# Impact of design strategy favoring powder removal on mechanical performance of bio-inspired porous architectures for laser-based powder-bed fusion additive manufacturing

Nicolas Cadoret, Julien Chaves-Jacob, Marc-Antoine Celli, Jean-Marc Linares

<figure><img src="image_1.png" /></figure>

## To cite this version:

Nicolas Cadoret, Julien Chaves-Jacob, Marc-Antoine Celli, Jean-Marc Linares. Impact of design strategy favoring powder removal on mechanical performance of bio-inspired porous architectures for laser-based powder-bed fusion additive manufacturing. Materials & Design, 2024, 245, pp.113240. 10.1016/j.matdes.2024.113240. hal-04965246

HAL Id: hal-04965246

https://hal.science/hal-04965246v1

Submitted on 25 Feb 2025

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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

Distributed under a Creative Commons Attribution 4.0 International License
**Title:** Impact of design strategy favouring powder removal on mechanical performance of bio-inspired porous architectures for laser-based powder-bed fusion additive manufacturing

**Authors:**

Nicolas Cadoret<sup>a,b</sup>, Julien Chaves-Jacob<sup>b,*</sup>, Marc-Antoine Celli<sup>a</sup>, Jean-Marc Linares<sup>b</sup>

<sup>a</sup>Airbus Helicopters, Aéroport de Marseille Provence, Marignane 13700, France

<sup>b</sup>Aix Marseille Université, CNRS, ISM, Inst Mouvement Sci, UMR 7287, Marseille, France

* Corresponding author: julien.chaves-jacob@univ-amu.fr (J. Chaves-Jacob)

**Abstract:**

Mass reduction is a concern both in mechanical engineering and living organisms to reduce energy and materials consumption. Lately, transfer of knowledge from biology to engineering has become easier thanks to additive manufacturing. Trabecular bone for example optimally adapts to the mechanical stress it undergoes. Mass reduction methods bio-inspired from trabecular architecture, proposed in the literature, are not always applicable for laser-based powder-bed fusion. In particular, bio-inspired mass reduction methods based on a material removal principle generate porosities throughout a solid initial part which can result in enclosed porosities that trap un-melted powders. Here, we propose a 3D depowderable bio-inspired mass reduction method. Powder removal was controlled in different samples through weighing and X-ray computed microtomography and the impact of the design strategy that ensures correct powder removal on mechanical performance (stiffness) was quantified. Mean performance difference is 10.35% (between 1.82% and 26.47%). For stiff and light parts loaded in bending, the proposed method outperforms parts made of bulk steel alloy by leveraging an architecture bio-inspired from trabecular bone.

**Keywords:** additive manufacturing, powder bed fusion, powder removal, bio inspiration, light-weight structure
# 1. Introduction

It is now thoroughly documented that there's a need to reduce human activities' global footprint to ensure their sustainability [1]. This includes, but is not limited to, reducing energy consumption, reducing material consumption, and increasing products' lifespan. From an engineering point of view, this translates to increasing parts' performances without compromising on their safety. In the aerospace industry, for example, it is unthinkable to reduce the weight of an aircraft simply by using thinner sheet metal. Instead, engineers are looking to improve performances, such as strength-to-weight ratio [2], often through advanced materials (titanium alloys, composite materials, technical ceramics, etc.).

In natural environments, constraints can be very similar [3]. Indeed, the energy available for a given organism is limited and is a function of its environment [4]. Additionally, the pool of materials that can be used by living beings is very limited, often restricted to a few polymers, ceramics or composites of both [5]. Metallic materials across species are confined to specific functions such as chemical transport [6]. Nonetheless, living organisms have evolved to meet these challenges [7]. In fact, it has long been documented that natural materials such as wood or bone often perform better than the sum of their chemical constituents (for example, collagen and hydroxyapatite in the case of bone), sometimes on par with engineering materials [8]. In materials science, hybrid materials (a combination of two or more materials in a predetermined geometry and scale [9]) have been shown to combine the best of their constituent materials' performance. This is true for composite materials, explaining their success for engineering purposes [10]. This is also true for cellular materials which can be considered as an association of a given material and air. Finally this is true for natural materials which owe their outstanding performance to their hierarchical organisation across length scales rather than their chemical composition [6,11]. Indeed, it is not unusual for natural materials to showcase a composite structure at the micro-scale, a cellular structure at the meso-scale and an optimised shape at the macro-scale. Bone, for example, is a tissue that shows multiple hierarchical levels of organisations [11]: it is a composite of collagen and hydroxyapatite that differentiates into compact cortical bone or porous trabecular bone. The trabecular architecture is locally adapted to the stress it undergoes through a phenomenon known as functional adaptation [12–14]. As a consequence of their performances, natural materials have seen increasing interest across industries in recent years, especially using bio-inspiration techniques [5,7].

This renewed interest has mostly been fuelled by advancements in additive manufacturing techniques that enable the manufacturing of parts regardless of their complexity [15]. In demanding industries, laser-based powder-bed fusion (LB-PBF) techniques especially have seen rapid growth because of their high resolution, high quality, self-supporting nature, static performances and batch production capabilities for a wide range of materials [16]. For LB-PBF techniques, proper post-processing after printing is essential to ensure that parts meet their requirements. Correctly removing all un-melted powders from printed parts in particular represents a challenge as these powders could become trapped or sintered inside parts [15,17]. Several control techniques have been tested to ensure the proper removal of these un-melted powders, with X-ray computed microtomography and weighing showing the best results [18]. In particular, some bio-inspired mass reduction methods are based on a material removal principle unlike most design strategies where porous architectures are obtained by addition of solid struts (e.g. lattices) or surface-based elements (e.g. triply periodic minimal surfaces) [19–21]. Material removal-based design strategies often generate
porosities throughout a solid initial part which can result in enclosed porosities that trap un-
melted powders. Hence, several design methodologies have been developed to ensure that
un-melted powders can be removed from parts with complex cellular architectures using a
network of powder removal canals (PRC) [22], based on a bio-inspired approach [23] or
leveraging topology optimisation techniques [17]. Unfortunately, adding PRCs or holes
weakens the mechanical part. It is essential to understand and control this influence to be
able to minimise the loss of performance [17].

Here, a 3D Depowderable Bio-Inspired Mass Reduction (3D²BIMR) method suitable for LB-PBF parts is proposed. It builds upon a previous method [21] that is first presented succinctly in section 2.1. Then, the design strategy that ensures correct powder removal is described and controlled using computed microtomography and weighing in section 2.2. The proposed method reduces mechanical parts' weight without excessive loss of stiffness, leveraging gradient porous architectures. To compare the performance of these architectures and study the influence of the powder removal design strategy, a performance index is proposed in section 2.3. A Finite Element (FE) process is presented in section 2.4 for calculating this performance index in porous samples. Finally, the effect of local parameters on the performance difference induced by the powder removal design strategy is analysed in section 2.5.

