Design framework for programmable three-dimensional woven metamaterials

Domain: physics
Published: 2026-01-26
Editorial outcome: minor_revision

Abstract
Mechanical metamaterials have continued to offer unprecedented tunability in mechanical properties, but most designs to date have prioritized attaining high stiffness and strength while sacrificing deformability. The emergence of woven lattices—three-dimensional networks of entangled fibers—has introduced a pathway to the largely overlooked compliant and stretchable regime of metamaterials. However, the design and implementation of these complex architectures has remained a primarily manual process, restricting identification and validation of their full achievable design and property space. Here, we present a geometric design framework that encodes woven topology using a graph structure, enabling the creation of woven lattices with tunable architectures, functional gradients, and arbitrary heterogeneity. Through use of microscale in situ tension experiments and computational mechanics models, we reveal highly tunable anisotropic stiffness (varying by over an order of magnitude) and extreme stretchability (up to a stretch of four) within the design space produced by the framework. Moreover, we demonstrate the ability of woven metamaterials to exhibit programmable failure patterns by leveraging tunability in the design process. This framework provides a design and modeling toolbox to access this previously unattainable high-compliance regime of mechanical metamaterials, enabling programmable large-deformation, nonlinear responses.

Introduction
The advancement of additive manufacturing technologies has solved many fabrication challenges, enabling the ability to print complex internal structures across length scales or the possibility to print highly compliant and slender three-dimensional (3D) structures. These new possibilities have driven interest in the development of increasingly complex 3D architected materials—or mechanical metamaterials—as evidenced by this field’s accelerated progress in recent years1–5.

Architected materials have been demonstrated to exhibit a variety of complex and unconventional behaviors, including high strength and stiffness at low densities6–8, energy absorption and impact resistance9–12, tunable Poisson’s ratio13,14, negative thermal expansion15, acoustic wave guiding16, and vibration absorption17. While thousands of microstructures have been proposed over the last decade, most design paradigms of architected materials can be broadly classified into three main categories, namely truss-6–8, shell-10,18, and plate-based19,20 architectures. These categories emerged naturally in the search for record-breaking stiff and strong materials, where optimized mechanical properties for a given material density were sought. In this race to achieve ultra-high stiffness and exceptional strength-to-weight ratios, the equally important pursuit of highly compliant and deformable architectures has remained largely unexplored.

Designing compliant architected materials presents opportunities to populate uncharted areas for this materials system, in a domain where they can offer promising solutions for soft-matter applications such as flexible sensors21 and bioscaffolds22. To achieve enhanced compliance and deformability, variations of conventional design categories—i.e., variations of truss-, shell-, or plate-based architectures—with functional gradients23–25 or heterogeneity26–29 are often proposed. For example, density-graded architectures have been used to reduce the initial stiffness of truss-based lattices while enabling a continuous increase in engineering stress, resulting in higher peak stress and dissipated energy density23. Functional gradients have also been utilized to inhibit the formation of shear bands by inducing layer-by-layer deformation, which prevents catastrophic failure25. In addition, heterogeneous lattices that combine multiple types of unit cells in the same sample have been implemented to tailor the deformation or failure of lattices, allowing for pre-determined failure paths by designing regions with different compliance26–29. Nevertheless, drastically improving the compliance and deformability of architected materials solely by varying the topological arrangement of classical architectures is challenging, especially under tension, owing to the intrinsically stiff nature of these designs. Although inverse and optimization-based design strategies have been demonstrated as powerful tools for exploring design spaces of gradient and heterogeneous architectures, these nevertheless require underlying models of structural classes (e.g., beam-based lattices)30–32 or are, conversely, stymied by extremely high dimensionality of design space (e.g., topology optimization)33,34.

Recently, new classes of architected materials, such as 3D helical35 and woven lattices36–38, have been proposed to achieve extreme combinations of compliance and deformability. These designs leverage the structural flexibility of coiled helical fibers, as opposed to the straight beams found in truss-based lattices, to reduce stiffness while enhancing stretchability through fiber straightening. In particular, woven lattices differ from conventional truss- or plate-based lattices with monolithic junctions at joints/nodes by instead forming entanglements of fibers with constant curvature at junctions. As a result, stress concentration is reduced at these entangled junctions during deformation, thus enabling large deformations (with stretch λ values exceeding 2) even with stiff and brittle constituent materials36,37. Moreover, frictional sliding39 and the evolution of entanglements between woven fibers during deformation provide additional energy dissipation mechanisms36–38,40–43, resulting in high toughness. However, current studies on woven lattices have been limited to periodic tessellations of two main architectures (octahedron and diamond)36 and their simpler variations37, restricting the full range of achievable properties in this materials system. Although these two architectures have been demonstrated to achieve significant toughness when integrated with truss lattices38, these double-networks alone offer limited options for tailoring lattice behavior. In particular, they are not well suited for achieving spatially graded or locally anisotropic stiffness distributions, nor for programming targeted deformation or failure pathways. The lack of variation and tunability in woven lattice designs can be attributed to the difficulty of parametrizing their complex hierarchical nature, consequently leading to the absence of robust design frameworks capable of efficiently generating woven lattices with arbitrary 3D topologies—without the need for manual design in computer-aided design software.

Here, we introduce a design and modeling framework for 3D woven metamaterials—validated by experiments on a range of newly achievable designs—that vastly expands the usable design space of these promising materials. We introduce a general route for converting arbitrary beam lattices into 3D woven lattices using their graph representation30. As a demonstration, we introduce three previously unachievable woven topologies. By defining the design space of a given topology with two strut-level parameters, i.e., the effective strut radius and the number of fiber revolutions within, we allow facile design of arbitrary functionally graded and spatially patterned lattices without loss of geometric compatibility between differing unit cells. Using computational mechanics models directly coupled to the design-generation framework, we establish the broad range of elastic behaviors achievable by solely varying unit cell and strut parameters. Most importantly, we present a reduced-order beam-element representation for nonlinear, large deformation, which allows rapid exploration of the full nonlinear design space for large tessellations of this metamaterial class, which is validated against in situ tension experiments on microscale woven lattices. Specifically, we investigate their characteristic deformation and the effects of fiber topology (i.e., the path of a given continuous fiber in a 3D domain) on the mechanical responses of the metamaterials, while identifying fiber straightening and entanglement evolution as key deformation mechanisms. This validated framework—provided herein as a freely available software design tool—enables broader implementation of woven metamaterials as a newly established compliant-material class within the mechanical metamaterials toolbox.

Results
A rational design framework for woven metamaterials
To fully determine the design space of 3D woven metamaterials and establish relations between design parameters and resulting macroscopic properties, we developed a systematic framework for their rational design (Fig. 1a). In short, the generation of woven lattices begins with a classification based on the parent monolithic-beam lattice, translated to a spatial-graph-based representation of the topology (i.e., the topology graph) as vertices and edges. The vertices in the topology graph are expanded into spherical node graphs, which dictate how fibers in a woven beam connect to adjacent woven beams. Each edge in the topology graph that meets the spherical node is then replaced with an effective woven beam—a construct formed by helical fiber bundles—characterized by an effective diameter and the revolutions of its constituent fibers. To connect effective woven beams, including those with dissimilar geometric parameters, the woven nodes are engineered using a chiral twist that seamlessly connects each effective beam to its topological neighbors, ensuring continuity across all fiber strands. Thus, this simple graph-based approach facilitates the design of large tessellations of woven unit cells by effectively capturing the topology of the woven lattice and embedding the complexity of the woven nodes and beams as scalar parameters of the graph’s vertices and edges.

Once the topology graph for an envisioned woven lattice is defined, the more involved step consists of generating the graph representation of its woven nodes. The vertices of these spherical node graphs are determined by the intersection of a woven beam’s centerline with the spherical node region. Joining all vertices with edges forms the faces of a polyhedron (as shown in Fig. 1a Node graph) within this spherical region. The resulting nodal graph formed by the vertices and edges (Fig. S1b) is the spherical dual of the topology graph, which represents connections between spatially neighboring edges44. Connecting neighboring beams accordingly enforces the fundamental criterion that each beam must interweave with all its nearest neighbors, reconstructing the full connectivity of the original beam lattice. Additionally, this representation introduces a structural constraint that the degree (i.e., connectivity) of each nodal graph must remain constant, which guarantees that each beam in the woven lattice is composed of an equal number of intertwined fibers (Fig. S2). In other words, a topology graph of n-degree produces a woven lattice whose beams are composed of n helical fibers. For the case of a cubic lattice, depicted in Fig. 1a, the spherical dual graph of a cube has the connectivity of an octahedron; each vertex has degree n = 4, and the resulting woven lattice is thus composed of 4-helices.

