# Architected materials for space applications: a computational tool for the parametric optimization of a three-dimensional lattice subjected to stiffness constraints

F. Ongaro, K. Mathis, F. Masson, Justin Dirrenberger

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## To cite this version:

F. Ongaro, K. Mathis, F. Masson, Justin Dirrenberger. Architected materials for space applications: a computational tool for the parametric optimization of a three-dimensional lattice subjected to stiffness constraints. 8th European Conference for Aeronautics and Space Sciences (EUCASS), Jul 2019, Madrid, Spain. 10.13009/EUCASS2019-167. [hal-04840779](https://HAL Id: hal-04840779)

Submitted on 18 Dec 2024

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# Architectured materials for space applications: a computational tool for the parametric optimization of a three-dimensional lattice subjected to stiffness constraints

F. Ongaro<sup>a</sup>, K. Mathis<sup>b</sup>, F. Masson<sup>b</sup> and J. Dirrenberger<sup>a</sup>

federica.ongaro@ensam.eu · kevin.mathis@cnes.fr · frederic.masson@cnes.fr · justin.dirrenberger@ensam.eu

†Corresponding author

## Abstract

ArchitECTured MAterials, whose discrete configuration provides unique combinations of enhanced struc-
tural properties at low weight, solved a variety of technical challenges in material science, architecture,
aerodynamics and mechanical engineering. This peculiar characteristic, together with a high degree of de-
sign freedom leading to the possibility to tailor their mechanical properties in each direction, makes them
very promising in a vast number of industries including aerospace, automotive, marine and constructions.
However, the use of architECTured materials is conditional upon the development of appropriate consti-
tutive models for revealing the complex relations between the parameters of the microstructure and the
macroscopic behavior. Notwithstanding a great variety of analytical and numerical techniques have been
proposed and discussed in recent years, explicit formulas for the effective mechanical properties are de-
rived in a very small number of investigations. To provide a contribution in this limitedly explored research
area, this paper describes the mathematical formulation and modelling technique leading to closed-form
expressions for the effective stiffness of a three-dimensional lattice composed of identical hexatruss cells.
The derived analytical relations, verified by performing experimental tests on a 3D-printed lattice, are
then integrated into a parametric optimization problem for finding the optimal microstructure's parameters
that meet a given set of stiffness requirements. This strategy offers a less computationally way to solve
optimization problems for architECTured materials and, as a practical example, the specific case of the
Spacecraft-Launcher Damping Interface is considered. The developed theory, however, is general enough
to be easily applied to different types of structures in the aerospace industry.

## 1. Introduction

ArchitECTured materials, defined as a composition of two or more materials and space,¹ offer unique combinations of
mechanical properties, like stiffness, strength or toughness, at low weight, providing the coexistence of what used to
be antagonistic performances within a single homogeneous material.

Due to their peculiar configuration, arising from their discrete nature, architECTured materials are very promising
for engineering applications in a variety of industries including aerospace, automotive, marine and constructions. For
example, the low density makes them ideal core materials in lightweight and high-performance sandwich panels used
in aerospace components and sporting equipments, while the low compressive strength and Also, high deformation
capacity provide excellent shock mitigation and energy absorption characteristics in impulsive phenomena.

However, the use of architECTured materials is conditional upon the development of appropriate constitutive
models for revealing their effective properties. From this point of view, two sets of parameters affect their mechanical
behavior: the first characterises the constituent material, the second is related to the geometric and topological prop-
erties of the microstruture. In addition, the formulation of a continuum model is hindered by the crucial passage from
the microscopic discrete description to the coarse continuous one and, to overcome this problem, energy equivalence
concepts and micro-macro relations in terms of forces and displacements are usually applied. Also, linear elasticity

<footnote><sup>a</sup>PIMM Lab, ENSAM-ParisTech/CNAM/CNRS UMR 8006, 151 Bd de l'Hôpital, 75013 Paris, France</footnote>

<footnote><sup>b</sup>CNES-Launchers Directorate, 52 Rue Jacques Hillairet, 75612 Paris Cedex, France</footnote>
and material isotropy, in conjunction with the underlying microstructure assumed to be governed by the classical beam theory,² are three commonly used simplifications that provide explicit stress-strain relations and help clarifying the basic mechanical aspects.

Many authors studied the mechanical modelling of architECTured materials and a great variety of techniques have been proposed and exhaustively discussed in recent years.³⁻⁶ However, the most known and widely used micromechanical model is due to the pioneering contribution of Gibson & Ashby,⁷,⁸ focusing on the deformation mechanism of a single cell subjected to different types of external loads. The authors, in particular, obtained first-order power-law relations between the microstructure's parameters and the macroscopic properties of a wide range of architECTured materials by assuming infinitesimal strains and applying the standard beam theory to model the lattice' edges.

An alternative approach to solve the crucial passage from micro to macro and to derive the constitutive model for two-dimensional microstructures subjected to in-plane deformations was adopted by Chen et al.⁹ and Kumar & McDowell¹⁰ where, as in Gibson & Ashby,⁷,⁸ the discrete lattice is idealised as a sequence of Euler-Bernoulli beams while, in this case, the macroscopic description follows from an energy-based technique.

In the framework of homogenization theory, Gonella & Ruzzene¹¹ interprets the discrete lattice according to the finite difference formalism and the equivalent continuum derives from the application of Taylor's series expansions of the nodal displacements and rotations, motivated by the multi-scale nature of the considered problem.

