Materials for Biomedical Engineering. Mohamed N. Rahaman

of ceramics is that their tensile or flexural strength is also much lower than their compressive strength, often by a factor of ~5–10 or more (Table 4.1). This is because cracks propagate more stably under compressive loads. Cracks have to twist out of their original orientation to propagate parallel to the direction of compression. Consequently, fracture does not occur by the rapid propagation of one crack (usually the largest crack) as in tensile loading. Instead, fracture in compression occurs by the slow extension of many cracks that eventually lead to crushing of the specimen.

      Overall, because of the difference in crack propagation:

       Metals (and other ductile materials) have approximately the same measured strength in compression and in tension

       Ceramics (and other brittle materials) have a measured compressive strength that is much higher than their tensile (or flexural) strength

      Consequently, proper design of structural ceramics is required to avoid their exposure to excessively high tensile stresses.

      The fracture surface of ductile metals is often considerably rougher than that of ceramics or glasses due to the high degree of plastic deformation. Ceramics show a smoother fracture surface because crack propagation involves little or no plastic deformation but instead, involves cleavage of atomic planes. Another characteristic difference is that ductile fracture in metals occurs more slowly than brittle fracture of ceramics due to the energy absorbing process of plastic deformation during crack propagation.

      Theoretical Analysis of Brittle Fracture

A criterion for fracture of brittle solids could be developed by determining the stress at a sharp crack tip and equating it to the theoretic strength. However, this is challenging because it requires detailed consideration of the atom arrangement at the crack tip. Instead, a widely recognized theory of fracture in brittle solids, called the Griffith theory, is based on energy considerations and assumes that the solid is a continuum, thus neglecting atomic considerations of the fracture process. An energy balance concept is assumed such that the limiting condition for crack growth occurs when the energy released by the elastic strain energy is equal to the energy required to create new crack surface. The analysis typically assumes a model composed of a thin wide plate containing a long thin elliptical‐shaped crack of length 2c through it which is subjected to a tensile stress σ in the direction perpendicular to the length of the crack (Figure 4.9). For this model, the theory predicts that fracture will occur when the stress at the crack tip becomes equal to σ_f, called the fracture stress, given by

(4.24)

where, E is the Young’s modulus and γ is the surface energy per unit area of the solid. Equation (4.24), commonly called the Griffith equation, is often taken as a necessary condition for fracture to occur. As the surface energy of many ceramics falls within a narrow range, ~0.5–2.0 J/m², Eq. (4.24) can be seen to provide theoretical underpinning for the strong effect of microstructural flaws on their tensile strength.

Figure 4.9 Geometrical model used in the Griffith theory of brittle fracture.

      Equation (4.24) is often written

      (4.25)

      where, G_c is called the toughness, equal to 2 γ (Section 4.2.6). Thus, we can say that brittle facture will occur when

(4.26)

According to Eq. (4.26), fracture will occur when, for a material subjected to a stress σ, a crack reaches a certain critical size c or, alternatively, when a material containing cracks of size c is subjected to some critical stress σ . The term on the left‐hand‐side of Eq. (4.26) occurs frequently in fracture mechanics and is often referred to as the stress intensification factor K. Thus, we can also say that fracture will occur when K = K_c where K_c is called the critical stress intensification factor or more commonly, the fracture toughness, given by

(4.27)

Using Eqs. (4.26) and (4.27), the tensile strength is given by

      (4.28)

      Brittle materials do not contain just one crack but many cracks that differ in size. In tension (or flexure), fracture occurs typically by rapid propagation of the largest crack of length 2c. On the other hand, fracture in compression typically occurs less rapidly by extension of many cracks. Thus, the fracture strength in compression is given as

      where, H is ~10 and is the average crack length.

      4.2.6 Toughness and Fracture Toughness

      Toughness refers to the ability of a material to withstand rapid propagation of a crack through it. The toughness G_c of a material is defined as the energy absorbed per unit area of crack (units J/m²). A high G_c means that it is difficult for a crack to propagate through a material, as in pure ductile metals such as aluminum and copper, for example, which have G_c values in the range 100–1000 kJ/m². In comparison, brittle materials such as ceramics have low G_c values, in the range 0.01–0.1 kJ/m², and, thus, it is easy for cracks to propagate through them.

The toughness of a material is difficult to measure and, consequently, more easily measured parameters are used to provide a measure of toughness. One such parameter is the area under the stress–strain curve in a given loading mode such as tension or flexure (Figure 4.10). This area can be used to compare the relative toughness of specimens with a similar geometry but it is not equal to the toughness G_c of the material. As Figure 4.10 indicates, there is often no correlation between strength and toughness. The area under the elastic region of the stress–strain curve is often referred to as the resilience because it gives a measure of the energy recovered upon unloading a specimen.