# Beyond the classical Hashin–Shtrikman bounds: the Hashin–Shtrikman principle revisited

Sébastien Brisard

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Sébastien Brisard. Beyond the classical Hashin–Shtrikman bounds: the Hashin–Shtrikman principle revisited. Séminaire du Laboratoire de Mécanique des Solides, École Polytechnique, Apr 2017, 91128 Palaiseau Cedex, France. [hal-01513354](https://HAL Id: hal-01513354)

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<figure><img src="image_1.png" /></figure>
# Beyond the classical Hashin-Shtrikman bounds
## The Hashin-Shtrikman principle revisited

Sébastien Brisard

Université Paris-Est, Laboratoire Navier, ENPC, IFSTTAR, CNRS UMR 8205

UNIVERSITÉ
— PARIS-EST

Navier

<figure><img src="image_2.png" /></figure>

École des Ponts
ParisTech

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

IFSTTAR

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

<figure><img src="image_5.png" /></figure>
# Experimental homogenization

## Effective moduli measured through simple uniaxial test

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

## Macroscopic strain & stress

$$ \bar{\epsilon} = \frac{\delta}{L} \qquad \bar{\sigma} = \frac{F}{S} $$

## Effective Young modulus

$$ E = \frac{\bar{\sigma}}{\bar{\epsilon}} $$

<figure><img src="image_7.png" /></figure>

Courtesy S. Bahafid, S. Ghabezloo

## Loading conditions are such that

$$ \bar{\epsilon} = \langle \epsilon \rangle \qquad \bar{\sigma} = \langle \sigma \rangle \qquad \bar{\sigma} \bar{\epsilon} = \langle \sigma \epsilon \rangle $$
# Numerical homogenization
## The corrector problem

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

<figure><img src="image_9.png" /></figure>
# The homogenization toolkit

**Pros**

* ✓ Simple

* ✓ Versatile

**Cons**

* X No correlations

* X Restricted to ellipsoids!

**Pros**

* ✓ Captures fine details of microstructure

**Cons**

* X Time consuming

* X Difficult to analyze

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>
# Outline of this talk

* Variational methods for homogenization

* Polarization techniques for homogenization:
  the Hashin–Shtrikman principle

* Realization-by-realization applications of the HS principle
  (FFT-based methods, EIM)

* Statistical applications of the HS principle: classical and
  improved HS bounds

<figure><img src="image_12.png" /><figcaption>20 avril 2017 ◊ Séminaire LMS ◊ S. Brisard ◊ Beyond Hashin–Shtrikman bounds</figcaption></figure>
Varational methods for
homogenization
# Variational methods

## The **true** solution of the corrector problem

* Minimizes some energy

* Trial fields must be **admissible**

## Construction of an **approximate** solution?

* Construct a **subspace** of admissible trial fields

* Minimize energy over this subspace

## How to select “good” trial fields?
# Minimum of potential energy

## Potential energy

$$ \Pi(\epsilon) = \frac{1}{2} \langle \epsilon : \mathbf{C} : \epsilon \rangle \geq \frac{1}{2} \bar{\epsilon} : \mathbf{C}^{\text{eff}} : \bar{\epsilon} $$

## Simple admissible trial strain?

<figure><img src="image_13.png" /></figure>

## Admissible trial strain

$$ \text{In } \Omega: \epsilon = \text{sym grad } u $$

$$ \text{On } \partial\Omega: u = \bar{\epsilon} \cdot x $$

* Try phase-wise constant strains

$$ u(\mathbf{x}) = \epsilon_{\alpha} \cdot \mathbf{x} \text{ for } \mathbf{x} \text{ in phase } \alpha $$

* Displacement is continuous

$$ \epsilon_1 \cdot \mathbf{x} = \epsilon_2 \cdot \mathbf{x} \text{ on the interface} $$

## Constant strain field!!!

$$ \mathbf{C}^{\text{eff}} \leq \langle \mathbf{C} \rangle \quad (\text{Voigt}) $$
# Minimum of complementary energy

