# On stress-gradient materials: formulation and homogenization

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

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## 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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Submitted on 5 Apr 2018

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Distributed under a Creative Commons Attribution 4.0 International License
# On stress-gradient materials: formulation and homogenization

S. Brisard¹, V.P. Tran¹,², K. Sab¹, J. Guilleminot³

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LABEX MMCD

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

4 April 2018

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École des Ponts
ParisTech

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IFSTTAR

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# Stress-gradient elasticity in a nutshell

## The Stress Principle of Cauchy

$$ \boldsymbol{\sigma} \cdot \nabla + \boldsymbol{b} = \mathbf{0} \quad \text{and} \quad [[\boldsymbol{\sigma}]] \cdot \boldsymbol{n} = \mathbf{0} \quad (\text{SA}) $$

## Cauchy elasticity

$$ W^c[\boldsymbol{\sigma}] = \int w^c(\boldsymbol{\sigma}) dV $$

## Stress-gradient elasticity

$$ W^c[\boldsymbol{\sigma}] = \int w^c(\boldsymbol{\sigma}, \boldsymbol{\sigma} \otimes \nabla) dV $$

## A mature model

- 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)

## Alternative model (not discussed here)

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

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

## Some open questions
# Derivation of the stress-gradient model for elasticity
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
# Decomposition of stress-gradient (1/2)

## Complementary stress energy of stress-gradient materials

$$W^c[\boldsymbol{\sigma}] = \int w^c(\boldsymbol{\sigma}, \boldsymbol{\sigma} \otimes \nabla) dV$$

## Classical equilibrium equation

$$(\boldsymbol{\sigma} \otimes \nabla) : \mathbf{I}_2 = \boldsymbol{\sigma} \cdot \nabla = -b$$

## Orthogonal decomposition of stress gradient

$$\boldsymbol{\sigma} \otimes \nabla = \mathbf{Q} + \mathbf{R} \quad \text{with} \quad \mathbf{R} : \mathbf{I}_2 = \mathbf{0} \quad \text{and} \quad \mathbf{Q} : \mathbf{R} = \mathbf{0}$$

$$\mathbf{R} = \mathbf{I}'_6 : (\boldsymbol{\sigma} \otimes \nabla)$$

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

## Equivalent expression of complementary stress energy

$$W^c[\boldsymbol{\sigma}] = \int w^c(\boldsymbol{\sigma}, \mathbf{Q}, \mathbf{R}) dV$$
# Decomposition of stress-gradient (2/2)

## Orthogonal decomposition of stress gradient

$$ \sigma \otimes \nabla = Q + R \quad \text{with} \quad R: I_2 = 0 \quad \text{and} \quad Q: R = 0 $$

Q is in fact fully prescribed

$$ Q = -\frac{1}{2} I_4 \cdot b $$

(straightforward linear algebra)

## No strain measure attached to Q (generalized prestress)

### Modeling assumption

$$ W^c[\sigma] = \int w^c(\sigma, \nabla, R) dV = \int w^c(\sigma, R) dV $$

with $R = I'_6 \quad (\sigma \otimes \nabla)$
# Constrained minimization of energy

## Initial problem

Minimize: $W^c[\boldsymbol{\sigma}] = \int w^c(\boldsymbol{\sigma}, \mathbf{I}'_6 : (\boldsymbol{\sigma} \otimes \nabla)) dV$

subject to: $\boldsymbol{\sigma} \cdot \nabla + \mathbf{b} = \mathbf{0}$

## Stress-gradient as independent variable

Minimize: $W^c[\boldsymbol{\sigma}, \mathbf{R}] = \int w^c(\boldsymbol{\sigma}, \mathbf{R}) dV$

$$ \text{subject to: } \begin{cases} \boldsymbol{\sigma} \cdot \nabla + \mathbf{b} = \mathbf{0} & \cdot \mathbf{u} \\ \mathbf{R} = \mathbf{I}'_6 : (\boldsymbol{\sigma} \otimes \nabla) & \therefore \mathbf{\Phi} \end{cases} $$

