In this article we will discuss about:- 1. Introduction to Fracture 2. Fundamentals of Fractures 3. Theoretical (Atomic) Fracture Strength of Brittle Material 4. Toughness 5. Prevention.
Introduction to Fracture:
The final stage of the tensile test is the process of fracture. Inspite of understanding of the behaviour of materials, service failures frequently occur. It is almost always an undesirable event. The sudden breakage of an axle may cause an automobile accident.
The sources of the failures include improper design, material selection, material processing and misuse, and the prevention of failures is difficult to guarantee. When the failure does occur, and often with disastrous consequences, the engineer must assess its causes to prevent future accidents. Preventing failures during service is one of the most important challenges facing the engineer.
Fracture is the separation of a part into two, or more pieces under a stress. The stress could be tensile, compressive, shear, or torsional. When the materials fail under cyclic load, it is called fatigue fracture, whereas the materials under service at high temperatures can fail due to creep fracture.
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The study of the modes of fracture is called fractography. It relates the features of fracture surface with processes responsible for stress-induced separation. Although visual examination, light microscopy are commonly used but high resolution and great depth of focus available in electron and scanning microscopes are preferred for fractographic studies.
The science of fracture mechanics now allows the quantitative assessment of growth of cracks in different situations. And thus, it is now frequently possible to predict the stress-level to which materials can be confidently subjected without the risk of catastrophe.
Fundamentals of Fractures:
Any fracture can be considered to be made up of two steps- crack initiation and crack propagation. Fractures in engineering materials can be broadly divided into two classes, based on the ability of the material to undergo plastic deformation as ductile fracture and brittle fracture. The mode of fracture is highly dependent on the mechanism of crack propagation.
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A ductile fracture is characterized by substantial plastic deformation prior to and during propagation of the crack. Large energy is thus absorbed before fracture occurs. The fractured surface shows good amount of plastic deformation. Furthermore, the process proceeds relatively slowly as the crack length is extended.
Such a crack is often called to be ‘stable’, that is, it resists any further extension unless there is an increase in the applied stress. A brittle fracture in metals is characterized by a fast rate of crack propagation with no macro deformation and very little micro deformation. Very low energy absorption accompanies a brittle fracture such as in glass, polystyrene and some of the cast irons.
Such cracks are called to be ‘unstable’, that is, once the crack nucleates, its propagation continues spontaneously without an increase in magnitude of the applied stress. Brittle fracture is akin to cleavage fracture familiar in many minerals and inorganic crystalline solids occurring in a crystallographic fashion along planes of low indices, i.e., high atomic density. At low temperatures, zinc cleaves along the basal-plane, while BCC-Fe cleaves along {100} planes as do all BCC-metals. It is now recognised that the propagation of the fracture is the important stage.
Ductile fracture is almost always preferred over brittle fracture, or rather, brittle fracture is almost always to be avoided at all costs, as the latter occurs suddenly and catastrophically without any warning due to rapidity of crack-propagation. The process of plastic deformation, such as necking of a section, gives enough warning to allow preventive measures to be taken.
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The tendency to brittle fracture increases with the fall of temperature, increasing strain rate and having triaxial stress-conditions (such as by having a notch). The boundary between a ductile and brittle fracture is arbitrary depending on the situation. S.G. iron is ductile as compared to gray iron, but is considered brittle when compared with mild steel.
As more strain energy is required to cause ductile fracture, in-as-much as ductile materials are generally tougher. Most metals, particularly under tensile stress, are ductile, but ceramics are always brittle. Polymers may show brittle as well as ductile fractures.
Theoretical (Atomic) Fracture Strength of Brittle Material:
A truly brittle solid (elastic solid) is put under a tensile force, F as illustrated in Fig. 15.9, where two rows of atoms which face each other across a plane of fracture (pp’) are shown. The equilibrium interatomic spacing of the atoms in the unstrained condition is Fracture is the breaking of bonds between these atoms which face each other.
Fig. 15.10 illustrates the effect of increasing the distance of separation between two neighbouring atoms on the cohesive force between them. As the separation between atoms under tensile force increases, the repulsive force decreases more rapidly than the attractive force.
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A point of separation is reached where the repulsive force is negligible, and the attractive force is still decreasing. This point corresponds to the maximum in the curve, Fig. 15.10. This is also equal to the theoretical cohesive strength of the material, i.e., theoretical fracture strength.
The curve in Fig. 15.10 between cohesive force and the distance of separation between atoms can be represented approximately by a sine curve of the form-
where, σ is the applied stress, σm is the theoretical cohesive strength, i.e., stress at the instant of fracture, x = a – a0 is the displacement in atomic spacing in a crystal lattice with sinusoidal wave length λ. As the elastic displacements are small, i.e., sin x ≈ x, the equation (15.1) becomes
As we are considering a brittle elastic solid, which follows the Hooke’s law.
σ = Ee = Ex / a0 … (15.3)
Combining equations (15.2) and (15.3),
σm = (Eλ) / (2 π a0) … (15.4)
According to Fig. 15.10, it can be assumed that a0 ≈ λ/2, thus,
σm = E / π … (15.5)
This predicts a high value of theoretical cohesive strength.
