Mechanics of reinforced concrete column confinement with architected auxetic steel lattices

Abstract

Negative Poisson’s ratio (auxetic) truss lattice metamaterials have recently emerged as highly effective reinforcements for brittle matrices, enabling strength and ductility levels that were previously unattainable. In this paper, we demonstrate how these architectures can be used to confine axially loaded structural elements, thereby achieving superior mechanical performance. We show that the enhancement arises primarily from exploiting the strain mismatch between the composite phases, which amplifies lateral confinement and induces higher hydrostatic stresses in the matrix. Experimental tests on high aspect ratio prismatic specimens confirm the reproducibility of this effect, extending prior findings from near-cubic samples to structural scale geometries. Through combined analytical and numerical studies, we quantify the differences between auxetic confinement and conventional schemes, and propose new predictive expressions for the load capacity of auxetically confined members. These results establish a direct link between reinforced concrete confinement theory and architected metamaterial design, opening new pathways for structural applications of auxetic lattices.

Introduction

The confinement of reinforced concrete columns has been a subject of extensive study for several decades, fundamentally shaping modern approaches to strength and ductility enhancement. By enabling access to higher stress and ductility domains that were previously unattainable with unconfined concrete, confinement reshaped the design and philosophy of reinforced concrete structures and became embedded in all major specifications, including ACI 318-19(22)¹. This breakthrough provided the foundation for the development of ductile structural systems able to sustain combined axial, shear, and bending actions, and continues to underpin seismic design provisions worldwide. From a mechanics perspective, confinement can be interpreted as the deliberate imposition of lateral stresses (σ₁, σ₂) on a brittle composite to increase its axial load-bearing capacity (σ₃) by shifting the stress state toward the hydrostatic axis (σ₁ = σ₂ = σ₃) of the three-dimensional ultimate strength surface.

The beneficial influence of lateral pressure on concrete was first documented through hydrostatic testing by Schickert-Winkler² and Kupfer-Hilsdorf-Rusch³, who mapped compressive and tensile meridians using fluid-induced stress states long before mechanical confinement was applied. These studies revealed that concrete strength is highly sensitive to confinement, thereby establishing the fundamental basis for mechanical confinement systems. The first systematic mechanical implementation was introduced by Richart et al.⁴ in 1929, who investigated spirally reinforced columns and identified an empirical relationship between spiral reinforcement and strength enhancement. Building upon this seminal finding, subsequent decades, particularly the 1970s and 1980s, produced a surge of research that extended the concept to flexural members, to behavior at low and high strain rates, and to high-strength concrete. Pioneering contributions by Balmer⁵, Kent and Park⁶, Sheikh and Uzumeri⁷, and Priestley, Park, and Scott^8,9, followed by Setunge and Attard^10,11 and more recently Paultre and Legeron¹², collectively established the breadth of confinement effects and design methods. Among these, the models of Mander et al.^13,14 provided one of the first fully consistent mathematical formulations for confined concrete, producing a widely adopted predictive framework that continues to guide design today.

Conventional reinforcement patterns act as passive confinement systems, in which lateral pressure develops only in reaction to the volumetric expansion of the concrete core. More recently, the idea of active confinement has been introduced, in which pre-stresses are externally induced in the confining cage. Shape memory alloys (SMAs), which contract under thermal or mechanical activation, have been proposed as active confining elements, enabling externally controllable confinement levels beyond those achievable with passive hoops^15,16,17,18. The ability to tune confinement dynamically represents a fundamental departure from conventional practice, with potential to create adaptive and resilient structural systems.

At the same time, advances in digital fabrication and additive manufacturing have enabled the realization of complex architected materials^19,20, including metallic truss lattices²¹ and interpenetrating phase composites (IPCs)^22,23,24. In the architected materials literature, the term truss is frequently used to describe lattice topologies composed of beam-node assemblies (e.g., octet, Kelvin, and body-centered cubic lattices), independent of the governing stress state²⁰. This usage functions primarily as a geometric descriptor, distinguishing lattices formed by straight struts and discrete nodes from architectures defined by surfaces or plates. Such lattices are commonly classified as stretching- or bending-dominated, reflecting the intended primary load-transfer mechanism²⁵. In limiting cases—such as stretching-dominated lattices or architectures with low nodal stiffness—the load path may approximate that of a pin-jointed truss, with predominantly axial force transfer. Of particular relevance is the family of negative Poisson’s ratio (auxetic) lattices, which can significantly enhance the strength and ductility of brittle matrices by harnessing the strain mismatch between phases (${\epsilon }_{m}={\epsilon }_{t}^{(T)}-{\epsilon }_{t}^{(M)}$)^26,27. Within this class, the three-dimensional re-entrant bowtie (BT) or auxetic honeycomb architecture has emerged as a leading and versatile candidate architecture that can illustrate a wide range of elastic properties within the auxetic and convex regimes. Specifically, in the bowtie architecture family considered here, the θ = 90^∘ configuration corresponds to a zero-Poisson’s ratio metamaterial, architectures with θ < 90^∘ display auxetic behavior, and those with θ > 90^∘ exhibit positive (convex) Poisson’s ratios. Previous studies on this architecture, including linear stiffness matrix decompositions, have demonstrated that the maximum enhancement of lateral (confining) pressure is achieved with moderately auxetic designs. This configuration provides an optimal balance between lateral contraction and the overall transverse and axial stiffness of the architecture. It also uniquely combines favorable geometric features such as strong transverse strut connectivity and low strut slenderness, which make it especially effective in the nonlinear regime²⁸. Computational investigations indicate that these lattices amplify lateral confinement pressures, thereby enhancing matrix hydrostatic stresses and shifting the failure response from bending-dominated to stretching-dominated modes^25,27. However, most prior experiments have focused on near-cubic specimens, leaving open fundamental questions regarding stress-strain response, failure mechanisms, and strengthening effects in high-aspect-ratio columns. Furthermore, understanding the mechanisms that drive the enhanced performance of auxetic composites requires disentangling the contributions of strain compatibility, stress redistribution, and strut-level failure propagation. Without this understanding, the development of predictive analytical models remains incomplete, limiting the translation of auxetic architectures into structural practice.

In this study, we address these challenges by designing and fabricating tall column-like auxetic composites using laser powder bed fusion, embedding them within cementitious matrices, and subjecting them to comprehensive monotonic and cyclic axial tests. We demonstrate that the significant strength and ductility enhancements previously observed in cubic specimens are reproducible in high-aspect-ratio members, and we extend the experimental evidence to regimes of cyclic loading. We then combine experimental data with finite element simulations to compare auxetic confinement directly against conventional reinforcement, quantifying fundamental differences in stress development and failure mechanisms. Finally, we propose a new analytical capacity model that adapts classical confinement theory to the auxetic regime, bridging the long-established field of reinforced concrete confinement with the rapidly evolving domain of architected metamaterials.

