetries

Generalized compliance M operates on trace-free tensors M=I'₆∴M∴I'₆

or

σ=C ∴e and R=L∴Φ

C=S⁻¹ and L=M⁺

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Isotropic linear elasticity

General isotropic linear stress-gradient elasticity

I'₆=J₆+K₆

Simplified model for isotropic linear stress-gradient elasticity 2 μ S=1−2 ν

1+ν J₄+K₄ 2 μ

ℓ² M=1−2 ν

1+ν J₆+K₆

0 J₆ 0

J₆

H₆ K₆

K₆ 0

H₆ H₆

H₆ 0

H₆ K₆

J₆ 2 μM=ℓ²_J J₆+ℓ²_K K₆+ℓ²_H H₆ ∴

M=I'₆∴M∴I'₆

Altan and Aifantis (1992), Scripta Metallurgica et Materialia 26(2)

Altan and Aifantis (1997), Journal of the Mechanical behavior of Materials 8(3)

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Homogenization

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Setting the stage for homogenization

The 3 cases

►Stress-gradient (micro)→stress-gradient (macro)

►Cauchy (micro)→stress-gradient (macro)

►Stress-gradient (micro)→Cauchy (macro) Separation of scales

►Classical condition

d≪L_meso≪L_macro

Microstructure RVE Structure

►Additional condition for stress-/strain- gradient materials ℓ∼d or ℓ≪d

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Effective and apparent properties

Effective behaviour: stress-gradient →Cauchy ₍ℓ∼d)

⟨e⟩=S^eff:⟨σ⟩

⟨σ⟩= 1 V ∫

Ω

σ d V

⟨e⟩= 1 V ∫

Ω

e d V = 1 V ∫

Ω

ε[u]dV + 1 V ∫

∂Ω

Φ⋅n d S

σ⋅∇=0

e=S∴σ Φ=M∴(σ⊗∇) e=Φ⋅∇ +ε[u]

+ boundary conditions!

⟨e⟩=S^app(Ω):⟨σ⟩

Local problem on SVE Ω

Apparent elastic properties of (large) SVE Ω

No body forces!

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Generalized Hill–Mandel lemma

⟨σ^*:e+(σ^*⊗∇)∴Φ⟩=⟨σ^*⟩:⟨e⟩

Uniform stress boundary conditions

σ|_∂_Ω=σ ⟨e⟩=S^σ (Ω):⟨σ⟩=S^σ (Ω):σ

Uniform strain boundary conditions

Φ⋅n+sym(u⊗n)=sym[(e⋅x)⊗n] ⟨σ⟩=C^ε(Ω):⟨e⟩=C^ε(Ω):e Uniform traction boundary conditions

σ|_∂Ω⋅n=σ⋅n ⟨e⟩=S^T(Ω):⟨σ⟩=S^T (Ω):σ a⋅[Φ⋅n+sym(u⊗n)]⋅a=0 a=I −n⊗n

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Variational formulation

Minimum of complementary stress energy

σ :S^σ (Ω):σ=inf {⟨σ:S:σ+(σ⊗∇)∴M∴(σ⊗∇ )⟩}

subject to: σ⋅∇=0 and σ|_∂Ω=σ Minimum of strain energy

e:C^ε(Ω):e=inf {⟨ε[u]:C :ε[u]+Φ∴L∴Φ⟩}

subject to: Φ⋅n+sym(u⊗n)=sym[(e⋅x)⊗n] Bounds for finite-size SVEs

C^σ (Ω_I)≤C^σ(Ω_II)≤C^eff≤C^ε(Ω_II)≤C^ε(Ω_I) for Ω_I⊂Ω_II

Huet (1990), Journal of the Mechanics and Physics of Solids 38(6)

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Softening size-effect

Variational definition of apparent compliance (uniform σ BC)

Fixed microstructure, variable material internal length

Scaled microstructure, constant material internal length If S_I≤S_II and M_I≤M_II everywhere in Ω , then S_I^eff≤S_II^eff

For S_I=S_II and M_I= ℓ_I²

ℓ_II² M_II: if ℓ_I≤ℓ_II then S_Iêff≤S_IIêff More generally: if ( ^d^ℓ )Î^≤( ^d^ℓ )ÎI ^then ^SÎêff^≤^SÎIêff

If d ≤d then C^eff≤C^eff Softening size-effect!