Notions of crystals' theory¹² helped Hutchinson & Fleck,¹³ Elsayed & Pasini,¹⁴ Vigliotti & Pasini¹⁵ to estimate the effective stiffness of different types of bi-dimensional lattice materials, represented as a pin-jointed infinite microtruss structure obtained by tessellating a unit cell into a periodic modular pattern. The suggested approach relies on the Cauchy-Born assumption¹⁶ to express the lattice's nodal displacements in terms of the macroscopic strain field applied and to derive the homogenized properties.

Force- and energy equivalence-based techniques have also been applied to characterise the mechanical behavior of three-dimensional lattice materials. Zhu et al.,¹⁷ for example, expressed the Young's modulus, shear modulus and Poisson's ratio of the tetrakeidecahedral lattice as a function of the relative density, while the mechanical performance of the octet-truss lattice is investigated in Deshpande et al.¹⁸ and Challapalli & Ju¹⁹ from a theoretical and numerical point of view. Both the effective elastic properties and collapse surfaces for buckling and plastic yielding are derived and verified by means of experimental observations. A more general analysis of three-dimensional periodic lattices is proposed in Vigliotti & Pasini²⁰ and Refai et al.,²¹ where the authors employed a finite element-based homogenization method to determine the macroscopic elastic properties of open- and closed-cell lattices with different known topologies of the unit cell, as BCC, FCC and octet.

In the last decade, due to the advent of additive manufacturing technologies, several attempts have been made to further explore the potential advantages of architECTured materials and special attention is given to the possibility to achieve optimised structures of actual use, in terms of tailored combinations of stiffness and strength. For instance, how to lower the value of the stiffness while increasing the strength, in order to obtain a lattice material that is easily deformed but resistant to rupture Hagpanah et al.²² Also, as reported in Vigliotti & Pasini²³ with reference to the design of a cantilever bracket subjected to a point load, a proper selection of the microstructure's parameters allows the tunability of the macroscopic mechanical properties in different directions. Starting from the geometry and boundary conditions of the examined component, the authors set up a multiobjective optimization problem (MOO) to determine the optimal lattice material providing the maximisation of the bracket's stiffness while reducing its mass. Four lattice topologies have been considered, kagome, equilateral triangular and rectangular triangular, and, for each one, the optimized parameters have been obtained by reformulating the original MOO problem into a sequence of single objective optimization problems, solved with a discrete gradient-based method.²⁴ As suggested by the name, gradient-based optimization techniques use gradient informations to find the optimal solution and, despite being very popular in engineering, have several drawbacks including the difficulty to be efficiently implemented and the fact that only a local optimum can be identified.²⁴,²⁵ A different way to solve optimization problems of lattice strutures, providing a better chance to find the global optimum, consists in adopting non-gradient optimization techniques, such as the evolutionary optimization algorithms.²⁴,²⁶ These well established methods mimic the process of natural evolution by following the four basic steps of reproduction, mutation, recombination and selection²⁵ and, among them, the genetic algorithm²⁷ has found to be extremely robust, easy to implement and well suited for numerous structural engineering applications.²⁴,²⁵ An abundance of literature exists in this field, as the work proposed by Vaissier et al.²⁸ focusing on the development of a genetic algorithm-based framework for the shape and topology optimization of a lattice structure used as a support in additive manufacturing. Referring the interested reader to Burczynski & Osyczka²⁹ for a more detailed discussion, two final examples include the application of the genetic algorithm to perform the parametric optimization of chiral lattices for flexible frames with various specifications and dimensions,³⁰ and the shape optimization of regular lattice structures, hexagonal, triangular and square, in order to obtain the maximum shear capacity.³¹

With reference to the applicability of architECTured materials to design certain components of the Ariane Launcher, this work presents a computational tool for the parametric optimization of a three-dimensional lattice composed of iden-
tical hexatruss unit cells, subjected to a given set of stiffness requirements. The paper is organised in 6 sections, including this introduction. Initially, by modelling the examined lattice as a sequence of three-dimensional Euler-Bernoulli beams, an energy-based approach coupled with the Cauchy-Born assumption provide, in Section 2, closed-form expressions for the macroscopic stiffness. Such relations, verified by comparison with the outcome of the experimental campaign described in Section 3, are then integrated in the parametric optimization problem described in Section 4. Here, in particular, the optimal parameters of the microstructure are derived by optimizing a system of three analytical equations, i.e., the effective stiffness in the three directions, and this strategy, based on the genetic algorithm, offers a less computationally expensive way to solve optimization problems for architected materials since a change in the parameters does not require a re-generation of the lattice. Finally, Section 5 summarises the main findings.

## 2. The hexatruss lattice: theoretical modelling and homogenization of the discrete system

### 2.1 Overview

This paper focuses on the three-dimensional auxetic microstructure presented by Dirrenberger et al.,³² generated by tessellating a hexatruss unit cell (Fig. 1) through the space along three independent periodic vectors $\mathbf{l}_1, \mathbf{l}_2, \mathbf{l}_3$. In the global reference system, defined by the unit orthonormal vectors $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ and by the coordinate system (X, Y, Z), the components of the lattice vectors are

$$ \mathbf{l}_1 = [L_1 \quad 0 \quad 0]^T, \quad \mathbf{l}_2 = [0 \quad L_2 \quad 0]^T, \quad \mathbf{l}_3 = [0 \quad 0 \quad L_3]^T, \qquad (1) $$

with $L_1, L_2, L_3$ the overall dimensions of the cell (the lattice size).