Results

Unit cell architecture and column design

To evaluate the confinement response of column-type auxetic composites, reinforced prisms were designed directly from the re-entrant bowtie unit cell architecture (Fig. 1). The selected configuration was an 80^∘ re-entrant lattice assembled as a 4 × 4 tessellation and fabricated using metal additive manufacturing. Building upon prior studies of near-cubic auxetic prototypes^27,28, the periodic unit cell relative density was fixed at ${\rho }_{uc}^{* }=5 \%$, with a minimum truss member diameter of 1 mm. Each unit cell measured 13.2 × 13.2 × 11.07 mm with uniform strut lengths of L = 6.7 mm. A total of N_tot = 18 unit cells were stacked vertically, resulting in an overall specimen aspect ratio of ~ 3.7, chosen to fall within the range of classical column confinement studies (typical values 3-4) and to match the maximum build height permitted by the fabrication process. This specimen aspect ratio also ensured no interference from elastic buckling. An Euler buckling check indicates that the critical buckling stress σ_cr is much higher than the experimentally observed capacities σ_peak, ensuring that the sections qualify as stocky and are expected to always fail inelastically due to fracture of their transverse reinforcement.

Fig. 1: Auxetic bowtie unit cell assembled into column geometry.

A 4 × 4 tessellation of the re-entrant unit cell stacked to N = 18 layers forms prismatic specimens with an aspect ratio of 3.7, comparable to classical confinement tests. Functional grading at the supports increases strut thickness and local stiffness, strengthening boundary regions and promoting failure localization within the column core for direct comparison with conventional confinement schemes.

The truss architecture was intentionally designed to mirror reinforcement strategies used in conventional reinforced concrete columns subjected to axial compression. Two such strategies are typically documented in the literature: (i) densification of transverse reinforcement near the supports, as reported by Scott et al. (1982) and Mander et al. (1988)^8,13, and (ii) widened supports, as demonstrated in the experiments of Sheikh et al. (1980) and Foster et al. (2001)^7,29. In the present design, the first strategy was adopted due to its fabrication efficiency and conceptual alignment with the lattice architecture. Since the bowtie unit cell inherently interconnects transverse and longitudinal elements, functional grading was introduced near both supports to ensure failure localization within the gauge section. This was achieved by increasing the relative density from ${\rho }_{uc}^{* }=5 \%$ in the column core to ${\rho }_{uc}^{* }=7 \%$ at the supports, corresponding to a strut diameter increase from t_truss = 1.0 mm to t_supports = 1.2 mm. The transition zone (shown with purple dotted lines in Fig. 1) is located in the third unit-cell layer at each end, with two layers on each side fabricated with increased local relative densities. This localized stiffening provided stronger boundary regions while preserving uniform confinement behavior in the column mid-height.

Monotonic axial compressive response

First, plain mortar cubes and prisms were tested to quantify the matrix strength and compare the unreinforced response to the reinforced specimens. The unconfined prismatic specimens (Fig. 2a) exhibited an average strength of f_c,prism = 40.9 MPa ± 6% and an average axial modulus of E_c = 9100 MPa ± 1.25 GPa. Similarly, the previously cubic specimens had a base strength of f_c,cube = 38.1 MPa ± 12% ²⁷.

Fig. 2: Monotonic compression behavior of auxetic columns.

a Stress-strain curves for confined and unconfined specimens. The notched stress-strain region at gauge strains in the range of 0.0075 to 0.016 (annotated with gray) corresponds to the cover failing near the unconfined mortar fracture strain. The points of the peak stress are indicated with stars. A computational verification of the experiment is also presented. b The failure mode of the experimental specimens was consistent in all experiments and is shown from different views (isometric front, isometric top, right side, left side). c Consistent shear-plane failures (56^∘--60^∘) across all tests reveal strong and uniform confinement, with negligible material loss outside the core.

The stress-strain responses obtained from the monotonic column tests under axial compression demonstrated a high degree of consistency across specimens, and aligned closely with previously reported results for the 80^∘ cuboid bowtie configuration²⁷, indicating a remarkably repeatable mechanical response in taller aspect ratio specimens. The onset of nonlinearity in the response of the columns was consistently identified between 60-90 MPa, coinciding with a distinct notch in the stress-strain curves. Experimental observations attributed this behavior to the spalling of the mortar cover and the subsequent exposure of the reinforcing truss architecture. This response is expected, as the concrete outside the effectively confined core exhibits mechanical behavior similar to that of unconfined plain mortar, a phenomenon that has been well documented in studies of conventional reinforced concrete columns¹³.

The results of the monotonic concentric axial compression experiments are presented in Fig. 2. The average peak strength of the confined column specimens was 142.5 MPa, compared to 139.3 MPa for the 80^∘ cuboid specimens²⁷, corresponding to a marginal difference of only 2.3%. The compressive strains observed in both the experiments and computational simulations conducted in this study can be classified based on the measurement location into two categories: (a) strain measured at the gauge section, and (b) total strain, which encompasses both the gauge section strain and the strain at the supports. The supports were designed with significantly higher relative densities (7% versus 5%), resulting in substantially higher effective composite stiffnesses. Consequently, strain localizes primarily at the gauge section, producing gauge strains in the range of 6 –7.5% and total strains of 4–5%. This behavior is consistent with the assumption of near-rigid supports, which would inherently reduce the total strain across the composite cross-section. From a structural materials perspective, the observed peak strains (e.g., ≈ 0.05) are relatively high when considering all structural components and may be impractical for typical applications. Geometrical modifications of the architectures investigated here are therefore the focus of future work, with the aim of increasing the stiffness of the response while maintaining overall performance. In Fig. 2a, the strains are calculated both in the gauge area and the total height of the specimen, including the support section, for the sake of completeness. By contrast, peak strain values showed greater sensitivity to specimen geometry. The average failure strain was 0.0704 for the cuboids (shown in²⁷) and 0.067 gauge strain (0.0404 total axial strain) for the columns, with the cuboids presenting an approximately 5% higher ductility at the peak. Notably, the confined column results indicate strength enhancement relative to the unconfined mortar baseline, consistent with the findings presented in previous studies with cuboids. The confined stiffnesses calculated from the total axial strain were 11.1 GPa ± 1.8 GPa. Overall, the columns exhibited a stress-strain behavior consistent with the cuboid specimens up to the region of their peak stress.

With respect to failure modes, all confined column specimens consistently exhibited well-defined shear failure planes. In the monotonic tests, the observed shear planes were inclined at 60^∘, 58^∘, and 56^∘ (measured clockwise from the horizontal, shown in Fig. 2c) for specimens ST1 through ST3, respectively, with a mean inclination of 58^∘. These consistent shear angles are indicative of a high degree of confinement. Aside from the primary failure plane, no additional damage was observed. The shear plane propagated through multiple layers of unit cells, intersecting both transverse and vertical structural elements. Except for cover spalling, negligible material loss was observed from the column perimeter, attributed to the 45^∘ arching stresses that developed along the boundaries. This phenomenon highlights the nearly complete utilization of the mortar matrix in the presence of the rough as-built stainless-steel truss, with minimal volumes of inefficiently confined material beyond the outer cover. In other words, the cohesive tensile capacity of the matrix was sufficient to resist the tensile stresses near the perimeter. However, this confinement efficiency is sensitive to tessellation size and lattice dimensions.

