Abstract

Next-generation protective systems require adaptive materials capable of reconfiguring their response to impact type and severity, thereby offering multiple force–displacement pathways. Here, the study introduces twisting metamaterials, a subclass of architected lattices whose mechanics are captured by micropolar elasticity. Derived from twisting operations on primitive lattices, these structures exhibit geometry-induced torsional actuation and nonlinear responses, enabling adaptive crashworthiness. A multiscale predictive framework—combining Cosserat continuum mechanics, finite element modeling, and experiments—demonstrates its viability. Twisting sheet-based gyroid structures (10% relative density) are additively manufactured in FE7131 steel and tested under quasi-static and dynamic compression with varied torsional constraints, revealing adaptive energy absorption. When rotation is constrained, the structures achieve high axial stiffness (4.8 GPa), collapse stress (21 MPa), and specific energy absorption (15.36 J g−1), while free-to-twist and over-rotation conditions reduce these values by up to 25%, 24%, and 33%, respectively. A macroscale model captures both axial and torsional responses, while SEM and µCT analyses of process-induced defects inform a parametric finite element study extended to 5% and 15% relative densities. Mapping their performance onto an Ashby chart highlights twisting metamaterials as a promising class of mechanically adaptive, crashworthy materials for advanced protection systems in automotive, rail, aerospace, and defence applications.

1 Introduction

The design of mechanical components for engineering applications has evolved significantly in recent years, driven by the demand for enhanced performance and versatility. Modern components must not only fulfill their primary mechanical functions but also meet a broad range of additional engineering requirements, including multifunctionality, ergonomics, sustainability, recyclability, comfort, safety, and compatibility with efficient manufacturing processes.[1, 2] Looking ahead, next-generation components are expected to transcend passive functionality by actively adapting to changing operational conditions and varying severity levels, while continuing to satisfy other engineering criteria.[3, 4] This shift in design philosophy has led to the emergence of active components composed of adaptive materials, capable of dynamically adjusting their mechanical responses to external stimuli such as temperature,[5] light,[6] electromagnetic field,[7] jamming,[8] or interlocking networks.[9]

In the domain of impact absorption, progress in mechanical design is exemplified by innovations in the automotive sector.[10] Early impact absorbers—such as crash boxes and bumpers—provided a single force–displacement response characteristic for all crash scenarios encountered during vehicle collisions. The early design philosophy emphasized material selection for the absorbing structure to strike a balance among competing performance requirements, seeking the optimum achievable compromise.[11] However, the growing need to comply with increasingly stringent and often conflicting safety standards has shifted the focus toward the structural optimization of sacrificial components, advancing the philosophy of mechanical design. For example, in frontal collisions, the front structure of a vehicle must remain sufficiently rigid to prevent intrusion and protect the driver, while also being compliant enough to absorb crash energy and minimize injuries to pedestrians. Optimization efforts have included modifying structure geometry (e.g., shape,[12] geometric gradients,[13] and geometric sweeps[14]) or incorporating heterogeneous materials (e.g., multi-material systems[15-17]) within proven energy absorbers to enhance energy dissipation. Despite their ingenuity, such approaches render the engineering solutions adaptable rather than adaptive, since the force–displacement response remains fixed—albeit optimized—for different impact scenarios.

By contrast, adaptive mechanical structures should respond to impacts of different types and severities by providing multiple force–displacement responses depending on external stimuli.[3, 10] Through external actuation, adaptive structures can actively reconfigure their mechanical behavior to meet specific impact conditions with optimized performance. This trend toward active control aligns with emerging engineering paradigms in real-time environmental monitoring[18] and intelligent decision-making systems that process contextual information and generate targeted mechanical responses.[19, 20] As such, the development of adaptive materials for next-generation protective systems is becoming increasingly essential, fuelling intense research across materials science, mechanics, and manufacturing disciplines.

Recent progress in adaptive materials has largely relied on experimental strategies, employing either bioinspired[21] or rationally designed architectures[22] that alter their mechanical response under external stimuli. However, most adaptive energy-absorbing materials remain unsuitable for demanding engineering applications: soft systems such as hydrogels are mechanically too compliant,[23] others depend on impractical actuation conditions such as intense electromagnetic or thermal fields,[24] and many lack robust mesoscale predictive models.

Recently, Zhao et al.[25] and Zang et al.[26] modulated the force–displacement compression response of modular origami using external torque, demonstrating that torsional actuation provides a practical route for tuning mechanical behavior. Zhao et al. exploited compression–torsion coupling mechanical (CTCM) metamaterials, which intrinsically couple rotational and translational deformations due to their topological layout: two parallel planes connected by axisymmetric inclined struts, such that geometric chirality induces rotation.[27-29] Comparable compression–torsion coupling can be realized using plate-based Kresling origami,[26, 30, 31] 3D chiral metamaterials with more sophisticated shapes,[32-34] or short soft cylindrical shells operating in controllable buckling modes.[35] While these studies introduced compression–torsion coupling through specific geometrical layouts (e.g., chirality, axisymmetric struts, origami folds, or cylindrical shells), here we propose twisting metamaterials as a new subclass of CTCM metamaterials. Twisting metamaterials can be derived by applying a twisting operation to primitive lattice geometries, providing a simple and generalizable design pathway to obtain CTCM metamaterials from generic lattices. As demonstrated by Frenzel et al,[36] all CTCM metamaterials are governed by micropolar elasticity because their complex compression–torsion behavior can be captured using Cosserat continuum theory. This framework treats rotations and displacements independently, enabling the formulation of micropolar mesoscale models that are directly compatible with numerical simulations and supporting design and topological optimization of materials via micropolar constitutive parameters tailored to specific engineering needs.[37-39] Therefore, Cosserat continuum mechanics offers a robust theoretical foundation to demonstrate the adaptive energy absorption performance of twisting metamaterials when actuation involves rotational motion, a practically implementable mechanical stimulus.

