Topological mechanical metamaterial for robust and ductile one-way fracturing

Domain: neuroscience
Published: 2026-02-06
Editorial outcome: accept

Abstract
Fracturing is unavoidable and threatens the reliability and functionality of materials. Therefore, regulating the propagation of cracks in a predictable and ductile manner is of paramount importance. Herein, we exploit the properties of topological mechanical metamaterials (TMMs) as a versatile mechanism to guide cracks unidirectionally and turn fracturing of lattices made of brittle materials into ductile events. Inspired by quantum topological states, recent discoveries of TMMs have uncovered varieties of unconventional mechanical phenomena, ranging from one-way wave propagation to polar elasticity. We show that polarized floppy modes occurring in TMMs lead to strongly asymmetric stress fields localizing around the notch tips, leading to ductile one-way fracturing, in sharp contrast to classical theories of fractures in brittle materials. Our work demonstrates the universality of this fracture unidirectionality feature, which is protected by TMM’s bulk topology, and provides robust solutions in programmable fracturing for a broad class of materials and structures.

Introduction
Fracture undermines the stability of structural and material systems across scales and physical realms, from rocks1, bones and teeth2,3, polymers4–12, to two-dimensional materials13–15 and microscalemolecules and fibers16–19. Fractures are inevitable; what one can do is design materials endowed with the ability to retard the manifestation of fracturing (crack initiation and propagation) or mitigate the severity of fracturing events to avoid the onset of catastrophic damage. To this end, understanding and subsequently controlling the parameters that guide fracture propagation remains a key challenge in theoretical and practical research in fracture mechanics20–32. Recently proposed methods to steer cracking direction include applying different external loading conditions23,33–35, controlling material anisotropy (e.g., using bio-inspired nacre36 and seashell)37, programming the micro-structure geometry38–41 and introducing local curvature42,43. However, it is a universal phenomenon in fracturing that sharp geometries of the initial notch, small defects and system imperfections can act as stress concentrators, influencing both crack initiation and propagation44, interfering with the control of cracking directions, leading to indeterminacy as observed in ref. 39. Despite this abundance of approaches, a more robust principle using geometry to guide the direction of fractures that is insensitive to notch geometries and imperfections is still lacking.

In the past decade, principles from quantum topological states of matter have been widely utilized to design mechanical metamaterials featuring topologically protected properties, such as one-way edge states and polarized elasticity45–49. Among them, Maxwell lattices have attracted extensive interest due to the ability to topologically control stress and strain fields, offering programmable soft boundaries45,50–52, transformable responses53–55, programmable stress distribution56–59, and low-frequency edge waves60,61. Earlier studies of TMMs have focused on exploring topological floppy modes49–54,60,61 rather than controlling stress fields. A handful of papers studied fracturing control using TMMs but they either focused on the capability of using states of self-stress to concentrate stress, predominately in small strain regimes before fracturing57–59, or only notches on the side62,63.

In this paper, we show that topological mechanics of Maxwell lattices can be used to robustly guide fractures to propagate in a unidirectional way even during the nonlinear fracturing process, and we demonstrate this effect using theories, simulations and experiments. We show that the topological protection of TMMs endow them with unidirectional cracking that is robust against geometric details. We find that the topological polarization, which controls the spatial patterns of the stress fields and specifically the regions of stress focusing, is responsible for strong asymmetries in the stress fields that localize around cracks, causing the lattice to fracture along a deterministic and predictable direction. Remarkably, this property is preserved even when the fracture progresses into the nonlinear regime. Moreover, the sequential activation of topological zero modes resulting from the onset of crack make the fracture process ductile, although the material the lattice is made from is brittle. Such geometry-induced effective ductility drastically reduces the occurrence of catastrophic failures.

We provide rationales for our results in light of topologically protected zero modes (ZMs) in Maxwell TMMs, and compare our results with linear elastic fracture mechanics (LEFM)64–67. Specifically, we find that topological ZMs generate stress distributions that significantly differ from those predicted by LEFM in terms of both asymmetries and angular profiles near the crack tip, thus providing drivers for crack propagation that differ qualitatively by those observed in classical media as well as in nonpolarized lattices.

In summary, the properties of TMMs enables several advances in fracture mechanics and engineering: (1) using the combination of purely periodic geometry and isotropic constituent materials, the initial cracks are guided in a unidirectional and robust way even considering the nonlinear deformation at crack tips and material nonlinearity, for a wide range crack shapes; (2) protected by topology, the TMM is less sensitive to defects from fabrication and variances of notch shapes. (3) the TMM turns a sudden, brittle and unpredictable failure into a ductile and predictable one with early indicators; (4) the TMM enables on-demand control of crack tip stress conditions and achieves diffusive and concentrated fracturing in the same material; (5) based on topological kagome lattices, the TMM is light-weight.

Results
Fracture of Maxwell lattice TMMs
In a homogeneous material, stress fields focus at the two tips of a crack symmetrically (Fig. 1A)69. As a result, in practical scenarios involving brittle materials (Supplementary Fig. 1), the propagation direction (to the left or to the right) that the crack eventually follows is highly sensitive to the material’s defects or imperfections and thus hard to predict or control. Moreover, the damage process is sudden and often without significant prior deformation (Supplementary Movie 1). The aim of crack-guiding functional materials35,38,39 is to break the symmetry of the geometry and guide the crack propagation in a unidirectional and predictable way.

A trivial approach to breaking this symmetry could be obtained by modulating material constants or disrupting mirror symmetry in micro-geometry. Such approaches suffer from material and strain dependence. Here, in contrast, we explore the architecture of polarized Maxwell lattice-based TMMs (Fig. 1B) as a means to manipulate the stress field and direct the crack propagation in a robust way based on topological protection. Maxwell lattices are mechanical lattices with an equal number of degrees of freedom and constraints in the bulk, thus operating on the verge of mechanical instability. They are known to exhibit topologically polarized elasticity, which manifests as the localization of states of self stress (SSS) and zero modes (ZMs) at opposite edges of a finite lattice domain45,50–52.

Topologically polarized elasticity predicts that, by replacing a solid continuum (Fig. 1A) with a TMM (Fig. 1B), an asymmetric stress field emerges in the neighborhood of a notch. Note that, in a lattice, a notch effectively consists of a sequence of severed ligaments (serving as structural hinges) along a line. This scenario is depicted in Fig. 1B via FEM simulations and experimental data70. For the experiment, we cut a Polyethylenimine (PEI) film into the TMM geometry in Fig. 1B and sandwich it between a pair of polarizers (see “Methods”, Supplementary Fig. 2). After subjecting it to slight stretching, it reveals an asymmetric stress field, indicated by the colormap.

We compare (1) a continuum medium (Fig. 1A) with a slim notch, (2) a topological kagome lattice TMM (Fig. 1B) and (3) a regular kagome lattice (Fig. 2A) - the latter two cases featuring a line of severed hinges acting as crack. Applying the same amount of total stress to all configurations, we extract stress profiles along horizontal paths between the immediate neighborhood of the notch tips and the free boundaries (Supplementary Figs. 3, 4). For the two lattice configurations, we adopt the average von Mises stress in the structural hinges as a stress indicator (Methods). Among the three, only the topological kagome shows a strongly asymmetric stress field above the notch. (1) For the continuum medium, the stress decays as 1/\sqrt{r}, where r is the distance from the notch tip (gray curve in Fig. 1C), consistent with classical theories64–67. (2) For the TMM (green curve), interestingly, the left side of the crack does not display a stress build-up at the effective notch tip, while on the right side the tip stress is three times the bulk value. The fluctuations at the left edge are a signature of the floppy modes available on that open edge due to polarization. The deformation mode in the bulk is a combination of stretching and bending of the hinges. (3) For the regular kagome (orange curve), an increase of stress from the bulk to the crack tip is observed, albeit smoother than in the continuum medium and the right side of topological kagome (Supplementary Figs. 6, 7), which can be explained by the blunting effect of kagome lattices71,72. For this lattice, the dominant deformation mode in the bulk is stretching. In addition, the sudden drop of the stress near the free boundaries (Fig. 1C)71 can be explained by noting that, at a free boundary, due to the lack of constraints, the deformation mode switches locally to bending-dominated and exhibits the deepest boundary layer of the three cases.

We further show that the trend of asymmetric tip stresses in TMMs continues deep into the fracturing process, beyond the linear elastic states of stress considered in previous studies56–58, injecting asymmetry into the actual damage propagation regime. Large rotations at the notch tip demonstrate geometric nonlinearity (circles at the tip in Fig. 1I–K) and the failure process includes material nonlinearity. We test several samples for each medium type (continuum, regular kagome and TMM) featuring the same length, width, and notch size to extract some statistics of failure (Fig. 1D) and document the existence or lack thereof of crack propagation directionality in these systems. Specifically, we count the number of instances where the crack propagates left/right. As shown in Fig. 1D, when it comes to realistic experimental conditions, for the continuum medium and the regular kagome it is impossible to predict (or enforce) a priori which way the crack will propagate or induce symmetric, simultaneous, two-way propagation from both tips of the crack. This is despite the fact that they both show perfectly symmetric stress fields in simulations (Fig. 1C). In Fig. 1E, we report normalized experimental force-displacement curves acquired during the tests (Supplementary Fig. 6). The curve for the continuum samples (gray line) clearly shows a sharp drop from the peak state, representative of canonical brittle behavior. An intermediate and final cracked state of a continuum sample are shown in Fig. 1G and H, respectively. The paper sample undergoes infinitesimal deformation before undergoing sudden failure (Supplementary Movie 1), and the direction of crack propagation is unpredictable from one sample to the next.

In contrast, the TMM presents a much slower manifestation of failure reminiscent of ductile failure, as indicated by the green line in Fig. 1E, which features a jagged profile that stretches the failure process over an elongation range 5–6 times larger than that observed for the continuum (Supplementary Movie 2). Moreover, driven by the significant stress difference between the two tips (Fig. 1C), the crack propagates consistently to the right, as indicated by the 100% count in Fig. 1D. The local physical processes involved in crack opening for the TMM are documented in the snapshots of Fig. 1I-M. Interestingly, before cracking, the TMM gives a warning: in both simulations and experiments, the circled right tip hinge (brown circle) shows an opening angle larger than the others by ≈10° (Fig. 1I). The breaking of the first hinge on the right side corresponds to the first drop in the jagged force profile of Fig. 1E. At 203 s, with the first hinge broken, the largest stress is found in the second right tip hinge (circled), as shown in Fig. 1J. After that, the crack keeps propagating to the right, reaching the state of Fig. 1K at 333 s, with the second hinge broken and the largest stress found in the third hinge, and that of Fig. 1K at 600 s, when the one-way cracking pattern is fully formed in simulations. In the experiments, due to setup imperfections (e.g., hinge fabrication and mixed-mode loading, see Methods) and proximity to the free boundary, hinges on the left started to break earlier than expected. The final state shown in Fig. 1M at 1300 s mark the collapse point where the sample loses integrity. Inspection of the deformation and stress fields reveals something additional about the mechanics of the process. As the first hinge breaks and the second one carries the largest stress, it experiences a more pronounced bending deformation, which is accompanied by an increase in stiffness (Fig. 1E, I-L). When the second hinge goes to the maximum allowed strain and breaks, we observe stress concentration in the third hinge on the right at larger strain. This nonlinear phenomenon is due to the sequential activation of topological ZMs in the nonlinear regime, as we discuss in the next section.

Topological zero modes control the asymmetric stress field
The strongly asymmetric distribution of stress induced by crack opening in TMMs can be understood via the analysis of the localized topological ZMs45,48–51 that are triggered during the deformation process. Here we first discuss this effect in the case of ideal Maxwell lattices, featuring only nearest-neighbor (NN) bonds and then generalize it to structural TMMs featuring additional weak bending constraints which can be characterized by next-nearest-neighbor (NNN) bonds.

In the NN-only lattice, the nearly-ZMs near the notch leads to a strongly asymmetric displacement field. In a topological kagome lattice as an NN-bond only ideal Maxwell lattice, each node has two degrees of freedom from the in-plane displacements and two constraints from the four bonds connecting to it (as each bond is shared by 2 nodes). Maxwell’s counting rule of equal numbers of degrees of freedom and constraints is satisfied at every wavenumber, and the system has two homogeneous ZMs (x, y translations) and two homogeneous SSS that provide the elastic moduli for the coarse-grained lattice. Assume removing 1 node and 4 bonds in the bulk to open a small crack (Fig. 3A). This operation removes two degrees of freedom and four constraints, and as a result, it lifts the two homogeneous SSS (Supplementary Fig. 14A) and creates two nearly-ZMs near the notch (the even mode is shown in Fig. 3A). Different from ZMs, nearly-ZMs exhibit not exactly but nearly zero frequency. The frequencies of these two nearly-ZMs vanish exponentially as a function of the system size. A simulation of the ideal lattice (under either periodic boundary conditions or fixed boundary conditions, if the lattice is large) reveals the spatial pattern of these two nearly-ZMs, which are localized around the right tip of the notch, as a result of the topological polarization R_T, following the predictions of topological mechanics theories45,48,49. The two nearly-ZMs are even u(e) and odd u(o) with respect to the horizontal mirror plane of the notch, respectively, and they have dynamical frequencies λ_e, λ_o (corresponding to mode frequencies λ_e = m, λ_0 = m with m being the mass of the nodes). These two nearly-ZMs exhibit two supersymmetry partners, which are nearly-SSSs45, s(e) and s(o).

The response of the lattice upon an external load of y-stretching, as we derive in detail in the Supplementary Note IID, is dominated by the even nearly-ZM u(e) (Fig. 3A) via

u_i = u_i^{(aff)} + \sum_e \frac{1}{\sqrt{\lambda_e}} s_e^{(b)} E_b^{(aff)} u_i^{(e)}, (1)

where i labels the nodes, b labels the bonds, u(aff), E(aff) are the affine (homogeneous strain) displacement and bond extension of the lattice under the load, and the nonaffine response is controlled by the even nearly-ZM u(e), with the coefficient given by the overlap of the affine strain E(aff) with the corresponding even nearly-SSS s(e).

A complimentary way to understand the topological nature of this strongly asymmetric displacement field is to realize that the bulk solution of ZMs in Maxwell lattices can be generally written in the form of e^{iq \cdot r}, where r is the position and the imaginary part of q controls the decay of the ZMs45,51,53. The horizontal topological polarization R_T of the lattices we consider here directly controls the imaginary part of q_x of the ZMs. For each given real q_y, there are two ZM solutions, both of which have the imaginary part q''_x > 0 for the topological kagome, but the two q''_x's have opposite signs for the twisted kagome which is unpolarized (Supplementary Fig. 11). The actual displacement field upon loading comes from a linear combination of these ZMs, with the coefficients determined by the boundary conditions. The notch in the middle allows for finite displacement amplitude around it. The displacement must decay towards the right in the topological kagome, consistent with the numerical results (Fig. 3). Importantly, if there is any displacement amplitude on the left tip, it must exponentially increase towards the bulk on the left, which is not allowed. This requires vanishing displacement on the left tip in the topological ideal Maxwell lattice case. In contrast, in the twisted kagome lattice, decay occurs in both ±x directions, giving a ratio of displacements of order unity between the two tips (Supplementary Fig. 11). This argument also generalizes this phenomenon from the one-node notch to other sizes and shapes of cut, as we mention below.

The discussions above describe the response of the NN-only topological Maxwell lattice under load, where due to the lack of SSSs48,49, the lattice moves along the linear combination of the affine strain and the nonaffine nearly-ZM (Eq. (1)), and there is no stress to the linear order (Supplementary Fig. 14A). The strong asymmetry of the nearly-ZM gives rise to the strong asymmetry of the displacement field of the lattice under load, leading to stress responses only at the nonlinear level (Supplementary Information IIE).

