Materials for Biomedical Engineering. Mohamed N. Rahaman

and performance of biomaterials.

4.2 Mechanical Properties of Materials

Biomaterials often have to withstand a complex combination of loads that may operate in more than one direction. An implant designed to heal large defects in the long bones of the limbs, for example, should be capable of withstanding appropriate physiological loads not only in compression but also in bending (flexure) and twisting (torsion). The implant should also be capable of withstanding loads due to walking, running, and jumping which are composed predominantly of repeated cyclic loads and not just a constant load. Thus, an understanding of the mechanical response of biomaterials when subjected to these different loading modes is essential.

      Different classes of materials, such as metals, ceramics, and polymers, show different characteristic responses when subjected to mechanical forces, such as the ductility of metals and the brittleness of ceramics. Unlike metals and ceramics, synthetic polymers and many natural materials show a property called viscoelasticity in which the mechanical response depends not just on the magnitude of the load but also on the time over which the load is applied. These different mechanical responses strongly influence the selection and design of biomaterials.

      Our discussion of mechanical properties is confined to materials composed of a single type of solid. These properties can be modified or improved by combining two or more different solids to form composites, a topic considered in Chapter 12. While the general aspects of mechanical testing procedures are discussed when appropriate, these procedures are not covered in detail because standard procedures are well described by organizations such as the American Society for Testing and Materials (ASTM) and the International Organization for Standardization (ISO).

      4.2.1 Mechanical Stress and Strain

      The mechanical behavior of a material is commonly determined by taking nominally identical specimens and deforming them in a mechanical testing machine according to a standard procedure. This type of experiment commonly gives data for the applied load and the deformation of the specimen. Load is actually a force and these two terms are often used interchangeably. Force, which has the unit newton (N), is commonly converted to a stress by dividing it by the area of the specimen over which it acts. This means that stress has units of N/m², also called pascal (Pa). The stresses encountered in testing of materials are often quite high and, thus, units of MPa (10⁶ Pa) or GPa (10⁹ Pa) are often used.

      The measured deformation of a specimen in response to a stress, such as its elongation, is normalized to the appropriate dimension of the specimen, such as its original length, thus converting it to a strain, expressed as a fraction or percent (that is, strain is unitless). Conversion of load and deformation to stress and strain, respectively, serves to normalize the data. This compensates for dimensional changes of the specimen during testing and the use of different specimen dimensions in different testing laboratories. A common way to show the mechanical response of a material is a diagram in which the measured stress (Y‐axis) is plotted versus the measured strain (X‐axis) to give what is called a stress versus strain curve, sometimes written stress–strain curve for short. However, in some experiments, it may also be required to show a plot of the applied force versus the deformation.

      In mechanical testing, the load can be applied in a few different ways, giving data for the mechanical response of a material under these different loading modes. These loading modes include uniaxial tension and compression, shear, torsion, and bending.

      Uniaxial Tension and Compression

In uniaxial tension, a common testing mode for metals and polymers, a specimen of uniform cross section is pulled in one direction along its length, that is, uniaxially (Figure 4.1a), whereas in uniaxial compression the specimen is compressed in one direction (Figure 4.1b). In tension, a load (force) F is applied at the larger ends of the specimen of circular or rectangular cross section, and the stress and strain are measured for the central portion, called the gage length. For a specimen of gage length l_o and uniform cross section A, the applied stress σ is defined as

      (4.1)

      and the strain ε is defined as

      (4.2)

      where, δ is the increase in the gage length from l_o to l.

Figure 4.1 Different loading modes in mechanical testing of materials: (a) tension, (b) compression, (c) shear, (d) torsion, and (e) four‐point bending (flexure).

      As the specimen does not have to be gripped at its ends in compression testing, the use of a cylindrical or tetragonal specimen of uniform cross section, that is, without the larger ends, is easier and less expensive. The common convention in mechanics is that tensile stresses and strains are positive in sign whereas compressive stresses and strains are negative in sign.

      Shear Deformation

Mechanical testing in shear is more difficult to perform when compared to tension or compression and, thus, it is of interest only when shear is a significant deformation mode in the intended application. For a specimen in the shape of a cube of length l, for example, a force F is applied transversely over two opposite faces of the specimen of area A (Figure 4.1c). The horizontal faces of the specimen tend to slide relative to each other, with their edge lengths remaining the same, and the vertical faces tilt to accommodate this motion. The shear stress τ is defined as

      (4.3)

      where, A is the area over which the shear force acts, equal to l ² for this example. The shear strain is defined as

      (4.4)

      where, δ is the displacement of one horizontal face of the cube relative to the other. The shear strain is also given by

      (4.5)

      where, θ is the angle (in degrees) by which the one vertical face has tilted relative to the other. For small displacements ( δ /l less than ~5%), the strain is also given to a good approximation as γ = θ, where θ is now expressed in radians.

      Torsional Deformation

      In torsional deformation, a torque is applied to one end of a specimen to twist it relative to the other end and the angle of twist is measured. Although the geometry is different from that described in the previous section, torsional deformation results in shearing one part of the specimen over another and, thus, we can also define a shear stress and shear strain for this mode of deformation. A force F applied to one end of a specimen in the shape of a cylinder of length l and radius r to twist it in the circumferential direction relative to the other end (