The resulting system can be conceived as a collection of discrete elements connected at nodes locations periodically arranged. In particular, two classes of nodes can be identified: the internal nodes and the boundary nodes. The first connect elements of only a single cell while the second connect elements of confining cells. Also, due to the periodicity of the lattice configuration, the boundary nodes are corresponding along the lattice vectors and, as outlined in the following sections, such nodes will be subjected to the imposition of kinematic constraints in order to preserve the periodicity of the deformed lattice.

Finally, from a mechanical point of view, the examined microstructure is treated as an interconnected network of three-dimensional Euler-Bernoulli beams having circular cross-section of radius $r$ and made of a linear elastic isotropic material with Young's modulus $E_s$, Poisson's ratio $\nu_s$ and shear modulus $G_s$, as explained in the next section.

<figure><img src="image_3.png" /><figcaption>Figure 1: The hexatruss unit cell</figcaption></figure>

### 2.2 The three-dimensional Euler-Bernoulli beam element

Let us focus on the $e$-th Euler-Bernoulli beam element illustrated in Figure 2.

In the global reference ($\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$), the configuration of the considered element is known by specifying the coordinates of its end nodes, $I$ and $J$. However, to analyse the generic component it is more convenient using a local reference system, specific to the considered beam and closely dependent to its geometry. Such reference is defined by three orthonormal unit vectors $\eta_1^e, \eta_2^e, \eta_3^e$ (Fig. 2), and by the coordinate reference system (x, y, z).

It should be noted that, hereinafter, the local notation will be adopted and the extreme nodes of the beam will be denoted by the indices $i$ and $j$. Similarly, lowercase letters will be used to differentiate the local variables from the global ones.
<figure><img src="image_4.png" /><figcaption>Figure 2: The Euler-Bernoulli beam element in the local reference system: (a) degrees of freedom, (b) forces and couples</figcaption></figure>

Regarding the kinematics of the three-dimensional Euler-Bernoulli beam, each node has six degrees of freedom: three translations, $u^e$, $v^e$ and $w^e$, and three rotations, $\phi_x$, $\phi_y$ and $\phi_z$. Thus, the following 12x1 vector completely describes the element nodal displacements:

$$ \mathbf{d}^e := \begin{bmatrix} \mathbf{d}_i^e \\ \mathbf{d}_j^e \end{bmatrix} = \begin{bmatrix} u_i^e & v_i^e & w_i^e & \phi_{xi}^e & \phi_{yi}^e & \phi_{zi}^e & u_j^e & v_j^e & w_j^e & \phi_{xj}^e & \phi_{yj}^e & \phi_{zj}^e \end{bmatrix}^T. \quad (2) $$

In terms of energetics, the elastic strain energy of the Euler-Bernoulli beam element takes the form

$$ w^e := \frac{1}{2} (\mathbf{d}^e)^T \cdot \mathbf{k}^e \mathbf{d}^e, \quad (3) $$

with $\mathbf{k}^e$ the element stiffness matrix expressed in the local reference system.³³

Finally, similarly to Equation (2), the forces and couples acting at the extreme nodes of the beam (Fig. 2b),

$$ \mathbf{f}^e := \begin{bmatrix} \mathbf{f}_i^e \\ \mathbf{f}_j^e \end{bmatrix} = \begin{bmatrix} f_{xi}^e & f_{yi}^e & f_{zi}^e & m_{xi}^e & m_{yi}^e & m_{zi}^e & f_{xj}^e & f_{yj}^e & f_{zj}^e & m_{xj}^e & m_{yj}^e & m_{zj}^e \end{bmatrix}^T, \quad (4) $$

are derived from the classical relation

$$ \mathbf{f}^e := \mathbf{k}^e \mathbf{d}^e. \quad (5) $$

## 2.3 The hexatruss unit cell: description and energetics

The typical approach to the continuum modelling of architectured materials includes the selection of a Representative Volume Element (RVE) that, in investigating materials with a periodic microstructure as in the present paper, coincides with the repetitive unit cell of the tessellation.³⁴

In the examined case, in particular, the unit cell is composed by the eight boundary nodes 1, 2, 3, 4, 5, 6, 7, 8 linked to the internal ones 9, 10, 11, 12, 13, 14 by the line elements (Fig. 3)

1 – 9, 2 – 9, 6 – 9, 5 – 9 (Face Front)

4 – 10, 3 – 10, 7 – 10, 8 – 10 (Face Back)
<table><tr><td>1 - 11, 2 - 11, 3 - 11, 4 - 11</td><td>(Face Bottom)</td></tr><tr><td>5 - 12, 6 - 12, 7 - 12, 8 - 12</td><td>(Face Top)</td></tr><tr><td>1 - 13, 4 - 13, 8 - 13, 5 - 13</td><td>(Face Left)</td></tr><tr><td>2 - 14, 3 - 14, 7 - 14, 6 - 14</td><td>(Face Right)</td></tr></table>

that, in the global reference system ($e_1, e_2, e_3$), are described by the vectors

$$ \mathbf{b}_{A-B} = \mathbf{p}_B - \mathbf{p}_A, \qquad (6) $$

being $\mathbf{p}_A$ and $\mathbf{p}_B$, respectively, the position vectors of the extreme nodes of the element, A and B the index listed in Table (1). For sake of clarity, it is worth noting that, in Equation (6), $\mathbf{p}_A$ and $\mathbf{p}_B$ are expressed in the global reference system so that, to differentiate the global variables from the local ones (cf. Section 2.2), uppercase letters, i.e., indices A and B, are used.

As anticipated in Section 2.1, the connecting elements are represented as three-dimensional Euler-Bernoulli beams and this assumption, along with an energy-based approach, provide, in the following section, closed-form expressions for the effective elastic moduli of the examined lattice.

