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@@ -32,33 +32,44 @@ analyze these tensors, simplifying workflows for materials scientists an enginee
 
 # Statement of Need
 
-Strain analysis is crucial in fields such as materials science and engineering. Elasticity 
-tensors, which govern the stress-strain relationships in materials, are complex to compute and analyze, especially when 
-accounting for crystal or material symmetries. Existing software solutions often lack accessibility or do not fully 
-support complex symmetry operations, making them challenging for non-specialist users or those seeking rapid prototyping
-and analysis.
+In continuum mechanics, the deformation of a material is described by the second-order strain tensor (usually denoted 
+$\mathbf{\varepsilon}$) whereas the stress is described by the second-order Cauchy's stress tensor ($\mathbf{\sigma}$).
+Under the linear elasticity assumption, the relationship between the $\mathbf{\varepsilon}$ and $\mathbf{\sigma}$ is 
+given through the fourth-order stiffness tensor $\mathbf{C}$ with:
 
-Elasticipy addresses this gap by providing:
+$$\sigma_{ij}=C_{ijk\ell}\varepsilon_{k\ell}$$
 
-  - Intuitive Python-based APIs for defining and manipulating second- and fourth-order tensors, such as strain, stress
+where $\mathbf{C}_{ijk\ell}$ denotes the $ijk\ell$-th component of $\mathbf{C}$. Its value 
+depends on the material, whereas its shape (set of zero-components, or linear relationships between them) depend on the 
+material's symmetry [@nye].
+
+Existing software solutions often lack accessibility or do not fully support complex symmetry operations, making them 
+challenging for non-specialist users or those seeking rapid prototyping and analysis. Elasticipy addresses this gap by 
+providing:
+
+  - intuitive Python-based APIs for defining and manipulating second- and fourth-order tensors, such as strain, stress
 and stiffness;
 
-  - Support for standard crystal symmetry groups [@nye] to facilitate the definition of stiffness/compliance components; 
+  - support for standard crystal symmetry groups [@nye] to facilitate the definition of stiffness/compliance components; 
+
+  - visualization tools for understanding directional elastic behavior (Young modulus, shear modulus and Poisson ratio);
 
-  - Visualization tools for understanding directional elastic behavior.
+  - a collection of built-in methods to easily and efficiently perform fundamental operations on tensors (rotations, 
+products, invariants, statistical analysis etc.).
 
 Unlike other software such as pymatgen [@pymatgen] or Elate [@elate], Elasticipy emphasizes ease of use, flexibility, 
 and integration with existing Python workflows. In addition, it introduces the concept of *tensor arrays*, in a similar 
-way as in MTEX [@MTEX], allowing to process thousands of tensors at once (e.g. rotation of tensors) with simple and 
-highly efficient commands. In order to highlight the performances of Elasticipy, \autoref{fig:compa} shows the wall-time 
-required to perform two basic operations on tensors, as functions of the number of considered tensors. This evidences 
-that, when processing large number of tensors ($>10^3$), basic operations on tensors are 1 to 2 orders of magnitude 
-faster when using Elasticipy than pymatgen.
+way as in MTEX [@MTEX], allowing to process thousands of tensors at once with simple and highly efficient commands. In 
+order to highlight the performances of Elasticipy, \autoref{fig:compa} shows the wall-time required to perform two basic 
+operations on tensors, as functions of the number of considered tensors. This evidences that, when processing large 
+number of tensors ($>10^3$), basic operations on tensors are 1 to 2 orders of magnitude faster when using Elasticipy 
+than pymatgen. These performances are reached by taking advantage of `numpy`'s array 
+broadcasting.
 
 ![Performance comparison between Elasticipy and pymatgen.\label{fig:compa}](ElasticipyVSpymatgen.png){ width=75% }
 
-Nevertheless, as tensor algebra is not the core of pymatgen, Elasticipy supports conversion to pymatgen, and vice versa. 
-It also allows direct imports of elastic data from 
+Nevertheless, as tensor algebra is not the core of pymatgen, Elasticipy supports conversion to pymatgen, and vice versa,
+allowing to work with these packages altogether. Elasticipy also allows direct imports of elastic data from 
 [the Materials Project](https://next-gen.materialsproject.org/) [@MaterialsProject].
 
 # References