To fully understand the stress response of the TMM, we theoretically analyze the topological kagome lattice with NNN bonds, and compare with numerical results on the NNN lattice characterized by a spring constant ratio k_NNN/k_NN, and the TMM which is a continuum model characterized by normalized hinge thickness h/|a_1|. Firstly, we verify that despite the deviation from the ideal Maxwell limit, the displacement responses of the NNN lattice (Fig. 3B) and the TMM (Fig. 3C) under load is indeed dominated by the even nearly-ZM shown in Fig. 3A, featuring drastically different rotations at hinges at the left and right tips (Fig. 3D). Secondly, a subtle difference arises, as the weak additional constraints due to the NNN bonds or the bending stiffness of the hinges induce homogeneous SSS even in the presence of the notch. This leads to weak stress responses at the linear level, mainly carried by the NNN bonds (Fig. 3E). In contrast, the NN stress increases nonlinearly at small k_NNN/k_NN, as determined by the ideal theory which applies to marginally stable ideal lattices. When the NNN spring constant or the hinge thickness becomes large, the lattice deviates from the ideal Maxwell case and crosses over to normal elastic media behavior where geometric details prevail, with the topological polarization effect vanishing (Fig. 3F, G). This result further confirms that the observed asymmetric features originate primarily from topology rather than geometry. As the NNN spring constant increases, the lattice shape remains geometrically asymmetric, maintaining one sharp side and one blunt side, yet the stress difference between crack tips gradually diminishes (Fig. 3F). The tip stress difference can also be tuned by varying the unit cell geometry which controls the localization length of the topological ZMs, characterizing the polarization magnitude (Supplementary Note IIH). Less polarized configurations exhibit a weaker asymmetric feature. In practice, finding the best configuration is essential. Reducing the polarization, either by lowering its magnitude or by increasing the hinge thickness, weakens control over one-way cracking.

Due to the robustness of the topological polarization, this strong asymmetry of strain and stress applies more generally beyond this simple scenario of a small notch (Supplementary Fig. 12). When the notch is longer, more nearly-ZMs are generated. However, due to the topological nature of the ZM profiles in the bulk, the excited combination of ZM is still characterized by strong stress on the right, as shown in Fig. 1, Supplementary Information IIE and Supplementary Fig. 13. This also applies to notches of other shapes. For simplicity, we will just use the term “ZMs” to refer to both exact ZMs and nearly-ZMs in other sections.

Furthermore, as the first hinge on the right breaks due to high stress, the lattice gains more ZMs which again asymmetrically localize more on the right than on the left due to the topological polarization R_T. The process of nonlinear deformation along the ZMs, generating high bending stress which evolves into stretching stress at higher strain at the right tip, occurs again until the next hinge breaks, and repeats until all hinges on the right break, as we show in Fig. 1. The nonlinear excitation of the topological ZMs is behind the ductility of this fracturing process, although the original material is brittle.

Comparison with linear elastic fracture mechanics
It is instructive to compare the asymmetric stress fields observed in TMMs to the classical solution of linear elastic fracture mechanics (LEFM). In LEFM, the tip stress is symmetric between left and right tips, and the dependence of the tip stress field upon the angle around, and distance from, the crack tip can be treated as separated variables64–67. Our tests and simulations suggest that the stress fields established in the Maxwell TMMs fundamentally differ from the LEFM cases in both respects. To quantitatively study this effect, we compare four cases: a topologically polarized kagome TMM, a regular kagome lattice, an elastic continuum, and the analytical solution from LEFM. For TMM and regular kagome, the stress fields are shown in Fig. 4A and B, respectively. Since most stress localizes at the hinges, here, for visualization purposes, we extract the hinge stresses (see Material and Methods) and interpolate them over the entire sample domain to obtain a continuous effective stress field. The deviations from LEFM predictions can be agilely appreciated by inspection. The left-right symmetry is broken at the center of the tip, and this behavior is observed robustly for different values of the hinge thickness parameter h, as shown in Fig. 4E. Moreover, the angular stress dependence (profiles of stress vs. angle γ or γ) is different when sampled at different distances (radii) from the tip, as shown in Fig. 4C, violating the independence of the radial and angular variables in LEFM. Figure 4C also shows that the left and right tip stress fields feature qualitatively different angular dependence, revealing an additional element of asymmetry due to polarization. For completeness, an analogous study is given for the regular kagome (green curve) and for the continuum (dark gray curve with markers for numerical simulations and light gray curve for analytical solution) in Fig. 4D. Here, all the curves apply to all radii, as the dependence upon γ or γ is decoupled from that upon distance.

We further study the case of elliptical notches in a continuum, for which finite LEFM analytical solutions are available, with emphasis on the dependence on the crack aspect ratio a/b. In Fig. 4F we show the stress field for a reference choice of a/b in a continuum. Not surprisingly, the stress field is symmetric close and far from the notch. From LEFM, the stress magnitude varies with a/b as σ_tip = σ_∞(1 + 2a/b), where σ_∞ is the far-field stress64–67. In Fig. 4G, we fix b and only vary a, de facto sweeping the aspect ratio between 1 and 4 and we monitor the ratio between the stress in the bulk, where the stress plateaus and can be taken as the far-field stress, and the tip stress. The trend of the gray curves confirm the expected decay. The slight deviation between simulation (dark gray with markers) and analytical solution (light gray) is due to numerics and progressively vanishes upon mesh refinement.

In Fig. 4G we also repeat the analysis for the two considered lattices (TMM and regular kagome). Interestingly, here the tip-to-bulk stress ratio appears to be independent of a/b, with the usual asymmetry between left and right tip exhibited by the TMM. The independence on defect length implies special mechanical features of the TMM: compared to normal media, the metamaterial shows a larger fracture toughness with delayed cracking due to slower increase of stress focusing as notch size increases.

Topology-controlled fracture: a portfolio of patterns
The one-way fracture mechanism discussed above is not limited to straight line notches located in the bulk. It also applies to initial notches of different shapes or differently located in the lattice domain. This robustness attribute proceed directly to the fact that the ZM-controlled fracture mechanism is topologically protected. The importance of notch-shape insensitivity lies in the fact that naturally occurring notches in materials take on diverse forms due to various types of damage. Achieving a metamaterial that guides cracks unidirectionally regardless of notch shape would greatly expand the applicability of fracture control in complex environments.

We demonstrate this through three representative cases (schematically depicted and color-coded in Fig. 5A): (I) a tilted cut in the bulk (green), (II) a cut at the soft edge (blue), and (III) a cut at the rigid edge (orange). For (I), we first check the distribution of ZMs predicted by theory for the idealized NN lattice under periodic boundary conditions (similar to Fig. 3A). After removing sites and bonds along the direction of lattice vector a_2 (Fig. 1B, Methods), all the ZMs are localized asymmetrically at only the right edge of the crack in Fig. 5D, as a result of topological polarization. We then examine the problem for the corresponding TMM in Fig. 5E. Simulations indicate that, under uniform stretching of the lattice, the deformation localizes at the right edge as well (see displacement colormap in Fig. 5E). Consequently, the stress developed in the hinges on the right edge is greater than that at the left edge, as depicted in Fig. 5B (left). A similar asymmetric pattern of ZMs and stress localization can also be found for other crack shapes (e.g., the triangular hole in Supplementary Fig. 19).

To quantify this asymmetry, we define n as the vector normal to the notch edge and α the angle between n and the polarization vector R_T. It is known from theory of polarized Maxwell lattices that the density of ZMs at an edge is proportional to n · R_T = cos α45. As described below, we find this quantity to be a good indicator of stress distribution around holes. In Fig. 5F, we consider holes of different orientations and plot the magnitude of ZMs at the edge and the bulk-to-edge stress ratio against the angle α. When α approaches π (or cos α → -1), the edge has no localized ZMs (the left side of the crack in Fig. 3A, Fig. 5D) and shows the same hinge stress as the bulk stress (Fig. 4A). On the other hand, as α goes to 0, the stress ratio approaches 0, implying a significant stress concentration at the edge. Here, the hinge thickness is set to h/|a_1| = 0.015 in Fig. 5B, C, F.

This asymmetry of stress distribution leads to robustly controllable crack propagation directions for a variety of hole orientations occurring in TMM domains, which we demonstrate numerically and experimentally in Fig. 5G, H. The initial state has a notch along a_2 in Fig. 5G. After being stretched, both nonlinear simulations and experiments show that the crack initially propagates to the right, emanating from the bottom-right tip and breaking sequentially the two adjacent hinges (Fig. 3G). After that, the crack also propagates to the left side, as shown in the Supplementary Movie 8, likely due to the proximity of the crack to the right boundary. The results emphasize that the unidirectional cracking mechanism is invariant to the shape of the bulk notch (see one-way cracking of a triangle notch in Supplementary Movie 9 and a rectangular notch in Supplementary Movie 10).

The same principle also controls stress fields and crack propagation driven by edge cuts, with some noteworthy differences between soft (cases II) and rigid (cases III) edges. In Fig. 5C, via simulations we calculate and plot the stress developed along the hinges in the same row along which the crack propagates, from the notch tip to the other side of the sample. We observe qualitatively different trends between the two cases. When the initial notch is at the soft edge, the stress drops sharply into the bulk, whereas when the initial notch is at the rigid edge, the stress decreases gradually. Overall, the decay of the soft edge curve follows a trend similar to the decay of hinge stress along the right tip path observed for the bulk crack in Fig. 4C. As shown in the experiment with initial notch at the soft edge (Fig. 5I–J), since the tip hinge has a much larger stress than the surrounding hinges (Fig. 5B, middle), the crack always propagates along the straight path (Fig. 5J). In contrast, the stress at the rigid edge is diffusive, uniformly distributed around the tip, as indicated by the color of the dots in the K inset (consistent with the numerical prediction in Fig. 5C). This results in multiple potential crack paths, as observed in the experiments documented in Fig. 5K–M. As a result, small random perturbations caused by experimental non-idealities can cause unexpected bifurcations of the cracking paths. In Fig. 5L, the crack turns directly into the direction along a_2. In the cases shown Fig. 5M, the crack propagates straightly first and then turns into the direction along a_2 at a later stage. Furthermore, in both simulations and experiments, the rigid edge exhibits delayed cracking compared to the soft edge (Supplementary Fig. 22E, F), despite similar force peaks in the force-displacement curves for cases (II) and (III).

These differences in fracturing behavior between the two edges suggest that crack tips in TMM display a variety of regimes, ranging from stable propagation at the soft tip to diffusive cracking at the rigid tip. Diffusive failures, characterized by fracturing without stress concentration, typically occur in systems with high disorder or low connectivity24,25,73. Previously, people achieve concentrated stress and diffusive stress tips in different systems by tuning network connectivity74 or rigidity9,24,25. The former appears in highly connected or rigid systems, while the latter shows up in systems with low connectivity. Here, we demonstrate that both concentrated and diffusive stress can emerge within the same over-constrained real system, allowing for on-demand fracturing control.

Programming fracture sequences and fracture resistance
Topologically controlled fracturing sequences can also be further programmed to control fracture resistance and mechanical stiffness in more versatile ways. By incorporating more polarization directions in the lattice, we can generate distinct cracking paths that result in tunable fracture resistance (Fig. 6). Here, we compare the single-domain TMM (with one polarization direction; see Fig. 1) to interface-TMMs (featuring two opposed polarization directions, Fig. 6A, C). In these systems, the programmed interfaces play a dominant role over the notch tips, due to the excited SSSs in the interfaces. In the photoelastic experiments, concentrated stress is clearly observed at the interfaces for both geometries (Fig. 6B, D), and this finding is consistent with the simulations. In Fig. 6E, F, cracks first originate in the interface regions (the gray box). Once the interface hinges fail, especially those in the zigzag interface near the notch and all hinges in the straight interface, the crack extends toward the notches. Owing to the topological polarization effect, the crack propagates to the right tip before reaching the left tip.

Furthermore, the shape of the interfaces influences not only the cracking path but also the force-displacement curves and fracture resistance (Fig. 6H). In the straight interface configuration, all interface units experience highly concentrated stress and break almost simultaneously (Fig. 6D, F–H), resulting in the earliest cracking and the most brittle behavior among the configurations. In contrast, the tortuous path in the zigzag interface geometry creates stress gradients, causing the interface units to break sequentially in a more gradual manner rather than all at once. As a result, the zigzag interface configuration absorbs more energy and exhibits the highest fracture toughness and linear elastic stiffness (Fig. 6H inset). These results highlight how topological features can be leveraged to precisely program stress distributions, crack sequences, and fracture resistance.

Methods
Experiments
We utilize Silhouette Cameo 4 Cutting Machine for cutting all the experiment samples. Our blade is the 0.2 mm Kraft Blade. We ensure that none of the hinges are broken before loading, to minimize initial imperfections. For the photoelastic setup in Fig. 1C and Supplementary Fig. 2, we set one polarizer in the front of the sample and the other behind the sample (with a 90 deg angle difference of the transmitting axis of the front one). The photo elastic thin film is PEI film with a thickness of 0.076 mm, purchased from McMASTER-CARR. The loading is performed quasi-statically, with a speed of 2 mm per minute. In Fig. 1, sample width w = 152.4 mm, sample length L = 157 mm, crack length 2a = 75 mm, and hinge thickness h ≈ 0.8 mm (h/|a_1| = 0.06) in Fig. 1H-L and Fig. 5G, H, I-M. Compared to idealized simulations, out-of-plane deformation in experimental tests of thin lattices can slow the rapid force drop and delays the cracking process77.

In Figs. 1 (D-M) and 5, the cardboard paper is the Wexford Folder 2-pocket (0.26 mm thick), purchased from Walgreens. Each paper sample weighs 3.6 g. The loading machine is Instron 4301 equipped with a 100 N loading cell and a 10 kN load cell. To allow uniform stretching/clamping at the top/bottom boundaries, we prepare four aluminum pieces (1.6 mm thick), sandwich each of the two sides between two pieces and glue them together. In Fig. 1D, for each of the three geometries, ten samples are loaded to observe the probabilities of fracturing direction.

In Fig. 2, we quantitatively characterize the triangular lattice, the regular kagome lattice, the twisted kagome lattice and the TMM. The ductility is measured as the elongation ratio at failure, Δ_c/l_0, where Δ_c is the critical displacement at the force peak and l_0 is the original sample length44. The fracture toughness is calculated as the critical stress intensity factor σ_c \sqrt{\pi a f(a/W)}, where σ_c is the critical stress applied, 2a is the full crack length, and f(a/W) is a specimen geometry dependent function of the crack length a and sample width W44. The consumed energy is ∫F dx, where F is the applied force and x is the displacement44. As shown in Supplementary Fig. 10H, fabricated hinge thickness shows a measure average 0.8 ± 0.2 mm. We measured 20% of the hinges in 10 representative samples (≈400 hinges in total). Notably, despite these minor fabrication errors, one-way cracking of the TMM is consistently observed across all specimens thanks to the topological protection, and error ranges of quantitative force-displacement curves remain within acceptable in Fig. 2B. The one-way propagation of cracks in the TMM is not affected by this variation, again demonstrating the robustness of topological mechanical effects.

In the material test, the strip samples have the size of 70 mm by 30 mm. Paper has two main axes (Figure)78, and in this work all the experimental stretching is along the length direction. In Supplementary Fig. 2D, the long edge of the strip is cut along the length direction of the paper folder.

The gripper rotated passively (side-notch results in Fig. 5) since it’s connected by a hinge. It mainly affects side cracks at the very end of the loading stage, due to the lack of constraints. For all the force-displacement measurements in this work, the cracks are in the bulk. The force-displacement curves measured in this work are all for bulk notches and therefore, the influence from top gripper rotation doesn’t affect the initial fracture events and plays a very minor role in the curves. For the sample in Supplementary Movie 2, after fracture initiates on the right, slight rotation of the grippers introduce mixed-mode loading conditions, leading to deviations in the crack path compared to idealized simulations. While this study focuses on the initial crack scenarios, the influence of boundary-induced mixed-mode effects is an important consideration for future work79.

Simulations
We use finite element software Abaqus/CAE 2023 for simulations80. Since we utilize cardboard paper as brittle material in experiments, we experimentally measured its stress-strain curve (Supplementary Fig. 2D, Supplementary Fig. 23A), and input that as the material parameters in the simulations. For spring-mass lattices, we apply truss elements (T2D3), which have no bending stiffness and can only carry axial stress. For continuous geometries, we choose reduced shell element (S4R).

We perform three types of simulations, linear frequency analysis (mode analysis in Figs. 3A, 5D), implicit static analysis (linear stress distribution in Figs. 1C, 3B, C, 4, and 5B, C, E, F) and explicit cracking simulations (Figs. 1 and 5H). For periodic boundary conditions (along x) in Figs. 3 and 5, we couple the degree of freedom of the left boundaries nodes with that of the right boundary nodes. The ZM weight is obtained via U_i = \sum ZM u_i^2, where i labels the nodes81. In explicit cracking simulations, we allow element deletion and strain failure criteria. For simplification and better visualization, the stress fields are normalized by comparing the stress to the largest stress in the system in Figs. 1A, H-L, 3C, 5. In all simulation results, the stress magnitude before normalization is between 0 and 7 × 10^7 Pa (Supplementary Fig. 23). In Fig. 1C, to compare the differences in stress distribution, we apply the displacement in the simulations, and then normalize the sum of extracted stresses in the three geometries to ensure that they exhibit the same total reaction stress.

