# Hybrid auxetics: Postponing failure with elastomer infiltration of Ti-6Al-4V lattices

Frédéric Albertini, Justin Dirrenberger, Cyrille Sollogoub, Andrey Molotnikov,

Jérôme Adrien, E Maire

► To cite this version:

Frédéric Albertini, Justin Dirrenberger, Cyrille Sollogoub, Andrey Molotnikov, Jérôme Adrien, et al. Hybrid auxetics: Postponing failure with elastomer infiltration of Ti-6Al-4V lattices. International Journal of Mechanical Sciences, 2025, 306, pp.110704. 10.1016/j.ijmecsci.2025.110704. hal-05253493

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Submitted on 14 Sep 2025

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# Hybrid auxetics: Postponing failure with elastomer infiltration of Ti-6Al-4V lattices

Frédéric Albertini<sup>a, ID</sup>, Justin Dirrenberger<sup>a,b ID, *</sup>, Cyrille Sollogoub<sup>a ID</sup>, Andrey Molotnikov<sup>c ID</sup>, Jérôme Adrien<sup>d ID</sup>, Eric Maire<sup>d ID</sup>

## ARTICLE INFO

### Keywords:

Architected materials

Lattice structures

Crashworthiness

Auxetics

Elastomer infiltration

X-ray microtomography

## ABSTRACT

This work examines how a soft elastomeric phase can be used to reinforce and delay the failure of additively manufactured metal lattice structures with auxetic geometry. A novel hexaround unit-cell was designed to ensure printability of Ti-6Al-4V via laser powder-bed fusion. Once infiltrated with a compliant polyurethane, these hybrid lattices showed a pronounced delay in strut-level failure and shear-band formation. Microtomography coupled with in-situ compression experiments confirmed that nodal cracking commonly observed in unfilled lattices is mitigated by the polymer phase. While hybrid lattices exhibited comparable or slightly increased stiffness only at large deformations, their enhanced integrity and stress redistribution suggest promise for crashworthiness and impact applications requiring high ductility or energy absorption. We discuss how the stiffness ratio between filler and metallic skeleton determines the extent of improvement, offering design guidelines for next-generation hybrid lattices. This is the first in-situ tomographic evidence that a compliant matrix can defer the critical shear-band in an additively manufactured metal auxetic lattice.

## 1. Introduction

Modern engineering applications demand materials with high specific mechanical properties, e.g. strength, stiffness, or crashworthiness, while remaining lightweight. Architected materials, which exploit deliberate morphological arrangements of one or more constituent phases, are excellent candidates for such demands [1]. Thanks to advances in additive manufacturing, sophisticated lattice structures can now be fabricated in metals and polymers alike [2–6]. These periodic frameworks, composed of struts connected by nodes, offer exceptional strength-to-weight ratios [7–9], potential for thermal management [10], biomedical integration [11], tailor-made instabilities [12,13] or quasi-periodic patterns for enhanced fracture toughness [14].

Among lattice designs, auxetics exhibit a negative Poisson's ratio, i.e. expanding laterally when stretched, while contracting when compressed, and have been largely studied in the literature [15–19]. Resulting from negative Poisson's ratio, numerous potential applications have been reported, as acoustic damping [20], improved indentation resistance compared to conventional cellular materials [21,22], improved fracture toughness [23], or as wave-guiding medium [24,25]. Moreover, they show promise for energy absorption and damage tolerance

when compared to conventional structures [26–31]. Yet, such lattices often suffer catastrophic or localized failure (e.g., nodal fracture, shear-band formation) under high loads [32], reducing their crashworthiness benefits. In biological systems, hard-and-soft composite architectures such as exoskeletons [33], bones [34] or nacre [35] circumvent similar issues by combining stiff and compliant constituents. This has inspired the concept of hybrid materials [36,37], in which the intricate combination of two materials outperforms each phase considered individually [38,39], potentially delaying fracture and improving toughness [40–44]. The term *hybrids* has recently been used in the field of lattice structures for mixing various lattice patterns together, we refer to such concept as *hybrid topologies*, as studied in [45–47]. Our work is focused on *hybrid materials*, as originally proposed by Ashby & Bréchet [36] in the field of structural materials. Hybrid materials are still an active topic of research, as they are an effective lever to improve the properties of materials [48–51].

Recent work has explored filling metal lattices with polymers to enhance mechanical performance. Li et al. [52] propose to fill metallic lattice with hard epoxy to reinforce the structure, increase its elastic properties and delay the buckling of the constitutive struts under a

<footnote><sup>a</sup> PIMM, Arts et Métiers, Cnam, CNRS UMR 8006, 151 bd de l'Hôpital, 75013 Paris, France</footnote>

<footnote><sup>b</sup> Institut universitaire de France (IUF), Paris, France</footnote>

<footnote><sup>c</sup> RMIT Centre for Additive Manufacturing, RMIT University, 124 La Trobe St, Melbourne VIC 3000, Australia</footnote>

<footnote><sup>d</sup> CNRS, INSA Lyon, Université Claude Bernard Lyon 1, MATEIS UMR 5510, Villeurbanne F-69621, France</footnote>

