# Experimental and modeling study on earthen sediment-based bricks with and without flax fibers: A finite element analysis

Alioune Nacro, Philippe Karamian-Surville, Daniel Levacher, Mazhar Hussain

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## To cite this version:

Alioune Nacro, Philippe Karamian-Surville, Daniel Levacher, Mazhar Hussain. Experimental and modeling study on earthen sediment-based bricks with and without flax fibers: A finite element analysis. Results in engineering, 2025, 25, pp.104536. 10.1016/j.rineng.2025.104536. hal-05014754

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Submitted on 31 Mar 2025

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Research paper

# Experimental and modeling study on earthen sediment-based bricks with and without flax fibers: A finite element analysis

Alioune Nacro<sup>a,*</sup>, Philippe Karamian-Surville<sup>a, ID,*</sup>, Daniel Levacher<sup>b</sup>, Mazhar Hussain<sup>c</sup>

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

## ARTICLE INFO

**Keywords:**

Dredged sediments

Flax fibers

Earthen bricks

Numerical simulation

Experimental data

Homogenization

Code_Aster

## ABSTRACT

This paper investigates the improvement of earthen sediment-based bricks through the use of natural flax fibers to enhance their mechanical properties. The experimental tests were designed to develop numerical modeling based on material guidelines that have been partially followed. To facilitate the precise placement of flax fibers within the earthen sediment, periodically distributed holes have been deliberately made on both sides of the molds, ensuring the proper positioning of the flax fibers. The three-point bending test was conducted to assess the mechanical strength of the earthen bricks, providing insights into their stiffness and resistance to fracture. Non-linear numerical modeling was employed to gain a deeper understanding of fatigue behavior and crack propagation in earthen bricks under stress. The comparison of raw experimental data with numerical simulations validated the consistency of the results without adjustments, while the application of the homogenization principle improved data approximation. These findings confirm that the numerical homogenization method is the most effective approach for ensuring the validity of the numerical model and should be systematically applied to this type of problem.

## 1. Introduction

Bricks are an essential component of civil engineering construction, widely used due to their durability, structural integrity, and availability. However, the traditional production process for bricks relies heavily on fossil fuels, primarily due to the firing stage, which is integral to achieving the required mechanical strength. This reliance contributes significantly to greenhouse gas emissions and environmental degradation, raising concerns about the sustainability of conventional brick manufacturing. To mitigate these environmental impacts, researchers and industries have proposed alternative methods for producing environmentally friendly bricks. These methods focus on integrating agricultural waste and local natural resources, such as flax, coconut, banana, and palm fibers, along with river sediments, into the production process. The use of these materials allows bricks to be dried in the open air, eliminating the need for energy-intensive firing processes. These innovations aim to enhance mechanical performance while simultaneously reducing reliance on non-renewable resources, as detailed in [1-4]. Nevertheless, despite their promise, these eco-friendly bricks face challenges

related to material consistency, mechanical reliability, and large-scale adoption, necessitating rigorous testing and validation.

Among natural reinforcements, flax fibers have emerged as a highly promising material for enhancing the mechanical and thermal properties of construction materials. Flax fibers are known for their lightweight nature, high tensile strength, and biodegradability, making them an attractive choice for sustainable construction applications. Studies have demonstrated that the incorporation of both short and long flax fibers into mortar and concrete significantly improves tensile strength [5,6], optimizes thermal behavior under diverse environmental conditions [7], and enhances durability under varying humidity levels [8]. These findings underscore the potential of flax fiber-reinforced composites as viable and sustainable alternatives to conventional building materials. Furthermore, extensive research has explored the integration of natural fibers as replacements for traditional reinforcements, with a focus on evaluating the mechanical properties of composites and their applicability in construction. Interested readers can refer to [9] for a comprehensive overview of these developments.

<footnote><sup>a</sup> Normandy University, UNICAEN, UMR 6139 CNRS, Nicolas Oresme Mathematics Laboratory, 14000, Caen, France</footnote>

<footnote><sup>b</sup> Normandy University, UNICAEN, UMR 6143 CNRS-M2C, 14000, Caen, France</footnote>

<footnote><sup>c</sup> CESI LINEACT UR 7527, Strasbourg, 67380, France</footnote>

<footnote>* Corresponding authors.</footnote>

<footnote>E-mail addresses: alioune.nacro@unicaen.fr (A. Nacro), philippe.karamian@unicaen.fr (P. Karamian-Surville), daniel.levacher@unicaen.fr (D. Levacher), mhussain@cesi.fr (M. Hussain).</footnote>
Incorporating flax fibers into earthen bricks has garnered significant attention as a sustainable approach to improving the mechanical performance of these materials while preserving their environmental benefits. Earthen bricks, which are unfired and made from natural sediments, are increasingly recognized as eco-friendly alternatives to traditional fired bricks. However, their mechanical properties often limit their practical applications. The addition of flax fibers addresses these limitations by enhancing tensile strength, reducing brittleness, and improving durability. To fully understand the behavior of such fiber-reinforced composites, researchers have employed a combination of experimental techniques [10–12] and advanced numerical simulations [13–15]. Homogenization techniques, particularly the full-field method, have been widely utilized to model these composites as periodic materials with representative volume elements. This approach provides accurate predictions of their mechanical properties while minimizing the need for extensive experimental testing.

Despite significant advancements in modeling natural fiber-reinforced earthen bricks, several gaps remain in understanding their mechanical behavior under real-world conditions. Existing studies have primarily focused on linear elasticity, which is insufficient for accurately simulating complex stress-strain relationships and crack propagation encountered in practical applications. These limitations underscore the need for innovative approaches that integrate experimental validation with advanced numerical modeling to bridge the gap between theoretical predictions and practical performance.

The literature provides substantial insights into the modeling and manufacturing of natural fiber-reinforced earthen bricks. For instance, [16] offers a detailed review of previous research, focusing on techniques for fabricating sediment-based bricks reinforced with natural fibers. This paper also explores finite element analysis (FEA) as a tool for modeling the mechanical behavior of such composites, highlighting its effectiveness in simulating both linear and nonlinear responses [17–19]. While the reviewed studies emphasize the importance of experimental validation [20,21], they also identify limitations in existing methodologies [22,17–21,23]. These gaps present an opportunity for advancements in modeling techniques, particularly through numerical homogenization methods. This study adopts asymptotic techniques to model earthen bricks reinforced with flax fibers as equivalent materials—homogeneous yet anisotropic—enabling comprehensive analysis under linear and nonlinear elasticity theories.

Homogenization techniques are integral to modeling composite materials. Two prominent approaches, the mean-field method [24–26] and the full-field method [27–29], are widely regarded as reliable. In this work, we employ the full-field method, leveraging asymptotic and multiscale techniques [30–33] to model fiber-reinforced earthen bricks as periodic materials. This methodology effectively captures the mechanical behavior of the composites, facilitating accurate property estimation while reducing the computational cost of simulations.