This study advances metamaterials research toward impact engineering by establishing twisting metamaterials as adaptive crashworthy materials through a predictive framework that integrates Cosserat continuum mechanics, finite element modeling, and experimental validation. Adaptive energy absorption is governed by torsional actuation, external torque, or rotational inertia, positioning twisting metamaterials as an engineering solution for next-generation protection systems. The adaptivity concept is first formulated within the Cosserat continuum framework, focusing on axial stiffness, initial collapse stress, and specific energy absorption of a micropolar elastic material subjected to externally controllable torque, introducing a torque ratio as a key control parameter. The adaptive crashworthiness of twisting metamaterials based on gyroid structures is then demonstrated under both quasi-static and dynamic compression, with direct comparison to primitive (untwisted) gyroid counterparts.[40] Three torsional boundary conditions were imposed—locking, free-to-twist, and over-rotated—so that the experimental setup enabled simultaneous measurement of axial (force–displacement) and torsional (torque–rotation) responses, even under dynamic loading. Both primitive and twisting gyroid structures were additively manufactured via powder bed fusion with 10% relative density using FE7131 steel. To extend understanding beyond this density, a parametric finite element study was conducted across 5–15% relative densities. Validation of the macroscale model required constitutive characterization of the parent material; therefore, bulk FE7131 properties were determined under quasi-static and dynamic conditions, modelled using the Johnson–Cook constitutive framework, and supported by microstructural, microhardness, and crystallographic analyses of both bulk and gyroid-sheet specimens. For accurate numerical–experimental alignment, geometric imperfections introduced by additive manufacturing were quantified by comparing CAD geometries with micro-CT reconstructions of the printed structures. Finally, the adaptive energy absorption performance of the twisting gyroid was mapped onto an Ashby chart and generalized through an ad hoc predictive scaling law, thereby extending classical cellular materials theory to incorporate adaptive effects.

2 Mechanics of Twisting Metamaterials

2.1 Cosserat Continuum Mechanics

Cosserat continuum theory, also known as the micropolar continuum, extends classical Cauchy elasticity by introducing independent rotational degrees of freedom at each material point, enabling a more complete representation of material behavior, including couple stresses and microstructural effects.[36] Cosserat theory introduces the microrotation field in addition to the displacement one, defining the micropolar strain tensor ( ε ${\varepsilon}$ ij) and curvature tensor (kij). These tensors are usually expressed component-wise using Eringen’s notation[41] as follows.

ε i j = ∂ u j ∂ x i + ∈ j i k ϕ k $$\begin{equation};{\varepsilon _{ij}} = \frac{{\partial {u_j}}}{{\partial {x_i}}} + { \in _{jik}}{\phi _k}\end{equation}$$ (1)

k i j = ∂ ϕ j ∂ x i $$\begin{equation};{k_{ij}} = \frac{{\partial {\phi _j}}}{{\partial {x_i}}};\end{equation}$$ (2)

where ϕi and ui are the components of the microrotation and displacement fields, respectively, while ∈jik represents the Levi-Civita tensor. Cosserat continuum theory introduces two distinct stress tensors: the force stress tensor (σij), analogous to the classical Cauchy stress tensor, and the couple stress tensor (mij) to account for local moment effects. The constitutive equation for the force and couple stresses can be derived from the strain energy density. For a physically linear micropolar-elastic material,[42] the strain energy density ( L $\mathcal{L}$ ) in elastic regime takes a quadratic form in terms of the micropolar strain and curvature tensors:

L = 1 2 ε i j C i j k l ε k l + ε i j D i j k l k k l + 1 2 k i j A i j k l k k l $$\begin{equation}\mathcal{L} = \frac{1}{2}{\varepsilon _{ij}}{C_{ijkl}}{\varepsilon _{kl}} + {\varepsilon _{ij}}{D_{ijkl}}{k_{kl}} + \frac{1}{2}{k_{ij}}{A_{ijkl}}{k_{kl}}\end{equation}$$ (3)

where Aijkl and Cijkl are 4th order elasticity tensors with symmetry properties: Cijkl = Cklij and Aijkl = Aklij . Dijkl is a rank-four pseudo-tensor that reverses its sign under space inversion, responsible for coupling strain and curvature, and capable of predicting material handedness due to architecture chirality.[37] The energy density in Equation (3) contains 171 independent material parameters since both ε ${\varepsilon}$ ij and kij are generally non-symmetric tensors. However, material symmetries, such as isotropy, greatly reduce the number of independent elastic constants. For instance, an isotropic micropolar-elastic material requires only six independent elastic constants.[42]

The constitutive equations for the force stress tensor (σij) and couple stress tensor (mij) are derived by differentiating the strain energy density ( L $\mathcal{L}$ ) in Equation (3) with respect to micropolar strain and curvature tensors:

σ i j = ∂ L ∂ ε i j = C i j k l ε k l + D i j k l k k l $$\begin{equation}{\sigma _{ij}} = {{\partial {\cal L}} \over {\partial {\varepsilon _{ij}}}}; = {C_{ijkl}};{\varepsilon _{kl}} + {D_{ijkl}}{k_{kl}}\end{equation}$$ (4)

m i j = ∂ L ∂ k i j = A i j k l k k l + D k l i j ε k l $$\begin{equation};{m_{ij}} = \frac{{\partial \mathcal{L}}}{{\partial {k_{ij}}}} = {A_{ijkl}};{k_{kl}} + {D_{klij}}{\varepsilon _{kl}}\end{equation}$$ (5)

In the special case, where Cijkl = Cklij = Cjikl and the coupling tensors vanish, i.e., Dijkl = Aijkl = 0, the Cosserat continuum theory reduces to the traditional Cauchy continuum theory. Therefore, the Cosserat theory offers a generalized continuum framework capable of capturing critical material features where traditional Cauchy theory falls short, such as material size-dependency,[36] chirality,[43] and anisotropy.[44] Cosserat theory can also pave the way for the development of innovative materials with advanced mechanical properties, as illustrated by this study’s focus on adaptive energy-absorbing materials.

2.2 Energy Absorption Capability of Twisting Metamaterial

The crashworthiness of materials is typically assessed under uniaxial compression because it is directly linked to the force transmitted to the component to protect and the amount of energy dissipated during an impact load. This implies focusing on key mechanical properties such as axial stiffness, initial collapse stress, and specific energy absorption. A twisting metamaterial exhibits force stress coupled with curvature under compression load, exhibiting the behavior of a micropolar-elastic material within the Cosserat continuum framework. To simplify the notation in uniaxial compression, repeated indices were replaced with a single subscript “1”, which corresponds to the loading direction, simplifying the expressions for force stress (σ1) and couple stress (m1) as follows: σ1 = C1 ε ${\varepsilon}$ 1 + D1k1 and m1 = A1 k1 + D1 ε ${\varepsilon}$ 1. Consequently, the force stress of micropolar-elastic material was rearranged from Equation (4) by incorporating the couple stress from Equation (5) as follows:

σ 1 = C 1 ε 1 + D 1 A 1 m 1 − D 1 ε 1 $$\begin{equation}{\sigma _1} = {C_1};{\varepsilon _1} + {{{D_1}} \over {{A_1}}}\left( {{m_1} - {D_1}{\varepsilon _1}} \right)\end{equation}$$ (6)

The Equation (6) shows that the force stress depends on both constitutive parameters C1,A1, and D1 and couple stress aligned with the axial direction (m1): a controllable external torque (M) applied over the micropolar-elastic material cross-section (A0). In the linear elastic regime, the external torque (M) applied to the material can be linearly approximated as follows:

m 1 = M A 0 = η ε 1 $$\begin{equation};{m_1} = \frac{M}{{{A_0}}} = \eta {\varepsilon _1}\end{equation}$$ (7)

where η determines how much torque is exerted to the material during the axial deformation. Therefore, the force stress in Equation (6) was rewritten using Equation (7) as follows:

σ 1 = C 1 + D 1 A 1 η − D 1 ε 1 $$\begin{equation};{\sigma _1} = \left[ {{C_1} + \frac{{{D_1}}}{{{A_1}}}\left( {\eta - {D_1}} \right)} \right];{\varepsilon _1}\end{equation}$$ (8)

Interestingly, preventing the micropolar-elastic material rotations during axial compression means the curvature (k1) becomes zero because of an external torque (ML) which is not constant, as inferable by Equation (5):

M L A 0 = η ε 1 = D 1 ε 1 $$\begin{equation}{{{M_L}} \over {{A_0}}} = ;\eta ;{\varepsilon _1} = {D_1};{\varepsilon _1}\end{equation}$$ (9)

That specific boundary condition allows introducing the dimensionless parameter M = M M L = η D 1 $\mathcal{M} = \frac{M}{{{M_L}}} = \frac{\eta }{{{D_1}}};$ , called torque ratio, capable of identifying the torsional configurations induced by external torques. For a better understanding, Figure 1 summarizes how variations in the torque ratio affect the torque directions, rotations, and torsional loading configurations in counterclockwise (CCW) twisting metamaterials: right-handed micropolar metamaterial (positive C1,A1, and D1 constitutive parameters).

Influence of torque ratio ( M $\mathcal{M}$ ) on the torque directions, rotations, and torsional loading configurations of CCW twisting metamaterial (right-handed micropolar metamaterial).