Cyclic loading and damage tolerance

Apart from the monotonic concentric axial compression, the cyclic loading regime presents a unique opportunity to further understand and evaluate the inelastic behavior of negative Poisson’s ratio interpenetrating phase composites. Two specimens similar to those tested under a monotonic regime were tested in cyclic loading-unloading to map their stiffness degradation and recovery throughout strain hardening up to the point of failure. The same displacement-controlled loading rate was used for the cyclic tests as in the monotonic tests. The cyclic loading protocol (Fig. 3a) was established such that the initial unloading branch commenced at a load of 200 kN (45,000 lbf), corresponding approximately to the yield point determined by companion monotonic tests. The loading history followed a sawtooth linear load-unload profile, with a constant minimum load of 22 kN (5,000 lbf) to ensure continuous contact between the specimen and the compressive platens. A dwell period of 1 second was imposed at both the maximum and minimum load levels in each cycle. The maximum load ceiling was progressively increased by 22kN (5000 lbf) per cycle until specimen failure was observed.

Fig. 3: Cyclic loading behavior of auxetic columns.

a The cyclic loading protocol utilizing a sawtooth loading pattern, (b1) and (b2) Confined core stress - total axial strain hysteresis loops for two specimens exhibiting stable cyclic hardening and peak strengths of 160.7 MPa and 154.9 MPa, exceeding monotonic capacities. Minimal stiffness degradation underscores the damage tolerance of the auxetic confinement mechanism.

The experimental findings of these tests are shown in Fig. 3b1 and b2. Following the analysis of the cyclic test data, the intersection points of the loading/unloading branches in the load-displacement response were identified and are indicated in the figure, along with the corresponding minimum and maximum forces achieved in each cycle. The peak strengths recorded under cyclic loading were consistently higher than those observed in monotonic tests, with values of 160.7 MPa and 154.9 MPa for the first and second specimens, respectively. This can be attributed to the strength variability of specimens cast in separate batches. The strength of the specimens tested in the cyclic loading regime was, on average, 9% higher than that of the specimens tested in monotonic axial compression. The initial stiffnesses were measured as 15,500 MPa and 16,700 MPa, respectively. In the cyclic loading experiments, two distinct mechanisms were identified that differentiate the behavior of the cementitious auxetic composite from that of a conventionally reinforced specimen: (a) The combination of densely interconnected struts under elevated hydrostatic stresses, $({\sigma }_{h}=\frac{1}{3}({\sigma }_{xx}+{\sigma }_{yy}+{\sigma }_{zz}))$, which suppresses crack initiation and slows crack propagation, and (b) The distributed lattice topology, which reduces section losses in the post-cracking regime, due to the increase in the effectively confined core area (k_e). Although cracked concrete is less effective at carrying mechanical loads, the minimized section loss promotes more efficient stress redistribution during repeated loading and unloading cycles.

An evaluation of stiffness degradation demonstrated a remarkable resistance to deterioration under repeated load-unload cycling (Fig. 4). Furthermore, the results exhibited notable repeatability across the two tested specimens. It is important to highlight that the initial stiffness, determined from the first loading segment, was 57% lower than the second-cycle stiffness for specimen 1 (24300 MPa) and 33% lower for specimen 2 (22300 MPa). These reduced first-cycle stiffnesses are expected and can be attributed to two primary mechanisms. First, significant crack propagation and stress redistribution occur near the column cover during the initial loading, which typically spalls at or shortly beyond the yield point. Second, in accordance with ASTM C496/C469M, accurate determination of elastic properties requires at least two cycles of loading and unloading to condition the specimen by eliminating loading-related compliance (e.g., pore collapse, edge breakage).

Fig. 4: Normalized stiffness degradation data over the maximum measured stiffness from the cyclic loading of two column specimens.

The results demonstrate exceptional damage tolerance and repeatability of auxetic confinement under repeated loading.

Consequently, the maximum stiffness measured at the end of the second load-unload cycle is considered the full stiffness, while subsequent repetitions beyond this point exhibit monotonically decreasing values. These reduced values are defined as the damage stiffnesses, indicating a remarkable damage tolerance of the negative Poisson’s ratio confined specimen even in large strains.

Comparison to conventional confinement of axially loaded members

To validate the experimental findings, facilitate comparison with conventional confinement schemes commonly employed in construction practice, assess the effects of varying parameters, and capture the internal stress states of the composite material, with the overarching objective of elucidating the fundamental mechanisms that govern and enhance the auxetic IPC response, finite element models were developed. Two distinct sets of material models were utilized to provide a comparison of responses using the experimentally used materials and common structural practice materials: the first set of materials modeled a mortar-stainless steel 15-5 additively fabricated composite, while the second included typical normal-strength concrete and A605 structural reinforcing steel. The Abaqus CAE/Explicit (Dassault Systèmes SE, France) finite element analysis suite was utilized with the use of 2.7 mil. C3D4 linear interpolation tetrahedral elements for the simulation of the truss and 280,000 C3D8R reduced integration hexahedral elements for the composite matrix simulation. The composite behavior assumed strain compatibility with a fully bonded approach. The assumption of full bond presumes properly tied hoops that do not slip, fabricated in accordance with the ACI 318-22 detailing provisions, which ensures effective composite action between concrete and reinforcement, while interfacial debonding is approximated through the concrete’s inelastic law. This assumption is consistent with experimental observations, which indicate that failure is not governed by the debonding of transverse reinforcement but by the large expansion forces mobilized within the concrete matrix. Under fully bonded composite action in three-dimensional compression, failure is governed by the yielding of the transverse reinforcement. In this idealized case, strain compatibility governs the overall response of the composite. The composite strength depends primarily on the yield strength of the lateral hoops, rather than on the strength of the connection between transverse and longitudinal bars. A minimum time increment of 0.004sec was implemented for the dynamic analysis using mass scaling.

The stainless steel 15-5 phase was modeled with an elastic modulus of E_s = 140 GPa and a Poisson’s ratio of ν = 0.3 ²⁸, neglecting elastic anisotropy across individual lattice features. A bilinear plasticity law with hardening was adopted, defined by a yield stress of σ_y = 850 MPa at zero plastic strain and an ultimate stress of σ_u = 1050 MPa at a plastic strain of 0.05. Fracture was incorporated through a ductile failure criterion at a plastic strain of 0.09, corresponding to a stress triaxiality of 0.33. For the mortar phase, the linear and inelastic responses were calibrated using experimental data obtained from unconfined prismatic specimens. A concrete damage plasticity (CDP) model was employed, characterized by a dilation angle of 40^∘, a ratio of fb₀/fc₀ = 1.16, and a viscosity parameter of K = 2/3. The parameters of the CDP model for the mortars in this study were calibrated using matrix experiments reported in this work and in previous studies²⁷. The inelastic parameters of the concrete damage plasticity model for concrete were calibrated against experiments and tables from the literature¹³.

