Fracture Toughness - Crack Growth As A Stability Problem

Crack Growth As A Stability Problem

Consider a body with flaws (cracks) that is subject to some loading; the stability of the crack can be assessed as follows. We can assume for simplicity that the loading is of constant displacement or displacement controlled type (such as loading with a screw jack); we can also simplify the discussion by characterizing the crack by its area, A. If we consider an adjacent state of the body as being one with a broader crack (area A+dA), we can then assess strain energy in the two states and evaluate strain energy release rate.

The rate is reckoned with respect to the change in crack area, so if we use U for strain energy, the strain energy release rate is numerically dU/dA. It may be noted that for a body loaded in constant displacement mode, the displacement is applied and the force level is dictated by stiffness (or compliance) of the body. If the crack grows in size, the stiffness decreases, so the force level will decrease. This decrease in force level under the same displacement (strain) level indicates that the elastic strain energy stored in the body is decreasing—is being released. Hence the term strain energy release rate which is usually denoted with symbol G.

The strain energy release rate is higher for higher loads and broader cracks. If the strain energy so released exceeds a critical value G_c, then the crack will grow spontaneously. For brittle materials, G_c can be equated to the surface energy of the (two) new crack surfaces; in other words, in brittle materials, a crack will grow spontaneously if the strain energy released is equal to or more than the energy required to grow the crack surface(s). The stability condition can be written as

elastic energy released = surface energy created.

If the elastic energy released is less than the critical value, then the crack will not grow; equality signifies neutral stability and if the strain energy release rate exceeds the critical value, the crack will start growing in an unstable manner. For ductile materials, energy associated with plastic deformation has to be taken into account. When there is plastic deformation at the crack tip (as occurs most often in metals) the energy to propagate the crack may increase by several orders of magnitude as the work related to plastic deformation may be much larger than the surface energy. In such cases, the stability criterion has to be restated as

elastic energy released = surface energy + plastic deformation energy.

Practically, this means a higher value for the critical value G_c. From the definition of G, we can deduce that it has dimensions of work (or energy) /area or force/length. For ductile metals G_Ic is around 50–200 kJ/m2, for brittle metals it is usually 1–5 and for glasses and brittle polymers it is almost always less than 0.5.

The problem can also be formulated in terms of stress instead of energy, leading to the terms stress intensity factor K (or K_I for mode I) and critical stress intensity factor K_c (and K_Ic). These K_c and K_Ic (etc.) quantities are commonly referred to as fracture toughness, though it is equivalent to use G_c. Typical values for K_Icare 150 MN/m3/2 for ductile (very tough) metals, 25 for brittle ones and 1–10 for glasses and brittle polymers. Notice the different units used by G_Ic and K_Ic. Engineers tend to use the latter as an indication of toughness.