Notably, for M < 0 $\mathcal{M} < 0$ the twisting metamaterial experiences over-rotation due to an external positive torque (driving torque) aligned with the twisting direction (CCW). On the contrary, for M > 1 $\mathcal{M} > 1$ , the twisting metamaterial experiences over-rotation in the opposite direction, clockwise (CW) direction, caused by an external negative torque (driving torque). 0 < M < 1 $0 < \mathcal{M} < 1$ is a peculiar range of twisting metamaterial delimitated by two specific torsional configurations: locking ( M = 1 ) $\mathcal{M} = 1)$ and free-to-twist ( M = 0 ) $\mathcal{M} = 0)$ conditions. The external torque within the range 0 < M < 1 $0 < \mathcal{M} < 1$ counters the material rotation induced by the geometric twisting. As a result, the twisting metamaterial experiences under-rotation in the twisting direction (CCW), with the torque reversing direction and acting as a resistive torque. It is noteworthy to highlight that the range 0 < M < 1 $0 < ;\mathcal{M} < 1$ is not exhibited by cellular materials governed by classical Cauchy continuum mechanics because the locking ( M = 1 ) $\mathcal{M} = 1)$ and free-to-twist ( M = 0 ) $\mathcal{M} = 0)$ conditions coincide. Additional details based on strain energy density were provided in Section S1 (Supporting Information).

Based on the torque ratio parameter, the Equation (8) was further rearranged to highlight the influence of external torque on the force stress of micropolar-elastic material:

σ 1 = C 1 + D 1 2 A 1 M − 1 ε 1 $$\begin{equation}{\sigma _1} = \left[ {{C_1} + {{{D_1}^2} \over {{A_1}}}\left( {{\cal M} - 1} \right)} \right];{\varepsilon _1}\end{equation}$$ (10)

Thus, the force stress defined in Equation (10) was substituted into the strain energy density expression, Equation (3), obtaining the micropolar-elastic energy density for uniaxial compression in elastic regime:

L = 1 2 C 1 + D 1 2 A 1 M 2 − 1 ε 1 2 = A 1 2 C 1 A 1 + D 1 2 M 2 − 1 C 1 A 1 + D 1 2 M − 1 2 σ 1 2 $$\begin{equation}\mathcal{L} = \frac{1}{2}\left[ {{C_1} + \frac{{{D_1}^2}}{{{A_1}}}\left( {{\mathcal{M}^2} - 1} \right)} \right];{\varepsilon _1}^2 = \frac{{{A_1}}}{2}\left[ {\frac{{{C_1}{A_1} + {D_1}^2\left( {{\mathcal{M}^2} - 1} \right)}}{{{{\left( {{C_1}{A_1} + {D_1}^2\left( {\mathcal{M} - 1} \right)} \right)}^2}}}} \right]{\sigma _1}^2\end{equation}$$ (11)

If micropolar-elastic materials were twisting metamaterial designed by applying a twisting operation on the primitive architected materials, they behave like primitive one under compression, exhibiting the characteristic compressive response composed by elastic regime, flat plateau in post-collapse regime, and densification. Therefore, the adaptive energy absorption capability of twisting metamaterial can be inferred by evaluating the initial collapse stress (σP) of micropolar-elastic material because the densification strain ( ε ${\varepsilon}$ d) is only dependent on material relative density. Assuming the initial collapse stress (σp) occurs once the material reaches a critical energy density value ( L c ${\mathcal{L}_c}$ ) in elastic regime, σp can be expressed as function of torque ratio ( M $\mathcal{M}$ ) using Equation (11) as follows:

L c = A 1 2 C 1 A 1 + D 1 2 M 2 − 1 C 1 A 1 + D 1 2 M − 1 2 σ p 2 $$\begin{equation}{{\cal L}_c} = ;{{{A_1}} \over 2}\left[ {{{{C_1}{A_1} + {D_1}^2\left( {{{\cal M}^2} - 1} \right)} \over {{{\left( {{C_1}{A_1} + {D_1}^2\left( {{\cal M} - 1} \right)} \right)}^2}}}} \right]{\sigma _p}^2\end{equation}$$ (12)

By selecting the locking condition ( M = 1 $\mathcal{M} = 1$ ) as the reference initial collapse stress (σPL), the influence of external torsional loading on the σp of micropolar-elastic materials was derived from Equation (12) as follows:

σ p σ P L = C 1 A 1 + D 1 2 M − 1 2 C 1 A 1 C 1 A 1 + D 1 2 M 2 − 1 $$\begin{equation}{{{\sigma _p}} \over {{\sigma _{PL}}}} = \sqrt {{{{{\left( {{C_1}{A_1} + {D_1}^2\left( {{\cal M} - 1} \right)} \right)}^2}} \over {;{C_1}{A_1}\left( {{C_1}{A_1} + {D_1}^2\left( {{{\cal M}^2} - 1} \right)} \right)}}} ;\end{equation}$$ (13)

σ P σ P L $\frac{{{\sigma _P}}}{{{\sigma _{PL}}}}$ reaches the maximum values when M = 1 $\mathcal{M} = 1$ and decreases applying a driving torque ( M > 1 , M < 0 $\mathcal{M} > 1, \mathcal{M} < 0$ ) or a resistive torque 0 < M < 1 $0 < \mathcal{M} < 1$ to the material.

Figure 2 summarizes the torque ratio ( M ) $( \mathcal{M} )$ influences on axial stiffness, initial collapse stress, and energy absorption capability of micropolar-elastic material, in which the mechanical performances are normalized to locking ( M = 1 ) $\mathcal{M} = 1)$ condition. The axial stiffness, obtained from the Equation (10) as ∂σ1/∂ ε ${\varepsilon}$ 1, linearly depends on the external torque applied during axial loadings, decreasing as the torque is aligned with the twisting direction, as shown in Figure 2a. The initial collapse stress reaches the maximum values at M = 1 $\mathcal{M} = 1$ , as show in Figure 2b, thus the characteristic compressive response exhibits a flat plateau in post-collapse regime which reduces as torque ratio decreases, as illustrated in Figure 2c where the stress-strain curves correspond to different torsional loading configurations: black for M = 1 $\mathcal{M} = 1$ , red for M = 0 $\mathcal{M} = 0$ , and blue for M < 0 $\mathcal{M} < 0$ .