After defining both the topology graph and the nodal graph, the first user-defined parameter of the lattice is the number of turns of a helix n_rev, within a woven beam. This parameter is a discrete, unitless number assigned to each woven beam, which dictates the helix pitch and ultimately how the n fibers of one beam twist and connect to the n fibers in adjacent beams. This twist determines the chirality of these helices, which is arbitrary, but remains constant over the entire lattice. At this stage, the representation contains the topological information required to define the woven lattice. The second user-defined parameter of the lattice is the effective radius R_eff of the woven-beam helices, which determines both the shape of the helix and the radius of the node, ensuring a smoothly varying curve path between neighboring helices (Fig. S3). This parameter is defined so that the original monolithic-beam lattice is recovered when R_eff goes to zero. At this stage, the lattice is represented as a set of three-dimensional space curves forming the centerlines of fibers, which can then be discretized by sampling points along the centerlines. To obtain a solid geometry, the curves are then swept with a circular profile of a specified fiber radius (details in Supplementary Information S1 and Fig. S1). We note that, for a given lattice unit cell and size, a finite fabricable (n_rev, R_eff) parameter space exists—being primarily dictated by fiber separation that either avoids crossing or avoids fusing of fibers during fabrication. While determining the fabricable space is non-trivial, it can be readily mapped for unit cell topologies once the resolution and limitations of a given manufacturing process are defined (see Fig. S4 for an example).

This geometric representation, therefore, preserves the defining features of woven lattices, i.e., an effective beam structure composed of fibers that are highly entangled in all directions at nodes, while allowing for high tunability. While R_eff controls the density and spacing of woven fibers, n_rev governs fiber entanglement since higher values result in more twisted fibers, while also determining the path by which a fiber traverses the lattice. For example, in the case of a cubic lattice, with a specified n_rev = 3n/4 for integer n, each fiber exits the node along the same cubic face it enters, causing the fiber path to follow the edges of a cubic face (Fig. S5). This configuration results in a lattice of interlocking square rings, where the integer value of n_rev dictates how many times each ring interlocks with its neighboring rings. Conversely, other values of constant n_rev create fiber paths that traverse multiple unit cells. Therefore, since n_rev determines the global fiber topology, it has a direct effect on the deformation of the woven lattice. A stochastic selection of n_rev generates a randomly entangled network that retains structured order at the lattice scale. The tunability of the described framework thus allows for the generation of spatially varying woven lattices by allowing n_rev and R_eff to vary as functions of position within the lattice (Fig. 1b, Supplementary Information S2 and Fig. S3). By spatially modulating n_rev, one can transition from highly entangled regions to more open, loosely connected domains, enabling controlled gradients in mechanical and functional properties. Similarly, varying R_eff allows for localized tuning of effective beam stiffness and node size, accommodating heterogeneous structural requirements. This approach is particularly useful for designing functionally graded materials, where different regions of a structure demand distinct mechanical behaviors. Furthermore, leveraging computational optimization techniques can facilitate the automated design of spatially varying woven lattices tailored to target performance criteria, opening new avenues for advanced metamaterial design and multifunctional applications. Towards facilitating generation of arbitrary woven-unit-cell tessellations, the design framework outlined in this section enabled the creation of an automated design tool45.

Linear property space and anisotropy
The mechanical response of woven lattices is inherently tunable through variations in lattice topology and geometric parameters, enabling a diverse range of anisotropic stiffness behaviors. To evaluate the mechanical tunability of various woven designs, their linear-elastic responses were determined using computational homogenization. By applying periodic boundary conditions at the edges of the unit cell and imposing a series of linearly independent strain inputs, the elasticity tensor for each woven structure was calculated (see Methods). The resulting omnidirectional stiffness, normalized by the Young’s modulus of the constituent material, was visualized as elastic surfaces to illustrate the anisotropic characteristics of each architecture (Figs. 2a and S6). We identify notable changes in anisotropy for some lattice topologies when converting them from monolithic to woven beams. For instance, lattices with strongly stretching-dominated orientations, such as the cubic and octahedron lattices (both in the [100] direction), undergo drastic anisotropy changes. In their woven forms, both lattices are no longer stretching-dominated in any direction, and this is visible as a distinct change in their elastic surfaces (Fig. S7).

In contrast to monolithic lattices, a given woven lattice topology can provide a diverse range of anisotropic responses through control of geometric parameters such as R_eff and n_rev. For instance, changing R_eff/L of a cubic woven lattice, where L is the unit cell size, from 1/30 to 1/5 results in a decrease of the normalized maximum specific modulus E_max = ρ' E_s — defined as the maximum directional stiffness E normalized by the product of the material’s relative density (or fill fraction) ρ' and the constituent materials’ Young’s modulus E_s — by a factor of more than 2 (Fig. 2b, top). This behavior is similar to that of a classic torsion spring, where an increase in the helix radius results in reduced stiffness. Since R_eff can be tuned continuously, it enables a continuous transition between different stiffness regimes. On the other hand, varying n_rev alters both stiffness and anisotropy (Fig. 2b, middle), which can be explained by the difference in fiber topology determined by the changing connections between beams (Fig. S5). To further demonstrate the design flexibility enabled by this framework, we also show that a parameter change can tune anisotropy within an individual unit cell (Fig. 2b, bottom). By selecting dissimilar R_eff values for the beams along one direction and those orthogonal to it (defined by a ratio R_eff = R_eff'), the unit cell develops preferentially stiff directions, as demonstrated by the change in shape of the elastic surface. This ability to tune geometric parameters at the sub-unit-cell scale demonstrates localized control over mechanical properties, expanding the property space of each lattice topology.

Plotting the maximum and minimum specific moduli, normalized by relative density ρ', for a range of lattice topologies further illustrates the tunability of woven lattices in the elastic regime (Fig. 2c). This stiffness parameter space reveals a distinct clustering of structures based on their underlying lattice topology, which sets the fundamental mechanical signature of a woven lattice. However, within each topology, moderate changes in R_eff, n_rev, and R_eff = R_eff' allow for smooth transitions across this property space, unlocking stiffness and anisotropy ranges that approach one order of magnitude. This tunability suggests that woven lattices, unlike their monolithic counterparts, occupy a continuous design space where mechanical performance can be selectively programmed over a range that spans across multiple orders of magnitude.

Nonlinear large-deformation response
The primary utility of woven lattices lies far beyond their linear-elastic responses, in a large-deformation regime where they demonstrate exceptional deformability and resilience. To experimentally characterize the range of nonlinear responses available in woven lattices, we conducted in situ uniaxial tension experiments on 2×2×2 tessellations of microscale woven lattices designed through our framework (Fig. 3). We also evaluated cyclic performance in compression and tension in larger 4×4×4 tesselations (Fig. S8 and Supplementary Information S3), as well as large deformation response of graded woven lattices (Fig. S9 and Supplementary Information S4). Using a two-photon lithography process with IP-Dip2 photoresist, each sample had unit cell sizes of L = 60 μm and fiber radii of 1 μm. To ensure that potential size effects on the response of the polymer did not affect the responses across samples or complex stress states within the fibers, all design variations maintained the same fiber radius. The experiments were performed in situ within a scanning electron microscope using a displacement-controlled nanoindenter (Alemnis AG) equipped with a custom tension gripper, ensuring precise control and real-time observation of stress-strain responses, deformation, and failure mechanisms (Fig. S10). We designed these experiments to directly assess the role of design parameters such as effective radius R_eff and fiber revolutions n_rev on the large-deformation responses of these architectures.

