Читать книгу Materials for Biomedical Engineering - Mohamed N. Rahaman - Страница 127

When metals and ceramics are subjected to a constant stress below their elastic limit at temperatures relevant to biomedical applications and the majority of engineering applications, the resulting strain remains constant with time. In comparison, if a constant tensile stress is applied to a polymer specimen, the resulting strain does not remain constant but will increase slowly with time (Figure 4.4a). This slow extension with time is called creep. It is due to rearrangement of the mobile sections of the polymer chain in response to the stress. If the stress is removed, the polymer chains return slowly to their original conformation and the strain decreases slowly to zero. This strain recovery in polymers has its origins in the overall entropy of the polymer chains. Rearrangement of the polymer chains during deformation leads to a decrease in their entropy. Consequently, upon removal of the stress, there is a driving force for the chains to return to their higher entropy conformation. If a polymer specimen is extended to a certain constant strain, the tensile stress required to maintain this strain is not constant but decreases slowly with time (Figure 4.4b). This effect is called stress relaxation.

Figure 4.4 Linear viscoelastic behavior of polymers. (a) In creep, a constant stress σ applied at t = 0 leads to time‐dependent strain ε (t); at a given time, the strain increases linearly with the applied stress. (b) In stress relaxation, a constant strain ε applied at t = 0 leads to a time‐dependent stress σ (t); at a given time, the stress increases with increasing strain.

Creep and stress relaxation are manifestations of a general property of polymers called viscoelasticity. The overall response is a combination of two responses:

 An elastic response because, given sufficient time, the solid recovers completely upon removal of the stress

 A viscous response that results in creep or stress relaxation, in much the same way that a very thick syrup of high viscosity will deform slowly under an applied stress.

Polymers 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

(4.21)

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 .