<table><caption>Table 1: The indices A and B identifying the unit cell elements</caption><thead><tr><th>B</th><th>A</th></tr></thead><tbody><tr><td>9</td><td>1, 2, 6, 5</td></tr><tr><td>10</td><td>4, 3, 7, 8</td></tr><tr><td>11</td><td>1, 2, 3, 4</td></tr><tr><td>12</td><td>5, 6, 7, 8</td></tr><tr><td>13</td><td>1, 4, 8, 5</td></tr><tr><td>14</td><td>2, 3, 7, 6</td></tr></tbody></table>

Finally, in terms of energetics, for any given deformation the elastic energy representative of the whole discrete structure, W, can be evaluated from that of the beams composing the unit cell of the periodic array. Specifically, after writing Equation (3) in the global reference system and summing the energetic contribution of each beam, it emerges

$$ W = \frac{1}{2} \mathbf{D}^T \cdot \mathbf{K} \mathbf{D}, \qquad (7) $$

with

$$ \mathbf{D} := \begin{bmatrix} \mathbf{D}_1 \\ \mathbf{D}_2 \\ \vdots \\ \mathbf{D}_{14} \end{bmatrix} = \begin{bmatrix} U_1 & V_1 & W_1 & \Phi_{X1} & \Phi_{Y1} & \Phi_{Z1} & \cdots & \Phi_{Z14} \end{bmatrix}^T \qquad (8) $$

the global displacements vector, collecting the displacements and rotations of all nodes of the unit cell and K the global stiffness matrix, assembled by adopting standard techniques of the finite element method.³³

## 2.4 Continuum modelling: effective stiffness

The continuum form of W can be derived by adopting the Cauchy-Born relation,²⁰,²³ providing a viable approach to express the displacements of each node of the unit cell as a function of the macroscopic strain field applied,

$$ \mathbf{E} := \begin{bmatrix} \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\ \varepsilon_{12} & \varepsilon_{22} & \varepsilon_{23} \\ \varepsilon_{13} & \varepsilon_{23} & \varepsilon_{33} \end{bmatrix}. \qquad (9) $$

Specifically, if we introduce the vector $\mathbf{D}_R$ collecting the displacements of the generic R-th node within the cell and the corresponding position vector $\mathbf{p}_R$, the Cauchy-Born relation, in its general form, can be expressed as

$$ \mathbf{D}_R = \mathbf{D}_0 + \mathbf{E} \cdot (\mathbf{p}_R - \mathbf{p}_0), \qquad (10) $$

with $\mathbf{D}_0$ and $\mathbf{p}_0$, in turn, the displacements vector and position vector of an appropriately selected unit cell's independent node. Also, by enforcing the equilibrium conditions on the nodal forces,²⁰ it is possible to calculate the displacements $\mathbf{D}_0$ in terms of $\mathbf{E}$.
<figure><img src="image_5.png" /><figcaption>Figure 3: The beams composing the hexatruss unit cell: (a) Face Front and Face Back, (b) Face Right and Face Left, (c) Face Top and Face Bottom</figcaption></figure>

Notwithstanding alternative choices of the independent node are possible, in the proposed study the boundary node 4 has been selected as independent node, leading to

$$D_R = D_4 + E \cdot (p_R - p_4), \quad R = 1, 2, \dots, 14. \qquad (11)$$

This assumption, in conjunction with Equation (11), provide an easy way to impose kinematic conditions on the boundary nodes of the cell that, as anticipated in Section 2.1, coincide with the application of periodic boundary conditions in order to enforce the periodicity of the deformed configuration.

Substituting Equation (11) into (7) and dividing the expression that turns out from the calculation by the volume of the unit cell, $V = L_1L_2L_3$, give the strain energy density in the continuum approximation, $w$. It emerges that $w$ is a quadratic form of the infinitesimal strains $\varepsilon_{ij}$,

$$w = w(\varepsilon_{ij}), \quad i, j = 1, 2, 3, \qquad (12)$$

and, as in classical continuum mechanics, enables to derive the components of the macroscopic stiffness tensor of the lattice,

$$E_{ijhk} = \frac{\partial^2 w}{\partial \varepsilon_{ij} \partial \varepsilon_{hk}}, \quad i, j, h, k = 1, 2, 3. \qquad (13)$$

and the corresponding effective elastic constants. In terms of effective Young's moduli, main focus of the present paper, Equation (13) gives

$$E_1 \equiv E_{1111} = \frac{\partial^2 w}{\partial \varepsilon_{11} \partial \varepsilon_{11}}, \quad E_2 \equiv E_{2222} = \frac{\partial^2 w}{\partial \varepsilon_{22} \partial \varepsilon_{22}}, \quad E_3 \equiv E_{3333} = \frac{\partial^2 w}{\partial \varepsilon_{33} \partial \varepsilon_{33}}, \qquad (14)$$

with $E_1, E_2, E_3$, on order, the effective Young's moduli in the directions $e_1, e_2$ and $e_3$.

As expected, the effective constants are strongly affected by the geometrical and mechanical properties of the microstructure and, from this point of view, the advantage of the proposed theory is that an explicit relation between
the microstructure’s parameters and the macroscopic properties can be obtained. Specifically, it emerges

$$ 
\begin{aligned}
E_1 &= E_1(L_1, L_2, L_3, r, d_1, d_2, d_3), \\
E_2 &= E_2(L_1, L_2, L_3, r, d_1, d_2, d_3), \\
E_3 &= E_3(L_1, L_2, L_3, r, d_1, d_2, d_3).
\end{aligned} 
\qquad (15) $$

To maintain the paper focused and not unnecessarily long, the complete expressions of $E_1, E_2, E_3$ is reported in Appendix A.

