Materials for Biomedical Engineering. Mohamed N. Rahaman

often shows considerable scatter because the size, position, and orientation of flaws such as microcracks vary from one specimen to another (Section 4.2.5). This has important consequences for design. As failure at any flaw can lead to failure of the entire specimen, the strength that can be safely used in design is often far lower than the average measured strength. A specimen size effect also has to be taken into account because there is a higher probability of a strength‐determining flaw present in a larger volume. Another factor is that the measured strength depends on the testing technique because each technique subjects a different volume of the specimen to a tensile stress. In order to account for all of these factors quantitatively, the design of brittle materials such as ceramics and glasses is often treated by a statistical method called Weibull statistics.

      Weibull statistical analysis employs a weakest link assumption that failure of a single element of the specimen results in failure of the entire specimen. Instead of the more common distribution functions used to describe experimental data, such as the normal distribution, Weibull analysis assumes a function, called the Weibull distribution, to describe the measured strength of brittle materials. One form of the distribution, called the two‐parameter distribution, is

(4.36)

      where, P_f ( σ ) is the cumulative probability of failure, that is, the probability that failure has occurred by a stress σ, m is the Weibull shape parameter, a number often called the Weibull modulus, which is an inverse measure of the width of the distribution, that is, a higher value of m corresponds to a narrower distribution, and σ_o is the Weibull scale parameter, a measure of the center of the distribution, equivalent to the probability of failure occurring at or below a stress σ_o is 0.63.

Weibull analysis of experimental data starts with measuring the strength of ~30 or more specimens in a given loading mode such as bending or compression using a standard procedure. Then the data are fitted, often using a least‐squares method, by the Weibull distribution. Taking the natural logarithm twice of Eq. (4.36), we get

(4.37)

The Weibull parameters m and σ_o are determined by fitting a straight line to a plot of the left‐hand‐side of Eq. (4.37) versus ln σ for the measured data. The general procedure for determining P_f from the data is to assign a rank to each value of the measured strength after sorting in an ascending order, by assigning a rank of 1 to the lowest value and n to the highest value. The failure probability assigned to the ith strength value is

      (4.38)

As an example of fitting strength data to obtain the Weibull parameters, Figure 4.15 shows data for porous bioactive glass specimens with a uniform grid‐like microstructure (similar to that shown in Figure 3.24b). The data were obtained by measuring the strength of 30 or more nominally identical specimens each, in their as‐fabricated state, in compression and in bending (Liu et al. 2013). Data taken from the literature for hydroxyapatite and β‐tricalcium phosphate specimens with a similar microstructure, tested in compression only, are shown for comparison. For the purpose of comparison, the data for the different materials are presented as a double logarithmic plot of ln[1/(1−P_f) versus σ, instead of a plot represented by Eq. (4.37).

Figure 4.15 Weibull plots for porous bioactive glass (BG) specimens in compression and flexural loading. Data for hydroxyapatite (HA) and β‐tricalcium phosphate (β‐TCP) specimens with a similar microstructure are shown for comparison.

      Source: From Liu et al. (2013) / with permission of Elsevier.

      These data illustrate features that are significant in the design of brittle materials for use in biomedical applications that are subjected to significant stresses, such as healing large defects in the long bones of the human limbs. At a specific applied stress, the bioactive glass specimens are more reliable in compression than in bending due to their higher m and σ_o values in compression. This may be attributed to the higher volume of the specimen subjected to a tensile stress in bending. For example, the maximum compressive stress on the human femur in walking or running is estimated at less than ~10 MPa. Based on the data in Figure 4.15, the probability P_f that one of these bioactive glass specimens will fail under a compressive stress of 10 MPa is estimated at less than 10⁻⁶, that is, less than one in a million specimens is predicted to fail. On the other hand, for the same stress (10 MPa) in bending, the probability of failure P_f is 0.3, or approximately 1 in 3 specimens will fail.

The data in Figure 4.15 also show that the Weibull parameters for β‐tricalcium phosphate are considerably lower than those of hydroxyapatite that, in turn, are lower than those of the bioactive glass. Due to its low mechanical reliability, β‐tricalcium phosphate should not be used in any application that is subjected to even a low compressive stress. In general, bioactive ceramics such as bioactive glass, hydroxyapatite and β‐tricalcium phosphate degrade in the biological environment in vivo (Chapter 7) and lose a significant fraction of their strength with time, an important factor that is not represented by the data in Figure 4.15.

      4.4.3 Designing with Polymers

      In designing with polymers, the property of viscoelasticity must be taken into account. For applications in which a tensile load is present, for example, data for the tensile strength and creep modulus over the appropriate range of conditions such as stress, time scale, and temperature are relevant. If the polymer degrades in vivo, the effects of environmental conditions should also be taken into account. Designing to avoid yielding of the polymer is often straightforward once the time and temperature dependence of the yield strength is accounted for. In comparison, designing to avoid brittle fracture is more difficult due to the presence of flaws such as microcracks and pores (Section 4.2.5).

4.5 Electrical Properties

      Electrical properties are important for biomaterials used in devices to deliver an electric current or an electrical signal. Electrically conducting metals such as platinum are used as electrodes in cardiac pacemakers and neural probes. On the other hand, electrically insulating materials such as polyurethane are used as coatings to isolate or insulate sensitive electronic devices from surrounding tissues and fluids. Whereas polymers are typically electrical insulators, several polymers have been synthesized recently which show a strong ability to conduct an electrical current.