,Among mechanical engineers there is a rule-of-thumb: avoid materials with fracture toughnesses K1c less than 15 MPam1/2. Almost all metals pass: they have values of K1c in the range of 20–100 in these units. White cast iron, and a few powder metallurgy products fail; they have values around 10 MPam1/2. Ordinary engineering ceramics have fracture toughnesses in the range 1 – 6 MPam1/2; mechanical engineers view them with deep suspicion. But engineering polymers are even less tough, with K1c values in the range 0.5 – 3 MPam1/2, and yet engineers use them all the time. What is going on here?
When a brittle material is deformed, it deflects elastically until it fractures. The stress at which this happens is
| , |
(M14.1) |
where Kc is an appropriate fracture toughness, ac is the length of the largest crack contained in the material and C is a constant which depends on geometry, but is usually about 1. In a Load-limited design (Figure 14.1(a)) — a tension member of a bridge, say — the part will fail in a brittle way if the stress exceeds that given by equation (M14.1). Here, obviously, we want materials with high values of Kc.
But not all designs are load-limited; some are energy-limited; others are deflection limited (Figure 14.1 (b) and (c)). Then the criterion for selection changes. Consider, then, the three scenarios created by the three alternative constraints of Table 14.1.
FUNCTION |
Resist brittle fracture |
OBJECTIVE |
Minimize volume (mass, cost...) |
CONSTRAINTS |
Design load specified |
Table 14.1 The design requirements
Figure 14.1 (a) Load-limited design; (b) energy-limited design and (c) displacement-limited design.
Figure 14.1 shows, schematically, three classes of fracture-limited design. The tie, the beam and the pressure vessel of Figure 14.1 (a) are examples of load-limited design: each must carry a specified load or pressure without fracturing. Then the local stress must not exceed that specified by equation (M14.1) and — for minimum volume — the best choice of materials are those with high values of
 . |
(M14.2) |
It is usual to identify Kc with the plane-strain fracture toughness, corresponding to the most highly constrained cracking conditions, because this is conservative. For load-limited design using thin sheet, a plane-stress fracture toughness may be more appropriate; and for multi-layer materials, it may be an interface fracture toughness that matters. The point, though, is clear enough: the best materials for load-limited design are those with large values of Kc.
But not all design is load-limited. Springs, and containment systems for turbines and flywheels (Figure 14.1(b)) are energy-limited. Take the spring as an example. The elastic energy per unit volume stored in the spring is the integral over the volume of
 . |
(M14.3) |
For an axial spring — a rubber band for instance — it is exactly this quantity. The stress is limited by the fracture stress of equation (M14.1) so that the maximum energy the spring can store is
 . |
(M14.4) |
For a given initial flaw size, energy is maximized by choosing materials with large values of
 . |
(M14.5) |
where Jc is the toughness (usual units: kJ/m2).
There is a third scenario: that of displacement-limited design (Figure 14.1(c)). Snap-on bottle tops, snap together fasteners and such like are displacement-limited: they must strain enough to allow the snap-action without failure. The strain is related to the stress by
 |
(M14.6) |
and the stress is limited by the fracture equation (M14.1). Thus the maximum strain is
 . |
(M14.7) |
The best materials for displacement-limited design are those with large values of
 . |
(M14.8) |
Figure 14.2 shows a Chart of fracture toughness, K1c, plotted against modulus E. It allows materials to be compared by values of fracture toughness, M1, by toughness, M2, and by values of the deflection-limited index M3. As the engineer's rule-of-thumb demands, almost all metals have values of K1c which lie above the 15 MPa.m1/2 acceptance level for load-limited design. Polymers and ceramics do not.
The line showing M2 on Figure 14.2 is placed at the value 1 kJ/m2. Materials with values of M2 greater than this have a degree of shock-resistance with which engineers feel comfortable (another rule-of-thumb). Metals, composites and some polymers qualify; ceramics do not. When we come to deflection-limited design, the picture changes again. The line shows the index M3 = K1c/E at the value 10-3m1/2. It illustrates why polymers find such wide application: when the design is deflection-limited, polymers — particularly nylons, polycarbonates and polystyrene — are as good as the best metals.
Figure 14.2(a) A chart of fracture toughness, K1c, against Young's modulus, E, showing the index M1.
Figure 14.2(b) A chart of fracture toughness, K1c, against Young's modulus, E, showing the index M2.
Figure 14.2(c) A chart of fracture toughness, K1c/E, against Young's modulus, E, showing the index M3.
Design type, and rule-of-thumb |
Material |
Load-limited design |
Metals, polymer-matrix composites. |
Energy-limited
design |
Metals, composites and some polymers. |
Displacement-limited
design |
Polymers, elastomers and some metals. |
Table 14.2 Materials fracture-limited design
The figure gives further insights. The mechanical engineers' love of metals (and, more recently, of composites) is inspired not merely by the beauty of their K1c-values. They are good by all three criteria (K1c, K1c2/E and K1c/E). Polymers have good values of K1c/E but not the other two. Ceramics are poor by all three criteria. Herein lie the deeper roots of the engineers' distrust of ceramics.
Brock, D (1984) 'Elementary Engineering Fracture Mechanics' Martinus Nijhoff, Boston, USA.