Fracture toughness

7.4.2 Fracture toughness

In engineering structures, particularly heat-treated steels, cracks are likely to arise from weld defects, inclusions, surface damage, etc. and it is necessary to design structures with the knowledge that cracks are already present and capable of propagation at stresses below the macroscopic yield stress as measured in a tensile test. Since different materials show different crack propagation characteristics (e.g. hard steel and glass) it is necessary for the design engineers to find the limiting design stress in terms of some property or parameter of the material. For this reason, a fracture toughness parameter is now being employed to measure the tendency of cracks of given dimensions to propagate under particular stress conditions.

In Section 7.4.1 it was shown that σ (πc) = (EG), which indicates that fast fracture will occur when a material is stressed to a value σ and the crack reaches some critical size, or alternatively when a material containing a crack is subjected to some critical stress σ, i.e. the critical combination of stress and crack length for fast fracture is a constant, (EG) for the material, where E is Young’s modulus and G is the strain energy release rate. The term σ (πc) is given the symbol K and is called the

stress intensity factor, with units MN m −3/2 . Fast fracture will then occur when K =K c , where K c [ = (EG c )] is the critical stress intensity factor, or more commonly the fracture toughness parameter. The fracture toughness of a material can alter markedly depending on whether the elastic–plastic field ahead of the crack approximates to plane strain or plane stress conditions, much larger values being obtained under plane stress conditions as operate in thin sheets. The important and critical factor is the size of the plastic zone in relation to the thickness of the section containing the crack.

When this is small, as in thick plates and forgings, plane strain conditions prevail and the hydrostatic tension ahead of the crack results in a semi-brittle ‘flat’ fracture. When the value is large, as in thin sheets of ductile metals, plane stress conditions operate and the tension at the crack front is smaller, giving rise to a more ductile mode of failure. At intermediate values a mixed fracture, with a flat center bordered by shear lips, is obtained. Thus, without changing the structure or properties of the materials in any way it is possible to produce a large difference in fracture toughness by changing the section thickness. For thick sections, when a state of complete constraint is more nearly approached,

the values of K c and G c approach minimum limiting values. These values are denoted by K 1c and

G 1c and are considered to be material constants; the subscript ‘1’ denotes the first mode of crack extension, i.e. the opening mode (see Figure 7.31). The general procedure in measuring the fracture toughness parameter is to introduce a crack of suitable size into a specimen of suitable dimension and geometry. The specimen is then loaded at

a slow rate and the crack extension measured up to the critical condition. The measurement of K 1c

Mechanical properties II – Strengthening and toughening 427 1000

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Strength ␴ y( MN m ⫺ 2 ) 2 Fracture-toughness, K IC , plotted against strength, ␴ y K . The contours show the value of 2

IC ,␲␴ y ––roughly, the diameter of the process zone at a crack tip (units: mm). The guidelines of constant K IC /␴ y and K 2 IC /␴ y are used in yield-before-break and leak-before-break design.

Figure 7.32 Ashby property chart of fracture toughness versus strength (Ashby, 1992).

will be valid if the plastic zone size is small (by a factor 10) in relation to the cross-section of the specimen. The zone size r y may be obtained by equating the stress field of the crack at r =r y to the strength σ y of the material and is given by

r y =K 2 1c / 2πσ 2 y. An Ashby property chart of fracture toughness versus strength, given in Figure 7.32, shows that

the size of the zone (broken lines) varies from atomic dimensions for brittle ceramics to tens of centimeters for ductile metals.

