When the local stress reaches the theoretical or ideal strength of the solid, equal to approximately E/3 to E/8, where E is the Young’s modulus, it is sufficient to break the interatomic bonds and the crack propagates rapidly between two atomic planes, a process described as cleavage. Ceramics, glasses and glass‐ceramics have low toughness or a low resistance to crack propagation when compared to ductile metals.
As ceramics almost invariably contain microstructural flaws, their measured strength in tensile or flexural loading is much lower than their theoretical strength, often by a factor of 102 to 103 or more, due to the high amplification of the applied stress at the sharp tips of microcracks and pores. Another feature 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 σf, called the fracture stress, given by
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, Gc is called the toughness, equal to 2 γ (Section 4.2.6). Thus, we can say that brittle facture will occur when
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 = Kc where Kc is called the critical stress intensification factor or more commonly, the fracture toughness, given by
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
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 Gc of a material is defined as the energy absorbed per unit area of crack (units J/m2). A high Gc 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 Gc values in the