In accordance with the linear perturbation simulations (Fig. 2b), the experiments confirmed a reduction in the lattice stiffness as R_eff increased (Fig. 3a–c and S11). For example, reducing the R_eff from 6 to 2 μm for a BCC woven lattice of unit cell size L = 60 μm results in an increase in stiffness from 40 kPa to 160 kPa. However, this comes at the cost of reduced stretchability before onset of failure (with failure stretches decreasing from ~2 to 1.5), as the shorter fiber segments limit the available deformation before full uncoiling. The R_eff parameter also determined the topology-dependent maximum stress prior to failure—e.g., providing the highest strength in the woven BCC lattice at R_eff = 4 μm, in contrast to R_eff = 2 μm for octahedral and diamond woven lattices. This topology-dependent behavior is in accordance with cyclic tension performance recorded in Supplementary Information S3 and Fig. S8. In addition, these experiments indicate that woven lattices undergo negligible failure up to 15% strain. Notably, the cyclic tensile experiments show essentially no degradation in mechanical response even after ten loading cycles.

The number of fiber revolutions per strut n_rev, also significantly alters the large-deformation responses, again leading to topology-dependent effects. In the woven BCC lattice, increasing n_rev enhances ductility, leading to higher ultimate stretch and more gradual failure response. For instance, at n_rev = 4/3, failure occurs abruptly with a sharp drop in stress. However, at n_rev = 5/3 or 6/3, stress decreases gradually, indicating a transition from brittle-like failure to a more ductile response. This behavior can be attributed to increased fiber length, which allows for greater energy dissipation before rupture. In contrast, for octahedral and diamond-based woven lattices, the influence of n_rev is less pronounced, likely due to fiber-topology-dependent entanglements, which can introduce localized stress concentrations that counteract the expected increase in ductility.

Overall, the results from experiments confirm that woven lattices provide an expansive mechanical property space, validating our design framework’s capability in creating lattices with a wide range of mechanical properties. While stiffness is primarily dictated by R_eff, large-strain behavior is strongly influenced by both R_eff and n_rev, demonstrating the tunability of mechanical response through geometric variation. Notably, unlike monolithic beam lattices that typically fail at low stretch values (λ < 1.5)46–50, all tested woven lattices retained remarkable stretchability, sustaining stretches of at least 2 and up to 4 before complete failure. Additionally, failure strain consistently increased with R_eff, likely due to the greater available fiber lengths enabling more progressive deformation.

Furthermore, we examine the relation between specific modulus and specific energy absorption to assess the mechanical performance of woven lattices (Fig. 3d). Specific energy absorption is defined as the area under the stress-stretch curve, W, normalized by the lattice density. Through this analysis, we experimentally validated the significantly expanded material property space enabled by woven lattices. Importantly, even with just four fundamental lattice topologies, we observe a vast range of mechanical responses, underscoring the potential for further design exploration. This range suggests that by leveraging additional topological variations, such as the inclusion of more architectures, variation in sub-unit-cell parameters, and creation of spatially graded lattices, the mechanical properties of woven lattices can be tailored for a diverse array of applications, from energy-absorbing structures to highly flexible metamaterials. Beyond the single-architecture space presented in Fig. 3d, we performed experiments on functionally graded architectures that exhibit predictable heterogeneous deformation, with strain concentrating in the most compliant unit cells and reduced deformation in regions with higher n_rev (Supplementary Information S4 and Fig. S9). These experiments showcase the potential for an even broader performance space if spatial grading is systematically implemented in the design of woven lattices.

Nonlinear modeling framework
To gain mechanistic insight on the nonlinear, large-deformation responses of the woven lattices—as well as predictive capabilities of their performance—accurate computational modeling tools are required. While finite element analysis with continuum elements provides the highest-fidelity discretization (Fig. S12), its high computational cost limits feasibility of large-tessellation modeling. Conversely, homogenization at the unit cell level fails to capture the intricate behavior of pseudoperiodic, functionally graded lattices, falling short in predicting the responses of parametric designs enabled by our design framework.

To enable efficient large-tessellation simulations of these woven architectures, we employ Timoshenko beam-element representations that incorporate a Coulomb friction contact model as well as a strain-hardening material model (see Methods). This method is well suited for simulating spatially complex, slender structures subjected to large strains. A comparison between beam-element and tetrahedral-element simulations reveals strong agreement in kinematic behavior and stress-stretch responses (Figs. 4a, b and S12), indicating that the beam-element framework effectively captures the physics associated with their deformation. Comparing the experimental and simulated (beam-element representation) stress-stretch response of a BCC woven lattice also shows that the simulations can predict both the linear and nonlinear responses obtained from the experiments (Figs. 4c and S13). Notably, the stretch at the onset of failure is accurately predicted by the simulations, while their higher peak stresses and more abrupt failure can be explained by the absence of manufacturing defects in our models, which are inevitable in experimental specimens. In the experiments, these defects result in a distribution of failure strains within the woven fibers, leading to a more sequential or gradual failure, rather than the more abrupt failure observed in the simulations. Incorporating a random variation in failure strain within the beam-element simulation reduced these peaks, suggesting that the differences between the simulations and experiments are due to over-idealization of the material properties in the beam-element approach, and that a more complex material model approach could more accurately replicate these effects in simulation (see Fig. S14). The beam-element approach drastically reduces computational cost compared to the tetrahedral models (from 125.7 h to 1 min and 30 s on the same hardware, for the unit cell in Fig. 4a), making it feasible to simulate large tessellations.

To demonstrate the applicability of the computational framework for larger woven-lattice tessellations, a 4×4×4 BCC woven-tessellation was simulated to a stretch of λ = 4 using the beam-element approach (Fig. 4d), revealing key mechanisms governing large-deformation behavior. As the lattice deforms, fibers initially straighten, as indicated by a negative change in curvature Δκ (defined as the norm of the second derivative along the centerline) with respect to the initial configuration. At the same time, the models identify curvature increases that become highly concentrated in the entangled regions. Likewise, contact pressure, generated by entanglements between the woven fibers, localizes at these regions. The change in curvature characterizes the bending behavior of the fibers, directly influencing global deformation patterns, while contact pressure highlights regions of concentrated frictional interactions that govern load transfer and energy dissipation (Fig. S15). Together, these parameters identify stress distribution pathways, revealing potential failure sites and high energy-dissipation zones. Understanding these mechanisms can enable the design of optimized fiber entanglements that enhance stretchability and toughness. Leveraging our computational framework to analyze the large-deformation behavior and mechanics of woven lattices, these findings highlight fiber topology as a crucial design parameter for tailoring and optimizing mechanical performance.

Tailored deformation and failure patterns
The proposed design framework enables the creation of woven lattices with precisely tailored deformation and failure behaviors (Fig. 5). As an example, Fig. 5a demonstrates a 30×10×2 BCC woven-lattice-sheet engineered to form a visually distinct text pattern under large strains. This effect is achieved by assigning different unit-cell parameters to the background, boundary, and text-pattern regions, creating a mechanically tunable system that enhances visual contrast through controlled deformation (Fig. S16). The woven lattice in the text region is designed with lower n_rev = 2/3 and higher R_eff/L = 1/10 (where L corresponds to the unit cell size), resulting in a highly compliant architecture with comparatively low entanglement that reshapes visibly under strain. This early deformation causes the fibers to straighten and the nodes to tighten, increasing local stretch and making the letters visually prominent. The background region, i.e., the lattice unit cells outside the text region, is instead assigned R_eff/L = 1/12 and n_rev = 7/3, creating a stiffer network that resists deformation, ensuring that the surrounding material provides a contrasting frame for the stretching text region. To further refine the deformation contrast and maintain pattern clarity, a boundary zone is introduced at the interface between the text and background regions. This boundary region is assigned R_eff = 1/15 and n_rev = 7/3, resulting in the lowest compliance. By deforming the least, these unit cells compensate for the large elongation of the text region unit cells and reduce blurring of the pattern at the edges of the text region.

Beyond controlled deformation, this framework also enables the strategic design of failure pathways. As shown in Fig. 5b, a 24×10×2 woven lattice is patterned with a sinusoidal failure path using similar design principles as those in the letter pattern. Additionally, a linear gradient in effective radius is introduced, reducing R_eff towards the chosen end of the lattice. This variation concentrates stress and directs failure propagation in a controlled manner, with failure initiating at the end with smaller radii due to the lower available length of fibers on that side. The gradient follows a stepwise linear reduction in R_eff at each unit cell boundary (up to 1/20 over the width of the lattice). In Fig. 5c, a 10×5×2 printed lattice—patterned similarly with an asymmetric path—demonstrates experimentally how deformation concentrates in the targeted regions. As the lattice deforms, failure initiates in these regions then coalesces along the targeted failure path. This controlled spatial variation in fiber topology enables predictable failure pathways, demonstrating how the woven lattice framework can be leveraged to program both deformation and fracture behavior.