## 2. Materials & Methods

1. Presentation of 3D bio-inspired mass reduction method

As stated in section 1, the proposed mass reduction method builds upon a previous 3D bio-
inspired mass reduction (3DBIMR) method, established based on bio-inspiration from
trabecular architecture [21]. Contrary to most design strategies where porous architectures
are obtained by addition of solid struts (e.g. lattices) or surface-based elements (e.g. triply
periodic minimal surfaces), 3DBIMR works by removing materials from a solid part. In
summary, this method takes an initial solid part and its loading in operation as a basis to
generate a porous architecture. First, the stress field in the solid part is simulated using FE
analysis. Then based on FE results, ellipsoid porosities are distributed along a face centred
cubic (FCC) pattern of size *p* as presented in Fig.1. Each porosity's size, aspect ratio and
orientation are determined based on estimated stress in its vicinity. Hence, a porous
architecture, locally adapted to the estimated stress the initial solid part undergoes, is
created. It reproduces several relationships observed in biological trabecular bone.
<figure><img src="image_3.png" /><figcaption>Figure 1: example of 3-point bending beam designed using 3DBIMR. The porous architecture is structured around a face centred cubic pattern of size p and varies locally to adapt to the stress the initial solid part undergoes, as apparent on the zoom view. There are some enclosed porosities inside the architecture as apparent with transparency.</figcaption></figure>

First, there is an allometric scaling of the trabecular pattern size [24], which varies depending on the size of the bone and the species' class. This relationship, adapted from [25] to be more accurate for small dimensions, is reproduced through Eq.1 and gives the architecture's pattern size p as a function of the initial part's characteristic dimension h.

$$p = \begin{cases} 0.3535 \left(\frac{h}{2}\right)^{0.2864}, & \text{for mammalian bio-inspiration} \\ 0.6243 \left(\frac{h}{2}\right)^{0.5648}, & \text{for bird bio-inspiration} \end{cases} \qquad (1)$$

Second, local volume fraction $\rho_f$ varies as a function of local Von Mises stress $\sigma_{VM}$ and the material's elastic limit $R_e$, as presented in Eq.2 [25].

$$\rho_f = 3.07 (\sigma_{VM}/R_e)^{0.59} \qquad (2)$$

Third, the local mean intercept length-based fabric tensor M, defined in [26] as a symmetric second rank tensor which characterizes the microstructure in a porous material, depends on local stress. In particular, M is oriented with its principal directions along the principal stress directions and its eigenvalues depend on the principal stresses' magnitudes. As such, the i-th fabric tensor eigenvalue $M_i$ is aligned with the i-th principal stress $\sigma_i$ (with $|\sigma_1| \ge |\sigma_2| \ge |\sigma_3|$) and can be calculated using Eq.3 [27,28].

$$M_i = \frac{|\sigma_i|^{1/4}}{(|\sigma_1|^{1/4} |\sigma_2|^{1/4} |\sigma_3|^{1/4})^{1/3}}, i = 1,2,3 \qquad (3)$$

3DBIMR has shown promising results for optimisation of mechanical parts, achieving more than twice the stiffness in the case of a 3-point bending beam as a topology optimisation of
equal mass [21]. Nonetheless, because of the design strategy that removes material from a solid part rather than adding material, the porous architecture presents enclosed porosities in the final design as apparent on the zoom view in Fig.1. As stated in section 1, for parts manufactured using LB-PBF, to achieve the mass reduction goal it is essential to ensure that no un-melted powders are trapped in the finished part. Hence, a new design strategy must be developed to ensure that powder removal in the manufactured part is possible and that no un-melted powders remain in enclosed porosities. This strategy is described in the next section.

## 2. Design strategy ensuring correct powder removal for laser-based powder-bed fusion additive manufacturing of 3D bio-inspired mass reduction method

Hence, porosities in 3DBIMR are distributed along an FCC pattern. The proposed powder removal design strategy leverages the properties of this pattern to connect enclosed porosities to the outside of the part. In an FCC pattern, each node is surrounded by twelve nearest neighbours, as shown in Fig.2 for the yellow central node. It should be noted that spheres in Fig.2 are not porosities but rather the positions at which nodes are found in a face-centred cubic pattern. For the central node (yellow in Fig.2), among the closest neighbouring nodes, 4 are found on the plane normal to the x-axis (in red in Fig.2), 4 on the plane normal to the y-axis (in green in Fig.2) and 4 on the plane normal to the z-axis (in blue in Fig.2). Going back to 3DBIMR, by connecting each enclosed porosities to some or all its closest neighbours in the FCC pattern, it would become possible to remove un-melted powders from the manufactured part, working from one porosity to the next. Here, it is proposed to connect neighbouring porosities through cylindrical holes, also called powder removal canals (PRC).

<figure><img src="image_4.png" /><figcaption>Figure 2: a) view of a node (yellow) in a face-centred cubic pattern b) zoom on the surrounding twelve closest neighbours with 4 neighbouring nodes found on the plane normal to the x-axis (red), 4 on the plane normal to the y-axis (green) and 4 on the plane normal to the z-axis (blue).</figcaption></figure>

Based on that observation, twelve samples were tested to control that the proposed design strategy ensures that all un-melted powders can be removed. These samples varied in the
radius of the PRCs, the size of the FCC pattern and the number of connected closest neighbours as shown in Table 1. Canal radius is the radius of the cylindrical hole between neighbouring porosities and varies between 0.5 and 1mm based on previous results [25] that showed 0.5mm radius to be the critical size needed for free flow of un-melted powders. Pattern size *p*, as presented in Fig.1, is the length of a side of the face-centred cubic pattern that underlies the architecture and it was selected as multiples of 30mm, which was the height and thickness of 3-point bending beams tested in [21]. “Connected neighbours” refers to the number of closest neighbours a given porosity is connected to in its vicinity, from the twelve possible described above. No powder removal samples involving canals to all twelve closest neighbours were tested because the authors estimated that it would not add significantly to powder removal rates and would have had an important influence on the part’s volume fraction and mechanical properties.