<footnote>* Corresponding author at: PIMM, Arts et Métiers, Cnam, CNRS UMR 8006, 151 bd de l'Hôpital, 75013 Paris, France. E-mail address: justin.dirrenberger@ensam.eu (J. Dirrenberger).</footnote>
compression loading, as was experimentally observed by Bernal-Ostos et al. [53] on lattice filled with polymeric foams. The presence of a filler can mitigate geometric defects in lattices, thereby increasing their load-bearing capacity [54,55]. Stiffer fillers can increase strength but risk premature densification [56–58], whereas more compliant fillers, e.g., elastomers, may supply strut stabilization without blocking deformation at low strain [59–61]. In previous studies on polymeric auxetic lattices, infiltration with a compliant elastomer improved energy absorption and delayed critical collapse [62]. However, the effect of a soft filler on metallic auxetic lattices has been comparatively underexplored, especially in terms of in-situ failure analysis via microtomography. X-ray computed microtomography is a well-established characterization method, especially in the field of materials science [63–65], this method can reveal the internal defects of a three-dimensional object. Nowadays, it is used quite commonly for quality control measurements for additively manufactured architecture materials [66–69], but its use combined with in-situ mechanical testing is still relatively seldom in the literature.

The association of auxetic lattice with hyperelastic elastomer is expected to create an opposition [70] between both behavior: the elastomeric incompressibility ($v \approx 0.5$) should provide an efficient reinforcement to lattices exhibiting negative Poisson's ratio [71–73]. This reinforcement effect has been observed both numerically [74,75] and experimentally [76–78]. Contrarily to hybridization with a stiff phase, compliant elastomer is expected not to create early densification and to provide support at high strain, as its stiffness increases with strain.

Yet, how a hyperelastic phase specifically affects the failure mode of a metallic auxetic lattice remains less understood, particularly regarding the delay of shear-band formation, nodal cracking, or overall crashworthiness. This study addresses that gap by designing and fabricating Ti-6Al-4V auxetic lattices via laser powder-bed fusion (L-PBF) and infiltrating them with a compliant polyurethane. We combine numerical homogenization, uniaxial compression, and in situ X-ray microtomography to reveal how elastomer infiltration modifies mechanical response and postpones catastrophic failure. Our findings highlight a straightforward design route for lightweight, damage-tolerant hybrid lattices suitable for impact and crash attenuation. Section 2 introduces the materials and methodological approaches implemented in this work, followed by experimental and computational results in Section 3. Conclusions are postponed to Section 4.

## 2. Materials and methods

### 2.1. Auxetic lattice design

The auxetic lattices considered in this work are derived from author's previous work [79]. Struts of the cell are arc-shaped, and negative Poisson's ratio is achieved through inward orientation of arcs. This cell can be described with 3 parameters: L, side length of the cubic cell, a, waviness amplitude, and D, diameter of the struts (see Fig. 1 right). The novel cell proposed in the present work presents a modified design to improve manufacturability using L-PBF process: struts presenting an angle inferior to 20° with the printing plateau have been subtracted. The resulting lattice called modified Hexaround is represented on Fig. 1. Such design modification was proven efficient to improve manufacturability using L-PBF in past works [2,80].

For the sake of comparison, two different cell configurations have been studied in the experimental part of this work, with L = 3 mm (named L3) and 4 mm (named L4). Geometric parameters for both cell sizes are detailed in Table 1, for 5 different lattice volume fractions $V_f$ (0.05, 0.10, 0.15, 0.20, 0.25). Lattice volume fractions are obtained by adjusting struts diameter D, keeping L and a constant. For each cell size, waviness amplitude a has been chosen for the cell to exhibit auxetic behavior, with a nearly equivalent L/a ratio. Hybrid lattices are obtained by filling voids in lattice geometry with a compliant

<table><caption>**Table 1**

Geometrical parameters of L3 and L4 unit-cells for different volume fractions.</caption><thead><tr><th></th><th colspan="3">L3</th><th colspan="6">L4</th></tr></thead><tbody><tr><td>$V_f$ (-)</td><td>0.05</td><td>0.10</td><td>0.15</td><td>0.20</td><td>0.25</td><td>0.05</td><td>0.10</td><td>0.15</td><td>0.20</td><td>0.25</td></tr><tr><td>L (mm)</td><td>3.00</td><td>3.00</td><td>3.00</td><td>3.00</td><td>3.00</td><td>4.00</td><td>4.00</td><td>4.00</td><td>4.00</td><td>4.00</td></tr><tr><td>a (mm)</td><td>0.60</td><td>0.60</td><td>0.60</td><td>0.60</td><td>0.60</td><td>0.75</td><td>0.75</td><td>0.75</td><td>0.75</td><td>0.75</td></tr><tr><td>D (mm)</td><td>0.24</td><td>0.35</td><td>0.43</td><td>0.51</td><td>0.58</td><td>0.32</td><td>0.46</td><td>0.58</td><td>0.68</td><td>0.77</td></tr></tbody></table>

<table><caption>**Table 2**

L-PBF printing parameters for Ti-6Al-4V samples.</caption><tr><td>Laser power (W)</td><td>175</td></tr><tr><td>Scanning speed (mm/s)</td><td>710</td></tr><tr><td>Hatch spacing (μm)</td><td>120</td></tr><tr><td>Layer thickness (μm)</td><td>30</td></tr></table>

polymer. In this work, lattices have been generated using Rhinoceros 3D CAD software with Grasshopper plug-in. CAD files have been exported to STEP and STL format for Finite Element (FE) and 3D printing respectively.

### 2.2. Lattice fabrication via L-PBF

Metallic lattice samples have been obtained employing L-PBF using a 250HL machine (SLM Solution GmbH) and using printing parameters displayed in Table 2. Metallic alloy used as constitutive material for lattices is Ti-6Al-4V (powder provided by TLS Technik). This alloy was selected due to its wide availability, well-known processability using L-PBF [81,82], and its low density.