### 3. Model validation: the experimental campaign

As seen in the previous section, the effective properties of the hexatruss lattice can be predicted by knowing the constitutive laws and spacial distribution of its components.

Before addressing the optimization problem in order to find the optimal microstructure’s parameters satisfying a given set of stiffness constraints (cf. Section 4), the proposed analytical results in Equation (15) are verified by comparison with those obtained by performing the experimental tests described in the following.

#### 3.1 Description of the samples

As illustrated in Figure (4), the experimental tests have been conducted on a 3D-printed 3×3×3 unit cell hexatruss lattice and, to obtain an estimation of the effective stiffness in the directions $e_1, e_2$ and $e_3$, coinciding with the three principal directions of the unit cell, the configurations listed below have been investigated (Fig. (5)):

- Configuration C1: to consider the direction $e_1$ and derive $E_1$,

- Configuration C2: to consider the direction $e_2$ and derive $E_2$,

- Configuration C3: to consider the direction $e_3$ and derive $E_3$.

Aiming at obtaining a good quality of the printed samples, a unit cell with overall dimensions ($L_1 \times L_2 \times L_3$) of 9.6 mm × 11.52 mm × 7.68 mm has been selected, leading to

- Configuration C1 (length × width × height): 34.56 mm × 23.04 mm × 28.8 mm,

- Configuration C2 (length × width × height): 28.8 mm × 23.04 mm × 34.56 mm.

- Configuration C3 (length × width × height): 28.8 mm × 34.56 mm × 23.04 mm,

while, in terms of radius of the beams, $r$, and position of the internal nodes, $d_1, d_2, d_3$, the following values have been adopted: $r = 0.45$ mm, $d_1 = 2.8$ mm, $d_2 = 2.2$ mm, $d_3 = 3.3$ mm.

In addition, to account for possible errors associated with the 3D-printing process, three copies of each configuration have been printed and tested.

Finally, regarding the constituent material, two different polymers have been used to print the samples: PLA (Polylactic Acid) and PET (Polyethylene Terephthalate) having, on order, Young’s modulus of

$$ E_{PLA} = 1.3 \text{ GPa}, \quad E_{PET} = 0.8 \text{ GPa}, \qquad (16) $$

values obtained by conducting traction tests according to the ASTM standards.35

<figure><img src="image_6.png" /><figcaption>Figure 4: Polymeric samples: compression tests</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 5: The three different configurations of the samples involved in the mechanical tests: (a) configuration C1, (b) configuration C2, (c) configuration C3</figcaption></figure>

## 3.2 Methodology

### 3.2.1 Type of tests

In order to mechanically characterise the 3D-printed samples, compression tests have been considered and, to validate the theoretical approach in Section 2, force-displacement and stress-strain curves have been analysed. Specifically, the following type of test have been performed:

- uniaxial compression tests at room temperature,

- displacement-controlled tests at a displacement rate of 0.5 mm/min,

- only elastic deformations involved, $\varepsilon_{max} = 1\%$,

- no interfaces between the sample and the test bench

and the testing machine Instron 5881 has been employed.

### 3.2.2 Objectives

The aim of the tests was to evaluate the effective stiffness of the hexatruss lattice in the three principal directions $e_1$, $e_2$, $e_3$, measured in terms of effective Young's moduli denoted, respectively, by $E_1$, $E_2$ and $E_3$. The latter, in particular, have been derived by:

i) recording, during the test, both the displacement, $\Delta L$, and resulting load, $F$, applied to the sample;

ii) generating, with the sample geometry, a stress-strain curve, being the engineering strain, $\varepsilon$, and the engineering stress, $\sigma$, given by

$$ \varepsilon := \frac{\Delta L}{L_0} \quad \text{and} \quad \sigma := \frac{F}{A_0}, \qquad (17) $$

with $L_0$ and $A_0$, in turn, the initial height of the sample and its initial cross-sectional area normal to the loading direction;

iii) finally, focusing on the slope of the linear portion of the stress-strain curve that coincides, in the elastic regime, with the Young's modulus $E$ of the considered material. Namely,

$$ E = \frac{\sigma}{\varepsilon}. \qquad (18) $$

## 3.3 Results and discussion

The outcome of the experimental campaign is presented in Tables 2 and 3, where the comparison between the experimental results and the theoretical predictions in Equation (15) is also provided.

For sake of clarity, it should be noted that the analytical values reported in Tables 2 and 3 are obtained by substituting in Equation (15) the quantities

$$ \begin{gathered} r = 0.45 \text{ mm}, \\ L_1 = 9.6 \text{ mm}, L_2 = 11.52 \text{ mm}, L_3 = 7.68 \text{ mm}, \\ d_1 = 2.8 \text{ mm}, d_2 = 2.2 \text{ mm}, d_3 = 3.3 \text{ mm} \end{gathered} \qquad (19) $$
and

$$ 
\begin{aligned}
E_s &\equiv E_{PLA} = 1.3 \text{ GPa}, (\text{Table 2}), \\
E_s &\equiv E_{PET} = 0.8 \text{ GPa}, (\text{Table 3}),
\end{aligned} 
\qquad (20) $$

coinciding with the geometrical, the first, and mechanical, the second, parameters defining the unit cell adopted in the experimental tests.

As it can be seen, it generally emerges a good agreement between the theoretical and experimental results and, as expected, the theoretical values overestimate the experimental ones. From this point of view, the difference is mainly related to the quality of the 3D-printed samples that, differently from the analytical model where the beams are assumed to be perfectly homogeneous and without defects, in some cases present some imperfections, i.e., small parts missing, not perfectly homogeneous or beams not perfectly connected at the nodes.