Modeling fiber-reinforced unfired bricks is crucial due to the challenges and costs associated with experimental testing. Numerical simulations provide an efficient, adaptable, and reliable alternative, capable of addressing complex phenomena such as non-linearity, fracture mechanics, and thermal dynamics. This study utilizes the finite element method, as demonstrated in [34], and the open-source Code_Aster software for conducting high-precision numerical simulations [35]. Code_Aster is well-suited for structural mechanics and thermal analysis, offering versatility and accessibility through its open-source framework [34]. Its capabilities make it an ideal tool for advancing the modeling and analysis of fiber-reinforced composites in construction.

This article presents an integrated approach combining state-of-the-art numerical modeling with experimental validation to advance the understanding of fiber-reinforced earthen bricks. Specifically, finite element simulations using Code_Aster are employed to investigate the impact of flax fiber reinforcement on the mechanical properties of bricks manufactured and dried without kilns, aiming to minimize greenhouse gas emissions. Experimental investigations include three-point bending

tests to assess mechanical strength, while numerical modeling focuses on simulating the nonlinear behavior of bricks under load. The use of numerical homogenization, including the Lippmann-Schwinger equation and fast Fourier transform (FFT)-based techniques, enables accurate predictions of mechanical behavior. This study highlights the importance of validating numerical models and proposes directions for future research to improve the durability and performance of eco-friendly building materials by incorporating natural fibers.

This work contributes to sustainable construction by introducing innovative methods to enhance the mechanical performance of earthen bricks. By developing a robust numerical simulation framework, this study aims to optimize the design and application of fiber-reinforced earthen bricks in environmentally conscious construction.

The structure of this paper is as follows: The initial section describes the experimental methodology, followed by an analysis of numerical modeling approaches. The numerical section is divided into finite element modeling using Code_Aster and homogenization techniques for property estimation. Finally, the conclusions and future research directions are presented.

## 2. Experimental study

According to the standard [36], the manufacturing of earthen bricks has been carried out in five steps, the details of which are provided in Section 2.2. A series of nine earthen bricks with dynamic compaction (three earthen bricks without fibers, three earthen bricks with 16 horizontal long fibers, and three earthen bricks with 64 horizontal long fibers) has been produced. Similarly, another series of nine earthen bricks with static compaction (three earthen bricks without fibers, three earthen bricks with 16 long fibers, and three earthen bricks with 64 long fibers) has also been made to explore the challenges of compaction in manufacturing earthen bricks. Some conclusions have been drawn from these experiments.

Finally, the earthen bricks, both with and without fibers, have been subjected to the three-point bending test in accordance with the AFNOR NF EN 1015-11 standard [37].

### 2.1. Materials

#### 2.1.1. Raw materials

Sediment dredged from the Usumacinta River in Jonuta, located in Tabasco province, Mexico, is utilized in this study with flax fibers to manufacture fiber-reinforced earthen bricks. Sediments are sampled from various locations in the city of Jonuta. The sediments used in this study are referenced as J3-9C, where J denotes Jonuta, 3 represents the sampling site, and 9C corresponds to the barrel number. In order to use these sediments for brick manufacturing according to standards [38,39], their various characteristics are analyzed to determine their suitability for earthen bricks. Some of these geotechnical characteristics are presented in Table 1.

Here are brief explanations for each of the symbols used in Table 1:

* Clay (%): Represents the percentage of clay particles in the sediment.

* Silt (%): Indicates the percentage of silt particles in the sediment.

* Sand (%): Denotes the percentage of sand particles in the sediment.

* $CaCO_3$ (%): Reflects the percentage of calcium carbonate ($CaCO_3$) in the sediment. Calcium carbonate is a common sediment component, often derived from the shells of marine organisms.

* $W_{opt}$ (%): Refers to the optimum moisture content of the sediment, which is the moisture level at which the sediment achieves its maximum dry density when compacted.

* Organic matter (%): Represents the percentage of organic material in the sediment.

* MBV (g/100 g): Stands for Methylene Blue Value, which measures the quantity of clay minerals in the sediment. It indicates the sediment's ability to absorb water and swell.
<table><caption>**Table 1**
Sediment characteristics.</caption><thead><tr><th>Sediments<br>Unity</th><th>Clay<br>(%)</th><th>Silt<br>(%)</th><th>Sand<br>(%)</th><th>CaCO<sub>3</sub><br>(%)</th><th>W<sub>opt</sub><br>(%)</th><th>Organic matter<br>(%)</th><th>MBV<br>(g/100 g)</th><th>pH</th><th>LL<br>(%)</th><th>PL<br>(%)</th><th>ρ<sub>sed</sub><br>(g/cm<sup>3</sup>)</th></tr></thead><tbody><tr><td>J3-9C</td><td>4.66</td><td>36.3</td><td>59</td><td>7.84</td><td>19.3</td><td>4.48</td><td>2.73</td><td>7.91</td><td>34.76</td><td>19.83</td><td>2.63</td></tr></tbody></table>

<table><caption>**Table 2**
Characteristics of flax fibers.</caption><thead><tr><th>Type of fibers<br>Unity</th><th>Density<br>(g/cm<sup>3</sup>)</th><th>Absorption Coefficient<br>(%)</th><th>Modulus of elasticity<br>(GPa)</th><th>Tensile strength<br>(MPa)</th></tr></thead><tbody><tr><td>Flax</td><td>1.19-1.55</td><td>63-330</td><td>4.4-110</td><td>93-2000</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Long raw flax fibers.</figcaption></figure>

**pH:** Represents the acidity or alkalinity of the sediment. A pH of 7 is neutral; values below 7 indicate acidity, while values above 7 indicate alkalinity.

**LL (%):** Liquid Limit, the water content at which the sediment transitions from a plastic to a liquid state. It is a key indicator of the sediment's consistency.

**PL (%):** Plastic Limit, the water content at which the sediment begins to exhibit plastic behavior. It represents the minimum moisture level at which the sediment can be molded.

**ρ<sub>sed</sub> (g/cm<sup>3</sup>):** The density of sediment particles, representing the mass per unit volume of the sediment material itself, excluding void spaces.

### 2.1.2. Flax fibers

Linen is a resistant plant fiber, as shown in Fig. 1. Flax fibers, used for manufacturing earthen bricks, measure 160 mm in length and 1 mm in diameter. The twist rate, expressed as the number of turns per meter, has been determined by the supplier Depestele in Rouen; however, this data is confidential. Table 2 summarizes some characteristics of flax fibers.

Here is an explanation of the symbols used in Table 2:

* **Density (g/cm<sup>3</sup>):** Indicates the mass per unit volume of flax fibers, measured in grams per cubic centimeter (g/cm<sup>3</sup>). This value varies depending on the type of flax and its processing.

* **Absorption coefficient (%):** Measures the ability of flax fibers to absorb moisture, expressed as a percentage. This coefficient reflects the maximum moisture the fibers can absorb relative to their dry weight.

* **Modulus of elasticity (GPa):** Indicates the stiffness of flax fibers, measured in gigapascals (GPa). A higher modulus implies greater stiffness and less deformation under stress.