All lattices were oriented such that their vertical axis (z on Fig. 1) aligned with the build direction to minimize internal supports. Post-processing involved mild ultrasonic cleaning to remove loosely bound powder from the lattice pores.

### 2.3. Lattice hybridization

A two-component polyurethane elastomer (subsequently referred to as PU) was prepared by mixing the base and curing agent in a 1:1 mass ratio. After degassing, the liquid mixture was poured into the lattice pores, with the assembly placed in a custom 3D-printed mold to confine the polymer. The mold and filled lattice were then transferred to a vacuum chamber (–0.2 bar) for 4 h to ensure complete infiltration and removal of entrapped air. The hybrid samples were cured at room temperature for 24 h before demolding. This procedure yielded hybrid auxetic lattices composed of a continuous elastomeric matrix interpenetrating the Ti-6Al-4V skeleton. Examples of hybridized lattices are shown in Fig. 2.

### 2.4. Constitutive material modeling

The mechanical behavior of the constitutive materials has been identified and modeled using experimental tests and FE analysis. Both experimental and FE model curves can be seen on Fig. 3. Lattice constitutive material Ti-6Al-4V is considered elasto-plastic, and has been modeled through an isotropic non-linear hardening model given by Eq. (1):

$$R = R_0 + \sum_{l}^{2} Q_l(1 - e^{-b_l p}). \quad (1)$$

Material parameters of Ti-6Al-4V model were identified from tensile tests using L-PBF printed dog-bone samples (ASTM E8M method), a stress vs. strain curve example is shown in Appendix A. Elastic properties were estimated from the linear elastic part of the curve, and material parameters for hardening model were identified from the plastic regime (see Fig. 3(a)): $E = 97.7$ GPa; $v = 0.30$; $\sigma_Y = 759.0$ MPa;
<figure><img src="image_4.png" /><figcaption>Fig. 1. Modified Hexaround unit-cell, with their periodicity vectors $x, y, z$ (left), and geometric parameters (right).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2. Manufactured samples: L3 lattice and corresponding hybrid (a), L4 lattice and hybrid (b).</figcaption></figure>

$Q_1 = 331.0$ MPa; $b_1 = 332.0$; $Q_2 = 259.0$ MPa; $b_2 = 5.8$. Ti-6Al-4V density is 4.3 g cm⁻³. The samples used for material parameters identification exhibited failure at 7% plastic strain.

PU is used as the compliant phase. Its behavior is hyperelastic and has been modeled using a modified version of the Arruda-Boyce model [83] taking into account compressibility as presented in [84], with an additional strain energy term accounting for volumetric changes as discussed in [85] and initially introduced by [86] in the context of finite deformation elasto-plasticity. This material model was implemented in the Z-Set FE package based on [87]. Fitting experimental results from compression testings, following parameters of Arruda-Boyce model have been identified (see Fig. 3(b)): $μ = 0.37$ MPa; $λ = 9.0$; $D = 0.40$. In this model, the additional parameter $D$ is used to account for material compressibility. The density of PU produced has been measured equal to 1.07 g cm⁻³, using the hydrostatic weighing method.

## 2.5. Finite element modeling

### 2.5.1. Computational homogenization (linear elastic regime)

To predict the effective Young's modulus and Poisson's ratio of both empty and filled lattices, the periodic unit cell geometry was meshed

with tetrahedral elements (C3D10). A mesh convergence analysis has been performed: for a hybrid unit-cell, an element size of 4.10⁻² mm was used for the cell with a dimension of L = 3 mm, and 5.10⁻² mm for the cell with a dimension of L = 4 mm (resulting in a total number of around 450,000 elements). A small-deformation elastic model was used with the following parameters for Ti-6Al-4V, $E_{Ti-6Al-4V} = 97.7$ GPa, $ν = 0.30$. For PU, $E_{PU} = 1.64$ MPa, $ν = 0.499$. The Young modulus for PU is estimated from the initial slope of the stress-strain curve (Fig. 3(b)). The Poisson ratio value is set to 0.499 in order to account for near-incompressibility. Periodic boundary conditions were prescribed, and six independent load cases (tension along x, y, z, plus three shear loadings) were applied to extract the full fourth-order stiffness tensor. The resulting data were used to compute directional elastic modulus and Poisson's ratio surfaces in 3D. Normalized volume fractions are obtained by varying the lattice strut diameter, while maintaining waveness amplitude and cell size. The hybrid lattices normalized modulus is computed considering a rule of mixture for constitutive material $E_0$. This corresponds to a theoretical maximum value for composites, hence making it adequate for normalizing the results:

$$E_0 = V_f E_{Ti-6Al-4V} + (1 - V_f) E_{PU}. \quad (2)$$
<figure><img src="image_6.png" /><figcaption>Fig. 3. Ti-6Al-4V (lattice) (a) and PU (filler) (b) stress/strain curves used for material parameter identification.</figcaption></figure>

Please refer to Appendix B for a complete description of the computational homogenization procedure.

