Consider a perfectly elastic material, such as glass, which contains a sharp crack. The crack grows when the atomic bonds at the crack tip are broken. Work must be done to break these bonds and separate the atomic planes. The total energy required to form a crack of length 2a is
W = 4a*(gamma)
where 4a is the total surface area of the crack per unit thickness and gamma is the surface energy of the material. As the crack grows, more surface area is created and so more work is done by the applied forces. The total energy per unit thickness required to produce a crack of length 2a under an applied tensile stress is
Wtotal = 4a*(gamma) - (S^2)*pi*a^2/E
where S is the applied stress, pi=3.1416 and E is the elastic modulus. The condition for unstable crack growth is obtained by taking the derivative of Wtotal with respect to crack length a and setting the resulting expression equal to zero (taking the derivative with respect to either a or 2a results in the same answer). We find
dWtotal/da = 0
4*(gamma) - 2*pi*S^2*a/E = 0
S = [2*E*(gamma)/(pi*a)]^0.5
or
S*[(pi*a)^0.5] = (2*E*gamma)^0.5
As the length of a pre-existing crack increases, the stress required for fracture decreases. This equation is known as the Griffith criterion for fracture. It often appears in the form
S = [E*Gc/(pi*a)]^0.5
or
S*[(pi*a)^0.5] = (E*Gc)^0.5
where Gc is called the critical strain energy release rate, or the total work of fracture. This equation can be used to predict the critical values of stress and crack length that are required for a crack to grow in an unstable manner. When the term S*[(pi*a)^0.5] reaches the critical value (E*Gc)^0.5, the crack will begin to grow. In this context, it is convenient to treat S*[(pi*a)^0.5] as a measure of the driving force for crack propagation. It is common practice to define
K = S*[(pi*a)^0.5]
as the stress intensity factor. Fracture occurs when the stress intensity factor K equals or exceeds the critical stress intensity factor KIC where
KIC = (E*Gc)^0.5
KIC is usually referred to as the fracture toughness.
For most metals, the measured values of the critical strain energy release rate Gc are two to four orders of magnitude greater than the surface energy term. The reason for this difference is that the stresses at the crack tip cause localized yielding to occur before fracture. Yielding extends over a region called the plastic zone, within which tensile stresses are comparable to the yield stress. The local stress ahead of a sharp crack in an elastic material is given by
Slocal = S + S*[(a/2r)^0.5]
where r is the radial distance from the crack tip, and S is the applied stress. We can estimate the radius of the plastic zone ry by setting Slocal equal to the yield stress Sy. If r << a, then to a good approximation,
ry = (S^2)*a/[2*(Sy^2)]
ry = (K^2)/[2*pi*(Sy^2)]
Note that the radius of the plastic zone decreases rapidly as the yield strength increases. Cracks in relatively soft materials produce a large plastic zone compared to cracks in hard ceramics which result in a small plastic zone, or none at all.
Metals often contain alloying elements or impurities that form non-metallic inclusions. If inclusions are contained inside the plastic zone, then plastic flow will occur around them. As plastic deformation continues, elongated cavities form around these inclusions ahead of the crack tip. As these cavities link up, the crack tip advances by means of this ductile tearing. The plastic flow at the crack tip turns the initial sharp crack into a blunt crack, and a great deal of energy must be expended to make the crack grow. This is why the measured value of Gc for metals is much higher than the theoretical values. Since Gc is high, so is KIC. This is why metals are so tough.
The fracture surface of a ceramic or glass tends to be very flat and shiny as opposed to the dull, rough fracture surface of a ductile metal. This occurs because ceramics and glasses have very high yield strengths. Very little plastic deformation takes place at the crack tip, and even if a small amount of crack tip blunting is allowed for, the stress at the crack tip still exceeds the ideal strength of the material. The resulting crack grows between a pair of atomic planes giving rise to an atomically flat surface. This is called cleavage. The energy required to break the atomic bonds is much less than that absorbed by ductile tearing, and this is the reason why materials like glass are so brittle. It also explains why some steels become brittle and fail like glass at low temperatures.
At low temperatures, metals with BCC and HCP microstructures become brittle and fail by cleavage even though they may be tough at room temperature. Only metals with an FCC microstructure are unaffected by temperature in this way. In metals that do not have the FCC microstructure the motion of the dislocations is affected by the thermal agitation of the atoms. As the temperature drops, the thermal agitation decreases and dislocations become much less mobile than they are at room temperature. This increases the yield strength, which causes the plastic zone at the crack tip to shrink until it becomes so small that the failure mechanism changes from ductile tearing to cleavage. This effect is called the ductile-to-brittle transition. This is the reason why many of the Liberty ships that were constructed during WW II failed. When the ships were exposed to the cold water temperatures in the North Atlantic the welds in the ship’s hulls underwent a ductile-to-brittle transition. Many of the ships split in two under their own weight while they were docked. I hope that this gives you some insight into this type of phenomenon.
Maui