<table><caption>Table 1: tested powder removal samples</caption><thead><tr><td>Sample number</td><td>Canal radius (mm)</td><td>Pattern size (mm)</td><td>Connected neighbours</td></tr></thead><tbody><tr><td>1</td><td>0.5</td><td>3</td><td>8</td></tr><tr><td>2</td><td>0.5</td><td>3.75</td><td>8</td></tr><tr><td>3</td><td>0.5</td><td>5</td><td>8</td></tr><tr><td>4</td><td>0.75</td><td>3.75</td><td>8</td></tr><tr><td>5</td><td>0.75</td><td>5</td><td>8</td></tr><tr><td>6</td><td>1</td><td>5</td><td>8</td></tr><tr><td>7</td><td>0.5</td><td>3</td><td>4</td></tr><tr><td>8</td><td>0.5</td><td>3.75</td><td>4</td></tr><tr><td>9</td><td>0.5</td><td>5</td><td>4</td></tr><tr><td>10</td><td>0.75</td><td>3.75</td><td>4</td></tr><tr><td>11</td><td>0.75</td><td>5</td><td>4</td></tr><tr><td>12</td><td>1</td><td>5</td><td>4</td></tr></tbody></table>

Hence, to test the correct powder removal all twelve cubic samples were manufactured using metallic LB-PBF on a 3D Systems PMX 300 machine with 3D System LaserForm® 17-4PH (A) powder, as shown in Fig.3. 17-4Ph stainless steel is a precipitation-hardened martensitic stainless steel designed primarily for the aerospace industry and commonly used for LB-PBF applications. 3D System LaserForm® 17-4PH (A) powder contains 15-17% chrome, 3-5% nickel and 3-5% copper with a mean powder size of 3.87 µm and maximum powder size of 26 µm. The 3D System PMX 300 machine has the particularity of spreading the powder bed using a roller. The printing parameters used were optimised to maximise material integrity for 17-4PH as reported in [29] and are the following:

- Laser travel speed: 1.045 m/s

- Laser power: 184 W

- Hatch spacing: 59 µm

- Laser defocusing: 4.8

- Layer thickness: 30 µm
<figure><img src="image_5.png" /><figcaption>Figure 3: samples manufactured using metallic LB-PBF to test the proposed powder removal design strategy with respective sample numbers as reported in Table 1</figcaption></figure>

After printing, un-melted powders were removed from the parts according to a two part process in accordance with [18,23]. The un-melted powder was first broken down using a vibrating shaker and free-flowing powders were then removed under a stream of water. Samples were then separated from the build plate using wire electrical discharge machining. After separation, un-melted powder removal was controlled using the protocol recommended in [18,23]. For that purpose, individual samples were first weighed and compared with theoretical mass predicted from CAD data. Mass percentage errors were calculated as in Eq.4.

$$ \Delta_{mass} = \frac{|theoretical\ mass - measured\ mass|}{theoretical\ mass} \qquad (4) $$

The sample with the largest mass percentage error was subjected to X-ray computed tomography to assess whether any powders remained trapped. Tomography was conducted with a voxel size of 53.05µm on a General Electrics Phoenix V|TOME|X M machine which is a CT system for 3D metrology and analysis with up to 300 kV / 500 W.

Based on visual inspection, weighing and tomography results, further presented in section 3, all tested samples showed complete removal of un-melted powders. Hence, from the various strategies presented in Table 1, the one with the lowest impact on volume fraction (PRCs to 4 closest neighbours and 0.5mm canal radius) was selected for the rest of the study. It was implemented in the 3DBIMR algorithm resulting in a 3D Depowderable Bio-Inspired Mass Reduction (3D²BIMR) method, as shown in Fig.4 for a 3-point bending beam. In this implementation, no PRCs were added between nodes without porosities or between porosities that already intersect, to help reduce redundancy. As with 3DBIMR, parts designed using 3D²BIMR are adapted to the stress the initial solid part undergoes through variations of the local porous architecture. Enclosed porosities are connected to the outside
through a network of PRCs which ensure that all un-melted powders can be removed from the part, as is apparent in the detailed view in Fig.4.

<figure><img src="image_6.png" /><figcaption>Figure 4: example of 3-point bending beam designed using 3D²BIMR. The porous architecture is locally adapted to the stress the initial solid part undergoes whilst allowing for all un-melted powders to be removed.</figcaption></figure>

### 3. Measuring the performance of a porous sample

The proposed design strategy was selected to limit the impact on volume fraction. Nonetheless, the PRC network will reduce the mechanical performance of the porous sample [17], as measured here through apparent Young's modulus $E^*$.

Here, we seek to compare samples presenting various architectures and properties. However, as with most cellular materials, a change of architecture will influence mechanical behaviour. To avoid this effect and allow direct comparison of performance per unit weight, a performance index ($PI$) is calculated as in Eq.5.

$$PI = \frac{E^*}{\rho_f} \qquad (5)$$

with $E^*$ the sample's apparent Young's modulus and $\rho_f$ the volume fraction of solid in the sample.

### 4. Finite element analysis of porous samples

Hence, it is necessary to establish a process to evaluate samples' mechanical properties, such as apparent Young's modulus $E^*$. For that purpose, samples are analysed using finite elements (FE). Using FE analysis allows to neglect some of the manufacturing process' influence on mechanical performance (roughness, defects, etc.) because geometries are idealized.

The FE process was built to limit border effects in the region of interest. 7x7x7 samples of repeated patterns, as shown in Fig.5, were simulated in uniaxial compression but only the centre-most region was studied. Compressive force was applied on the top face of the
sample. Symmetry boundary conditions, as shown in Fig.5, were applied on three of the sample's faces to simulate the insertion in an infinite solid. Hence, the analysis' region of interest was distant from the zones where boundary conditions applied.

<figure><img src="image_7.png" /><figcaption>Figure 5: finite element model used to measure the mechanical properties of 3D²BIMR samples.</figcaption></figure>

Cubic samples ranging from 21mm to 35mm on a side were meshed using tetrahedral elements of size 0.5mm with material properties for 17-4PH stainless steel ($E = 210 GPa$ and $\nu = 0.3$). The size of element was selected as it allowed adequate shape representation, as visible in the zoom view in Fig.5, with PRCs of radius 0.5mm and ellipsoid porosities' semi-axes length between 0.5mm and 2.7mm. Because of samples' geometric complexity, further reducing element size led to drastic increase in computation time. Furthermore, a preliminary mesh convergence study showed that a 0.5mm mesh size allowed precise results in the analysis region of interest even with complex architectures. Further reducing the size of the mesh changed results by less than 1%. $E^*$ is calculated as in Eq.6.

$$E^* = \frac{F}{S\varepsilon} \qquad (6)$$

with $F$ the applied compressive force, $S$ the section and $\varepsilon$ the mean deformation of the centre-most region. $\rho_f$ is predicted from CAD data and $PI$ is calculated accordingly.

## 5. Design of Experiment to study the effect of pattern parameters on mechanical performance difference induced by powder removal design strategy

Thus, the FE process established in the previous section can be used to measure performance for a given porous sample. From there, it becomes possible to compare the mechanical performance of a given 3DBIMR porous sample (performance index, Eq.5: $PI_{3DBIMR}$) and the performance of the same sample with added PRC network ($PI_{3D^2BIMR}$). This comparison sheds light on the difference of mechanical performance induced by the powder removal design strategy. This performance difference $\Delta_{PI}$ can be quantified as in Eq.7.
$$ \Delta_{PI} = \frac{|PI_{3D^2BIMR} - PI_{3DBIMR}|}{PI_{3DBIMR}} \qquad (7) $$

As stated previously in section 2.2 and apparent in Fig.4, presented porous architectures are locally adapted to the stress the initial solid part undergoes and material is removed from low-stressed areas where it is least needed. This local adaptation is achieved by adjusting different pattern parameters (pattern size *p*, porosity aspect ratio *A*<sub>R</sub>, porosity orientation *θ* and volume fraction *ρ*<sub>f</sub>). Because the architecture adapts to the stress it undergoes, it is expected to showcase a range of mechanical properties as a function of pattern parameters and *Δ*<sub>PI</sub> is expected to vary. The various pattern parameters and their relationships are presented below.