## Complementary energy

$$ \Pi^*(\sigma) = \frac{1}{2} \langle \sigma : \mathbf{S} : \sigma \rangle \geq \frac{1}{2} \bar{\sigma} : \mathbf{S}^{\text{eff}} : \bar{\sigma} $$

## Simple admissible trial stress?

<figure><img src="image_14.png" /></figure>

## Admissible trial stress

$$ \text{In } \Omega: \text{div } \sigma = 0 $$

$$ \text{On } \partial\Omega: \sigma \cdot \mathbf{n} = \bar{\sigma} \cdot \mathbf{n} $$

* Try phase-wise constant stresses

$$ \sigma(\mathbf{x}) = \sigma_{\alpha} \text{ for } \mathbf{x} \text{ in phase } \alpha $$

* Traction is continuous

$$ \sigma_1 \cdot \mathbf{n} = \sigma_2 \cdot \mathbf{n} \text{ on the interface} $$

## Constant trial stress!!!

$$ \mathbf{C}^{\text{eff}} \geq \langle \mathbf{S}^{-1} \rangle^{-1} \quad (\text{Reuss}) $$

<figure><img src="image_15.png" /></figure>
Polarization techniques for
homogenization
# The Green operator for strains

The initial, corrector problem

$$ \mathrm{div} \sigma = 0 $$

$$ \sigma = C : \epsilon $$

$$ \epsilon = \mathrm{symgrad} u $$

$$ \epsilon \in KA(\bar{\epsilon}) $$

A new, auxiliary problem

$$ \mathrm{div} \sigma = 0 $$

$$ \sigma = C_0 : \epsilon + \tau $$

$$ \epsilon = \mathrm{symgrad} u $$

$$ \epsilon \in KA(\bar{\epsilon}) $$

* Reference material $C_0$ is homogeneous

* Stress-polarization $\tau$ is heterogeneous

* Solution depends linearly on $\tau, \bar{\epsilon}$

Korringa, J. Math. Phys. 14(4), 1973

Zeller & Dederichs, Physica Status Solidi (B) 55(2), 1973

$$ \epsilon = \bar{\epsilon} - \Gamma_0[\tau] $$
# The Lippmann-Schwinger equation

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

Korringa, J. Math. Phys. 14(4), 1973

Zeller & Dederichs, Physica Status Solidi (B) 55(2), 1973
# The Hashin–Shtrikman principle (1/3)

## The Lippmann–Schwinger equation

$$ (\mathbf{C}-\mathbf{C}_0)^{-1}:\boldsymbol{\tau}+\Gamma_0[\boldsymbol{\tau}]=\bar{\epsilon} $$

## Variational form of the Lippmann–Schwinger equation

Find $\boldsymbol{\tau}$ such that $\langle \boldsymbol{\omega}: (\mathbf{C}-\mathbf{C}_0)^{-1}: \boldsymbol{\tau} \rangle + \langle \boldsymbol{\omega}: \Gamma_0[\boldsymbol{\tau}] \rangle = \langle \bar{\epsilon}: \boldsymbol{\omega} \rangle$ for all $\boldsymbol{\omega}$

## The functional of Hashin–Shtrikman

$$ \mathrm{HS}(\boldsymbol{\omega}; \bar{\epsilon}) = \frac{1}{2} \bar{\epsilon} : \mathbf{C}_0 : \bar{\epsilon} + \langle \bar{\epsilon} : \boldsymbol{\omega} \rangle - \frac{1}{2} \langle \boldsymbol{\omega} : (\mathbf{C}-\mathbf{C}_0)^{-1} : \boldsymbol{\omega} \rangle - \frac{1}{2} \langle \boldsymbol{\omega} : \Gamma_0[\boldsymbol{\omega}] \rangle $$

Hashin & Shtrikman, JMPS 10(4), 1962

Willis, JMPS 25(3), 1977
# The Hashin–Shtrikman principle (2/3)

## Stationarity principle

Hashin & Shtrikman, JMPS 10(4), 1962
Willis, JMPS 25(3), 1977

$$ \text{For } (\mathbf{C}-\mathbf{C}_0)^{-1}: \tau+\Gamma_0[\tau]=\bar{\epsilon} $$

$$ \frac{\partial \text{HS}}{\partial \boldsymbol{\omega}}(\tau; \bar{\epsilon}) = 0 $$

$$ \text{HS}(\tau; \bar{\epsilon}) = \frac{1}{2} \bar{\epsilon} : \mathbf{C}^{\text{eff}} : \bar{\epsilon} $$

