Griffith attributed the discrepancy between the observed fracture strength of crystals and the theoretical cohesive strength to the presences of flaws in brittle materials. This theory is applicable only to perfectly brittle material such as glass, and cannot be used directly to metals. However, Griffith’s ideas formed a base to understand the fracture in metals.

Griffith proposed that a brittle material contains a large number of fine cracks. He postulated a criterion for the propagation of such a crack in a brittle material. During propagation, there is a release of what is called the elastic strain energy, some of the energy that is stored in the material as it is elastically deformed.

Further-more, during crack propagation, new free surfaces are created as the faces of a crack. This requires energy to overcome the cohesive force of the atoms, that is, it requires an increase in surface energy. Griffith developed a criterion for crack propagation by performing an energy balance using these two energies as- a crack propagates when the decrease in elastic strain energy is atleast equal to the energy required to create the new crack surface. The thermodynamic relationship between these two energies determines the magnitude of the tensile stress needed to propagate a crack of a certain size to produce a brittle fracture.

Consider a wide sheet of unit thickness of a brittle material. It has a lens-shaped crack of 2 c length which runs from the front to the back face as illustrated in Fig. 15.12. When a longitudinal tensile stress, r is applied, the crack tends to increase its length in the transverse direction (i.e., in a perpendicular direction to the tensile stress).

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When the crack spreads, the surface area of the crack increases, but the elastic train energy stored in the thin sheet decreases because the elastic strains are not continuous across the cracked region. The elastic strain energy released per unit of plate thickness is given by-

where, E is Young’s modulus. The negative sign indicates the release of elastic strain energy as the crack grows. As the crack propagates, two surfaces are created. If γ is the surface energy per unit area of the material and as 2 c is the crack length, the surface energy of the crack is

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U_{s} = 4 c γ … (15.15)

Now, it is possible to write an energy equation as the crack forms. The change in energy, ∆U as the crack propagates is given by-

Griffith postulated that the crack propagates under a constant applied stress, σ, if an incremental increase in crack length produces no change in the total energy of the system, that is, the Griffith condition for fracture is obtained if the rate at which strain energy is released balances the rate at which energy is required to create the new surfaces, i.e., the critical value is obtained by setting-

As the applied tensile stress is the external variable for a given material having a crack of length 2c, it is more appropriate to express the critical condition as a critical fracture stress, σ_{c}. Here, σ_{c} is the critical value of stress required for the propagation of the flaw of length 2c. This is the Griffith equation, a corner stone of modern fracture theory.

It is a go/no-go condition; that is, the flaw does not propagate until the critical value of stress is reached. Once the critical stress is applied to a brittle material, the pre-existing crack propagates spontaneously with a decrease in energy, or rather its rate of growth accelerates, since the strain-energy released as it lengthens is increasingly in excess of that required for the creation of new surfaces. This acceleration continues until the crack acquires a terminal velocity of the order of one-half the velocity of the longitudinal sound wave in the material.

The Griffith equation (15.17) produces an important result as it gives the value of critical stress required to propagate a crack in a brittle material as a function of the size of the micro-crack. It indicates that the fracture stress is inversely proportional to the square root of the crack length. For example, if the crack length increases by a factor of 9 decreases the fracture stress by one-third.

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**Problem 1: **

Two samples A and B of a brittle material have crack length in the ratio 3:1. The ratio of tensile strengths (measured normal to the cracks) of A and B will be- (i) 1:3; (ii) √3: 1; (iii) 1:√3; (iv) 1:9

**Solution: **

As the fracture stress is inversely proportional to the square root of the crack length,

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σ_{A} / σ_{B }= 1 / √3

i.e., answer (iii) is correct.

**Problem 2: **

A sodium silicate glass has no surface defects as etching has removed them, but has cracks inside from 2 µm to 5 µm in length. Calculate the surface energy of glass if given. Fracture strength: 100 MNm^{-2}; Young’s modulus = 70 GNm^{-2}.

**Solution: **

As the longest crack is most damaging, thus, 2 c = 5 µ m

**Using Griffith’s equation: **

The Fig. 15.12 also illustrates a surface crack of length c. The Griffith equation (15.17) is also valid for such a crack, that is, a surface crack of depth c is as effective as an internal crack of length 2 c, because the rate of change of energy in both cases is same. It is an important result that surface defects in a material are more damaging than internal cracks.

Engineering materials have statistical distribution of cracks in them. The longest crack which is most favourably oriented (with respect to the stress-axis) is the first one to propagate to cause the fracture as the applied stress is increased.

**Problem 3: **

(a) Show with the help of Griffith equation low the longest of the largest crack in a brittle material effects the fracture strength?

