Homogenization of Inconel 625 based periodic auxetic lattice structures with varying strut thickness

Architected materials, which have precisely defined architectures at the micro- and nanoscales, are a new and important class of advanced engineered materials.^1–3 Unprecedented combinations of qualities, such as strength, stiffness, toughness, and multifunctionality, can be obtained by intentionally modifying the geometry of the internal material structure from the microscopic to the macroscopic level, overcoming limits inherent in traditional materials. Significant research has concentrated on lightweight lattice-based structures with extraordinary mechanical efficiency, such as microlattices and nanolattices. The micro- and nano-architectures’ high degree of control and tunability gives mechanisms for engineering materials with specific target attributes. Such complicated interior material patterns can be produced at scale using additive manufacturing methods.^4–7 Because their possible property space far exceeds that of traditional materials, architected materials are poised to be a highly disruptive approach in materials science and engineering, opening up new frontiers in structural and multifunctional materials’ design.

Topology, which is concerned with the organization and geometric arrangement of these interior formations, acts as the foundational framework determining the operational properties of the material. The intricacies of topology also have an impact on qualities such as energy absorption capacities, customization potential, and the ability to fulfill many purposes concurrently. Importantly, the growing convergence of topology optimization and cutting-edge manufacturing processes, such as 3D printing, accelerates the realization of complicated interior structures. The capacity of auxetic materials—especially those in the form of lattice structures—to absorb impact energy more effectively than that of conventional materials is being studied. This feature is useful in situations where reducing the impact force is essential for user safety, such as sports equipment, helmets, and protective gear. The application of auxetic structures in tissue engineering and biomedical implants is being investigated. These materials may be appropriate for implants and scaffolds that closely resemble the mechanical properties of genuine tissues because the negative Poisson’s ratio can imitate some characteristics of biological tissues. Auxetic materials can be used to make elastic and flexible parts for wearables and electronics. These lattice structures are appropriate for applications such as stretchy conductors and sensors because they permit deformation without sacrificing the material’s integrity. It is possible to create auxetic lattice structures that function well as sieves or filters. Researchers can alter the size of the gaps in the structure by varying the lattice’s characteristics, which makes it beneficial for applications such as liquid and air filtration. Auxetic materials can be used to create lightweight structures with better energy absorption capabilities in aeronautical applications. This may aid in the creation of stronger and more effective aerospace components, such as impact-resistant materials and aircraft parts. Auxetic lattice structures can be included into mechanical joints and seals to improve deformation resistance and sealing effectiveness under different loading scenarios. In mechanical systems, this may result in increased longevity and dependability. Auxetic materials’ distinct mechanical properties can be used to reduce vibration in many applications. Researchers hope to improve the overall stability and performance of structures by lowering vibrations in mechanical systems through the use of auxetic lattice structures. Gaining knowledge of auxetic lattice structures enhances the comprehension of materials science and creative design concepts. Investigating materials with negative Poisson’s ratios creates new opportunities to modify material characteristics to suit particular uses, advancing materials science and engineering.^8–11 

The importance of auxetic metamaterials stems from their extraordinary mechanical properties, which contradict conventional norms. These materials have the unusual capacity to expand under tension, bringing a variety of benefits with far-reaching ramifications. This feature correlates with exceptional toughness and resilience, making them particularly ideal for applications requiring good shock absorption, such as protective gear, sports equipment, and cushioning solutions. What distinguishes auxetic materials is their ability to provide customized mechanical reactions via intentional structural design, allowing for precise customization aligned with specific requirements. Shen et al.¹² took the concept of pattern alteration triggered by buckling to an advanced level, applying it to the creation of a novel collection of three-dimensional metamaterials. These materials were engineered to exhibit a negative Poisson’s ratio across a wide range of strains. Lei et al.¹³ introduced an original auxetic metamaterial design. By harnessing the shape memory characteristics of its constituent elements, they achieved the ability to dynamically customize the in-plane moduli and Poisson’s ratios. Cheng et al.¹⁴ devised re-entrant unit cells featuring diverse variable stiffness factors (VSFs) with the aim of manipulating stiffness adjustability by tuning densification strain. This study revealed that the compaction points of these re-entrant structures could be precisely adjusted through the utilization of the specified VSF, thereby introducing a novel approach for the strategic design of unit cells with negative Poisson’s ratio.