## “Soft” reference material

$\mathbf{C}_0 \le \mathbf{C}$ everywhere in $\Omega$

$\frac{1}{2} \bar{\epsilon} : \mathbf{C}^{\text{eff}} : \bar{\epsilon} \le \text{HS}(\boldsymbol{\omega}; \bar{\epsilon})$ for all $\boldsymbol{\omega}$

## “Stiff” reference material

$\mathbf{C}_0 \ge \mathbf{C}$ everywhere in $\Omega$

$\frac{1}{2} \bar{\epsilon} : \mathbf{C}^{\text{eff}} : \bar{\epsilon} \ge \text{HS}(\boldsymbol{\omega}; \bar{\epsilon})$ for all $\boldsymbol{\omega}$

<figure><img src="image_17.png" /></figure>
# The Hashin–Shtrikman principle (3/3)

## General form of trial stress-polarization

$$ \tau(\mathbf{x}) = \mathbf{T}(\mathbf{x}):\bar{\epsilon} $$

HS is a quadratic form!

$$ \mathrm{HS}(\tau; \bar{\epsilon}) = \mathrm{HS}(\mathbf{T}:\bar{\epsilon}; \bar{\epsilon}) = \frac{1}{2} \bar{\epsilon} : \mathbf{C}^{\mathrm{HS}} : \bar{\epsilon} $$

$\mathbf{C}^{\mathrm{HS}}$: estimate or bound on $\mathbf{C}^{\mathrm{eff}}$

**Any stress-polarization is admissible!!!**
# The problem with bounded domains

Closed-form expressions of the Green operator
are generally not known!!!

## Heuristic approach

$$ \Gamma_0[\tau] \approx \Gamma_0^\infty[\tau - \chi\langle\tau\rangle] $$

Willis, JMPS 25(3), 1977

## Rigorous justification

* Consider the above relation as the definition of a "modified Green operator"

* The associated Lippmann-Schwinger equation solves **another, well-posed and consistent** corrector problem

Brisard, Sab & Dormieux, JMPS 61(12), 2013
Realization-by-realization applications:
bounds on the apparent properties
# FFT-based methods (1/2)

## Features

* approximate full-field solver: requires **realizations**

* dedicated to pixel/voxel geometries: ideally suited to **imaging**

* periodic boundary conditions

* arguably more efficient than FEM

Moulinec & Suquet, CMAME 157(1-2), 1998

## FFT-based methods revisited

* **cell-wise constant trial stress-polarization**

* optimization of functional of H&S leads to linear system

* (matrix-free) linear system can be solved iteratively

Brisard & Dormieux, Comp. Mat. Sci. 49(3), 2010

Brisard & Dormieux, CMAME 217-220, 2012
# FFT-based methods (2/2)

## A small, but active community

* more efficient linear solvers
  * Brisard & Dormieux, Comp. Mat. Sci. 49(3), 2010
  * Zeman et al., J. Comp. Phys. 229(21), 2010

<figure><img src="image_18.png" /></figure>

* improved high-frequency filtering
  * Brisard & Dormieux, CMAME 217-220, 2012
  * Willot, CR Mécanique 343(3), 2015

* error estimates (a priori, a posteriori)
  * Schneider, Math. Meth. Appl. Sci. 38(13), 2014
  * Brisard, IJNME 109(4), 2017

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

* other discretizations
  * Vondřejc et al., CMAME 297, 2015

* non-linearities (material, geometric)
  * Moulinec & Suquet, CMAME 157(1-2), 1998
  * Gélébart & Mondon-Cancel, Comp. Mat. Sci (77), 2013
  * Kabel et al., Comp. Mech. 54(6), 2014

<figure><img src="image_20.png" /></figure>
# Equivalent Inclusion Method (1/2)