Methods
Selection of geometric parameters
Geometric parameters in experiments were varied within bounds determined by manufacturability considerations. All samples were fabricated with a unit cell size L of 60 μm (except the more spatially complex tetrakaidecahedron, which was printed with a unit cell size of 120 μm), to obtain a well-defined unit cell while minimizing lattice stitching required by workspace limits. The effective radius R_eff was varied between 2 and 6 μm, leading to R_eff/L ratios between 1/30 and 1/10. The number of turns n_rev, limited by manufacturable strand spacing (Fig. S4), ranged between 4 and 7 (for lattices consisting of 3-helices: diamond, octahedron, BCC, and tetrakaidecahedron) and 4 and 6 for the 4-helix cubic lattice. Fiber radius was fixed at 1 μm. Relative densities ρ' ranged between 1% and 5%, determined by the spatial density of the generating lattice and fiber packing density as determined by both R_eff and n_rev. To compare these manufactured lattices, the effective stiffness and the energy absorption capacity (computed as the area under the stress-strain curve when subjected to tension to a set strain) were normalized by relative density ρ' and the density of the constituent IP-Dip2 polymer ρ_s.

Fabrication and mechanical characterization of microscale lattices
Microscale woven lattice samples were fabricated on a silicon substrate using two-photon lithography (Photonic Professional GT2, Nanoscribe GmbH) out of IP-Dip2, an acrylate-based photoresist. The architected structures were printed at 25 mW laser power with a 10 mm s−1 scan speed, while monolithic tensile fixtures were fabricated at 20 mW to reduce cavitation. Printing parameters remained consistent, with a uniform hatching and slicing distance of 0.2 μm for the lattices and pillars and 0.3 μm for the tensile fixtures. Post-processing involved immersion in propylene glycol monomethyl ether acetate for up to 2 h to remove residual resin, followed by a 5-min isopropanol rinse and drying via critical point drying (Autosamdri 931, Tousimis). To enhance visualization for in situ mechanical characterization, a 10 nm gold coating was applied using sputtering (SCD040, Balzers).

For mechanical characterization, uniaxial tension experiments were performed using a custom gripper mounted on a displacement-controlled nanoindenter (Alemnis AG) inside a scanning electron microscope (Gemini 450, ZEISS), enabling real-time imaging. Specimens were loaded at a strain rate of 5 × 10−2 s−1. Stress-strain data were derived from load-displacement curves, normalized by nominal cross-sectional area and height. The dissipated energy density was determined by integrating the area under the stress-stretch curve.

Finite element modeling
To model complex large-deformation responses, finite element models employing beam elements were conducted using ABAQUS/Explicit 2023. The centerline geometry was directly used to create meshes using linear Timoshenko beam elements (B31) with general contact using an anisotropic Coulomb friction model with an empirically matched dynamic friction coefficient of 0.5 and a hard-contact criterion in the normal direction. Failure was modeled through element deletion based on a maximum strain criterion of 0.0377. The material model consisted of an anisotropic elastic-plastic model experimentally derived from the experimental response of an IP-Dip2 micropillar under uniaxial tension.

