Stress-gradient materials: an analytical exploration

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Stress-gradient materials: an analytical exploration

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

To cite this version:

Vinh Phuc Tran, Sébastien Brisard, J Guilleminot, Karam Sab. Stress-gradient materials: an analytical exploration. Workshop on Computational Mechanics of Generalized Continua and Applications to Materials with Microstructure, The François COSSERAT–Tullio LEVI CIVITA (Coss &Vita) In- ternational Associated Laboratory, Oct 2015, Catania, Italy. �hal-01226290�

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Workgroup on Computational Mechanics of Generalized Continua and Applications to Materials with Microstructure

Stress-gradient materials:

an analytical exploration

V.P.Tran^1,2, S.Brisard¹, J.Guilleminot², K.Sab¹

1Laboratoire NAVIER (UMR CNRS 8205), Universit´e Paris-Est, ENPC, IFSTTAR

2Laboratoire MSME (UMR 8208 CNRS), Universit´e Paris-Est

Catania, 29-31 October, 2015

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Motivation (1)

At first glance...

Strain gradient elasticity theory

w = w(ε,∇ε)

strain energy density

strain

strain gradient

Mindlin, R.D., Eshel, N.N., 4, pp. 109–124, International Journal of Solids and Structures, 1968

V.P.Tran, S.Brisard, J.Guilleminot, K.Sab Motivation Catania, 29-31 October, 2015 1 / 13

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Motivation (2)

At first glance...

w = w(ε,∇ε) Stress gradient elasticity theory

w^∗= w^∗(σ,∇σ)

stress energy density

stress stress gradient

S.Forest, K. SabMechanics Research Communications, 40, pp. 16–25, 2012

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Motivation (3)

At first glance...

w = w(ε,∇ε) Stress gradient elasticity theory

w^∗= w^∗(σ,∇σ)

Is stress gradient elasticity equivalent or complementary to strain gradient elasticity?

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Objective

Outline

? How does stress gradient elasticity theory (Forest and Sab, 2012) differ from strain gradient one?

Stress gradient elasticity theory (Forest and Sab, 2012).

Closed form solution to Eshelby spherical inhomogeneity problem.

Homogenization of heterogeneous stress-gradient materials (Mori Tanaka estimate)

S. Forest, K. SabMechanics Research Communications, 40, pp. 16–25, 2012 K. Sab, F. Legoll, S. Forestaccepted for publication, Journal of Elasticity, 2015

V.P.Tran, S.Brisard, J.Guilleminot, K.Sab Objective Catania, 29-31 October, 2015 2 / 13

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Decomposition of stress gradient tensor

The spherical and deviatoric part of a third order tensor

Strain gradient tensor

”free”

Stress gradient tensor constrained

∇ ·σ+b= 0 ⇔(∇σ) : δ+b= 0 Decomposition of stress gradient tensor

∇σ = J ∴∇σ +K ∴ ∇σ

spherical projector deviatoric projector

Jijklmn = 1

8(δikδjlδmn+δikδjmδln+δilδjkδmn+δimδjkδln) Kijklmn = Iijklmn−Jijklmn

V.P.Tran, S.Brisard, J.Guilleminot, K.Sab Decomposition of third order tensor Catania, 29-31 October, 2015 3 / 13

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Stress gradient elasticity theory

Linear, isotropic stress gradient materials

Stress energy density function

w^∗ =w^∗(σ,K∴ ∇σ

| {z }

R

)

For isotropic stress gradient materials:

w^∗(σ,R) = 1

2σ :S:σ+ 1

2R∴ M∴ R Constitutive laws

e= ∂w^∗

∂σ = S: σ, Φ = ∂w^∗

∂R =M∴R NOTA: Φ,R have to be deviatoric tensors!

M=K ∴M∴ K

V.P.Tran, S.Brisard, J.Guilleminot, K.Sab Stress gradient elasticity theory Catania, 29-31 October, 2015 4 / 13

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Simplified stress / strain gradient elasticity

A comparison of field equations (no body force)

Simplified stress gradient Simplified strain gradient

Equilibrium σ_ij,j = 0 τ_ij,j = 0

Gradient Rijk=σij,k κijk=εij,k

Constitutive e_ij=S_ijklσ_kl σ_ij=C_ijklε_kl

laws e_ij=ε_ij+ Φ_ijk,k τ_ij=σ_ij−µ_ijk,k

Φ_ijk =`²K_ijkpqrS_pqmnR_mnr µ_ijk=`²C_ijmnκ_mnk

SUBC σij= Σij σijnj−(µijknk),j+(µijknknl),lnj=ti

on∂Ω(smooth part)

