Separation of a solid into two or more parts under applied load or stress is called fracture. Depending on the type of load, fracture may be termed as tensile fracture, compressive fracture, shear fracture, fatigue fracture, creep fracture and cleavage fracture etc. However, these fractures are mainly characterized by either a
- Ductile fracture, or
- Brittle fracture.
The process of fracture basically involves crack initiation, and crack propagation.
Ductile fracture occurs after prolonged plastic deformation. The crack initiates from formation of the voids, and propagates slowly.
Brittle fracture occurs after little or no plastic deformation. The propagation of crack is rapid in this case.
Ductile materials fail showing the character of ductile fracture in normal conditions. However they may fail as brittle fracture at much lower temperatures and at higher rates of straining.
Brittle fractures can cause disastrous failure without prior notice, hence their understanding is even more essential.
Ductile Fracture in Metals
In case of ductile fracture in metals , a they absorb large energy prior to failure. Successive stages in the mechanism of ductile fracture are, shown in Figures 1 (a – f), and explained as follows.
Figure 1: Mechanism of ductile fracture shows (a) original specimen, (b) neck formation, (c) crack nucleation, (d) nucleation growth, (e) tiny cracks, and (f) fracture.
- When a specimen is loaded in tension, a neck forms beyond the ultimate strength point. The neck has reduced cross-sectional area but enhanced true stress.
- Brittle and natural particles such as cementite in steel or oxide inclusions in copper are invariably present in the material. The crack nucleates at these brittle particles due to incompatibility with the surrounding region.
- With increasing stress and deformation, the nucleation growth results into cracks of about one mm size.
- The crack proceeds outwards towards the surface in perpendicular direction to the applied stress.
- The crack spreads in a direction 45° to the tensile axis resulting in cup and cone fracture.
Types of Ductile Fracture Ends
Various configurations of fractured ends are noticed when materials fail in ductile fracture. These fractured ends are of the following types, and are shown in Figures 2(a – d). 1. Cup and cone fracture, 2. Fibrous fracture, 3. Star fracture, and 4. Granular fracture.
The cup and cone fracture is common in plain carbon steels. Percentage of carbon in steel has an impact on the profile of cup and cone. Mild steel develops deeper cup and cone, while with increasing percentage of carbon it becomes shallower. It may disappear altogether in high carbon steel.
Wrought iron shows fibrous fracture ends. Brittle cast iron fails with a plane surface having granular appearance.
The failure of a material with rapid rate of crack propagation and negligible plastic deformation is called brittle fracture. Broken pieces of material can be assembled together to get unbroken shape and size.
In ideal brittle materials such as silicate, glass and glassy polymers, the materials break at tensile stress of about E/6 where E is the Young’s modulus.
In crystalline brittle materials, this tensile stress is about E/1000 only. In either case, brittle fracture propagates normal to the applied tensile stress. It often occurs unpredictably as the cracks propagate suddenly.
The reason for such a happening is the pre-existence of extremely small cracks in tile material where stress remains concentrated. Theory of propagation of these cracks has been given by Griffith.
Griffith’s Theory of Fracture
The propagation of a pre-existing crack in a brittle material has been explained by Griffith. He assumes that there are too many fine elliptical cracks in a brittle material. Consider one such crack of 2l length extending through the thickness t of the material as shown in Figure 3.
Length of the crack extends in transverse direction on application of uniaxial tensile stress σ. This causes an increase in the surface area A of the crack and decrease in the elastic strain energy Ue stored in the material. If the elastic surface energy per unit area of the material is γe then the total surface energy Us of the crack will be
Us = γeA
Where A is total surface area of crack ABCDEF of thickness. Considering AB and DE surfaces each of ‘2lt’ and neglecting curved surfaces BCD and EFA, we can write
Us = γe(2l + 2l)t approximately
= 4γelt approximately ………… (equation 1)
This energy is gained due to extension of cracks, hence is taken positive. Loss in the elastic strain energy Ue is taken as negative and is given as
Ue = – (σ2 ÷ 2E)V = – (σ2 ÷ 2E)( πl2 + πl2)t
= –(πσ2l2t ÷ E) approximately ………… (equation 2)
Where E is Young’s modulus and V is the volume of the material.
We can see from Figure 3 that the spread of crack creats two surfaces. The energy change U is thus obtained as
U = Us + Ue = 4γelt – (πσ2l2t ÷ E) approximately………… (equation 3)
Equation 3 is known as Griffith’s energy balance criterion for crack propagation. According to Griffith, such a crack will propagate when an incremental increase in its length does not change the net energy of the system. So
This shows that the stress required to cause brittle fracture varies inversely as square root of half crack length. Griffith further says that a crack may not propagate at all if sufficient stress concentration at the crack tip does not exist.
Infact, atomic bonds at the crack tip break during crack propagation. On the tip of such a crack, there always remains a strong concentration of stresses.
