# Design, simulation, and experimental validation of metamaterials with direction-dependent stiffness

Hossein Rahimi, Mahdi Khajepour, Davood Rahmatabadi, Ghader Faraji, Mostafa Baghani, Daniel George, Majid Baniassadi

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## To cite this version:

Hossein Rahimi, Mahdi Khajepour, Davood Rahmatabadi, Ghader Faraji, Mostafa Baghani, et al. Design, simulation, and experimental validation of metamaterials with direction-dependent stiffness. Alexandria Engineering Journal, 2026, 134, pp.460-472. 10.1016/j.aej.2025.12.031. hal-05426870

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# Design, simulation, and experimental validation of metamaterials with direction-dependent stiffness

Hossein Rahimi<sup>a, ∗</sup>, Mahdi Khajepour<sup>a, †</sup>, Davood Rahmatabadi<sup>a</sup>, Ghader Faraji<sup>a</sup>, Mostafa Baghani<sup>a, ‡</sup>, Daniel George<sup>b, ∗, ‡</sup>, Majid Baniassadi<sup>a, ∗</sup>

<figure><img src="image_4.png" /></figure>

ARTICLE INFO

Keywords:

Direction-Dependent Stiffness (DDS)

Mechanical Metamaterials

Anisotropic Metamaterials

Finite Element Analysis (FEA)

Additive Manufacturing (AM)

Digital Light Processing (DLP)

A B S T R A C T

Mechanical metamaterials derive their functionality from geometry rather than composition, yet achieving experimentally validated three-dimensional direction-dependent stiffness (DDS) has remained challenging. This work introduces a 3D metamaterial unit cell with strong, tunable DDS, developed through a heuristic, gradient-free optimization framework that iteratively generated and refined geometries using finite element analysis. The final design features asymmetric internal struts and orientation-specific contact surfaces that activate bending- or stretching-dominated modes depending on loading direction. The unit cell and corresponding lattices were fabricated using high-resolution digital light processing (DLP) additive manufacturing and tested under compression along all three orthogonal axes. Simulations and experiments show excellent agreement (<1 % deviation), confirming distinct mechanical responses in the X, Y, and Z directions. The structure exhibits a stiffness increase above 350 % across deformation regions and up to 80 % contrast between its stiffest and most compliant orientations, while maintaining smooth force-displacement behavior with negligible stress concentrations. It also demonstrates substantial direction-dependent energy absorption, reaching several hundred joules in the stiffest orientation. These results establish a practical and manufacturable pathway toward architected materials with true 3D DDS, offering strong potential for exoskeletons, crash-protection systems, biomedical implants, and seismic-resistant components.

1. Introduction

Metamaterials have emerged in recent decades as one of the most
transformative classes of engineered materials, enabling properties that
extend far beyond those observed in conventional systems. Unlike
traditional materials, whose performance is dictated largely by chemical
composition, metamaterials derive their unusual properties from
deliberately engineered architectures and periodic geometrical patterns.
This paradigm toward geometry-driven property tailoring has given rise
to a wide variety of novel behaviors, including negative Poisson's ratio
(auxeticity), programmable anisotropy, negative refractive index, and
ultrahigh specific stiffness [1-3]. Owing to this flexibility, mechanical
metamaterials are now widely explored in fields as diverse as aerospace,
civil engineering, biomedicine, energy absorption, and vibration control
[4-8].

Within the broad domain of mechanical metamaterials, tunability of
stiffness has attracted particularly strong attention in recent years.

Conventional engineering materials display stiffness values that remain essentially fixed once manufactured, which limits their adaptability to varying service conditions. By contrast, stiffness-tunable metamaterials offer the ability to tailor resistance to deformation either passively through architectural design or actively through reconfiguration mechanisms [9–11]. This capability is attractive for applications requiring a balance of rigidity and flexibility, such as automotive crashworthiness [12–15], biomedical implants or tissue scaffolds mimicking anisotropic natural tissues like bone and cartilage [16–18], and rehabilitation devices or exoskeletons providing load-bearing support along one axis while allowing natural movement in others [19,20].

A variety of strategies have been proposed to realize stiffness
tunability. Early work often relied on re-entrant or honeycomb geome-
tries. Feng et al. [21] demonstrated that hierarchical honeycomb met-
astructures incorporating star-shaped reinforcements could increase the
normalized elastic modulus by 3.62, although the design remained
limited to planar configurations. Cheng et al. [22] adjusted stiffness in

<footnote><sup>a</sup> School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran</footnote>

<footnote><sup>b</sup> University of Strasbourg, CNRS, ICube, Strasbourg 67000, France</footnote>

<footnote>* Corresponding authors.</footnote>

<footnote>E-mail addresses: george@unistra.fr (D. George), m.baniassadi@ut.ac.ir (M. Baniassadi).</footnote>
auxetic structures by integrating solid elements into re-entrant unit cells.

Other approaches include perforated unit cells, such as Li et al. [23]
butterfly-shaped holes for tunable stiffness and Poisson's ratio, or Zhang
et al.'s [24] tubular frameworks with lamellar auxetic reinforcements.
More recent 3D efforts include Lyu et al. [25] metamaterial combining
adjustable Poisson's ratio with tunable stiffness, Xiao et al. [26]
butterfly-inspired lattices, and Churchill et al.'s [27] dynamically vari-
able negative stiffness structures. However, most designs are restricted
to single-axis responses or simplified deformation mechanisms, with
limited exploration of fully three-directional stiffness variation and
scarce experimental validation under multi-axial loading. Classical
cellular solids theory, particularly the framework established by Gibson
and Ashby [28], provides the foundational distinction between
bending-dominated and stretching-dominated architectures, which
directly supports the deformation mechanisms activated in the DDS
design explored in this study.