Based on the developed simulations, in the finite element models developed for reinforced concrete composites, a bilinear plasticity law was adopted for the reinforcing steel, defined by a yield strength of σ_y = 420 MPa at zero plastic strain, an ultimate strength of σ_u = 620 MPa at a plastic strain of 0.088, and fracture modeled using a ductile failure criterion at a plastic strain of 0.11 under a stress triaxiality of 0.33. The concrete material was modeled as a normal-strength mix with an elastic modulus of E_c = 26 GPa, a dilation angle of 38^∘, a strength ratio of fb₀/fc₀ = 1.14, and a flow potential eccentricity of K = 2/3, in agreement with typical values seen in the literature^30,31. The choice of E_c = 26 GPa is consistent with values commonly reported in reinforced concrete confinement experiments, such as those by Mander et al. The accuracy of computations was validated by observing good agreement between the concrete damage plasticity parameters used in this work and the experimental stress-strain results from Mander et al.’s octagonally tied column tests, in computations of identical specimens¹³.

Axial loading stress-strain response

The peak strength of conventional confining schemes in the analytical expression provided by Mander et al.¹⁴ is governed primarily by the volumetric ratio of transverse reinforcement. To extend this understanding, computational predictions of the peak stress response for the 80^∘ bowtie specimens were evaluated in both mortar and concrete composites and compared with Mander’s analytical prediction. The analysis considered a range of strut radii corresponding to different ratios of transverse-to-longitudinal reinforcement (ρ_t/ρ_l) (achieved by varying the diameters of the two bar groups), while maintaining a constant total relative density of ρ_tot = ρ_t + ρ_l = 0.057 ± 0.01 calculated from the centerline dimensions of the transverse bars. This framework enabled the investigation of the effect of redistributing reinforcement between transverse and longitudinal directions to explore whether differences in internal steel configurations cause the increases observed in the load-bearing capacities of the composites. The range of reinforcement ratios corresponding to typical confining steel in the experiments from Scott et al.⁸ is indicated as a benchmark reference.

A unifying interpretation may be achieved by splitting the reinforcing steel in two primary directions: one parallel to the load and one perpendicular to the loading direction (Fig. 5a). The quasi-horizontal elements of the bowtie lattice are treated as transverse reinforcement and the vertical elements as longitudinal reinforcement, although the re-entrant bowtie unit cell functions structurally as a monocoque system.

Fig. 5: Comparison of the achieved peak stresses in conventional confinement schemes (rectangular, octagonal rhombic hoop configurations) to the attained peak strength for an auxetic bowtie 80-degree architecture with the same volumetric steel ratios.

a An analogous representation between the longitudinal and lateral reinforcements is presented for both conventional and auxetic reinforcing schemes. b The negative-Poisson’s -ratio lattice expands the achievable strength domain beyond classical confinement models. The pronounced strength increases observed at high transverse-to-longitudinal reinforcement ratios (ρ_t/ρ_l) arise from the auxetic lattice, leading to an increase in lateral confining pressures rather than from the magnitude of ρ_t/ρ_l itself or internal steel volumetric fraction distribution. In contrast, conventional confinement exhibits only marginal capacity gains for ρ_t/ρ_l > 1.5, underscoring the efficiency of the auxetic design.

The results of this investigation, presented in Fig. 5b, demonstrate a pronounced enhancement in peak stress capacity when transitioning from traditional confinement schemes to the negative-Poisson’s ratio re-entrant bowtie configuration. This trend is consistently observed in both 3D-printed mortar prototypes and conventional reinforced concrete composites, indicating that the auxetic confining scheme unveils a new area of peak strengths that aren’t captured with the current analytical approach. For mortar composites, analytical capacity predictions are slightly unconservative relative to finite element (FE) simulations, whereas in concrete composites, the analytical predictions are conservative as expected.

For conventional confinement, the redistribution of reinforcement from longitudinal to transverse directions is effective only near ρ_t/ρ_l ≈ 1, a ratio consistent with typical values reported in conventional confinement experiments. For auxetics, the bowtie confinement scheme achieves remarkable performance at ρ_t/ρ_l ≈ 3.35, corresponding to struts of uniform radius. For ρ_t/ρ_l < 1.5, transverse reinforcement yields prior to longitudinal reinforcement, whereas for ρ_t/ρ_l > 1.5, the longitudinal reinforcement yields first. The transition point occurs near ρ_t/ρ_l ≈ 1.5. The high gains in capacity for the high ρ_t/ρ_l are therefore attributed to the auxetic design and not to the high values of the ρ_t/ρ_l. In conventional confinement, the gains in capacity for ρ_t/ρ_l > 1.5 are minimal.

These enhancements in peak stress capacities observed in small-scale prototypes translate well to concrete composites, with increases of approximately 85% and 61% for the respective cases, despite the fact that linear elastic analyses predict diminished lateral pressure gains at higher elastic contrast ratios (E_m/E_t), as suggested in previous work by Tzortzinis et al.²⁶ on periodic representative volume elements. This discrepancy indicates that the proposed linear elastic decomposition based on the confinement parameter P_c, while useful for unit cell selection, cannot be directly applied to predict lateral pressures or to project principal stresses onto ultimate strength surfaces in finite tessellations due to the complex transverse compliance present in such composites, and the overall response being highly dependent on the effective edge boundary conditions.

This limitation can partly originate from the derivation of P_c through stiffness matrices of homogenized representative volume elements (RVEs) under periodic boundary conditions. In finite-sized truss lattices, confinement stresses develop with a gradient dependent on tessellation size. This observation underscores the necessity of a size- and geometry-dependent amplification function, inspired by this first linear decomposition, that scales similarly to the shape of the previously proposed P_c function decomposition.

In Fig. 6, the inelastic stress-strain response of a conventional confinement scheme utilizing octagonal hoops is compared with that of an auxetic composite incorporating an 80^∘ re-entrant bowtie geometry. The analysis focuses on characteristic points within the plastic regime of the auxetic lattice, which correspond to the onset of plasticity and subsequent fracture in distinct groups of struts.

Fig. 6: Axial stress-strain evolution of auxetic versus conventional confinement schemes.

Four characteristic points (A–D) mark key stages: A—yield of vertical struts, B—yield of quasi-horizontal struts, C—peak stress in vertical members, D—fracture onset in horizontal members. The auxetic system shows progressive plasticity and stable post-yield equilibria, in contrast with the brittle post-yield decay of conventional hoops.

For the auxetic, point A denotes the initiation of plasticity in the vertical strut group. This transition occurs rapidly for most vertical members, with the exception of the corner struts, which experience significant bending. At Point B, plastic deformation emerges in the quasi-horizontal struts located near the mid-height of the cross-section. This plasticity propagates progressively until the composite reaches the yield plateau, approximately at an axial strain of 0.01 mm/mm. Beyond yielding, the response exhibits a gradual increase in load-carrying capacity due to strain hardening governed by the bilinear steel constitutive law. Point C corresponds to the attainment of peak stress in the vertical struts, while Point D marks the initiation and progressive fracture of the quasi-horizontal members, which ultimately drives the post-peak load reduction.

In contrast, the conventional confinement scheme demonstrates markedly less ductile behavior. Its peak capacity aligns with the yield point of the transverse hoops, followed by a modest decline in load with increasing strain until fracture of the first hoop, which signifies complete loss of confinement capacity.

From an energy balance perspective, the auxetic system exhibits multiple stable post-yield equilibria associated with the progressive response of transverse reinforcement. Conversely, the conventional scheme lacks stable equilibria beyond the yield point.