Lagrange multipliers
# Elastic equilibrium of fixed, SG bodies

## Field equations

$$ \sigma \cdot \nabla + b = 0 $$

$$ \mathbf{R} = \mathbf{I}'_6 \circled{\left( \sigma \otimes \nabla \right)} $$

Generalized equilibrium equations

$$ \mathbf{e} = \partial_{\sigma} \mathbf{w}^{c} $$

$$ \mathbf{\Phi} = \partial_{\mathbf{R}} \mathbf{w}^{c} $$

Generalized stress-strain relations

$$ \mathbf{e} = \mathbf{\Phi} \cdot \nabla + \varepsilon [\mathbf{u}] $$

Generalized strain-displacement relations

## Boundary conditions

$$ \mathbf{\Phi} \cdot \mathbf{n} + \text{sym}(\mathbf{u} \otimes \mathbf{n}) = 0 $$

6 scalar boundary conditions

## Continuity conditions

$$ \left[ \mathbf{\Phi} \cdot \mathbf{n} + \text{sym}(\mathbf{u} \otimes \mathbf{n}) \right] = 0 $$

Weaker continuity of “displacements”

$$ \left[ \sigma \right] = 0 $$

Stronger continuity of “stresses”
# Linear elasticity

## Complementary stress energy density

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

No σ–R coupling for centrosymmetric materials!

S and M have major and minor symmetries

## Generalized compliance M operates on trace-free tensors

$$ \mathbf{M} = \mathbf{I}'_{6} : \mathbf{M} : \mathbf{I}'_{6} $$

## Stress-strain relationships

$$ \mathbf{e} = \mathbf{S} : \boldsymbol{\sigma} \quad \text{and} \quad \boldsymbol{\Phi} = \mathbf{M} : \mathbf{R} $$

or

$$ \boldsymbol{\sigma} = \mathbf{C} : \mathbf{e} \quad \text{and} \quad \mathbf{R} = \mathbf{L} : \boldsymbol{\Phi} $$

$$ \mathbf{C} = \mathbf{S}^{-1} \quad \text{and} \quad \mathbf{L} = \mathbf{M}^{+} $$
# Isotropic linear elasticity

## General isotropic linear stress-gradient elasticity

$$M=I'_{6} \therefore M \therefore I'_{6}$$

$$2 \mu M = \ell_{J}^{2} J_{6} + \ell_{K}^{2} K_{6} + \ell_{H}^{2} H_{6}$$

$$I'_{6} = J_{6} + K_{6}$$

<table><tbody><tr><td>∴</td><td>J<sub>6</sub></td><td>K<sub>6</sub></td><td>H<sub>6</sub></td></tr><tr><td>J<sub>6</sub></td><td>J<sub>6</sub></td><td>0</td><td>0</td></tr><tr><td>K<sub>6</sub></td><td>0</td><td>K<sub>6</sub></td><td>H<sub>6</sub></td></tr><tr><td>H<sub>6</sub></td><td>0</td><td>H<sub>6</sub></td><td>H<sub>6</sub></td></tr></tbody></table>

## Simplified model for isotropic linear stress-gradient elasticity

$$2 \mu S = \frac{1 - 2 \nu}{1 + \nu} J_{4} + K_{4}$$

$$\frac{2\mu}{\ell^2} M = \frac{1 - 2\nu}{1 + \nu} J_{6} + K_{6}$$

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

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

Gao and Park (2007), International Journal of Solids and Structures 44
Homogenization
# 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

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* Additional condition for stress-/strain- gradient materials

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# Effective and apparent properties

Effective behaviour: stress-gradient → Cauchy ($ℓ \sim d$)

$$ \langle \mathbf{e} \rangle = \mathbf{S}^{\text{eff}} : \langle \boldsymbol{\sigma} \rangle $$