However, taking into account the very close comparison in Tables 2 and 3, it can be said that the experimental tests validate the theoretical expressions in Equation (15) and can thus be employed to formulate the optimization problem in the next section.

<table><caption>Table 2: PLA samples, Young's moduli. Comparison between the theoretical and experimental results</caption><thead><tr><th></th><th>Experiments</th><th>Theoretical results</th></tr></thead><tbody><tr><td>E<sub>1</sub> (MPa)</td><td>2.72</td><td>3.09</td></tr><tr><td>E<sub>2</sub> (MPa)</td><td>4.55</td><td>4.79</td></tr><tr><td>E<sub>3</sub> (MPa)</td><td>1.32</td><td>1.34</td></tr></tbody></table>

<table><caption>Table 3: PET samples, Young's moduli. Comparison between the theoretical and experimental results</caption><thead><tr><th></th><th>Experiments</th><th>Theoretical results</th></tr></thead><tbody><tr><td>E<sub>1</sub> (MPa)</td><td>1.81</td><td>1.90</td></tr><tr><td>E<sub>2</sub> (MPa)</td><td>2.71</td><td>2.95</td></tr><tr><td>E<sub>3</sub> (MPa)</td><td>0.90</td><td>0.92</td></tr></tbody></table>

# 4. Parametric optimization

## 4.1 Problem description

As anticipated in Section 1, in this paper the parametric optimization of the hexatruss lattice is performed with reference to the design of the Ariane Launcher and, in particular, the specific case of the Spacecraft-Launcher Damping Interface (LALS) is considered. An extended set of requirements is expected for the LALS, in terms of geometry, maximal density, stiffness, mechanical damping and load withstanding. Leaving the other aspects for future research investigations, here, for simplicity, only the maximal density,

$$ \rho_{req} \le \rho_{max} = 1.9 \text{ g/cm}^3 \qquad (21) $$

and stiffness,

$$ E_1^{req} = 8 \text{ MPa} \pm 0.8 \text{ MPa}, $$

$$ E_2^{req} = 15 \text{ MPa} \pm 3 \text{ MPa}, \qquad (22) $$

$$ E_3^{req} = 0.4 \text{ MPa} \pm 0.08 \text{ MPa} $$

requirements are considered, being $\rho_{req}$ and $E_1^{req}$, $E_2^{req}$, $E_3^{req}$, respectively, the required values of the density and of the effective Young's moduli in the directions $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$.

Our aim consists in finding the optimal parameters of the microstructure that meet the specifications listed in Equations (21),(22) and, to do so, the following multiobjective optimization problem has been formulated:

$$ \text{Find } \mathcal{P} = \{L_1, L_2, L_3, r, d_1, d_2, d_3\} $$
$$ \text{To minimize } f(\mathcal{P}) := \begin{cases} E_1(\mathcal{P}) - E_1^{\text{req}} \\ E_2(\mathcal{P}) - E_2^{\text{req}} \\ E_3(\mathcal{P}) - E_3^{\text{req}} \end{cases} $$

$$ \text{Subjected to} \quad \begin{cases} \mathcal{G}_1 \equiv r \in [r_{\min}, r_{\max}] \\ \mathcal{G}_2 \equiv L_1, L_2, L_3 \in [L_{\min}, L_{\max}] \\ \mathcal{G}_3 \equiv \begin{cases} d_1 \in [d_1^{\min}, d_1^{\max}] \\ d_2 \in [d_2^{\min}, d_2^{\max}] \\ d_3 \in [d_3^{\min}, d_3^{\max}] \end{cases} \end{cases} $$

with $\mathcal{P}$ the design variables, defined in Section 2, that have to be optimized, $f(\mathcal{P})$ the objective function, expressed in terms of the analytical relations derived in Equation (15), $\mathcal{G}_1, \mathcal{G}_2$ and $\mathcal{G}_3$ the constraints introduced to limit the parameters' value to a range of realistic significance. Specifically, $\mathcal{G}_1$ and $\mathcal{G}_2$ are related to manufacturing limits, $r_{\min}$ and $L_{\min}$, and to the multiscale approach of the present paper, where the length of scale of the microstructure is assumed to be much smaller than the length of scale of the component, $r_{\max}$ and $L_{\max}$. Finally, $\mathcal{G}_3$ assures the suitability of the microstructure, where all the hexatruss cells must have all the nodes in the correct position.

In formulating the optimization problem, the maximal density requirement has been neglected, being easily satisfied just by employing architecture materials (cf. Section 4.3).

## 4.2 Solution method

The parametric optimization problem previously described has been solved by using the genetic algorithm and, in particular, the in-built MATLAB function *gamultiobj* has been employed. Such function is based on the genetic algorithm and allows the user to efficiently solve a multiobjective optimization problem with several optimization variables, as in the present case.

Being a detailed description of the working principle of the genetic algorithm well beyond the scope of the present paper, in Figure 6 only a schematic representation of the method is reported. As illustrated in Figure 6, the genetic algorithm starts by creating an initial random population of potential solutions, $P(t)$, whose individuals are then ranked according to their fitness, i.e., their value of the objective function that, for a minimization problem, has to be as smaller as possible. If the termination condition is not satisfied, for example, if the maximum number of iterations is not reached, the three operators of reproduction, crossover and mutation are applied to $P(t)$ in order to create a new generation of individuals, $P(t + 1)$. The reproduction operator, in particular, selects the 'most fit' individuals of $P(t)$ to form a matching pool so that, similarly to the principles of natural evolution, the individuals with the best traits have a higher probability to survive. The second operator, crossover, generates the individuals of $P(t + 1)$ by randomly mixing the best traits of the individuals in the matching pool while, for maintaining diversity in the population, the mutation operator alters the traits of some individuals in $P(t + 1)$ in order to create, hopefully, a better individual. Similarly to $P(t)$, the generation $P(t + 1)$ is then ranked and the process described above is repeated until the termination condition is met and the algorithm converges to the optimal solution of the problem.