## Features

* approximate full-field solver: requires **realizations**

* dedicated to simple geometries: spheres, ellipsoids

* **interactions** are explicitly taken into account

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

Moschovidis & Mura, J. App. Mech. 42, 1975

## EIM revisited

* **inclusion-wise polynomial trial stress-polarizations**

* linear system is small, **but**

* assembly of the matrix requires off-line (tricky) calculations

Brisard, Sab & Dormieux, JMPS 61(12), 2013

Brisard, Dormieux & Sab, IJSS 51(3-4), 2014
# Equivalent Inclusion Method (2/2)

## 2D (plane strain) application

* 160 monosized pores

* porosity: 40%

* average over 1000 realizations

Computation provides upper-bounds on $\mu^{\text{eff}}$

<table><thead><tr><td>Degree</td><td>Number of DOFs</td></tr></thead><tbody><tr><td>0</td><td>480</td></tr><tr><td>1</td><td>1440</td></tr><tr><td>2</td><td>2880</td></tr><tr><td>3</td><td>4800</td></tr><tr><td>FEM</td><td>≈ 320,000</td></tr></tbody></table>

<figure><img src="image_21.png" /></figure>

* Exponential convergence!

* Not so good in 3D

<figure><img src="image_22.png" /></figure>
Statistical applications (1/2)
The classical Hashin-Shtrikman bounds
# Outline of the statistical approach

1. Pick a (random) trial stress-polarization field

► $\tau(\mathbf{x}, \omega)$ defined over whole space. $\omega$: realization of random material.

2. Apply the Hashin–Shtrikman principle to a finite size “SVE”

► $\frac{1}{2} \bar{\epsilon}: \mathbf{C}^{\text{app}}(\omega): \bar{\epsilon} \le \text{HS}(\chi \tau(\cdot, \omega); \bar{\epsilon})$ $\chi$: indicator function of SVE $\Omega$.

3. Take the ensemble average over all SVEs of given size

► $\frac{1}{2} \bar{\epsilon}: \text{E}[\mathbf{C}^{\text{app}}]: \bar{\epsilon} \le \text{E}[\text{HS}(\chi \tau; \bar{\epsilon})]$

4. Take the limit for large SVEs

► $\frac{1}{2} \bar{\epsilon}: \mathbf{C}^{\text{eff}}: \bar{\epsilon} \le \lim_{|\Omega| \to \infty} \text{E}[\text{HS}(\chi \tau; \bar{\epsilon})]$ 	Willis, JMPS 25(3), 1977
# What should we expect?

$$ \frac{1}{2} \bar{\epsilon} : \mathbf{C}^{\text{eff}} : \bar{\epsilon} \le \lim_{|\Omega| \to \infty} \mathrm{E}[\mathrm{HS}(\chi \tau; \bar{\epsilon})] $$

$$ \mathrm{HS}(\tau; \bar{\epsilon}) = \frac{1}{2} \bar{\epsilon} : \mathbf{C}_0 : \bar{\epsilon} + \langle \bar{\epsilon} : \tau \rangle - \frac{1}{2} \langle \tau : (\mathbf{C} - \mathbf{C}_0)^{-1} : \tau \rangle - \frac{1}{2} \langle \tau : \Gamma_0[\tau] \rangle $$

## One-point information

$$ \langle \tau : (\mathbf{C} - \mathbf{C}_0)^{-1} : \tau \rangle = \frac{1}{V} \int_{\mathbf{x} \in \Omega} \tau(\mathbf{x}) : [\mathbf{C}(\mathbf{x}) - \mathbf{C}_0] : \tau(\mathbf{x}) dV_{\mathbf{x}} $$

## Two-point information: correlations!

$$ \langle \tau : \Gamma_0[\tau] \rangle = \frac{1}{V} \int_{\mathbf{x}, \mathbf{y} \in \Omega} \tau(\mathbf{x}) : \Gamma_0(\mathbf{x}, \mathbf{y}) : \tau(\mathbf{y}) dV_{\mathbf{x}} dV_{\mathbf{y}} $$
# Trial stress-polarization

## Phase-wise constant trial stress-polarization

$$ \tau(\mathbf{x}, \omega) = \sum_{\alpha} \chi_{\alpha}(\mathbf{x}, \omega) \tau_{\alpha} $$