The primary purpose of this research is to gain a full understanding of the mechanical behavior and properties presented by these complex materials at a macroscopic level. Because of their intricate microstructural arrangements, re-entrant-based auxetic materials exhibit strange and illogical deformation patterns. Homogenization procedures were generally employed to bridge the gap between the material’s overall macroscopic reactivity and its microscale behavior, which is produced from individual structural components. The goal of this investigation was to see how varying strut thicknesses affected the homogenized properties of these materials. These findings are expected to result in increased performance across a wide range of real-world applications and the advancement of optimal material design.

Topology optimization is a mathematical technique that optimizes the distribution of material inside a specific design area based on a number of loads, constraints, and performance requirements. The best material arrangement, or “topology,” must be determined in order to maximize a performance metric, such as stiffness or strength, while also conforming to production and weight constraints. It entails discretizing the design space into finite elements and iteratively altering each element’s density between solid and void until the best material arrangement is reached. This iterative process is powered by numerical optimization algorithms that simulate different material distributions and update densities to improve the objective function. The resulting topologies are usually characterized by holes and thin, organic features that improve structural efficiency. The mechanism of topology optimization is depicted in Fig. 1.

Topology optimization generates compact, high-performance systems by adjusting the material architecture to the physics of the challenge. However, making the intricate forms it creates may be difficult. Geometric constraints and manufacturing considerations must be incorporated into the formulation in order to develop designs that are suitable for production. Because of advancements in additive manufacturing, topology optimization is becoming more prominent as an automated design process in engineering domains, such as aerospace, automotive, and biomechanics. The purpose of current topology optimization method research is to shorten solution durations, widen their scope, and make them more manufacturing-ready.

Topology has a direct impact on the bulk physical qualities and performance of these cutting-edge engineered materials, making it essential when designing architected materials. Rather than their fundamental chemical makeup, architect materials acquire their specific properties from the internal organization, connection, and geometry that have been painstakingly constructed. Architected materials can be divided into three different types according to the degree of control over geometric features and the reliance on additive manufacturing processes, as shown in Fig. 2 of this study. The two-dimensional honeycomb and three-dimensional foam arrangements stand out among these varieties due to their popularity and basic characteristics. The exceptional mechanical characteristics of two-dimensional honeycomb structures, characterized by regularly spaced hexagonal cells on a flat lattice, are particularly remarkable. These characteristics include high strength-to-weight ratios and efficient material consumption. These structures have a wide range of applications in aviation components, sandwich panels, and lightweight composites. They are similar to natural examples, such as beehives, which serve as sturdy yet mobile frames for honey storage and larval development. They are incredibly useful to engineers because of their capacity to distribute loads efficiently, absorb energy, and provide exact geometric control. However, the first class of cellular structures, which includes honeycombs and foams, has inherent restrictions that force the development of a second class, represented by lattices with nodes and struts. In honeycombs and foams, the fixed cell configuration limits the design flexibility and prevents customization to satisfy particular engineering requirements. Contrarily, lattices with nodes and struts provide for greater design flexibility, giving engineers the ability to optimize connections and geometry for improved mechanical properties. In addition, this second class may have better load-bearing capacities than honeycombs and foams, making it appropriate for applications requiring increased strength and load-bearing capacities. More design freedom is actually possible with the second class of cellular structures, which are characterized by lattices with struts and nodes. Lattice systems show stress concentration in response to external stresses, leading to concentrated areas of higher stress, particularly at strut and node connections. When there are acute angles or abrupt transitions in the lattice geometry, this phenomenon might result in a localized collapse and diminished structural integrity. The issue of stress concentration is addressed by the third class of cellular structures, known as Triply Periodic Minimal Surfaces (TPMS). Small surface areas and continuous, smooth curves of TPMS aid in more evenly dispersing stress throughout the material. The interconnected and regular structure of TPMS allows for gradual transitions between cells, as opposed to lattices with sharp edges, which lowers stress concentrations. The improved mechanical properties and material efficiency of TPMS are due to their increased surface area-to-volume ratio. TPMS are thus appropriate in circumstances needing an enhanced load-bearing capacity and a higher structural performance.