## 4.3 Results and discussion

The results of our investigations are reported in Table 5, where titanium Ti64 with Young's modulus $E_s = 97$ GPa, Poisson's ratio $\nu_s = 0.3$ and density $\rho_s = 4.5$ g/cm³ has been assumed as constituent material of the microstructure. 6 different cases have been considered by changing the size of the population or the bounds of the parameters' value, as summarised in Table 4 where, for conciseness, the limits $\ell_1$ and $\ell_2$ are given by:

$$ \ell_1 := \begin{cases} 3 \text{ mm} \le L_1, L_2, L_3 \le 20 \text{ mm} \\ 0.13 \text{ mm} \le r \le 0.5 \text{ mm} \\ 0 < d_1 < L_3/2 \\ 0 < d_2 < L_2/2 \\ 0 < d_3 < L_1/2 \end{cases} \qquad \ell_2 := \begin{cases} 3 \text{ mm} \le L_1, L_2, L_3 \le 15 \text{ mm} \\ 0.13 \text{ mm} \le r \le 0.5 \text{ mm} \\ 0 < d_1 < L_3/2 \\ 0 < d_2 < L_2/2 \\ 0 < d_3 < L_1/2 \end{cases} \qquad (23) $$
<figure><img src="image_8.png" /><figcaption>Figure 6: The genetic algorithm: working principle</figcaption></figure>

As a general observation, it can be said that Table 5 suggests the possibility, for the hexatruss lattice, to satisfy the three stiffness requirements expected for the LALS. This is a great advantage of the examined microstructure since the values of the required stiffness in the three directions are very different from each other, making their simultaneous satisfaction very challenging. Slightly different solutions emerge in the investigated cases. However, comparing the values of $L_1, L_2, L_3$ and of $d_1, d_2, d_3$, reveals that, in each example, the following relations hold:

$$L_2 \approx 1.42 L_1, \quad L_3 \approx 0.46 L_1, \qquad (24)$$

$$d_2 \approx 0.45 d_1, \quad d_3 \approx 1.4 d_1. \qquad (25)$$

In a practical context, this finding, in conjunction with Equation (15), could assist the designer in the selection of the microstructure's parameters in order to reach a prescribed value of macroscopic stiffness. Finally, as anticipated in Section 4.1, all the configurations listed in Table 5 have a density of $\rho \approx 0.1 \text{ g/cm}^3$, value that is well beyond the required limit in Equation (21).

<table><caption>Table 4: Parametric optimization of the hexatruss: considered cases</caption><thead><tr><td>Case</td><td>Limits</td><td>Population's size</td></tr></thead><tbody><tr><td>1</td><td>l<sub>1</sub></td><td>2000</td></tr><tr><td>2</td><td>l<sub>1</sub></td><td>4000</td></tr><tr><td>3</td><td>l<sub>1</sub></td><td>6000</td></tr><tr><td>4</td><td>l<sub>1</sub></td><td>8000</td></tr><tr><td>5</td><td>l<sub>2</sub></td><td>8000</td></tr><tr><td>6</td><td>l<sub>2</sub></td><td>4000</td></tr></tbody></table>

## 5. Conclusions

Because of their discrete configuration, architectured materials offer unique combinations of improved mechanical properties, like stiffness, strength or toughness, at low weight. This unusual characteristic, in conjunction with the possibility to tailor their mechanical properties in each direction, makes them very promising in a vast number of engineering fields including aerospace, automotive, marine and constructions. Differently from the traditional homogeneous materials, architectured materials require appropriate constitutive models for revealing the complex relations between the microstructure's parameters and the macroscopic properties. A great deal of effort has gone in this direction and, in the literature, a variety of techniques have been recently proposed and discussed. Different methods and assumptions have been suggested, however, only in a very limited number of investigations, closed-form expressions
<table><caption>Table 5: Parametric optimization of the hexatruss lattice: results</caption><thead><tr><th></th><th>case 1</th><th>case 2</th><th>case 3</th><th>case 4</th><th>case 5</th><th>case 6</th></tr></thead><tbody><tr><td>L<sub>1</sub> (mm)</td><td>12.16</td><td>12.22</td><td>12.04</td><td>12.13</td><td>10.2</td><td>10.1</td></tr><tr><td>L<sub>2</sub> (mm)</td><td>17.17</td><td>17.43</td><td>17.03</td><td>17.36</td><td>14</td><td>14.1</td></tr><tr><td>L<sub>3</sub> (mm)</td><td>5.63</td><td>5.68</td><td>5.58</td><td>5.64</td><td>4.7</td><td>4.7</td></tr><tr><td>r (mm)</td><td>0.19</td><td>0.19</td><td>0.19</td><td>0.19</td><td>0.15</td><td>0.15</td></tr><tr><td>d<sub>1</sub> (mm)</td><td>2.54</td><td>2.80</td><td>2.6</td><td>2.74</td><td>1.42</td><td>1.9</td></tr><tr><td>d<sub>2</sub> (mm)</td><td>1.18</td><td>1.32</td><td>1.2</td><td>1.2</td><td>0.66</td><td>0.89</td></tr><tr><td>d<sub>3</sub> (mm)</td><td>3.6</td><td>4.0</td><td>3.66</td><td>3.73</td><td>1.96</td><td>2.68</td></tr><tr><td>E<sub>1</sub> (MPa)</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td></tr><tr><td>E<sub>2</sub> (MPa)</td><td>15</td><td>15</td><td>15</td><td>15</td><td>15</td><td>15</td></tr><tr><td>E<sub>3</sub> (MPa)</td><td>0.4</td><td>0.4</td><td>0.4</td><td>0.4</td><td>0.4</td><td>0.4</td></tr><tr><td>ρ (g/cm<sup>3</sup>)</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.09</td><td>0.09</td></tr></tbody></table>