For the implicit static analyses in Figs. 1C, 3G, 4 and 5B, C, F, we obtain a single value of stress out of a hinge with finite width. Since the stresses are concentrated at the hinges, we distribute extra fine meshes at all the hinges. For each hinge, we average the stresses over the elements at its thinnest cross-section (Supplementary Fig. 4).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In this manuscript, the authors investigated the fracture behaviors of topological mechanical metamaterials (TMMs) and compared their behavior with that of continuum material and regular Kagome lattices. They demonstrated the directional asymmetry in the fracture behavior of TMMs, which contrasts with the symmetry behaviors of continuum materials and regular Kagome lattices. Overall, the novelty of the current study is limited, as the authors primarily examined the fracture behaviors of an existing structure TMM, which has already been extensively studied (e.g., refs. [48-52]) including its fracture properties (ref. [56]). Furthermore, the current study lacks the depth, as it merely presents qualitative phenomena without providing a quantitative comparison of fracture toughness. The reported phenomena of directional asymmetry in fracture behavior are also not novel (PRL, 126, 025503, 2021). Furthermore, some claims appear to be overstated (see comment #4 below). Therefore, the current manuscript cannot be recommended for publication in Nature Communications. Specific comments are given below:\n1. The novelty of the study is limited. There are many existing studies on TMMs, including those from the same authors (refs. [48-52, 56-60]). Regarding fracture properties, it is not surprising that such an asymmetric structure exhibits directional asymmetry in fracture behavior, as in one direction the crack tip is sharp and the structure is weaker while in the other direction the crack tip is blunt and the structure is stronger. The concept of “fracture unidirectionality feature” is not new either (ref. [39] PRL, 126, 025503, 2021). The current study simply demonstrates the directional asymmetry in fracture behavior for an existing structure without providing new insights into how such designs could achieve superior fracture resistance.\n2. The current study does not provide any quantitative comparison beyond showing the phenomena. For example, in Fig. 1E and Fig. S6, only the normalized force (force/peak force) is reported. The authors are suggested to provide the absolute force-displacement curves and calculate the fracture toughness of TMMs, and compare these results with those of regular Kagome lattices and other benchmark lattices, such as triangular lattices. Because it would be better to confirm whether TMMs achieve such fracture properties at significant expense of load-bearing capacity.\n3. Even though using the normalized force, it seems that the regular Kagome lattices experimentally outperforms the TMMs in terms of fracture resistance (Fig. S6C). Furthermore, the discrepancy between simulations and experiments should be addressed.\n4. In the Introduction Line 26, the claim of “Despite this abundance of approaches, a robust general principle for using geometry to guide the direction of fractures is still lacking.” seems overstated. The current study does not seem to provide a robust general principle either. In Line 50, it is stated that “the TMM is light-weighted and features material-invariant and scale-invariant properties.” Although it may be true for “material-invariant and scale-invariant”, there are no results in the manuscript that directly support this.\n5. In Fig. 4, it is seen that the upper gripper rotated during loading. The authors are suggested to improve the mechanical testing setup to ensure uniaxial loading and provide the loading-displacement curves to make the results more quantitative. Moreover, the demonstration in Fig. 4 is too simple. It would be better to show how to engineer a tortuous crack path that leads to superior toughness through such asymmetry design.", "reviewer_2": "The paper by Wang et al. uses topological metamaterials to control fracture. In particular, since topological metamaterials are polarized---they host asymmetric floppy modes, the fracture propagation becomes unidirectional and leads to large deformations before failure. As a consequence, the metamaterial appears ductile even though it is made from a brittle material.\nAs clearly described in the manuscript, the mechanism at stake is stress asymmetry, that comes from the topological polarization, the distorted Kagome unit cells are floppy on one side, therefore tend to deform without stretching any of the bonds on this side. In contrast, the opposite size exhibits no such zero energy deformation and the bonds tend to stretch when load is applied. What the authors show is that this simplified picture of the balls-and-springs holds even when the metamaterial is made from a continuous material rather than discrete springs, i.e. when the geometry is made from real hinges. Whereas the floppy hinges exhibit low stress, the opposite hinges undergo high stresses and hence break first. This is the reason why the fractures tends to propagate in one direction and, as the authors show, even turn around when the notch is created on the side with high stresses.\nThe main take away from this work is that topological metamaterials offer a new paradigm to control fracture, in a way that is very different from classical fracture mechanics. Fracture metamaterials have recently emerged as a promising area of research, but so far broadly applicable design concepts are missing. The present work establishes topology as a promising approach and hold the potential to become a seminal work. Therefore I think this paper is potentially an excellent fit for Nature Communications. I however think that the control of fracture displayed in the manuscript in its present form remains rather limited. The key attribute of metamaterials is their tunability. I think that the manuscript would reach a much higher impact if the authors were able to provide more compelling demonstrations of fracture control by utilizing further the design freedom at their disposal and topological floppy modes as a guide. Besides this main concern (which I believe the authors are well equipped to address), I list below some comments, which might help the authors further improve their manuscript.\n*1 The authors argue that the unidirectionality of the fracture is protected by topology and is therefore less sensitive to defects and imperfection. It would perhaps be worth to back up that statement, especially since fracture is a phenomenon that is known to be extremely sensitive to imperfections. In particular, to which extend is the approach more robust that the asymmetric fracture reported in ref [39]?\n*2 Is topology really essential here? What happens if one take a twisted (undistorted) Kagome instead? Would such metamaterial exhibit symmetric fracture instead?\n*3 Likewise, does the magnitude of the topological polarization matter for unidirectional facture propagation? I would expect a topological metamaterials with a small polarization/asymmetry to be less robust to imperfections?\n*4 Section 2.2 can be understood by physicists well versed in topological metamaterials but less so by scientists with no background in this area. The authors might want to make that section a bit more pedagogical to extend the reach of their paper.\n*5 The authors often refer to supplementary figures from the Main Text, which make the read somewhat cumbersome and gives the impression that key information is given in the SI rather than in the Main Text. For instance, the comparison with continuous media in Fig. 1 is somewhat of a naive benchmark. From the text it seems that comparing with the undistorted Kagome would be more meaningful (and would allow less back and forth between Main Text and Supplementary Information).\n*6 Topological metamaterials can host states of self-stress. What is the role of those states of self stress on the fracture propagation?\n*7 Higher order topological metamaterials can host corner modes, would such corner modes be beneficial for the control of fracture?\n*8 More generally, to reflect my main concern above, it would be important to know what the range of control one can achieve with topological metamaterials as well at the limitations."}, "author_response": "Point-by-point response to the reviewer comments (NCOMMS-24-72040)\nThe authors would like to thank the reviewers for their constructive comments and suggestions, which have greatly improved the quality of this manuscript. The manuscript has undergone a thorough revision according to the editor and reviewers’ comments. We include substantial new experimental and computational results in the new version. Please see below our responses. For the reviewers’ convenience, we have highlighted significant changes in the revised manuscript and SIinblue. (Figuresmentionedintheresponserefertofiguresinthisfileunlessotherwisespecified.)\nReviewer 1\nIn this manuscript, the authors investigated the fracture behaviors of topological mechanical metamaterials (TMMs) and compared their behavior with that of continuum material and regular kagome lattices. They demonstrated the directional asymmetry in the fracture behavior of TMMs, which contrasts with the symmetry behaviors of continuum materials and regular Kagome lattices. Overall, the novelty of the current study is limited, as the authors primarily examined the fracture behaviors of an existing structure TMM, which has already been extensively studied (e.g., refs. [48-52]) including its fracture properties (ref. [56]). Furthermore, the current study lacks the depth, as it merely presents qualitative phenomena without providing a quantitative comparison of fracture toughness. The reported phenomena of directional asymmetry in fracture behavior are also not novel (PRL, 126, 025503, 2021). Furthermore, some claims appear to be overstated (see comment 4 below). Therefore, the current manuscript cannot be recommended for publication in Nature Communications. Specific comments are given below:\nReply: We would like to thank Reviewer 1 for their invaluable suggestions, which have greatly contributed to refining and strengthening our paper. We have implemented all recommendations, added extensive results, and thoroughly revised the manuscript. In particular, we highlight the novelty of our study by comparing it with previously reported TMMs and the noted directional fracture phenomena, as detailed in our responses to each of the reviewer’s comments.\nReviewer Comment1.1 — Thenoveltyofthestudyislimited. Therearemanyexistingstudies on TMMs, including those from the same authors (refs. [48-52, 56-60]). Regarding fracture properties, it is not surprising that such an asymmetric structure exhibits directional asymmetry in fracture behavior, as in one direction the crack tip is sharp and the structure is weaker while in the other direction the crack tip is blunt and the structure is stronger. Theconceptof“fractureunidirectionalityfeature”isnotneweither(ref. [39]PRL,126,025503, 2021). The current study simply demonstrates the directional asymmetry in fracture behavior for an existing structure without providing new insights into how such designs could achieve superior fracture resistance.\nReply: Thank you for your comments. In the revised version, we have clarified the novel aspects of this paperby(1)comparingitwithourearlierTMMwork(nowrefs. [49-53, 57-61]), (2)demonstratingthat an asymmetric feature can be observed even when the two tips are similar in terms of visual sharpness, provided that the geometry is topological, (3) contrasting it with another ’fracture unidirectionality feature’ (ref. [39] PRL, 126, 025503, 2021) and twisted kagome (another asymmetric lattice) and (4) the fracture resistance of the TMM.\n(1)InearlierTMMstudies,mostworkhasfocusedonexploringfloppymodes(nowRefs. [49–53,60, 61]) rather than controlling stress fields. While Refs. [57–59] demonstrate that TMMs can concentrate stress along a narrow interface, they do not address stress control at notch tips or the direction of crack propagation. Moreover, these studies primarily examine pre-cracking behavior and do not analyze the fracturing process in realistic TMM applications. As a result, the potential of TMMs for fracture control remains largely unexplored in the existing literature.\nBelowisaconcisebulletlistsummarizingthegeneralfocusoftheearlierTMMpapers(Refs. [49–53, 57–61]):\n• Ref[49] Maxwell lattices and topological mechanics (2018): the review paper gives theoretical derivation of the topological mechanics in ideal periodic spring system, without any discussion of fracturing.\n• Ref[50] Topological edge floppy modes in disordered fiber networks (2018): this paper explores floppy edge modes in disordered fiber network, which is still based on the spring lattice and focuses on floppy modes rather than stress control. There is also no discussion of stress focusing or fracturing.\n• Ref[51]Continuumtheoryfortopologicaledgesoftmodes(2020): thispapertheoreticallyextends spring lattice into continuum mechanics, but it does not contain any discussion of stress focusing or fracturing.\n• Ref[52] Backscattering-free edge states below all bands in two-dimensional auxetic media (2023): this paper proposes a system for one-way wave propagation, and has nothing to do with stress focusing or fracturing.\n• Ref[53] Transformable topological mechanical metamaterials (2017): This paper is concerned with topological transitions in topological lattices, without any discussion of stress focusing or fracturing.\n• Ref[57] Fracturing of topological Maxwell lattices (2018): this paper only theoretically demonstrates that TMMs can concentrate stress and initiate cracks along vertical interfaces in ideal spring-based TMM models. In contrast, our current work focuses on controlling stress near notch tipsandthroughouttheentiresampleinreal-worldTMMs. Inaddition,fracturinginrealmaterials have many qualitatively different features compared to ideal lattices.\n• Ref[58]StressfocusinganddamageprotectionintopologicalMaxwellmetamaterials(2023): based on Ref. [57], this paper first extends the ideal spring-based system to a real-world kagome lattice constructed with steel triangles. However, the study primarily employs linear elastic analysis and does not investigate the actual cracking process, which involves a collection of fundamentally new phenomena including one-way fracturing, as we discover in the current manuscript.\n• Ref[59]Robustnessofstressfocusinginsoftlatticesundertopology-switchingdeformation(2024): thispaperextendstheidealspring-basedsystemtoareal-worldkagomelatticemadeofsoftrubber, 2 demonstratingthetopologicalpolarizationunderfinitedeformations. However,therubbermaterial is too ductile, and an actual fracturing process is still not investigated.\n• Ref[60] Edge modes and asymmetric wave transport in topological lattices: Experimental characterization at finite frequencies (2018): this paper experimentally observes the propagation of topological floppy modes in real-world TMMs, whereas our new manuscript focuses on stress and fracturing control using TMMs.\n• Ref[61] Omnimodal topological polarization of bilayer networks: Analysis in the Maxwell limit and experiments on a 3D-printed prototype (2022): this paper discusses the fabrication of bilayer TMMs and shows their polarized floppy modes, while our current one analyzes fracture control.\nBeyond the existing TMM studies, we propose using TMMs to tune the stress field and guide crack propagation, in the full nonlinear regime, throughout the entire lattice structure, rather than confining it to a narrow interface. Additionally, we demonstrate the superior fracture properties of TMMs: they can modify a material’s mechanical properties to absorb more energy and enable controlled, ductile cracking behaviors in otherwise brittle materials.\nThe following sentence has been included in the main text:\nLine 36: Earlier studies of TMMs have focused on exploring topological floppy modes (Refs. [49–53, 60, 61]) rather than controlling stress fields. A handful of papers studied fracturing control using TMMs but all focused on the capability of using states of self stress to concentrate stress, predominately in small strain regimes before fracturing [57–59].\n(2) The reviewer stated, ”it is not surprising that such an asymmetric structure exhibits directional asymmetry in fracture behavior; in one direction the crack tip is sharp, and the structure is blunt, while in the other direction the crack tip is blunt and the structure is stronger.” The reviewer is right that an asymmetric structure usually exhibit directional fracture, e.g., ref[39]. In our results, twisted kagome is also able to guide cracks along one direction (Fig. 4). The difference is the TMM can guide cracks more robustly despite defects, imperfections, and shapes of the initial crack, which is different from other geometrically asymmetric structures.\nWe acknowledge that the TMM geometry in the draft indeed features a sharp tip on the right and a blunt tip on the left, as shown in Fig. 4(A, bottom). However this is more of a coincidence than the cause of the topological stress focusing we report. To demonstrate this point, we conducted new computational studies of the topological square lattice, which does not exhibit geometrically distinct tips, but still generates an asymmetric stress field that governs the crack propagation according to the topological polarization (Fig. 1). Similar to the polarized kagome lattice, topological zero modes produceanasymmetricstressdistributionaroundthecrack,resultingintherobustunidirectionalcracking behavior.\nMoreover, even in the topological kagome lattice, the effect of tip geometry on fracture asymmetry is relatively limited compared to the influence of topological polarization. For a straight crack, as the spring stiffness ratio increases or the hinges become thicker, the left and right crack tips remain rounded and sharp, respectively. However, the effect of topological polarization is significantly reduced, with the stressdifferencebetweenthetwotipsapproaching1,avalueattributablepurelytogeometricasymmetry. This observation indicates that geometric differences play only a minor role in determining the overall asymmetric fracture behavior.\nFigure 2: Geometrical distinction between two tips only play a minor role in generating the asym- metric stress field. Scaled tip stress ratios from Fig. 3 in the main text. (F) In the next-nearest neighbor lattice (NNN), the stress difference at the tip decreases as the difference between spring next-nearest neighbor spring (k ) and nearest neighbor spring (k ) gets smaller. (G) In the NNN NN TMM (real-world solid structure), the stress difference at the tip decreases as the hinge thickness increases.\nThe following sentences have been included in the main text:\nLine 222: This result further confirms that the observed asymmetric features originate primarily fromtopologyratherthangeometry. AsNNNspringconstantincreases,thelatticeshaperemains geometricallyasymmetric, maintainingonesharpsideandonebluntside, yetthestressdifference between crack tips gradually diminishes (Fig. 3F).