## 2.5.2. Nonlinear unit-cell simulation (large strain + plasticity)

Full-field finite element simulation within a finite deformation framework is performed, with the objective to simulate the compression experiments for comparison. A perfect bonding assumption between the lattice and the filler material is made. FE stress–strain curves are presented with complete experimental curves in Appendix C. The stress measured is the Boussinesq or 1st Piola–Kirchhoff stress, while the corresponding strain measure is the Green–Lagrange strain. Computations are performed on unit-cells, applying periodic boundary conditions and prescribing macroscopic uniaxial strain component. The choice of working on a periodic unit-cell instead of the full sample geometry is driven by the computational cost of such calculation. Non-linear material models used for computational experiments are described in Section 2.2 and presented in Fig. 3. Damage modeling was not considered in the present work, nor was the effect of lattice densification, i.e. contact between struts.

## 2.6. Quasi-static compression testing

Macroscopic compression tests were performed on an Instron 5581 universal testing machine equipped with a 5 kN load cell. Cube-like specimens consisting of a 3 × 3 × 3 arrangement of unit cells (for each

lattice type) were tested using lubricated plates in order to reduce friction, using common machine grease. The compression axis corresponds to the z axis on Fig. 1. A constant crosshead speed corresponding to a nominal strain rate of $\dot{\epsilon} = 10^{-3}$ s$^{-1}$ was used. The engineering stress $\sigma$ was derived from the measured load over the nominal cross-sectional area $(3L)^2$. Tests were stopped at the final strain $\epsilon_f = 0.75$. Densification strain $\epsilon_D$ was identified from the stress–strain curve, where the stress reached the same value as the initial peak stress. Experimental crashworthiness evaluation is made using experimental stress–strain curves, computed as described above. The Energy Absorption (EA) is calculated as a mechanical work density by computing the area under the stress–strain curve until final strain $\epsilon_f$ is attained, such that:

$$EA (\epsilon_f) = \int_{0}^{\epsilon_f} \sigma(\epsilon) d\epsilon. \qquad (3)$$

In order to fairly compare lattice and hybrid lattice crashworthiness, EA must be balanced with their corresponding mass. Specific Energy Absorption (SEA) is thereby obtained in the following way:

$$SEA (\epsilon_f) = \frac{1}{m} \int_{0}^{\epsilon_f} \sigma(\epsilon) d\epsilon. \qquad (4)$$

## 2.7. In-situ X-ray microtomography

To visualize internal failure modes in real time, a custom 2 kN mini-compression stage was placed inside a GE v|tome|x s tomograph. Each
<figure><img src="image_7.png" /><figcaption>Fig. 4. Strut diameter measures: L3 lattice (a), L4 lattice (b).</figcaption></figure>

<table><caption>**Table 3**

Theoretical and experimental strut diameter and sample mass for L3 and L4 samples.</caption><thead><tr><th>Sample</th><th>ø CAD (mm)</th><th>ø measured (mm)</th><th>m CAD (g)</th><th>m measured (g)</th></tr></thead><tbody><tr><td>L3</td><td>0.346</td><td>0.370 (±0.006)</td><td>0.31</td><td>0.48 (±0.03)</td></tr><tr><td>L3 hybrids</td><td>0.346</td><td>0.370 (±0.006)</td><td>1.03</td><td>1.19 (±0.05)</td></tr><tr><td>L4</td><td>0.315</td><td>0.315 (±0.005)</td><td>0.38</td><td>0.58 (±0.04)</td></tr><tr><td>L4 hybrids</td><td>0.315</td><td>0.315 (±0.005)</td><td>2.15</td><td>2.22 (±0.05)</td></tr></tbody></table>

specimen was compressed incrementally, pausing at selected strain levels to collect full 3D scans (9 µm voxel size, 100 kV beam). The reconstructed volumes were analyzed to identify crack initiation, shear-band formation, and failure progression in both empty and hybrid lattices. Comparing images at corresponding strain values revealed how the elastomeric phase modifies local strut deformations and delays nodal fractures. It is noteworthy that the custom compression stage is considerably more compliant than the Instron 5581, therefore yielding discrepancies in compression stress-strain curves as strain gauges could not be used.

Samples were compressed between two lubricated cylindrical platens at the same displacement speed used in the Instron tests. The compression was paused as needed to acquire tomographic scans. Three scans were performed for unfilled lattice samples: at initial state (ε = 0.00), during the elastic regime (ε = 0.05), and just after the first peak stress (ε = 0.17). For hybrid samples, a fourth scan was also taken at a higher strain (ε = 0.35).

# 3. Results and discussion

## 3.1. Strut dimensions and lattice morphology

All additively manufactured lattices exhibited strut diameters slightly larger than the nominal CAD values, most likely due to partially melted powder adhering to strut surfaces. Optical microscopy confirmed this trend, see Fig. 4, with measured diameters exceeding the design by approximately 10%–20%. Theoretical and actual manufactured strut diameter and total mass are shown in Table 3. Consequently, the actual mass of each lattice was correspondingly higher, up to +55% and +53% respectively for L3 and L4 samples. The general geometry was nonetheless preserved, and no macroscopic porosity or defects were observed. Additional mass may come from partially melted powder adhered to lattices surface, as shown on Fig. 4.

The 2 kN load capacity of the tomograph limited both the size and volume fraction of the manufactured lattices. As a result, the L3 samples

were produced at a nominal volume fraction $V_f = 0.1$, while the L4 samples were set at $V_f = 0.05$. A full factorial approach was adopted, yielding 12 total specimens: three repeats of each lattice type (L3 or L4) in either empty or hybrid configurations. Each specimen comprised a 3 × 3 × 3 array of unit cells, with overall dimensions of 9 × 9 × 9 mm for L3 and 12 × 12 × 12 mm for L4.

