Together with our post-doctoral fellow Othmane Zerhouni, my colleague
Kostas Danas
(LMS, École
polytechnique) and I have recently
published a new paper in *International Journal of Engineering
Science*. The paper is titled “Quantifying the effect of two-point
correlations on the effective elasticity of specific classes of random
porous materials with …

# Welcome!

You’ve reached the blog of Sébastien Brisard. My most recent posts are listed below. A full list of posts can be found in the “Archives” section. If you are wondering why this blog is relatively “low-tech”, read the “About this blog” section.

While you’re at it, you can also visit the documentations of some of my projects

- https://sbrisard.github.io/janus: the documentation of the
`Janus`

library, - https://sbrisard.github.io/pw85: the
documentation of the
`PW85`

library.

I hope you will enjoy reading this blog!

## 13 May 2021 — Numerical analysis of a spring mesh

In this post, we compute numeric values of the apparent stiffness introduced in this post. Our goal is to find the solution to the general problem depicted below.

We use a numeric approach that is akin to the finite element method. More precisely, we minimize the total potential energy of …

## 12 May 2021 — Symbolic analysis of a spring mesh

In this post, we compute symbolic expressions of the apparent stiffness introduced in this post. Our goal is to find the solution to the general problem depicted below.

## Outline of the derivation

We consider a simplified case, where the mesh is square and vertical and horizontal springs have equal stiffnesses …

## 09 May 2021 — What is homogenization? Part 5: introducing the representative volume element

In the previous instalment of this series on homogenization, we derived the homogenized properties of a rectangular spring mesh. Our goal in this post is to analyse the “convergence” towards the homogenized stiffness; meanwhile, we will discuss the size of the so-called representative volume element (RVE).

## In the previous episode …

## 08 December 2020 — What is homogenization? Part 4: a first example

In the previous instalment of
this series on homogenization, we discussed volume and ensemble averages for
random heterogeneous materials. In view of introducing *statistical* and
*representative* volume elements, we will introduce in this post our first
homogenization example, namely a two-dimensional, rectangular mesh of springs
that is loaded in its …

## 25 November 2020 — On the stiffness matrix of a linear spring

This post is not going to be the most exciting I have ever written. We will derive the expression of the elastic energy of a linear spring as a function of the displacement of its two end-points. I will need this expression in my series on “What is homogenization”. Having …

## 21 April 2020 — What is homogenization? Part 3: ensemble averages vs. volume averages

In the previous instalment of this series on homogenization, we discussed homogenization of a distribution of black dots on a white background to a uniform shade of gray. In that example, homogenization boils down to evaluating the fraction of the total area occupied by the dots: this is the so-called …

## 08 April 2020 — What is homogenization? Part 2: experimental vs. theoretical homogenization

In the previous instalment of this series, we introduced homogenization and the separation of scales. In the present post, we will discuss two strategies to carry out homogenization: the experimental and the theoretical approaches. To do so, we will again use the analogy with the halftoning technique.

Remember that we …

## 02 April 2020 — What is homogenization? Part 1: on the separation of scales

Most of my research activities deal with upscaling the mechanical
properties of heterogeneous materials. This is also known as
homogenization. So what *is* homogenization? This series explores this
question; it is largely based on the introduction to my Habilitation
defense.

In order to introduce homogenization, we will draw inspiration from …

## 26 March 2018 — On the periodic-plus-smooth decomposition of an image, part 7: improved implementation of Moisan's algorithm

In the previous instalment of this series, we implemented Moisan's (2011) efficient algorithm to compute the periodic-plus-smooth decomposition of an image. This algorithm relies heavily on the discrete Fourier transform, and already improves quite a lot over our previous conjugate gradient-based implementation. In the present post, we will show that …

