Characterization of auxetic polyurethanes foam for biomedical implants

1.1 Auxetic materials

Most of the materials out there have a positive Poisson’s ratio, and when stretched in one direction they thin out in the other direction. However, auxetic materials do the opposite as they become wider when stretched because they possess a negative Poisson’s ratio, which has been an accepted concept in classical elasticity theory for over a 100 years (4, 5). They were not given special attention initially, being treated as an accident or a curiosity, but later on they were studied in depth by Lakes (6) and other authors during the 1980s (7, 8). Since then, a variety of auxetic materials and structures have been discovered, fabricated or synthesized, ranging from the microscopic down to the molecular levels in all major classes of materials such as polymers, composites, metals and ceramics (9–13). Also, a variety of auxetic PU foams have been manufactured using different techniques (4, 14, 15). The influence of the manufacturing process on the final properties of the processed foams (16, 17), the acoustic absorption, the dynamic behavior and the applications to viscoelastic damping were discussed in Refs. (18–20). Auxetic materials do exist naturally such as arsenic, cadmium, α-cristobalite and many cubic elemental metals (21–23). Some biological materials are also auxetic including cat skin (24), cow teat skin (25), salamander skin (26) and load-bearing cancellous bone from human shins (27). The arterial endothelium when subjected to both wall shear stresses and a cyclic circumferential strain due to pulsatile blood flow exhibits auxetic effects as well (28). Furthermore, it has been demonstrated through studies and experiments that the same equipartition of mechanical stresses and the property of a material to resume its original shape or position after being bent, stretched or compressed have potential applications in the biomedical arena (29, 30), such as for knee prosthetics, artificial blood vessels, ophthalmic devices, compression bandages and artificial intervertebral disc, to name a few (31). Auxetic materials wrap around the indenter, providing a more uniform stress distribution, with peak stresses lowered by an average factor of 3 (32–35). Their enhanced indentation approves their good candidature for medical devices, scaffolds, stents, implants and prostheses (36–43), for example, as an implant for articular cartilage repair to provide a cushion between the surfaces of the bones and for meniscal repair or meniscal replacement, where they actually wrap around the round end of the upper bone to fill the space between it and the flat shinbone to help distribute the weight from the femur to the tibia (44, 45).

Here, we present a characterization of biocompatible auxetic PU foam as a soft tissue implant. Only limited work has been done in the theoretical analysis of auxetic PU foam as a tissue implant. To our knowledge, mathematical modeling for an auxetic PU soft tissue implant has not yet been developed. In this study, we will present a mathematical model for studying the compressible behavior of auxetic soft tissue under bi-axial loading in the context of nonlinear elasticity. With the data obtained from the experiments, numerical results were computed and discussed to study the auxetic vs. nonauxetic behavior of soft tissues.

2.1 Selection of materials and fabrication of auxetic PU foam samples

The composition of alternating polydisperse blocks of soft and stiff segments combined with excellent biocompatibility makes polyurethanes one of the most promising synthetic biomaterials (2). However, studies and experiments demonstrated that auxetic PUs tailored with enhanced mechanical properties and a unique deformation mechanism offer a huge potential in the biomedical industry. Therefore, after a literature survey, we selected auxetic PU foam to study the elastic and mechanical behavior of soft tissue implants. Based on the method given in Ref. (46), an auxetic PU foam sample was fabricated, which is shown in Figure 2, along with the microscopic microstructures of both auxetic and nonauxetic PU foams.

Figure 2

(A) Nonauxetic and (B) fabricated auxetic PU foam samples. Microstructure of (C) a conventional foam and (D) an auxetic foam.

2.2 Mathematical modeling

We consider a soft tissue as a rectangular block of auxetic and nonauxetic PU foams, and their deformation is given by

where λ₁, λ₂, λ₃ are the stretch ratios in the coordinate directions and the coordinates x_i and X_I referred to the same Cartesian system.

Next, the Lagrangian E_i,j and Eulerian e_i,j strain components in the matrix form given in Ref. (47) are

The deformation gradient is

Also, the constitutive relations are given by the first Piola-Kirchhoff stress tensor

where t_II=0 for I≠J, W is the strain energy density function, p is the Lagrange multiplier, and the Cauchy stresses are

In the further discussion, for convenience, the subscript and superscript and the lowercase and uppercase indices are not taken into account.

Next, when the tissue is transversely isotropic, relative to the x₁ direction, a strain-energy density function W in the case of compressibility reduces to

where W is the function of the strains I₁, I₂, I₃, I₄ given by

I1=λ12+λ22+λ32I2=λ12λ22+λ22λ32+λ32λ12I3=λ12λ22λ32I4=12(λ12-1)

and characterized by the Blatz-Ko strain-energy function as

where μ is the shear modulus.

The constitutive relations given by the first Piola-Kirchhoff and Cauchy stress components, upon using Equations 7–11, are given as

Furthermore, the plane stress condition in the x₃ directions gives t₃₃=0; therefore, from Equation 7 we have

and in the isotropic case Equations 12 and 13 along with Equation 18 give

3.1 Compressive testing

The fabricated foam specimen was subjected to an axial compression using an Instron 3369 testing machine (Norwood, MA, USA), and the schematic is shown in Figure 3A and B, respectively. The foam sample was placed between the parallel compression plates of the machine and compressed at a constant cross-head speed of 2 mm/min. Different types of conventional and auxetic foams were tested five times for each specimen for better accuracy of the result, and the compressive stress-strain curves for the nonauxetic and auxetic foams are shown in Figure 3C and D, respectively. Furthermore, the values of Poisson’s ratio and Young’s modulus calculated from experimental data are given in Table 1.

Figure 3

(A) The foam sample placed in the Instron machine; (B) schematic and compressive stress-strain curves of (C) a nonauxetic foam and (D) an auxetic foam.

3.2 Morphology

In Figure 2C and D, the microstructures of nonauxetic and auxetic foam show a clear transition from the open cell foam structure characteristic of a conventional foam with a positive Poisson’s ratio to the higher density reentrant foam showing a negative Poisson’s ratio response.

3.3 Analytical results

In Figure 3C and D, the compressive stress-strain curves show that the auxetic foam behaves more like an isotropic material. The mechanical behavior in the three directions is more similar for an auxetic foam, the orientation of the cell ribs is more random and the reentrant cell structure is similar in each dimension. The results for the stress-stretch curves for equibiaxial stretching of an isotropic compressible auxetic and nonauxetic soft tissue were computed using data obtained experimentally, which are given in Table 1. The depicted graphs in Figures 4 and 5 show the compressibility effect for different values of Poisson’s ratio to study the auxetic vs. nonauxetic behavior of tissues.

Figure 4

Stress-stretch curves of the nonauxetic PU sample.

Figure 5

Stress-stretch curves of the auxetic PU sample.

For positive values of Poisson’s ratio, v in Figure 4, the stiffness increases with increasing Poisson’s ratio, especially as v=0.5, and maximum stress is observed due to the incompressibility effect. In contrast, in Figure 5, which shows the stress-stretch ratio curves for negative values of Poisson’s ratio, the effects of compressibility are quite low and the stiffness decreases in auxetic materials compared to that in conventional materials.