Apparent elastic properties of (large) SVE $\Omega$

$$ \langle \boldsymbol{\sigma} \rangle = \frac{1}{V} \int_{\Omega} \boldsymbol{\sigma} \, dV $$

$$ \langle \mathbf{e} \rangle = \mathbf{S}^{\text{app}}(\Omega) : \langle \boldsymbol{\sigma} \rangle $$

$$ \langle \mathbf{e} \rangle = \frac{1}{V} \int_{\Omega} \mathbf{e} \, dV = \frac{1}{V} \int_{\Omega} \varepsilon[\mathbf{u}] \, dV + \frac{1}{V} \int_{\partial\Omega} \mathbf{\Phi} \cdot \mathbf{n} \, dS $$

Local problem on SVE $\Omega$

$$ \boldsymbol{\sigma} \cdot \nabla = \mathbf{0} $$

No body forces!

$$ \mathbf{e} = \mathbf{S} : \boldsymbol{\sigma} $$

$$ \mathbf{\Phi} = \mathbf{M} : (\boldsymbol{\sigma} \otimes \nabla) $$

$$ \mathbf{e} = \mathbf{\Phi} \cdot \nabla + \varepsilon[\mathbf{u}] $$

+ boundary conditions!
# Boundary conditions

## Generalized Hill–Mandel lemma

$$ \langle \boldsymbol{\sigma}^* : \mathbf{e} + (\boldsymbol{\sigma}^* \otimes \nabla) : \boldsymbol{\Phi} \rangle = \langle \boldsymbol{\sigma}^* \rangle : \langle \mathbf{e} \rangle $$

## Uniform stress boundary conditions

$$ \boldsymbol{\sigma}|_{\partial\Omega} = \bar{\boldsymbol{\sigma}} $$

$$ \langle \mathbf{e} \rangle = \mathbf{S}^{\sigma}(\Omega) : \langle \boldsymbol{\sigma} \rangle = \mathbf{S}^{\sigma}(\Omega) : \bar{\boldsymbol{\sigma}} $$

## Uniform strain boundary conditions

$$ \boldsymbol{\Phi} \cdot \mathbf{n} + \text{sym}(\mathbf{u} \otimes \mathbf{n}) = \text{sym}[(\bar{\mathbf{e}} \cdot \mathbf{x}) \otimes \mathbf{n}] $$

$$ \langle \boldsymbol{\sigma} \rangle = \mathbf{C}^{\varepsilon}(\Omega) : \langle \mathbf{e} \rangle = \mathbf{C}^{\varepsilon}(\Omega) : \bar{\mathbf{e}} $$

## Uniform traction boundary conditions

$$ \boldsymbol{\sigma}|_{\partial\Omega} \cdot \mathbf{n} = \bar{\boldsymbol{\sigma}} \cdot \mathbf{n} $$

$$ \langle \mathbf{e} \rangle = \mathbf{S}^T(\Omega) : \langle \boldsymbol{\sigma} \rangle = \mathbf{S}^T(\Omega) : \bar{\boldsymbol{\sigma}} $$

$$ \mathbf{a} \cdot [ \boldsymbol{\Phi} \cdot \mathbf{n} + \text{sym}(\mathbf{u} \otimes \mathbf{n}) ] \cdot \mathbf{a} = \mathbf{0} $$

$$ \mathbf{a} = \mathbf{l}_2 - \mathbf{n} \otimes \mathbf{n} $$
# Variational formulation

## Minimum of complementary stress energy

$$ \bar{\sigma}:\mathbf{S}^{\sigma}(\Omega):\bar{\sigma}=\inf\{\langle\boldsymbol{\sigma}:\mathbf{S}:\boldsymbol{\sigma}+(\boldsymbol{\sigma}\otimes\nabla):\cdot, \mathbf{M}:\cdot:(\boldsymbol{\sigma}\otimes\nabla)\rangle\} $$

subject to: $\boldsymbol{\sigma}\cdot\nabla=0$ and $\boldsymbol{\sigma}|_{\partial\Omega}=\bar{\sigma}$