The limitations of current approaches can be summarized in several
ways. First, the dimensionality of many studies restricts them to 2D or
quasi-3D systems, which do not accurately represent real-world struc-
tural demands [21–24]. Second, tunability is often confined to a single
axis, which severely limits multifunctionality. Third, fabrication re-
mains a bottleneck: intricate unit-cell architectures are challenging to
realize with conventional manufacturing, and even additive
manufacturing (AM) methods struggle with resolution, material limi-
tations, and scalability [29–32]. Fourth, many reported lattices suffer
from severe stress concentrations at strut junctions, which compromise
mechanical integrity and reliability [33,34]. Finally, while numerical
studies are abundant, rigorous experimental validation particularly
under multi-axial loading is scarce. Together, these challenges highlight
the need for practical, experimentally validated metamaterials demon-
strating true 3D DDS.

Additive manufacturing has proven to be a decisive enabler in this context. Conventional fabrication methods such as machining or molding cannot replicate the fine, periodic features required for advanced metamaterials. By contrast, AM provides unparalleled design freedom, enabling the fabrication of intricate unit cells with controlled geometrical complexity [29,30,35]. Among AM techniques, digital light processing (DLP) has become especially valuable for metamaterials research because of its ability to produce high-resolution structures with complex internal features [36]. Recent studies have shown how resin chemistry, printing resolution, and process parameters affect the precision and mechanical fidelity of printed lattices [31]. Yet despite these advances, achieving consistent DDS in fabricated structures remains a substantial challenge. Many designs display significant discrepancies between computational predictions and experimental results, often due to printing defects, resin variability, or unintended stress concentrations. Thus, careful integration of both design and fabrication parameters is therefore required to unlock the potential of AM-enabled metamaterials.

Against this backdrop, DDS emerges as a critical frontier for multi-
functional responses, such as in exoskeletons, crash-protection struc-
tures, biomedical implants reducing stress shielding [16,37,38], and
seismic-resistant systems absorbing vibrations selectively [39,40].
Several recent studies offer context: Fan et al. [41] demonstrated
anisotropic wave propagation through a twisted heterobilayer meta-
surface, while Zha and Zhang [42] engineered a lattice exhibiting
reversible negative compressibility. However, these works primarily
address wave or stability phenomena rather than bulk mechanical
stiffness tuning under compressive loading. In contrast, the present work
focuses directly on directional stiffness control in a load-bearing 3D
architected structure, achieving quantitative stiffness gains of up to
~358 % and directional contrasts of up to ~80 %, with close agreement
between experiments and simulations.

The present study addresses these gaps by introducing a novel three-
dimensional metamaterial unit cell exhibiting DDS. Using a heuristic
optimization framework, a geometry was developed that achieves

distinct stiffness values along the X, Y, and Z axes. The structure was fabricated using high-resolution DLP additive manufacturing with an ABS-like resin and validated experimentally through uniaxial compression along all principal directions. Finite element simulations closely matched experimental results (≤1 % discrepancy). Two deformation mechanisms, mode A and mode B were identified as governing directional tunability. Importantly, the design exhibits smooth force-displacement curves with reduced stress concentration compared with conventional lattices. By integrating computational modeling, additive manufacturing, and multi-directional experimental testing, this work demonstrates a practical pathway toward 3D metamaterials with independently tunable stiffness along orthogonal axes.

## 2. Materials and methods

2.1. *Design and optimization*

The development of the DDS unit cell was performed using a heuristic optimization strategy aimed at obtaining distinct stiffness values along the three orthogonal axes. In this study, the heuristic process consisted of a gradient-free iterative search, where candidate unit-cell geometries were manually generated, evaluated through FEA for directional stiffness contrast, and refined based on predefined criteria including a minimum 20 % stiffness variation between axes, manufacturability constraints, and geometric symmetry considerations. The overall workflow including geometry generation, numerical screening, and iterative refinement, is shown in Fig. 1.

The process began with a survey of existing metamaterial topologies
to identify recurring limitations, such as confinement to two-
dimensional planar designs or tunability along only one axis. Based on
these observations, twenty candidate UC geometries were generated
using CATIA V5 software. Each model was derived from a cubic tem-
plate and modified by incorporating grooves, cut-outs, or anisotropic
lattice members to induce directional stiffness variation. Representative
discarded designs that failed to meet the required anisotropy thresholds
are shown in Fig. 2.

Finite element analysis (FEA) was employed to evaluate the mechanical performance of each candidate UC under uniaxial compression along the X, Y, and Z directions. Only geometries achieving a minimum of 20 % stiffness variation between axes were retained. The optimization objective was to maximize directional stiffness contrast while maintaining structural integrity. Evaluation criteria focused on a minimum 20 % relative stiffness difference between axes, quantified as the variation in slope of the linear regions of force-displacement curves from FEA simulations under uniaxial compression. Constraints included manufacturability via DLP (minimum feature size $≥$ 0.5 mm to prevent printing defects) and fixed overall unit cell dimensions of $12 × 12 × 12$ mm³ for lattice scalability and compatibility with testing equipment.

Through iterative refinement, an optimized UC was obtained, as depicted in Fig. 3. The final design, with external dimensions of $12 × 12 × 12$ mm³, incorporated two deformation mechanisms, later identified as mode A and mode B, which could be selectively activated depending on loading orientation. These modes form the basis of the observed direction-dependent stiffness.

## 2.2. *Experimental procedure*

The optimized UC and its lattice assemblies were fabricated using
DLP 3D printing, which offers the resolution necessary for reproducing
intricate geometrical features. Printing was performed with an Anycubic
Photon Mono X system using a commercial ABS-like photopolymer
resin. This resin was selected because of its toughness and dimensional
stability, which are critical to sustaining compressive loads without
brittle failure.