Hashin & Shtrikman, JMPS 10(4), 1962

Willis, JMPS 25(3), 1977

**Essential assumption: $τ_α$ constant and deterministic**

“Thermodynamic limit” of Hashin–Shtrikman functional

$$ \lim_{|\Omega| \to \infty} E[\mathrm{HS}(\chi \tau(\cdot, \omega); \bar{\epsilon})] = \frac{1}{2} \bar{\epsilon} : \mathbf{C}_0 : \bar{\epsilon} + \sum_{\alpha} f_{\alpha} \tau_{\alpha} : \bar{\epsilon} - \frac{1}{2} \sum_{\alpha} f_{\alpha} \tau_{\alpha} : (\mathbf{C}_{\alpha} - \mathbf{C}_0)^{-1} : \tau_{\alpha} - \frac{1}{2} \sum_{\alpha, \beta} \tau_{\alpha} : \mathbf{P}_{\alpha\beta} : \tau_{\beta} $$

**Optimization with respect to the $τ_α$ delivers the**
**classical Hashin–Shtrikman bounds**

<figure><img src="image_23.png" /></figure>
# The P-tensors

General expression (proof not so trivial!)

$$ \mathbf{P}_{\alpha\beta} = \hat{S}_{\alpha\beta}(\mathbf{0})\mathbf{P}_0 + \lim_{\delta \to 0} \int_{\|\mathbf{r}\| \ge \delta} \hat{S}_{\alpha\beta}(\mathbf{r}) \mathbf{\Gamma}_0^\infty(\mathbf{r}) \, dV_r $$

P₀: Hill tensor of spheres

Two-point correlation function (covariance)

$$ S_{\alpha\beta}(\mathbf{r}) = E[\chi_{\alpha}(\mathbf{0})\chi_{\beta}(\mathbf{r})] $$

$$ \hat{S}_{\alpha\beta}(\mathbf{r}) = S_{\alpha\beta}(\mathbf{r}) - f_{\alpha}f_{\beta} $$

Hashin–Shtrikman bounds are expected to account
for two-point correlations!

<figure><img src="image_24.png" /></figure>
# The isotropic case

## Angular average of Green operator

$$ \int_{||\mathbf{n}||=1} \Gamma_0^\infty(\mathbf{n}) \, dS_n = 0 $$

## The P-tensors

$$ \mathbf{P}_{\alpha\beta} = f_{\alpha}(\delta_{\alpha\beta} - f_{\beta})\mathbf{P}_{0} $$

Two-point information is lost!

## Expression of the classical bounds

$$ \mathbf{C}^{\text{eff}} \le \mathbf{C}^{\text{HS}} = \sum_{\alpha} f_{\alpha} \mathbf{C}_{\alpha} : \mathbf{A}_{\alpha}^{\text{HS}} $$

$$ \mathbf{A}_{\alpha}^{\text{HS}} = \mathbf{A}_{\alpha}^{\infty} : \left( \sum_{\beta} f_{\beta} \mathbf{A}_{\beta}^{\infty} \right)^{-1} $$

$$ \mathbf{A}_{\alpha}^{\infty} = [\mathbf{I} + \mathbf{P}_{0} : (\mathbf{C}_{\alpha} - \mathbf{C}_{0})]^{-1} $$

Hashin–Shtrikman bounds account for
volume fractions only!
# Hashin–Shtrikman vs. Voigt & Reuss

<figure><img src="image_25.png" /></figure>
Statistical applications (2/2)
Towards improved Hashin-Shtrikman bounds?
# Probes and statistical descriptors

## Reminder: Hashin–Shtrikman trial stress-polarization

$$ \tau(\mathbf{x}, \omega) = \sum_{\alpha} \chi_{\alpha}(\mathbf{x}, \omega) \tau_{\alpha} $$

* $\chi_\alpha$ will be called a “probe” of the microstructure.

## After optimization

* The resulting bounds depend on $f_\alpha$ only.

* $f_\alpha$ will be called a “statistical descriptor” of the microstructure.