Influence of torque ratio ( M $\mathcal{M}$ ) on the mechanical properties and energy absorption capability of CCW twisting metamaterial (right-handed micropolar metamaterial). a) Axial stiffness. b) Initial collapse stress. c) Axial stress-strain curve. d) Energy absorption capability. Black, red, and blue dots and lines correspond to different torsional loading configurations: M = 1 $\mathcal{M} = 1$ , M = 0 $\mathcal{M} = 0$ , and M < 0 $\mathcal{M} < 0$ , respectively.

Therefore, twisting metamaterials exhibit the highest initial collapse stress and energy absorption values when their rotations are fully restricted ( M = 1 $\mathcal{M} = 1$ ), while the application of an external resistive or driving torque results in decreasing energy absorption performances. The analytical description leads to the inference that the twisting metamaterials adapt their stiffness, collapse stress, and energy absorption capabilities in response to torsional constraints, actively controllable. Indeed, the stress-strain curves can also be integrated to yield energy absorption diagram, Figure 2d, to visualize the absorption capacity area covered by the effect of torque ratio on normalized energy absorption capacity E A = 1 σ P L ∫ 0 ε σ d ε $EA = \frac{1}{{{\sigma _{PL}}}}\mathop \smallint \nolimits_0^\varepsilon \sigma ;d\varepsilon $ . A peculiar advantage of twisting metamaterials lies in the operational range 0 < M < 1 $0 < ;\mathcal{M} < 1$ . From an engineering perspective, the endpoints of this range can be practically realized using a rotational base located between the material and the component to protect. Equipping the rotational base with locking ( M = 1 $\mathcal{M} = 1$ ) and unlocking ( M = 0 $\mathcal{M} = 0$ ) rotations device, these two configurations can be readily implemented. Furthermore, it is worth noting that operating within the 0 < M < 1 $0 < ;\mathcal{M} < 1$ regime necessitates the application of a resistive torque, which can be induced through braking mechanisms or rotational inertia devices, such as flywheels. Therefore, twisting metamaterials offer new functional designs for energy harvesting applications where impact energy can be partially converted into rotational kinetic energy, stored and, later, released through torsional motion.

3 Results and Discussion

3.1 Mechanical Response of Twisting Gyroid Structures

3.1.1 Quasi-Static Compression

The compressive response of both primitive and twisting gyroid structures with a relative density ρ ¯ $\bar \rho $ = 10% was investigated under quasi-static loading. Figure 3 summarizes their corresponding macroscopic compressive stress–strain responses, deformation and failure modes, captured at three distinct torsional boundary conditions: locking condition “TwGyL” ( M = 1 $\mathcal{M} = 1$ ), free-to-twist condition “TwGyF” ( M = 0 $\mathcal{M} = 0$ ), and rotative condition “TwGyR” ( M = − 1 $\mathcal{M} = - 1$ ). Each stress–strain curve for the gyroid structures exhibited an initial linear-elastic region, followed by a sudden stress drop, a plateau regime characterized by post-collapse stress oscillations, and a final densification phase. The collapse mechanism was dominated by yielding and widespread plastic deformation, leading to the formation of macroscopic crush bands. Following the first crush band (i.e., the initial collapse stress), the ductility of the gyroid-sheet material enabled a stable compressive response without catastrophic failure, resulting in a progressive, band-by-band collapse approaching full densification (see Videos S2 and S3, Supporting Information).

Quasi-static and dynamic compression behavior of gyroid structures with ρ ¯ $\bar \rho $ = 10%. a) Characteristic engineering stress-strain responses and b) deformation maps at various stages, including collapse modes, rotations, and torsional constraints. The torsional constraints are defined by “TwGyF” ( M = 0 $\mathcal{M} = 0$ ), “TwGyL” ( M = 1 $\mathcal{M} = 1$ ), and “TwGyR” ( M = − 1 $\mathcal{M} = - 1$ ) loading configurations.

The rotational behavior of the twisting gyroid structures is illustrated in Figure 3b by connecting a dotted white line between two reference points located at the upper and lower boundaries of the structures. During compression, this line inclines, visually indicating the degree of rotation in the vertical plane. It remains vertical when no rotation occurs—either due to constrained motion, as in the “TwGyL” ( M = 1 $\mathcal{M} = 1$ ) configuration, or due to geometric restrictions, as in the “Gy” configuration. As the inclination increases, it reflects enhanced rotation, most notably in the “TwGyF” ( M = 0 $\mathcal{M} = 0$ ) configuration (ϕ≅28° at ε ${\varepsilon}$ = 0.6), and even more significantly in the “TwGyR” ( M = − 1 $\mathcal{M} = - 1$ ) configuration (ϕ≅90° at ε ${\varepsilon}$ = 0.6).