Linear perturbation simulations were performed using nTop (nTop Inc.) and in ABAQUS/Standard; an orthogonal set of linear perturbations was performed with periodic boundary conditions to obtain the full elasticity and compliance tensors for each unit cell. For these homogenization simulations, periodic tetrahedral meshes were used, generated using the open source finite element mesh generator Gmsh54. Mesh convergence (see Supplementary Information) suggested a consistent mesh size of 0.25 of the fiber radius, generating meshes with approximately 105–106 quadratic tetrahedral elements (C3D10). Timing comparison (Fig. 4a) was performed on Intel® Xeon® W-2245 CPU @ 3.90 GHz with 8 cores and 128 GB of RAM.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The paper presents a geometric design framework for programmable three-dimensional woven metamaterials, encoding woven topology in a graph structure. This approach significantly expands the design space, enabling tunable architectures, functional gradients, and arbitrary heterogeneity. However, several critical issues prevent this reviewer to recommend its publication in current version.\nComment 1: The unit cell side length L (mentioned in Figure 1) should be explicitly labeled in the figure to improve clarity and facilitate reader comprehension.\nComment 2: While the manuscript thoroughly examines short-term mechanical properties, long-term performance and durability are essential for practical applications. The authors should incorporate a preliminary analysis of fatigue resistance under cyclic loading and stability under varying environmental conditions.\nComment 3: Mechanical testing is currently limited to 2×2×2 microscale lattices (L = 60 μm). Experimental validation is lacking for functionally graded designs (Fig. 1b) and larger tessellations (e.g., 4×4×4), which are crucial for demonstrating scalability.\nComment 4: Programmable failure patterns (Fig. 5b) are only supported by simulations. Experimental verification of controlled fracture paths is necessary to strengthen the findings.\nComment 5: The discussion section highlights potential applications (e.g., biological stents, impact protection) but lacks quantitative evidence for key performance metrics, such as permeability and specific energy absorption rates. These should be substantiated with relevant data.\nComment 6: The authors must clarify the fundamental distinctions between their work and the closely related study (10.1038/s41563-025-02219-5) to better position their contributions.", "reviewer_2": "This manuscript presents an interesting investigation about design strategy of three-dimensional woven metamaterials. A series of programmable woven metamaterials with functional gradients and arbitrary heterogeneity are presented and analyzed systematically by parameter study. Mechanical properties of proposed structures are tested and discussed in detail by experiments and simulations. I believe that the work will be sufficient to merit publication after some minor modifications.\nSome particular comments are listed as follow:\n1. The geometric configuration of woven metamaterials can be controlled by two parameters, Reff and nrev. For a given lattice configuration, considering fabrication and performance, is there a relatively optimal solution for the two parameters? More discussion is needed on the selection of parameters within the design space.\n2. In compression experiments, both the fiber diameter and cell size are selected at the micrometer scale. At this scale, when altering design parameters, has the impact of material size effects on structural mechanical properties been considered?\n3. Figure 3 compares the specific energy absorption and specific modulus of woven metamaterials with different design parameters across various lattice configurations. However, it is essential and intuitive to benchmark these properties against existing literature on woven metamaterials and other lightweight structures. A more detailed Ashby plot is recommended.\n4. Despite the forward design methods discussed in the manuscript, inverse design strategies for energy-absorbing metamaterials have concurrently emerged. A brief discussion of these advances is therefore encouraged, for example adma.202405567, advs.202204977 and s42256-025-01067-x.\n5. In Figure 4c, the peak values of the simulation curve exhibit a significant discrepancy compared to the experimental results. The statements about “absence of manufacturing defects in our models which are inevitable in experimental specimens” is too vague. It is suggested to carefully verify the comparison of the experimental and simulation results.\n6. The regulatory effect on deformation modes and failure paths is remarkably significant. However, only simulation fitting results are presented in the paper. Have experimental attempts been made to achieve the specified failure paths? Under manufacturing tolerances, substantial deviations from idealized results may occur.", "reviewer_3": "This manuscript focuses on the framework to design woven structures and demonstrates their tunable mechanical properties and programmable failure patterns. Generally, it is interesting by exploring the stretchable regime of mechanical metamaterials. In contrast, plenty of works have been done regarding the compressive mechanical behavior while ignoring the stretching counterparts. This manuscript paths a way for researchers to the compliance and stretchability of mechanical metamaterials. I have no doubts on the correctness of this work, but have some concerns on the superiority, applicability and engineering potentials. Before recommending for publication, I have following questions.\n1) The node design is interesting. Is the number of strands in woven beams pre-determined by the parent monolithic-beam lattice? For example, the beams lattices in fig 1a can only be replaced by woven structures with 4 fibers due to connectivity. Are there cases where three or other different number of fibers is applied?\n2) In page 10, the authors demonstrated that “all tested woven lattices retained remarkable stretchability, sustaining stretches of at least 2 and up to 4 before complete failure.” This is remarkable. However, it is not solid convincing to compare with other structures in literature. The comparative study should be: How good is the large-deformation performance compared with conventional monolithic beam lattices counterparts that occupy identical institutive material and relative density?\n3) Similar to the former question, is the energy absorption performance better or worse than their monolithic beam-and-junction-based counterparts? Especially in the case of stretching conditions.\n4) With regard to energy absorption, is it derived from tension-displacement curves for woven lattices while obtained from compression-displacement curves for beam lattices, respectively?\n5) It is for sure that woven lattices have significantly improved the extensibility than monolithic beam lattices. However, the latter was mainly designed for superior compressive behavior which woven lattices fall short of. This is like a coin that has two sides. I am concerned on the negative side of woven lattices, i.e., the compression performances.\n6) In fig. 3a, is the peak position of the stress-strain curve where fiber fracture starts to appear?\n7) The lattice microstructure is to some extent complicated in its geometry, where each beam is replaced by three or four helical fibers. There has been reported structures with only one fiber and demonstrates 230% extension, i.e., (2020) 11:1180, https://doi.org/10.1038/s41467-020-14996-5. What is the unique advantage or significant progress of this proposed structure?\n8) It is easy to envisage that monolithic metamaterials can have superior energy absorption behavior applied in crushing and impact of protective equipment, automotive, civil structures, et al. What are the major potential engineering applications for such stretching-featured mechanical metamaterials?\n9) As far as I understand, the different fiber strands are detached and not in contact with each other in their original state. In the very initial stage, the contact pressure should be zero. Can the authors present the approximate contact pressure (real value or magnitude, not max/min ratio) during the stretching process?"}, "author_response": "We thank the reviewer for their careful and detailed review of our work. The following are point-by-point responses to the reviewer’s comments and suggestions.\nComment 1.1: The unit cell side length L (mentioned in Figure 1) should be explicitly labeled in the figure to improve clarity and facilitate reader comprehension.\nResponse: We appreciate the reviewer’s suggestion, and have updated Figure 1a to include the unit cell side length L in the monolithic and woven representations of the unit cell. This edit (along with minor cosmetic edits) improves the clarity of this figure, helping the reader connect the graph to the resulting woven unit cell. For convenience, the updated version of Fig. 1 is reproduced below.\nComment 1.2: While the manuscript thoroughly examines short-term mechanical properties, long-term performance and durability are essential for practical applications. The authors should incorporate a preliminary analysis of fatigue resistance under cyclic loading and stability under varying environmental conditions.\nResponse: We fully agree that an understanding of these metamaterials’ long-term performance will be essential to enable applications. To that end, we fabricated additional samples and performed new cyclic tension and compression experiments, providing insight on the evolution of their performance to up to ten cycles. Specifically, we fabricated 4×4×4 lattices of two representative geometries—BCC and octahedron architectures—at similar relative densities of ρ¯ = 2.3 − 2.8%. The woven octahedron lattice was compressed to 15% strain for ten cycles, while woven BCC lattices were characterized in compression and tension, both to 15% strain for ten cycles each. These results, shown in Fig. R2, indicate an architecture- and loading-dependent multi-cycle response. Namely, while all architectures exhibited the same qualitative response, quantifying the energy dissipated per cycle showcased differences.\nIn compression, the octahedron architecture exhibited a modest degradation in energy dissipation, dropping by 45% over ten cycles, while the BCC architecture exhibited a similar reduction of 47% through ten cycles. This is consistent with behavior previously reported for woven lattices1, where the octahedron lattice, when subjected to cyclic tension and compression, exhibited moderate reduction in normalized energy absorption without catastrophic failure. However, in tension, the newly introduced BCC architecture exhibited minimal degradation. This difference is likely due to the lower entanglement of the woven BCC lattice unit cell for the same parameters, determined by its lower number of constituent beams compared to the octahedron lattice. This reduction in entanglement allows the fibers to reconfigure more readily in tension, aligning with the tensile load but returning to close to their undeformed configuration upon unloading, and thus reducing permanent deformation.\nThis additional characterization and discussion on these results is now included in the supplementary information, as Fig. S8 and corresponding Supplementary Text section S3 and referenced in the main manuscript text under the subheading Nonlinear Large-Deformation Response.\nAlthough the lattice’s stability under varied environmental conditions is important for future applications, because the experiments shown in this manuscript were performed in situ within a scanning electron microscope, it remains impossible to vary the environmental conditions (e.g., humidity, temperature) using the experimental capabilities available to us. However, the IP-Dip2 photoresist used in fabricating these lattices is known in the literature to be relatively stable up to approximately 150◦C2. Nevertheless, other polymer systems with appropriate thermal and mechanical properties could also be employed for two-photon polymerization (TPP) or other vat photopolymerization techniques. Moreover, the same design framework can be extended to other additive manufacturing routes—such as selective laser melting or selective laser sintering—to create metallic or ceramic woven architectures. In this work, because we are interested in comparing structural behavior, we have normalized experimental values by the material properties of the constituent resin.