Porosity aspect ratio *A*<sub>R</sub>, expressed in Eq.8, describes the shape of the ellipsoid porosity (spherical or elongated) and is a function of *a*, *b* and *c* the ellipsoid semi-axes length. Here, compared to [21], to simplify further calculations, *a* = *b* which describes an ellipsoid of revolution.

$$ A_R = \frac{c}{a} = \frac{c}{b} \qquad (8) $$

In a face-centred cubic pattern, following crystallographic theory, there are 4 nodes per pattern. The volume fraction of solid *ρ*<sub>f</sub>, as expressed in Eq.9, will then depend on pattern size *p* and porosities' semi-axes length *a*, *b* and *c*.

$$ \rho_f = \frac{V_{cube} - 4 V_{porosities}}{V_{cube}} = 1 - \frac{4 \left( \frac{4}{3} \pi a^2 c \right)}{p^3} = 1 - \left( \frac{16 \pi a^2 c}{3p^3} \right) \qquad (9) $$

<figure><img src="image_8.png" /><figcaption>Figure 6: x-z plane view of 3D²BIMR pattern showing local parameters that vary based on stress.</figcaption></figure>

Fig.6 shows a 3D²BIMR pattern in the x-z plane. As we can see, pattern size *p* controls the spacing between the nodes of the FCC pattern. This impacts the length of the PRCs but not their radius, kept at 0.5mm. Volume fraction *ρ*<sub>f</sub> controls the porosities' size and the amount of material in the pattern. Porosity aspect ratio *A*<sub>R</sub> controls the porosities' shape and thus the anisotropy of the pattern (porosities pronounced in one direction induce better
performance in that direction). Finally, porosity orientation $\theta$ controls the porosities'
alignment with principal stresses directions.

<figure><img src="image_9.png" /><figcaption>Figure 7: effect of each individual pattern parameters on the resulting architecture for samples of 7x7x7 repeated patterns; a) sample with $p = 5mm, \rho_f = 0.9, A_R = 2, \theta = 0^{\circ}$, used as a reference; b) sample with $p = 3mm, \rho_f = 0.9, A_R = 2, \theta = 0^{\circ}$ showcasing the effect of pattern size; c) sample with $p = 5mm, \rho_f = 0.3, A_R = 2, \theta = 0^{\circ}$ showcasing the effect of volume fraction; d) sample with $p = 5mm, \rho_f = 0.9, A_R = 1, \theta = 0^{\circ}$ showcasing the effect of porosity aspect ratio; e) sample with $p = 5mm, \rho_f = 0.9, A_R = 2, \theta = 45^{\circ}$ showcasing the effect of porosity orientation</figcaption></figure>
Fig.7 illustrates the effect of each individual pattern parameters on the resulting architecture for samples of 7x7x7 repeated patterns. For example, comparing Fig.7a) and Fig.7b), we can see the effect of pattern size *p* on the resulting sample which becomes smaller in Fig.7b) and will undergo higher deformation at a given compressive force. Comparing samples presented in Fig.7a) and Fig.7c) shows the effect of volume fraction *ρ*<sub>f</sub> and they are not expected to behave in the same way from a mechanical point of view because the sample in Fig.7a) presents a higher volume fraction. Similarly, the sample in Fig.7d) showcases the effect of aspect ratio *A*<sub>R</sub> and is expected to behave in a more isotropic manner as its porosities are spherical. To study the impact of pattern parameters *p*, *ρ*<sub>f</sub>, *A*<sub>R</sub> and *θ* on the difference in mechanical performance induced by the PRC network Δ<sub>PI</sub>, a Design Of Experiment (DOE) approach was used. In particular a set of 25 experiments were generated with the help of Nemrodw software [30,31], describing a cubic experimental domain.

For each experiment, two CAD samples are created with 3DBIMR and 3D<sup>2</sup>BIMR, using the same parameters. FE models of both samples are built, PI<sub>3DBIMR</sub> and PI<sub>3D<sup>2</sup>BIMR</sub> are measured and Δ<sub>PI</sub> is calculated accordingly. The DOE structure is given in Table 3. Based on this DOE structure, a model of the performance difference Δ<sub>PI</sub> as a function of *p*, *ρ*<sub>f</sub>, *A*<sub>R</sub> and *θ* is built using the Moore-Penrose inverse method [32]. The model is then refined to improve the fit of the response surface and reduce the standard deviation between predictions and results (see [33] for more details on the process of obtaining and refining the response surface).

<table><caption>Table 2: DOE structure</caption><thead><tr><th>Experiment number</th><th>p (mm)</th><th>ρ<sub>f</sub> (%)</th><th>A<sub>R</sub></th><th>θ (°)</th></tr></thead><tbody><tr><td>1</td><td>3</td><td>30%</td><td>1</td><td>0</td></tr><tr><td>2</td><td>5</td><td>30%</td><td>1</td><td>0</td></tr><tr><td>3</td><td>3</td><td>90%</td><td>1</td><td>0</td></tr><tr><td>4</td><td>5</td><td>90%</td><td>1</td><td>0</td></tr><tr><td>5</td><td>3</td><td>30%</td><td>2</td><td>0</td></tr><tr><td>6</td><td>5</td><td>30%</td><td>2</td><td>0</td></tr><tr><td>7</td><td>3</td><td>90%</td><td>2</td><td>0</td></tr><tr><td>8</td><td>5</td><td>90%</td><td>2</td><td>0</td></tr><tr><td>9</td><td>3</td><td>30%</td><td>1</td><td>45</td></tr><tr><td>10</td><td>5</td><td>30%</td><td>1</td><td>45</td></tr><tr><td>11</td><td>3</td><td>90%</td><td>1</td><td>45</td></tr><tr><td>12</td><td>5</td><td>90%</td><td>1</td><td>45</td></tr><tr><td>13</td><td>3</td><td>30%</td><td>2</td><td>45</td></tr><tr><td>14</td><td>5</td><td>30%</td><td>2</td><td>45</td></tr><tr><td>15</td><td>3</td><td>90%</td><td>2</td><td>45</td></tr><tr><td>16</td><td>5</td><td>90%</td><td>2</td><td>45</td></tr><tr><td>17</td><td>3</td><td>60%</td><td>1.5</td><td>22.5</td></tr><tr><td>18</td><td>5</td><td>60%</td><td>1.5</td><td>22.5</td></tr><tr><td>19</td><td>4</td><td>30%</td><td>1.5</td><td>22.5</td></tr><tr><td>20</td><td>4</td><td>90%</td><td>1.5</td><td>22.5</td></tr><tr><td>21</td><td>4</td><td>60%</td><td>1</td><td>22.5</td></tr><tr><td>22</td><td>4</td><td>60%</td><td>2</td><td>22.5</td></tr></tbody></table>
<table><tr><td>23</td><td>4</td><td>60%</td><td>1.5</td><td>0</td></tr><tr><td>24</td><td>4</td><td>60%</td><td>1.5</td><td>45</td></tr><tr><td>25</td><td>4</td><td>60%</td><td>1.5</td><td>22.5</td></tr></table>