\nThe following contents have been included in the SI IID (Page 15):\nIn this work, the kagome TMM geometry in the draft features a sharp tip on the right and a blunt tip on the left. However this is more of a coincidence than the cause of the topological stress focusing we report. We conduct computational studies of the topological square lattice, which does not exhibit geometrically distinct tips, but still generates an asymmetric stress field that governs the crack propagation according to the topological polarization (Fig. 1). Similar to the polarized kagome lattice, topological zero modes produce an asymmetric stress distribution around the crack, resulting in the robust unidirectional cracking behavior. However, fabricating a polarizedsquarelatticerequiresmoreadvancedtechniques, asthereisnostraightforwardmethod to convert a spring-like square lattice into a partially filled square lattice, as is possible with the kagome structure. We leave this development for future work.\n(3) Compared with other fracture-control approaches (e.g., Ref. [39]), the TMM is more general and robust. Its crack propagation direction is less sensitive to unavoidable fabrication imperfections and remains unaffected by notch shapes. In fact, Ref. [39] (PRL, 126, 025503, 2021) presents a unidirectional cracking mechanism that utilizes geometric asymmetry induced by an array of voids (triangular inclusions). Following the geometry shown in Fig. 4(c) of Ref. [39], we performed finite elementanalysis (FEM)inthelinear elasticregime; theresultingstressdistributionispresentedinFig.3. As illustrated in Fig.3(A), the left notch tip experiences higher stress than the right tip, which explains why forward cracking is systematically favored.\nSmall defects and system imperfections can act as stress concentrators, influencing both crack initiation and propagation (T. L. Anderson, Fracture Mechanics: Fundamentals and Applications). This sensitivity might explain the indeterminacy observed in PRL 126, 025503 (2021), where approximately one-quarter of the specimens showed cracks initiating in the bulk rather than at the edge notch. To accountforpotentialfabricationimperfections,weintroduceaslightgeometricmodificationbyrounding a sharp corner, as illustrated in Fig. 3B. In this modified case, the highest stress concentrates at the rounded corner, where crack initiation is expected. This demonstrates that even minor fabrication imperfections can significantly alter the crack propagation path. In contrast, TMMs exhibit notably more robust behavior experimentally. Despite uncontrollable fabrication imperfections in our TMM experiments,cracksconsistentlyinitiateontherightsideofthenotchinall40testedspecimens(provided thatnohingeswereaccidentallybrokenbeforetesting), asclearlyillustratedbythehistograminFig.3C. Thus, TMMsarelesssensitivetofabricationerrorsandprovideamorerobustapproachtocrackcontrol.\nMoreover, TMM exhibits topologically protected properties that remain invariant of crack shapes. If the directional asymmetry arises solely from the geometry, the directional cracking can be easily altered. Fig. 3C and D show two different notch shapes applied to the same specimen: Fig. 3C can preserve the predicted cracking direction, whereas Fig. 3C does not. In addition, the twisted kagome structure, though not topological, also features asymmetric geometry. Like the TMM, twisted kagome is blunt on the right tip and sharp on the left tip (Fig. 4A), which explains why it can guide a straight crack to the right. However, for other crack shapes, twisted kagome sometimes directs the crack to the left (Fig. 4F). Only the TMM consistently guides cracks to the right in a more robust manner despite of imperfections and crack shapes, as shown in Figs. 4D, E.\n(4)TheTMMshowssuperiorfractureresistanceinbringingductilebehaviourfromabrittlematerial andabsorbingthelargestamountofenergyduringcracking(amongthefourlattices, Fig.4). Themate- 6 rialresistancemayincludethesurfaceenergy, plasticwork, orothertypeofenergydissipationassociated with a propagating crack (T. L. Anderson, Fracture Mechanics: Fundamentals and Applications).\nTo better address the TMM is robust against crack shapes with superior fracture resistance, the following discussions have been included in the main text (figures in the box below refer to figures in the main text):\nLine 26: However, it is a universal phenomenon in fracturing that sharp geometries of the initial notch, small defects and system imperfections can act as stress concentrators, influencing both crack initiation and propagation (T. L. Anderson, Fracture Mechanics: Fundamentals and Applications), interfering with the control of cracking directions, leading to indeterminacy as observed in Ref[39].\nUnlike the the triangular and regular kagome lattices, both the twisted kagome lattice and the TMM effectively direct the propagation of a straight notch to the right (see Movie S6, left panel, and Fig. 1). However, the twisted kagome lattice’s directional cracking is easily compromised whenthecrackshapechanges(MovieS6). AsshowninFig.2panelsD–G,theTMMconsistently guides cracks to the right, whereas the twisted kagome lattice does not. Moreover, regarding fracture toughness, both the TMM and twisted kagome lattices exhibit similar values—each displayingonlyhalfthetoughnessoftheregularkagomelattice,suggestingthefracturetoughness of the TMM is comparable to other asymmetric lattices.\nImportantly, the TMM is significantly more ductile than the other lattice—its elongation is 0.066 compared to the twisted kagome (0.030), regular kagome lattice (0.019) and the triangular lattice (0.015). Moreover, the TMM dissipates the most energy while it fractures, about two times as high as the other lattices. This controlled deformation absorbs and dissipates energy that would otherwise accelerate crack propagation, enhancing the structure’s tolerance to defects and reducing the risk of sudden failure. Thereby, the TMM displays the most stable and gradual cracking behavior without a significant sacrifice of the fracture toughness.\nIn summary, the crack propagation in the TMM can be considered predictable, unidirectional and ductile. This further suggests the peculiarity of the sharp, asymmetric stress field and the uniqueness of the one-way cracking achieved working with TMMs.\nLine 398: In Fig. 2, we quantitatively characterize the the triangular lattice, the regular kagome lattice, the twisted kagome lattice and the TMM. The ductility is measured as the elongation ratio, ∆ /l , where ∆ is the critical displacement at the force peak and l is the original sample c 0 c 0 √ length. The fracture toughness is calculated as the critical stress intensity factor σ πaf(a/W), c where σ is the critical stress applied, 2a is the full crack length, and f(a/W) is a specimen c geometry dependent function of the crack length a and sample width W. The consumed energy (cid:82) is Fdx, where F is the applied force and x is the displacement.\nNew supplementary movies have been added to illustrate the fracturing processes of the triangular 8 lattice (Movie S3), the twisted kagome lattice with three different crack configurations (Movie S6), and the TMM with two triangular cracks (Movie S7).\nTo show the TMM is less sensitive to defects, the discussions between TMM and the structure in PRL 126, 025503 (2021) has been incorporated in the Supplement Information IA (Page 4).\nInconclusion,thisworkextendstheapplicationsofTMM,andisanoveladditiontothecrackguiding metamaterials. The asymmetric features originates topology and are more robust against fabrication imperfections and cracking conditions.\nReviewer Comment1.2 — The current study does not provide any quantitative comparison beyond showing the phenomena. For example, in Fig. 1E and Fig. S6, only the normalized force (force/peak force) is reported. The authors are suggested to provide the absolute force-displacement curves and calculate the fracture toughness of TMMs, and compare these results with those of regular Kagome lattices and other benchmark lattices, such as triangular lattices. Because itwouldbebettertoconfirmwhetherTMMsachievesuchfracturepropertiesatsignificantexpense of load-bearing capacity.\nReply: Thank you for your constructive feedback. Following the reviewer’s suggestion, we have conductedadditionalexperimentstoquantitativelycomparetheTMMwithotherlatticestructures. Our results confirm that the TMM not only robustly guides crack propagation but also exhibits the most stable fracture behavior and outstanding energy dissipation properties without significantly sacrificing load-bearing capacity. The fracture toughness of the TMM is comparable to that of the twist kagome lattice, which also features sharp tips but not as ductile as the TMM. We have revised the manuscript accordingly, now including absolute force–displacement curves and calculated fracture toughness values for the triangular lattice, regular kagome, twisted kagome, and TMM structures in Fig. 4 (Fig. 2 in the main text).\nAs illustrated in Fig. 4A, the triangular lattice structure has half of its total area filled with solid material, whereas each of the three kagome-based lattices contains only one-quarter. Consequently, the triangular lattice exhibits the highest peak force and stiffness among the four structures (Fig. 4B). Moreover, its fracture behavior closely resembles that of continuum materials, characterized by a sharp drop in force immediately after the peak—indicative of a brittle and unstable fracturing process.\nWith the same coordination number, the force peak of the regular kagome lattice is nearly twice that of the twisted kagome and TMM. Previous studies have shown that the regular kagome structure possesses high fracture toughness due to its blunt crack tips and diffused stress distribution (Fleck, N.A. and Qiu, X., JPMS, 55(3), 62–588, JMPS). Moreover, analysis of low-frequency density maps (Mao X, Lubensky TC, Annu. Rev. Condens. Matter Phys., 9(1):413–33, 2018) reveals that while the lowest-frequency band in the regular kagome extends throughout the bulk, the twisted and polarized kagome lattices exhibit localized zero modes (Fig. 5). Both twisted kagome and TMM display localized and concentrated stress at the crack tip, resulting in the lowest fracture toughness (the critical stress intensity; see Materials and methods in the main text for more details) among the three kagome-based lattices. The resulting asymmetric stress field leads to a sharp tip and guides crack propagation at the expense of the maximum load-bearing capacity.\nTwisted kagome and TMM exhibit the same fracture toughness; however, the energy dissipation of twisted kagome is only half that of TMM (Fig. 4C). Due to the excited Guest-Hutchinson mode in twisted kagome (Mao X, Lubensky TC, Annu. Rev. Condens. Matter Phys., 9(1):413–33, 2018), the units in twisted kagome tend to rotate back toward a regular kagome configuration. Consequently, 9 twisted kagome fails in a manner similar to regular kagome, making the fracturing process more brittle and less stable than TMM (Figs. 4B, C).\nFigure 5: The ZMs of regular kagome, twisted kagome and polarized kagome spring-mass lattice whenonenodeanditsfourconnectedareremoved. Theboundarynodesofthethreespringlattices are all fixed. (A) regular kagome lattice (unpolarized), (B) twisted kagome lattice (unpolarized), (C) polarized kagome lattice with a short, straight hole. The same as polarized kagome, the ZMs of twisted kagome lattice are not symmetric around the hole due to its broken symmetry, but its ZMs are distributed around all the edges of the crack. Regardless of the shape of the crack, the topological ZMs of polarized kagome never localize to the left side of the crack. This implies the robustness of the ZM distribution in the polarized kagome is due to topological factors rather than purely geometric ones.\nIn summary, although the TMM sacrifices some strength to guide fracture, it exhibits the most ductile cracking behavior and dissipates, the greatest amount of energy during fracture. Moreover, its fracture toughness remains comparable to that of the twisted kagome lattice, which displays a less stablecrackingbehavior. Fractureresistanceisnotonlylimitedtoinitialcrackingtoughness. Highload- bearing capacity is desirable, but catastrophic failure must be avoided in most structure components. We present an approach that promotes gradual, ductile cracking, providing early warning signs—such as visible deformation and increased strain—that enable timely intervention or safe shutdown. This controlleddeformationabsorbsanddissipatesenergythatwouldotherwiseacceleratecrackpropagation, enhancing the structure’s tolerance to defects and reducing the risk of sudden failure.\nAnd the following discussions have been included in the main text (figures in the box below refer to figures in the main text):\nLine 139: To provide a more comprehensive analysis of fracture properties of the TMMs, we quantitatively compare the cracking behaviors of four lattice configurations in our experiments: the triangular lattice, the regular kagome lattice, the twisted kagome lattice, and the TMM (see Fig. 2). Fig. 2A presents schematics of these four lattices. For consistency with the TMM, the twisted kagome lattice is designed with a blunt right tip and a sharp left tip. All lattices are fabricated from the same material, shared identical sample sizes, and have a crack length of 2a. Their force-displacement curves are shown in Fig. 2B. The the triangular lattice is generated by making all pointing-down triangles in a regular triangular lattice filled as solid triangles and all pointing-up triangles empty. It has a higher coordination number (6) compared to the kagome lattices(4). Thesedifferencesaccountforthethetriangularlatticeexhibitingthehigheststiffness andstrength. However,thethetriangularlatticeshowsasudden,brittle,andunpredictablefailure 10 mode (Movie S3). The regular kagome lattice features the highest fracture toughness among the three kagome configurations due to the blunting effect (Fig. 1C, Fig. 2B). Notably, the regular kagome also displays somewhat erratic propagation behavior. In ideal simulations, its symmetric geometry leads to two-way symmetric propagation resulting in brittle failure (Movie S4). Experimentally,nevertheless,thebluntedtipstress(Fig.1C,Fig.2B)producesunpredictable crack paths that may even veer in other directions due to uncontrollable imperfections in the sample (Movie S5).\nUnlike the the triangular lattice and regular kagome lattice, both the twisted kagome lattice and the TMM effectively direct the propagation of a straight notch to the right (see Movie S6, left panel, and Fig. 1). However, the twisted kagome lattice’s directional cracking is easily compromised when the crack shape changes (Movie S6). As shown in Fig. 2 panels D–G, the TMM consistently guides cracks to the right, whereas the twisted kagome lattice does not. Moreover, regarding fracture toughness, both the TMM and twisted kagome lattices exhibit similar values—each displaying only half the toughness of the regular kagome lattice, suggesting the fracture toughness of the TMM is comparable to other asymmetric lattices.\nImportantly, the TMM is significantly more ductile than the other lattice—its elongation is 0.066 comparedtothetwistedkagome(0.030),regularkagomelattice(0.019)andthetriangularlattice (0.015). Moreover,theenergydissipationoftheotherlatticesisonlyhalfthatofTMM(Fig.2C). This controlled deformation absorbs and dissipates energy that would otherwise accelerate crack propagation,enhancingthestructure’stolerancetodefectsandreducingtheriskofsuddenfailure. Thereby, the TMM displays the most stable and gradual cracking behavior without a significant sacrifice of the fracture toughness.\nIn summary, the crack propagation in the TMM can be considered predictable, unidirectional and ductile. This further suggests the peculiarity of the sharp, asymmetric stress field and the uniqueness of the one-way cracking achieved working with TMMs.\nNew supplementary movies have been added to illustrate the fracturing processes of the triangular lattice (Movie S3), the twisted kagome lattice with three different crack configurations (Movie S6), and the TMM with two triangular cracks (Movie S7).\nReviewer Comment1.3 — Even though using the normalized force, it seems that the regular Kagome lattices experimentally outperforms the TMMs in terms of fracture resistance (Fig. S6C). Furthermore, the discrepancy between simulations and experiments should be addressed.\nReply: We are grateful for your careful review and the helpful recommendations provided. Fracture resistance is defined as material’s resistance for crack growth. The material resistance may include the surface energy, plastic work, or other type of energy dissipation associated with a propagating crack (T. L. Anderson, Fracture Mechanics: Fundamentals and Applications). We have included more experimental results to quantitatively show the fracture resistance of TMMs (Fig. 4) and other lattices. Also, more results and discussions about the discrepancy in Fig. S6 are added.\nRegular kagome shows a higher stiffness and fracture toughness than polarized kagome, while TMM outperforms regular kagome in terms of determined cracking path, ductile fracturing, energy consumption, stable cracking and defect tolerance. In practice, when minimal deformation is critical (e.g., precision components), higher fracture toughness might be more important even if the material is more brittle. In other conditions, a ductile failure mode is generally preferred because it can redistribute stresses and provide noticeable deformation before failure, allowing for preventative measures.\nThe following discussion about the discrepancy has been added in the main text.\nLine 391: Compared to idealized simulations, out-of-plane deformation in experimental tests of thin lattices can slow the rapid force drop and delays the cracking process (Gu, H., Pavier, M., and Shterenlikht, A., IJSS, 152, 207–216, 2018).\nThe following contents have been included in the SI IB (Page 6):\nThe sensitivity to defects is also observed in regular kagome lattice (Fig. 6). In Fig. 6A, we show the scaled simulation results for topological kagome (TMM), perfect regular kagome and imperfect regular kagome (with randomly varied hinge width). The discrepancy between perfect regular kagome in simulations (red solid curve in Fig. 6A) and regular kagome in experiments (Fig. 6C) arises from a combination of out-of-plane deformation (nonlinearity) and unavoidable fabrication errors. When loading thin lattice structures, the free edges above and below the crack tend to bend out of plane. Compared to idealized simulations, out-of-plane deformation in experimental tests of thin lattices slows the rapid force drop and delays the cracking process (Gu, H., Pavier, M., and Shterenlikht, A., IJSS, 152, 207–216, 2018).\nToaccountforfabricationimperfections,werandomlyreducethethicknessofhingesintheregular kagome lattice and compare its cracking behavior with that of a perfect regular kagome structure (red dashed curve in Fig. 6). When geometric imperfections are introduced, the cracking pattern is not symmetric and the force is redirected. The force-displacement curves exhibit a lower peak force and a more ductile fracture process.