\nHowever, because the properties of the constituent material do contribute to the resulting behavior of the woven lattice, we share with the reviewers unpublished results demonstrating lattice behavior under varied material and thermal conditions (Fig. R3) in a parallel effort that attempts to address this question. This work, building on both the present manuscript and the double-network paradigm3, discusses the interplay between lattice and constituent material behavior under varied environmental conditions. In short, although the precise behavior in large deformation varies, the woven lattices continue to achieve relatively high extensibility.\nComment 1.3: Mechanical testing is currently limited to 2×2×2 microscale lattices (L = 60 μm). Experimental validation is lacking for functionally graded designs (Fig. 1b) and larger tessellations (e.g., 4×4×4), which are crucial for demonstrating scalability.\nResponse: We appreciate the reviewer’s suggestion to expand experimental validation of the design framework to larger and functionally graded specimens, and agree that demonstrating these capabilities would strengthen the work. In addition to the cyclic loading study on 4×4×4 lattices shown in Fig. R2, the revised manuscript now also explores experimental responses of functionally graded lattices—presented in a new supplementary Figure (Fig. R4) and an accompanying Supplementary Text section S4, and referenced in the main text with the following:\n“Beyond the single-architecture space presented in Fig. 3d, we performed experiments on functionally graded architectures which exhibit predictable heterogeneous deformation, with strain concentrating in the most compliant unit cells and reduced deformation in regions with higher nrev (Supplementary Information S4, Fig. S9). These experiments showcase the potential for an even broader performance space if spatial grading is systematically implemented in the design of woven lattices.”\nAs discussed in the Supplementary Information section S4, these experiments validate the predicted response from constant-parameter lattice experiments, where heterogeneous deformation emerges within the lattice due to higher deformation being carried by the most compliant unit cells. Large nonlinear deformation of lattices graded in n agrees with experimental behavior demonstrated in small-scale constant lattices, where stiffness is minimally altered but unit cells with greater values of n deform less at higher strain than those with smaller n due to their higher entanglement.\nComment 1.4: Programmable failure patterns (Fig. 5b) are only supported by simulations. Experimental verification of controlled fracture paths is necessary to strengthen the findings.\nResponse: To strengthen the point of programmable failure patterns in the manuscript, we performed additional experiments on a 10×2×5 BCC tessellation with a failure pattern determined by varying Reff and n simultaneously. The fabricated lattice has a sinusoidal pattern of designed failure upon uniaxial tensile loading. The results of this experiment, shown in the updated Fig. 5 (reproduced below as Fig. R5) and new supplemental video 5, provide initial demonstration that the design of a failure paths is indeed achievable in fabricated specimens. An additional Supplementary Information section S5 provides supporting information.\nThe main text now references this modified figure, providing the reader with experimental validation as support of the proposed design principles for failure. The following is an excerpt from the revised text:\n“In Fig. 5c, a 10×5×2 printed lattice—patterned similarly with an asymmetric path—demonstrates experimentally how deformation concentrates in the targeted regions. As the lattice deforms, failure initiates in these regions then coalesces along the targeted failure path.”\nComment 1.5: The discussion section highlights potential applications (e.g., biological stents, impact protection) but lacks quantitative evidence for key performance metrics, such as permeability and specific energy absorption rates. These should be substantiated with relevant data.\nResponse: We appreciate the reviewer’s request for quantitive, application-related metrics. Our intention with mentioning these potential applications in the last sentences of the manuscript is to highlight future research directions—some of which are ongoing—based on the property and design spaces that this work maps. To clarify this intention, highlighting to the reader that the primary contribution of the manuscript the the foundational framework for applications to come, we have revised the wording in that last paragraph of the manuscript. We believe it is important to connect the framework to future applications, even though these endeavors are not at a level of completion that would merit their inclusion in the manuscript.\nRegarding biological applications, we believe that the woven scaffolds’ low density, high permeability, and spatial tunability make them well suited for fundamental mechanobiological studies. They could also enable applications requiring precise control of mechanical properties, such as promoting cell polarization and alignment during wound healing through stiffness gradients. We further highlight that the stiffness range attainable by these scaffolds (∼ 30 − 500 kPa) is representative of a wide range of biological tissues. As an example of this particular realization, we would like to share unpublished (ongoing) cell culture experiments on woven and monolithic lattices, successfully demonstrating mouse fibroblast attachment on each (Fig. R6). The scaffolds were first treated with Poly-L-Lysine to enhance cell adhesion, after which BALB/3T3 mouse fibroblasts were introduced and cultured for two days. Following the culture period, the cells were fixed with 3% paraformaldehyde for SEM imaging. The results revealed that, after the same culture duration, a significantly higher number of cells adhered to and proliferated on the woven than the monolithic scaffolds. This highlights that the mechanical tunability of our woven scaffolds can indeed significantly influence cellular behavior.\nRegarding impact protection, though all experiments in the present study were performed quasi-statically, the specific energy dissipation of our scaffolds in compression (see Fig. R2) ranges from 6–13 Jg−1, which is on par with traditional structural and energy-absorbing foams albeit at a much lower density. Exploration in this direction of high-rate impact experiments is a key research focus of our group4,5, and we intend to answer these questions in future work.\nComment 1.6: The authors must clarify the fundamental distinctions between their work and the closely related study (10.1038/s41563-025-02219-5) to better position their contributions.\nResponse: We thank the reviewer for their attention to our prior work on double-network-inspired (DNI) metamaterials. Several key differences exist between these works:\n• Universal design framework: double-networks are merely a subset of the design space presented in this work—i.e., this work’s proposed framework generalizes to the design of monolithic lattices (Reff → 0) and woven latices with arbitrary geometry. The DNI work did not explore the role of geometric parameters within a given DNI network—instead, a single manual CAD design was presented for each of the two DNI concepts.\n• Novel topologies: The present work expands the design library of woven metamaterials to new topologies (unit cells) and also enables the generation of a bespoke unit cell beyond the ones proposed in Fig. 2. In prior woven-lattice works over the last 5 years3,6,7—including the DNI manuscript—the same two topologies had been employed.\n• Open source code for automated generation: By inclusion of the open-source Python code, this manuscript is the first to present a truly scalable framework for the design of 3D woven metamaterials at any length scale.\n• Functional variation: This work is the first to present a route to smoothly vary parameters throughout a tessellation, enabled by the simplicity of the design framework (and easily implemented via the code).\n• Design of failure: This manuscript also is the first to propose the design of failure paths in 3D woven metamaterials. This design-by-failure approach leverages the systematic study of the effect of geometric parameters on the response, two concepts that are non-existent in the DNI manuscript.\nTo clarify this point to the reader, we have expanded on the comparison with this related work in the manuscript. The following is an excerpt from the revised manuscript:\n”Although these two architectures have been demonstrated to achieve significant toughness when integrated with truss lattices3, these double-networks alone offer limited options for tailoring lattice behavior. In particular, they are not well suited for achieving spatially graded or locally anisotropic stiffness distributions, nor for programming targeted deformation or failure pathways.”\nREVIEWER #2\nGeneral comment: This manuscript presents an interesting investigation about design strategy of three-dimensional woven metamaterials. A series of programmable woven metamaterials with functional gradients and arbitrary heterogeneity are presented and analyzed systematically by parameter study. Mechanical properties of proposed structures are tested and discussed in detail by experiments and simulations. I believe that the work will be sufficient to merit publication after some minor modifications. Some particular comments are listed as follow:\nResponse: We thank the reviewer for their careful review of our work, in addition to the positive feedback provided.\nComment 2.1: The geometric configuration of woven metamaterials can be controlled by two parameters, Reff and nrev. For a given lattice configuration, considering fabrication and performance, is there a relatively optimal solution for the two parameters? More discussion is needed on the selection of parameters within the design space.\nResponse: We thank the reviewer for this suggestion, which prompted us to create a new supplementary figure that provides the reader a better sense of the manufacturable bounds for some morphologies (see Fig. R7, reproduced below).\nFirst, within the design space of a unit cell, there are indeed limits to both Reff and n beyond which fabrication is infeasible due to strands fusing or crossing. In particular, because of the fabrication constraints of the two-photon lithography process used to create the samples in this work, the strands of the lattice must be sufficiently spaced to prevent unintentional fusing—about one strand’s diameter apart. This limits both Reff and n, in a coupled manner. For example, in the case of 60 μm unit cells with a strand diameter of 2 μm, Reff is limited to about a minimum of 2 μm and a maximum of 8 μm. However, both Reff and n affect this spacing; due to the expansion of nodes required by larger Reff (see, for example, the rightmost column of unit cells in Fig. R7), n is more strictly limited for larger Reff, as increasing n packs more strands into the remaining effective beam length between nodes.\nTo provide the reader with a sense of the manufacturable regions of the searchitectures where values of n and Reff are appropriately balanced, Fig. R7 presents the design-spaces of two morphologies (BCC and cubic), indicating in green the parameter regions that are most readily fabricable. In addition, the revised manuscript now explicitly mentions the fabricability constraints, and points the reader to the example for the set two architectures in the SI.\nWhile optimization of these parameter combinations for a target property is a valuable research direction, this endeavor is challenging due to the complexity of the nonlinear responses in these architectures (e.g., requiring accounting for contact, friction, and a nonlinear material model). As such, these optimal parameters might be non-unique for a given topology (due to nonlinear response variability), and they will also vary across topologies and relative densities. We agree with the reviewer that this direction is worthy of its own separate study (to be done rigorously and systematically), and is indeed an opportunity for future work.