The printing parameters used, summarized in Table. 1 include a
<figure><img src="image_5.png" /><figcaption>Fig. 1. The heuristic optimization approach to designing and achieving a desired DDS unit cell.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 2. Initially designed UCs were discarded because they did not meet the desired objectives.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 3. a) DDS structure, b) 3D design of the final UC, c) dimensions and details of the final UC with DDS.</figcaption></figure>
<table><caption>**Table 1**
Printing parameters for the DLP 3D printed parts.</caption><thead><tr><th>Parameter</th><th>Value</th><th>Additional Parameter</th><th>Value</th></tr></thead><tbody><tr><td>Layer height</td><td>0.05 mm</td><td>Rest time after</td><td>4 s</td></tr><tr><td>Bottom layer count</td><td>6</td><td>Retraction</td><td>8 mm</td></tr><tr><td>Exposure time</td><td>2 s</td><td>Bottom lift distance</td><td>8 mm</td></tr><tr><td>Bottom exposure time</td><td>30 s</td><td>Lifting distance</td><td>8 mm</td></tr><tr><td>Transition layer count</td><td>6</td><td>Bottom retract distance</td><td>8 mm</td></tr><tr><td>Transition type</td><td>Linear</td><td>Retract distance</td><td>8 mm</td></tr><tr><td>Transition time</td><td>4 s</td><td>Bottom lift speed</td><td>60 mm/min</td></tr><tr><td>Decrement</td><td></td><td>Lifting speed</td><td>60 mm/min</td></tr><tr><td>Rest time before lift</td><td>0 s</td><td>Bottom retract speed</td><td>150 mm/min</td></tr><tr><td>Rest time after lift</td><td>0 s</td><td>Retract speed</td><td>150 mm/min</td></tr></tbody></table>

0.05 mm layer height for high resolution, 30 s bottom exposure for adhesion, 2 s normal exposure time, and six transition layers to minimize delamination. Lifting and retraction distances of 8 mm and a 60 mm/min lift speed were selected to minimize distortion during separation.

The fabricated UCs are shown in Fig. 4, illustrating the structures printed in the three orthogonal orientations (X, Y, and Z). Post-processing included rinsing in isopropyl alcohol (IPA) for 5 min to remove uncured resin, followed by UV curing for 15 min to ensure complete polymerization. To ensure reproducibility, three identical specimens were fabricated and tested for each principal direction. Compression test results represent the mean of three independent replicates, with standard deviations < 3 % for measured force values and < 2 % for calculated stiffness, confirming high repeatability.

The printing parameters (Table. 1) were selected based on the manufacturer data-sheet for the resin to achieve high resolution (0.05 mm layer height for accurate reproduction of intricate features), strong adhesion (30 s bottom exposure and 6 bottom layers), and minimal structural distortion (60 mm/min lift speed and 8 mm lift/retract distances)(Fig. 4).

The mechanical response of the printed UCs was characterized through uniaxial compression tests carried out on a SANTAM STM50 universal testing machine at room temperature (22 ± 2 °C). The loading rate was maintained at 3 mm/min, in accordance with ASTM D695 guidelines for polymeric materials under compression. Each specimen was compressed along Y-direction until either failure or a nominal strain of 60 % was reached. The compression sequence for a representative UC in the Y-direction is shown in Fig. 5, which illustrates the deformation process at time intervals of 15, 30, 45, and 60 s. Force-displacement data. The stiffness values were determined from the slopes of the linear regions of the force-displacement curves, while energy absorption was calculated as the area under the curves up to 60 % strain. To ensure statistical reliability, all results represent the mean values of three independent tests (Fig. 5).

Repeatability was evaluated using the three-sample sets printed for each direction. This approach captures potential variations arising from DLP printing resolution, resin curing conditions, and dimensional

deviations of small structural features. The force-displacement responses of the three samples showed minimal spread, with standard deviations below three percent for recorded forces and below two percent for calculated stiffness values in both linear deformation regions. These low variations confirm that the direction-dependent stiffness behavior is robust and not significantly influenced by process-related uncertainties or material inhomogeneity, thereby supporting the validity of the experimental confirmation of the DDS design.

## 2.3. Finite element analysis

To complement the experiments, finite element simulations were conducted using ABAQUS/CAE 2022. The resin was modeled as a linear elastic material with an elastic modulus of 1180 MPa and a Poisson's ratio of 0.3, based on manufacturer data-sheet, and the model was calibrated through iterative adjustments to match the experimental force-displacement curves, achieving < 1 % deviation across the deformation range. A dynamic/explicit solver was employed to simulate quasi-static compression with high computational efficiency.

To ensure consistent evaluation of direction-dependent stiffness in both simulation and experiment, the same loading configuration was imposed across all three principal directions. In the finite element models, the bottom face of the unit cell or lattice assembly was fully constrained in all translational degrees of freedom, while a rigid compression plate was positioned at the top and displaced downward to simulate quasi-static loading. The plate motion applied a prescribed displacement at a uniform rate, equivalent to the 3 mm/min crosshead speed used in physical compression tests. Contact between the upper platen and the structure was defined as frictionless, and general contact with zero tangential resistance was assigned to allow free lateral deformation without artificial constraint. The corresponding experimental setup replicated these conditions by placing each specimen in universal testing machine. This alignment of constraints and loading methods in both FEA and testing ensures that the measured differences in stiffness arise solely from the geometry and deformation mechanisms of the unit cell rather than variations in boundary conditions.

The boundary conditions involved fixing the bottom surface of the model and applying displacement to the top surface via a rigid plate. Contact was modeled using a general contact algorithm with zero tangential friction. To ensure accuracy, a mesh sensitivity study was conducted. The final optimized UC was discretized using tetrahedral C3D4 elements with a global size of 0.8 mm, yielding approximately 665,000 elements and 142,000 nodes. The meshed model with applied boundary conditions is shown in Fig. 6.