## 19 March 2018 — On the periodic-plus-smooth decomposition of an image, part 6: minimizing the energy, the clever way

In the previous instalment of this series, we computed Moisan's (2011) periodic-plus-smooth decomposition of an image by means of the conjugate gradient method. This worked like a charm, but was fairly inefficient, owing to the iterative nature of the method. Moisan actually showed that the whole decomposition could be computed …

## 12 March 2018 — On the periodic-plus-smooth decomposition of an image, part 5: minimizing the energy, the clumsy way

In this post, we will compute Moisan's (2011) periodic-plus-smooth decomposition of an image by direct minimization of the energy introduced in the second instalment of this series. More precisely, $u$ being a $m\times n$ image, we will minimize the function $F(v, u)$ over the space of $m\times …

## 05 March 2018 — On the periodic-plus-smooth decomposition of an image, part 4: implementing the linear operators

In the previous instalment of this series, we introduced the linear operators $Q_1$ and $Q$ that allow to define Moisan's (2011) periodic-plus-smooth decomposition $(p, s)$ of an image $u$ as follows

$$s=\operatorname*{arg\,min}_v F(v, u)\quad\text{and}\quad p=u-s,$$

with

$$F(v, u)=\langle …

## 26 February 2018 — On the periodic-plus-smooth decomposition of an image, part 3: the energy as a quadratic form

In the previous instalment of this series, we introduced the periodic-plus-smooth decomposition of an image as a pair of images which minimizes an energy functional. Observing that this energy is a quadratic form, the purpose of this post is to derive closed form expressions of the underlying linear operators. These …

## 19 February 2018 — On the periodic-plus-smooth decomposition of an image, part 2: defining the decomposition

In the previous
instalment
of this series, we discussed the need for periodic images. Although not all
images are periodic, some image analysis techniques are best performed in
Fourier space (using the fast Fourier transform). Applying Fourier-based
techniques to images that are *not* periodic (as is often the case) generates …

## 12 February 2018 — On the periodic-plus-smooth decomposition of an image, part 1: introduction

In this new series, we will explore the so-called *periodic-plus-smooth*
decomposition of an image, introduced by Moisan in 2011. This series is largely
based on Moisan's paper, called Periodic plus Smooth Image
Decomposition (the author's version
can be found on HAL). Besides
introducing a quite smart technique (that I do …

## 17 November 2017 — Announcing the defense of my “Habilitation à Diriger des Recherches”

On monday 20th, november 2017 (at 13:30), I will be defending my "Habilitation à Diriger des Recherches" at École des Ponts ParisTech in front of the following jury

- Renald Brenner (Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7190, Institut Jean le Rond d\'Alembert, F-75005, Paris, France),
- Samuel …

## 07 August 2017 — New conference paper on variational estimates of poroelastic coefficients

I recently had the pleasure to give a talk on “Variational estimates of the poroelastic coefficients” at the 6th Biot Conference on Poromechanics that took place at École des Ponts ParisTech on July 9-13, 2017. This is joint work with my colleague Siavash GHABEZLOO. It has been ongoing for years …

## 31 May 2017 — Scrapy-ing the NIST X-ray Attenuation Databases

I am currently preparing with two other colleagues a review paper on X-ray tomography of cementitious materials, for which I need to retrieve tabulated values of the X-ray mass attenuation coefficients of all elements. NIST provides such data (see X-Ray Mass Attenuation Coefficients by J. H. Hubbell and S. M …

## 28 April 2017 — New paper on one-dimensional compression of granular materials

My student Mohamed Hassan Khalili has recently had one of his papers
published in *Physical Review E*. The paper is called “A numerical study
of one-dimensional compression of granular materials”. I have only been
very marginally involved in this DEM study, which is divided in two
parts.