A higher strength-to-weight ratio is frequently correlated with a larger surface area-to-volume ratio. Weight is a function of volume, and strength in materials and constructions is the capacity to bear an applied force without cracking or deforming. The strength of the structure will be enhanced in relation to its weight if you increase the surface area while maintaining the volume constant. This is because the applied forces are more evenly distributed over a greater area. Structures with less material can be created by increasing the surface area-to-volume ratio. This is crucial for a lightweight design when cutting down on material usage is a top priority. Because of their complex patterns, TPMS structures use less material to attain a large surface area, which increases their efficiency in using resources. An increased ratio of surface area to volume may be a factor in the increased stiffness. Increased stiffness and structural integrity are the results of reducing the susceptibility to deformation by distributing the forces over a wider surface area.

Auxetic metamaterials are a class of materials that, when stretched longitudinally, expand transversely due to a negative Poisson’s ratio as shown in Fig. 3. When stretched, most common materials, such as rubber or metals, have a positive Poisson’s ratio and normally compress laterally; meanwhile, auxetic materials, or those with a negative Poisson’s ratio, expand laterally. Because of their distinct mechanical characteristics, auxetic materials are frequently used in the design of structures, such as lattice systems. Auxetic metamaterials have special properties, such as increased fracture toughness, indentation resistance, shear resistance, and synclastic curvature, because of this counterintuitive behavior. Various geometries and structures, including re-entrant models, rotating polygonal models, chiral models, crumpled sheets, and perforated sheets, can be used to create auxetic metamaterials. They can also be found naturally in a few materials, such as cat skin, cancellous bone, and some cubic metals. The development of auxetic metamaterials—metals and composites—to enhance their mechanical properties is still being researched.

In order to address the inherent conflicts between the strength and toughness of architected mechanical metamaterials, Wang et al.¹⁶ designed and produced a novel hybrid auxetic re-entrant metal–ceramic lattice. The synergistic effects of auxetic re-entrant metal honeycombs and infilled ceramic materials are studied experimentally and numerically, and auxetic deformation features and failure modes are characterized using the digital image correlation (DIC) technique as well as illustrated in Fig. 4.

In terms of stress and micromotion distributions, the new porous femoral hip meta-implant with a graded Poisson’s ratio distribution by Ghavidelnia et al.¹⁷ was compared to three other femoral hip implants (one solid implant and two porous meta-implants, one with a positive distribution of Poisson’s ratio and the other with a negative distribution) shown in Fig. 5.

Auxetic metamaterials are a distinct class of materials distinguished by the paradoxical property of becoming thinner (decreasing in the cross-sectional area) when compressed and thicker (increasing in the cross-sectional area) when stretched, in contrast to conventional materials that behave in the opposite way. Their unique internal structure leads to a negative Poisson’s ratio, which results in this peculiar behavior as shown in Fig. 6. The present work utilizes the Inconel 625 material for the development of 2 × 2 × 2 based periodic auxetic metamaterial using nTopology software. Figure 7 shows the cell map of the dimension 10 × 10 × 10 mm³. In the context of lattice systems, the term “cell map” refers to a graphic representation or a diagram that shows the configuration of individual cells within the lattice. Repeating unit cells make up lattice structures, which have patterns that fill the entire space. A cell map’s individual cells each stand in for a basic lattice repeating unit. The cell’s geometry is depicted on the map, including its size, shape, and connections to nearby cells. Understanding the mechanical behavior of the lattice and how it responds to stresses, pressures, and other stimuli requires knowledge of this information.