Comprehensive details regarding the mesh type, total number of elements, and number of nodes are provided in Table 2. Both single UCs and LS composed of multiple UCs were simulated under compression along the X, Y, and Z directions. Outputs included force-displacement curves, stiffness evolution, von Mises stress distributions, and energy absorption capacities. These results were subsequently compared with

<figure><img src="image_8.png" /><figcaption>**Fig. 4.** DLP printed UC with DDS shown in three directions.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 5. Experimental sequence of the UC in the compression test.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 6. Meshed model of the optimal UC with DDS.</figcaption></figure>

<table><caption>**Table 2**
FEA configuration for both the UC and LS.</caption><thead><tr><th>Property and interaction</th><th>Step</th></tr></thead><tbody><tr><td><strong>Elastic modulus</strong><br>1180 MPa</td><td><strong>Target time increment</strong><br>0.001 s</td></tr><tr><td><strong>Poisson ratio</strong><br>0.3<br><strong>General contact</strong><br>tangential (0)</td><td><strong>Total time</strong><br>60-200 s<br><strong>Amplitude</strong><br>smooth step</td></tr><tr><td colspan="2"><strong>Mesh property</strong></td></tr><tr><td><strong>Model</strong></td><td><strong>Approximate global size</strong><br>4900</td></tr><tr><td><strong>Rigid body</strong></td><td><strong>Number of mesh</strong><br>5041</td></tr><tr><td><strong>UC_x</strong></td><td><strong>Number of nodes</strong><br>R3D4</td></tr><tr><td><strong>UC_y</strong></td><td><strong>Number of nodes</strong><br>C3D4</td></tr><tr><td><strong>UC_z</strong></td><td><strong>Number of nodes</strong><br>C3D4</td></tr><tr><td><strong>S_x</strong></td><td><strong>Number of nodes</strong><br>C3D4</td></tr><tr><td><strong>S_y</strong></td><td><strong>Number of nodes</strong><br>C3D4</td></tr><tr><td><strong>S_z</strong></td><td><strong>Number of nodes</strong><br>C3D4</td></tr></tbody></table>

experimental data to validate the computational framework.

### 3. Results and discussion

#### 3.1. FEA data validation

Validating the reliability of FEA results is a key step in this study. To achieve this, UCs with DDS were fabricated using DLP 3D printing, as shown in Fig. 4, and tested under compression. The experimental data collected from these tests were then compared with FEA simulation results to assess the simulation's accuracy. The force-displacement curves for UCs in the Y-direction (UC-Y), derived from both the experimental test and the FEA simulation, are presented together in Fig. 7a. These curves track the relationship between applied force and displacement across two distinct regions, region A with steeper slope and region B with sharper slope. Fig. 7b complements this by comparing the stiffness values, calculated as the slope of the force-displacement curves for both datasets in these regions. Analysis of the data reveals strong consistency between the experimental and FEA results. In region

<figure><img src="image_11.png" /><figcaption>Fig. 7. a) Force-displacement curve, and b) stiffness values for experimental and FEA data for UC in the Y direction.</figcaption></figure>
A, the force-displacement curves from both sources overlap perfectly, reflecting identical linear behavior. As loading increases and the structure transitions into region B, the curves remain closely aligned, with only a minor divergence at higher displacements. This close agreement is further supported by the stiffness values in Fig. 7b, where differences between experimental and FEA data are negligible across both regions. Quantitatively, the maximum discrepancy between the experimental and FEA results is less than 1% throughout the entire deformation range. This minimal error underscores the high accuracy of the FEA

simulations.

The error reported between simulation and experiment was calculated using stiffness values extracted from the linear sections of their respective force-displacement curves. For each loading direction and deformation region, the slope of the force-displacement response was evaluated in both the experimental and FEA datasets. The percentage error was then obtained as the absolute difference between the two stiffness values divided by the experimental stiffness. This approach ensures that the comparison reflects the mechanical response of the

<figure><img src="image_12.png" /><figcaption>Fig. 8. deformation mechanisms of the UC structure with DDS. a) deformation mode A, showing under compression. b) deformation mode B, under compression. c) UC with all four vertical sides exhibiting deformation mode A. d) UC with a combination of deformation modes A and B. e) UC with all four vertical sides exhibiting deformation mode B. f) deformation mode A and g) deformation mode B after loading.</figcaption></figure>
structure rather than point-wise differences in displacement or stress. Applying this method across all three loading directions and evaluated regions resulted in a maximum stiffness deviation of less than one percent, confirming the strong correspondence between simulation and experimental measurements. Consequently, these findings confirm that the FEA approach is reliable for further analysis and design optimization of LS with DDS.

## 3.2. Deformation mechanisms driving DDS

The design process of the DDS structure has been elaborated in previous sections. This subsection examines the deformation mechanisms of the final UC. The UC exhibits two primary deformation mechanisms, indicated as deformation mode A and deformation mode B, which are illustrated in Figs. 8a and 8b, respectively. In deformation mode A (Fig. 8a), the UC undergoes compression, resulting in contact between only two surfaces, highlighted by red lines. These surfaces, located centrally within the UC's internal structure, form a cross-like pattern that governs the deformation behavior under vertical compressive loading. In contrast, deformation mode B (Fig. 8b) is characterized by a more complex response, where eight distinct contact surfaces, marked by red lines and blue arrows, interact during deformation. This mode arises from an intricate network of internal elements, including additional diagonal and vertical connections, leading to a higher number of contact points and a distinct stiffness response compared to mode A. When a compressive force is applied along the vertical axis of the UC, the four vertical sides of the cubic structure play a pivotal role in manifesting the DDS effect. This loading condition gives rise to three possible configurations of deformation modes across these sides, as depicted in Figs. 8c-8e:

*   **Uniform Mode A Configuration:** All four vertical sides exhibit deformation mode A, as shown in Fig. 8c. This uniform deformation pattern results in a consistent stiffness response across the UC, driven by the two-contact-surface behavior observed in mode A.