* **Tensile strength (MPa):** Represents the maximum stress flax fibers can withstand when stretched before breaking, expressed in megapascals (MPa). This strength varies with the fiber type and processing method.

### 2.2. Brick making process

According to the standard [36], the manufacturing of earthen bricks involves five distinct steps. The objective is to produce earthen bricks that undergo a natural drying process, ensuring the mechanical resistance recommended by the standards.

#### 2.2.1. The sediment grinding

The grinding operation consists in making the mass of sediment homogeneous (see Fig. 2). The sediment is transported from Mexico in sealed barrels to the laboratory in France. The sediments have been sieved with a 2 mm sieve to obtain homogeneous sediments, as recommended by the [38] standard.

#### 2.2.2. The sediment mixing

The percentage of water used to mix the sediments was determined using normal Proctor test parameters and amounts to 19.3% in accordance with standard [36], as illustrated in Fig. 3. The next step was to mix the sediments using a machine for five minutes.

#### 2.2.3. The molding and compaction

The molding process prepares three earthen bricks simultaneously. The dimensions of the prismatic mold used for manufacturing raw bricks are 40 mm in width, 40 mm in height, and 160 mm in length, conforming to standard [40]. The first series of samples includes the manufacturing of earthen bricks without fibers, bricks with 16 horizontal long fibers, as shown in Fig. 4, and 64 horizontal long fibers, as shown in Fig. 5. To incorporate the 16 or 64 rows of fibers, two molds with 16 or 64 holes distributed across 4 or 8 rows and 4 or 8 columns are used, as shown in Figs. 4 and 5. The spacing between holes varies for each mold. Specifically, the mold with 16 holes features holes of 1.5 mm in diameter with a spacing of 5 mm. The first and last rows and columns of holes are spaced 2.5 mm apart, as shown in Fig. 5.

To manufacture earthen bricks, two types of compaction methods have been used: dynamic and static compaction. A total of nine earthen bricks have been made using each method, divided into three groups: three bricks without fibers, three bricks with 16 long fibers, and three bricks with 64 long fibers. The aim of the study is to explore the challenges associated with each compaction method in the manufacture of earthen bricks and to draw relevant conclusions.

#### 2.2.3.1. *Dynamic compaction*

Dynamic compaction is a commonly used method to densify soils. The process involves dropping a weight several times onto the bulk sediment, respecting a regularly spaced mesh. The weight and drop height applied to the raw brick depend on the degree of compaction required. In this case study, the weight was 10.231 N, and the drop height was 178 mm. The impact of the falling weight generated waves that contributed to the densification of the sediments. These impacts have been able to compact the sediment in two layers at optimal dry density, as shown in Fig. 6. The tamper size is 40 mm by 40 mm. Therefore, the area compacted by each tamping is 1600 mm². The energy supplied E is given by the following formula (equation (1)).
<figure><img src="image_5.png" /><figcaption>(a) Raw sediments from Mexico</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(b) 2 mm sieve</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(c) Homogeneous sediments</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(a) Mixing sediment with water</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(b) The mixer</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(c) Sediments mixed</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 4. Specific three samples wooden mold with 16 fibers.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 5. Specific three samples wooden mold with 64 fibers.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Fig. 6. Dynamic compaction of three earthen bricks without fibers.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 7. Dynamic compaction of three earthen bricks with 16 long fibers.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 8. Dynamic compaction of three earthen bricks with 64 long fibers.</figcaption></figure>

$$E = \frac{M \times H \times n \times N}{V} \qquad (1)$$

where:

M is the mass of the element dropped on the bulk sediment;

H, the drop height;

n, represents the number of layers, which is 2 in this case;

N, the number of drops, 42 in this case study; and

V, the volume of the brick is (40 x 40 x 160 mm³).

So, the delivered energy E is 600 kN.m.m⁻³, equivalent to the energy applied in the normal Proctor test [40], commonly used in geotechnical applications and in accordance with standards [41,42].

The mass, number of drops, and drop height applied to the earthen bricks depend on the degree of compaction required. To obtain the energy, a weight of 10.231 N is lifted to a height of 178 mm and dropped 42 times on the earthen bricks. The manufacturing of these bricks is a two-stage process: First, the mold is half-filled with sediment, then dynamic compaction is applied, as shown in Fig. 6a. Next, the mold is filled with sediment and dynamically compacted once more, as shown in Fig. 6b. Finally, the earthen bricks are ready for demolding.

Earthen bricks containing 16 and 64 long fibers have been molded using similar methods. For the bricks with 16 long fibers, fibers were passed through holes using a needle to place them inside the molds, as shown in Fig. 9. Initially, 8 long fibers were arranged in 2 rows of 4 columns (see Fig. 7a), followed by sediment filling and dynamic compaction (see Fig. 7b). After placing the remaining 8 fibers, which were stretched and then covered with sediment, a final dynamic compaction was applied with the same weight (see Fig. 7c).

For the bricks with 64 long fibers, the process first involved placing 32 long fibers in 4 rows of 8 columns, as shown in Fig. 8a, then adding bulk sediment and performing dynamic compaction (see Fig. 8b). After the last 32 long fibers had been placed and the same sediment filling steps were repeated, final dynamic compaction was also applied, in the

<figure><img src="image_16.png" /><figcaption>Fig. 9. Different needles used for long fibers.</figcaption></figure>

same way with the same weight (see Fig. 8c). The two types of bricks are illustrated in Figs. 7 and 8.

2.2.3.2. Static compaction Static compaction, illustrated in Fig. 10, is a method also used to increase sediment density. The process involves depositing a 50 kg mass on the bricks for 16 hours (see Fig. 10b). No standards have been established for earthen bricks compressed by static compaction. The static compaction method has been used to manufacture these earthen bricks in the laboratory, as indicated by Hussain in reference [43].

The manufacturing of earthen bricks with 16 long fibers, as shown in Fig. 11, or 64 long fibers, as illustrated in Fig. 12 in the case of static compaction, involved first placing 16 or 64 fibers in the mold, then filling it with sediment and statically compacting it with a 50 kg mass placed on the mold in three layers for 16 hours, as illustrated in Fig. 10b.
<figure><img src="image_17.png" /><figcaption>(a) Mold filled with sediments</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>(b) Static 50 kg loading</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>(a) Placing of 16 long fibers in the mold</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>(b) Filling with bulk sediment</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>(a) Placing of 64 long fibers in the mold</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>(b) Filling with bulk sediment</figcaption></figure>

<caption>Fig. 12. Static compaction of three earthen bricks with 64 long fibers.</caption>

### 2.2.4. The demolding

After the earthen brick manufacturing process, demolding was carried out as follows: cutting the external long fibers, unscrewing the front of the molds to remove the earthen brick samples, as shown in Fig. 13, for the different types of earthen bricks.

Dynamic compaction is a superior technique to static compaction, particularly for producing earthen bricks with improved structural properties. This method involves active movements and consistent pressure that lead to better densification of the earthen brick material, ensuring a more uniform distribution of sediment particles within the earthen bricks. This uniformity translates into increased density and homogeneity, which are essential for improving the structural integrity and durability of the bricks.