(b) If a material has microscopic crack of 2µm in length, what is the fracture strength as compared to theoretical cohesive strength?

**Solution: **

(a) Equation 15.9 and 15.17 are used

If a_{0} ≈ 2 c, σ_{c} = σ_{m}, i.e., if crack length is of atomic dimension, then the real fracture strength is equal to the theoretical cohesive strength, otherwise, it decreases.

i.e., fracture strength is reduced by a factor of 100.

We have obtained two equations for fracture stress based on stress-concentration factor (15.12) and Griffith equation (15.17). Equation (15.12) is rewritten as-

and for r = 3 a_{0}, this equation becomes Griffith equation. As the fracture stress, σ_{f} cannot become zero for r approaching to become zero, the stress to produce brittle fracture is obtained from equation (15.17) for r < 3 a_{0}. For r > 3 a_{0}, the fracture stress is obtained by using equation (15.12).

In brittle materials such as silicate glass, no plastic deformation occurs. The stress concentration at the tip of the crack is not relaxed due to plastic deformation, i.e., the crack-tip remains sharp. The surface energy term, y, in the Griffith equation truly represents the surface energy of crack faces as no other additional work such as plastic deformation takes place during the propagation of the crack.

Griffith theory satisfactorily predicts the fracture strength of such a completely brittle material. In glass, reasonable values of crack length of about 1 µm have been calculated from equation (15.17).

Fracture of brittle materials can occur under compressive loading also, but compressive strength of such materials is usually an order of magnitude greater than tensile strength, and separation occurs by shearing process. Concrete and cast irons are similar examples.

Metals which have fractured in a completely brittle manner, and where the broken pieces can be fitted together to get the original shape and dimensions of the specimen, have been seen on a microscopic scale, to have undergone some plastic deformation, at least in a few grains in the vicinity of the crack-tip where the stress is concentrated before fracture.

Thus, the Griffith’s equation for fracture stress does not apply to metals. The plastic deformation effects in two ways. It increases the work that is necessary to propagate a crack as the work now includes not only the surface energy of the crack-faces that are created but also the energy consumed in plastically deforming the matrix in the vicinity of the crack-tip.

The second effect is to relax the stress concentration at the tip to blunt the crack. The plastic deformation at the root of the crack would also blunt the tip of the crack and increase r, the radius of the crack, which thus increases the fracture stress.

Orowan modified the Griffith equation to make it useful for brittle fractures in metals by the inclusion of a term γ_{p} expressing the plastic work necessary to extend the crack wall as localised plastic deformation occurs during fracture. The x-ray studies have confirmed the presence of heavily distorted layer immediately beneath the fracture surface. The modified form of equation is-

**Since the work of plastic deformation, ****γ**_{p}** (about 10 ^{2} – 10^{3} Jm^{-2}) is much larger than **

**γ**

**(about 1-2 Jm**

^{-2}), the equation (15.19) can be written as:as the work, γ, can be neglected. This modification is important because γ_{p} can always be increased by proper changes in the microstructure of the material. If the value of γ_{p} is increased, a higher value of stress can be applied with cracks of a given size, or flaws of a larger size can be tolerated for the same applied stress.

**Crack Velocity in Brittle Materials****: **

More and more strain energy is released as the crack length (2c) increases, though a part of this energy is used in forming the surfaces of the cracks. The remaining energy is changed into the kinetic energy. As a crack propagates, the material at the sides of the crack moves apart with a finite velocity. Kinetic energy is associated with this movement of the material near the end of the crack.

**The energy balance can be written as- **

The solution of this equation has been found to be-

where, ν_{c} is the velocity of the crack, v is the longitudinal velocity of sound in the material, C_{0} is the Griffith’s critical half crack length at any instant, k is a dimensionless constant having a value of 0.4-0.5. As the size of the crack becomes more than half critical crack length, the crack gains speed, and the velocity approaches a maximum value (0.5 ν), when the crack length becomes very large. A crack can move in a brittle material at 0:38 the velocity of sound in that material.

**Initiation of Brittle Crack****: **

Inorganic glasses contain inherently some bulk flaws, but the cracks which lower the strengths of the glasses probably are on the surface such as fine scratches, which may be introduced in the manufacturing process. Moreover, a surface crack of depth c is equivalent to an interior crack of length 2c.

The initiation of fracture in polymers too is associated with prior presence of the flaws. Griffith theory has been verified by experiments on glasses and polymers at low temperatures, where a simple process of fracture by the propagation of elastic cracks occurs (i.e., no plastic deformation occurs). Most industrial ceramics contain porosity and flaws of varying sizes, and thus the strength varies from specimen to specimen.

