Читать книгу Materials for Biomedical Engineering - Mohamed N. Rahaman - Страница 132

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.