When fibers are incorporated into the bricks, dynamic compaction becomes even more essential. It allows the fibers to be evenly distributed in the sedimentary matrix, which considerably improves the mechanical properties of earthen bricks and their performance under load. This compaction method also improves the adhesion of the fibers to the matrix, further enhancing the strength of the earthen bricks.

Moreover, dynamic compaction effectively reduces voids and imperfections in the brick structure. This is particularly important for fiber-reinforced bricks to ensure optimal adhesion between the fibers and the matrix, thus enhancing the brick's durability and resistance to environmental factors such as humidity and temperature changes.

Finally, dynamic compaction not only optimizes the quality and performance of earthen bricks, whether reinforced with fibers or not, but also offers significant advantages over static compaction in terms of strength, durability, and reliability.

### 2.2.5. The air-drying process

Earthen bricks must still lose a large proportion of their water content after manufacture. Air-drying of earthen bricks takes place at room temperature (20° ± 2°C), with relative humidity varying according to outside weather conditions, usually measured between 40%-60%. The bricks need three weeks to dry and be ready for experimental testing. The behavior of the bricks during drying, whether dynamically or statically compacted, is illustrated in Fig. 14, which shows the water weight loss of the samples.

Dynamically compacted earthen bricks showed no cracks at the end of manufacture, making them suitable for testing.

Static compaction, on the other hand, does not allow energy to be controlled, and above all, bricks compacted in this way show cracks, making them unsuitable for three-point bending and crack detection tests. After these two series of nine earthen bricks, one with dynamic compaction and the other with static compaction, respectively, mechanical testing was performed.

## 2.3. Mechanical tests

The three-point bending test has been carried out in accordance with AFNOR standard NF EN 1015-11 [37]. According to this standard, three samples were used for each bending test to obtain the load-displacement curve (maximum load and corresponding displacement). The series of nine earthen bricks resulting from dynamic compaction were divided into three groups: three earthen bricks without fibers, three earthen bricks with 16 long fibers, and three earthen bricks with 64 long fibers. All the tests were performed at a deformation rate of 0.5 mm/min.
<figure><img src="image_23.png" /><figcaption>Molding</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Unscrewing</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>Removal</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>No fibers</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>16 long fibers</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>64 long fibers</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>No fibers</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>16 long fibers</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>64 long fibers</figcaption></figure>

<caption>(c) Static compaction mode</caption>

<figure><img src="image_32.png" /><figcaption>**Fig. 14.** The weight loss versus time.</figcaption></figure>
<figure><img src="image_33.png" /><figcaption>Fig. 15. The three-point bending test.</figcaption></figure>

The three-point bending test [44] involves placing the earthen bricks on two supports and applying a specific load, as shown in Fig. 15, to obtain the force versus displacement, as illustrated in Fig. 16.

The three-point bending test was performed using a universal testing machine and comprised several key steps: the sample (earthen brick with and without fibers) was placed between two supports, vertical displacements were measured by a motorized transducer, and displacement data were recorded using Trapezium X software. An indentation was lowered until it came into contact with the earthen brick with and without fibers, applying a bending force to the middle of the earthen brick, and the resulting force was measured and recorded. Force and displacement data were used to plot a force-displacement curve, essential for analyzing the mechanical properties of earthen brick. Other features include an adjustable spring and damping system to protect the earthen brick sample and minimize fragment dispersion in the event of breakage. The simplicity of sample preparation is a notable advantage of this method, making the testing process fast and efficient. The three-point bending test, which has been used to evaluate the earthen bricks, has measured their mechanical strength, providing valuable information on their stiffness, resistance to fracture, and overall behavior under external forces. The tests were conducted under similar conditions on three earthen bricks for each case: the first with no fibers, the second incorporating 16 long fibers, and the third enriched with 64 long fibers. This test has been particularly relevant for assessing the structural performance of earthen bricks under real load conditions and for identifying weaknesses, such as brittleness or a tendency to crack.

However, this method also has some disadvantages [45,46]: the results of the test method are sensitive to geometry, loading, and strain rate of the samples. The experimental curves have been obtained respectively for earthen brick with no fibers, with 16 long fibers, and with 64 long fibers (see Figs. 16a, 16b, 16c), and the three graphs in a single representation, as shown in Fig. 17.

The experimental three-point bending tests of the earthen bricks have provided typical curves showing the evolution of the force applied as a function of displacement. In order to clarify the understanding of the different curves, some details of the behavior of each earthen brick need to be explained.

The flexural strength $R$, calculated using the formula (equation (2)), is also presented in this article.

$$R = \frac{3FL}{2bd^2} \qquad (2)$$

where:

$R$ is the maximum flexural stress (MPa);

$F$ is the maximum load (N);

$L$ is the span length between the supports (mm);

$b$ is the width of the brick (mm);

$d$ is the height of the brick (mm).

The stiffness $k$, calculated using the formula (equation (3)), is also presented in this article.

$$k = \frac{F}{\delta} \qquad (3)$$

where:

$F$ is the applied force (N);

$\delta$ is the displacement caused by this force (mm).

The type of earthen brick without fibers, shown in Fig. 16a, exhibits a distinct linear model until it reaches a critical limit. This limit is defined by a maximum force of 670 N and a maximum displacement of 0.69 mm. Up to this point, the brick maintains its structural integrity, effectively resisting the applied load without undergoing permanent damage. This linear phase indicates a proportional response of the earthen brick to the applied force, characteristic of materials that retain their shape and size under moderate loads. However, once the thresholds of force and displacement are exceeded, the earthen brick exhibits a sudden rupture. This phenomenon, known as brittle behavior, occurs when the material suddenly gives way without prior plastic deformation. At this stage, the curve provides important information about the earthen brick's capacity to withstand forces, and its flexural strength is 1.26 MPa, which is crucial for construction applications.

The graph representing the earthen brick reinforced with 16 long fibers, as illustrated in Fig. 16b, shows an initial linear behavior until it reaches a maximum force of 844 N and a maximum displacement of 1.02 mm. This linear phase indicates stable resistance of the brick up to these critical points. Beyond these limits, degradation of the reinforced brick is observed, showing a transition to brittle behavior, where rupture can occur suddenly without warning. The incorporation of fibers into the brick's matrix has played a crucial role in preventing sudden and brittle rupture. Indeed, when the maximum load is applied, the
<figure><img src="image_34.png" /><figcaption>Fig. 16. Load-displacement curves.</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>Fig. 17. Superposition of the three Load-displacement curves from experimental test.</figcaption></figure>

fibers gradually release from the matrix, contributing to the distribu-
tion of tension within the brick. This phenomenon helps maintain the
structural integrity of the brick even under high loads, delaying the on-
set of degradation. This behavior illustrates the effectiveness of fibers
in enhancing the flexural strength (1.58 MPa) of such earthen bricks,
increasing their ability to withstand significant forces without abrupt
failure. The presence of fibers not only improves stress distribution but
also enhances adhesion within the brick's matrix, resulting in a signifi-
cant improvement in its mechanical properties.