By examining the complete stress-strain responses in Fig. 7 for the cases presented in Fig. 5, a consistent trend emerges: the re-entrant bowtie composite sustains higher loads at larger strains across varying transverse-to-horizontal volumetric relative density ratios. Notably, a substantial elevation in the baseline post-yield stress is observed—near a strain of 0.01 in concrete composites and approximately 0.03 in mortar composites, indicating that the improved strength response initiates as early as the linear region.

Fig. 7: Influence of transverse-to-longitudinal reinforcement ratio (ρ_t/ρ_l) on composite response.

Under a constant total volumetric ratio ρ_tot = 0.057 ± 0.01, the auxetic bowtie architecture sustains higher loads and larger strains than conventional confinement across all ratios. High-strength steel lattices shift secondary stress peaks to larger strains, reflecting enhanced lateral confinement and post-yield hardening.

The transition from high-strength additively manufactured stainless steel with mortar to conventional reinforcement steel with normal-strength concrete produces marked differences in stress-strain behavior. In particular, the incorporation of high-strength steel shifts the location of the secondary stress peak to higher values, with this effect being most pronounced in the mortar composites. As shown, the stronger steel grade enables the development of greater confining (lateral) pressures within the matrix, which in turn increases the slope of the strain-hardening branch in the transverse reinforcement post-yield region, up to the attainment of peak strength.

Progressive failure response

To understand the non-linear response of the reinforcing truss inside the IPC, validate the previous observations in the global stress-strain response, and to describe the three-dimensional triaxial stress-states of each strut, a detailed study of the stress components in the auxetic truss was performed. The approach facilitates the identification of stress concentration regions and enables mapping of the lattice’s progressive yielding (Fig. 8a) and collapse sequence (Fig. 8b) under axial loading using the von Mises criterion (σ_vm) for ductile metals.

Fig. 8: Fracture sequence of various struts in a one-eighth symmetric area at the midheight of the truss lattice.

The strain at which the average Von-Mises stresses reach (a) the yield or (b) the fracture stress is captured to formulate the failure sequences at the yield and peak stress points. Clear clusters are formed for the vertical and horizontal struts at yield, while a mixed-mode failure is observed near the point of fracture. c The Lode angle is used as a proxy for the deviatoric stress in each strut. d The numbering indicating the stress-probing locations is shown at a section cut of the lattice.

Given the large number of struts in the lattice, explicit probing was limited to the one-eighth symmetric region where fracture was observed to initiate, shown in Fig. 8d, at the mid-height of the section where stresses concentrate. Although non-exhaustive, this targeted evaluation provides representative behavioral patterns for each strut group. The lode angle, serving as a predictor of stress state, was additionally examined, where 60^∘ corresponds to pure compression, 0^∘ to pure tension, and 30^∘ to pure shear. Yielding and fracture displacements were estimated by averaging the von Mises stress evolution across elements in a slice of each strut and identifying characteristic yield and peak coordinates. Owing to the symmetry of both the geometry and loading, similar stress evolution is expected across the lattice up to peak stress, with divergence primarily occurring in the post-peak region.

Observations in the yield and fracture sequences study show that the interior vertical struts yield first, forming a distinct cluster at the onset of yield, followed by the external vertical struts, with the corner struts yielding last. The horizontal elements exhibit near-synchronous yielding, initiating at locations 8, 3, and 6 on the innermost side of the first unit cell, and subsequently propagating to all horizontal members with minimal strain increments (Fig. 8d). The fracture sequence differs, involving a mixed-mode failure of vertical and horizontal elements. Examination of fracture relative to the lode angle (Fig. 8c) indicates that horizontal elements fail in a shear-dominated mode, transitioning toward tensile fracture deeper into the post-peak regime.

Two mechanical behaviors inferred from this analysis are consistent with earlier trends observed in Figs. 7 and 8, and can inform the development of analytical capacity models for cementitious auxetic composites. First, the sharp reduction in stiffness at yield (σ_y), occurring just prior to 0.01 strain, aligns with the simultaneous yielding of the quasi-horizontal struts. Notably, the monocoque lattice architecture maintains favorable stress states in the matrix beyond this point, preventing the appearance of a negative-stiffness branch typically associated with diagonal strut softening. Instead, the bilinear hardening law of the steel drives strain-hardening behavior, culminating in peak strength at large strains, concurrent with fracture of the struts at stress σ_u.

Second, the shear-dominated failure mechanism is consistent with the column test observations shown in Fig. 2. The interconnected architecture promotes mixed-mode failure, with both vertical and horizontal struts participating in the fracture plane. While the vertical struts surpass their compressive strength earlier, they remain connected due to the load vector’s direction and eventually fail in shear together with the diagonal struts. This indicates a greater degree of reinforcing steel participation than is characteristic of conventional confinement schemes, where vertical elements typically buckle shortly after the initial hoop fracture in the transverse reinforcement.

Examining the stresses developing within the brittle matrix is equally instructive, in addition to the truss-level stress investigation. Additional probes into the stress components of the concrete matrix, specifically the absolute maximum principal stresses, equivalent shear stresses, and hydrostatic pressure, were performed to validate the behaviors postulated previously in the literature^26,27 (Fig. 9). Four matrix points were selected in both the conventionally confined composite (ρ_t/ρ_l = 1.1) and the bowtie architecture with uniform radii at the same reinforcement ratio (ρ_t/ρ_l = 1.1). These points were chosen to ensure approximate equivalency in position and proximity to reinforcement elements across both confinement schemes, while avoiding regions of ineffective confinement.

Fig. 9: Stresses in the concrete matrix.

a Absolute maximum principal, equivalent shear, and pressure component stress probing in four separate key matrix locations. b, c Point A - Located in the innermost unit cell, far from any steel element and near the center of the matrix. Point B—Placed in the effectively confined region of the exterior unit cell, adjacent to a quasi-horizontal strut (auxetic) or a transverse hoop (conventional). Point C—Selected in another effectively confined area of the exterior unit cell, close to a vertical strut (auxetic) or a longitudinal bar (conventional). Point D—Chosen at a crack-initiation location on the free surface of the confined concrete core. Points B and C were located at the first hexahedral matrix element after the edge of the reinforcement, and point D was located at the first hexahedral matrix element inside the effectively confined core.

As shown in Fig. 9, the stress development at Point A is comparable in magnitude for both confinement schemes up to approximately 0.04 axial strain. Beyond this threshold, the conventional scheme exhibits a sharp stress drop. In contrast, the bowtie-confined composite maintains and further increases the stress fields due to the favorable confinement-induced stress state in the matrix.

This distinction becomes more pronounced at Points B and C, where the auxetic confinement scheme significantly increases both shear and hydrostatic stresses. In contrast, the conventionally confined composite shows a rapid decline in attainable stresses, accompanied by matrix plastic flow. Consequently, the hydrostatic stress at all points other than A is markedly lower under conventional confinement. This confirms that the auxetic lattice’s inward deformation tendency generates additional lateral pressure, which promotes higher volumetric stresses and mitigates the deviatoric components.