## Minimum of strain energy

$$ \bar{e}:\mathbf{C}^{\varepsilon}(\Omega):\bar{e}=\inf\{\langle\boldsymbol{\varepsilon}[\mathbf{u}]:\mathbf{C}:\boldsymbol{\varepsilon}[\mathbf{u}]+\boldsymbol{\Phi}:\cdot, \mathbf{L}:\cdot, \boldsymbol{\Phi}\rangle\} $$

subject to: $\boldsymbol{\Phi}\cdot\mathbf{n}+\text{sym}(\mathbf{u}\otimes\mathbf{n})=\text{sym}[(\bar{e}\cdot\mathbf{x})\otimes\mathbf{n}]$

## Bounds for finite-size SVEs

$$ \mathbf{C}^{\sigma}(\Omega_{\mathrm{I}})\leq\mathbf{C}^{\sigma}(\Omega_{\mathrm{II}})\leq\mathbf{C}^{\mathrm{eff}}\leq\mathbf{C}^{\varepsilon}(\Omega_{\mathrm{II}})\leq\mathbf{C}^{\varepsilon}(\Omega_{\mathrm{I}}) \quad \text{for } \Omega_{\mathrm{I}}\subset\Omega_{\mathrm{II}} $$
# Softening size-effect

## Variational definition of apparent compliance (uniform $\sigma$ BC)

If $\mathbf{S}_I \le \mathbf{S}_{II}$ and $\mathbf{M}_I \le \mathbf{M}_{II}$ everywhere in $\Omega$, then $\mathbf{S}_I^{\text{eff}} \le \mathbf{S}_{II}^{\text{eff}}$

## Fixed microstructure, variable material internal length

For $\mathbf{S}_I = \mathbf{S}_{II}$ and $\mathbf{M}_I = \frac{\ell_I^2}{\ell_{II}^2} \mathbf{M}_{II}$: if $\ell_I \le \ell_{II}$ then $\mathbf{S}_I^{\text{eff}} \le \mathbf{S}_{II}^{\text{eff}}$

More generally: if $\left(\frac{\ell}{d}\right)_I \le \left(\frac{\ell}{d}\right)_{II}$ then $\mathbf{S}_I^{\text{eff}} \le \mathbf{S}_{II}^{\text{eff}}$

## Scaled microstructure, constant material internal length

If $d_I \le d_{II}$ then $\mathbf{C}_I^{\text{eff}} \le \mathbf{C}_{II}^{\text{eff}}$

**Softening size-effect!**
# Eshelby's inhomogeneity problem for stress-gradient elasticity
Problem setting

$\sigma^\infty e_z$

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Matrix: $\mu_m, v_m, \ell_m$

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$v_i = v_m = 0.25$

$\mu_i = 10 \mu_m$

$-\sigma^\infty e_z$

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Some tedious calcs later...

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http://www.sympy.org

Hat tip to VP Tran!

<figure><img src="image_16.png" /><figcaption>Axial stress along vertical axis</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Axial stress at origin</figcaption></figure>

**Theorem of Eshelby (1957) does not hold!**
# Dilute stress concentration tensor

<figure><img src="image_18.png" /><figcaption>Axial stress along vertical axis</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Axial stress at origin</figcaption></figure>

$$ \langle \boldsymbol{\sigma} \rangle_i = \frac{1}{V_i} \int_{\Omega_i} \boldsymbol{\sigma} \, dV = \mathbf{B}^\infty : \boldsymbol{\sigma}^\infty $$
Mori–Tanaka estimates
for composites
with spherical inclusions
Problem setting

<figure><img src="image_20.png" /></figure>

Volume fraction of inclusions: f
# Estimates of bulk and shear moduli

<figure><img src="image_21.png" /><figcaption>Effective bulk modulus</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Effective shear modulus</figcaption></figure>

$$ \mathbf{S}_{\text{eff}} = \mathbf{S}_{\text{m}} + f (\mathbf{S}_{\text{i}} - \mathbf{S}_{\text{m}}) : \mathbf{B}^{\infty} : [(1-f) \mathbf{I}_{4} + f \mathbf{B}^{\infty}]^{-1} $$