The compressive mechanical response of the twisting gyroid structures was strongly influenced by the applied torsional boundary conditions, as illustrated by stress–strain curves in Figure 3a. By evaluating the slope of the initial elastic regime (defined as Ep = ∂σ1/∂ ε ${\varepsilon}$ 1 ), it is evident that the axial stiffness of “TwGy” structures declines as the torque ratio M $\mathcal{M}$ decreases. Specifically, the “TwGyL” ( M = 1 $\mathcal{M} = 1$ ) configuration exhibited the highest axial stiffness (4.8 GPa), while this value dropped to 4.1 GPa under free-to-rotate conditions (“TwGyF”, M = 0 $\mathcal{M} = 0$ ), and further to 3.6 GPa in the over-constrained counterclockwise case (“TwGyR”, M = − 1 $\mathcal{M} = - 1$ ). These elastic responses under various torsional constraints were accurately modeled using a mesoscale micropolar-elastic material formulation grounded in Cosserat continuum mechanics. The loading cases used for calibrating the constitute parameters are summarized in Table S1 (Supporting Information), employing Equations (4) and (5). From the experimental results, the elastic constants were determined as: C1 = 4812 MPa, D1 = 2.11 N/mm, and A1 = 0.0074 N/rad. All constants were positive, confirming that the twisting gyroid behaves as a right-handed micropolar metamaterial.

During the plateau regime, the initial collapse stresses (σp) and post-collapse stress fluctuations were also affected by the torsional boundary conditions, even if the collapse mechanism was the same: crush bands triggered by local yielding. Both σp and post-collapse stress plateau of “TwGy” structure reduces as the torque ratio ( M $\mathcal{M}$ ) reduces. Notably, the σp decreases from a maximum value of 21 MPa in “TwGyL” configuration to a minimum of 16 MPa in “TwGyR” configuration. The theoretical collapse stress values can be defined by Cosserat continuum theory using Equation (13) and the already calibrated constitutive parameters C1, D1, and A1.

Table 1 shows that the theoretical values are consistent with experimental ones, confirming that the mechanical properties of twisting gyroid structure under different torsional loading ( − 1 < M < 1 $ - 1 < \mathcal{M} < 1$ ) can be effectively predicted with the mesoscale micropolar-elastic material model in Cosserat continuum mechanics. The energy absorption performance was experimentally evaluated by comparing the onset of densification strain ( ε ${\varepsilon}$ d), the initial collapse stresses (σp), the energy absorption capacity ( W = ∫ 0 ε d σ d ε $W = \mathop \smallint \nolimits_0^{{\varepsilon _d}} \sigma ;d\varepsilon $ ), and the specific energy absorption ( S E A = W ρ $SEA = \frac{W}{\rho };$ ) where ρ is the average density of the unit-cell of gyroid structures. The SEA computed at the onset of densification, identified with the maximum efficiency procedure, takes the name SEAd. Table 1 shows that the SEAd of twisting gyroid structure with ρ ¯ = 10 % $\bar \rho = 10{\mathrm{% }}$ reduces as the torque ratio reduces. The “TwGy” structure exhibited a maximum value of 15.36 J/g in “TwGyL” configuration and decreased up to 10.32 J/g in “TwGyR” configuration. Therefore, “TwGy” demonstrates adaptive energy absorption properties because of external torsional conditions. Assuming the free-to-twist condition ( M = 0 $\mathcal{M} = 0$ ) as neutral position, the “TwGy” structure changes SEAd from -26.2% to +9.3% into the operation range − 1 < M < 1 $ - 1 < \mathcal{M} < 1$ , respectively.

Table 1. Theoretical, experimental, and numerical values of quasi-static mechanical and energy absorption performances of gyroid structures.

Sample Code Unit cell topology M $\mathcal{M};$ [−] Ep [GPa] σp [MPa] ε ${\varepsilon}$ d [mm mm−1] SEAd [J g−1] Exp. Gy Gyroid 1÷0 5.4 ± 0.28 24.22 ± 0.21 0.54 ± 0.01 15.41 ± 0.32 TwGyL Twist Gy. 1 4.8 ± 0.30 21.02 ± 0.25 0.55 ± 0.01 15.36 ± 0.35 TwGyF Twist Gy. 0 4.1 ± 0.31 19.52 ± 0.31 0.54 ± 0.01 14.05 ± 0.29 TwGyR Twist Gy. −1 3.6 ± 0.33 16.25 ± 0.28 0.53 ± 0.01 10.32 ± 0.23 FEA Gy Gyroid 1÷0 5.5 24.05 0.61 15.71 TwGyL Twist Gy. 1 4.9 21.80 0.61 15.64 TwGyF Twist Gy. 0 4.2 19.85 0.60 14.46 TwGyR Twist Gy. −1 3.7 16.53 0.59 11.01 Theory Gy Gyroid 1÷0 – – – – TwGyL Twist Gy. 1 4.8a) 21.02b) – 15.41c) TwGyF Twist Gy. 0 4.2a) 19.65b) – 14.21c) TwGyR Twist Gy. −1 3.6a) 16.60b) – 10.57c)

Comparing the primitive and twisting gyroid structure mechanical properties (Table 1), the “Gy” structure shows high stiffness and initial collapse stress compared to “TwGy”, whereas the energy absorption capabilities were comparable in “TwGyL” configuration ( M = 1 $\mathcal{M} = 1$ ). That happens because the post-collapse stresses of both structures were similar, as well as the onset of densification strains, as shown in Figure 3, inferring that ε ${\varepsilon}$ d is only function of relative density. Decreasing the torque ratio, the ε ${\varepsilon}$ d slightly decreases as shown in Table 1.