The resulting distribution of equivalent Tresca stresses within the auxetically confined matrix highlights a distinctive confinement mechanism, suggesting promising directions for further investigations of auxetic composites under non-axial loading conditions and alternative stress states.

Simple shear response

Motivated by the results presented in Fig. 9, a computational shear evaluation was conducted for both confining schemes, as shown in Fig. 10. The same lattice architectures previously examined under axial compression were fixed to shear plates subjected to constant shear displacement. This loading condition induced both normal and shear reactions due to the dilatant behavior of the concrete.

Fig. 10: Computational evaluation of a simple shear stress state for a uniform diameter near cubic bowtie 80^∘ specimen and a typical near cubic conventional specimen.

ρ_t/ρ_l = 1.1 compared to the uniform diameter auxetic with ρ_t/ρ_l = 3.35. Both designs have a similar relative density of ρ_tot = 0.057 ± 0.01, E_c = 26 GPa, and ψ = 38^∘. The relationship between the shear and normal reactions generated represents the dilatancy of the concrete matrix $\psi =\arctan \left(d{\varepsilon }_{v}^{p}/d{\varepsilon }_{s}^{p}\right)$ that connects the plastic volumetric strain with the plastic shear strain of the concrete matrix.

The results demonstrate the superior performance of the auxetic confining scheme under shear loading. Both the shear and normal dilatational components are significantly larger for the auxetic composite compared with the conventional scheme, despite an identical overall relative density. This favorable shear response is particularly relevant given its direct coupling to applied moments, which are central to nearly all structural applications of materials.

Although a detailed investigation of the coupled moment-shear behavior of auxetic composites lies beyond the scope of this work, the findings indicate a promising direction for future research.

Analytical formulation for capacity prediction

The experimental and computational findings discussed above provide the foundation for extending classical confinement theory. In particular, the analytical confinement model proposed by Mander et al.^14,32, in combination with the G. Shickert - H. Winkler (SW)² and H. Kupfer, H.K. Hilsdorf, H. Rusch (KHR)³ triaxial ultimate strength surfaces, can be adapted to formulate new predictive equations tailored for auxetically reinforced composites. These formulations enable estimation of the axial load capacity of this novel class of materials. The analytical framework is calibrated against numerical simulations and validated against experimental results, thereby improving robustness with respect to the variability in concrete strength that is inherent in experimental testing. A closed-loop validation is carried out qualitatively regarding the numerically expected estimated lateral stresses and finally against the experimental database to quantify predictive accuracy.

For simplicity, this work considers a reinforcing truss lattice with a tessellation size of n × n for square cross-sections. Nevertheless, the same methodology can be extended to more general cases, such as n × m tessellations.

First, the triaxial stresses are expressed as functions of the tensile and compressive meridians, as has been previously done in the literature for the prediction of the capacities of conventionally confined columns. The stress components σ₁, σ₂ represent the lateral stresses applied to the cross-section through mechanical confinement, and component σ₃ represents the vertically applied load, which corresponds to the axial capacity of the cross-section at the meridian surface.

$${\sigma }_{{\rm{oct}}}=\frac{1}{3}({\sigma }_{1}+{\sigma }_{2}+{\sigma }_{3})$$

(1)

Similarly, the shear octahedral stress (Von-Mises stress) is introduced as:

$${\tau }_{{\rm{oct}}}=\frac{1}{3}{\left[{({\sigma }_{1}-{\sigma }_{2})}^{2}+{({\sigma }_{2}-{\sigma }_{3})}^{2}+{({\sigma }_{3}-{\sigma }_{1})}^{2}\right]}^{1/2}$$

(2)

The normalized normal octahedral and shear octahedral stresses are also introduced:

$${\bar{\sigma }}_{{\rm{oct}}}=\frac{{\sigma }_{{\rm{oct}}}}{{f}_{c}^{{\prime} }},\,{\bar{\tau }}_{{\rm{oct}}}=\frac{{\tau }_{{\rm{oct}}}}{{f}_{c}^{{\prime} }}$$

(3)

The Lode angle is used in the parabolic interpolation function:

$$\cos \vartheta =\left[\frac{{\sigma }_{1}-{\sigma }_{{\rm{oct}}}}{\sqrt{2}\,{\tau }_{{\rm{oct}}}}\right]$$

(4)

where, ϑ = 0 corresponds to tension and ϑ = 60^∘ corresponds to compression and:

$${0}^{\circ } < \vartheta < 6{0}^{\circ }$$

(5)

The SW² and KHR³ dataset constants are defined as T and C for the tensile and compressive meridian, respectively:

SW: ${f}_{co}^{{\prime} }=30.6\,{\rm{MPa}}\,,\,{a}_{c}=1.21,\,{a}_{t}=0.1$

$$T=0.069232-0.661091{\bar{\sigma }}_{oct}-0.04935{\bar{\sigma }}_{oct}^{2}$$

(6)

$$C=0.122965-1.150502{\bar{\sigma }}_{oct}-0.315545{\bar{\sigma }}_{oct}^{2}$$

(7)

KHR: ${f}_{co}^{{\prime} }=32\,{\rm{MPa}}\,,\,{a}_{c}=1.15,\,{a}_{t}=0.091$

$$T=0.063046-0.662701{\bar{\sigma }}_{oct}-0.049435{\bar{\sigma }}_{oct}^{2}$$

(8)

$$C=0.113560-1.173709{\bar{\sigma }}_{oct}-0.300524{\bar{\sigma }}_{oct}^{2}$$

(9)

The William-Warnke³³ parabolic interpolation function is used to interpolate the ultimate stress surface in locations other than the tensile and compressive meridians, for mixed-mode stress states:

$${\bar{\tau }}_{oct}=\frac{C\cdot {\cos }^{\vartheta }\left(\frac{0.5D}{{\cos }^{\vartheta }+(2T-C)}{\left[D+5{T}^{2}-4TC\right]}^{1/2}\right)}{D+{(2T-C)}^{2}}$$

(10)

$$\,{\rm{where}}\,D=4({C}^{2}-{T}^{2}){\cos }^{2}\vartheta$$

(11)

A key step in predicting the triaxial strength of auxetic cementitious composites obtained by projecting the induced lateral stresses onto an ultimate strength surface is the determination of the lateral pressure component. To achieve this, the confinement effectiveness coefficient k_e is defined geometrically as the ratio of the effectively confined core area to the total truss centerline area, excluding the ineffectively confined regions formed by 45^∘ arching stresses:

$${k}_{e}=\left(1-\frac{2{(a-d)}^{2}}{3n{a}^{2}}\right){\left(1-\frac{{h}_{uc}}{4na}\right)}^{2}\,(n\times n\,{\rm{tessellation}})$$

(12)

$${k}_{e}=\left(1-\frac{(n+m){(a-d)}^{2}}{3nm{a}^{2}}\right)\left(1-\frac{{h}_{uc}}{4na}\right)\left(1-\frac{{h}_{uc}}{4ma}\right)\,(n\times m\,{\rm{tessellation}})$$

(13)