\nComment 2.2: In compression experiments, both the fiber diameter and cell size are selected at the micrometer scale. At this scale, when altering design parameters, has the impact of material size effects on structural mechanical properties been considered?\nResponse: We thank the reviewer for raising the potential impact of material size effects in these materials. First, as confirmed from other studies—including our own throughout the past 5+ years8—no size effects are present in these acrylate-based polymers at the ∼2 μm scale. That is, the stiffness and strength properties of these microscale samples match those of macroscopic specimens made of the same photoresists9.\nIn the process of varying the design parameters Reff and n, the woven fiber radius was kept constant at 1 μm and the unit cell size at 60 μm for all experiments, ensuring consistent material properties across designs even if size effects were present. Therefore, differences in mechanical properties due to feature size should be marginal, and do not impact the conclusions presented in the manuscript. To further substantiate this point, we conducted additional uniaxial tension experiments on monolithic polymer pillars with varying diameters (Fig. R8). These experiments revealed only marginal variations in mechanical properties. This consistency can be attributed to the high laser power used during the fabrication process. Notably, laser powers exceeding 10% above the selected level induced significant cavitation, indicating the system was operating near the upper limit of the resin’s processing window. The selected laser power ensured a high but homogeneous degree of crosslinking across all fabricated struts or fibers.\nComment 2.3: Figure 3 compares the specific energy absorption and specific modulus of woven metamaterials with different design parameters across various lattice configurations. However, it is essential and intuitive to benchmark these properties against existing literature on woven metamaterials and other lightweight structures. A more detailed Ashby plot is recommended.\nResponse: We fully agree with the reviewer’s suggestion to benchmark the specific energy absorption and modulus values reported in Fig. 3d. We have added benchmark values for woven and equivalent monolithic lattices tested in tension from works in the existing literature3,6 to the Ashby plot in this figure, better illustrating the expansion of the woven metamaterial property space reported in this work. This revised Ashby chart highlights new high-specific-energy-absorption designs (right-most point), while adding to the extreme compliance property space (bottom space).\nComment 2.4: Despite the forward design methods discussed in the manuscript, inverse design strategies for energy-absorbing metamaterials have concurrently emerged. A brief discussion of these advances is therefore encouraged, for example adma.202405567, advs.202204977 and s42256-025-01067-x.\nResponse: We thank the reviewer for this valuable suggestion. We have expanded our discussion to place our forward-design framework in the broader context of recent advances in inverse design strategies for energy-absorbing metamaterials. In the revised manuscript, we now highlight that while our method provides a deterministic, geometry-based pathway to generate heterogeneous and spatially varying woven lattices, inverse design approaches such as data-driven optimization have concurrently emerged as powerful tools for discovering metamaterial architectures with targeted mechanical responses10–12.\nWe further emphasize that these approaches are complementary. Unified frameworks for the parametric design and computational modeling of woven lattices, such as the one proposed in this work, provide a compact and physically interpretable design space. This can substantially benefit inverse design algorithms by reducing the dimensionality, improving search efficiency, and enabling optimization over architectures that would be impractical to construct manually. In turn, inverse design methods offer a promising route to take full advantage of our framework by automatically identifying spatially varying weaving patterns that optimize deformation modes, failure pathways, or energy-dissipation characteristics.\nTo reflect this perspective, we have added the following statements to the revised discussion section:\n”This unified forward approach also unlocks opportunities for studying the inverse design of woven lattices, including data-driven strategies that have recently emerged for energy-absorbing metamaterials. Integrating our parametrization with such inverse methods could enable systematic discovery of woven architectures with tailored large-deformation and failure responses10–12. To overcome potential challenges in the creation of sufficiently large computational datasets—where nonlinear processes such as failure and contact substantially increase the cost of each simulation—our framework could also allow for fully experimental data sets that use our parametrization for inverse design13.”\nWe hope that this expanded discussion satisfactorily addresses the reviewer’s comment and strengthens the manuscript’s connection to current developments in the field.\nComment 2.5: In Figure 4c, the peak values of the simulation curve exhibit a significant discrepancy compared to the experimental results. The statements about “absence of manufacturing defects in our models which are inevitable in experimental specimens” is too vague. It is suggested to carefully verify the comparison of the experimental and simulation results.\nResponse: We agree with the reviewer that this statement needs to be better supported and explained. To verify our understanding of the comparison between beam simulation and experimental results, we performed additional simulations on the sample lattice compared in Fig. 4c (Fig. R10, reproduced below). In order to clarify the effect of possible fabrication-related defects, we performed three additional studies adjusting the material model used for the beam-element simulations. In essence, the argument is that there exists an inevitable distribution in the effective material properties throughout the lattice—due to micro/nanoscopic defects or irregularities in the crosslinking—which is an approach that has been successfully adopted in the literature of architected materials14.\nIn each study, we assigned each beam element a randomized failure criterion (i.e., plastic strain at failure) within a binned normal distribution whose mean was the original, experimentally derived value and a standard deviation set to three different percentages of the mean (Fig. R10b). Upon distributing these material properties randomly, each beam element had material properties independent from its adjacent elements.\nThe results suggest that the differences between the simulations and experiments are due to over-idealization of the material properties in the beam-element approach. A larger standard deviation of plastic strain at failure, corresponding to a broader range of representative values, reduces the maximum stress achieved by the lattice (Fig. S14a), in a manner that more accurately captures the experiments.\nThese simulations still represent an approximation to the experimental behavior. More advanced modeling approaches, including further refinement of the material model and contact interaction, may be required to address this remaining discrepancy.\nComment 2.6: The regulatory effect on deformation modes and failure paths is remarkably significant. However, only simulation fitting results are presented in the paper. Have experimental attempts been made to achieve the specified failure paths? Under manufacturing tolerances, substantial deviations from idealized results may occur.\nResponse: We wholeheartedly agree with the reviewer that experimental validation of this result would strengthen this work. To that end, we have performed an additional experiment on a lattice with a sinusoidal pattern of crack path and performed uniaxial tension to failure. The results of this experiment, shown in the updated Fig. 5 (reproduced below), supplemental video 5, and in section S5, provide initial demonstration that fabricating failure paths is achievable in experiment—with remarkable agreement to the idealized designed failure.\nReviewer #2 (Remarks on code availability): I have carefully reviewed the code and can confirm that all results presented in the manuscript are reproducible. The codebase is well-structured and will serve as a valuable resource for the broader metamaterials community in future research.\nResponse: We thank the reviewer for their thorough review and for their positive comments.\nREVIEWER #3\nGeneral comment: This manuscript focuses on the framework to design woven structures and demonstrates their tunable mechanical properties and programmable failure patterns. Generally, it is interesting by exploring the stretchable regime of mechanical metamaterials. In contrast, plenty of works have been done regarding the compressive mechanical behavior while ignoring the stretching counterparts. This manuscript paths a way for researchers to the compliance and stretchability of mechanical metamaterials. I have no doubts on the correctness of this work, but have some concerns on the superiority, applicability and engineering potentials. Before recommending for publication, I have following questions.\nResponse: We thank the reviewer for their careful review of our work and constructive comments. The following are point-by-point responses to the reviewer’s comments and suggestions.\nComment 3.1: The node design is interesting. Is the number of strands in woven beams pre-determined by the parent monolithic-beam lattice? For example, the beams lattices in fig 1a can only be replaced by woven structures with 4 fibers due to connectivity. Are there cases where three or other different number of fibers is applied?\nResponse: The number of strands is indeed determined by the parent lattice. In further detail, the number of strands is defined by the connectivity of the spherical dual graph used to construct the node (see Figure 1a, inset under the heading “Node graph”; Figure S1b). This graph represents the strands that connect neighboring beams. In the cubic lattice shown in Figure 1, as well as the tetrakaidecahedron lattice, this dual graph has a degree of four, and thus four strands make up each woven beam. In the BCC, diamond, and octahedron lattices, the degree is three, and three strands make up each woven beam. These important details are now presented in a new Supplementary Figure S2, reproduced below for convenience.\nComment 3.2: In page 10, the authors demonstrated that “all tested woven lattices retained remarkable stretchability, sustaining stretches of at least 2 and up to 4 before complete failure.” This is remarkable. However, it is not solid convincing to compare with other structures in literature. The comparative study should be: How good is the large-deformation performance compared with conventional monolithic beam lattices counterparts that occupy identical institutive material and relative density?\nResponse: We appreciate the valuable feedback concerning the comparison between woven and monolithic lattices of identical material and relative density. To address this point, we have performed an additional study directly comparing identically simulated lattices of matching relative density, and the results are shown in Fig. R13, reproduced below. In short, the woven lattices as shown in Fig. 3 demonstrate considerably higher stretch and much lower stiffness than the monolithic beam lattices in the matching range of relative densities. Therefore, the various 3D woven lattices presented in the main text expand the parameter space of architected materials into an extreme-compliance and high-stretchability space.\nComment 3.3: Similar to the former question, is the energy absorption performance better or worse than their monolithic beam-and-junction-based counterparts? Especially in the case of stretching conditions.