*   **Mixed Mode Configuration:** Two of the vertical sides display deformation mode A, while the other two exhibit deformation mode B, as illustrated in Fig. 8d. This hybrid configuration highlights the UC's ability to combine distinct deformation mechanisms, potentially tailoring the DDS effect to specific loading conditions.

*   **Uniform Mode B Configuration:** All four vertical sides exhibit deformation mode B, as depicted in Fig. 8e. With eight contact surfaces active on each face, this configuration suggests an enhanced stiffness response, differing significantly from the other two arrangements.

While previous studies by CB Churchill [27] and X Cheng [22] investigated dynamically variable and tunable stiffnesses, these configurations underscore the versatility of the UC design in achieving direction-dependent mechanical properties, a critical aspect of the DDS structure. The observed deformation modes and their combinations, validated through FEA, provide insight into the structural behavior under compressive loading, informing potential applications and further optimization of the design.

From a mechanics standpoint, the observed anisotropy arises from the fundamental difference between bending-dominated and stretching-dominated deformation responses in the two identified modes. In deformation mode A, the load is mainly carried through bending of the inclined internal members, resulting in larger rotations, lower stiffness, and delayed engagement of load-bearing surfaces. This mode corresponds to a classical bending-dominated lattice response, where structural compliance is governed by flexural deformation. In contrast, deformation mode B transitions the load path toward axial stretching of the internal struts, which significantly increases stiffness. The higher number of activated contact surfaces and the more direct force transmission through the axial members make mode B characteristically stretching-dominated. The directional anisotropy of the unit cell

therefore originates from the selective activation of bending versus stretching mechanisms depending on the loading orientation.

## 3.3. Mechanical characterization

Simulation results demonstrate that the proposed UC (Fig. 9a) exhibits distinct mechanical behaviors along the X, Y, and Z directions. The forces sustained in these directions, based on a final 3 mm displacement, reflect the proportional distribution of the UC's stiffness. It is also worth mentioning that Fig. 9 Illustrates negligible stress concentration throughout the compressed structure, irrespective of the geometry of the structure or the cell. Numerous studies highlight that stress concentration is a significant drawback in metamaterial structures for industrial applications [33,34,43]. The designed structure in this study, despite having relatively low stress concentration, demonstrates the potential for use in a wide variety of applications.

The stress contour maps shown in Fig. 9 substantiate the claim of negligible stress concentration, as the von Mises stress fields remain smooth and evenly distributed across the lattice during loading. No isolated peaks or abrupt stress gradients appear at critical intersections or contact regions, indicating that the geometry effectively disperses applied loads and prevents localized structural overstressing. This behavior supports the suitability of the design for applications where stress uniformity and load reliability are required.

It is clear from Fig. 10A, the force increases steadily with displacement across all directions. Among the three directions, the Y direction exhibits the highest force toleration, sustaining a force of 3166 N. In this direction, the structure responds to the applied force exclusively through deformation mode A. As mentioned earlier, this mode exhibits a higher capacity to withstand force and stiffness, making this direction the stiffest among all. This significant resistance makes it ideal for transferring large forces in structural applications [44]. In the X direction, the applied force reaches 2674 N, indicating moderate force endurance due to activation of both deformation modes A and B, which is beneficial for withstanding axial stresses. In contrast, the Z direction sustains the lowest force, 2118 N, highlighting its greater flexibility with activating deformation mode A. This characteristic can be advantageous in applications requiring energy absorption, such as impact-resistant designs. The absence of abrupt peaks, dips, or discontinuities in these curves highlights an optimized stress distribution within the structure, effectively minimizing stress concentration points that are often a vulnerability in conventional designs. To further investigate the mechanical behavior of the LS with DDS using the final UC, FEA data has been utilized. The results (Fig. 10b) confirm that the overall structural behavior aligns closely with that of the individual UC. The maximum forces sustained by the structure are 89.5 kN in the Y direction by activating deformation mode A, 75.6 kN in the X direction by engaging deformation mode A and B, and 59.4 kN in the Z direction through deformation mode B. Once again, the smooth, continuous nature of these curves, free of sudden changes or irregularities, confirms the structure's ability to distribute stress uniformly, even under substantial static loading conditions.

To clearly illustrate the anisotropic mechanical behavior, the simulated load-displacement curves for the X, Y, and Z loading directions are presented in Fig. 10. These curves show the evolution of mechanical response throughout the compression process and visually confirm the varying slopes that represent direction-dependent stiffness. The Y direction exhibits the steepest curve, indicating the highest resistance to deformation, while the X and Z directions show progressively lower slopes consistent with their lower stiffness levels. These visual trends correlate directly with the extracted stiffness values reported later in this section, confirming that the DDS behavior is not limited to isolated numerical values but is clearly observable in the full deformation response of the structure.

These values maintain a similar load distribution pattern as observed in the single UC, demonstrating that the structure effectively inherits the
<figure><img src="image_13.png" /><figcaption>Fig. 9. Mises stress contour for the FEA applied to the a) final cell, and b) LS in the X, Y, and Z directions at compression loading.</figcaption></figure>

mechanical characteristics of its fundamental unit. The force-displacement curves can be divided into three central regions. In the initial stage, all three directions exhibit similar behavior within region A and the same stiffness values. However, after the separation point between areas A and B, each direction begins to behave differently, resulting in distinct linear trends that represent the different stiffness levels in area B. Despite Fig. 10 in region C, the stiffness of the DDS

structures begins to decrease. This matter is related to the initiation of the plastic deformation in the structures.