Following the rational explained earlier in Fig. 5, the transverse ρ_t and longitudinal reinforcement ratios ρ_l can be approximately calculated as seen below in Eqs. (14) and (16):

$${\rho }_{t}=\frac{\pi {d}^{2}(2n+1)\left(\frac{3an}{sin(\theta )}-2(2n+1)d\right)}{3{a}^{2}{n}^{2}{h}_{uc}}\,(n\times n\,{\rm{tessellation}})$$

(14)

$${\rho }_{t}=\frac{\pi {d}^{2}}{nm{a}^{2}{h}_{uc}}\left[\frac{a}{2sin(\theta )}(4nm+n+m)-\frac{2d}{3}(2n+1)(2m+1)\right]\,(n\times m\,{\rm{tessellation}})$$

(15)

$${\rho }_{l}=\frac{\pi {d}^{2}\,(3h+4d)\,\left({(n+1)}^{2}+{n}^{2}\right)}{12\,{h}_{uc}\,{n}^{2}\,{a}^{2}}\,(n\times n\,{\rm{tessellation}})$$

(16)

$${\rho }_{l}=\frac{\pi {d}^{2}\,(3h+4d)\,\left((n+1)(m+1)+nm\right)}{12\,{h}_{uc}\,n\,m\,{a}^{2}}\,(n\times m\,{\rm{tessellation}})$$

(17)

The analytical ratios ρ_t and ρ_tot are calculated from the finite-sized truss lattice centerlines. In the evaluation of transverse reinforcement, nodal regions are excluded, as their high connectivity provides additional strength; yielding is shown to localize primarily in the middle span of the quasi-horizontal struts. Computational results support this approach, showing that nodal regions yield only at later stages of strain propagation. For the longitudinal reinforcement ratio, overlapping nodal areas are treated as hemispherical and corrected accordingly. The total ratio, ρ_tot, is obtained as the sum of ρ_t and ρ_l, and for low volumetric relative density lattices agrees closely with CAD-based measurements.

The numerical analyses discussed earlier demonstrate that auxetic confinement enhances the lateral pressure within the effectively confined core. This increase is not linear when multiple layers of unit cells are stacked in a 4 × 4 tessellation, as shown in Fig. 9. Instead, a pressure gradient develops within the matrix due to the presence of multiple horizontal diagonal confining elements. Although this stress distribution is not explicitly defined, it can be approximated using an equivalent lateral hoop stress, consistent with conventional confinement models. The amplification of lateral pressure is captured by a numerically derived and experimentally validated effective amplification function w, which quantifies the negative Poisson’s ratio interaction between the two phases of the composite Eq. (18). The function w was fitted to the computational results of composites shown in Fig. 11 with an R² = 0.9962. While the amplification function is obtained through numerical fitting and validated against a substantial set of experimental data, it is also supported by prior analytical work. The linear decomposition presented by Tzortzinis et al.²⁶ indicates that the increase in matrix pressures follows a trend similar to that of the fitted function introduced here. Building on this foundation, this function represents a tessellation-specific equation that is experimentally defined, since the lateral stiffness component of each tessellation configuration can strongly influence the resulting lateral pressure enhancement. Consequently, the magnitude of the increase in lateral pressure differs between the two models: the present work considers a finite tessellation, whereas Tzortzinis et al. analyzed composite unit cells with infinite periodicity (periodic boundary conditions).

$$w=100\,{e}^{-0.0486\theta }-1017\cdot 1{0}^{3}\,{e}^{-0.186\theta }\,(4\times 4\,{\rm{tessellation}}),\theta \in [70,130]$$

(18)

This approach represents a first step toward more sophisticated formulations, quantifying concrete confinement and describing how different characteristic angles of the bowtie architecture influence the increase in lateral confinement, as outlined in previous work²⁷. The amplification function is introduced to capture the increase in lateral stresses arising from the negative Poisson’s ratio interaction, while implicitly accounting for transverse stiffness effects. Its purpose is to simplify the representation of the nonlinear increase in lateral pressure observed at different characteristic angles. Since the amplification effect depends on both tessellation hierarchy, higher- or lower-order tessellations require tailored amplification functions.

Fig. 11: Simulated and analytically predicted capacities of re-entrant bowtie interpenetrating phase composites composed from cementitious and concrete matrices.

a, c Are computationally derived stress-strain curves depicting the confined core strength against the applied axial strain for varying characteristic re-entrant angles. b, d Are simulation and analytical results illustrating the agreement between the calculated and predicted yield stresses and peak stresses in the composite. e Depicts the agreement of the proposed model to the experiments of columns shown in Figs. 2 and 3 and previously published experiments of cuboid specimens²⁷.

Compared to an auxetic truss lattice, the reinforcing cage of a conventionally confined column is expected to exhibit slightly positive Poisson’s ratios (ν). According to the linear stiffness matrix decomposition based on homogenized RVEs, previously presented by Tzortzinis et al.²⁶, the 90^∘ bowtie is expected to illustrate a near-zero Poisson’s ratio, which indicates an auxetic behavior at large strains with a transition to ν < 0 during strain progression. Similarly, the 100^∘ specimen exhibits Poisson’s ratios similar to the mortar phase (ν = 0.18), thus providing zero lateral pressure increase. Experiments and simulations conducted on bare trusses²⁸ have shown an around 5^∘ reduction of the characteristic geometric angle at the peak stress. Thus, from the observed behaviors mentioned and considering both the linear and non-linear response, the amplification function is expected to provide a near w ~ 1.0 amplification at around 95^∘, which marks the transition point where an auxetic lattice boosts the strength of the composite, or conversely, a convex lattice reduces the strength due to the engagement of an unfavorable lateral stress state.

For normal-strength concrete, accurate capacity prediction requires the combined use of Eqs. 5.1 - 5.18. Care must be taken when applying these equations under high lateral stresses, as singularities may arise from square-root terms. In such cases, the nearest numerically stable solution can be selected, which can often be identified through graphical inspection.

Based on the experimental and numerical findings previously described, two predictive equations are developed: one for the yield plateau capacity (Eq. (19)) and one for the peak capacity (Eq. (20)). These provide the y-coordinates of a bilinear stress-strain curve. The corresponding x-coordinates can be easily obtained from the full-scale stress-strain responses at different angles, as shown in Figs. 7 and 11, since the analytical calculation of such using energy methods can be particularly cumbersome and impractical.

$$f{l}_{yield}^{{\prime} }=-{f}_{y}\,{\rho }_{t}\,{k}_{e}\,w$$

(19)

$$f{l}_{ult}^{{\prime} }=-{f}_{u}\,{\rho }_{tot}\,{k}_{e}\,w$$

(20)

Eq. (19) defines the yield plateau stress immediately following the yielding of all quasi-horizontal elements at f_y. This expression provides reliable predictions for normal-strength concrete (Fig. 11d), but tends to overestimate the capacity of convex 4 × 4 truss lattices in mortar composites (Fig. 11b). A similar overprediction is observed in simulations of conventionally confined mortars. This discrepancy arises because the William-Warnke interpolation function and the associated triaxial ultimate strength surfaces were originally calibrated against experimental data on normal-strength concretes.