Strain- and stress- gradient models are **not equivalent!**

Benveniste (1987), Mechanics of Materials 6(2)

Ma and Gao (2014), Acta Mechanica 225(4-5)
Some questions
# The total complementary energy

**Assumption 1: Stress Principle of Cauchy**

$$ \sigma \cdot \nabla = 0 \quad \text{and} \quad \sigma \cdot n \Big|_{\partial \Omega} = \bar{T} $$

**Assumption 2: complementary work of prescribed displacements**

$$ V^c = \int_{\partial \Omega_u} \bar{u} \cdot \sigma \cdot n \, dS $$

**Minimization of complementary energy**

$$ \text{Minimize: } \Pi^c = W^c - V^c $$

$$ \text{subject to: } \sigma \cdot \nabla = 0 \quad \text{and} \quad \sigma \cdot n \Big|_{\partial \Omega} = \bar{T} $$

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# The general BVP

## Field equations

$$ \boldsymbol{\sigma} \cdot \nabla + b = 0 $$

$$ \mathbf{e} = \partial_{\boldsymbol{\sigma}} w^c $$

$$ \mathbf{e} = \mathbf{\Phi} \cdot \nabla + \varepsilon[u] $$

$$ \mathbf{R} = \mathbf{l}'_6 : (\boldsymbol{\sigma} \otimes \nabla) $$

$$ \mathbf{\Phi} = \partial_{\mathbf{R}} w^c $$

## Boundary conditions

$$ \partial_{\Omega_T}: \left\{ \begin{array}{l} \boldsymbol{\sigma} \cdot \mathbf{n}|_{\partial \Omega} = \bar{T} \\ \mathbf{a} \cdot [\mathbf{\Phi} \cdot \mathbf{n} + \text{sym}(\mathbf{u} \otimes \mathbf{n})] \cdot \mathbf{a} = \mathbf{0} \end{array} \right. $$

$$ \partial_{\Omega_u}: \mathbf{\Phi} \cdot \mathbf{n} + \text{sym}(\mathbf{u} \otimes \mathbf{n}) = \text{sym}(\bar{\mathbf{u}} \otimes \mathbf{n}) $$

$$ \mathbf{a} = \mathbf{l}_2 - \mathbf{n} \otimes \mathbf{n} $$

## Continuity conditions

$$ [[\boldsymbol{\sigma}] = \mathbf{0}] $$

$$ [[\mathbf{\Phi} \cdot \mathbf{n} + \text{sym}(\mathbf{u} \otimes \mathbf{n})] = \mathbf{0}] $$
# The total potential energy

## Potential strain energy

$$W = \int w(\mathbf{e}, \mathbf{\Phi}) dV$$

## Work of prescribed body forces and tractions

$$V = \int_{\Omega} b \cdot u dV + \int_{\partial \Omega_T} \bar{T} \cdot [u + 2\mathbf{\Phi}:n \otimes n - (\mathbf{\Phi} \cdot n \otimes n \otimes n)n] dS$$

## Minimization of total potential energy

$$\text{Minimize: } \Pi = W - V$$

$$u + 2\mathbf{\Phi}:n \otimes n - (\mathbf{\Phi} \cdot n \otimes n \otimes n)n = \bar{u}$$

$$\text{subject to:} \quad \begin{cases} \partial \Omega_T: \mathbf{a} \cdot [\mathbf{\Phi} \cdot n + \text{sym}(u \otimes n)] \cdot \mathbf{a} = 0 \\ \partial \Omega_u: \mathbf{\Phi} \cdot n + \text{sym}(u \otimes n) = \text{sym}(\bar{u} \otimes n) \end{cases}$$

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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)
Thanks for your attention!

Sebastien.brisard@ifsttar.fr
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.

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