### 3. Results

Measures of mass of all twelve samples presented in section 2.2 are shown in Table 2. Based on these results, all sample showcase a mass higher than the theoretical one predicted from CAD data. As mentioned in section 2.2 the sample with highest mass percent error, sample 12, was submitted to X-ray computed tomography to make sure that all un-melted powders are correctly removed and that mass discrepancies do not come from trapped un-melted powders.

<table><caption>Table 3: results of the measure of mass of all twelve 3D²BIMR samples manufactured.</caption><thead><tr><td>Sample number</td><td>Theoretical mass (g)</td><td>Measured mass (g)</td><td>Percent error</td></tr></thead><tbody><tr><td>1</td><td>9.256</td><td>11.732</td><td>26.74%</td></tr><tr><td>2</td><td>27.731</td><td>31.184</td><td>12.45%</td></tr><tr><td>3</td><td>86.978</td><td>91.318</td><td>4.99%</td></tr><tr><td>4</td><td>9.221</td><td>12.678</td><td>37.49%</td></tr><tr><td>5</td><td>55.017</td><td>61.496</td><td>11.78%</td></tr><tr><td>6</td><td>20.328</td><td>27.100</td><td>33.31%</td></tr><tr><td>7</td><td>12.222</td><td>14.220</td><td>16.34%</td></tr><tr><td>8</td><td>33.924</td><td>36.052</td><td>6.27%</td></tr><tr><td>9</td><td>98.551</td><td>100.118</td><td>1.59%</td></tr><tr><td>10</td><td>12.820</td><td>17.348</td><td>35.32%</td></tr><tr><td>11</td><td>70.720</td><td>75.659</td><td>6.98%</td></tr><tr><td>12</td><td>24.362</td><td>34.574</td><td>41.92%</td></tr></tbody></table>

Pictures from the X-ray computed tomography of sample 12 were stacked, segmented and converted into an STL file using the Fiji software [34]. The obtained STL file was compared to nominal CAD data using the open-source software CloudCompare v2.12.4 to compute the deviation between the two. The deviation of a given mesh to a reference represents the orthogonal distance for each point in the studied mesh (here, tomography results) to the boundary of the reference (here, CAD data). The sign is determined by whether a point is in the interior of the reference (negative values inside, zero on the boundary and positive values outside the reference). Tomography results and deviation field between nominal and measured geometries are presented in Fig.8. Based on visual inspection of all tomography sections (a few are represented in Fig.8a), no un-melted powders remain trapped in the sample after printing and separation from the build plate. Deviations follow a normal distribution of mean 0.072mm and standard deviation 0.080mm, showing a substantial difference between the nominal and manufactured surfaces.
<figure><img src="image_10.png" /><figcaption>Figure 8: Geometry results from manufactured sample 12, a) tomography results, b) Deviation field between measured and nominal geometries.</figcaption></figure>

The DOE results obtained from FE simulations presented in section 2.5 are given in Table 4.

<table><caption>Table 4: DOE results</caption><thead><tr><th>Experiment number</th><th>PI<sub>3D<sup>2</sup>BIMR</sub><br>(MPa)</th><th>PI<sub>3DBIMR</sub><br>(MPa)</th><th>Δ<sub>PI</sub><br>(%)</th></tr></thead><tbody><tr><td>1</td><td>136045</td><td>171953</td><td>20.88%</td></tr><tr><td>2</td><td>103092</td><td>133494</td><td>22.77%</td></tr><tr><td>3</td><td>182175</td><td>191316</td><td>4.78%</td></tr><tr><td>4</td><td>181477</td><td>184887</td><td>1.84%</td></tr><tr><td>5</td><td>199819</td><td>219011</td><td>8.76%</td></tr><tr><td>6</td><td>182319</td><td>199000</td><td>8.38%</td></tr><tr><td>7</td><td>179458</td><td>200457</td><td>10.48%</td></tr><tr><td>8</td><td>186070</td><td>197404</td><td>5.74%</td></tr><tr><td>9</td><td>138770</td><td>171953</td><td>19.30%</td></tr><tr><td>10</td><td>114392</td><td>133494</td><td>14.31%</td></tr><tr><td>11</td><td>178458</td><td>191316</td><td>6.72%</td></tr><tr><td>12</td><td>177256</td><td>184887</td><td>4.13%</td></tr><tr><td>13</td><td>185804</td><td>220348</td><td>15.68%</td></tr><tr><td>14</td><td>188984</td><td>208884</td><td>9.53%</td></tr><tr><td>15</td><td>175262</td><td>200022</td><td>12.38%</td></tr><tr><td>16</td><td>177234</td><td>199322</td><td>11.08%</td></tr><tr><td>17</td><td>170774</td><td>173947</td><td>1.82%</td></tr><tr><td>18</td><td>168182</td><td>177163</td><td>5.07%</td></tr><tr><td>19</td><td>158703</td><td>175729</td><td>9.69%</td></tr><tr><td>20</td><td>187136</td><td>194460</td><td>3.77%</td></tr><tr><td>21</td><td>148034</td><td>201318</td><td>26.47%</td></tr><tr><td>22</td><td>179752</td><td>191701</td><td>6.23%</td></tr><tr><td>23</td><td>166792</td><td>179548</td><td>7.10%</td></tr><tr><td>24</td><td>169727</td><td>193753</td><td>12.40%</td></tr><tr><td>25</td><td>163418</td><td>180590</td><td>9.51%</td></tr></tbody></table>
Based on these results, as outlined in section 2.5, a model was established and refined to predict performance difference $\Delta_{PI}$ as a function of pattern parameters (Eq.10).

$$ \Delta_{PI} = 0.442 - 0.127\rho_f - 0.052p^2 + 0.311p + 0.308A_R^2 - 1.204A_R + 0.07pA_R \quad (10) $$

## 4. Discussion

Based on the experimental results shown in Table 2, all manufactured samples showcase a mass higher than the mass predicted from CAD data, with the lowest percent error in mass 1.59% and the highest 41.92%. This is not surprising as AM parts often show roughness and surface deviation from nominal dimensions [23]. Indeed, when the laser melts powders in LB-PBF, there is a heat-affected zone around the scanning area. This heat-affected zone can lead to powder grains melting onto the surface when they were not originally expected to. This extra material can be removed after printing through post-processing of AM parts [15]. In this work, no further post-processing (for example, heat treatment, chemical etching, or grit blasting) is conducted after separation from plate to avoid the risk of trapping un-melted powders inside the sample. Hence, deviation from nominal CAD data due excess material being melted onto samples' exposed surfaces could explain the mass percent error for manufactured samples. The differences in mass percentage error between samples may come from the fact that samples have varying degrees of exposed surface-area per unit volume, as quantified through the surface-area to volume ratio. Samples with larger surface areas per unit volume will have more excess material and therefore higher mass percent error. This is confirmed in Fig.9 which shows the surface-area to volume ratio of all twelve samples, predicted from CAD data, plotted against the mass percent error.