There is no reliable evidence to prove that Griffith cracks exist in metals in the unstressed condition. However, even when a metal fails by brittle cleavage, a certain amount of plastic deformation almost always occurs prior to fracture. Enough experimental evidences are available to prove that micro-cracks can be produced by plastic deformation.

Micro-cracks can form at non-metallic inclusions or brittle second phase in steels by decohesion from matrix, or by cracking of the particle as a result of plastic deformation as illustrated in Fig. 15.4, but such a micro-crack need not necessarily produce brittle fracture. The experiments have shown that cracks responsible for brittle-cleavage type fracture are produced by plastic deformation.

**There are three stages in the process of brittle fracture: **

1. Plastic deformation to produce pile-up of dislocations at an obstacle. Such an obstacle could be an inclusion, or second phase particle or even grain boundary or twin interface or at the junction of two intersecting slip planes. Because slip is often observed even when the specimen fractures in a brittle manner, it is clear that slip dislocation must play some part in the fracture process.

2. Micro-crack-initiation occurs due to buildup of shear stress at the head of the dislocation pile-up. In steels, the fracture of the carbide particle by the stress field of a pile up is an essential intermediate event between the formation of dislocation pile-up and cleavage of the ferrite as illustrated in Smith’s model of micro-crack formation in grain boundary carbide plate as illustrated in Fig. 15.13.

The formation of a crack in a carbide plate can initiate cleavage fracture in adjacent ferrite if stress concentration is relatively high. It has been seen if the dispersed second-phase particle is readily cut by the dislocations, then at some obstacle large pile-up of dislocations occurs. It leads to the development of high stresses, easy initiation of micro-cracks, and the brittle fracture.

In crystalline materials, as some plastic deformation always occurs, then it is the slip dislocations generated immediately prior to fracture which give rise in certain regions of the crystal to micro-cracks, even without the presence of brittle particles. Several models have been put formed for the process whereby slip dislocations are converted into micro-cracks. Zener was the first who advanced the idea that the high stresses produced at the head of a dislocation pile-up could produce the fracture as illustrated in Fig. 15.14.

A second mechanism of crack formation, suggested by Cottrell is the one arising at the junction of two intersecting slip planes as illustrated in Fig. 15.15. This mechanism is able to account for cleavage on {100} cube planes as result of slip on {110} <111> slip systems. Here two slip dislocations combine to form an edge dislocation. A slip dislocation gliding in the (101) plane coalesce with another slip dislocation gliding in (101̅) plane to form a crack dislocation which lies in the (001) plane according to the reaction-

1/2 [1̅ 1̅ 1] + ½ [1 1 1] → [001]

The new dislocation having a Burgers vector of a [001], is a pure edge dislocation. This dislocation may be considered as a wedge. A crack opens in the (001) plane as more such dislocations coalesce. Although this mechanism readily accounts for the observed (100) cleavage plane of BCC metals, but has not been directly observed in BCC metals.

The crack nucleation at the intersection of slip bands similar to Cottrell’s mechanism has been observed in a ceramic material, magnesium oxide. Even intersecting deformation twin bands have been seen to create cracks.

**Factors Effecting the Initiation of Cleavage Fracture****: **

**1. Temperature Dependence of Yield Stress: **

All BCC-metals show increase of yield stress with the fall of temperature, that is, the stress to move the dislocation, the Peierls-Nabarro stress increases with the fall of temperature. But as the velocity of dislocation-motion is proportional to the stress, the first dislocations move very rapidly with the fall of temperature. Thus, the dislocation squeezing as illustrated in Fig. 15.14 and 15.15 can occur to initiate a crack.

**2. Grain Size: **

Smaller the grain size, smaller is the number of dislocations in the pile-ups at grain boundaries, and smaller is the shear stress developed at the head of the pile-up. Thus, the local stress-concentration at the grain boundaries becomes less resulting in less crack-nuclei. Even the ductile-brittle transition temperature is lowered.

**3. Effect of Sharp Yield Point: **

Whenever it occurs, a very rapid localised movement of dislocations occurs due to high stresses. This can nucleate cracks by coalescence of dislocations.

**4. Nucleation of Micro-Cracks at Twins and Carbide Particles: **

As the temperature decreases, twinning becomes the main deformation mode in BCC metals including iron. Cracks nucleate preferentially at twin intersections, at points where twins contact grain boundaries. Cleavage of twins occurs more commonly in materials of larger grain size.