The curve illustrating the behavior of the earthen brick reinforced
with 64 long fibers, as shown in Fig. 16c, exhibits an initial linear be-
havior until it reaches a maximum force of 1500 N and a displacement
of 2.389 mm. At this point, degradation of the earthen brick is observed,
indicating a transition to fragile behavior. However, thanks to the pres-
ence of the fibers, sudden and abrupt rupture is avoided. When the
maximum load is applied, the fibers gradually detach from the matrix of
the brick and then slide inside it. This process contributes significantly
to maintaining the structural integrity of the reinforced brick, delaying
the onset of degradation. This behavior highlights the effectiveness of
fibers in improving the flexural strength (2.81 MPa) and durability of
these earthen bricks, particularly under high loads. The fibers ensure
uniform stress distribution in the brick matrix and improve adhesion,
resulting in a significant improvement in mechanical properties.

3. Numerical modeling by finite elements

There are several commercial and non-commercial numerical codes
using finite element methods to evaluate displacements, strains, and
stresses in order to quantify the response of complex structures and
ensure more accurate and safer structural construction. In this article,
Code_Aster has been chosen because it is a non-commercial and open-
source code that is easy to manipulate. It is capable of exchanging CAD

(Computer-Aided Design) data and contains a powerful library of cal-
culations, specially dedicated to the non-linear approach. Non-linear
numerical modeling of earthen bricks has led to a better understand-
ing of fatigue and crack propagation. This study focused on improving
earthen bricks, using them as a matrix, by adding natural fibers such
as flax to enhance their mechanical properties. The approach was par-
ticularly innovative, as it relied on the development of a numerical
simulation tool capable of reproducing the experimental behavior of
earthen bricks, with and without fibers. This model made it possible to
anticipate and optimize the mechanical performance of earthen bricks,
while reducing the need for costly experimental testing.

To operate, Code_Aster needs a command file containing the char-
acteristics of earthen bricks, the boundary conditions, the elements we
want to calculate, and a file describing the mesh [47].

For the numerical simulation carried out with Code_Aster, precise
mechanical characteristics were established for the materials analyzed.
The Young's modulus of the earthen material was set at 1.05 GPa, while
that of the flax fibers was set at 80 GPa, according to the Rouen-based
supplier Depestele, as shown in Table 2. The Poisson's ratio for both
materials was set at 0.3 by Depestele. A maximum bending force of
1500 N, limited to stopping the model analysis, was used to assess the
strength of the bricks in different configurations - without fibers, with 16
long fibers, and with 64 long fibers, in line with experimental data. This
method made it possible to assess the impact of adding fibers on brick
strength under bending loads and provided the information needed to
evaluate the performance of the different configurations.

3.1. *Creation of the geometry with Gmsh*

The implementation of a mesh for the finite element analysis of earthen bricks, whether they are fiber-free or reinforced with 16 or 64 long fibers, requires a meticulous approach, which is essential to guarantee the accuracy and reliability of the numerical simulations. This process begins with the preparation of geometric data, including the precise modeling of the earthen brick geometry, with or without fiber reinforcement. It is essential to integrate all relevant structural characteristics, such as fiber dimensions and specifications, in accordance with the principles described by Owen and Hinton in their study on finite element methods [48].

The Code_Aster software was used to simulate the earthen bricks,
opting for a TETRA4 mesh, ideal for its ability to adapt precisely to the
geometric complexity and heterogeneous composition of earthen bricks.
The tetrahedral TETRA4 elements are particularly effective for model-
ing interfaces and transitions, which are crucial aspects for analyzing
the impact of fibers on the mechanical properties of earthen bricks. This
choice also allows for efficient and economical simulation, reducing cal-
culation time while maintaining high precision, which is beneficial for
analyzing the diversity and variability of earthen bricks.

The refinement of the mesh, the second essential stage, plays a de-
cisive role in improving the results obtained. According to Zienkiewicz
and Taylor [49], this refinement is adaptive, adjusting the size of the
elements according to local variations in geometry for configurations
without fibers and with 16 or 64 long fibers, thus optimizing the mesh
<figure><img src="image_36.png" /><figcaption>Fig. 18. Designing of meshes.</figcaption></figure>

in areas of high stress or geometric discontinuities. The geometry of the earthen bricks without fibers was designed with a regular TETRA4 mesh, composed of 6480 nodes and 9030 elements, as shown in Fig. 18a. Additionally, the earthen bricks reinforced with 16 long fibers were developed with a regular TETRA4 mesh, comprising 38,832 nodes and 235,767 elements, as shown in Fig. 18b, and the earthen bricks with 64 long fibers were meshed with a total of 304,009 nodes and 402,407 elements, as shown in Fig. 18b.

Finally, it is crucial to precisely define the contact conditions and boundary conditions. The contact conditions model the interactions between different surfaces, including friction aspects, while the boundary conditions specify how the model is supported, particularly with surfaces dedicated to supports and the application of loads. The rigorous application of these conditions is essential for the model to accurately reflect the expected behavior under various loads and interactions, in accordance with the numerical simulation standards defined by Bathe [50].

Each of these steps, from the preparation of geometric data to the refinement of the mesh, including the application of boundary conditions, is crucial for the development of a finite element model capable of providing accurate and reliable analyses, suitable for a wide range of engineering applications. For this, three different files are generated: one for earthen bricks with no fibers and two for earthen bricks with 16 and 64 long fibers, as shown in Fig. 18.

Once the `.geo` file has been created, it is necessary to convert the mesh file to `.med` format, which is the appropriate format for performing calculations with `Code_Aster`. The content of the file with the `.med` extension remains identical to that of the initial extension, which contains the mesh information for the earthen brick [51].

## 3.2. Creation of the command file

To use `Code_Aster` effectively, it is necessary to prepare a command file that orchestrates the entire finite element simulation process for a brick, both with and without long fibers. This crucial file begins by detailing the mesh used in the analysis. It also specifies the material properties, such as the modulus of elasticity and Poisson's ratio, which are essential for the model. In this case, the mechanical modeling is specifically tailored to the required analysis. The command file also controls the application of boundary conditions, including supports (represented by two supports in the three-point bending test) and loads, which include the forces applied to the bricks. These elements directly influence the behavior of the brick, both in configurations without and with long fibers. It also defines control parameters such as time steps, with increased precision. In conclusion, the post-processing of the results is configured to extract key information such as force and displacement, thus facilitating accurate interpretation and effective visualization of the results of the numerical simulation.

Several tasks have been carried out, taking into account both linear and non-linear approaches. First, numerical simulations were performed for the linear approach. For the non-linear approach, a specific `Code_Aster` [52] library was used, and several numerical simulations were performed.