\nReviewer Comment1.4 — IntheIntroductionLine26, theclaimof“Despitethisabundanceof approaches, a robust general principle for using geometry to guide the direction of fractures is still lacking.” seems overstated. The current study does not seem to provide a robust general principle either.\nIn Line 50, it is stated that “the TMM is light-weighted and features material-invariant and scale-invariant properties.” Although it may be true for “material-invariant and scale-invariant”, there are no results in the manuscript that directly support this.\nReply: We are grateful for your careful review and the helpful recommendations provided.\n1) ”Despite this abundance of approaches, a robust general principle for using geometry to guide the direction of fractures is still lacking.” The sentence intended to show the limitation of the current 12 Figure 6: Force-displacement curves of regular kagome lattices in simulations and experiments. (A) In the ideal simulations, the regular kagome breaks in a brittle way ((Movie S4)), comparing to the topological kagome (simulation in (Movie S2). (Top) The schematic of regular in simula- tions. (Bottom) The dependence of normalized force (force/force peak) on the applied boundary displacements. (B) The initial geometry of the regular kagome in the experiment. (C) However, in the experiments, due to the blunting effect at the tips, the tip stress concentration is not as strong as in the simulations and the crack goes into side-way directions easily, resulting in unpredictable fracturing paths (Movie S5). Scale bar: 12.5 mm.\nresearch: although geometry-controlled or material-controlled cracking approaches are powerful, they remain relatively sensitive to defects and variations in notch shapes (see our discussion above about Fig. 3, page 6, the answer to Reviewer Comment1.1). What our manuscript brings to the table is that we provide a TMM framework for robust unidirectional crack initiation and propagation, which extends beyond the existing literature (also as detailed in our response to Reviewer Comment1.1), and we are aware that this framework has not been proven to be general enough for all the fracture control, at least not for ductile materials. To avoid any potential confusion, we have revised the sentence as: “Despite this abundance of approaches, a more robust principle using geometry to guide the direction of fractures that is less insensitive to notch geometries and imperfections is still lacking. ”\n2) We have removed the statement “light-weighted and features material-invariant and scale- invariant properties” and included a discussion on material selection. Although the TMM itself does not depend on specific materials, achieving unidirectional cracking with the TMM does impose certain material constraints (brittle material is required).\nLine 29: Despite this abundance of approaches, a more robust principle using geometry to guide the direction of fractures that is insensitive to notch geometries and imperfections is still lacking.\nLine 58: (4) based on topological kagome lattices, the TMM is light-weight.\nLine 366: Additionally, topological polarization is only maintained in the small deformation regime, restricting the choice of fabrication materials to brittle ones with limited plastic deformation.\nReviewer Comment1.5 — In Fig. 4, it is seen that the upper gripper rotated during loading. The authors are suggested to improve the mechanical testing setup to ensure uniaxial loading and provide the loading-displacement curves to make the results more quantitative.\nMoreover, thedemonstrationinFig. 4istoosimple. Itwouldbebettertoshowhowtoengineer a tortuous crack path that leads to superior toughness through such asymmetry design.\nReply: Your careful review and detailed suggestions are sincerely appreciated.\n1) In Fig. 5 I-M (Fig. 4 in the initial submission) in the main text, the upper gripper is able to passively rotate during experiments since it is connected to the load cell via a hinge, but its influence on the force–displacement curves is negligible.\nSnapshots comparing the loading process for side-notch and bulk-notch samples are shown in Fig. 7. Rotation primarily occurs for samples with side notches (Fig. 7 A-E), and only at the very end of the loading stage due to the lack of constraints (≈ 5◦, Fig. 7E). The setup allows further propagation of cracks and the deformation kinematics resembles what happens in a three point bending test where bending occurs to some extent. Also, even in that case, the full external stretching applied is still mainly projected to the vertical stretching.\nAll force–displacement data presented in this work correspond to cracks initiated from bulk notches, with samples remaining firmly connected at the left and right edges throughout the entire loading period. As our measurements focus exclusively on bulk-notch samples, the effect of gripper rotation on the reported force–displacement curves is minimal.\nThe following paragraph has been added in the Materials and Methods in the main text (figures in the box below refer to figures in the main text):\nLine 407: The gripper rotated passively (side- notch results in Fig. 5) since it’s connected by a hinge. It only happens for side cracks at the very end of the loading stage, due to the lack of constraints. For all the force-displacement measurements in this work, the cracks are in the bulk. And the samples can remain connected at left and right edge for the full loading period.\nThe force-displacement curves measured in this work are all for bulk notches and therefore, the influence from top gripper rotation plays a very minor role.\n2) The reviewer comments, ”It would be better to show how to engineer a tortuous crack path that leads to superior toughness through such asymmetry design.” By manipulating the polarization direction, we can promote various cracking sequences and paths to tune fracture resistance (including fracture toughness) (Fig. 8). To address this question, we compare a single-domain TMM (featuring a single polarization direction, see Fig. 1 in the main text) with interface TMMs (containing two opposite polarization directions, indicated by R in Fig. 8A, C). When interfaces are introduced, these 14 Figure 7: Snapshots illustrating the rotation of the top gripper relative to the load cell during the loading process. (A–E) Sample with a notch at the right edge: (A) Initial undeformed state; (B) pre-cracking condition; (C–D) early-stage crack propagation; and (E) final fractured state. (F–J) Sample with a triangular notch located in the bulk: (F) Initial undeformed state; (G) pre-cracking condition; (H–I) early-stage crack propagation; and (J) final fractured state. Note that visible gripper rotation occurs only when the notch is located at the edge (A–E), whereas no significant rotation is observed for samples with notches in the bulk (F–J). Scale bar: 12.5 mm.\nprogrammed domain interfaces carry larger stress and dominate crack initiation over the notch tips. In our photoelastic experiments, the concentrated stress clearly emerges at these interfaces for both configurations (Fig. 8B, D), consistent with numerical simulations. As illustrated in Fig. 8E, F, the first cracking phase initiates within the interface regions (cracking stage-I, highlighted by gray boxes). After the hinges at the interfaces fail—particularly those forming the zigzag interface near the notch, and all hinges in the straight interface—the crack then progresses toward the notches. Then, the second 15 cracking phase begins near the notch (blue box in Fig. 8 E, F) and the crack consistently propagates toward the right notch tip before reaching the left due to the topological polarization.\nFurthermore, thegeometryof theinterfaces influencesnot only thecrack propagationpaths butalso the force–displacement response and fracture resistance(Fig. 8H). Initially, in the linear elastic regime, both straight and zigzag interfaces stiffen the structure (Fig. 8H, inset). For the straight-interface configuration, stress concentrates uniformly in the interface, causing nearly simultaneous fracture of all interface units (Fig. 8D, F–H). As a result, the straight interface configuration presents the earliest crack initiation and the most brittle fracture behavior among the studied configurations. In contrast, the tortuous interface of the zigzag-interface generates a stress gradient (Fig. 8B), leading interface units to break sequentially and more gradually. Consequently, the zigzag configuration absorbs more energy during fracturing and exhibits the highest fracture toughness and linear elastic stiffness (Fig. 8H, inset). These results highlight how topological features can be leveraged to precisely program stress distributions, crack sequences, and fracture resistance.\nFigure 8: Cracking paths and tunable fracture responses in TMMs with topological interfaces. (A) Zigzag topological interface schematic. (B) Photoelastic stress images of the zigzag interface at the right tip. Brighter areas carry larger stress. Pink lines indicate the crack. (C) Straight interface schematic. (D) Photoelastic stress images of the straight interface at the right tip. Cracking order of (E) zigzag interface and (F) straight interface in the simulations. The crack initiates in the gray box (cracking stage I) and then go to the blue box (cracking stage II). The cracking order inside two boxes is indicated by the colored dots. The notch length and sample dimensions are fixed in experiments and simulations. (G) Linear stress profile across the system along the horizontal paths including the notch of zigzag interface geometry, straight interface geometry and single domain geometry in the simulations. (H) Force-displacement curves of zigzag interface geometry, straight interface geometry and single domain geometry in the simulations. Scale bar: 12.5 mm.\nThe following paragraphs are included in the main text. Figures in the box below refer to figures in the main text.\nFurthermore, the shape of the interfaces influences not only the cracking path but also the force- displacement behavior and fracture resistance (Fig. 6H). For the straight interface configuration, allinterfaceunitsexperiencehighlyconcentratedstressandbreakalmostsimultaneously(Fig.6D, F–H), leading to the earliest cracking and the most brittle behavior among the configurations. In contrast,thetortuouspathinthezigzaginterfacegeometrycreatesstressgradients,andtherefore the interface units break sequentially in a more gradual manner rather than all at once. As a result, the zigzag interface configuration absorbs more energy and exhibits the highest fracture toughness (Fig. 6H) and linear elastic stiffness (Fig. 6H inset). These results highlight how topological features can be leveraged to precisely program stress distributions, crack sequences, and fracture resistance.\nReviewer 2\nThe paper by Wang et al. uses topological metamaterials to control fracture. In particular, since topological metamaterials are polarized—they host asymmetric floppy modes, the fracture propagation becomes unidirectional and leads to large deformations before failure. As a consequence, the metamaterial appears ductile even though it is made from a brittle material.\nAs clearly described in the manuscript, the mechanism at stake is stress asymmetry, that comes from the topological polarization, the distorted Kagome unit cells are floppy on one side, therefore tend to deform without stretching any of the bonds on this side. In contrast, the opposite size exhibits no such zero energy deformation and the bonds tend to stretch when load is applied. What the authors show is that this simplified picture of the balls-and-springs holds even when the metamaterial is made from a continuous material rather than discrete springs, i.e. when the geometry is made from real hinges. Whereas the floppy hinges exhibit low stress, the opposite hinges undergo high stresses and hence break first. This is the reason why the fractures tends to propagate in one direction and, as the authors show, even turn around when the notch is created on the side with high stresses.\nThe main take away from this work is that topological metamaterials offer a new paradigm to control fracture, in a way that is very different from classical fracture mechanics. Fracture metamaterials have recently emerged as a promising area of research, but so far broadly applicable design concepts are missing. The present work establishes topology as a promising approach and 17 holdthepotentialtobecomeaseminalwork. ThereforeIthinkthispaperispotentiallyanexcellent fit for Nature Communications. I however think that the control of fracture displayed in the manuscript in its present form remains rather limited. The key attribute of metamaterials is their tunability. I think that the manuscript would reach a much higher impact if the authors were able to provide more compelling demonstrations of fracture control by utilizing further the design freedom at their disposal and topological floppy modes as a guide. Besides this main concern (which I believe the authors are well equipped to address), I list below some comments, which might help the authors further improve their manuscript.\nReply: We thank this reviewer for supporting us to publish in Nature Communications, and we appreciate the positive comments! We have carefully revised our manuscript according to the constructive suggestions from the reviewer.\nWe have conducted additional extensive experimental and computational studies to provide more compelling demonstrations of fracture control by further leveraging the design freedom of the TMMs. We demonstrate that crack propagation paths, which are governed by the underlying topology, can be programmably shaped to achieve tunable mechanical properties, including stiffness, energy absorption, and fracture resistance. These results are presented in Fig. 8 (corresponding to Fig. 6 in the main text), and the associated analysis has been incorporated into the revised manuscript.\nLine 326: Topologically controlled fracturing sequences can also be further programmed to control fracture resistance and mechanical stiffness in more versatile ways. By manipulating the polarization direction, we can generate distinct cracking paths that result in tunable fracture resistance (Fig. 6). Here, we compare the single-domain TMM (with one polarization direction; seeFig.1)tointerface-TMMs(featuringtwooppositepolarizationdirections,Fig.6A,C).Inthese systems, the programmed interfaces play a dominant role over the notch tips. In the photoelastic experiments,concentratedstressisclearlyobservedattheinterfacesforbothgeometries(Fig.6B, D), and this finding is consistent with the simulations (Fig. 6G). In Fig. 6E, F, cracks first originate in the interface regions (cracking stage-I in the gray box). Once the interface hinges fail—specifically, those in the zigzag interface near the notch and all hinges in the straight interface—the crack extends toward the notches. Owing to the topological polarization effect, the crack propagates to the right tip before reaching the left tip (cracking stage-II in Fig. 8E, F).\nFurthermore, the shape of the interfaces influences not only the cracking path but also the force- displacement behavior and fracture resistance (Fig. 6H). For the straight interface configuration, allinterfaceunitsexperiencehighlyconcentratedstressandbreakalmostsimultaneously(Fig.6D, F–H), leading to the earliest cracking and the most brittle behavior among the configurations. In contrast,thetortuouspathinthezigzaginterfacegeometrycreatesstressgradients,andtherefore the interface units break sequentially in a more gradual manner rather than all at once. As a result, the zigzag interface configuration absorbs more energy and exhibits the highest fracture toughness (Fig. 6H) and linear elastic stiffness (Fig. 6H inset). These results highlight how topological features can be leveraged to precisely program stress distributions, crack sequences, and fracture resistance.\nReviewer Comment2.1 — The authors argue that the unidirectionality of the fracture is protected by topology and is therefore less sensitive to defects and imperfection. It would perhaps be worth to back up that statement, especially since fracture is a phenomenon that is known to be 18 extremely sensitive to imperfections. In particular, to which extend is the approach more robust that the asymmetric fracture reported in ref [39]?\nReply: We are grateful for the excellent suggestion. The TMM is less sensitive to unavoidable fabrication defects and imperfections, and is not affected by notch shapes, comparing to other fracture control approaches (e.g. Ref[39]). In the following, we present a detailed comparison of the TMM and Ref. [39] with respect to their robustness in directing crack propagation.\nRef[39] describes a unidirectional cracking mechanism achieved through geometric asymmetry introduced by triangular inclusions (void arrays). Using the geometry presented in Ref. [39], we performed finite element analysis (FEM) within the linear elastic regime; the resulting stress distribution is shown in Fig.3. As illustrated in Fig. 3A, the left notch tip experiences higher stress compared to the right tip, explaining why forward cracking is consistently favored. To better account for potential fabrication imperfections, we introduce a small geometric modification in Fig. 3B by rounding a sharp angle. In this case, the highest stress is observed at the modified corner, where a crack is expected to initiate.\nIn contrast, TMM demonstrates more robust unidirectional cracking behavior in experiments. In our TMM experiments, despite the presence of unavoidable fabrication imperfections, cracks consistently initiate on the right side of the notch in all 40 specimens (provided that no hinges accidentally break), in Fig. 3C. The TMM is therefore less sensitive to fabrication defects.\nMoreover, TMM exhibits topologically protected properties that is not influenced by crack shapes. If the directional asymmetry arises solely from the geometry, then the directional cracking can be easily altered. Fig. 3D and E show two different notch shapes applied to the same specimen: Fig. 3D preserves the predicted cracking direction, whereas Fig. 3E does not. In conclusion, the structure in Ref[39] (PRL, 126, 025503, 2021) remains sensitive to minor fabrication defects and notch variations, whereas our proposed TMM configuration reliably maintains crack directionality, independent of such imperfections.\nThe following sentence has been added in the introduction to address the robustness of the TMM.\nLine 26: However, it is a universal phenomenon in fracturing that sharp geometries of the initial notch, small defects and system imperfections can act as stress concentrators, influencing both crack initiation and propagation, interfering with the control of cracking directions, leading to indeterminacy as observed in Ref[39].\nLine 50: (2) protected by topology, the TMM is less sensitive to defects from fabrication and invariant to notch shapes.\nThefollowingdiscussionsregardingthecomparisonbetweenourTMMandPRL126, 025503(2021) have been added to SI IA (Page 6).