Moreover, the lowering of temperature makes plastic deformation more difficult at the tip of the moving crack. As less plastic blunting of crack-tip occurs, the propagation of crack becomes easier. When a second phase such as cementite or brittle inclusions such as alumina or silicate particles are present, it is still easier to nucleate cracks as these particles can crack to produce micro-cracks which in certain circumstances propagate to cause catastrophic cleavage fracture.

Nucleation of a cleavage crack occurs when a critical value of effective shear stress is attained due to a critical grouping (ideally a pile-up) of dislocations which can create a crack-nucleus, i.e., by fracturing a carbide particle. Whereas, the propagation of a cleavage crack depends on the magnitude of the local tensile stress, which must reach a critical value.

This critical stress – is not temperature-dependent. At low temperatures, the yield stress is higher so that the crack propagates when the plastic zone ahead of the crack is small, whereas at higher temperatures, the yield stress is smaller, a larger plastic zone is required to achieve the critical local tensile stress.

**Propagation of Cleavage Cracks****: **

In a truly elastic (brittle) solid, the crack propagates releasing the elastic strain energy which is used in creating the fracture surfaces and for the movement of material at the sides of the crack. But as localised plastic deformation accompanies the motion of cracks in crystalline solids as x-ray studies confirm the presence of a heavily distorted layer immediately beneath the fracture surface, a part of the released strain energy is used up in it. The equation (15.19) is the modified form for fracture stress which includes the energy used in plastic deformation.

Lowering of the temperature and use of high strain rates try to reduce the amount of this energy. Plastic deformation occurs just ahead of a moving crack. The metal in this region is under large shear stresses to nucleate dislocations on favourably oriented slip planes in advance of the crack as seen in single crystal of LiF.

There is a critical velocity of crack below which dislocations are nucleated and not above that. The maximum velocity of crack propagation in LiF has been measured as 0.31 the velocity of sound, which compares well as in equation (15.22).

When the plastic deformation occurs during the crack propagation, energy is used in nucleating and moving dislocations. This energy is supplied by the elastic strain energy. If a larger amount of energy is used in plastic deformation, the crack may actually decelerate and stop. There is thus a minimum critical velocity which must be attained before a crack can move freely.

Below this velocity, work done in plastic deformation may be so large that some metals may not cleave. As in well-known that FCC metals do not show brittle cleavage. It is because metals require less energy to fracture by a mechanism involving shear than by cleavage.

The stress to cause fracture by shear in FCC metals is only about 5 percent of the stress needed to cause cleavage. This value is about 25 percent in BCC metals. If work done for plastic deformation is increased by suitable changes in the microstructure of the material then a higher value of stress can be applied with flaws of a given size, or flaws of larger size can be tolerated at a given applied stress.

A crack grows by the plastic deformation i.e. the dislocation source continues to force the dislocations into the pile-up. As only shearing stresses cause the dislocation to move, tensile stresses are not involved in the micro-crack nucleation process, but are needed to make the micro-crack propagate. Thus, crack- propagation stage is ordinarily more difficult than crack initiation in metals.

Cleavage cracks are more difficult to propagate through a polycrystalline material than through a single crystal, as it is difficult for the crack to pass through a large angle grain boundary. Actually, the grain size limits the initial size to which a cleavage crack can grow, i.e., crack can grow too equal to the size of the grain diameter.

At sufficiently high stress, the crack can spread catastrophically i.e., fail by brittle nature. If the average micro-crack has the length equal to the diameter of the grain, then the Griffith criterion for crack propagation as given by equation (15.17) can be used to an approximation by putting 2c = D (grain diameter), ϒ = ϒ_{p}, where ϒ_{p} term includes true surface energy and energy of plastic deformation forcing the crack through a polycrystalline solid as,

The strength of the polycrystalline metals, which undergo brittle cleavage, varies as the reciprocal of the square-root of average grain diameter. Fig. 15.16 illustrates the effect of grain size on fracture and yield strength of low carbon steels tested at -195°C. In the range of small grain size, the steel is fairly ductile as there are two separate curves for yield stress and fracture stress.

As the grain size becomes larger, a critical diameter is reached above which the stress needed to expand a micro-crack becomes smaller than the stress to form, or nucleate a crack inside a crystal.

At the critical size, as the first crack forms, it causes the brittle fracture, i.e., as soon as the crack assumes the size equal to the grain diameter, it becomes large enough to pass through the boundary and continues to grow.

Above the critical diameter, fracture stress is controlled by the stress required to nucleate the cracks in the solid, and as the cleavage cracks are not nucleated until the yield point is reached, the fracture should occur almost simultaneously with yielding. But when the grain size is smaller, fracture stress is controlled by the stress required to propagate the crack through the polycrystalline solid. The curve in Fig. 15.16 has been divided into two regions as nucleation limited and as propagation limited.