In contrast, Eq. (20) predicts the peak strength at the end of the strain-hardening regime, where the reinforcement system reaches its ultimate stress f_u. The formulation assumes that all steel struts contribute to resisting the lateral dilation of the confined matrix: vertical struts provide shear resistance, while quasi-horizontal struts exhibit shear-tensile resistance. Under the assumption of bonded behavior in the effectively confined core, this mechanism elevates the predicted capacity in the plastic regime up to the initiation of fracture in the quasi-horizontal elements.

Both Eqs. (19) and (20) incorporate the auxetic amplification function w, which is taken to be linearly dependent on ρ_t and ρ_tot. For simplicity, these ratios are evaluated assuming a 90^∘ bowtie configuration, with ρ_t analytically obtainable from Eq. (14). This assumption holds near ${\rho }_{uc}^{* }\approx 5 \%$, where the amplification function was derived, but deviations are expected at higher relative densities where the Hashin-Shtrikman bounds exhibit strong nonlinearity. As ${\rho }_{uc}^{* }$ and the volumetric phase fractions vary, the two-phase interaction characteristics also change significantly, particularly through altered nodal stiffness and modified transverse and axial moduli.

Finally, the model’s predictive accuracy is benchmarked against experimental data (a total number of 13 experiments), with results summarized in Fig. 11e. The cylinder concrete strength f_co is estimated to be ~ 80% of the cube strength f_c,cube.

Discussion

In this manuscript, we design and fabricate auxetic column specimens and showcase experimental findings from concentric monotonic and cyclic tests under axial compression. We show that the experimental results obtained from the smaller cuboid specimens were successfully reproduced in larger column-scale prototypes, where comparable capacity gains were observed. The loading-unloading response of the auxetic columns demonstrated excellent recovery within the plastic region, showing neither significant nor rapid stiffness degradation up to peak load. Even in the post-yield regime, the columns retained their stability and maintained at least 60% of their maximum stiffness at all times, delineating a highly reproducible and predictable structural response.

The experimental campaign motivated a deeper investigation into the mechanics underlying the strength and ductility enhancement provided by the bowtie architecture. A direct comparison with classical confinement strategies for reinforced concrete columns carried out using analytical calculations and numerical simulations calibrated against both in-house experiments and the benchmark data of Mander et al., highlighted the significant advantages of the auxetic scheme. The favorable stress states produced by the monocoque composite architecture were responsible for substantial gains in both yield and ultimate strength. These improvements were attributed to enhanced hydrostatic pressures within the core due to the mismatch strain effect under strain compatibility, increased steel participation due to interconnectivity, and larger effectively confined areas due to the sparsely distributed steel reinforcement. Across all bowtie composites studied, strength improvements reached up to ~ 85% in mortar specimens and ~ 61% in normal-strength concrete. Importantly, these increases were linked directly to auxetic pressure effects rather than steel rearrangement at constant relative density. The distributed reinforcement further contributed to nearly doubling ductility at fracture by alleviating localized strain concentrations in the reinforcing phase.

The stress analysis of bowtie lattices also revealed markedly improved shear performance. The combined enhancement in axial and shear capacity makes these architectures strong candidates for use in structural columns, where simultaneous axial, moment, and shear demands are common—particularly under dynamic lateral loading. Consequently, extending the investigation to axial-moment and moment-shear interaction is a logical next step toward structural adoption of auxetic confinement.

Failure mechanisms of the bowtie configuration were further examined numerically, revealing coordinated strut-group behavior during strain evolution. A clear distinction emerged between the pseudo-transverse and pseudo-longitudinal reinforcement, simplifying principal stress analysis. The observed progression began with vertical strut yielding, followed by yielding of the quasi-horizontal elements. Similarly, vertical members reached ultimate stress first, with diagonals failing subsequently. Post-yield, the interaction between struts sustained a stable strain hardening until the peak, after which vertical and diagonal elements fractured rapidly under shear-dominated conditions, with vertical struts continuing to resist core expansion.

The yield and ultimate strengths were predicted using a modified analytical approach similar to the one used for traditionally confined columns¹⁴. Triaxial ultimate stress surfaces, specifically those of Schickert-Winkler and Kupfer-Hildsdorf-Rusch, were used to predict the axial strength from the mechanically provided lateral stresses. The reinforcing phase was analytically modeled to provide lateral resisting stresses at yield and failure, while the auxetic effect arising from the mismatch strain of the two phases was captured through an amplification factor derived from numerical analyses and experimentally validated as a function of the characteristic angle. Building on these insights, novel geometry-based predictive equations were proposed for the bowtie lattice, inspired by conventional confinement formulations.

Overall, the predictive equations, calibrated on numerical concrete data, validated the mortar experiments with high accuracy, yielding a mean error of 2.5% and a maximum overprediction of − − 10%. This demonstrates strong predictive capability for auxetic confinement in cementitious composites, with the largest deviations expected in the yield strength predictions of highly convex mortar models under the currently proposed unified model.

Methods

Additive manufacturing and casting

The reinforcing lattice was fabricated using laser powder bed fusion (LPBF) with a prealloyed 15-5 stainless steel powder. The selective laser melting exposure parameters, material properties, and support strategy (Fig. S1a) followed previously established protocols that have been characterized in detail²⁸. After fabrication, support structures were removed by electrical discharge machining (EDM) (Fig. S1b). All printed lattices were free of macroscopic imperfections.

Custom molds were produced by fused deposition modeling (FDM) using polylactic acid (PLA) filament extruded through a 0.4 mm nozzle at 200^∘C, with a uniform wall thickness of 2.25 mm (Fig. S1c). A 4 mm tolerance was provided between the lattice and mold walls to achieve a uniform mortar cover, with 3D-printed spacers employed to replicate conventional rebar placement practice. This design allowed consistent cover thickness and helped mitigate post-curing shrinkage.

The matrix phase consisted of a sand-based cementitious mortar cast into the molds and compacted by vibration, following procedures detailed in previous work²⁷ (Fig. S1d). Material characterization was performed according to ASTM C109/C109M. Alongside the reinforced specimens, plain mortar prisms were prepared to serve as unconfined controls. In total, six 2-in. ASTM-compliant cubes, three unreinforced prisms, and five reinforced composite columns were fabricated. The average density of plain mortar prisms was 2.28 g/cm³ ( ± 0.01), while the reinforced composites exhibited a slightly higher density of 2.38 g/cm³ ( ± 0.06), attributed to reduced porosity from internal vibration of the lattice. Specimens were demolded after 24 h of air curing, then submerged in water until testing. All tests were conducted at 253 days of age, under quasi-static displacement control at a rate of 0.024 mm/s (0.00095 in./s), following the application of a 2.2 kN (500 lbf) preload.

Mechanical testing was carried out using a Forney FHS-400-VFD high-stiffness compression machine, equipped with additional instrumentation (Fig. S1e). Displacements of the platens were monitored and controlled using an AEP TLDT50MM linear displacement transducer ( < 1μm accuracy, 150 Hz response). Two Novotechnik 100 mm LVDTs measured axial displacements. Loads were recorded using an Omega 2.5D through-hole 200,000 lbf load cell connected to a Vishay P3 strain indicator, with data acquisition through a DataQ DI-710 system.