Materials for Biomedical Engineering. Mohamed N. Rahaman
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are viscoelastic over a wide range of temperature, starting from as low as approximately −100 to −200 °C. Consequently, in designing polymers for applications in which they are subjected to a mechanical stress, we must take into account that the strain is dependent not only on the magnitude of the stress but also on the time over which it is applied. The mechanical response of polymers is also strongly dependent on temperature. Although the majority of biomaterials are used within a narrow range of temperature, such as near room temperature or the physiological temperature (~37 °C), in general, the effect of temperature must also be taken into account when polymers are used in applications that cover a wider temperature range.
For small strains (less than ~0.5%), the strain at a specific time increases linearly with the applied stress and this type of behavior is called linear viscoelastic (Figure 4.4). On the other hand, at higher strains, the strain at a specific time increases faster with applied stress than predicted by extrapolation of the linear relationship, a behavior described as nonlinear viscoelastic. In the linear viscoelastic range, the relationship between applied stress, such as an applied tensile stress σ, and the time‐dependent strain ε(t) is
where J1(t) is the time‐dependent creep modulus. This relation is similar to Eq. (4.15) for a perfectly elastic solid except that the elastic modulus and strain are now time‐dependent. The stress relaxation modulus J2(t) can be defined in a similar way, that is
(4.22)
The creep modulus J1(t) can be measured by subjecting a specimen to a constant load and measuring the tensile strain as a function of time. In the linear viscoelastic region, as J1(t) is independent of the applied stress σ, it can be determined by measuring ε(t) for a single specific applied stress. Once J1(t) is known for a given time range, ε(t) can be determined for any stress within this time range from Eq. (4.21). On the other hand, for nonlinear viscoelastic deformation, each value of the stress leads to a time‐dependent strain that can be obtained only by experiment performed at that stress. However, it is not necessary to measure ε(t) for every conceivable stress of interest. Interpolation between a few curves measured at appropriate stresses can often provide sufficiently precise data for design.
Mechanical models composed of various combinations of elastic springs and viscous dashpots are sometimes used to provide a phenomenological description of linear viscoelastic behavior. While these models do not provide a description of viscoelastic behavior at a molecular level, they are useful for fitting experimental data and, subsequently, for use in design to predict linear viscoelastic behavior under certain circumstances. One such model is the Zener model, also referred to as the standard linear solid. This model can be depicted in two equivalent ways, as a spring in series with a Kelvin model (Figure 4.5a) or a spring in parallel with a Maxwell model (Figure 4.5b). The springs account for the elastic contribution to the deformation whereas the dashpot accounts for the viscous contribution. Typically, the models are used to develop an equation for the creep modulus J1(t) or the stress relaxation modulus J2(t) (McCrum et al. 1997). Then, by fitting the model equation to the appropriate experimental data for creep or stress relaxation, model parameters relating to the elastic modulus of the springs, E1 and E2, and the viscosity of the dashpot η are determined. Once these parameters are determined from data for a specific polymer, the phenomenological equations can be used to predict linear viscoelastic behavior of the same polymer with reasonable accuracy under appropriate conditions.
Figure 4.5 Two alternative versions of the Zener model, also called the standard linear solid, used to provide a phenomenological theory of linear viscoelastic behavior. The important parameters of the models are the elastic modulus of the two springs E1 and E2, and the viscosity of the dashpot η .
4.2.4 Stress–Strain Behavior of Metals, Ceramics, and Polymers
Ceramics (including glasses and glass‐ceramics) are brittle and, thus, they undergo only elastic deformation prior to fracture (Figure 4.6). The majority of ceramics used as biomaterials such as alumina, zirconia‐toughened alumina (ZTA), yttria‐stabilized zirconia (YSZ), and silicon nitride typically have a higher Young’s modulus than most metals such as titanium and its alloys, stainless steel and cobalt–chromium alloys (Table 4.1). The strength of these ceramics, often determined in flexural testing, is also typically higher than the tensile strength of many metals and alloys. Consequently, the stress–strain curve consists of straight line and the stress at failure is often larger than the yield stress of many metals. Glasses typically have a lower Young’s modulus and flexural strength than most ceramics used in biomedical and engineering applications. The key parameters obtained from the stress–strain curves for brittle materials are their Young’s modulus, strength and strain to failure in flexural loading. In comparison, metals show a stress–strain behavior characterized by an elastic deformation followed by plastic deformation, and the key parameters are their Young’s modulus, yield strength, elastic limit, and elongation to failure.
Figure 4.6 Schematic stress–strain curves to illustrate the characteristic response of brittle ceramics and ductile metals.
At room temperature, polymers can show a brittle, ductile, or elastic response depending on their composition and structure (Figure 4.7). Some amorphous polymers such as polystyrene, for example, show a brittle response under some appropriate rate of loading, fracturing at tensile strains of less than approximately 0.5%. In comparison, semicrystalline polymers typically show a ductile response. Polyethylene, for example, shows a ductile response in which the strain to failure can be up to a few hundred percent. This difference in mechanical response between polystyrene and polyethylene is attributed to differences in their structure (see Section 8.2.6). Amorphous polymers with a crosslinked structure such as natural rubber show an elastic response over a very large strain (a few hundred percent for natural rubber) followed by failure, but the stress–strain curve is nonlinear. The strength and elastic modulus of polymers are far lower than those for ceramics and metals (Figure 1.5).