\nTheTMMislesssensitivetounavoidablefabricationdefectsandimperfections,andisnotaffected by notch shapes, comparing to other fracture control approaches. In this subsection, based on thegeometrypresentedinthatpaper(Fig.4(c)inPRL126, 025503), weconductedfiniteelement analysis (FEM) in the linear regime to show that fractures are sensitive to imperfections. The resulting stress distribution is shown in Fig. 3. In Fig. 3A, the left notch tip experiences higher stress than the right, explaining why forward cracking is systematically favored.\nThe sensitivity to defects may explain the indeterminacy observed in PRL 126, 025503, where approximately one-quarter of the specimens of Fig.4(c) in PRL 126, 025503 exhibited cracks 19 initiating in the bulk rather than at the edge notch. To better account for potential fabrication imperfections, we introduce a small geometric modification in Fig. 3B by rounding a sharp angle. In this case, the highest stress is observed at the modified corner, where a crack is expected to initiate. In contrast, TMM demonstrates more robust behavior in experiments. In our TMM experiments, despite the presence of fabrication imperfections, cracks consistently initiate on the right side of the notch in all 40 specimens (provided that no hinges accidentally break before testing) in Fig. 3C. TMMs are less sensitive to fabrication errors and present a more robust crack control behavior.\nReviewer Comment2.2 — Istopologyreallyessentialhere? Whathappensifonetakeatwisted (undistorted) Kagome instead? Would such metamaterial exhibit symmetric fracture instead?\nReply: Thank you for your insightful comment. Yes, topology is essential for the robust asymmetry feature. Twisted kagome doesn’t show symmetric fracture feature, but its asymmetric feature can be easily altered. Only the TMM with topological polarization show robust one-way fracturing.\nFigure 9: Breaking a twisted kagome lattice in simulations and experiments. (A) Setup. Red lines indicate the bulk crack. The left tip is boxed in the blue box on the left. The right tip is boxed by the light blue box on the right tip. Scale bar: 12.5 mm.\nThe twisted kagome does break mirror symmetry and thus exhibits an asymmetric stress field (Fig. 9B), leading the crack propagation to the right. However, this geometric asymmetry in twisted kagome is not robust against the initial crack shapes. As shown in Fig. 4F, G, the cracking direction of the twisted kagome can be altered easily. Also, the tip units of the twisted kagome rotate and untwist (Fig. 9C, due to the excited Guest-hutchinson mode). Therefore, the twist kagome gradually losing their asymmetric character as the crack develops. This behavior explains why twisted kagome is more brittle, similar to regular kagome (Fig. 4B).\nFigure 10: The ZMs of regular kagome, twisted kagome and polarized kagome spring-mass lattice with two types of holes. The horizontal boundary nodes of the three lattices are connected. The top and bottom nodes of the three lattices are all pinned. (A) regular kagome lattice (unpolarized), (B) twisted kagome lattice (unpolarized), (C) polarized kagome lattice with a short, straight hole. (D) regular kagome lattice (unpolarized), (E) twisted kagome lattice (unpolarized), (F) polarized kagomelatticewithatiltedhole. Regardlessoftheshapeofthecrack, theZMsofpolarizedkagome never localize to the left side of the crack. This implies the robustness of the ZM distribution in the polarized kagome is due to topological factors rather than purely geometric ones.\nTherefore,asprotectedbythetopology,theTMMfeaturesamorerobustZMdistributionagainstthe crack shape, which leads to a robust asymmetric stress field and one-way cracking behavior. Topology is essential here for unidirectional crack propagation.\nToemphasizethattopologyisessentialhere, thecomparisonbetweentheTMMandtwistedkagome (and also regular kagome) is included in the main text as follows.\nLine 153: Unlike the triangular and regular kagome lattices, both the twisted kagome lattice and the TMM effectively direct the propagation of a straight notch to the right (see Movie S6, left panel, and Fig. 1). However, the twisted kagome lattice’s directional cracking is easily compromised when the crack shape changes (Movie S6). As shown in Fig. 4 panels D–G, the TMM consistently guides cracks to the right, whereas the twisted kagome lattice does not.\nLine 164: In summary, the crack propagation in the TMM can be considered predictable, unidirectional and ductile. Topological is essential here. This further suggests the peculiarity of the sharp, asymmetric stress field and the uniqueness of the one-way cracking achieved working with 21 TMMs.\nNew supplementary movies have been added to compare the fracturing processes of the twisted kagome lattice with three different crack configurations (Movie S6) and the TMM with two triangular cracks (Movie S7).\nThe asymmetric fracturing process of twisted kagome with a straight notch is added in SI IB (Page 6)\nReviewer Comment2.3 — Likewise, does the magnitude of the topological polarization matter for unidirectional facture propagation? I would expect a topological metamaterial with a small polarization/asymmetry to be less robust to imperfections?\nReply: This is a great point. Indeed, a more polarized geometry leads to greater robustness. We find that stronger asymmetric stress fields promote more reliable and robust unidirectional cracking during experiments. To quantify this effect, we use the unit cell geometry presented in this work as the baseline geometry. As the middle node moves along the horizontal axis to the right, the localization length of the topological edge modes decreases, which can be considered as being “more polarized”. Thus we use the geometric parameter ∆ which is the rightward shift of the middle node (Fig.11A), to denote the polarization magnitude. The stress difference between the left and right crack tips, corresponding to varying polarization magnitudes, is plotted in Fig.11B. When the polarization magnitude decreases to ∆ = −0.65, the stress difference approaches unity, indicating nearly symmetric stress distribution. Although this transition is clearly evident in simulations, such weak polarization results in significantly reduced robustness during experimental testing.\nFigure 11: Tuning the stress asymmetry through polarization magnitude in simulations. (A) The connectingnodesbetweentwotrianglesaremovedalongthehorizontallinebyadistance,∆. ∆ = 0 configuration is the original geometry in this work. (B) The dependence of the stress ratio (left tip hinge to right tip hinge) on the hinge position in continuous geometries (von Mises stress). Smaller σ /σ ratio correspond to stronger polarization.\nL R\nThe above discussion has been incorporated in the SI IIE (Page 16), and the following sentences are added in the main text.\nReviewerComment2.4 — Section2.2canbeunderstoodbyphysicistswellversedintopological metamaterials but less so by scientists with no background in this area. The authors might want to make that section a bit more pedagogical to extend the reach of their paper.\nReply: We thank the reviewer for the valuable comment regarding Section 2.2. We appreciate the suggestion to make this section more pedagogical, which helps broaden the accessibility of our paper to a wider scientific audience. We have accordingly revised Section 2.2 to include clearer explanations. Given the length constraints, we have also added additional references to earlier studies in Section 2.2, where detailed theoretical frameworks have been thoroughly derived and discussed.\nReviewer Comment2.5 — The authors often refer to supplementary figures from the Main Text, which make the read somewhat cumbersome and gives the impression that key information is given in the SI rather than in the Main Text. For instance, the comparison with continuous media in Fig. 1 is somewhat of a naive benchmark. From the text it seems that comparing with the undistorted Kagome would be more meaningful (and would allow less back and forth between Main Text and Supplementary Information).\nReply: We thank the reviewer for the comments. We rearranged the texts so that we mainly refer to the contents in the main text. We also include a new main figure (Fig. 2 in the revised main text) with four lattices in the experiments. Here are the modified paragraphs in this section. The figures in the box refer to the labels in the main text.\nLine 140: To provide a comprehensive analysis of fracture protection of the TMMs, we quan- titatively compare the cracking behaviors of four lattice configurations in our experiments: the triangular lattice, the regular kagome lattice, the twisted kagome lattice, and the TMM (see Fig. 2). Fig. 2A presents schematics of these four lattices. For consistency with the TMM, the twisted kagome lattice is designed with a blunt tip (right) and a sharp tip (left). All lattices are fabricated from the same material, share identical sample sizes, and have a crack length of 2a. Their force-displacement curves are shown in Fig. 2B. The triangular lattice is generated by making all pointing-down triangles in a regular triangle lattice filled as solid triangles and all pointing-up triangles empty. It has a higher coordination number (6) compared to the kagome lattices(4). Thesedifferencesaccountforthetriangularlatticeexhibitingthehigheststiffness andstrength. However, the triangular lattice shows a sudden, asymmetric, brittle, and unpredictable failure mode (Movie S3). The regular kagome lattice features the highest fracture toughness among the three kagome configurations due to the blunting effect (Fig. 1C, Fig. 4B). Notably, the regular kagome also displays somewhat erratic propagation behavior. In ideal simulations, its symmetric geometry leads to two-way symmetric propagation resulting in brittle failure (Movie S4, Fig.S6C).Experimentally, nevertheless, thebluntedtipstress(Fig.1C,Fig.5B)producesun- predictablecrackpathsthatmayevenveerinotherdirectionsduetouncontrollableimperfections in the sample (Movie S5).\nUnlike the triangular and regular kagome lattices, both the twisted kagome lattice and the TMM 23 effectively direct the propagation of a straight notch to the right (see Movie S6, left panel, and Fig.1). However,thetwistedkagomelattice’sdirectionalcrackingiseasilycompromisedwhenthe crack shape changes (Movie S6). As shown in Fig. 2 panels D–G, the TMM consistently guides cracks to the right, whereas the twisted kagome lattice does not. Moreover, regarding fracture toughness, both the TMM and twisted kagome lattices exhibit similar values—each displaying only half the toughness of the regular kagome lattice, suggesting the fracture toughness of the TMM is comparable to other asymmetric lattices.\nImportantly, the TMM is significantly more ductile than the other lattice—its elongation is 0.066 comparedtothetwistedkagome(0.030),regularkagomelattice(0.019)andthetriangularlattice (0.015). Moreover,theenergydissipationoftheotherlatticesisonlyhalfthatofTMM(Fig.2C). This controlled deformation absorbs and dissipates energy that would otherwise accelerate crack propagation,enhancingthestructure’stolerancetodefectsandreducingtheriskofsuddenfailure. Thereby, the TMM displays the most stable and gradual cracking behavior without a significant sacrifice of the fracture toughness.\nIn summary, the crack propagation in the TMM can be considered predictable, unidirectional and ductile. Topology is essential here. This further suggests the peculiarity of the sharp, asymmetric stress field and the uniqueness of the one-way cracking achieved working with TMMs.\nReviewer Comment2.6 — Topological metamaterials can host states of self-stress. What is the role of those states of self stress on the fracture propagation?\nReply: Thank you so much for your constructive comments. Three kinds of states of self stress (SSSs) are involved in this work. For a single domain, there are two kinds of SSSs: (1) SSSs in the NN lattice near the crack, (2) SSSs in the NNN lattice and real-world TMM near the crack. For two or more domains and the crack doesn’t cross the interface, there is (3) SSSs in the interface. For the two near crack SSSs ((1) and (2)), since our fracture results are obtained in the small deformation regime, only the second type of SSS plays a role, arising as a secondary effect of the near-crack topological soft modes for guiding cracks.\nIn the resubmitted manuscript, we add new case studies of TMMs with multiple domains, so type (3) of SSSs become relevant (Fig. 8). In this case, we harness it to program cracking path and fracture behaviours. The third kind of SSS can be excited in NN, NNN and real-world TMMs. More detailed explanations are as follows.\n(1) For the NN only lattice, in this ideal case, the SSS only appears in the nonlinear regime, when the applied displacement is large enough. In an ideal nearest-neighbor (NN) spring lattice, SSSs are locally removed (Fig.12A) near the crack tips. Instead, zero modes (ZMs) near the crack (Fig.12B) cause hinge rotation, which does not generate stress due to the lack of bending stiffness. Due to the lack of constraints, there’s no SSS near the crack in the linear, small deformation regime.\n(2) For the NNN lattice and real-world TMM, the SSSs can appear in the small deformation regime, with the existence hinge rotation constraints (NNN bonds or hinge bending stiffness). The topological soft mode in the NN bonds, the primary effect, results in hinge rotations, which excite the SSSs (in the NNN bonds or hinges) as the secondary effect for the asymmetric stress and crack propagation.\nAs detailed in Fig. 12, we show why SSS is the secondary effect for asymmetric stress concentration. In the NNN lattice, the rotation of NN springs on the right side of the crack stretches the NNN springs (Fig. 12C, E). In another word, the hinge rotation (Fig. 12B) is projected onto the state of self-stress in the NNN springs, localizing stress at the right tip. The NNN springs correspond to the hinges in a solid lattice with bending stiffness (real-world TMM). Therefore, zero modes or soft modes primarily lead to stress concentration and ultimately nucleate cracks.\nNNN NN\n(3) When two or more domains are present, SSSs can emerge at the interfaces. As illustrated in Fig. 8, these excited SSSs concentrate stresses at the interfaces under external loading. In this scenario, the SSSs dominate crack initiation and early-stage propagation within the interfaces. However, after the interface breaks and cracks advance toward the notch tips, ZMs become the governing effect again and controls the subsequent crack propagation near the notch.\nThe following sentences are added to explain the SSS in the main text.\nLine206: ThediscussionsabovedescribetheresponseoftheNN-onlytopologicalMaxwelllattice under loading, where due to the lack of SSSs, the lattice moves along the linear combination of the affine strain and the nonaffine nearly-ZM, and there is no stress to the linear order (SI IIB). The strong asymmetry of the nearly-ZM gives rise to the strong asymmetry of the displacement field of the lattice under load, leading to stress responses only at the nonlinear level.\nLine 216: Secondly, asubtledifferencearises, astheweakadditionalconstraintsduetotheNNN bonds or the bending stiffness of the hinges induce homogeneous SSSs even in presence of the 25 crack. This leads to weak stress responses at the linear level, mainly carried by the NNN bonds (Fig. 3E).\nLine 330: In these systems, the programmed interfaces play a dominant role over the notch tips, due to the excited SSSs. In the photoelastic experiments, concentrated stress is clearly observed at the interfaces for both geometries (Fig. 8B, D), and this finding is consistent with the simulations. In Fig. 8E, F, cracks first originate in the interface regions (the gray box). Once the interface hinges fail—specifically, those in the zigzag interface near the notch and all hinges in the straight interface—the crack extends toward the notches. Owing to the topological polarization effect, the crack propagates to the right tip before reaching the left tip.\nWe have included Fig. 12 and more detailed discussions in the SI IIB (Page 13). The figures below refer to the figures in this response file.\nThree kinds of states of self stress (SSSs) are involved in this work. For a single domain, there are two kinds of SSSs: (1) SSSs in the NN lattice near the crack, (2) SSSs in the NNN lattice and real-world TMM near the crack. For two or more domains and the crack doesn’t cross the interface, there is (3) SSSs in the interface. For the two near crack SSSs ((1) and (2)), since our fracture results are obtained in the small deformation regime, only the second type of SSS plays a role, arising as a secondary effect of the near-crack topological soft modes for guiding cracks.\nThe case studies of TMMs with multiple domains are only applied in Fig. 6. In this case, we harness it to program cracking path and fracture behaviours. The third kind of SSS can be excited in NN, NNN and real-world TMMs. More detailed explanations are as follows.\nIn an ideal nearest-neighbor (NN) spring lattice, SSSs are locally removed (Fig. 12A) and do not affect crack tips. Meanwhile, zero modes (ZMs) induce hinge rotation near the crack (Fig. 12B). Due to the lacks of bending stiffness (or constraints) in the NN lattice, these rotations do not generate stress in the linear order.\nIncontrast, inanext-nearestneighbor(NNN)lattice, therotationofNNspringsontherightside of the crack stretches the NNN springs (Fig. 12C, E). In this case, the hinge rotation (Fig. 12B) is projected onto the state of self-stress in the NNN springs, localizing stress at the right tip. The role of NNN springs resembles the hinges in a solid lattice with bending stiffness (TMM). Therefore, zero modes or soft modes in the NN bonds (or rigid triangles in the real-world TMM) lead to stress concentration and ultimately nucleate cracks in the NNN lattice and real-world TMMs.\nIn Fig. 8, the SSSs are distributed in the interfaces. Upon stretching, the SSSs are excited as locallyconcentratedstress. Sinceinterfacesplayagoverningroleinthesystem,thecracksinitiate in the interfaces first, before it goes to the tips of the notch and governed by the topological soft modes again.\nReviewerComment2.7 —Higherordertopologicalmetamaterialscanhostcornermodes,would such corner modes be beneficial for the control of fracture?\nReply: Wethankthe reviewerforproposingthis interestingidea. Following thissuggestion weexplored stress distributions in topological mechanical lattices with corner modes. However, in our results, soft corner modes around the crack appear to not concentrate stress or guide cracks. A low frequency corner 26 mode we are aware of is introduced in Phys. Rev. B 98, 180102(R). We reproduce the low frequency corner modes in the upper right corner of the crack (Fig. 13). However, for all the cases of uniform staticloadingattheboundaries, theupperrightcornershowsthesoftmodewithoutconcentratedstress at the hinges. In Fig. 13B, C, we show that applied external boundary loadings cannot be projected to the state of self-stress at the corner. The three other corners all have a larger stress than the upper right corner (corner 2). Therefore, we have not found a way to use those modes to localize stress and nucleate fractures at the corner.