The purpose of the [53] module of non-linear `Code_Aster` is to perform quasi-static analyses of structures with non-linear behavior. These non-linearities can result from contact friction interactions, geometric effects, or material properties. This module incorporates a set of equations and physical laws, in particular mechanical equilibrium equations, which ensure that the sum of the forces and moments in each part of the structure is equal to zero, and constitutive material behavior laws, which describe the response of materials under load (for example, elastic, plastic, etc.).

To solve these equations, non-linear [54–56] uses the finite element method, which transforms the equations into a numerically solvable algebraic system. The discretization process consists of dividing the domain of the structure into small finite elements, where the equations are applied locally. Although the analysis is quasi-static, the operator can manage time-dependent loads using temporal integration techniques.

The challenges of non-linearity are addressed using iterative solvers, such as the Newton-Raphson method [57–59], which effectively resolve non-linear problems by iteratively refining the approximations of the solution until convergence is reached. This approach allows problems related to significant deformations, complex material behavior, and contact interactions between structural elements to be addressed. As a result, the non-linear method is a robust and accurate tool for analyzing structures subject to various requirements and constraints, whether in industrial applications or in research contexts.

Non-linear ordering in `Code_Aster` is crucial for performing non-linear static analyses. It is suitable for a variety of structures, including the study of complex materials such as earthen bricks, with or without long fibers. This feature allows the calculation model to be precisely configured by specifying parameters such as the mechanical model (adapted to fiber-reinforced earthen bricks) and material properties (such as Young's modulus, Poisson's ratio, and density of earthen bricks and fibers). It is particularly effective in managing non-linearities, offering options for adjusting the behavior of earthen bricks, with or without long fibers, including elastic, plastic behavior, and geometric non-linearity. It also offers parameters for managing small deformations, loading precision, and time steps, detailed in the following section.

## 3.3. The non-linear approach

This article focuses on a non-linear approach to improving the understanding of the behavior of earthen bricks with or without natural fibers. Due to the lack of studies on the modeling of earthen bricks in this context, a specific approach is presented to enable accurate and efficient modeling of their behavior. For this purpose, a specific command line in `Code_Aster` has been used. In `Code_Aster`, the non-linear command plays a central role in simulating and understanding the non-linear behavior of mechanical structures, specifically applied to earthen bricks in this case study. The aim of the numerical simulations with `Code_Aster` is to compare the results with experimental data.

For efficient numerical simulation with `Code_Aster`, it is essential to define an appropriate time step to reproduce a deformation of 0.5 mm per minute, as shown by the results of an experimental study. The first step is to convert this deformation rate into the standard units of time, i.e., seconds. Thus, a speed of 0.5 mm per minute is equivalent
<figure><img src="image_37.png" /><figcaption>Fig. 19. Non-linear numerical load-displacement curves.</figcaption></figure>

to 0.00833 mm/s [60]. The objective is to study the material's response over a total duration of 10 minutes, i.e., 600 seconds. Choosing the right time step is crucial to ensure accurate simulation results. Adopting a time step corresponding to a displacement of 0.1 mm, it will take approximately 12 seconds to achieve this displacement at the specified speed. Therefore, a time step of 12 seconds is recommended to examine how each 0.1 mm increment affects material behavior [61]. For this specific configuration in Code_Aster, this involves defining the loading conditions at 890 N, divided into increments to better capture the material's response to the applied conditions [62]. The temporal analysis uses a time step of 12 seconds with a displacement increment of 0.1 mm for the three configurations tested: earthen bricks without fibers, with 16 long fibers, and with 64 long fibers. The results are illustrated by graphs showing the relationship between the applied force and the corresponding displacement, as shown in Fig. 19.

Regarding the numerical simulation of the three-point bending test on earthen bricks, Fig. 19 illustrates typical curves showing the evolution of applied force versus displacement. To clarify the understanding of the different graphs, the behavior of each curve is detailed below.

The curve shown in Fig. 19a represents the non-linear behavior of earthen brick without fibers. It can be seen that the response begins with linear behavior, followed by hardening behavior. This response is common to all elasto-plastic materials with hardening. The linear part of the curve reaches a maximum force value of 680 N, corresponding to a maximum displacement of 0.698 mm, before the brick begins to undergo plastic deformation. This observation highlights the strength limits of earthen brick without fiber reinforcement. The linear phase of the curve indicates a proportional relationship between applied force and displacement, up to the threshold where the brick begins to deform irreversibly. This point marks the brick's transition from the elastic to the plastic domain, illustrating its intrinsic mechanical properties and ability to withstand specific loads. In summary, this curve provides a clear understanding of the performance characteristics of fiber-free earthen brick under load, with a flexural strength of 1.27 MPa.

The curve shown in Fig. 19b represents the non-linear behavior of earthen brick with 16 long fibers. It can be seen that the response begins with linear behavior, followed by plastic behavior. This is a natural response for any elasto-plastic material with strain-hardening, but here there is decay. The linear part of the curve reaches a maximum force value of 850 N, corresponding to a maximum displacement of 1.03 mm, before the brick begins to undergo plastic deformation. This observation highlights the strength limits of the brick with 16 long fibers reinforcement. The linear phase of the curve indicates a proportional relationship between applied force and displacement, up to the threshold where the brick begins to deform irreversibly. This point marks the brick's transition from the elastic to the plastic domain, illustrating its intrinsic mechanical properties and ability to withstand specific loads. This curve provides a clear understanding of the performance characteristics of earthen bricks reinforced with 16 long fibers under load. This behavior reveals the importance of fibers in stabilizing and strengthening the brick structure under high loads, with a flexural strength of 1.59 MPa. It can be seen that the curve can be broken down into 4 stages: the first stage is linear behavior, the second is a non-linear decrease

<figure><img src="image_38.png" /><figcaption>Fig. 20. Superposition of the three load-displacement curves from numerical simulation.</figcaption></figure>

corresponding to the sliding of certain fibers with the matrix, i.e., the sediments, the third stage, which is an inflection, could correspond to the rupture of certain fibers, and the last stage corresponds to the degradation of all the fibers.

The curve shown in Fig. 19c represents the non-linear behavior of earthen brick with 64 long fibers. It can be seen that the response also starts with linear behavior, followed by plastic behavior. As mentioned before, this is a natural response for any elasto-plastic material with strain-hardening, but here there is decay. The linear part of the curve reaches a maximum force value of 1480 N, corresponding to a maximum displacement of 2.4 mm, before the brick begins to undergo plastic deformation. This observation highlights the strength of earthen bricks reinforced with 64 long fibers, with a flexural strength of 2.77 MPa. The linear phase of the curve indicates a proportional relationship between applied force and displacement, up to the threshold where the brick begins to deform irreversibly. This point marks the brick's transition from the elastic to the plastic domain, illustrating its intrinsic mechanical properties and ability to withstand specific loads. The same type of behavior can be observed as for the brick with 16 long fibers, with the difference that deformation has increased; everything now requires a certain level of effort.