<figure><img src="image_11.png" /><figcaption>Figure 9: surface-area to volume ratio of all manufactured samples plotted against mass percent error, as calculated between theoretical CAD mass and measured mass. Tomographied sample 12 is showcased as the sample with highest mass percent error.</figcaption></figure>

With the increase of surface-area to volume ratio, the mass percent error increases as expected. This confirms the hypothesis of mass percent error being due to excess material melted onto the samples' exposed surfaces rather than un-melted powders staying trapped. This is further confirmed by the tomography results in Fig.8 which showed no visible trapped powders and a 0.072 ± 0.080mm deviation between nominal and manufactured surface for sample 12. Correcting the theoretical mass to account for that deviation from CAD data (adding a volume equal to the sample's surface-area times the mean deviation between meshes of 0.072 mm), mass percent error for sample 12 falls from 41.92% to 0.54%. This
means that all un-melted powders are correctly removed, and mass differences are not
caused by trapped un-melted powders that would not appear in tomography images.

With regards to FE simulation results, no increase of mechanical performance was identified after introduction of the PRC network. The proposed powder removal design strategy reduces mechanical performance of the porous samples, similar to results from [17,23]. But with regards to performance per unit weight, that loss is counterbalanced by a reduction of volume fraction ρf. Hence, the mean performance difference in stiffness-to-weight ratio was only 10.35% with some samples showing loss as low as 1.82%. Through proper understanding of the sample's mechanical behaviour, it becomes possible to minimise the performance difference. Porous bio-inspired architectures designed using 3D²BIMR can thus be used in combination with LB-PBF technologies, making them relevant in demanding sectors such as the aerospace, automotive or biomedical industries.

With regards to the mathematical model (Eq.10), derived from FE simulation results to predict Δ<sub>Pi</sub> based on pattern parameters, it is worth noting that orientation θ is removed during the refinement of the model. Removing θ in the model reduces the standard deviation of the modelling error between predicted and measured Δ<sub>Pi</sub>. One possible explanation is that PRCs may produce a quasi-isotropic difference of mechanical performance, thus being of influence on the mean performance difference rather than as a parameter of Δ<sub>Pi</sub>. The 4D response surface for Eq.10 is provided in Fig.10. In this 4D map, the x-y-z axes represent the pattern parameters *p*, ρ<sub>f</sub> and A<sub>R</sub> respectively. The 4-th dimension of performance difference Δ<sub>Pi</sub> is represented through the size and colour of the spheres.

<figure><img src="image_12.png" /><figcaption>Figure 10: 4D response surface of the mathematical model linking Δ<sub>Pi</sub> to pattern parameters p, ρ<sub>f</sub>, A<sub>R</sub></figcaption></figure>

In Fig.10, it is apparent that the difference in mechanical performance induced by the PRC
network varies throughout the considered range of parameters. The performance difference
for samples with high volume fraction seems overall lower than the difference for samples
with low volume fractions. This is surprising as lower volume fraction samples have bigger
porosities and shorter PRCs. Shorter PRCs should induce a lower difference of mechanical performance than longer ones. It is possible that samples with lower volume fraction (shorter PRCs) are loaded more evenly making them more sensitive to changes in their architecture. Another hypothesis is that cellular materials, in particular open-celled foams, see their Young's modulus scale as a function of their volume fraction squared [8]. Hence, the same 5% difference in volume fraction will have more effect on the performance of a low volume fraction sample compared to a high volume fraction sample. Intuitively, one might have thought that pattern size *p* would have a greater influence. Indeed, smaller samples mean smaller porosities. Because PRCs are of fixed radius, they could have induced a higher performance difference on smaller samples, which doesn't seem to be the case judging from results shown in Fig. 10.

Overall, the behaviour of $\Delta_{PI}$ as a function of $p, \rho_f$ and $A_R$ corresponds to the behaviour of trabecular bone, whose mechanical performance correlates highly to volume fraction and fabric tensor [35,36]. This is a consequence of the bio-inspiration process used to establish 3DBIMR. As prescribed in [3], general principles were abstracted from trabecular architecture (namely allometric scaling and local adaptation to stress through gradient volume fraction and anisotropy) and transfered to technological solutions. Details of the bio-inspired structural design process can be found in [21].

Increasing the performance of parts is often one of the main objectives in engineering, along with guaranteeing safety and keeping costs reasonable. Hence, engineers regularly have to select a material that best suits their application, often using a material selection graph (also called an Ahsby plot) [37]. This type of graph plots one or more material properties in relation to each other, representing a slice in the material property space [10]. Selecting the best material then comes down to choosing the material with best combination of properties. Hybrid materials (composites, natural materials, cellular materials, etc.) can help expand the material property space into areas that were previously unreachable [9]. As defined in [9] "a hybrid material is a combination of two or more materials in a predetermined geometry and scale, optimally serving a specific engineering purpose". Cellular materials are hybrids of their constituent material and air, taking advantage of their architecture to achieve superior properties. In this context, 3DBIMR and 3D²BIMR samples leverage a bio-inspired architecture to outperform the material they are made from. This is apparent in Fig. 11 which presents an Ahsby plot of Young's modulus versus density with envelopes drawn based on data obtained for 3DBIMR and 3D²BIMR samples. This type of charts help select the best material for stiff and light parts. In the case of a bending beam for example, the performance of a material is represented by its position with regards to the guidelines of slope 2, which represents a selection criteria of $E^{1/2}/\rho$. As can be seen on Fig. 11, samples outperform the 17-4PH stainless steel they are made from for bending applications. These mass reduction methods open up new possibilities to design high performance parts.
<figure><img src="image_13.png" /><figcaption>Figure 11: Ashby plot presenting the position of 17-4PH stainless steel 3DBIMR and 3D²BIMR samples compared to the material they are made from for optimisation of stiff and light beams loaded in bending, based on DOE results from Table 4. The parallel lines of slope 2 indicate the selection criteria E¹/²/ρ.</figcaption></figure>