Further analysis through energy absorption diagrams (see Figure 4) revealed that the “TwGy” structure displayed a shaded region representing attainable SEA values within the operational torque range. The blue-shaded region corresponds to scenarios with externally applied driving torque, while the red-shaded region reflects SEA under resistive torque. This visual clearly demonstrates the twisting metamaterial’s capability to modulate both force transmission and energy dissipation under impact loads. In high-energy dissipation applications, the structure should be torsionally fixed to maximize absorption, while in load-limiting applications, free rotation can be permitted to reduce transmitted force. It is noteworthy that external torque can also be applied to primitive gyroid structures, generating a corresponding shaded SEA region in Figure 4a. However, in this case, only the blue-shaded region applies, as the free-to-twist and locked states coincide due to structural symmetry. The red-shaded region is a distinguishing feature exclusive to twisting gyroid structures—or more broadly, to twisting metamaterials.

Energy absorption diagrams of primitive and twisting gyroid structures with relative density ρ ¯ = 10 % $\bar \rho = 10% $ . a) Effect of external torque on energy absorption under quasi-static loading. b) Specific energy absorption at the onset of densification (SEAd) measured under quasi-static and dynamic conditions. Torsional boundary conditions are defined as “TwGyF” ( M = 0 $\mathcal{M} = 0$ ), “TwGyL” ( M = 1 $\mathcal{M} = 1$ ), and “TwGyR” ( M = − 1 $\mathcal{M} = - 1$ ).

3.1.2 Dynamic Compression

The compression behavior of both primitive and twisting gyroid structures with two different torsional boundary conditions was evaluated at high strain rates. The macroscopic stress–strain responses of gyroid structures with a relative density of ρ ¯ $\bar \rho $ = 10% at an engineering strain rate of 500 s−1 are shown in Figure 3a. These dynamic stress–strain curves closely follow the quasi-static behavior, with a slightly more pronounced stress drop post-collapse. The strain rate sensitivity of the parent material did not significantly affect the initial collapse stress of the gyroid structures. Notably, the twisting gyroid structure continued to rotate freely under high-speed impact conditions (≈6 m s−1). Experimental results confirm that the distinctive rotational behavior of twisting metamaterials is retained under dynamic loading (see Video S1, Supporting Information). The SEAd values measured under dynamic loading are consistent with those obtained under quasi-static conditions, as illustrated in Figure 4b. This confirms that the adaptive energy absorption behavior of the twisting gyroid structure is preserved at high strain rates. Figure 4b also highlights that the twisting gyroid structure in the “TwGyL” configuration exhibits a lower initial collapse stress than the “Gy” configuration, despite their comparable SEAd values. Compared to primitive gyroid structure, the twisting metamaterial reduces the peak force transmitted to the protected component while maintaining equivalent energy absorption. This result demonstrates that incorporating geometric twisting into the design of architected materials can significantly enhance energy absorption performance and protective efficiency relative to untwisted structures.

3.2 Bulk and Gyroid-Sheet Material Characterization

The quasi-static and dynamic compression tests on the FE7131 bulk samples revealed high strength and ductility, as shown by the true stress-strain curve in Figure S1 (Supporting Information). Under quasi-static condition, the compressive yield stress, estimated as σ0.2%, was ≈780 MPa followed by strain hardening, which led to a stable compression stress of ≈1110 MPa at large deformation; no failures were exhibited. Increasing the strain rate up to 1000 s−1, the bulk material showed a moderate strain rate sensitivity, as σ0.2% increased to 850 MPa. The Young’s modulus was ≈195 GPa, determined via Digital Image Correlation (DIC) analysis. The mechanical behavior was implemented into Abaqus environment using the Johnson-Cook material model; hence, strain rate sensitivity and thermal softening of the gyroid-sheet material were accounted for in FE simulations of gyroid structures. The Johnson-Cook model was calibrated following the procedure described in previous work;[45] its parameters are listed in Table S2 (Supporting Information). The results of EDS microanalysis are reported in Table S3 (Supporting Information), confirming consistent chemical composition between bulk and gyroid-sheet material. The XRD patterns in Figure S2 (Supporting Information) revealed peaks corresponding to the α-Fe (ferrite) body-centred cubic (bcc) phase (ICDD 65–4899) with nominal lattice parameter a = 2.8670 nm. Peak broadening analysis, summarized in Table S4 (Supporting Information), shows a slightly finer crystallite size in gyroid-sheet material than bulk material, attributed to the faster cooling rates in PBF-LB manufacturing of gyroid structure. Both lattice parameters were consistent with the nominal value (Δa/anom < 0.25%), demonstrating minimal crystallography variations and consistent fully ferritic microstructure of bulk and gyroid-sheet material. XµCT analysis quantified the percent of closed porosity (CP) equal to 0.5% ± 0.1%, and 0.12 ± 0.04% were found for the bulk and gyroid-sheet materials, respectively. However, even if the gyroid-sheet material displayed a denser microstructure, both materials exceeded 99% relative density, demonstrating high structural integrity. The Vicker microhardness (HV) measurements in three different areas of gyroid-sheet material (lines S1-316HV, S2-330HV, and S3-324HV in Figure S3, Supporting Information) demonstrated homogeneity in mechanical response across the twisting gyroid structure. Despite the slight microstructural differences between gyroid-sheet and bulk material due to the manufacturing process, their mechanical properties can be considered equivalent for the purposes of this study and homogeneously distributed, confirming the Johnson-Cook model implementation into the macroscale model.

