Poisson’s ratio plays an important role in providing a unique strain performance in the transverse direction relative to axial direction. It measures the relative deformation of the material in the perpendicular direction, relative to the deformation at the applied load direction. If the force load is applied in tension, the material will increase in length in the direction of the loading, (axial), while the perpendicular direction (transverse) will decrease in length to conserve the volume change of the material. If the material is exposed to compression, the phenomena will hold, but the deformation direction will reverse. The mathematical equation for Poisson ratio has a negative sign to account for the decrease in the dimension of the strain, as shown in Equation (1).

v = − ϵ Transverse ϵ Axial (1)

If the material dimension increases in all directions in tensile loading or decreases in dimension in all directions in compressive loading, the result of the Poisson ratio equation will be negative. Materials with a negative Poisson’s ratio are classified as auxetics.

Auxetic materials exhibit synclastic behavior, where the material displays a dome shape when subjected to bending, in contrast to the typical saddle shape for materials with a positive Poisson’s ratio [1]. Auxetic materials also present higher amounts of indentation resistance when compressive loads are applied, in addition to higher fracture propagation [1]. Another observed property of auxetic materials is vibration damping of various patterns of auxetic structure, which can dampen up to 10 times more if the same material did not have an auxetic structure design [1] [2]. This leads to the use of auxetic structure for acoustic absorption, and various sound-related applications [2]. Another important feature of auxetic materials is their unique anisotropy, which can achieve different properties in distinct directions that are impossible by a material that has a positive Poisson’s ratio [2].

Due to the anisotropic nature of the auxetic material, loading in the axial direction will yield different stresses, strains, and deformation patterns compared with loading along the transverse direction [3]. Additionally, auxetic properties are lost after yield stress is observed within the material due to deformation beyond the elastic region [4]. However, materials that have a low negative Poisson’s ratio often have fewer other favorable mechanical properties, creating a fine line between mechanical and auxetic properties for most applications [4]. Thus, it is important to consider the direction, magnitude, and type of loading when analyzing auxetic materials to maximize the utilization of the auxetic function of the material.

Physiological aspects of various fields can be improved through the implementation of auxetic properties into already-existing devices. One common example includes the synclastic behavior of helmets, which increases the protection provided to the user’s head upon impact [1]. These properties can also be expanded to a wide range of other protective equipment to allow for lighter weight and increased comfort [2].

K. Saxeen et al., 2016 [2] have illustrated numerous biomedical applications for auxetic structures. The uses can be divided into various categories such as implant devices, medical or surgical devices. Some of the most noteworthy implant applications include auxetic stents that reduce inflammation inside the body, and artificial blood vessels. Some procedures include dental floss, surgical sutures, and even smart bandages.

Tissue engineering aims to restore damaged tissue functions using biomaterials. Tissues engineered with an auxetic internal structure can potentially better mimic the mechanical properties of the original tissues than the ordinary engineered tissues. Tendons, vasculature, cancellous bone, embryonic epithelial tissues, arteries, and the annulus fibrosus of the intervertebral disc all have auxetic properties [10] [13].

It is a common practice for researchers to use computer-aided design software and finite element method to evaluate mechanical properties and compare them with experimental data. Warner et al., 2017 [10] found that modified unit-cell architecture with stabilized rounded hinges is effective at transmitting displacement throughout the structure and can minimize out-of-plane deflections using FEA. Combined with further research, these findings can potentially be used in tendon-muscle tissue regeneration. Kapnisi M. et al., 2018 [7] compared the theoretical resultant effective stiffness and anisotropic ratio of effective stiffness using an analytical model established by I. G. Masters and K. E. Evans, 1996 [14] with the experimental data collected, and an agreement was discovered between the two researchers.

R. Ajdary et al., 2022 [15] fabricated bacterial nanocellulose (BNC) with triangular, circular and star unit cell shapes using molds designed in SolidWorks. The Poisson’s ratios of −0.19, −0.36, and −0.13 are found for the triangular, circular, and star unit cells, respectively. The auxetic BNC meshes obtained after cell culturing can have a tensile strength up to 456 MPa depending on the cell culturing time. The reversible structural expansion under tension minimizes the tissue damage that commonly occurs by shrinkage of plastic-based mesh implants, and is expected to maximize comfort, minimize material fatigue, and thus improve the overall success of future long-term mesh implants.