Stiffness is a fundamental mechanical property in metamaterials, governing their resistance to deformation under applied loads [45]. This study aims to develop a DDS structure with maximized tunability, enabling tailored mechanical responses across different directions. The stiffness of both the UC and the LS is evaluated through the slope of their
<figure><img src="image_14.png" /><figcaption>Fig. 10. The force-displacement graph of the simulated a) final unit cell, b) structure under compressive loading in the three directions X, Y, and Z, and c) energy absorption for three directions in the LS.</figcaption></figure>

force-displacement curves, which reveal how stiffness evolves under compressive loading. As shown in Fig. 10, the transition from region A to region B is defined at approximately 0.8 mm displacement (corresponding to ~7% nominal strain), where the force-displacement curve shifts from the initial linear slope (lower stiffness dominated by deformation mode A) to a steeper slope (higher stiffness due to activation of deformation mode B and increased contact surfaces). The force-displacement curves for both the UC and LS exhibit a clear transition between two distinct regions, region A and region B. In region A, the initial linear slope reflects a consistent, lower stiffness as the structure begins to deform. As deformation progresses into region B, the slope steepens, indicating a significant increase in stiffness. This shift highlights the DDS structure's ability to adapt its mechanical response, a key feature for applications requiring variable stiffness.

Fig. 11 provides a detailed breakdown of specific stiffness values for the UC and LS in each region and direction (X, Y, and Z), showcasing the structure's direction-dependent behavior. In the UC, region A exhibits a uniform specific stiffness of 18.7 N/mm.g across all directions, indicative of an isotropic initial response. However, in region B, the stiffness increases dramatically and becomes anisotropic: 60 N/mm.g in the Z direction, 85.5 N/mm.g in the X direction, and 104.6 N/mm.g in the Y direction. This variation demonstrates the UC's capacity to tune its specific stiffness based on the loading direction, with the Y direction showing the highest resistance to deformation.

On the other hand, for the LS, region A stiffness is measured at 3.2 N/mm.g, lower than the UC. In region B, the stiffness escalates further: 8.1 N/mm.g in the Z direction, 11.3 N/mm.g in the X direction, and 14.5 N/mm.g in the Y direction. The Y direction again exhibits the highest stiffness, while the Z direction remains the lowest, consistent with the UC trends.

In the simulations, the mechanical load was applied through a prescribed strain, and the corresponding stress response was extracted directly from the software. Under these conditions, a single unit cell exhibits an upper-bound estimate of the specific stiffness and therefore deviates from the true representative volume element (RVE) behavior of the metamaterial. As the number of unit cells in the simulated structure increases, the response progressively converges toward the RVE properties, which act as a lower bound in this case due to the boundary constraints imposed during simulation. Consequently, the lower specific stiffness observed in the lattice assemblies compared with the single-unit-cell model is attributed to this convergence effect and the influence of boundary conditions.

Notably, the LS stiffness in the Y direction in region B reflects a 358% increase compared to region A, underscoring the design's exceptional tunability. Additionally, the stiffness in region B varies by at least 40% between directions, emphasizing the pronounced anisotropy of the LS. The substantial increase in stiffness from region A to region B, coupled with significant directional variation, highlights the effectiveness of the DDS design. The 358% stiffness rise in the LS's Y direction demonstrates its potential for applications requiring high load-bearing capacity. In comparison, the minimum 40% directional difference in region B enhances its adaptability for tailored mechanical responses. These characteristics position the proposed metamaterial as a versatile solution for diverse engineering challenges.

The stiffness values and reported percentage improvements represent internal comparisons within the same metamaterial architecture rather than against an external isotropic reference lattice. The 358% increase reflects the change in stiffness of the structure as it transitions from region A, dominated by bending deformation mechanisms, to region B, where stretching-dominated load paths and increased contact
<figure><img src="image_15.png" /><figcaption>Fig. 11. Stiffness levels for a) UC, and b) LS models; energy absorption for c) UC, and d) LS.</figcaption></figure>

surfaces enhance rigidity. Likewise, the 80% directional contrast is calculated from the difference in stiffness between the Y and Z loading directions in region B, where the Y direction exhibits the highest resistance to deformation and the Z direction the lowest. These comparisons therefore quantify the tunable and directionally dependent behavior intrinsic to the proposed unit cell and serve to demonstrate the range of mechanical responses achievable within a single architected geometry.

The energy absorption capacity of the proposed metamaterial, a critical feature for applications demanding controlled deformation and impact resistance, was thoroughly evaluated for both the UC and the LS through FEA. The specific absorbed energy, calculated as the area under the force-displacement curves shown in Fig. 10a for the UC and Fig. 10b for the LS, reveals pronounced directional dependence, with specific values detailed in Fig. 11c for the UC and Fig. 11d for the LS. The UC absorbs 173.6 mJ/g in the Y direction, 151.5 mJ/g in the X direction, and 125.8 mJ/g in the Z direction, while the LS, benefiting from its interconnected architecture, demonstrates higher absorption at 215.3 mJ/g in the Y direction, 193.5 mJ/g in the X direction, and 169.2 mJ/g in the Z direction.

This variability stems from the structure's innovative design, which activates distinct deformation mechanisms, particularly deformation mode A and deformation mode B, along each axis to tailor energy dissipation. In the Y direction, where deformation mode A predominates, enhanced stiffness and resistance to compressive forces yield higher energy absorption, making it well-suited for load-bearing scenarios. Conversely, the Z direction, influenced by the flexibility of deformation mode B, exhibits lower energy absorption but excels in energy dissipation, ideal for impact-resistant applications. Notably, the design minimizes stress concentration across all loading directions, ensuring robust structural integrity. This combination of direction-

dependent energy absorption and mechanical stability positions the metamaterial as a versatile solution for multifunctional engineering challenges.