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Eshelby’s inhomogeneity problem for stress-gradient elasticity

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Problem setting

σ^∞ e_z

−σ^∞ e_z

Matrix: μ_m , ν_m , ℓ_m μ_i, ν_i , ℓ_i

ν_i=ν_m=0.25 μ_i=10 μ_m

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Some tedious calcs later…

Hat tip to VP Tran!

http://www.sympy.org

Axial stress along vertical axis Axial stress at origin

Theorem of Eshelby (1957) does not hold!

Eshelby (1957). Proc. Royal Soc. A 241(1226)

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Dilute stress concentration tensor

⟨σ⟩_i= 1

V ∫^σ ^d ^V ⁼^B^∞^:^σ^∞

Axial stress along vertical axis Axial stress at origin

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Mori–Tanaka estimates for composites with spherical inclusions

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Problem setting

μ_m, ν_m=0.25, ℓ_m

μ_i=10 μ_m , ν_i=0.25 , ℓ_i

Volume fraction of inclusions: f

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Estimates of bulk and shear moduli

Strain- and stress- gradient models are not equivalent!

Effective bulk modulus Effective shear modulus

S_eff=S_m+f (S_i−S_m):B^∞:[(1−f )I₄+f B^∞]⁻¹

Benveniste (1987), Mechanics of Materials 6(2) Ma and Gao (2014), Acta Mechanica 225(4-5)

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Some questions

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The total complementary energy

Ω

u b T

∂Ω_u ∂Ω_T Assumption 1: Stress Principle of Cauchy

σ⋅∇=0 and σ⋅n|_∂_Ω=T

Assumption 2: complementary work of prescribed displacements

V^c=∫

∂Ω_u

u⋅σ⋅nd S

Minimization of complementary energy Minimize: Π^c=W^c−V ^c

subject to: σ⋅∇=0 and σ⋅n| =T

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The general BVP

Field equations

R=I'₆∴(σ⊗∇) Φ=∂_R w^c

e=∂_σ w^c σ⋅∇+b=0

e=Φ⋅∇ +ε[u]

∂Ω_T :

{

â^⋅[^Φ^⋅ⁿ⁺^σ^sym^⋅ⁿ^|^∂^Ω⁽û⁼^⊗^Tⁿ^)]⋅â⁼⁰

Continuity conditions

⟦Φ⋅n+sym(u⊗n)⟧=0

⟦σ⟧=0

∂Ω_u: Φ⋅n+sym(u⊗n)=sym(u⊗n)

a=I₂−n⊗n

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The total potential energy

Potential strain energy

Work of prescribed body forces and tractions V =∫

Ω

b⋅u d V + ∫

∂Ω_T

T⋅[u+2Φ :n⊗n−(Φ∴ n⊗n⊗n)n]d S

Minimization of total potential energy Minimize: Π=W −V

W=∫ ^w⁽^e ^,^Φ⁾^d ^V

subject to:

{

^∂^Ω^∂û^Ω^:^T ^:^Φ^⋅âⁿ^⋅[⁺^Φ^sym^⋅ⁿ⁺⁽^symû^⊗ⁿ⁽⁾⁼û^⊗^symⁿ^)]⋅â⁽û⁼^⊗⁰ⁿ⁾

u+2Φ:n⊗n−(Φ∴ n⊗n⊗n)n=u

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Conclusion & perspectives

Summary

►Simplified model for isotropic linear elasticity

►General framework for stress-gradient→Cauchy homogenization

►Softening size-effect

►Mori–Tanaka estimates for spherical inclusions Outlook

►Hashin–Shtrikman (size-dependent) bounds on effective properties

►Cauchy→stress-gradient homogenization

►Understand the meaning of u (use above)

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Thanks for your attention!

[email protected] http://navier.enpc.fr/BRISARD-Sebastien http://sbrisard.github.io

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This work has benefited from a French government grant managed by ANR within the frame of the national program Investments for the Future ANR-11-LABX-022-01.