M. N. Ali et al., 2014 [8] simulated the auxetic esophageal stent using a single square FEA model and an auxetic-ring FEA model in ABAQUS. The auxetic polyurethane film is compared with the auxetic film model derived from a single square FEA model extracted from the auxetic film for simplicity and accuracy, shown in Figure 2. The rigid auxetic polyurethane stent is comparable with the auxetic-ring FEA model. The two FEA models are concluded to be in good agreement with the experiment.

The M. N. Ali et al., 2014 [8] expected the auxetic esophageal stent with anisotropic mechanical behavior to conform well to the multilayered esophageal wall with a non-linear anisotropic mechanical response. Additionally, the small auxetic stent graft can be delivered orally into the esophagus without the use of an expensive dedicated delivery device using a balloon catheter [16]. With further study, such auxetic esophageal stents can potentially replace the current esophageal stents.

The auxetic pedicle screw proposed by Y. Yao et al., 2020 [17] is reported to have an improved fixation strength, while the femur auxetic meta-implant proposed by N. Ghavidelnia et al., 2021 [18] is reported to reduce the stress shielding and micromotions.

Y. Yao et al., 2021 [19] conducted thorough mechanical tests experimentally and computationally using FEA on the designed auxetic pedicle screw. Predictions by the FEA simulations are consistent with experimental results, but there are discrepancies between FEA and experimental results, including higher tensile stiffness, tensile strength and Poisson’s ratio [17]. The auxetic pedicle screw can improve bone-screw fixation by its radial expansion of the screw body under tensile force to resist pulling out. The pullout force was increased by 6.29% - 14.46% compared to that of the solid screw; however, a few auxetic screws have fewer pullout forces than the solid screw [19]. N. Ghavidelnia et al., 2021 [18] focused on developing an analytical solution with FEA and found the femur auxetic meta-implants to have enhanced stress and strain distributions and a more compatible and uniform distribution of micromotion.

Biomaterials with auxetic properties usually have complex geometry, making it difficult for researchers to design and analyze. However, with the increasing computation power in recent decades, researchers can incorporate auxetics in some other biomedical applications.

A. Bonfanti and A. Bhaskar, 2019 [20] proposed a wrinkle-resistant auxetic film to tackle the wrinkling problem of synthetic skin. The auxetic films showed the ability to be stretched without forming wrinkles with experiments, analytical solutions, and FEA. The wrinkle-resistant auxetic film can be a promising candidate for future applications such as artificial skin, cardiac patches, and biomedical stretchable sensors.

A. Arjunan et al., 2021 [21] developed a one size fits all auxetic nasopharyngeal swab that can be easily 3D printed. The FE model overestimates the yield and peak forces by 4.30% and 3.87% respectively, but the force-displacement curve of the swab heads shows that the FE model agrees with experimental results. The swab head can reduce patient discomfort by shrinking inward while navigating the nasal cavity, thus exerting a reduced stress on surrounding tissue compared to a regular swab.

V. Gupta and A. Chanda, 2023 [22] developed skin graft computational models using I-shaped auxetic incision structures and tested them under uniaxial and biaxial loads to evaluate their expansion potentials. The finite element model and testing of the auxetic skin graft using an isotropic and elastic material to represent skin demonstrated a high expansion and skin cover.

F. Tsegay et al., 2023 [12] presented an auxetic hydrogel skin pH indicator wound dressing as a smart band-aid or dressing. Using a phenol red dye in the hydrogel matrix of the material could help monitor pH changes in the vicinity of the wound to better indicate what treatment is required for the patient.

As mentioned before, a large constraint to auxetic unit cell design is the high stress zones of the structure, which limits the use cases for these structures, especially when manufactured from biomaterials. It is possible that auxetic biomaterials can be further optimized, which would involve exploring composite materials such as nano-size particles. However, it is beyond the scope of this paper.