## 4. Potential application for LS with DDS

The LSs designed in this study exhibit a unique combination of high stiffness in one direction and flexibility in others, which makes them particularly suitable for multifunctional applications where anisotropy is advantageous. A primary example is in exoskeleton design, where structural elements must resist vertical loading while permitting lateral mobility. The results in Fig. 11 demonstrate that the LS achieves a maximum compressive strength of 89.5 kN in the Y-direction, with stiffness values reaching 14.5 N/mm.g. By comparison, the stiffness values in the X and Z directions are 11.3 N/mm.g and 8.1 N/mm.g, respectively. This directional difference provides sufficient rigidity for weight-bearing tasks while enabling flexibility for human movement.

To place these results in context, previous tunable mechanical metamaterials have achieved substantially lower directional stiffness contrasts. For example, Feng et al. [21] reported a hierarchical honeycomb metastructure with a normalized elastic modulus increase of 3.62x, but their design was limited to planar geometries and exhibited tunability along a single axis. Similarly, Cheng et al. [22] demonstrated auxetic re-entrant lattices with stiffness variation introduced by solid elements, though their maximum stiffness difference between directions did not exceed 25%. More recent works by Li et al. [23] and Zhang et al. [24] achieved tunable stiffness through perforated and tubular frameworks, but the anisotropy remained confined to a single dominant axis.

In contrast, the DDS lattice presented here provides a minimum of 40% stiffness variation between axes in region B, with the Y-direction
stiffness being 79% higher than the Z-direction. Compared to the butterfly-inspired designs of Xiao et al. [26], which offered variable stiffness but did not exceed a two-fold anisotropy ratio, our structure demonstrates a tunability of up to 3.5x between the weakest and strongest directions. The inverse-designed sequential metamaterials reported by Han et al. [45] achieved extreme stiffness of up to ~10,000 N/mm, but their design lacked directional tunability. The present DDS design not only reaches comparable stiffness levels (20,292 N/mm in the LS) but also provides directional control, a feature rarely achieved in prior literature.

Energy absorption capacity further highlights the potential of the DDS lattice. The LS absorbs 300 J in the Y-direction, compared with 270 J in the X-direction and 236 J in the Z-direction. By comparison, bio-inspired metamaterials studied by Yan et al. [13] absorbed up to 100 J under comparable loading conditions, while typical auxetic lattices in the literature rarely exceed 50-80 J [22,23]. Thus, the DDS structure demonstrates three- to five-fold higher energy absorption than conventional tunable designs, making it highly attractive for protective systems.

The mechanical properties achieved by the proposed DDS lattice correspond well with the performance requirements reported for several application domains. In exoskeleton systems, vertical load-bearing components typically require directional stiffness values in the range of 10,000-25,000 N/mm to safely transmit user and external loads without excessive deformation, while lateral stiffness is often preferred to remain below 50-60% of the axial stiffness to preserve natural mobility. The lattice developed in this work reaches 20,292 N/mm in the Y direction and significantly lower stiffness in the X and Z directions, aligning with these criteria. In crash-protection structures, effective energy absorption modules commonly target 150-250 J before structural collapse under quasi-static loading; the DDS lattice surpasses this with absorption reaching 300 J in the stiff direction. Orthopedic implants and biomimetic scaffolds, particularly those intended to replace anisotropic tissues such as trabecular bone, generally require stiffness values between 100-2000 N/mm in compliant directions and higher resistance along the principal load axis. The directional contrast demonstrated by the DDS architecture falls within these ranges and offers tunability unavailable in monolithic implant materials. These comparisons demonstrate that the proposed lattice not only achieves high stiffness anisotropy but does so in a manner compatible with real engineering performance targets.

The mechanical failure modes observed during compression testing also reflect the direction-dependent deformation mechanisms embedded within the unit cell geometry. In the Z direction, where bending-dominated deformation is more prominent, local buckling of inclined struts was the primary mode of collapse, leading to gradual load reduction and high energy dissipation without sudden catastrophic fracture. In contrast, loading in the Y direction activated stretching-

dominated paths, causing the structure to carry higher loads but eventually fail through progressive crushing and localized plastic collapse near the main contact surfaces. The X direction demonstrated a mixed failure response, combining moderate lateral buckling with limited shear fracture as both deformation modes were engaged. These observations confirm that the anisotropic geometry not only produces direction-dependent stiffness but also governs the progression of failure, enabling tailored responses ranging from ductile-like controlled collapse to stiffer load-bearing behavior depending on the loading axis.

To further emphasize the novelty of the proposed design, a comparative summary with representative prior works is presented in Table 3. The table highlights how the DDS lattice surpasses existing tunable metamaterials in terms of directional stiffness, anisotropy ratio, and energy absorption capacity.

Such performance is particularly relevant to wearable exoskeletons, where the vertical direction must support the user's body weight or external loads, often exceeding 70-100 kN in industrial applications [20]. The DDS lattice is capable of withstanding these loads without

<figure><img src="image_16.png" /><figcaption>Fig. 12. Illustration of DDS lattice integration in exoskeleton main frames, demonstrating high vertical stiffness for load-bearing and lateral flexibility for natural mobility.</figcaption></figure>

**Table 3**

Comparison of the proposed DDS lattice with representative tunable metamaterials reported in the literature, highlighting differences in stiffness, anisotropy/ tunability, energy absorption, and design limitations.