## 3.2. Effective elastic properties from computational homogenization

A small-strain linear elastic computational homogenization was performed on single unit cells, both empty and elastomer-filled, to predict their effective Young's modulus $E^*$ and effective Poisson's ratio $v^*$. As all lattices considered exhibit tetragonal symmetry, the elastic behavior is only represented on 1/4 of the plane, the angle considered is described in Fig. B.13. A complete 3D map for elastic stiffness of the L3 lattice is given in Appendix D.

For lower lattice volume fractions ($V_f = 0.05$), the addition of the compliant PU filler yielded a noticeable increase in the minimum directional modulus (up to 66%). In contrast, at higher lattice volume fractions ($V_f = 0.25$), the intrinsic stiffness of the Ti-6Al-4V skeleton dominated, and filler infiltration produced only marginal (1%) stiffness increase. These trends are summarized in Fig. 5, underscoring that the ratio of metal-to-polymer stiffness is pivotal in determining whether the polymer can effectively supplement the lattice's load-bearing capacity. However, some works [60,78] on metallic lattices filled with compliant elastomer demonstrated an important increase of Young's modulus resulting from the hybridization. Conversely, when the lattice structure is filled with a material stiffer than the lattice constitutive material, a decrease or no change in specific Young's modulus have been reported [52,57].

Computational homogenization also revealed that the lattices retained their auxetic character when filled, as shown on Fig. 6. Although it was expected from the literature [73,76,78,88] that the presence of the elastomer would hinder the deformation of the struts, thus decreasing the lattice auxeticity, it was observed in the present work that the influence of the PU filler on the re-entrant auxetic mechanism is rather limited.

## 3.3. Mechanical response under compression

Results of compression tests are presented in Fig. 7. Lattices and hybrids experimental and FE properties are summarized in Table 4 $\sigma_{peak}$ being the experimentally measured peak stress and $\epsilon_D$ the densification strain. $E_{hom}$ is the homogenized Young's modulus (using linear material
<figure><img src="image_8.png" /><figcaption>Fig. 5. Computed normalized modulus of L3 (a) and L4 (c) lattices and hybrids for different lattice volume fraction, as well as the evolution of minimal and maximal elastic modulus values from hybridization for L3 (b) and L4 (d).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Computed effective Poisson's ratio of L3 (a) and L4 (c) lattices and hybrids for different lattice volume fraction, as well as the evolution of minimal Poisson's ratio value from hybridization for L3 (b) and L4 (d).</figcaption></figure>

models) and $E_{simul}$ the computed Young's modulus using non-linear material models.

During compression tests, we observed that L3 lattices endured shear band failure followed by individual struts breaking. One of the lattice sample endured a catastrophic failure after the onset of shear band collapsing, roughly at a strain of $\epsilon = 0.10$. No densification can

be clearly observed at the final compression strain of $\epsilon = 0.75$. L3 hybrid lattices showed similar behavior with shear band breaking, but densification is observed at large strain (see $\epsilon_D$ in Table 4) with stress pursuing an increasing trend from $\epsilon = 0.32$.

On the contrary, we observed that L4 lattices endured a layerwise failure, with each stress drop corresponding to the failure of one layer.
<figure><img src="image_10.png" /><figcaption>Fig. 7. Stress/strain curves from experimental compression tests on (a) L3 and (b) L4 samples.</figcaption></figure>

<table><caption>**Table 4**

Summary of lattice and hybrid properties obtained via simulation and experiments.</caption><thead><tr><th>Sample</th><th>E<sub>hom</sub> (MPa)</th><th>E<sub>simul</sub> (MPa)</th><th>σ<sup>exp</sup><sub>peak</sub> (MPa)</th><th>ε<sub>D</sub> (-)</th></tr></thead><tbody><tr><td>L3-1</td><td>224</td><td>224</td><td>15.2</td><td>-</td></tr><tr><td>L3-2</td><td>224</td><td>224</td><td>15.2</td><td>-</td></tr><tr><td>L3 hybrid-1</td><td>269</td><td>228</td><td>15.5</td><td>0.69</td></tr><tr><td>L3 hybrid-2</td><td>269</td><td>228</td><td>13.2</td><td>0.72</td></tr><tr><td>L4-1</td><td>48</td><td>48</td><td>4.4</td><td>-</td></tr><tr><td>L4-2</td><td>48</td><td>48</td><td>3.2</td><td>-</td></tr><tr><td>L4 hybrid-1</td><td>80</td><td>52</td><td>4.8</td><td>0.66</td></tr><tr><td>L4 hybrid-2</td><td>80</td><td>52</td><td>3.0</td><td>0.48</td></tr></tbody></table>

No densification has been clearly observed. L4 hybrid lattices exhibited a shear band failure before densification at large strain. In a similar fashion to the L3 hybrids, the stress level follows an increasing trend after first stress peak, from $\epsilon = 0.31$.

Upon infiltration with the elastomer, the overall stiffness measured in the low-strain region changed only slightly for L3 lattices, consistent with the homogenization results. L4 hybrid lattices, however, displayed

a more pronounced increase in stress levels after yielding, highlighting the effect of stiffness ratio between both phases, as it has previously been noted in the literature [59]. Nevertheless, in both L3 and L4 hybrids, the polymer phase supported the lattice struts at large deformations, resulting in a secondary stress plateau that was absent from the empty-lattice curves.