Part I is …

## 26 April 2017 — New paper on Discrete Digital Projections Correlation

Our paper “Discrete Digital Projections Correlation: a
reconstruction-free method to quantify local kinematics in granular
media by X-ray tomography” has recently been published in *Experimental
Mechanics*. The abstract is reproduced below

We propose a new method to measure the translations and rotations of each individual grain in a granular material …

## 23 February 2017 — New paper out!

My paper “Towards improved Hashin--Shtrikman bounds on the effective
moduli of random composites” has recently been published in *Mechanics &
Industry*. The abstract is reproduced below

The celebrated bounds of Hashin and Shtrikman on the effective properties of composites are valid for a very wide class of materials. However, they incorporate …

## 27 June 2016 — Orientation correlations among rice grains, part 8: estimating the correlations

In the previous
instalment
of this series, we have analyzed the morphology of the rice grains. In
particular, we have defined their *orientation* as that of the major axis of
inertia. We are now in a position to quantify the statistics of the
orientations. We will first discuss one-point statistics …

## 13 June 2016 — New paper

My paper “Reconstructing displacements from the solution to the periodic
Lippmann--Schwinger equation discretized on a uniform grid” has recently
been published in *International Journal of Numerical Methods in
Engineering*. The abstract is reproduced below

Uniform grid solvers of the periodic Lippmann--Schwinger equation have been introduced by Moulinec and Suquet for …

## 19 February 2016 — Orientation correlations among rice grains, part 7: analysis of the shape of the grains

$\newcommand{\tens}{\mathbf}\newcommand{\D}{\mathrm{d}}$

In the previous instalment of this series, we have segmented the 3D image of the assembly of rice grains. In other words, each voxel of the image is attributed the label of the grain to which it belongs. Remember the initial goal of …

## 08 December 2015 — New paper published

My student Vinh Phuc Tran has recently had one of his papers published
in *Mechanics of Materials*. The paper is called *Stochastic modeling of
mesoscopic elasticity random field*. The abstract is reproduced below

In the homogenization setting, the effective properties of a heterogeneous material can be retrieved from the solution …

## 30 September 2015 — Orientation correlations among rice grains, part 6: segmentation

In the previous
instalment
of this series, I showed that a convincing binary image could be produced from
the gray level 3D reconstruction of the assembly of rice grains, using Otsu's
threshold. However, I intend to carry out statistical analyses of the *grains*
themselves in the subsequent instalments. Therefore, instead …

## 09 July 2015 — Orientation correlations among rice grains, part 5: thresholding

In the previous instalment of this series, we used the circle Hough transform to find the boundary of the sample and define the circular ROI. Within this ROI, we now need to segment the rice grains. In other words, starting from a gray-level image (Fig. 1, left), we want to …

## 06 July 2015 — When a thin shell is not so thin, part 3: the thick shell solution

Finding the stress resultant and stress couple in a spherical pressure vessel seems easy enough. Well, this apparently simple problem allows us to highlight subtle thickness effects within shells. It should be emphasized again that these thickness effects have nothing to do with shear corrections. Indeed, shear stresses are null …

## 01 July 2015 — When a thin shell is not so thin, part 2: the 3D, exact solution

In the previous instalment of this series, we analysed a spherical pressure vessel by means of Koiter's linear theory of thin shells. We found the somewhat unexpected result that the stress couple was not null. Besides, we also found that the stress resultant was slightly different from the well-known value …

## 08 June 2015 — When a thin shell is not so thin, part 1: Koiter's linear theory

In structural analysis, thick beams (resp. plates) usually refer to shear
deformability, and the Timoshenko beam
theory (resp.
Mindlin–Reissner plate
theory).
With curved elements however (e.g. curved beams or shells), the situation is
more subtle, as thickness corrections may be necessary even in shells *where the
shear stress …*

## 29 May 2015 — Orientation correlations among rice grains, part 4: defining the ROI

In the previous instalment of this series, we obtained binned slices of the sample. Fig. 1 below displays a typical example of these binned slices. We now want to segment the rice grains. However, the analysis (in particular, Otsu thresholding) might be perturbed by the fact that the walls of …