Fig.12 presents the same type of Ashby plot with a large scope of materials represented, giving a global view of material design space for stiff and light parts. As can be seen in Fig.12, technical ceramics and hybrid materials (composites and natural materials) show better performance than metals and alloys, explaining their use in high-performance industries. In practice, metals & alloys are still widely used because of their multiple other properties (electric conductor, high strength, easy to shape and manufacture, etc.). 3D²BIMR architectures bridge the gap between natural and metallic materials by leveraging the bio-inspiration process. Hence, steel 3D²BIMR samples present better performance than bulk 17-4PH stainless steel whilst maintaining some of steel's interesting mechanical, thermal and electrical properties that cannot be met by ceramics, composites or natural materials. In this regard, 3D²BIMR fills up a hole in the material design space.
<figure><img src="image_14.png" /><figcaption>Figure 12: Ashby plot presenting the position of 17-4PH stainless steel 3DBIMR and 3D²BIMR samples in the global material design space for optimisation of stiff and light parts, based on DOE results from Table 4 and data from the CES Selector software. The parallel lines of slope 2 indicate the selection criteria $E^{1/2}/\rho$.</figcaption></figure>

## 5. Conclusion

In this work, we propose and study the mechanical impact of a design strategy that ensures correct powder removal in porous architectures designed using a 3D bio-inspired mass reduction method and manufactured using laser-based powder-bed fusion. The powder removal design strategy leverages the properties of the face centred cubic pattern that underlies the porous architecture to create a powder removal canal (PRC) network. In this network, for each internal porosity, cylindrical holes of radius 0.5 mm are created to 4 of the porosity's twelve closest neighbours. This network allows for complete removal of un-melted powders in the porous architecture, as measured through weighing and microtomography.

Finite elements analysis was used to study the mechanical performance of porous samples. For that purpose, uniaxial compression tests were simulated. A performance index was defined as the ratio of apparent Young's modulus to volume fraction.

Integrating PRCs in a porous sample induces a difference of performance. It was hypothesized that this difference varied based on the porous pattern's parameters. Hence, to study the influence of pattern parameters $p, \rho_f, A_R$ and $\theta$ (pattern size, volume fraction, porosity aspect ratio and porosity orientation respectively) on the performance difference $\Delta_{PI}$, a Design Of Experiment methodology was used. Based on these results, a mathematical model was fitted and refined to link pattern parameters to performance difference. Overall, the mean performance difference induced by the PRC network is 10.35% with some architectures as low as 1.82% and some as high as 26.47%.

3D²BIMR architectures manufactured with 17-4PH stainless steel powder opens up new performance for metallic materials, bridging the gap between natural and metallic materials by leveraging the bio-inspiration process. Steel 3D²BIMR samples present better
performance for stiff and light applications than bulk steel parts whilst maintaining some of
steel's other interesting mechanical, thermal and electrical properties that cannot be met by
ceramics, composites or natural materials.

CRediT authorship contribution statement:

**Nicolas Cadoret:** Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Visualization, Writing – original draft, Writing – review & editing. **Julien Chaves-Jacob:** Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – review & editing. **Marc-Antoine Celli:** Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – review & editing. **Jean-Marc Linares:** Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Supervision, Validation, Visualization, Writing – review & editing.

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.

Acknowledgment

The experimental equipment was funded by the European Community, French Ministry of
Research and Education and Aix-Marseille Conurbation Community. Authors acknowledge
the Association Nationale Recherche Technologie (France) for financial support through a
Convention Industrielle de Formation par la Recherche fund (Cifre thesis n°2022/0012).
Authors would like to thank Thibaut Pouly who carried out the finite element analysis
presented in this work. Authors would also like to thank the anonymous reviewers who
helped to make this article clearer and more thorough. For the purpose of Open Access, a
CC-BY public copyright licence has been applied by the authors to the present document and
will be applied to all subsequent versions up to the Author Accepted Manuscript arising from
this submission.

Data availability

The data used in this study will be made available upon reasonable request.

References

[1] IPCC, Climate Change 2023: Synthesis Report. A Report of the Intergovernmental Panel on Climate Change. Contribution of Working Groups I, II and III to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change, IPCC, Geneva, Switzerland, 2023. https://doi.org/10.59327/IPCC/AR6-9789291691647.

[2] L.R. Meza, S. Das, J.R. Greer, Strong, lightweight, and recoverable three-dimensional ceramic nanolattices, Science 345 (2014) 1322–1326.
https://doi.org/10.1126/science.1255908.

[3] ISO 18458:2015 Biomimetics - Terminology, concepts and methodology, (2015).
https://www.iso.org/standard/62500.html.
[4] A. Alsheghri, N. Reznikov, N. Piché, M.D. McKee, F. Tamimi, J. Song, Optimization of 3D network topology for bioinspired design of stiff and lightweight bone-like structures, Mater. Sci. Eng. C 123 (2021) 112010. https://doi.org/10.1016/j.msec.2021.112010.

[5] C. Broeckhoven, A. du Plessis, Escaping the Labyrinth of Bioinspiration: Biodiversity as Key to Successful Product Innovation, Adv. Funct. Mater. 32 (2022) 2110235. https://doi.org/10.1002/adfm.202110235.

[6] P. Fratzl, R. Weinkamer, Nature's hierarchical materials, Prog. Mater. Sci. 52 (2007) 1263–1334. https://doi.org/10.1016/j.pmatsci.2007.06.001.

[7] G. Byrne, D. Dimitrov, L. Monostori, R. Teti, F. van Houten, R. Wertheim, Biologicalisation: Biological transformation in manufacturing, CIRP J. Manuf. Sci. Technol. 21 (2018) 1–32. https://doi.org/10.1016/j.cirpj.2018.03.003.

[8] L.J. Gibson, Biomechanics of cellular solids, J. Biomech. 38 (2005) 377–399. https://doi.org/10.1016/j.jbiomech.2004.09.027.

[9] M.F. Ashby, Y.J.M. Bréchet, Designing hybrid materials, Acta Mater. 51 (2003) 5801–5821. https://doi.org/10.1016/S1359-6454(03)00441-5.

[10] N.A. Fleck, V.S. Deshpande, M.F. Ashby, Micro-archituctured materials: past, present and future, Proc. R. Soc. Math. Phys. Eng. Sci. 466 (2010) 2495–2516. https://doi.org/10.1098/rspa.2010.0215.

[11] N. Reznikov, R. Shahar, S. Weiner, Bone hierarchical structure in three dimensions, Acta Biomater. 10 (2014) 3815–3826. https://doi.org/10.1016/j.actbio.2014.05.024.

[12] H.M. Frost, Bone mass and the mechanostat: A proposal, Anat. Rec. 219 (1987) 1–9. https://doi.org/10.1002/ar.1092190104.

[13] H.M. Frost, Bone's mechanostat: A 2003 update, Anat. Rec. 275A (2003) 1081–1101. https://doi.org/10.1002/ar.a.10119.