In designing safe structures for a given load, the structure is required to yield before it breaks. For

a minimum detectable crack size of 2c this condition is given by (K 1c /σ y ) √ ≥ π c. The safest material is the one with the greatest value of K 1c /σ y . Clearly, a high yield stress σ y is also required. The chart

428 Physical Metallurgy and Advanced Materials shows that steel satisfies both these requirements and indicates why it is still the best material for

highly stressed structures where weight is not important. Fracture toughness requirements are now written into the general specification of high-technology alloys and hence it is necessary to determine the effect of heat treatment and alloying additions on fracture toughness parameters. Processes such as ausforming and controlled rolling improve the fracture toughness of certain steels. Carbon has a considerable effect and there are advantages in reducing the C-level below 0.1% where possible. High-strength low-alloy (HSLA) steels have

C < $ 0.1%, and the Nb, V and Ti additions form fine carbides which, together with the small grain sizes, enable good strength levels and acceptable fracture toughness values to be achieved. Maraging steels with high alloy and low carbon (<0.04%) give very high strength combined with high toughness (see Chapter 8).

The brittleness of ceramics is directly linked to their high notch sensitivity. The fracture toughness of most ceramics is low: expressed in quantitative terms, it is commonly less than 8 MN m −3/2 . Flaws, often very minute, are almost invariably present in ceramics and act as stress-concentrating notches. It is extremely difficult to prevent these flaws from forming during manufacture and service. In terms of engineering practice, brittle ceramic components are intolerant of misalignment and poor fits within assemblies. The presence of flaws is also responsible for the variability or ‘spread’ of results from mechanical tests and introduces uncertainty into the design process. (The ‘spread’ is much less for metallic materials.) Design procedures have moved well beyond the principle that ceramics are only safe when compressive stresses are dominant. Probabilistic assessments of mechanical test results from ceramics now tackle the difficult task of allowing for randomness in the size, shape and distribution of flaws (see Chapter 10).

Despite these underlying problems, progress has been made in producing tough ‘ductile ceram- ics’. Fracture toughness values above 20 MN m −3/2 have been achieved. Typical measures include elimination of flaws (microcracks, pores), incorporation of crack-retarding phases (in zirconia) and reduction of average grain size (below 1 mm).

At ambient temperature, fracture surfaces of ceramics are conchoidal (glassy), intergranular or cleavage in character, depending on the material. The strain at fracture is very small, being about 0.001. At elevated temperatures, under creep conditions, fracture strain is greater. Conventional ceramics have little or no capacity for slip and crack tips therefore tend to remain sharp. As the crack propagates, the load-supporting cross-section gets smaller and the general level of stress increases so that failure can be sudden. These remarks refer to tensile stresses, since compressive stresses will, of course, tend to close the crack.

In some polycrystalline ceramics, such as magnesia, the von Mises criterion for maintenance of cohesions is not satisfied. Slip is limited and cracks are not effectively blunted. However, raising the temperature often enables the necessary minimum of five independent deformation modes to operate, leading to ductility. Any treatments which eliminate surface flaws will naturally enhance this ductility. (Against this background, one can readily appreciate why fabrication methods for polycrystalline ceramic components, in contrast to those for metals, are not based upon bulk deformation.) At these higher temperatures the environment becomes increasingly more likely to react with surfaces of the ceramic: it may even penetrate an open texture and cause crack blunting. The mechanisms of flow in polycrystalline ceramics at elevated temperatures are similar to those encountered in metallic systems (e.g. glide and climb of dislocations, vacancy diffusion, grain boundary sliding). The similarity between deformation processes in polycrystalline ceramics and metals is evident if one compares the layout of the corresponding deformation mechanism maps. A minor phase is usually present at grain boundary surfaces in ceramics, functioning as a ceramic bond, and if it softens with rise in temperature, then the grains may be able to slide past each other. These regions tend to liquefy before the actual grains so that it is advisable not to exceed the solidus temperature. Grain boundary surfaces are particularly susceptible to the nucleation of cracks. As in the final accelerating stage of metallic creep, these cracks form in planes normal to the direction of applied stress.

Mechanical properties II – Strengthening and toughening 429 s

(101) plane (101

s (001) cleavage

(001) plane

plane 2c

Figure 7.33 Formation of a crack: (a) by the piling up of dislocations against a grain boundary after strain and (b) by the dislocations on (1 0 1) and (1 0 ¯1) planes running together (after Cottrell, 1958, courtesy of American Institute of Mechanical Engineers).