\nFigure 13: Soft corner modes at the crack delocalize stress. (A) The lowest frequency mode localizes in the upper right corner of the checkerboard lattice. a and a are two lattice vectors, 1 2 a = (1.242,0.067) and a = (0.480,1.313). The positions of the lattice sites inside one unit cell of 1 2 the lattices are r = (−0.350,1.038), r = (0.480,1.313), r = (0.892,1.104) and r = (0,0). (B) In 1 2 3 4 the static analysis, biaxial stretching is applied to the four edges. Four interior corners of the crack are boxed. The red arrow points to the localized corner mode, corner 2. (C) The distribution of stress at the four deformed corners.\nReviewer Comment2.8 — More generally, to reflect my main concern above, it would be important to know what the range of control one can achieve with topological metamaterials as well at the limitations.\nReply: We are grateful to the reviewer for this insightful suggestion, which has significantly enriched the depth of our manuscript. Thanks to the reviewer’s suggestion, we now show that we can tune the configuration and topological effect by changing topological polarization magnitude (shifting node coordinates) and hinge thickness, so that we control the asymmetric feature (tip stress difference). In practice, it’s crucial to find the optimal configuration. Diluting the polarization by either decreasing the polarization magnitude or increasing the hinge thickness results in a more symmetric stress field and a less robust control over cracking. On the other hand, although very thin hinges and highly polarized 27 geometries guarantee a huge stress difference between two sides, the fabrication can be extremely challenging.\nThe related discussion has been added in the main text.\nLine 226: In practice, finding the best configuration is essential. Reducing the polarization, either by shifting the unit cell geometry or by increasing the hinge thickness, weakens control over one-way cracking. On the other hand, although very thin hinges and strongly polarized geometries produce a large stress difference between the two sides, they are much harder to manufacture.\nWe also discuss the limitations of TMM in terms of geometries and materials. The following paragraph has been added to the conclusion and discussion section in the main text.\nLine 365: While TMM provides a robust method for fracture control, it also has notable lim- itations. Because topological polarization is a concept in linear elasticity, it is maintained only in the small deformation regime (less than 30% for the TMMs studied here [59]). This means that the phenomena discussed here are limited to TMMs made of brittle materials with minimal plastic deformation.\nThe authors appreciate the reviewers and the editor for their constructive comments and sugges- tions, which have greatly improved the quality of this manuscript. The manuscript has undergone a thorough revision according to the editor and reviewers’ comments. We include substantial re- sults in the new version. Please see below our responses. For the reviewers’ convenience, we have highlighted significant changes in the revised manuscript and SI in blue. (Figures mentioned in the response refer to figures in this file unless otherwise specified.)\nReviewer 1\nI appreciated the authors’ efforts in revising the manuscript and carefully addressing the concerns raised in the first-round review. The revision demonstrates substantial work and improvements. However, some fundamental concerns remain unresolved, particularly regarding (1) the level of novelty and significance, and (2) overstated claims. Given the high standards and broad scope of Nature Communications, the manuscript cannot be recommended for publication in its current form. Detailed comments are provided below.\nReply: We sincerely thank the reviewer for carefully re-evaluating our revised manuscript and for acknowledgingtheimprovementsmade. Weappreciatetheopportunitytofurtherclarifyourcontribution and address the remaining concerns. In this second revision, we have aimed to better articulate the novelty and significance of our findings while refining the presentation of our claims.\nOur work introduces a topological mechanical metamaterial (TMM) that achieves robust, notch- insensitivecrackguidanceunderfabricationimperfections, whichcannotbeachievedinothergeometric- induced fracturing systems as demonstrated below. We show that our work constitutes a meaningful advance in the context of both topological mechanics and mechanical metamaterials. To respond to the reviewer’s points, we have revised the manuscript to clarify the engineering relevance of this robustness, qualify the scope of our claims, and strengthen the experimental and comparative analyses. We hope these revisions address the remaining concerns and present the core contributions of our work with both rigor and clarity.\nReviewer Comment1.1 — Novelty and significance. The concept of unidirectional fracture is not new and has already been demonstrated in the literature.\nIt can also be easily achieved in twisted Kagome as shown in this manuscript. The authors try to emphasize the novelty of the TMM in enabling “robust” one-way fracturing that is insensitive to notch geometries and imperfections.\nHowever, this claimed novelty seems very limited. It is not surprising that fracture propagates preferentially in one direction when one side undergoes higher stress. This is achieved at the cost of substantial reduction in fracture toughness. Moreover, the significance of achieving such insensitivity is also limited. (see detailed comment below)\nReply: We thank the reviewer for the comments. However, the robustness of directional fracture and stress localization in the TMM is can not be easily achieved by systems like twisted kagome. And the 1 fracture toughness of TMM is comparable with lattices of similar weight density. We respectfully clarify that the novelty and significance of our work go beyond the four points raised, as outlined below.\n(1)-”It can also be easily achieved in twisted Kagome as shown in this manuscript.” Although the twisted kagome makes the fracture propagate to the right in perfect conditions (straight notch), it is not robust at all. Its stress field and crack propagation direction are sensitive to notch shapes in Fig. 1E, F (Fig . 2E, F in the main text). In contrast, the TMMs provide robustness from the topological protection.\n(2)—”It is not surprising that fracture propagates preferentially in one direction when one side undergoes higher stress.” The comment is correct. However, obtaining higher stress on one side in a robust way is not trivial. Doing this with local geometry, such as earlier designs we compared to ref. [39] PRL, 126, 025503, 2021 in our last response (Fig. S5 in the SI), gives stress around the cracks that is highly dependent on the crack shape. In contrast, topological mechanics offers a robust way to achieve asymmetric stress, which is directed by the bulk of the system, instead of local effects. This unique feature is invariant to crack shapes and fabrication imperfections.\n(3)—”This is achieved at the cost of substantial reduction in fracture toughness.” A direct comparison between the TMM and the triangular lattice in terms of load-bearing capacity and fracture toughness is not fair, as these structures differ fundamentally in both connectivity and material usage. The triangular lattice features six bonds per node in the spring–mass representation and occupies roughly half the area in Fig. 1A (Fig. 2A in the main text). In contrast, the kagome lattice has only four bonds per node and occupies approximately one-quarter of the area. In the triangular lattice, the higher connectivity provides more redundant load paths, which naturally results in greater stiffness and fracture toughness (Calladine, 1978; Lubensky et al., 2015).\nThe kagome lattice, by design, is marginally connected (4 bonds per nodes in 2-dimensions) and intended to enable controllable, defect-tolerant fracture propagation rather than maximizing strength. We compare regular kagome lattice, twisted kagome lattice and TMM in Fig. 1 (Fig. 2 in the main text). Both the twisted kagome lattice and the TMM effectively direct the propagation of a straight notch to the right (see Movie S6, left panel). However, the twisted kagome lattice’s directional cracking is easily compromised when the crack shape changes in Fig. 1E, F (Fig. 2E, F in the main text).\nWe added these arguments in the main text.\nLine 169: A more comprehensive discussion of notch shapes will be given in Sec 2.4.\nLine 157: The triangular lattice features six bonds per node in the spring–mass representation and occupies roughly half of the bulk area. In contrast, the kagome lattice has only four bonds per node and occupies approximately one-quarter of the area.\nLine 179: As a result, with the same material density, the TMM demonstrates the most stable and gradual fracture behavior, without a significant reduction in fracture toughness.\n(4)—”Moreover, the significance of achieving such insensitivity is also limited.” We thank the re- viewer for noticing that the significance of achieving notch shape insensitivity is not clearly articulated in the manuscript. The significance stems from the fact that naturally occurring cracks in materials come in all kinds of shapes due to different types of damage. Obtaining a metamaterial that can control cracks in a unidirectional way for any shaped cracks will greatly broaden the applicability of fracture control in complex environments. In this work we demonstrated such insensitivity to other notch shapes (straight, tilted, triangular, trapezoid and rectangular notch shapes), see Fig. 3 below.\nAs detailed in the answer to Reviewer Comment 1.2 (1). We added these arguments in the main text. 2 Figure 1: (Revised Fig.2 in the main text) Comparing the fracturing behaviors of triangular lattice (generatedbymakingallpointing-downtrianglesinaregulartrianglelatticefilledassolidtriangles andallpointing-uptrianglesempty),regularkagomelattice,twistedkagomelatticeandTMMinthe experiments: onlytheTMMdemonstratesrobustnesstonotchshapesandhighenergyconsumption during fracturing. (A)-(B) Straight notch results. (A) Lattice geometries of the four lattices. The straight notch has length 2a. (B) Experimental force-displacement curves of the four lattices. The cracking direction of triangular lattice and regular kagome lattice exhibit is undetermined. Cracks in both the twisted kagome and the TMM propagate to the right for all the tested samples. (C-D) Triangular notch results. Cracking snapshot of TMM with (C) sharp side (of the triangle notch) pointing to the left and (D) sharp side (of the triangle notch) pointing to the right. The crack propagates to the right in both cases. Cracking snapshot of twisted kagome with (E) sharp side pointing to the left and (F) sharp side pointing to the right, leading to opposite cracking directions. Circled hinges are broken. Hinge thickness h is shown in the insert in (C) and (D). Scale bar: 12.5 mm.\nLine 295: The importance of notch-shape insensitivity lies in the fact that naturally occurring notches in materials take on diverse forms due to various types of damage. Achieving a meta- material that guides cracks unidirectionally regardless of notch shape would greatly expand the applicability of fracture control in complex environments.\nTitleofFig.2inthemaintext: Comparingbehaviorsoftriangularlattice(generatedbymaking all pointing-down triangles in a regular triangle lattice filled as solid triangles and all pointing-up triangles empty), regular kagome lattice, twisted kagome lattice and TMM in the experiments: only the TMM demonstrates robustness to notch shapes and high energy consumption during fracturing.\nReviewer Comment1.2 — (1) To demonstrate the “robust” one-way fracturing of TMM, the authors introduced “triangular cracks” (Fig. 2D-G), which is an unusual and rarely used configuration in fracture studies. If the TMM only shows an advantage in this special case, it means the significance is limited.\n(1.1)-In Movie S7, in TMM with “triangular crack”, after a hinge on the right breaks, the subsequent fracture occurs on the left side—this contradicts the claimed robustness.\n(2) In Movie S6 and Fig. 2F, the twisted Kagome sample appears to have thicker hinges.\n(2.1)-If the hinge thicknesses were reduced, would the fracture direction change?\n(2.2)-When comparing different structures in Fig. 2, were the hinge thicknesses controlled consistently across the samples?\n(2.3)-Can the fabrication method consistently produce the desired hinge thickness?\n(3) Additionally, in the context of fracture mechanics, “crack” typically refers to sharp flaws or narrow slits. Using the term “triangular cracks” to describe these holes is misleading. It would be more appropriate to refer to them as defects or notches.\nReply: We thank the reviewer for the constructive comments. We showed general advantages of the TMM in different types of crack shapes (straight, tilted, triangular, trapezoid and rectangular notch shapes are experimentally verified in Fig. 3 in this file), and it is NOT limited to triangular ones. In addition, wehaverevised”triangularcracks”as”triangularnotches”followingthereviewer’ssuggestion.\n(1) The TMM’s robustness is general and not limited to the triangular notch case. We first theo- retically explained the robustness via the topologically-protected zero frequency modes (main text Sec 2.2), and then tested different notch shapes through zero-mode analysis and finite element simulations Fig. 3A-D (this file), supported by five experimental validation (shown in Fig. 3E-I in this file). Triangu- larnotcheswereselectedasextremelyasymmetricexamples, whereonemightexpectthesharpersideto break first. Despite this, the TMM consistently guided fracture to the right, demonstrating directional control even under asymmetric conditions.\n(1.1) We thank the reviewer for rising this comment. The robustness claim is valid: in our new computational results in Fig. 2 (now Fig. S9 in the SI) using thinner hinges and a more brittle material, we show that more hinges on the right side break before the subsequent fracturing on the left. The 4 new specimen with thinner hinges in (Fig. 2C, which is Fig. S9C in the SI) and made of a more brittle material can be better approximated by the ideal spring-mass lattice (Fig. 2B, dashed grey curve, which is Fig. S9B in the SI), comparing with the less ideal material in our experiments (see solid green curve in Fig. 2B and Fig. 2D, which are Fig. S9B and C in the SI; main text Fig. 2C). Since the topological effect is derived in ideal spring lattice with small deformation, a better approximation leads to a more pronounced robustness. This supports the topological origin of the one-way crack propagation phenom- ena we observe: specimen closer to the ideal theory exhibits cleaner one-way propagation throughout the fracturing process, whereas the effect is more diluted in specimen further away from this limit (Jolly, J. C., et. al, Advanced Science 10, no. 22 (2023): 2302475 and Widstrand, C., et. al, International Journal of Solids and Structures 2023, 274, p.112268).\nThe triangular notch case is similar to the rectangular notch case (Fig. 3H, which is Fig. S21 in the SI). Breaking a vertical open edge on the right side of the notch (Fig. 3H and I) poses a bigger challenge for the current experiment setup: the hinges in free edge at the right tip side experience stress-free deformation. As our new one-way fracturing results in Fig. 2B (now Fig. S9C in the SI) and the lattice shape in Fig. 2C (now Fig. S9C), the experimental realization of full one-way cracking for triangular and rectangular cracks calls for more brittle materials and more advanced fabrication methods in the following work.\nOur current, non-ideal experiments supported initial one-way propagation with localized nonlinear deformation at crack tip, not the full nonlinear regime. Furthermore, our primary focus is on the initial fracture propagation in this case, as the subsequent events depend on many factors such as nonlinear deformations (e.g., mixed-mode loading), real material constants and system size effects (after the breaking of the first hinge, the fracture front is close to the edge of the system). Quantitatively studying the the nonlinear fracturing process and finite size effects will be interesting topics for future work, but it is beyond the scope of the current paper.\nThe following sentences have been added in the main text. The SI file is also modified accordingly (see P7-10).\nLine388: Therobustnessoftheone-wayfracturingalsoextendsdeeperintothenonlinearprocess after the breaking of the first hinge in general, supported by the topological mechanics theory as we discuss in Sec. 2.2. This is observed in our experiments to varying extent depending on the type of the notch (man text Fig. 1, 2 and 5), where straight cracks displays the clearest one-way propagation for subsequent breaking events (main text Fig. 1). However, due to limitations of the fabrication method we currently use, full quantitative investigation of this effect will be left for future work. Nevertheless, we support this by numerical simulations with thinner hinges and more brittle materials which are closer to the ideal theory limit (Fig. 2B, C, which are from Fig. S9 in the SI). In this case, the one-way propagation of the crack is robust throughout the fracturing process (Fig. 2B, C, which are from Fig. S9 in the SI). This effect is diluted in experimental specimen due to hinge thickness and material constants, similar to how topological mechanical effects are diluted as the TMM deviate from the ideal limit in earlier studies.\n(2) Because the triangular units in the polarized and twisted kagome lattices differ in geometry, an identical hinge thickness can appear visually larger in the twisted kagome, giving the impression of thicker hinges. See the inserts in Fig. 1 C and D (main text Fig. 2). Moreover, reducing or increasing the hinge thickness does not affect its fracture direction of twisted kagome lattice with triangular notch: 5 Figure 2: (Fig. S9 in the revised SI) Crack propagation for TMM and twisted kagome with the same triangular notch demonstrating robust one-way fracturing as hinge thickness is reduced, ap- proaching ideal theory. (A) Details of unit cell geometries for (C-F). (B) Stress-strain relations in simulations. Tuned material (dashed curve) is only applied in (C); for all the other fracturing calculations, paper material is applied (solid green curve). Initial fracture events for (C-D) polar- ized kagome lattice using different hinge thickness and (E-F) twisted kagome lattice using different hinge thickness. 