µijknjnk=qi on∂Ω(smooth part) [[µijkmjnk]] = 0on the edge of∂Ω

Altan, B.S., Aifantis, E.C., Journal of the Mechanical Behavior of Materials, 8(3), pp. 231–282, 1997 Gao, X.-L., Park, S.K.International Journal of Solids and Structures, 44, pp. 7486–7499, 2007

V.P.Tran, S.Brisard, J.Guilleminot, K.Sab Simplified stress / strain gradient elasticity Catania, 29-31 October, 2015 5 / 13

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Eshelby’s spherical inhomogeneity problem (1)

Geometry + mechanical properties

µi, κi, `i

µm, κm, `m

σ^∞=Σ

2a

Stress gradient (this work):

Uniform isotropic stress:

Σ =e_x⊗e_x+e_y⊗e_y+e_z⊗e_z

Uniform axial stress:

Σ =e_z ⊗e_z Strain gradient:

Gao, X.-L., Ma, H.MActa Mechanica, 207 (3), pp. 163-181, 2009

Gao, X.-L., Ma, H.MJournal of the Mechanics and Physics of Solids, 58 (5), pp. 779-797, 2010

V.P.Tran, S.Brisard, J.Guilleminot, K.Sab Eshelby’s inhomogeneity problem Catania, 29-31 October, 2015 6 / 13

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Eshelby’s spherical inhomogeneity problem (2)

Closed-form solution

Postulated stress form in spherical coordinates (similar to Love’s approach)

Uniform isotropic stress prescribed at boundary:

σ =σ(r) Uniform axial stress prescribed at boundary:

σ =σ(r, θ)

Continuity conditions at the surface matrix-inclusion σm(r =a)= σi(r = a)

[u⊗^s n+Φ·n]_m(r =a) = [u⊗^sn+Φ·n]_i(r =a)

Love, A.E.H.Dover Publications, New York, 1944

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Uniaxial problem Σ = e_z ⊗ e_z

Radial stress field σ_rr(r, θ= 0)

µ_i= 27µ_m, ν_i= 0.4, ν_m= 0.08, `_i=`_m=`

0 1 2 3 4 5

r/a

1.0 1.2 1.4 1.6 1.8 2.0

σrr(r,θ=0)/Σ

`/a= 1

`/a= 0

Classical elasticity Stress gradient

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Effect of material internal length (1)

Effective shear moduli (Mori Tanaka estimate)

µi= 27µm, νi= 0.4, νm= 0.08, `i=`m=`

0.0 0.2 0.4 0.6 0.8 1.0

volume fraction of inclusion 0

5 10 15 20 25

µeff/µm

Strain gradient`↓

Stress gradient`↓ Voigt

Reuss Classical elasticity

Stress gradient (this work) Strain gradient (Ma & Gao)

Stress gradient is complementary to strain gradient elasticity!

Ma, H.M, Gao, X.-L.Acta Mechanica, 225 (4-5), pp. 1075-1091, 2014

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Effect of material internal length (2)

Effective shear moduli: f = 20%

µi= 27µm, νi= 0.4, νm= 0.08, `i=`m=`

0 2 4 6 8 10

radius of inclusionsa 1.20

1.25 1.30 1.35 1.40 1.45 1.50 1.55

µeff/µm

`/a= 1

Classical elasticity Stress gradient

Stress gradient elasticity: capture size effects.

Fora`, stress gradient effects vanish as expected.

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Softening size effect

... has been reported elsewhere (molecular dynamics simulations)!

To cite some:

Polyimide matrix ”reinforced” by silica spherical nanoparticles with various surface treatment.

Odegard, G.M., Clancy, T.C., Gates, T.S.Polymer, 46, pp.553–562, 2005

Polystyrene matrix ”reinforced” by silica spherical nanoparticles.

Davydov, D., et alSoft Materials, 12, S142–S151, 2014

etc

V.P.Tran, S.Brisard, J.Guilleminot, K.Sab Softening size effect Catania, 29-31 October, 2015 11 / 13

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Conclusion & Perspective

Conclusion

Solution to Eshelby’s spherical inhomogeneity problem Mori Tanaka estimate for stress gradient materials

Stress gradient elasticity differs from strain gradient elasticity

Strain gradient: stiffening size effect Stress gradient: softening size effect

Perspective

Numerical implementation: FEM, FFT

Physical interpretation of material internal length?

Acknowledgments

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.

V.P.Tran, S.Brisard, J.Guilleminot, K.Sab Conclusion & Perspective Catania, 29-31 October, 2015 12 / 13

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Thank you for your attention!

[email protected]