Effect of this stress concentration is to develop a higher value of stress than the applied stress. In a schematic crack tip shown in Figure 4, the effect of applied stress a is to produce
σm = 2σ √(l/r) ……….(equation 6)
Where σm is maximum stress and r the radius of curvature at the crack tip. Ratio of maximum stress induced over applied stress called stress concentration factor Kt. Thus
Kt = σm/σ = 2√(l/r) ……….(equation 7)
Salient Features of Griffith Theory
- Some of the salient observations of Griffith’s theory are as follows Surface cracks on materials are twice as effective as the internal cracks. Hence surfaces of machine components should be finely finished.
- A material fractures at a certain value of applied stress. One of its broken pieces will require more applied stress than the previously applied stress to fracture. It is because the most effective cracks are eliminated in the first fracture.
- A scratch on the surface is of the order of r = 0.1 oA or less in equation 6. Assuming l = 10-6 m, the σm approximately will be equal to 630 σ. Thus from equation 7 the stress concentration factor Kt = σm/σ = 630 is too high and will fracture the material even under a low applied stress.
- During tearing of a piece of paper, we fold it and press. Folding and pressing results in creating a scratch that induces higher stress concentration. The paper is then teared apart very easily along the scratch.
- A window glass panel is scratched by a mechanical nail to introduce stress concentration. Then the glass breaks very easily in two pieces along the scratch.
- Fracture resistance of a material increases by making blunt cracks, fillets or notches of some radius. In doing so the stress concentration at the tip is relaxed.
Example: A glass piece has a crack length of 3 µm. Its Young’s modulus is 70 GPa. If the specific surface energy is 1.05 J/m2, estimate its fracture strength. Also find the ratio of this strength to Young’s modulus and explain the cause of difference from the theoretical estimates.
Solution: The given values are
l = 3/2 µm = 1.5 x 10-6 m, E = 70 x 109 N/m2, γe = 1.05 J/m2
After substituting these values in equation 5 and solving, we obtain
σf = 176.45 Mpa
The ratio of σl/E = 176.45/70 x 10-3 = 2.52 x 10-3
The difference in the ratio of actual and theoretical values of E/σf is large, and it confirms the Griffith’s criterion that the observed values are much less than the ideal values.
Methods of Protection Against Fracture
Fracture of materials can be delayed by various protective methods given as follows.
Surface treatment involves the processes of etching and sizing etc. Etching of glass in hydrofluoric acid removes the surface cracks and thus improves the strength of glass. Glass fibers are sized by use of starch to help protection against abrasion among themselves.
Introduction of compressive stress on the surface of a material makes the surface cracks ineffective. Plexiglass windscreen of auto-vehicles and tempered glasses require a higher tensile stress to initiate crack propagation as the tensile stress has to overcome the introduced compressive stress.
Tempering process of heat treatment introduces compressive stress in the interior part of the silicate glass, and thus improves the fracture strength by many times.
Chemical strengthening involves ion exchange method by which sodium cations are replaced by potassium anions on the surface of sodium silicate glass.
Fine grain control is made to obtain finer grain sizes in glass and ceramics. The surface cracks are minimized due to fine grains and hence the fracture strength improves.
Ductile-brittle transition is a limiting (demarcating) state between ductile and brittle behavior of a material. This transition between ductile and brittle behavior is demarcated by a temperature, called ductile-brittle transition temperature tdb.
A ductile material shows brittle nature below this temperature whereas a brittle material exhibits ductile nature above this temperature. A ductile metal may behave as brittle metal and tendency to brittle fracture is increased with certain conditions. These conditions are:
- a low or decreasing temperature,
- high rate of straining,
- large grain size of material,
- high stress concentration,
- rough surface conditions, and
- triaxial stress conditions.
Transition temperature depends on various factors. Recall Hall-Petch, according to which yield strength increases with decreasing grain size. Thus, fine grained materials possess lower transition temperature than the coarse grained materials.
Transition temperature is raised due to stress concentration such as on sharp notches. Effect of higher straining rate is to cause increased transition temperature.
Most of the ductile BCC metals behave as brittle materials at low temperatures and at a very high rate of straining, whereas many FCC metals behave as ductile materials at very low temperatures.
It is because a higher yield stress σy is required to move dislocations in BCC metals than FCC metals. This increases rapidly when temperature lowers down but this is not the case with the stress required to propagate a crack σf.
Therefore the material is brittle below tdb and ductile above it. The following relations exist at tdb and at temperature t above or below it.
σy = σf at t ≤ tdb
σy = σf at t > tdb
The effect of ductile-brittle transition may be seen on impact energy (or fracture energy) of materials in Figure 5. This energy is substantially lower below tdb where material exhibits brittle nature.
For wrought iron, tdb is about —75°C and it increases with increasing percentage of carbon in different steels.
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