3.3 Predicting Mechanical Response of Twisting Gyroid Structures

3.3.1 Effect of AM-Induced Shape Variations

In additively manufactured architected materials, validating numerical models requires accurate information about the printed geometry, as the AM process typically introduces defects such as geometric and density variations, lack of fusion, internal porosity, trapped material, and other imperfections.[46] Therefore, the numerical model of real 3D-printed twisting gyroid was reconstructed from CT analysis and then used for a preliminary FE analysis.

CT analysis of the 3D-printed twisting gyroid structure revealed no porosity within the gyroid walls and only minor geometric deviations compared to the CAD model, as shown in Figure 5a where the CT scan is overlaid over the CAD model. Figure 5a highlights excess material deposited along the gyroid waves down skins due to the additive manufacturing process, which altered the geometry and increased the relative density to ≈11.8%. Therefore, the CT-scan model was meshed (Figure 5b) and used in preliminary FE simulations to account for the effect of the AM process on the effective geometry of the twisting gyroid structure in the “TwGyL” configuration. The stress-strain responses for the CT-scan and CAD models were numerically predicted. The CT-based model exhibited a mechanical response 50% higher than the experimental results, as shown in Figure 5c, whereas CAD model closely aligned with the experimental data. This indicates that the CAD-based predictions more accurately represent the actual mechanical response of the twisting gyroid structure, despite the CT-scan model having a higher relative density because of manufacturing-induced shape variations.

Effect of AM-induced shape variations on the twisting gyroid structure. a) CT-scan model superimposed on CAD geometry. b) Mesh generated from CT-scan model. c) Experimental versus numerical compression response for the “TwGyL” ( M = 1 $\mathcal{M} = 1$ ) configuration. d,e) SEM images of gyroid wave surfaces. f) Area ratios along the build direction from CAD and CT-scan models. g) Segmented layers of CAD and CT-scan models. h) Comparison of CT and CAD geometry beneath a gyroid wave.

To investigate the underlying cause, a layer-by-layer comparison between the CT-scan and CAD models was performed. Segmenting layers of both models, as shown in Figure 5, the cross-sectional area (A) was measured and normalized by the reference area (A0). The resulting area ratios (A/A0) were plotted along the vertical direction of the gyroid structure (Z-direction of the 3D printer) in Figure 5f, allowing the localization of excess material, as further illustrated by SEM images of gyroid waves down skins in Figure 5c,d. Significant differences in area ratios were observed beneath the gyroid wave connections, primarily due to residual powder adhering to inclined surfaces and critical unsupported regions, as illustrated in Figure 5h. The area ratios deviation can be attributed to manufacturing-induced shape variations because of trapped metal powders beneath the gyroid wave connections, which are common imperfections in AM structures, known as overhang and bridging issues. In contrast, the area ratios above the gyroid wave connections matched between the CT-scan and CAD models, indicating that the gyroid wall thickness remained consistent with the original design. As a consequence, the excess material observed beneath the connections was residual powder, which does not contribute to mechanical deformation, as supported by experimental/numerical findings from previous study[47] and other researchers.[48, 49] Thus, despite the higher relative density and minor shape variations in the CT-scan model of 3D-printed gyroid, the mechanical response of the twisting gyroid structure can be accurately predicted using the CAD model. This is because the structural regions of 3D printed gyroid are fully solid and geometrically consistent with the CAD model, whereas the excessive material (residual, partially sintered powders) merely increases the apparent density of the gyroid without contributing to its mechanical performance.

3.3.2 FE Validation

Using the CAD models, numerical predictions of both primitive and twisting gyroid structures ( ρ ¯ = 10 % $\bar \rho = 10% $ ) with distinct torsional boundary conditions are compared with the experimental results in Figure 6 (see Videos S2 and S4, Supporting Information). The comparison shows characteristic engineering stress–strain curves, deformation maps, and rotations at various stages.

Experimental versus FE-predicted quasi-static compression behavior of gyroid structures with relative density ρ ¯ $\bar \rho $ = 10%. a) Engineering stress–strain responses. b) Deformation maps at different stages, highlighting collapse modes, rotations, and torsional constraints. Torsional boundary conditions are defined as “TwGyF” ( M = 0 $\mathcal{M} = 0$ ), “TwGyL” ( M = 1 $\mathcal{M} = 1$ ), and “TwGyR” ( M = − 1 $\mathcal{M} = - 1$ ).

The numerical models effectively capture the compressive behavior of the gyroid structures, accurately predicting the onset of the initial crush band and the progressive post-collapse response up to near-complete densification. Additional details of collapse mechanism associated with the crush band are provided in Section S2 (Supporting Information). The predicted mechanical properties and energy absorption characteristics are summarized in Table 1, demonstrating the accuracy of the finite element (FE) model in predicting axial stiffness, initial collapse stress, and specific energy absorption (SEAd).

The validation of the twisting gyroid structure’s mechanical behavior using FE simulations extends beyond conventional stress–strain curves to include rotational responses (