In the present work, four lattice structures having a unit cell dimension of 2 × 2 × 2 of strut thickness 0.2, 0.3, 0.4, and 0.4 mm are taken into account as shown in Fig. 8.

A combination of analytical models, numerical simulations (such as finite element analysis), and experimental testing is used to analyze the mechanical behavior of lattice systems. An analytical model includes honeycomb and truss models, periodic cell models, and homogenization techniques. A numerical simulation based on finite element analysis includes mesh based simulations, material modeling, and parametric studies. Experimental studies include compression and tensile testing, 3D printing and prototyping, and digital image correlation.

Homogenization simulations were performed on a re-entrant auxetic lattice structure consisting of a 2 × 2 × 2 unit cell made of Inconel 625. The strut thickness was varied from 0.2 to 0.4 mm to study its effect on the overall auxetic behavior and stiffness as shown in Figs. 9(a)–9(c). Periodic boundary conditions were applied to allow the unit cell response to represent the effective homogenized properties. The displacements in the principal X, Y, and Z directions along with XY, XZ, and YZ shear, respectively, were extracted for each thickness case as shown in Table I and are further illustrated in Figs. 10–15. It was observed that increasing the strut thickness from 0.2 to 0.4 mm reduced displacements in all normal and shear directions. This indicates an enhancement in stiffness and auxetic effects, as the lattice expands laterally when loaded axially. The periodic boundary conditions enable the efficient prediction of the macroscale auxetic behavior by modeling a single repeating unit cell. Varying the geometry and thickness allows tuning the design for desired stiffness and Poisson’s ratio. The reduced displacements with thicker struts demonstrate that tuning strut thickness is an effective approach for tailoring the performance of re-entrant auxetic lattices.

The material’s behavior after homogenization is thoroughly explained in Table II, with special attention paid to the variation in Young’s modulus across various thicknesses. A crucial mechanical measure called Young’s modulus reveals a material’s stiffness shown in Fig. 16 and tendency to respond to outside forces. The results show a distinct pattern: the increase in material thickness is accompanied by a perceptible rise in Young’s modulus. This pattern suggests that bigger specimens exhibit a higher deformation resistance, highlighting their superior stiffness compared to slimmer specimens.

The key findings of this study include that the rigidity of the structure tends to rise with increasing lattice strut thickness. This is so that larger loads can be supported without causing severe bending or buckling since thicker struts have a stronger resistance to deformation. The strut thickness is not the only geometric aspect that affects the lattice’s rigidity; the length and placement of the struts are also important factors. Generally speaking, though, thicker struts tend to be stiffer.

In conclusion, this study effectively used homogenization approaches to investigate the mechanical characteristics of auxetic re-entrant-based metamaterials that have characteristics similar to those of Inconel 625. A thorough examination of the displacement patterns and Young’s modulus values over a range of material thicknesses has provided important new information about how these materials react to external forces. In particular, the special qualities of Inconel 625 have improved our comprehension of their behavior. The promise of these metamaterials in applications needing specialized mechanical responses and increased stiffness is demonstrated by findings. From these findings, several intriguing directions for future research are apparent. Analyzing how microstructural changes affect homogenized qualities may help us comprehend the material’s behavior better. In addition, the reliability and legitimacy of the results would be improved by experimental validation of our findings. The range of potential applications would be increased by expanding this study to include dynamic loading situations and taking into account additional material properties in addition to Young’s modulus. Advanced computer methods could also make it possible to analyze complicated structures in greater detail.

The authors have no conflicts to disclose.

Ethics approval is not required.

All the authors have read and agreed to the published version of the manuscript.

Akshansh Mishra: Conceptualization (equal); Data curation (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Vijaykumar S Jatti: Data curation (equal); Supervision (equal). Eyob Messele Sefene: Conceptualization (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Emad Makki: Conceptualization (equal); Investigation (equal); Project administration (equal); Supervision (equal).

The data that support the findings of this study are available within the article.