In an ideal brittle elastic fracture, all the energy or the work done in producing the fracture is absorbed in the creation of two new surfaces, that is, none is absorbed in plastic deformation. If each of these surfaces has a surface energy of ϒ Jm-2, the total energy of these two surfaces = 2 ϒ Jm-2. The work done per unit area of the fracture surface (W), which is expended in bringing the specimen to the point of fracture, is also the area under the cohesive force-displacement curve, which is,
But this energy is used in creating the two new fracture surfaces whose energy is 2 ϒ, i.e.,
(λ σm)/ π = 2 ϒ … (15.7)
Where, E is Young’s modulus, ϒ is the surface energy, (Table 15.2 gives values of surface energies of some materials), is the mean interatomic distance in unstrained state across the fracture plane. If more refined method is used than the above approximate derivation, the value of σm could lie between E/4 to E/15. On an average, it may be taken as, σm = 0.1 E
When the Young’s modulus is 210 GPa for steel, the fracture strength of steel should be 21 GPa, but as is well known that fracture strength of high-strength steels in excess of 2 GPa is exceptional. Real materials always have strength far below the theoretical cohesive strengths.
Table 15.3 compares actual strength of some materials with corresponding Young’s modulus. Engineering materials have been seen to have fracture strength that are 10 to 1000 times lower than the theoretical value except in rare cases in whiskers (defect-free metal crystals of diameter 1 µm) and in silica-fibres. This difference of two orders of magnitude between the theoretical and the observed fracture strength is due to the presence of flaws or cracks in them. Flaws (cracks), pre-existent or produced during service, are decisive in the process of fracture.
Problem 1:
Determine the fracture strength of steel, if E = 210 GPa. ϒ= 1.4 Jm-2 and a0 = 2.481 Å.
Stress Concentration:
In the 1920s, Griffith proposed that the discrepancy between theoretical cohesive strength and the actual fracture strength could be explained by the presence of very small, microscopic flaws or cracks that are always present under normal conditions at the surface and in the interior of the material.
Because the extremities of the crack act as stress raisers, Griffith assumed that the theoretical cohesive strength is achieved at the ends of the crack, even though the average stress was still far below the theoretical cohesive strength. Postponing until later the question of where the cracks come from, these cracks are detriment to the fracture strength because an applied stress is amplified or concentrated at the tip of the crack, the magnitude of this amplification depends on the crack orientation and geometry.
Fig. 15.11 (a) shows a thin elliptical crack in an infinite wide plate, having length of 2 c and a radius of curvature at the lip of crack of ρ. Fig. 15.11 (b) illustrates the stress-profile. The maximum stress, at the tip of the crack is σm, and it diminishes with distance away from the crack-tip to become equal to the nominal stress, σnom at positions far removed. Due to the ability to amplify an applied stress in its vicinity, such a crack is sometimes called stress-raiser. According to Ingles, the maximum stress σm at the tip of the elliptical crack is given by,
where, c is the length of surface crack, or half of length of an internal crack. For relatively long micro-crack having smaller tip-radius, this equation reduces to-
Griffith said that fracture occurs when, upon application of a tensile stress, σnom, amplification causes the theoretical cohesive strength of the material to be exceeded at the tip of the crack. This leads to then quick propagation of the crack. If no flaws are present, the fracture stress would be equal to the cohesive strength of the material as happens in metallic and ceramic whiskers.
The stress concentration also occurs at macroscopic internal defects (holes), at sharp corner and at notches. The effect of defects is more damagingly significant in brittle than in ductile materials as plastic deformation occurs when the maximum amplified stress exceeds the yield strength.
This results in a more uniform distribution of stress in the vicinity of the stress-raiser and to the development of a maximum stress-concentration factor less than the theoretical value. But this does not happen in brittle material. Now if σm of equations 15.9 and 15.11 are equated, then σnom becomes the fracture strength, of the material containing the crack,
This equation is Orowan’s version of Griffith criterion for brittle fracture. The sharpest possible crack can have r = a0, so that equation (15.12) becomes-
Here as the crack length increases, the stress to keep it increasing in length decreases, i.e., once the crack starts moving, it is able to accelerate to high velocities of propagation.
Fracture Toughness:
All engineering materials contain some defects or flaws, which are produced during casting, fabrication or heat treatment stage of the material. When a machine component or a structure has the defects, it is of great importance to know the maximum stress that the component can withstand if it contains flaws of a certain size and geometry, i.e., the fracture resisting capability must be evaluated in presence of the flaws. Fracture mechanics is the discipline that studies the behaviour of materials containing cracks or other flaws, and fracture-toughness is the fracture resistance of a material in the presence of cracks or flaws.