<table><thead><tr><td>Reference</td><td>Geometry / Approach</td><td>Stiffness (N/mm)</td><td>Anisotropy / Tunability</td><td>Energy Absorption</td><td>Limitations</td></tr></thead><tbody><tr><td>Feng et al. [21]</td><td>Hierarchical honeycomb metastructure</td><td>~500 (normalized modulus ×3.6 increase)</td><td>Single-axis, planar only</td><td>~50-70 J</td><td>Planar geometry, not 3D</td></tr><tr><td>Cheng et al. [22]</td><td>Auxetic re-entrant structure</td><td>~300-400 (single-axis tunability)</td><td>~25% directional variation</td><td>~80 J</td><td>Limited directional control</td></tr><tr><td>Li et al. [23]</td><td>3D lattice with butterfly-shaped perforations</td><td>~600 (single unit cell scale)</td><td>Confined to one axis</td><td>~60-100 J</td><td>Single unit-cell focus, no large lattice</td></tr><tr><td>Zhang et al. [24]</td><td>Tubular auxetic framework</td><td>~700-900 (tunable tubular)</td><td>Single-axis tunability</td><td>~100 J</td><td>Directional constraints</td></tr><tr><td>Xiao et al. [26]</td><td>Butterfly-inspired lattices</td><td>~1000-1200 (variable but limited anisotropy)</td><td>Up to 2 × difference between directions</td><td>≤ 120 J</td><td>Moderate anisotropy only</td></tr><tr><td>Han et al. [45]</td><td>Inverse-designed sequential metamaterials</td><td>~10,000 (extreme stiffness, isotropic)</td><td>No tunability (isotropic)</td><td>~150 J (high stiffness but brittle)</td><td>High stiffness but no DDS</td></tr><tr><td>This study (DDS LS)</td><td>Optimized DDS lattice structure</td><td>11,375 (Z) - 20,292 (Y)</td><td>Up to 3.5 × difference between axes</td><td>236 J (Z) - 300 J (Y)</td><td>Demonstrates validated 3D DDS with high energy absorption</td></tr></tbody></table>
deformation in the Y-axis while offering reduced stiffness in the X- and Z-directions, thereby allowing natural gait and joint motion. This concept is illustrated in Fig. 12, where the DDS lattice is integrated into the main frames of an exoskeleton, highlighting its ability to combine high vertical stiffness with lateral flexibility for enhanced ergonomics.

Beyond exoskeletons, this anisotropic energy absorption could be applied in automotive crash structures, where forces must be dissipated preferentially along safe axes, or in biomedical implants, where mimicking the anisotropic stiffness of bone reduces stress shielding. In civil engineering, the LS could be used in seismic damping systems, where high stiffness ensures structural stability while lateral flexibility and energy dissipation mitigate earthquake-induced vibrations.

## 5. Limitations and future work

Although the proposed DDS metamaterial shows strong tunability and close simulation-experiment agreement, several limitations remain. DLP printing introduces geometric imperfections such as surface roughness, over-curing in thin features, and trapped resin that may affect local stiffness. The ABS-like photopolymer also has brittleness and batch variability that could influence energy absorption and long-term durability. Scaling the design to larger lattices may accumulate printing tolerances and increase post-processing sensitivity. Moreover, this study evaluated only quasi-static loading; dynamic, fatigue, and environmental responses remain unexamined. Future work will explore tougher materials, integrate topology optimization, and develop scalable manufacturing approaches for larger, reliable DDS structures.

The proposed geometry is compatible with diverse materials, including tougher photopolymers, elastomers for damping, and metallic alloys produced via powder bed fusion or binder jetting for high-load applications. While material selection changes absolute stiffness, the underlying DDS mechanisms remain geometry-driven. The unit cell is inherently modular and scalable, with limitations mainly related to print resolution for thin features, dimensional tolerances, and post-processing uniformity. Advances in industrial AM systems, such as higher voxel resolution and multi-material capability will further enhance the feasibility of fabricating large DDS structures for exoskeletons, crash absorbers, and seismic damping components.

## 6. Conclusion

This work presents the first experimentally validated three-dimensional metamaterial that achieves strong direction-dependent stiffness (DDS) through geometry-driven design. The unit cell, derived from a heuristic optimization workflow and fabricated using high-resolution DLP printing, demonstrates highly controlled anisotropy enabled by the selective activation of bending-dominated (mode A) and stretching-dominated (mode B) deformation mechanisms. The experimental results align with FEA predictions within less than 1 % error, confirming both the fidelity of the manufacturing process and the reliability of the numerical framework.

The proposed DDS architecture achieves a stiffness contrast of up to 3.5 × between the stiffest and most compliant directions, and a stiffness increase of 358 % between deformation regions. The lattice also exhibits substantial directional energy absorption, reaching 300 J in the stiffest orientation—three to five times higher than values reported for comparable tunable metamaterials. Importantly, the structure maintains smooth stress distributions without sharp concentration zones, a key requirement for practical deployment in load-bearing or protective systems.

While the present design demonstrates robust DDS behavior, future optimization pathways are abundant. Multi-material additive manufacturing could provide additional freedom to independently tune stiffness, damping, and failure behavior. Likewise, topology refinement or inverse-designed variants of the current unit cell could broaden the achievable anisotropy range or tailor deformation sequencing for

specific applications. Material innovations such as elastomeric or shape-memory resins may also enable adaptive or programmable DDS responses.

More broadly, the design principles and validation framework established here, coupled bending/stretching mechanisms, orientation-specific stiffness control, and seamless FEA-experiment convergence offer a platform that may inspire the next generation of anisotropic mechanical systems. These include soft robotic joints, architected biomedical implants, impact-damping layers with directional response, and multifunctional smart materials capable of reconfigurable stiffness. The demonstrated performance and generality of the approach mark a significant step toward practical, application-ready DDS metamaterials.

## CRediT authorship contribution statement

**Daniel George:** Writing – review & editing, Supervision, Funding acquisition. **Majid Baniassadi:** Writing – review & editing, Validation, Supervision, Project administration. **Ghader Faraji:** Writing – review & editing, Supervision. **Mostafa Baghani:** Writing – review & editing, Supervision, Conceptualization. **Mahdi Khajepour:** Methodology, Investigation, Data curation. **Davood Rahmatabadi:** Writing – review & editing, Supervision, Methodology, Investigation. **Hossein Rahimi:** Writing – original draft, Software, Methodology, Data curation, 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.

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