\nResponse: The comparison of specific energy absorption, as demonstrated in the revised Fig. 3 with reference to the published literature and also shown in Fig. S17, shows comparable performance in tension for most monolithic and woven lattices. The monolithic cubic lattice, when loaded along the direction of its component members, stands as an outlier in this analysis; the range is otherwise similar for all simulated lattices. This indicates that the decrease in stiffness in the woven lattices is offset by the increase in stretch before failure.\nComment 3.4: With regard to energy absorption, is it derived from tension-displacement curves for woven lattices while obtained from compression-displacement curves for beam lattices, respectively?\nResponse: For both woven and beam lattices, the energy absorption values noted are in tension. We have clarified this point in Methods under the heading Selection of Geometric Parameters. However, we have also considered compression performance, as noted in the following response.\nComment 3.5: It is for sure that woven lattices have significantly improved the extensibility than monolithic beam lattices. However, the latter was mainly designed for superior compressive behavior which woven lattices fall short of. This is like a coin that has two sides. I am concerned on the negative side of woven lattices, i.e., the compression performances.\nResponse: We appreciate the reviewer’s concern regarding the compressive performance of woven lattices. Although woven architectures are often highlighted for their high extensibility in tension, prior studies from our group have also shown that they can sustain large compressive strains with good resilience 6—i.e., minimal permanent deformation. To further verify this, we conducted cyclic compression experiments on two representative woven lattices (BCC and octahedron), using 4×4×4 lattice specimens compressed to 15% strain for ten consecutive cycles.\nAs presented in Fig. S8 and detailed in Section S3, both lattices deform stably under repeated loading and show minimal changes in their overall stress–strain response. The modest reduction observed after each cycle is primarily due to the viscoelastic behavior of the photoresist, rather than permanent structural damage. Consistent with this interpretation, both woven architectures retain similar levels of normalized energy absorption, decreasing to approximately 43–45% of the first-cycle value after ten cycles, and we observe no evidence of fiber breakage or irreversible deformation. These results indicate that woven lattices still provide robust compressive performance within the tested strain range, despite their enhanced tensile extensibility.\nComment 3.6: In fig. 3a, is the peak position of the stress-strain curve where fiber fracture starts to appear?\nResponse: That is correct, although multiple peaks may occur due individual fibers straightening and the stick-slip behavior of the entangled fibers. We have adjusted the placement of callouts in Figure 3a to more accurately reflect this relationship and highlighted this in the figure caption.\nComment 3.7: The lattice microstructure is to some extent complicated in its geometry, where each beam is replaced by three or four helical fibers. There has been reported structures with only one fiber and demonstrates 230% extension, i.e., (2020) 11:1180, https://doi.org/10.1038/s41467-020-14996-5. What is the unique advantage or significant progress of this proposed structure?\nResponse: We thank the reviewer for raising this important point. The structure reported in the cited study indeed achieves large tensile extension by using single helically arranged fibers. However, the woven lattices proposed in our work provide several distinct advantages that represent a meaningful step forward.\nFirst, woven lattices eliminate the nodal stress concentrations that arise when individual fibers terminate or merge at discrete junctions. In our architecture, fibers continuously pass through the lattice without forming traditional nodes, which avoids localized stress buildup and promotes more uniform load distribution throughout the structure.\nSecond, the interlocking of fibers at the nominal “nodes” produces beneficial self-contact during deformation. This interaction introduces frictional energy dissipation through sliding and entanglements and diffuses damage over a broader region, as demonstrated in Fig. 4 and Fig. S15. These mechanisms, which enable enhanced energy dissipation, are absent in architectures constructed from single independent fibers.\nThird, the redundant network created by fibers that traverse multiple unit cells enables extreme extensibility in the very large deformation regime. When the lattice is highly stretched or partially damaged, long continuous fibers are able to slide through the structure while remaining engaged. This behavior supports extensions greater than 200 percent and is evident in both our simulations (Fig. 4 and Fig. S15) and experiments (Fig. 3). The ability to maintain structural connectivity under severe deformation is a key differentiator from single-fiber designs, which lack this type of redundancy. This feature could be particularly valuable in functional applications that require uninterrupted conductive pathways.\nTaken together, these features show that the proposed woven architecture introduces important advantages in stress distribution, energy dissipation, and large-strain robustness that complement and extend beyond the capabilities of previously reported single-fiber systems.\nComment 3.8: It is easy to envisage that monolithic metamaterials can have superior energy absorption behavior applied in crushing and impact of protective equipment, automotive, civil structures, et al. What are the major potential engineering applications for such stretching-featured mechanical metamaterials?\nResponse: We consider these highly compliant, highly tunable metamaterials to have applications in multiple disciplines, such as biological scaffolds and compliant sensors and actuators.\nThe low density, high porosity, and stiffness range of these compliant materials (∼ 30 − 500 kPa) are comparable to a range of biological tissues. Their high tunability and fibrous network structure allows them to mimic a diversity of living tissues, making them well suited to biomedical engineering applications. For further quantitative support for this application area, we refer the reviewer to our response to comment 1.5. In addition, because of their high spatial tunability, scaffolds built of these mechanical metamaterials could be tailored to enable such applications as guiding cell alignment through grading of stiffness.\nTheir extensibility and spatially tunable structure also provides opportunities for resilient and defect-tolerant actuators and sensors. Their high self-contact and tailorable large-deformation behavior, without requiring highly elastic constituent materials, make them suitable for highly stretchable device components, both for applications such as wearables and for pressure sensors with wide sensing ranges. With extreme extensibility and tunable reconfiguration and failure responses, they could be used in lightweight impact and tension mitigation.\nComment 3.9: As far as I understand, the different fiber strands are detached and not in contact with each other in their original state. In the very initial stage, the contact pressure should be zero. Can the authors present the approximate contact pressure (real value or magnitude, not max/min ratio) during the stretching process?\nResponse: We thank the reviewer for highlighting this point. Indeed, in the inital state of the lattice, the contact pressure is zero. We have updated the illustration of contact pressure in Figure S15 and its caption to clarify this point, including per-element contact magnitude values and total contact pressure.\nReviewer #3 (Remarks on code availability): Finite Element Modeling is well presented in the Methods section. The simulaiton is completed using commercial available software ABAQUS/Explicit 2023. Details like periodic boundary conditions, mesh types and mesh sizes are all depicted.\nResponse: We thank the reviewer for their careful review of our methods."}, {"round": 2, "reviewer_comments": {"reviewer_1": "Since all comments are addressed point by point, this reviewer recommends its publication in current version.", "reviewer_2": "The authors have addressed most of my comments in their revised manuscript. Regarding the discussion of size effect in comment 2.2, it seems insufficiently rigorous to conclude that the material exhibits no size effect based on tensile tests. For instance, size effect might become apparent under bending conditions. For example JMPS 51(2003)1477-1508. The reviewer’s intention is not to ask for additional experiments, but to encourage the authors to revise the wording so that to make the discussion more rigorous.", "reviewer_3": "Considering that all my previous concerns have been clarified, i recommend it for publication."}, "author_response": "REVIEWER #1\nRemarks to the Author: Since all comments are addressed point by point, this reviewer recommends its publication in current version.\nResponse:\nWe thank the reviewer for their positive recommendation and for their thorough and constructive feedback, which has helped us strengthen the manuscript.\nREVIEWER #2\nRemarks to the Author: The authors have addressed most of my comments in their revised manuscript. Regarding the discussion of size effect in comment 2.2, it seems insufficiently rigorous to conclude that the material exhibits no size effect based on tensile tests. For instance, size effect might become apparent under bending conditions. For example JMPS 51(2003)1477-1508. The reviewer’s intention is not to ask for additional experiments, but to encourage the authors to revise the wording so that to make the discussion more rigorous.\nResponse:\nWe thank the reviewer for highlighting this remaining point. We have carefully considered the reviewer’s comment as well as the suggested work. Although we agree that unconsidered size effects may be present at the scale of our material, because the fiber radius is kept constant throughout the experiments, these effects should not affect the relative behavior and the conclusions in the manuscript. However, this is a point that we should determine rigorously in subsequent work, specifically in the presence of strain gradients and not simple uniaxial loading—we thank the reviewer for bringing this to our attention.\nTo address this point and improve the rigor of our discussion, we have revised the main text to include the following when introducing the samples we fabricated:\n“ To ensure that potential size effects on the response of the polymer did not affect the responses across samples or complex stress states within the fibers, all design variations maintained the same fiber radius.”\nFurthermore, in the discussion, we have enhanced this point via the following wording:\n“While this study focuses on five fundamental woven unit-cell topologies with realizations at the microscale, future extensions of this framework could explore multi-scale hierarchical lattices that integrate stochastic and periodic elements to achieve tailored mechanical responses. To ensure applicability of this framework to woven topologies at larger scales, comparison between micro- and macroscale samples will benefit from rigorous identification of potential size effects, emerging from changing tribological or elastic properties under complex strain gradients.”\nREVIEWER #3\nRemark to the Author: Considering that all my previous concerns have been clarified, i recommend it for publication.\nResponse:\nWe thank the viewer for their kind recommendation and for their constructive feedback, which has helped us improve the quality of this work. We are glad that the updated manuscript has addressed all of the reviewer’s concerns.\nRemark on code availability: From my view, they are in details and are clear enough. The finite element simulations are well reproducible.\nResponse:\nWe sincerely thank the reviewer for their thorough evaluation of our simulation work."}]

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