# 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) International Associated Laboratory, Oct 2015, Catania, Italy. hal-01226290

HAL Id: hal-01226290

https://enpc.hal.science/hal-01226290

Submitted on 9 Nov 2015

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Distributed under a Creative Commons Attribution 4.0 International License
Stress-gradient materials:
an analytical exploration

V.P.Tran¹,², S.Brisard¹, J.Guilleminot², K.Sab¹

¹Laboratoire NAVIER (UMR CNRS 8205), Université Paris-Est, ENPC, IFSTTAR

²Laboratoire MSME (UMR 8208 CNRS), Université Paris-Est

Catania, 29-31 October, 2015
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## Strain gradient elasticity theory

<figure><img src="image_3.png" /></figure>
<figure><img src="image_4.png" /></figure>

Strain gradient elasticity theory

$$w = w(\varepsilon, \nabla\varepsilon)$$

Stress gradient elasticity theory

$$w^* = w^*(\sigma, \nabla\sigma)$$

<figure><img src="image_5.png" /></figure>
<figure><img src="image_6.png" /></figure>

Strain gradient elasticity theory

$$w = w(\varepsilon, \nabla\varepsilon)$$

Stress gradient elasticity theory

$$w^* = w^*(\sigma, \nabla\sigma)$$

Is stress gradient elasticity **equivalent** or **complementary** to strain gradient elasticity?
<figure><img src="image_7.png" /></figure>

? 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)
<figure><img src="image_8.png" /></figure>

## Strain gradient tensor

- "free"

## Stress gradient tensor

- constrained

$$ \nabla \cdot \sigma + b = 0 \Leftrightarrow (\nabla \sigma) : \delta + b = 0 $$

## Decomposition of stress gradient tensor

<figure><img src="image_9.png" /></figure>

$$ J_{ijklmn} = \frac{1}{8} (\delta_{ik}\delta_{jl}\delta_{mn} + \delta_{ik}\delta_{jm}\delta_{ln} + \delta_{il}\delta_{jk}\delta_{mn} + \delta_{im}\delta_{jk}\delta_{ln}) $$

$$ K_{ijklmn} = I_{ijklmn} - J_{ijklmn} $$
## Stress energy density function

$$w^* = w^*(\boldsymbol{\sigma}, \underbrace{\mathbf{K} \cdot \nabla \boldsymbol{\sigma}}_{\mathbf{R}})$$

For isotropic stress gradient materials:

$$w^*(\boldsymbol{\sigma}, \mathbf{R}) = \frac{1}{2} \boldsymbol{\sigma} : \mathbf{S} : \boldsymbol{\sigma} + \frac{1}{2} \mathbf{R} : \mathbf{M} : \mathbf{R}$$

## Constitutive laws

$$e = \frac{\partial w^*}{\partial \boldsymbol{\sigma}} = \mathbf{S} : \boldsymbol{\sigma}, \quad \boldsymbol{\Phi} = \frac{\partial w^*}{\partial \mathbf{R}} = \mathbf{M} : \mathbf{R}$$

NOTA: $\boldsymbol{\Phi}$, $\mathbf{R}$ have to be deviatoric tensors!

$$\mathbf{M} = \mathbf{K} : \mathbf{M} : \mathbf{K}$$
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# Simplified stress / strain gradient elasticity

## A comparison of field equations (no body force)

<table><thead><tr><th></th><th>Simplified stress gradient</th><th>Simplified strain gradient</th></tr></thead><tbody><tr><td>Equilibrium</td><td>σ<sub>ij,j</sub> = 0</td><td>τ<sub>ij,j</sub> = 0</td></tr><tr><td>Gradient</td><td>R<sub>ijk</sub> = σ<sub>ij,k</sub></td><td>κ<sub>ijk</sub> = ε<sub>ij,k</sub></td></tr><tr><td>Constitutive laws</td><td>e<sub>ij</sub> = S<sub>ijkl</sub>σ<sub>kl</sub><br>e<sub>ij</sub> = ε<sub>ij</sub> + Φ<sub>ijk,k</sub><br>Φ<sub>ijk</sub> = l<sup>2</sup>K<sub>ijklqr</sub>S<sub>pqmn</sub>R<sub>mnr</sub></td><td>σ<sub>ij</sub> = C<sub>ijkl</sub>ε<sub>kl</sub><br>τ<sub>ij</sub> = σ<sub>ij</sub> - μ<sub>ijk,k</sub><br>μ<sub>ijk</sub> = l<sup>2</sup>C<sub>ijmn</sub>κ<sub>mnk</sub></td></tr><tr><td>SUBC</td><td>σ<sub>ij</sub> = Σ<sub>ij</sub></td><td>σ<sub>ij</sub>n<sub>j</sub> - (μ<sub>ijk</sub>n<sub>k</sub>),j + (μ<sub>ijk</sub>n<sub>k</sub>n<sub>l</sub>),l n<sub>j</sub> = t<sub>i</sub><br>μ<sub>ijk</sub>n<sub>j</sub>n<sub>k</sub> = q<sub>i</sub> on ∂Ω (smooth part)<br>[[μ<sub>ijk</sub>m<sub>j</sub>n<sub>k</sub>]] = 0 on the edge of ∂Ω</td></tr></tbody></table>

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

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
<figure><img src="image_11.png" /></figure>

## Stress gradient (this work):

* Uniform isotropic stress:

$$ \Sigma = e_x \otimes e_x + e_y \otimes e_y + e_z \otimes e_z $$

* Uniform axial stress:

$$ \Sigma = e_z \otimes e_z $$

## Strain gradient:

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

* Gao, X.-L., Ma, H.M Journal of the Mechanics and Physics of Solids, 58 (5), pp. 779-797, 2010
<figure><img src="image_12.png" /></figure>

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

- Uniform isotropic stress prescribed at boundary:

$$ \sigma = \sigma(r) $$

- Uniform axial stress prescribed at boundary:

$$ \sigma = \sigma(r, \theta) $$

## Continuity conditions at the surface matrix-inclusion

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$$ \sigma_m(r = a) = \sigma_i(r = a) $$

$$ [\mathbf{u} \otimes^{\mathbf{s}} \mathbf{n} + \mathbf{\Phi} \cdot \mathbf{n}]_m (r = a) = [\mathbf{u} \otimes^{\mathbf{s}} \mathbf{n} + \mathbf{\Phi} \cdot \mathbf{n}]_i (r = a) $$
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Uniaxial problem $\Sigma = e_z \otimes e_z$

Radial stress field $\sigma_{rr}(r, \theta = 0)$

$$ \mu_i = 27\mu_m, \nu_i = 0.4, \nu_m = 0.08, \ell_i = \ell_m = \ell $$

<figure><img src="image_15.png" /></figure>
$$ \mu_i = 27\mu_m, \nu_i = 0.4, \nu_m = 0.08, \ell_i = \ell_m = \ell $$

<figure><img src="image_16.png" /></figure>

Stress gradient is complementary to strain gradient elasticity!

Ma, H.M, Gao, X.-L. Acta Mechanica, 225 (4-5), pp. 1075-1091, 2014
$$ \mu_i = 27\mu_m, \nu_i = 0.4, \nu_m = 0.08, \ell_i = \ell_m = \ell $$

<figure><img src="image_17.png" /></figure>

* Stress gradient elasticity: capture size effects.

* For $a \gg \ell$, stress gradient effects vanish as expected.
<figure><img src="image_18.png" /></figure>

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 al Soft Materials, 12, S142-S151, 2014

- etc
<figure><img src="image_19.png" /></figure>

**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.
Thank you for your attention!

vinh-phuc.tran@enpc.fr