Finite element analysis using non-linear material modeling showed a good agreement with experimental curves concerning in the elastic regime but failed to reproduce the peak stress as can be seen in Appendix C, Fig. C.14, as the onset of plasticity took place at a considerably lower stress than observed experimentally. Several reasons can be invoked: the microstructure and mechanical behavior of a L-PBF produced strut might strongly differ from a dog-bone sample due to a different thermal history, resulting in a higher yield stress than expected; the boundary conditions used in the simulation (periodic boundary conditions) might not accurately reflect the stress state of the lattice samples which are only 3 × 3 × 3 unit-cells, hence not being representative of an infinite equivalent medium. Finally, the finite element analysis did not account for any type of material damage, therefore limiting its accuracy. Future work will include proper damage modeling in order to describe the material failure accurately.
<figure><img src="image_11.png" /><figcaption>Fig. 8. L3 Stress/strain curves obtained in situ and corresponding microtomography images.</figcaption></figure>

### 3.4. Failure mechanisms and microtomography

In Section 3.3, discrepancies were observed between lattices and hybrids in terms of compressive behavior. These discrepancies were explained by the reinforcement effect of the elastomeric phase on the structure, delaying buckling of individual struts, thus changing the deformation behavior of the whole structure. To confirm this hypothesis, in-situ testing using microtomography was conducted, allowing a comparison between lattice deformation at different compression levels, with and without filler. Stress-strain curves and corresponding microtomographs are presented in Figs. 8 and 9. Failure mechanisms can be more clearly observed on Fig. 10, such that for empty lattices:

* Nodal fracture and shear-band initiation were already evident near the first stress peak (③), as highlighted with red dotted line in Fig. 10(a) and (c).

* Cracks propagated along weak nodes, ultimately causing the entire structure to collapse.

By contrast, in hybrid samples:

* The first visible shear band formed at a higher strain (after ③) than in empty lattices.

* At peak stress, they did not present any sign of initiation of the failure, see Fig. 10(b) and (c).

* The elastomeric matrix enhanced structural integrity, mitigating isolated nodal fracture and allowing larger overall deformation before significant collapse.

* Even after the onset of shear-band localization, the polymer phase continued to bridge local cracks, postponing total failure, see ④.

These observations confirm the stabilizing effect of the compliant filler: it better distributes stress during compression, diminishes the chance of abrupt strut failure, and yields a more progressive collapse pattern. These results agree with experimental [53] and numerical [70] observations found in the literature.

### 3.5. Practical implications for crashworthiness

To quantify the impact of hybridization on crashworthiness, the energy absorption (EA) was calculated as the area under each stress-strain curve and is plotted in Fig. 11 (left). While hybrid lattices absorbed more absolute energy (up to 5% higher for L3 and 40% higher for L4 at 70% strain), their specific energy absorption (SEA), i.e. normalized by specimen mass, generally decreased, as shown on Fig. 11 (right). The added polymer phase inevitably increased total mass without proportionally boosting low-strain stiffness, thus lowering SEA. Computed EA and SEA at a high strain of $\epsilon = 0.70$ are summarized on Table 5. It shows that hybridization increased EA by 5% for L3 and by 41% for L4, but decreased SEA by 52% for L3 and 63% for L4. These trends have been found in the literature in the case of too strong reinforcement effect resulting in early densification either from epoxy reinforcement [57] or from elastomer reinforcement [56].

Nevertheless, in scenarios where overall structural integrity, delayed failure, and higher total energy dissipation are prioritized over minimal mass, such as protective barriers, crash attenuators, or blast-resistant panels, these hybrid lattices present a significant advantage. The infiltration-derived support mechanism postpones the onset of localized strut fractures and allows a more gradual collapse. This not only avoids abrupt stress drops but also keeps the load-bearing capability at a more consistent level throughout compression, which can
<figure><img src="image_12.png" /><figcaption>Fig. 9. L4 Stress/strain curves obtained in situ and corresponding microtomography image.</figcaption></figure>

<table><caption>**Table 5**

EA and SEA comparison from experimental tests at high strain $\epsilon$ = 0.70.</caption><thead><tr><th></th><th>L3</th><th>L3 hybrid</th><th>L4</th><th>L4 hybrid</th></tr></thead><tbody><tr><td>EA (J cm<sup>-3</sup>)</td><td>5.80</td><td>6.10 (+5.2%)</td><td>1.65</td><td>2.33 (+41.2%)</td></tr><tr><td>SEA (J g<sup>-1</sup> cm<sup>-3</sup>)</td><td>12.07</td><td>5.15 (-51.9%)</td><td>2.84</td><td>1.06 (-62.7%)</td></tr></tbody></table>

be beneficial in crash scenarios requiring controlled energy dissipation rather than peak-specific efficiency.

Moreover, the PU phase maintains contact with fracturing struts, bridging cracks and delaying separation. Observations from tomography confirm that the polymer effectively redistributes stress, delaying the propagation of damage. As a result, the combined system exhibits more ductile-like crushing behavior. The trade-off, however, is the increased mass and slight reduction in SEA.

Ongoing work aims to optimize these hybrid lattices by refining relative density, filler stiffness, and lattice architecture in order to achieve an optimal balance between total energy absorption and weight-specific crashworthiness. Future work should address the strain rate sensitivity of the elastomer filler, as its dynamic behavior is expected to play a key role in the crash attenuation performance of hybrid lattices [89,90].