It can be seen that the 64 long fibers reinforced brick behaves similarly at higher loads, with a more pronounced behavior. Furthermore, this configuration shows that bricks with 64 long fibers can bear twice as much load for twice as much displacement with linear behavior. In short, the brick with 64 long fibers behaves twice as well as a brick with 16 long fibers and 3.4 times better than an earthen brick without fibers. This incorporation of long fibers considerably transforms the mechanical properties of the bricks, increasing their ability to withstand various stresses. The natural characteristics of natural fibers such as flax give bricks greater strength and durability (Fig. 20).
<figure><img src="image_39.png" /><figcaption>Fig. 21. Numerical and experimental results comparison for load-displacement relationship.</figcaption></figure>

<figure><img src="image_40.png" /><figcaption>Fig. 22. Relative error versus displacement for all types of earthen bricks.</figcaption></figure>

### 3.3.1. Comparison with experimental data

Fig. 21 shows the superposition of numerical simulation results (red curves) and experimental data (blue curves). The relationship between force and displacement for the earthen brick without fibers shows linear behavior up to a maximum force, after which sudden failure occurs. In contrast, the earthen brick with 16 and 64 long fibers shows linear behavior up to a maximum force, before degradation occurs (brittle behavior). Sudden brittle fracture is avoided thanks to the fibers. The difference between the experimental data and the numerical simulation using the finite element code (Code_Aster) needs to be justified, as the curves are very close to each other.

The numerical simulation carried out with Code_Aster reproduces the experimental results. The comparison criterion between experimental data and numerical simulation has been based on the $L^\infty$ norm, to prove that errors are limited and relatively small. It can be said that the incorporation of flax fibers in earthen bricks considerably increases the strength factors of earthen bricks.

### 3.3.2. Analysis

Fig. 22 shows the typical curves of the evolution of the $L^\infty$ norm applied as a function of displacement, to prove that the errors are limited and relatively small. These curves have been derived from both numerical simulations and experimental data. The central element of this analysis revolves around the evaluation of the error, which is the criterion for comparison between the experimental data and the numerical simulation based on the $L^\infty$ norm.

The curve shown in Fig. 22a represents the relative error based on the $L^\infty$ norm for the earthen brick without fibers. A comparison between numerical simulation and experimental data has revealed remarkable convergence, with a maximum relative error of no more than 1%. This indicates a good correlation between the numerical simulation and experimental data, reinforcing the validity of the numerical model.

The curve shown in Figs. 22b, 22c presents the relative error based on the $L^\infty$ norm for earthen bricks with 16 and 64 long fibers. A comparison between numerical simulation and experimental data has revealed remarkable convergence, with a maximum relative error of less than 5% for the 16 long fibers, as shown in Fig. 22b, and less than 6% for the 64 long fibers, as shown in Fig. 22c. This indicates a good correlation between the numerical simulation and experimental data, reinforcing the behavior law of a composite material, which is optimal when the ratio between the inclusion of the reinforced material and the dimension of the structure under study tends towards 0, with a new constitutive law representing a fictitious material equivalent to the studied medium. Several techniques are available, of which the mean-field method and the full-field method are the most reliable. In this article, the full-field technique is preferred, using the asymptotic and multiscale techniques. Indeed, fiber-reinforced earthen bricks are periodic materials and are assimilated to representative volume elements, which are commonly used to estimate the effective properties of composite materials.

This study highlighted the significant impact of adding natural fibers such as flax to earthen bricks, helping to reinforce their strength, a crucial aspect in the construction field. The remarkable consistency between numerical simulation and experimental data consolidates the reliability of the methodology that has been employed, confirming the validity of the conclusions put forward. However, an increase in relative error has been observed following the incorporation of natural fibers, although this remains relatively acceptable. To reduce this error, an approach based on the principle of homogenization has been adopted to calculate the $C^{hom}$ tensor, and a new numerical simulation has been carried out using it.

### 3.4. Homogenization

The homogenization process consists of replacing the behavior law of a composite material, which is optimal when the ratio between the inclusion of the reinforced material and the dimension of the structure under study tends towards 0, with a new constitutive law representing a fictitious material equivalent to the studied medium. Several techniques are available, of which the mean-field method and the full-field method are the most reliable. In this article, the full-field technique is preferred, using the asymptotic and multiscale techniques. Indeed, fiber-reinforced earthen bricks are periodic materials and are assimilated to representative volume elements, which are commonly used to estimate the effective properties of composite materials.

#### 3.4.1. Formulation of the local problem

Let's denote $\epsilon$ as the scaling parameter, which is the ratio between the diameter of the reinforced fibers and the characteristic dimension
<figure><img src="image_41.png" /><figcaption>Fig. 23. Heterogeneous material in a periodic medium Ω with ε = dl / L .</figcaption></figure>

<figure><img src="image_42.png" /></figure>

<figure><img src="image_43.png" /></figure>

$$
\left\{
\begin{array}{l}
\mathrm{div}(\sigma^{\epsilon}) + f = 0 \\
\sigma^{\epsilon} = \mathbf{C}^{\epsilon} \mathrm{grad}(u^{\epsilon})
\end{array}
\right.
\qquad (5)
$$

<figure><img src="image_44.png" /></figure>

$$
\left\{
\begin{array}{l}
\mathrm{div}(\sigma^0) + f = 0 \\
\sigma^0 = \mathbf{C}^{\mathrm{hom}} \mathrm{grad}(u^0)
\end{array}
\right.
\qquad (6)
$$

<caption>Fig. 24. The principle of the homogenization technique.</caption>

of the earthen bricks (see Fig. 23), assumed to be small and tending
towards 0 in the asymptotic process. The equations (4) satisfied by the
earthen bricks are:

$$
\left\{
\begin{array}{l}
\mathrm{div}(\sigma^{\epsilon}) + f = 0 \quad \text{within } \Omega \\
\sigma^{\epsilon} = \mathbf{C}^{\epsilon} \mathrm{grad} u^{\epsilon} \\
u^{\epsilon}_{\Gamma_D} = 0
\end{array}
\right.
\qquad (4)
$$

In these equations (4), σ^ε is the stress tensor; f is the volume force; C^ε is the stiffness tensor of the heterogeneous medium, and ∇(u^ε) is the strain tensor.

The homogenization process [63,30] involves scaling ε → 0 to ob-
tain the macroscopic equations (5), (6), as shown in Fig. 24, and the
stiffness tensor C<sup>hom</sup>, which is constant throughout the medium, to ex-
press the stress tensor σ<sup>0</sup> = C<sup>hom</sup>∇(u<sup>0</sup>). The homogenization principle
allows for determining the effective tensor C<sup>hom</sup> via first-order gradient
theory with an asymptotic expansion.

A home-made code has been developed to obtain the stiffness ten-
sor C<sup>hom</sup> based on the modified Green's kernel using the Fast Fourier
Transform (FFT) with the accelerated scheme algorithm.