**$\chi_\alpha$ is probably a very poor probe!**

<figure><img src="image_26.png" /></figure>
# Enriched trial fields

## Abstraction of the classical HS trial fields

$$ 
\begin{aligned}
\tau(\mathbf{x}, \omega) &= \sum_{\alpha} \chi_{\alpha}(\mathbf{x}, \omega) \tau_{\alpha} \\
&= \sum_{p} \text{Probe}_p(\mathbf{x}, \omega) \times \text{Deterministic const.}_p
\end{aligned}
$$

## Introducing new, general probes

$$ \tau(\mathbf{x}, \omega) = \sum_{\alpha} \sum_{p} \chi_{\alpha}(\mathbf{x}, \omega) \psi_{p}(\mathbf{x}, \omega) \tau_{\alpha p} $$

* $\psi_0, \dots, \psi_M$: new probes (not yet specified).

* $\psi_0 \equiv 1$: ensures that we effectively enrich the classical approach.
Limit of Hashin–Shtrikman functional

$$
\lim_{|\Omega| \to \infty} \mathrm{E}[\mathrm{HS}(\chi \tau(\cdot, \omega); \bar{\epsilon})] = \\
\frac{1}{2} \bar{\epsilon} : \mathbf{C}_0 : \bar{\epsilon} + \sum_{\alpha, p} \Psi_{\alpha p} \tau_{\alpha p} : \bar{\epsilon} - \frac{1}{2} \sum_{\alpha, p, q} \Psi_{\alpha p \alpha q}(\mathbf{0}) \tau_{\alpha p} : (\mathbf{C}_{\alpha} - \mathbf{C}_0)^{-1} : \tau_{\alpha q} \\
- \frac{1}{2} \sum_{\alpha, \beta, p, q} \tau_{\alpha p} : \mathbf{P}_{\alpha p \beta q} : \tau_{\beta q}
$$

$$
\frac{1}{2} \bar{\epsilon} : \mathbf{C}_0 : \bar{\epsilon} + \sum_{\alpha} f_{\alpha} \tau_{\alpha} : \bar{\epsilon} - \frac{1}{2} \sum_{\alpha} f_{\alpha} \tau_{\alpha} : (\mathbf{C}_{\alpha} - \mathbf{C}_0)^{-1} : \tau_{\alpha} - \frac{1}{2} \sum_{\alpha, \beta} \tau_{\alpha} : \mathbf{P}_{\alpha \beta} : \tau_{\beta}
$$

<figure><img src="image_27.png" /></figure>
Enriched statistical descriptors

$$
\Psi_{\alpha\rho} = \mathrm{E}[\chi_{\alpha}(\mathbf{0})\psi_{\rho}(\mathbf{0})] \qquad f_{\alpha} = \mathrm{E}[\chi_{\alpha}(\mathbf{0})]
$$

$$
\Psi_{\alpha p \beta q}(\mathbf{r}) = \mathrm{E}[\chi_{\alpha}(\mathbf{0}) \psi_{p}(\mathbf{0}) \chi_{\beta}(\mathbf{r}) \psi_{q}(\mathbf{r})] \qquad S_{\alpha\beta}(\mathbf{r}) = \mathrm{E}[\chi_{\alpha}(\mathbf{0}) \chi_{\beta}(\mathbf{r})]
$$

$$
\hat{\Psi}_{\alpha p \beta q}(\mathbf{r}) = \Psi_{\alpha p \beta q}(\mathbf{r}) - \Psi_{\alpha p} \Psi_{\beta q} \qquad \hat{S}_{\alpha\beta}(\mathbf{r}) = S_{\alpha\beta}(\mathbf{r}) - f_{\alpha} f_{\beta}
$$

$$
\mathbf{P}_{\alpha p \beta q} = \hat{\Psi}_{\alpha p \beta q}(\mathbf{0}) \mathbf{P}_0 + \lim_{\delta \to 0} \int_{\|\mathbf{r}\| \ge \delta} \hat{\Psi}_{\alpha p \beta q}(\mathbf{r}) \mathbf{\Gamma}_0^{\infty}(\mathbf{r}) \, dV_r
$$

$$
\mathbf{P}_{\alpha\beta} = \hat{S}_{\alpha\beta}(\mathbf{0})\mathbf{P}_0 + \lim_{\delta \to 0} \int_{\|\mathbf{r}\| \ge \delta} \hat{S}_{\alpha\beta}(\mathbf{r}) \Gamma_0^\infty(\mathbf{r}) \, dV_r
$$
# Problem solved?

