Page 305 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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Fracture Mechanisms in Nonmetals 285

critical energy release rate for crack propagation is given by

G = c J c f = ∫ c δ y y d σ δ (6.22)
0
The sections that follow outline several specific toughening mechanisms in modern ceramics.
6.2.1 MICROCRACK TOUGHENING

Although the formation of cracks in a material is generally considered deleterious, microcracking
can sometimes lead to improved toughness. Consider a material sample of volume V that forms N
microcracks when subject to a particular stress. If these cracks are penny shaped with an average
radius a, the total work required to form these microcracks is equal to the surface energy times the
total area created:

W c N = a2 πγ s (6.23)
2

The formation of microcracks releases the strain energy from the sample, which results in an
increase in compliance. If this change in compliance is gradual, as existing microcracks grow and
new cracks form, a nonlinear stress-strain curve results. The change in strain-energy density due
to the microcrack formation is given by

∆ = 2ρπ γ s (6.24)
a
w
2
whereρ ≡ NV/ is the microcrack density. For a macroscopic crack that produces a process zone of
microcracks, the increment of toughening due to microcrack formation can be inferred by inserting
Equation (6.24) into Equation (6.21).
A major problem with the above scenario is that stable microcrack growth does not usually
occur in a brittle solid. Preexisting flaws in the material remain stationary until they satisfy the
Griffith criterion, at which time they become unstable. Stable crack advance normally requires
either a rising R curve, where the fracture work w (Figure 2.6) increases with crack extension, or
f
physical barriers in the material that inhibit crack growth. Stable microcracking occurs in concrete
because aggregates act as crack arresters (see Section 6.3).
Certain multiphase ceramics have the potential for microcrack toughening. Figure 6.31 sche-
matically illustrates this toughening mechanism [40]. Second-phase particles often are subject to
residual stress due to thermal expansion mismatch or transformation. If the residual stress in the
8
particle is tensile and the local stress in the matrix is compressive, the particle cracks. If the signs
on the stresses are reversed, the matrix material cracks at the interface. In both cases there is a residual
opening of the microcracks, which leads to an increase in volume in the sample. Figure 6.31 illustrates
the stress-strain response of such a material. The material begins to crack at a critical stress σ ,
c
and the stress-strain curve becomes nonlinear, due to a combination of compliance increase and
dilatational strain. If the material is unloaded prior to total failure, the relative contributions of
dilatational effects (residual microcrack opening) and modulus effects (due to the release of strain
energy) are readily apparent.
A number of multiphase ceramic materials exhibit trends in toughness with particle size and
temperature that are consistent with the microcracking mechanism, but this phenomenon has been
directly observed only in aluminum oxide toughened with monoclinic zirconium dioxide [41].

8 The residual stresses in the matrix and particle must balance in order to satisfy equilibrium.