[14] R. Huiskes, If bone is the answer, then what is the question?, J. Anat. 197 (2000) 145–156. https://doi.org/10.1046/j.1469-7580.2000.19720145.x.

[15] M. Schmidt, M. Merklein, D. Bourell, D. Dimitrov, T. Hausotte, K. Wegener, L. Overmeyer, F. Vollertsen, G.N. Levy, Laser based additive manufacturing in industry and academia, CIRP Ann. 66 (2017) 561–583. https://doi.org/10.1016/j.cirp.2017.05.011.

[16] B. Blakey-Milner, P. Gradl, G. Snedden, M. Brooks, J. Pitot, E. Lopez, M. Leary, F. Berto, A. du Plessis, Metal additive manufacturing in aerospace: A review, Mater. Des. 209 (2021) 110008. https://doi.org/10.1016/j.matdes.2021.110008.

[17] W. Wang, Y. Xia, Topology optimization based channel design for powder-bed additive manufacturing, Addit. Manuf. 54 (2022) 102717. https://doi.org/10.1016/j.addma.2022.102717.

[18] L.W. Hunter, D. Brackett, N. Brierley, J. Yang, M.M. Attallah, Assessment of trapped powder removal and inspection strategies for powder bed fusion techniques, Int. J. Adv. Manuf. Technol. 106 (2020) 4521–4532. https://doi.org/10.1007/s00170-020-04930-w.

[19] M. Bilhère-Dieuzeide, J. Chaves-Jacob, E. Buhon, G. Biguet-Mermet, J.-M. Linares, Performance domains of bio-inspired and triangular lattice patterns to optimize the structures' stiffness, Heliyon 10 (2024). https://doi.org/10.1016/j.heliyon.2024.e26001.

[20] M. Bilhère-Dieuzeide, J. Chaves-Jacob, E. Buhon, G. Biguet-Mermet, J.-M. Linares, Stress-driven method bio-inspired by long bone structure for mechanical part mass reduction by removing geometry at macro and cell-unit scales, Mater. Des. 213 (2022) 110318. https://doi.org/10.1016/j.matdes.2021.110318.
[21] N. Cadoret, J. Chaves-Jacob, J.-M. Linares, Structural additive manufacturing parts bio-inspired from trabecular bone form-function relationship, Mater. Des. 231 (2023) 112029. https://doi.org/10.1016/j.matdes.2023.112029.

[22] X. Wei, X. Li, S. Wen, Y. Zheng, Y. Tian, Channel design for 3D models with applications in powder-bed additive manufacturing, Rapid Prototyp. J. 25 (2019) 1536–1544. https://doi.org/10.1108/RPJ-06-2018-0156.

[23] S. Verma, A. Kumar, S.-C. Lin, J.-Y. Jeng, A bio-inspired design strategy for easy powder removal in powder-bed based additive manufactured lattice structure, Virtual Phys. Prototyp. 17 (2022) 468–488. https://doi.org/10.1080/17452759.2022.2039071.

[24] M. Doube, M.M. Kłosowski, A.M. Wiktorowicz-Conroy, J.R. Hutchinson, S.J. Shefelbine, Trabecular bone scales allometrically in mammals and birds, Proc. R. Soc. B Biol. Sci. 278 (2011) 3067–3073. https://doi.org/10.1098/rspb.2011.0069.

[25] C. Audibert, J. Chaves-Jacob, J.-M. Linares, Q.-A. Lopez, Bio-inspired method based on bone architecture to optimize the structure of mechanical workspieces, Mater. Des. 160 (2018) 708–717. https://doi.org/10.1016/j.matdes.2018.10.013.

[26] S.C. Cowin, The relationship between the elasticity tensor and the fabric tensor, Mech. Mater. 4 (1985) 137–147. https://doi.org/10.1016/0167-6636(85)90012-2.

[27] J. Hazrati Marangalou, K. Ito, B. van Rietbergen, A novel approach to estimate trabecular bone anisotropy from stress tensors, Biomech. Model. Mechanobiol. 14 (2015) 39–48. https://doi.org/10.1007/s10237-014-0584-6.

[28] N. Cadoret, J. Chaves-Jacob, J.-M. Linares, A. Houssaye, Form-function relationship between trabecular bone architecture and biomechanical function in the horse humerus, in: Procedia CIRP, 2022: pp. 299–304. https://doi.org/10.1016/j.procir.2022.06.054.

[29] J. Linares, J. Chaves-Jacob, Q. Lopez, J.-M. Spruel, Fatigue life optimization for 17-4Ph steel produced by selective laser melting, Rapid Prototyp. J. 28 (2022) 1182–1192. https://doi.org/10.1108/RPJ-03-2021-0062.

[30] G.A. Lewis, D. Mathieu, R. Phan-Tan-Luu, Pharmaceutical experimental design, CRC press, 1998.

[31] S.D. Brown, R. Tauler, B. Walczak, Comprehensive chemometrics: chemical and biochemical data analysis, Elsevier, 2020.

[32] P. Courrieu, Fast Computation of Moore-Penrose Inverse Matrices, (2008). http://arxiv.org/abs/0804.4809 (accessed July 16, 2021).

[33] F. Ezedine, J.-M. Linares, J.-M. Spruel, J. Chaves-Jacob, Smart sequential multilateration measurement strategy for volumetric error compensation of an extra-small machine tool, Precis. Eng. 43 (2016) 178–186. https://doi.org/10.1016/j.precisioneng.2015.07.007.

[34] J. Schindelin, I. Arganda-Carreras, E. Frise, V. Kaynig, M. Longair, T. Pietzsch, S. Preibisch, C. Rueden, S. Saalfeld, B. Schmid, J.-Y. Tinevez, D.J. White, V. Hartenstein, K. Eliceiri, P. Tomancak, A. Cardona, Fiji: an open-source platform for biological-image analysis, Nat. Methods 9 (2012) 676–682. https://doi.org/10.1038/nmeth.2019.

[35] G. Maquer, S.N. Musy, J. Wandel, T. Gross, P.K. Zysset, Bone Volume Fraction and Fabric Anisotropy Are Better Determinants of Trabecular Bone Stiffness Than Other Morphological Variables, J. Bone Miner. Res. 30 (2015) 1000–1008. https://doi.org/10.1002/jbmr.2437.

[36] S.N. Musy, G. Maquer, J. Panyasantisuk, J. Wandel, P.K. Zysset, Not only stiffness, but also yield strength of the trabecular structure determined by non-linear µFE is best
predicted by bone volume fraction and fabric tensor, J. Mech. Behav. Biomed. Mater.
65 (2017) 808-813. https://doi.org/10.1016/j.jmbbm.2016.10.004.

[37] M.F. Ashby, Materials selection in mechanical design, Elsevier Science, 2016.
https://www.elsevier.com/books/materials-selection-in-mechanical-design/ashby/978-0-08-100599-6.