Following the Griffith approach, in 1950s, Irwin incorporated both the terms, ϒ and ϒp in a single term of equation as-
Gc = 2 (ϒ + ϒp) …(15.24)
where Gc defines the fracture toughness and gives the value of strain energy release per unit area of the crack surface when unstable crack extends to cause fracture. Equation (15.24) and (15.19) can be rearranged to provide another expression for Griffith cracking criterion as-
Gc = (π σ2 c) / E …(15.25)
The crack extends when π σ2 c/E exceeds the value of Gc for the particular material under consideration. In a truly brittle material Gc = 2ϒ, but in materials where plastic deformation does occur during the crack initiation. Gc is much larger than 2 ϒ. If the work done for plastic deformation is much larger as given by equation (15.20) than the true surface energy, then G, = 2 ϒp.
Fracture toughness is more commonly expressed by a parameter called critical stress intensity factor, K1C (pronounced as kay-one-cee).
For a sharp crack in an infinitely wide plate (plane strain fracture toughness), when the applied tensile stress is normal to the crack-faces, the K/c is:
Fracture starts in a material as soon as critical value of K1C is reached cither by increasing the stress (σ) or by increasing ‘c’ or both. If the width of the material is finite, or the loading geometry is not the same as illustrated above, a geometrical factor a is added as
K1C is a property of a material and can be estimated from the fracture tests. Table 15.3 gives values for some materials. While using a material for design, if the size of the most damaging crack is known, then the design stress can be calculated from K1C or vice-versa. Fracture toughness of high strength materials decreases as the strength increases. Higher the strength, more is susceptibility to Haws. K1C decreases because of decrease of plastic work done in initiating fracture at the tip of the flaw.
Problem 2:
The yield strength of high-strength steel is 1.46 GNm-2 and K1C of 98 MPa √m. Find out the size of the surface crack that shall cause sudden failure at half its yield strength.
Solution:
Using equation (15.27), and taking α = 1
Problem 3:
A steel has tensile strength of 1.6 GPa. A large tensile piece of such a steel has crack of length 7 mm in the interior and fractures at 0.6 GPa. Calculate its fracture toughness, K1C.
Solution:
Taking this as a plain strain fracture, and α = 1 in equation (15,27). As 2c = 7 mm, thus, c = 3.5 mm.
Some Factors Affecting Ability to Resist Growth of Crack:
1. Increasing the strength of a material reduces its fracture toughness due to its lower ductility.
2. The stress intensity factor is higher in thin sections, hut as the section thickness approaches the size of the cracks, its section size sensitivity increases, making the fracture toughness less predictable.
3. Greater the size of the defect, lesser is the permissible stress. Special methods must be used to control or reduce the defect sizes.
4. As is well known that FCC metal do not fail by brittle nature. Increase in temperature increases the fracture toughness of BCC and HCP metals, while FCC metals remain almost unaffected.
5. The material should undergo plastic deformation so that energy is consumed as the crack-tip deforms to blunt it to further growth. Truly brittle material such as glass docs not undergo plastic deformation and thus crack propagates catastrophically with little consumption of energy.
Prevention of Fracture:
The surface cracks are much more damaging. Glass samples are commonly etched to remove surface cracks and then given a coating of a resin to protect further against scratches by mechanical abrasion. If the surface cracks are going to be present in a material, then process the material in such a way to develop compressive stresses on the surface.
This makes the surface cracks ineffective. A heated silicate glass, to above its softening temperature and after long soaking, is cooled by a blast of cold air. The surface layers cool faster, contract faster to become rigid earlier. As the interior cools later, compressive stresses are introduced in surface layers, while the interior is under tensile stresses, resulting in increase of fracture strength of glass by two to three times.
Sodium silicate glass can be put under surface compressive stresses by another method called ion-exchange method, where smaller cations such as of Na+ in the surface layers are replaced by a larger cation such as of K+ with the help of a chemical reaction there. The crystalline glass can be strengthened by having a fine grain size of about 0.1µm, as these have excellent thermal and mechanical shock resistance.
Moreover, while grinding the brittle ceramic materials, surface cracks get introduced, but these cracks could be of depth of one grain diameter, i.e., fine grain sizes effectively decrease the size of the surface cracks, and thus are required for improved strength of ceramics.
It is not only that a surface crack of half the length of crack in the interior is damagingly effective, but such a surface crack if chemically adsorbs the atmospheric molecules to reduce effectively the surface energy, the critical fracture stress decreases.
Boundary effects of several kinds can affect the fracture behaviour. In steels, intergranular fracture occurs if a continuous brittle phase is present such as iron sulphide. Addition of more manganese can prevent formation of iron sulphide and the manganese sulphide formed is not present at grain boundaries, but as spherical particles within the grains. Temper embrittlement of alloy steels is due to co-segregation of elements such as As, Sb, Bi alongwith Ni, Mn, at the grain boundaries. Addition of 0.5% Mo helps in avoiding such intergranular fracture.
Problem 4:
A glass specimen has a surface crack 2µm deep. Given Young’s modulus = 70 GNm-2, surface energy = 1 Jm-2, radius of curvature at the tip of the crack = 2Å. What is the Griffith stress for this critical length?
Solution:
As ρ, the radius of curvature is less than 3a0, Griffith criterion is to be used for the fracture to occur. Thus,