* Compute $\Psi_{\alpha p}$ and $\Psi_{\alpha p \beta q}$ (e.g. image analysis)

* Compute $\mathbf{P}_{\alpha p \beta q}$ (singular integrals)

* Find the optimum $\tau_{\alpha p}$

* Evaluate the bound $\frac{1}{2} \bar{\epsilon} : \mathbf{C}^{\text{eff}} : \bar{\epsilon} \le \lim_{|\Omega| \to \infty} E[\text{HS}(\chi \tau; \bar{\epsilon})]$

**We need to select the probes!**
# Selecting a probe (1/2)

## Why are the classical HS bounds not so good?

$$ \tau(\mathbf{x}, \omega) = \sum_{\alpha} \chi_{\alpha}(\mathbf{x}, \omega) \tau_{\alpha} $$

* Depends on phase at point only.

* Does not account for neighborhood.

<figure><img src="image_28.png" /></figure>

## Local volume fraction

* Accounts for neighborhood!

* Introduces a length-scale!

$$ f_{\alpha}^{\text{loc}}(\mathbf{x}, \omega) = \frac{1}{W} \int w(\mathbf{y}-\mathbf{x}) \chi_{\alpha}(\mathbf{x}, \omega) dV_{\mathbf{y}} $$

<figure><img src="image_29.png" /></figure>

<figure><img src="image_30.png" /></figure>
# Selecting a probe (2/2)

## Polynomial expansion

$$ \psi_{\alpha p}(\mathbf{x}, \omega) = [f_{\alpha}^{\text{loc}}(\mathbf{x}, \omega)]^{p} $$

* Motivated by the $f_{\alpha}(1-f_{\alpha})$ prefactor in classical HS bounds

* Makes computations tractable.

## The case of spherical windows

**No improvement on classical HS bounds!**

Brisard, Mechanics & Industry 18(2), 2017

<figure><img src="image_31.png" /></figure>
# Where did we fail?

## For isotropic microstructures and spherical windows

$$ \Psi_{\alpha p \beta q}(\mathbf{r}) = \Psi_{\alpha p \beta q}(\|\mathbf{r}\|) $$

**Isotropic probe!**

## The P-tensors become trivial

$$ \begin{aligned} \mathbf{P}_{\alpha p \beta q} &= \hat{\Psi}_{\alpha p \beta q}(\mathbf{0}) \mathbf{P}_0 + \lim_{\delta \to 0} \int_{\|\mathbf{r}\| \ge \delta} \hat{\Psi}_{\alpha p \beta q}(\|\mathbf{r}\|) \mathbf{\Gamma}_0^{\infty}(\mathbf{r}) \, dV_r \\ &= \hat{\Psi}_{\alpha p \beta q}(\mathbf{0}) \mathbf{P}_0 \end{aligned} $$

**Physical interpretation**

**Same trial stress-polarization!**
# What can be done?

**Use anisotropic probes (work in progress)**

$\Psi_{\alpha\rho\beta q}(\mathbf{r})$ must have an angular dependency!

**Anisotropic local volume fraction**

<figure><img src="image_32.png" /></figure>
# As a conclusion

## Polarization techniques

* Based on a variational principle

* No admissibility condition for the trial stress-polarization fields

* Can provide bounds or estimates of the effective properties

* Span the whole range of homogenization techniques

## The quest for improved bounds

* Highlights connections between “probes” and stat. descriptors

* Does not work (yet!)

* Sheds a new light on the “curse of isotropy”

## Perspectives

* Improved bounds with anisotropic probes

* Non-linearities (OK for FFT-based methods)
Thank you for your attention!

sebastien.brisard@ifsttar.fr
http://navier.enpc.fr/BRISARD-Sebastien
http://sbrisard.github.io

<figure><img src="image_33.png" /></figure>

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