3.4.2. Evaluation of C<sup>hom</sup>

This section deals with fiber-reinforced earthen bricks for which the
volume fraction of natural fibers is set at 1.9% for earthen bricks with 16
long fibers, 12.5% for earthen bricks with 64 long fibers, and for which
the contrast ratio between the moduli of flax fibers and sediments is
evaluated at 100. All these data have been obtained from a survey of
the literature [63,64]. Due to the periodicity of earthen bricks, a cubic
sample is sufficient to determine the stiffness tensor C<sup>hom</sup>, as it is a
representative volume element (RVE) of statistical earthen bricks, as
shown in Fig. 25.

First-order gradient theory with a home-made code has enabled us
to determine the C<sup>hom</sup> tensor.

Numerical simulation with a home-made code of the earthen brick
with 16 and 64 long fibers has enabled us to obtain the following C<sup>hom</sup>
tensor.

$$
C_{16 \text{fibers}}^{\text{hom}} =
$$

$$
\begin{bmatrix}
+1.54288 & +0.76504 & +0.77061 & +0.00000 & +0.00000 & -0.00000 \\
+0.76504 & +1.54288 & +0.77061 & +0.00000 & +0.00000 & -0.00000 \\
+0.76931 & +0.76931 & +21.0254 & +0.00000 & +0.00000 & -0.00000 \\
+0.00000 & +0.00000 & +0.00000 & +0.39114 & +0.00000 & +0.00000 \\
+0.00000 & +0.00000 & +0.00000 & +0.00000 & +0.39114 & +0.00000 \\
+0.00000 & +0.00000 & +0.00000 & +0.00000 & +0.00000 & +0.38716
\end{bmatrix}
$$

$$
C_{64 \text{fibers}}^{\text{hom}} =
$$

$$
\begin{bmatrix}
+1.82376 & +0.85787 & +0.90460 & +0.00000 & +0.00000 & -0.00001 \\
+0.85787 & +1.82376 & +0.90460 & +0.00000 & +0.00000 & -0.00001 \\
+0.89388 & +0.89388 & +127.209 & +0.00000 & +0.00000 & -0.00001 \\
+0.00000 & +0.00000 & +0.00000 & +0.48819 & -0.00004 & +0.00000 \\
+0.00000 & +0.00000 & +0.00000 & -0.00004 & +0.48819 & +0.00000 \\
+0.00001 & +0.00001 & -0.00008 & +0.00000 & +0.00000 & +0.45343
\end{bmatrix}
$$

Numerical simulations have been conducted with the help of the
C<sup>hom</sup> tensor, representing the homogenized earthen bricks, according to
the first-order gradient theory. These results have then been compared
with the experimental data presented in the next section.

3.4.3. Comparison between the experimental data and the numerical simulation with first-order gradient theory

Figs. 26a and 26b show the superposition of numerical simulation
results (red curves) and experimental data (blue curves). The relation-
ship between force and displacement for the earthen brick with 16 and
64 reinforced long fibers shows linear behavior up to a maximum force
before degradation occurs (brittle behavior). The discrepancy between
the experimental data and the numerical simulations within the frame-
<figure><img src="image_45.png" /><figcaption>Fig. 25. Models of reinforced earthen bricks.</figcaption></figure>

<figure><img src="image_46.png" /><figcaption>Fig. 26. Load-displacement of earthen bricks with fibers in numerical simulation with first-order gradient theory and experimental test.</figcaption></figure>

<figure><img src="image_47.png" /><figcaption>Fig. 27. Relative error versus displacement with first-order gradient theory.</figcaption></figure>

work of first-order gradient theory, obtained by the finite element code (Code_Aster with $C^{hom}$), needs to be justified. The criterion for comparison between experimental data and numerical simulations is based on the $L^\infty$ norm to demonstrate that the relative errors are less than 3%, as shown in Fig. 27a for earthen bricks reinforced with 16 long fibers, while for earthen bricks reinforced with 64 long fibers, they are less than 4%, as shown in Fig. 27b.

The curves shown in Fig. 27 present the relative error based on the $L^\infty$ norm for earthen bricks reinforced with 16 and 64 long fibers, respectively. A comparison of the experimental data with numerical simulations using first-order gradient theory reveals that the relative errors are limited and relatively small. They are reduced to 3% and 4%, respectively, for the configurations reinforced with 16 and 64 long fibers, compared with the numerical simulations without the use of first-order gradient theory, which present relative errors of 5% and 6%, respectively, for the same configurations with reinforced long fibers, as shown in Figs. 22b and 22c.

In summary, numerical simulations using the first-order gradient theory tend to overestimate the experimental data, but they remain an efficient method for reducing relative error. This is easily explained by the fact that, in the homogenization process, the small parameter $\epsilon$ is intended to tend towards zero, whereas during numerical simulations $\epsilon = \frac{dl}{L}$, this parameter is fixed at 0.001, which is different from 0 and

explains the slight divergence. Convergence would have been observed if and only if the fiber diameter had been very fine, which means mathematically that $\epsilon \to 0$.

## 4. Conclusion and perspectives

Natural fibers, such as flax, are increasingly used to reinforce earthen bricks in construction, offering both environmental benefits and significant improvements in mechanical and thermal properties. While this practice is well-established, the novel contribution of this work lies in the advanced simulation methodologies developed to model the behavior of earthen bricks with and without natural fibers.

This study highlights the advantages of dynamic compaction over static compaction, resulting in higher-quality bricks with notable improvements in strength, durability, and reliability. Numerical simulations conducted using the Code Aster software successfully reproduced experimental results, achieving a maximum relative error of less than 6%, which is considered acceptable in terms of accuracy. To refine these results, homogenization principles were employed to calculate the $C^{hom}$ tensor, facilitating the development of enhanced numerical simulations.

A comparison of experimental and numerical data, within the framework of first-order gradient theory and using the $L^\infty$ norm, demonstrated the accuracy of this approach, with relative errors reduced to less
than 3% and 4% for bricks reinforced with 16 and 64 long fibers, respectively. These results underscore the effectiveness of first-order gradient theory in reducing relative error.

The introduction of a refined simulation methodology represents the primary innovative aspect of this study. While the integration of natural fibers enhances the mechanical properties of earthen bricks, making them more robust and suitable for construction applications, the focus on advanced simulation techniques offers a valuable tool for improving the design and analysis of sustainable construction materials. Furthermore, their potential contribution to sustainable housing, capable of enduring climate change and seismic risks, is discussed in the work of I. Djeran-Maigre [65]. Future work will aim to incorporate second-order tensors within the framework of second-order gradient theory to further enhance numerical simulations, minimize error margins, and explore fracture mechanisms, such as fiber arrangement and matrix cracking, in greater detail in a subsequent article.

## CRediT authorship contribution statement

**Alioune Nacro:** Writing – original draft, Software, Investigation, Data curation. **Philippe Karamian-Surville:** Writing – review & editing, Supervision, Software, Project administration, Formal analysis. **Daniel Levacher:** Resources, Methodology, Data curation. **Mazhar Hussain:** Data curation.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

Most of the computations detailed in this paper have been carried out on the cluster at the Center for Informatics Resources of Higher Normandy (CRIANN - Centre Régional Informatique et d'Applications Numériques de Normandie).

## Data availability

Data will be made available on request.

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