Figure 4.10 Area under the stress–strain curve used as measure of the relative toughness between materials of the same geometry.

      A more widely used parameter of toughness is the fracture toughness K_c. It is determined experimentally using standard techniques by inserting a crack of length c into a specimen and loading it until fracture occurs. K_c (units MN/m^3/2or MPa m^1/2) is related to G_c by Eq. (4.27).

      Pure metals have K_c values in the range ~100–350 MN/m^3/2 (Table 4.1). The K_c values of the majority of ceramics and glasses are in the range ~0.5–5 MN/m^3/2. On the other hand, a few ceramics such as YSZ and silicon nitride (Si₃N₄) can have K_c values equal to ~10 MN/m^3/2. The higher fracture toughness of YSZ is due to the occurrence of a phase transformation in the region of the crack tip, which dissipates some of the energy available for crack propagation. This process is described as transformation toughening (Chapter 7). On the other hand, a unique fibrous structure of elongated grains coupled with an appropriate thin glass phase at the grain boundaries is responsible for the higher fracture toughness of Si₃N₄. A crack propagates along the weaker glass phase at the grain boundaries, making the crack path more tortuous when compared to a straighter path directly across the grains. This tortuous path consumes more energy than a straighter path. Polymers have G_c values between those for metals and ceramics but low K_c values because their Young’s modulus is low.

      4.2.7 Fatigue

      The resistance of a material to fatigue, a process of slow crack growth under repeated stress cycles, is important in several biomedical and engineering applications. Implants used in total joint replacement or in repairing large defects in the long bones of the limbs, for example, are subjected not just to a constant or slowly varying stress but to repeated cyclic stresses as well during normal activities of walking, running, and jumping. Stents used to keep coronary arteries open have to withstand the pressure pulsations of blood flow through the arterial vessels.

Fatigue occurs by a process of slow crack growth under cyclic stresses that are often well below the strength of the material. Cracks formed during cyclic loading or accidentally present in the original material can grow slowly and, when they have propagated sufficiently far, failure occurs, often in a brittle manner, at stresses well below the strength of the material (Figure 4.11).

Figure 4.11 Schematic representation of the fracture surface of a ductile metal after fatigue failure in one‐way bending (a) and two‐way bending (b).

      Fatigue behavior is commonly studied by subjecting specimens to cyclic loads often sinusoidal in nature, in the requisite loading mode such as tension, compression or bending. Specimens, commonly of a geometry similar to those used to measure the strength of the material (Section 4.2.1), are loaded for the requisite number of cycles of until they fail.

      4.2.8 Hardness

Hardness provides a measure of the resistance of a material to penetration by a sharp object. Loading a material with a sharp indenter results in the creation of a permanent irreversible impression that is indicative of plastic deformation (Figure 4.12). This occurs even for brittle materials such as ceramics because the stresses due to a sharp indenter are sufficiently high to cause local plastic deformation. The hardness H is determined from the equation

(4.29)

where, F is the applied load (force) and A is the area of the residual deformation. While indenters of various geometry are available, the Vickers indenter consisting of a square‐shaped pyramid usually made of diamond is often used. The area A used in Eq. (4.29) is the actual area of the residual indent, that is, the area between the four faces of the pyramid and the surface of the solid after the indenter has been removed. Thus, for the Vickers indenter

(4.30)

where, 2a is the length of the diagonal of the indenter (Figure 4.12). The hardness determined from Eq. (4.30) is called the Vickers hardness. In another measure of hardness, sometimes called the true hardness, the projected area of indent is used in Eq. (4.29), giving

      (4.31)

Figure 4.12 Geometry of hardness test using a Vickers indenter consisting of a square‐shaped pyramid. (a) Side view of the indentation test. (b) View of the indent looking directly at the surface of the material.

      The true hardness is ~8% higher than the Vickers hardness and, thus, it is useful to state which hardness is being reported.

      For some ductile metals, H is related to the yield strength σ_y by the equation

(4.32)

but a correction factor is needed for other metals that become stronger with increasing plastic deformation (a process called work hardening or strain hardening). As seen from Eq. (4.32), measuring the hardness provides a simple and nondestructive test for determining the yield strength of a material. There is no need to make tensile specimens and the indenter is so small that several measurements can be performed on the same material sample without severely damaging it.

      Hardness controls the resistance to abrasive wear between the articulating surfaces of two materials. The harder the material, the less prone it is to wear. Wear is particularly important, for example, in implants used for total joint replacement (Chapter

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