## 4. Conclusions

This study set out to determine whether infiltrating a compliant polyurethane matrix into an additively manufactured Ti-6Al-4V auxetic lattice can postpone failure and improve crash-relevant performance. By combining linear-elastic homogenization, large-strain finite-element simulations, quasi-static compression tests and in situ X-ray microtomography, we have shown that the polymer phase fundamentally

alters the way the lattice deforms and collapses. Empty lattices fail early through nodal cracking or abrupt shear-banding; once the voids are filled, these localized damage mechanisms are suppressed and the first shear bands appear at significantly higher strains. The elastomer bridges cracks, redistributes stress and keeps struts engaged, so that the hybrid structure sustains higher loads well into the post-yield regime.

Although the added polymer increases specimen mass and therefore lowers the specific energy absorption (SEA), the total energy absorbed before densification rises for the more compliant L4 geometry by more than 40% at 70% global strain. Linear elastic homogenization clarifies why the stiffness benefit is modest: at higher relative densities the Ti-6Al-4V skeleton dominates the response, whereas at lower densities the soft filler meaningfully augments the weakest directions of the re-entrant network. These results highlight the importance of tuning both the lattice volume fraction and the filler modulus when balancing stiffness, mass and crashworthiness.

From an application standpoint, the hybrid concept is most attractive wherever controlled, ductile-like collapse and high absolute energy dissipation outweigh strict mass constraints, examples include sacrificial crash attenuators, blast shields or biomedical implants that must maintain integrity under overload. The synergy between auxetic geometry and hyperelastic filling also offers a design lever for tailoring deformation modes: by adjusting the stiffness ratio or introducing graded infill, one can shift the onset of shear-banding and tune the force plateau to specific loading envelopes.

Future work should extend the present in situ methodology to dynamic and cyclic loading, incorporate damageable constitutive laws in the FE models, and explore fillers of intermediate stiffness or multifunctional character, e.g. self-healing or damping polymers. Together, our findings demonstrate a straightforward route for engineering light
<figure><img src="image_13.png" /><figcaption>Fig. 10. Microtomographic comparison of L3 lattice (a) and hybrid (b) at $\epsilon = 0.17$ ; comparison of L4 lattice (c) and hybrid (d) at $\epsilon = 0.18$.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 11. EA of L3/L4 lattices and hybrids (left) ; SEA of L3/L4 lattices and hybrids (right).</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Fig. A.12. Stress vs. strain curves from tensile tests of additively manufactured Ti-6Al-4V specimens.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. B.13. Euler-Bunge angles.</figcaption></figure>

yet damage-tolerant hybrid lattices and provide quantitative guidelines for their optimization in next-generation crash-resistant structures.

### CRediT authorship contribution statement

**Frédéric Albertini:** Writing – review & editing, Writing – original draft, Software, Methodology, Investigation, Data curation, Conceptualization. **Justin Dirrenberger:** Writing – review & editing, Writing – original draft, Supervision, Software, Methodology, Funding acquisition, Data curation, Conceptualization. **Cyrille Sollogoub:** Writing – review & editing, Writing – original draft, Supervision, Methodology, Conceptualization. **Andrey Molotnikov:** Writing – review & editing, Writing – original draft, Supervision, Methodology, Funding acquisition, Conceptualization. **Jérôme Adrien:** Writing – review & editing, Writing – original draft, Software, Methodology, Investigation, Conceptualization. **Eric Maire:** Writing – review & editing, Writing – original draft, Software, Methodology, Conceptualization.

### 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.

### Acknowledgments

F. Albertini and J. Dirrenberger acknowledge funding by the France-Australia Science Innovation Collaboration program as well as funding from Académie Francaise, France. Financial support from the French Department of Higher Education, Research and Innovation is gratefully acknowledged for funding F. Albertini's PhD scholarship. J. Dirrenberger gratefully acknowledge Agence Nationale de la Recherche, France through the ANR JCJC SCOLASTIC project (grant no. 16-CE08-0009) and Institut Universitaire de France (IUF) for financial support. A. Molotnikov would like to acknowledge the financial support from the Rod Rickards Fellowship from the Australian Academy of Science.

### Appendix A. Stress vs. strain curve for additively manufactured Ti-6Al-4V

Tensile tests were carried out on Ti-6Al-4V dog-bone specimens fabricated using the same additive manufacturing process as described in Section 2.2. The tests were carried out according to ASTM F3122-14, at a crosshead speed of 0.5 mm/min, corresponding to a strain rate of $\dot{\epsilon} = 1.85 \times 10^{-4} s^{-1}$. The linear elastic region was used to identify the elastic properties, while the plastic region allowed identification of the hardening model material parameters (see Fig. A.12).

### Appendix B. Description of the computational homogenization procedure

In this work, computational homogenization using Periodic Boundary Conditions (PBC) is employed to predict the effective elastic properties (normalized Young's modulus $E^*$ and Poisson's ratio $\nu$) of lattices and hybrid lattices. Starting from CAD files (STEP format), geometries are meshed using GMSH software [91] with tetrahedral quadratic elements (C3D10). Then, FE analysis is conducted using Z-Set FE package.<fnref n="1" /> To ensure numerical accuracy, a mesh sensitivity study has been performed. Lattices and hybrids studied in this work are heterogeneous objects, and their effective properties can be identified by computing the volume averaged response under a physical stimulus over a virtual sample that is considered as a representative volume element (RVE) [92–94] on which PBC are applied.