for the effective mechanical properties are provided. Such relations, if available, would facilitate the designer in the selection of the material that best suits a given requirement.

To help filling this research gap, this paper deals with the analysis of a three-dimensional lattice made of identical hexatruss cells. In the framework of linear elasticity and by modelling the microstructure as a sequence of three-dimensional Euler-Bernoulli beams, an energetic approach leads to explicit expressions for the effective elastic properties. To obtain a more mathematically tractable problem, the Cauchy-Born assumption is used to represent the nodal degrees of freedom as a function of the applied macroscopic strain components. This allows to write the elastic energy of the discrete system as a quadratic function of them and, as in classical mechanics, enables to derive the macroscopic stiffness tensor of the lattice and the corresponding elastic moduli. The predicted results are then verified by comparison with the outcome of the experimental tests performed and it emerges a very good agreement.

With reference to the specific case of the Spacecraft-Launcher Damping Interface (LALS), a parametric optimization of the hexatruss lattice is also reported. The aim is to obtain the optimal parameters of the microstructure satisfying a given set of stiffness requirements expected for the LALS. To do so, a multiobjective optimization problem has been formulated by including the analytical equations previously derived and the genetic algorithm has been selected to solve the problem. This approach, easily applied to different types of structures in the aerospace industry, offers an efficient and less computationally expensive way to derive the optimal parameters since the re-generation and corresponding analysis of the lattice is not required at each iteration.

## Appendix A

$$ 
\begin{aligned}
E_1 &= \frac{16A \sqrt{p_1} + \sqrt{p_2} + \sqrt{p_3}}{L_2^2 L_3^2 p_1^{5/2} p_2^{5/2} p_3^{5/2}} ((EAL_1^2 p_5 + 48EI (p_6 + 4d_2^2 p_1^{5/2})) p_3^{5/2} + 64d_3^6 EA p_1^{5/2} p_2^{5/2} + 16d_3^4 EA p_4 + 192d_3^2 EI p_4), \\
E_2 &= \frac{16A \sqrt{p_1} + \sqrt{p_2} + \sqrt{p_3}}{L_2^2 L_3^2 p_1^{5/2} p_2^{5/2} p_3^{5/2}} (4d_3^2 (48EI + EAL_2^2) p_1^{5/2} p_2^{5/2} + 48EI p_7 + EA ((p_8 + p_9))), \\
E_3 &= \frac{16A \sqrt{p_1} + \sqrt{p_2} + \sqrt{p_3}}{L_2^2 L_3^2 p_1^{5/2} p_2^{5/2} p_3^{5/2}} (192d_3^2 EI p_1^{5/2} p_2^{5/2} + EA p_1 p_2 p_3 p_{10} + 48EI p_{11}),
\end{aligned} 
\quad (26) $$

where, to simplify the notation, $EA := E_s \pi r^2$, $EI := E_s \pi r^4 / 4$, $E_s$ the Young's modulus of the lattice's constituent material and
$$p_1 = 4d_1^2 + L_1^2 + L_2^2, \quad p_2 = 4d_2^2 + L_1^2 + L_3^2, \quad p_3 = 4d_3^2 + L_2^2 + L_3^2,$$

$$p_4 = p_1^{5/2} p_2^{5/2} (L_2^2 + L_3^2), \quad p_5 = p_2^{5/2} (4d_1^2 + L_2^2) + L_1^2 (p_1^{5/2} + p_2^{5/2}) + p_1^{5/2} (L_3^2 + 4d_2^2),$$

$$p_6 = (p_2^{5/2} (4d_1^2 + L_2^2) + L_3^2 p_1^{5/2}), \quad p_7 = (4d_2^2 L_1^2 p_1^{5/2} + (4d_1^2 + L_1^2) p_2^{5/2}) p_3^{5/2} + L_3^2 p_1^{5/2} (p_2^{5/2} + 4d_2^2 p_3^{5/2}), \quad (27)$$

$$p_8 = 16d_2^4 (4d_2^2 + L_1^2 + L_3^2) p_1^{5/2} p_3^{5/2} + L_2^4 p_2^{5/2} (p_1^{5/2} + p_3^{5/2}), \quad p_9 = L_2^2 p_2^{5/2} (L_3^2 p_1^{5/2} + (4d_1^2 + L_1^2) p_3^{5/2}),$$

$$p_{10} = 16d_1^4 p_2^{3/2} p_3^{3/2} + L_3^2 p_1^{3/2} (p_2^{3/2} + p_3^{3/2}),$$

$$p_{11} = (4d_2^2 p_1^{5/2} + L_1^2 (p_1^{5/2} + 4d_1^2 p_2^{5/2})) p_3^{5/2} + L_2^2 p_2^{5/2} (p_1^{5/2} + 4d_1^2 p_3^{5/2}).$$

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