6 7 Figure 3: TMM enables unidirectional cracking for various notch shapes. (A-D) Panels in Fig.5A- D: generalizing to all bulk notch shapes. (A) Schematic of (I) the cut in the bulk, (II) the cut at the soft edge, and (III) the cut at the rigid edge. (B) ZM weight U in the topological kagome lattice with a tilted cut along a . α is the angle between the cut edge normal n and R . Stretching 2 T the corresponding continuous TMM, we see (C) the displacement field. (D) Dependence of the magnitude of ZMs and bulk-to-edge stress ratio on the angle α. One-way fracturing for symmetric notches: (E) Panels in main text Fig. 1; (F) Panels in main text Fig.5; (G) Fig. S20; (H) Fig. S21. One-way fracturing for asymmetric notches: (I) Main text Fig. 2C, E. the left side consistently experiences higher stress than the right side (Fig. 4, which is Fig. S10 in the SI).\n(2.1) Reducing the hinge thickness of twisted kagome can’t change the fracture direction (Fig. 2E and F, which are from Fig. S9 in the SI), as long as the specimen enters the regime controlled by the theoretical prediction (Fig. 4D, which is from Fig. S10 in the SI). The theoretical prediction is made for the limit of hinge stiffness going to zero (Fig. 4A, D, which are from Fig. S10 in the SI). We first show frequencyanalysistoexplaintheoreticaloriginofstressdistributionnearthenotch, andthennumerically alter hinge thickness of polarized and twisted kagome in Fig. 4 B, C, E-G (now Fig. S10 in the SI).\nFirst, to explain static behaviors of the two isostatic lattices with small displacements, we perform mode analysis using spring-mass frame (zero hinge stiffness) in Fig. 4A, D (now Fig. S10 in the SI). In the polarized kagome lattice, zero-frequency modes (ZMs) localize on the right side of the notch due to topological polarization, whereas in the twisted kagome lattice they spread along all notch edges. Thus, strong asymmetry arises only in the polarized kagome.\nAs for static stress, the thin hinge scenario can be better approximated by the ideal spring-mass model and is governed by the ZMs: the side with larger zero modes shows larger hinge rotation angles and therefore larger stress. In Fig. 4B (now Fig. S10 in the SI), comparing to the left edge of the notch, the right edge of the polarized lattice go through much larger hinge rotation and therefore generate larger stresses. For the twisted kagome lattice in Fig. 4E (now Fig. S10 in the SI), since the hinge rotation modes are not localized at just one edge and stronger on the left edge, the stress is slightly larger on the left, σ > σ (orange curve in Fig. 4G, which is from Fig. S10 in the SI). In L,max R,max Fig. 2E and F (now Fig. S9 in the SI), we numerically analyze the fracturing behavior of twisted kagome lattices with varying hinge thicknesses. In both cases, cracks initiate on the left side. The fracture patterns agree well with the static results in Fig. 4G (now Fig. S10 in the SI).\nAs the hinges become much thicker in Fig. 4C and F (now Fig. S10 in the SI), the hinge rotation or hinge stress is not determined the ZMs. With thicker hinges, the two lattices take on similar geometries, and their stress asymmetries become nearly identical. The dependence is plotted in Fig. 4G (now Fig. S10 in the SI). When the stress difference between two sides is lower than 1, the stress-guiding effect is not robust to the experiment errors.\nThe following changes have been added in the main text. The SI file is also revised (see page 7).\nLine 151: Because the triangular units in the polarized and twisted kagome lattices differ in geometry, an identical hinge thickness can appear visually larger in the twisted kagome, giving the impression of thicker hinges (see main text Fig. 2C and E insert). Moreover, reducing or increasing the hinge thickness does not affect the fracture direction of twisted kagome lattice: 8 with the triangular notch in Fig. 2E (main text), the left side consistently experiences higher stress than the right side (Fig. 4G).\n(2.2 - 2.3) Due to fabrication limitations, we cannot guarantee uniform hinge thickness across all samples; as shown in Fig. 4H (now Fig. S10 in the SI), fabricated hinge thickness shows a measured average 0.8 ± 0.2 mm. Additionally, paper is a laminated and inhomogeneous material, and some variation in hinges inevitably exists. Notably, despite the slight hinge thickness variance and material non-uniformity,itserrorregionsinquantitativeforce-displacementscurvesinFig.1B(maintextFig.2B) remains acceptable.\nRemarkably, the one-way propagation of cracks in the TMMs is not affected by this variation, again demonstrating the robustness of topological mechanical effects.\nSince we cannot control hinge thickness perfectly in the experiments, we numerically study the effects of hinge thickness in twisted kagome, as demonstrated above (Fig. 4, which is now Fig. S10 in the SI).\nLine 432: As shown in Fig. 4H (now Fig. S10H in the SI), fabricated hinge thickness shows a measured average 0.8±0.2 mm. We measured 20% of hinges in 10 samples. We measured 20% of the hinges in 10 representative samples (≈ 400 hinges in total). Notably, despite these minor fabrication errors, one-way cracking of the TMM is consistently observed across all specimens thanks to the topological protection, and error ranges of quantitative force–displacement curves remain within acceptable in Fig. 1B (main text Fig. 2). The one-way propagation of cracks in the TMMs is not affected by this variation, again demonstrating the robustness of topological mechanical effects.\n(3)Wethankthereviewerforthishelpfulclarificationregardingterminology. Weagreethattheterm “triangular cracks” may be misleading in the context of fracture mechanics. In the revised manuscript, we have replaced it with “notches” or “defects” where appropriate to more accurately describe the geometry and maintain consistency with standard terminology.\nReviewer Comment1.3 — The experimental setup is not fully convincing. Although the authorsclaimminimalinfluencefromgripperrotation,suchdeformationcanintroducemixed-mode loading, which is known to affect fracture paths. The authors argued that the force-displacement curves were measured using center-cracked samples and “the samples can remain connected at left and right edge for the full loading period.” which is not accurate. As shown in Movie S2, after failure on the right, the upper gripper rotates, and the crack does not propagate straightly on the left side (likely due to the loading condition), in contrast to the simulation results.\nReply: We appreciate the reviewer’s careful attention to details, especially in closely examining the experimental videos. This level of scrutiny is deeply valued. The deviation between the experimental and simulation crack paths after initial fracture is indeed likely due to mixed-mode loading introduced by slight gripper rotation. While our current work focuses primarily on the directionality and control of the initial fracture events, we agree that the influence of mixed boundary conditions is important and deserves further investigation. We have now clarified this point in the revised manuscript, and thank the reviewer for helping us improve the rigor of our presentation.\nThe following sentences have been modified or added in the main text: 9 Figure 4: (Fig. S10 in the revised SI) Hinge thickness and stress distribution of the TMM and twisted kagome. (A-C) TMM. (A) Distribution of mode weights for the zero-frequency modes. The left side and right side of the notch are denoted. Truss element is applied in the FEM. Left side and right side of the lattices are connected to form a periodic lattice while top and bottom nodes are fixed. (B) Static stress distribution of thin hinge configuration. h is hinge thickness. (C) Static stress distribution of thick hinge configuration. (D-F) Twisted kagome. (A) Distribution of mode weights for the zero-frequency modes. (B) Static stress distribution of thin hinge configuration. (C) Static stress distribution of thick hinge configuration. (G) The dependence of stress ratio on the hinge thickness. The stress ratio is measured as the largest hinge stress on the left side to the largest hinge stress on the right side. (H) Fabricated hinge thickness shows a measured average 0.8±0.2 mm. We measured 20% of hinges in 10 samples. We measured 20% of the hinges in 10 representative samples (≈ 400 hinges in total). 10 Line440: Thegripperrotatedpassively(side-notchresultsinmaintextFig.5)sinceit’sconnected by a hinge. It mainly affects side cracks at the very end of the loading stage, due to the lack of constraints. For all the force-displacement measurements in this work, the cracks are in the bulk. And the samples can remain connected at left and right edge for the full loading period.\nThe force-displacement curves measured in this work are all for bulk notches and therefore, the influencefromtopgripperrotationdoesn’taffecttheinitial fractureeventsandplaysaveryminor role in the curves. For the sample in Movie S2, after fracture initiates on the right, slight rotation of the grippers introduce mixed-mode loading conditions, leading to deviations in the crack path compared to idealized simulations. While this study focuses on the initial crack scenarios, the influence of boundary-induced mixed-mode effects is an important consideration for future work (Hutchinson, J.W. and Sui, Z., 1991. Advances in applied mechanics, 29, pp.63-191.).\nReviewer Comment1.4 — Overstatements. The manuscript contains several claims that are overstated or insufficiently supported, which reduce the overall credibility. Below are just a few illustrative examples:\n(1) The authors state that the TMM “can be used to robustly guide fractures to propagate in a unidirectional and sequential way throughout the nonlinear fracturing process”. However, in Fig. 1 and Movie S2, after two hinges break on the right, a hinge on the left also fractures, indicating that the fracturing behavior is not entirely robust in the propagation stage. Nevertheless, lines 130-133 still describe the fracture as a fully formed one-way cracking pattern, ignoring the broken hinges on the left.\n(2) Statements such as “regardless of the crack shape” and “invariant to notch shapes” are too strong, especially considering that only triangular notches were tested.\n(3) The TMM achieves directional fracturing at the cost of significantly reduced load-bearing capacity and fracture toughness. Fig. 2 shows that, compared to the triangular lattice, the peak force and fracture toughness of the TMM drop by over 80%. Despite this, the manuscript claims “without significantly sacrificing load-bearing capacity” and “without a significant reduction in fracture toughness,” which are clearly misleading.\n(4)Inline60,theauthorsclaim“Thisstudyalsoaimsatfillingimportantgapsineffortstoapply TMMs to real-world engineering problems.” This is overly ambitious, as no practical application or engineering integration is presented in the paper.\nReply:\nWe thank the reviewer for highlighting potential overstatements in our wording. We have revised the manuscript to clarify the scope of our findings, avoid unsupported generalizations, and more accurately describe performance trade-offs. These changes help improve the precision and balance of the presentation.\n(1) The reviewer is right to point out that the left hinges eventually break in Fig. 1 (main text) and Movie S2. Due to the experimental limitations, our current setup is not close enough to the ideal one-way fracturing limit, and therefore the left side breaks earlier than expected from the ideal theory. We carefully rewrite “can be used to robustly guide fractures to propagate in a unidirectional and sequential waythroughoutthenonlinearfracturingprocess”(nowLine39inthemaintext)and“whentheone-way cracking pattern is fully formed” (now Line 139 in the main text).\nAsweexplainedintheanswertoReviwer Comment1.2(1.1)(page4-5inthisfile), thefull nonlinear regime is rich and will be left for the future work. In this paper, we intended to highlight:\ni)Asourprimaryfocusinthiswork, theinitial fracturingeventsareone-wayandconsidersgeometric nonlinearity at tips (indicated by circles in Fig. 5, which is from Fig. 1 in the main text) and material nonlinearity.\nii) For specimens we tested, in general, we observe one-way crack propagation even after the first hinge breaking. This is strongly supported by our simulation where when the hinge stiffness is very small, subsequent hinge breaking events also occur on the right until the crack approaches the boundary (Fig. 2B, C, which are now Fig. S9B, C in the SI). In experiments, however, our current fabrication method limits out testing from approaching this ideal robustness limit (Fig. 2B, D, which are now Fig. S9B, C in the SI; Fig. 1, which is Fig. 2 in the main text), so the full quantitative study of the whole crack path will require future studies.\nTo avoid ambiguity, besides the two sentences pointed out by the reviewers (now Line 39 and 139), the following modifications have been modified in the main text:\nLine 39: can be used to robustly guide fractures to propagate in a unidirectional and sequential way throughout the nonlinear fracturing process can be used to robustly guide fractures to propagate in a unidirectional way even during the nonlinear fracturing process\nLine 55: using the combination of purely periodic geometry and isotropic constituent materials, the initial cracks are guided in a unidirectional and robust way even considering the nonlinear deformation at crack tips and material nonlinearity, for a wide range of notch shapes;\nLine 116: Large rotations at the notch tip demonstrates geometric nonlinearity (circles at the tip in main text Fig. 1I-K) and the failure process includes material nonlinearity.\nTitle of Fig. 1 in the main text: ...(L) after breaking all hinges on the right side in the 12 simulations (except the special last one) and (M) final state of the TMM. In experiments, one left hinge breaks earlier due to hinge imperfection and proximity to the free boundary...\nLine 139: ... when the one-way cracking pattern is fully formed in simulations. In the experi- ments, due to setup imperfections (e.g., hinge fabrication and mixed-mode loading, see Materials and Methods) and proximity to the free boundary, hinges on the left started to break earlier than expected.\nLine 388: The robustness of the one-way fracturing also extends deeper into the nonlinear process after the breaking of the first hinge in general, supported by the topological mechanics theory as we discuss in Sec. 2.2 (main text). This is observed in our experiments to varying extent depending on the type of the notch (Movie S2, S7-10), where straight cracks display the clearest one-way propagation for subsequent breaking events (main text Fig. 1). However, due to limitations of the fabrication method we currently use, full quantitative investigation of this effect will be left for future work. Nevertheless, we support this by numerical simulations with thinner hinges and more brittle materials which are closer to the ideal theory limit (Fig. 2B, C, , which are now Fig. S9B, C in the SI). In this case, the one-way propagation of the crack is robust throughout the fracturing process (Fig. 2B, C, which are now Fig. S9B, C in the SI). This effect is diluted in experimental specimen due to hinge thickness and material constants, similar to how topological mechanical effects are diluted as the TMM deviate from the ideal limit in earlier studies.\n(2) While we agree that wording such as “regardless of the crack shape” and ”invariant to notch shapes” may be too strong, we would like to clarify that our study includes a range of notch geome- tries beyond the triangular shape. Specifically, Fig. 5E (main text) numerically shows the bulk-to-edge (or left-to-right) stress ratio of different crack shapes, characterized by angle α (Fig. 5A, main text). Experimental tests and fracturing simulations in Fig. 5G, F (main text) and Fig. S20-21 (trapezoid and rectangular) in the SI demonstrate consistent unidirectional fracturing behavior under notches with different shapes and orientations. We have revised the text to avoid overgeneralization and now ex- plicitly describe the range of notch types studied. We hope this provides a clearer and more accurate representation of the robustness demonstrated across these cases. The following modifications have been modified in the main text\nLine 55: regardless of crack shapes for a wide range of notch shapes\nLine 56: protected by topology, the TMM is less sensitive to defects from fabrication and invari- ant to variances of notch shapes.\nLine 63: as verified in straight, tilted, triangular, trapezoid and rectangular notch shapes exper- imentally\n(3) We thank the reviewer for raising this point and would like to clarify two aspects:\ni)AdirectcomparisonbetweentheTMMandthetriangularlatticeintermsofload-bearingcapacity and fracture toughness is probably not appropriate, as these structures differ fundamentally in both connectivity and material usage. The triangular lattice has six bonds per node in the spring–mass representation and occupies roughly half the area in Fig. 2A (main text), while the TMM has four bonds per node and occupies approximately one-quarter of the area. The higher connectivity of the triangular lattice provides greater load-path redundancy, leading to increased stiffness and fracture toughness (Calladine, 1978; Lubensky et al., 2015). In contrast, the TMM is marginally connected by design 13 and aims to enable controllable, defect-tolerant fracture propagation rather than maximize mechanical strength.\nii) A more relevant comparison is among the three Kagome-based lattices in our study. Both the TMM and the twisted kagome can guide cracks directionally, but only the TMM exhibits robust and consistent guidance across varying notch geometries and imperfections.\nWe have revised the manuscript to clarify this structural distinction and to avoid overgeneralized comparisons.\nLine 157: The triangular lattice features six bonds per node in the spring–mass representation and occupies roughly half the area in Fig. 2A (main text). In contrast, the TMM has only four bonds per node and occupies approximately one-quarter of the area. Line 179: As a result, with thesamematerialdensity, theTMMdemonstratesthemoststableandgradualfracturebehavior, without a significant reduction in fracture toughness.\n(4) The reviewer is right that no real application or product is shown in this work. Our work first considers material nonlinearity and failure scenarios of TMM, and we intended to show it as a step towards real applications. We revised the sentence to avoid any confusion.\nLine 64: This study considers real fracturing process in the TMM and serves as an important step to apply TMMs to real-world engineering problems.\nReviewer Comment1.5 — Fig. 2C, please avoid including a data table as a figure panel. Consider using a column chart or another graphical representation.\nReply: The figure is modified (Fig. 1 in this file). The new table can be found on Page 12 revised main test.\nReviewer 2\nReviewer 2 (Remarks to the Author):\nThe authors performed a massive amount of work that successfully addresses the comments of the reviewers. I therefore support publication of the manuscript in its present form. The data presented in there now convincingly establishes the potential of topological metamaterials to guide fracture in a robust fashion.\nReply: We thank this reviewer for supporting us to publish in Nature Communications, and we appre- ciate the positive comments!\n14"}]

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