<footnote n="1">¹ https://www.zset-software.com.</footnote>
<figure><img src="image_17.png" /><figcaption>Fig. C.14. Stress/strain curves from complete experimental compression tests (instron), L3 samples (a) and L4 samples (b).</figcaption></figure>

Considering the small deformation hypothesis, constitutive relations are expressed locally in a linear elasticity framework using the generalized Hooke law:

$$ \sigma(\mathbf{x}) = \tilde{c}(\mathbf{x}) : \varepsilon(\mathbf{x}) \qquad (B.1) $$

with $\sigma$ second-order symmetric Cauchy stress tensor, $\varepsilon$ second-order symmetric engineering strain tensor, and $\tilde{c}$ fourth-order positive definite tensor of elastic moduli. Considering a volume element $V$, the macroscopic stress and strain tensors $\Sigma$ and $\mathcal{E}$ are defined by the spatial averages over $V$ of local stress $\sigma$ and strain $\varepsilon$ fields as follows:

$$ \Sigma := \langle \sigma \rangle = \frac{1}{V} \int_V \sigma \, dV \qquad (B.2) $$

$$ \mathcal{E} := \langle \varepsilon \rangle = \frac{1}{V} \int_V \varepsilon \, dV \qquad (B.3) $$

From Eqs. (B.1)–(B.3), it yields the tensor of effective elastic moduli $\tilde{C}$:

$$ \Sigma = \tilde{C} : \tilde{\mathcal{E}} \qquad (B.4) $$

The elastic moduli tensor $\tilde{C}$ can be fully identified by applying successively 6 fundamental macroscopic loadings, i.e. 3 pure extensions, and 3 pure shears.

Here, RVE consists in lattice unit-cell that presents tetragonal symmetry. The elastic moduli tensor is identified by applying the macroscopic strain $\mathcal{E}$, and measuring the corresponding macroscopic stress components. Using Voigt's notation, here is an example of tetragonal elasticity:

$$ \begin{bmatrix} \Sigma_{11} \\ \Sigma_{22} \\ \Sigma_{33} \\ \Sigma_{23} \\ \Sigma_{31} \\ \Sigma_{12} \end{bmatrix} = \begin{bmatrix} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ C_{12} & C_{11} & C_{13} & 0 & 0 & 0 \\ C_{13} & C_{13} & C_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{66} \end{bmatrix} \begin{bmatrix} E_{11} \\ E_{22} \\ E_{33} \\ 2E_{23} \\ 2E_{31} \\ 2E_{12} \end{bmatrix} \qquad (B.5) $$

After identification of elastic moduli tensor, Young's modulus and Poisson's ratio can be computed for every loading orientation and plotted into 2D or 3D property map, following the approach described
<figure><img src="image_18.png" /><figcaption>Fig. D.15. 3D representation of the normalized directional elastic stiffness (Young's modulus as a function of orientation) for L3 cell at 0.05 and 0.25 volume fraction for both truss and hybrid lattices.</figcaption></figure>

in [79]. Euler-Bunge [95] angles $\phi, \theta$, and $\psi$ are defined as shown on Fig. B.13, generating a system of 3 orthogonal vectors $\underline{l}, \underline{m}$, and $\underline{n}$, such that:

$$ \underline{l} = \begin{bmatrix} \cos \phi \cos \psi - \sin \phi \sin \psi \cos \theta \\ \sin \phi \cos \psi + \cos \phi \sin \psi \cos \theta \\ \sin \psi \sin \theta \end{bmatrix} \qquad (B.6) $$

$$ \underline{m} = \begin{bmatrix} -\cos \phi \sin \psi - \sin \phi \cos \psi \cos \theta \\ -\sin \phi \cos \psi + \cos \phi \sin \psi \cos \theta \\ \cos \psi \sin \theta \end{bmatrix} \qquad (B.7) $$

$$ \underline{n} = \begin{bmatrix} \sin \phi \sin \theta \\ -\cos \phi \sin \theta \\ \cos \theta \end{bmatrix} \qquad (B.8) $$

Using the macroscopic stress $\Sigma(\phi, \theta, \psi)$ and strain $E(\phi, \theta, \psi)$ tensors, the effective Young modulus $E(\underline{l})$ and Poisson ratio $v^*(\underline{l}, \underline{m})$ can be defined such that:

$$ E = \frac{\underline{l} \cdot \Sigma \cdot \underline{l}}{\underline{l} \cdot \underline{E} \cdot \underline{l}} \qquad (B.9) $$

$$ v^* = - \frac{\underline{m} \cdot \underline{E} \cdot \underline{m}}{\underline{l} \cdot \underline{E} \cdot \underline{l}} \qquad (B.10) $$

Finally, the normalized modulus $E^*$ of the studied architecture materials can be defined with respect to its constitutive material Young's modulus $E_0$, and its volume fraction $V_f$, being defined as the volume occupied by a periodic cell $V_{\text{lattice}}$ divided by the cubic volume circumscribed by the cell ($V_{\text{cubic}} = L^3$):

$$ E^* = \frac{1}{E_0 V_f} E. \qquad (B.11) $$

## Appendix C. Complete experimental and FEA stress-strain plot

This section presents all the stress-strain curves obtained from the experimental tests (compression tests performed on the Instron machine, see Section 3.3) and from the numerical simulations using the finite element method. The Fig. C.14 shows the complete experimental results of compression tests for L3 samples C.14(a) and L4 samples C.14(b) using solid lines. FE results are presented using dotted lines.

## Appendix D. Elastic stiffness maps for lattices

The Fig. D.15 presents the 3D maps of effective elastic stiffness of the L3 lattices. The influence of the lattice volume fraction and polymer infiltration is represented on the right.

## Data availability

Data will be made available on request.

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