## 30 March 2015 — Orientation correlations among rice grains, part 3: intermezzo – Binning images

The full reconstructed image resulting from the tomography experiment described in the second instalment of this series is a 1747×1751×688 stack. The voxel size is about 0.030 mm. This is far too much for the purpose of the present study, since all we are interested in is …

## 10 March 2015 — Orientation correlations among rice grains, part 2: acquisition of tomography images

In the first instalment of this series, I briefly introduced orientational order in assemblies of anisotropic particles, and proposed an illustration based on synthetic microstructures. But what I really intend to do in this series is to show how orientational order can be quantified in a “real-life” sample, namely a …

## 23 February 2015 — Orientation correlations among rice grains, part 1: introduction

In this series, I will explore the notion of orientational order in random
packings of anisotropic (flat or elongated), hard particles. By orientational
order, I mean that particles which are close to each other tend to adopt the
same orientation. This leads to strong *local* anisotropy, while the packing may …

## 26 February 2014 — Decomposition of transverse isotropic, fourth-rank tensors

In a previous post, I introduced the fourth-rank spherical and deviatoric projection tensors. Any isotropic fourth-rank tensor is a linear combination of these two tensors; in other words, the space of isotropic fourth-rank tensors (with minor and major symmetries) is of dimension 2. Similarly, it can be shown (Walpole, 1984 …

## 19 February 2014 — On the double dot product

The double dot product of two tensors is the contraction of these tensors with
respect to the last two indices of the first one, and the first two indices of
the second one. Whether or not this contraction is performed on the *closest*
indices is a matter of convention. In …

## 31 January 2014 — The elastic acoustic tensor and its inverse

In this post, I will introduce the acoustic tensor of linearly elastic
materials. Closed-form expressions of the inverse of this tensor can be derived
in the case of *isotropic* materials. This will later come in handy to derive
closed-form expressions of the periodic Green operator for strains.

We consider a …

## 12 January 2014 — Elastic constants of an isotropic material, part 3: putting it all together

In the previous instalments of this series (see Part 1 and Part 2), I have shown that regardless of the dimensionality (3D or plane strain elasticity), the constitutive law of an isotropically elastic material reads

$$\sigma_{ij}=\kappa\varepsilon_{kk}\delta_{ij}+2\mu\left(\varepsilon_{ij}-\frac{\varepsilon_{kk …

## 02 January 2014 — Extract images from PDF files

I recently scanned a document as a PDF file, and I wanted to retrieve
the embedded images. I really mean *extract* the images, and not
convert the pages of the document to images (which would entail loss
of data). There is a nice command-line tool to do that: `pdfimages`

,
which …

## 29 December 2013 — Elastic constants of an isotropic material, part 2: plane strain elasticity

In the previous instalment of this series, I introduced the constitutive law of an isotropically elastic material, and the related material constants, within the framework of 3D elasticity. I will now address the case of plane strain elasticity.

Problems of plane strain elasticity are defined as problems where the following …

## 05 December 2013 — Elastic constants of an isotropic material, part 1: 3D elasticity

Among all classes of materials, the class of linearly elastic and isotropic materials is probably the simplest. Such materials are defined by two elastic constants: for example, Young\'s modulus $E$ and Poisson\'s ratio $\nu$. However, depending on the situation, this pair of constants might not be the most …

## 05 November 2013 — New accepted manuscript

One of my manuscripts has just been accepted for publication in the
International Journal of Solids and Structures. It is called *A
variational form of the equivalent inclusion method for numerical
homogenization*. The abstract is reproduced below

Due to its relatively low computational cost, the equivalent inclusion method is an …

## 31 August 2013 — New accepted manuscript

It is my pleasure to announce that one of my papers has recently been
accepted for publication in the Journal of the Mechanics and Physics of
Solids. It's called *New boundary conditions for the computation of the
apparent stiffness of statistical volume elements*.

It introduces a new set of boundary …