About fracture mechanics

Template:Cleanup-date Fracture mechanics is a method for predicting failure of a structure containing a crack. It uses methods of analytical Solid Mechanics to calculate the driving force on a crack and those of experimental Solid Mechanics to characterize the material's resistance to fracture.

In modern Materials Science, Fracture Mechanics is an important tool in improving the mechanical performance of materials and components. It applies the physics of stress and strain, in particular the theories of elasticity and plasticity, to the microscopic crystallographic defects found in real materials in order to predict the macroscopic mechanical failure of bodies.

An excellent introduction to Fracture Mechanics is: Adrian Demaid Fail Safe Open University (2004)

Fracture Mechanics

Fracture Mechanics was invented during World War I by English aeronautical engineer, A.A.Griffith, to explain the failure of brittle materials. Griffith was faced with the problem that theoretical calculations showed that the stress at the tip of a sharp crack approaches infinity. Accordingly, any structure containing a crack should fail, no matter how small the crack or how light the load. To solve this dilemma, Griffith developed a thermodynamic approach. He assumed that growth of a crack requires creation of surface energy, which is supplied by the loss of strain energy accompanying the relaxation of local stresses as the crack advances. Failure occurs when the loss of strain energy is sufficient to provide the increase in surface energy. For the simple case of a rectangular plate with a crack perpendicular to the load Griffith’s theory becomes:

   G = σ2πa/E = Gc            (1)

where G is the strain energy release rate, σ the applied stress, a half the crack length, E Young’s modulus, and G_{c the surface energy.}

Griffith’s work was ignored for over twenty years until a group under G. R. Irwin at the U.S. Naval Research Laboratory took it up during World War II. Their problem was that naval materials, e.g. ship-plate steel, are not perfectly elastic but undergo plastic deformation at the tip of a crack. Eventually a modification of Griffith’s theory emerged from this work; a term called stress intensity replaced strain energy release rate and a term called fracture toughness replaced surface energy. Both of these terms are simply related to the energy terms that Griffith used:

   σ(πa)0.5 = K         (2)

and:

   [EGc/(1 - ν2)]0.5 = Kc    (3)

where K is the stress intensity, K_{c the fracture toughness, and ν is Poisson’s ratio. Fracture occurs when K = Kc. Note that the expression for K in Eq. (2) will be different for geometries other than the center cracked plate, as discussed in the article on Stress Intensity. For the special case of plane strain deformation, Kc becomes KIc and is considered a material property.}

Arising from the manufacturing process, interior and surface flaws are found in all metal structures. Not all such flaws are unstable under service conditions. Fracture mechanics is the analysis of flaws to discover those that are safe (that is, do not grow) and those that are liable to propagate as cracks and so cause failure of the flawed structure.

Internal and external flaws act as stress raisers, raising the effective stress normal to the crack plane. More important for the purpose of predicting instability, we can estimate the elastic stress field in the neighbourhood of a crack tip and thereby determine the elastic strain energy that would be released if the crack were to grow. In order for the crack to propagate spontaneously this energy must exceed the surface energy of the (two) new crack surfaces. When there is plastic deformation at the crack tip (as occurs most often in metals) the energy to propagate the crack may increase by several orders of magnitude. This total energy release rate is given the symbol G_Ic, where the Roman number I (II or III) indicates the way the crack opens (e.g. whether there is a twisting component or not) and c denotes criticality — the crack will only propagate when G_Ic exceeds some critical value. In short, when

elastic energy released = surface energy created

For ductile metals G_Ic is around 50 to 200 kJ/m², for brittle metals it is usually 1-5 and for glasses and brittle polymers it is almost always less than 0.5.

More significant adjustments to this kind of calculation take account of the shape of the flaw and its relationship to the shape and size of the structure. This has led to the concept of a stress intensity factor. Analogously to G_Ic, at criticality this is given the symbol K_Ic. Typical values are 150 MN/m^3/2 for ductile (very tough) metals, 25 for brittle ones and 1-10 for glasses and brittle polymers. Notice the different units used by G_Ic and K_Ic. Engineers tend to use the latter as an indication of toughness. Where there is extensive plastic deformation this treatment has to be modified because the plastic zone at the tip of the crack consumes the major part of the work of fracture and modifies the elastic strain field in the crack tip region:

elastic energy released = surface energy + plastic deformation energy

Conjoint Action

There are number of instances where this picture of a critical crack is modified by corrosion. Thus, fretting corrosion occurs when a corrosive medium is present at the interface between two rubbing surfaces. Fretting (in the absence of corrosion) results from the disruption of very small areas that bond and break as the surfaces undergo friction, often under vibrating conditions. The bonding contact areas deform under the localised pressure and the two surfaces gradually wear away. Fracture mechanics dictates that each minute localised fracture has to satisfy the general rule that the elastic energy released as the bond fractures has to exceed the work done in plastically deforming it and in creating the (very tiny) fracture surfaces. This process is enhanced when corrosion is present, not least because the corrosion products act as an abrasive between the rubbing surfaces.

Fatigue is another instance where cyclical stressing, this time of a bulk lump of metal, causes small flaws to develop. Ultimately one such flaw exceeds the critical condition and fracture propagates across the whole structure. The 'fatigue life' of a component is the time it takes for criticality to be reached, for a given regime of cyclical stress. Corrosion fatigue is what happens when a cyclically stressed structure is subjected to a corrosive environment at the same time. This not only serves to initiate surface cracks but (see below) actually modifies the crack growth process. As a result the fatigue life is shortened, often considerably.

Stress-Corrosion Cracking (SCC)

Stress corrosion cracking

This phenomenon is the unexpected sudden failure of normally ductile metals subjected to a constant tensile stress in a corrosive environment. Certain austenitic stainless steels and aluminium alloys crack in the presence of chlorides, mild steel cracks in the present of alkali (boiler cracking) and copper alloys crack in ammoniacal solutions (season cracking). Worse still, high-tensile structural steels crack in an unexpectedly brittle manner in a whole variety of aqueous environments, especially chloride. With the possible exception of the latter, which is a special example of hydrogen cracking, all the others display the phenomenon of subcritical crack growth, i.e. small surface flaws propagate (usually smoothly) under conditions where fracture mechanics predicts that failure should not occur. That is, in the presence of a corrodent, cracks develop and propagate well below K_Ic. In fact, the subcritical value of the stress intensity, designated as K_Iscc, may be less than 1% of K_Ic, as the following table shows:

    Alloy	          KIc     SCC environment	 KIscc
                     MN/m3/2                      MN/m3/2
  ------------------------------------------------------

   13Cr steel	      60     3% NaCl	       12
   18Cr-8Ni	        200	42% MgCl2         10
   Cu-30Zn	        200	NH4OH, pH7	      1
   Al-3Mg-7Zn	         25	Aqueous halides	   5
   Ti-6Al-1V	         60	0.6M KCl	  20

The subcritical nature of propagation may be attributed to the chemical energy released as the crack propagates. That is,

elastic energy released + chemical energy = surface energy + deformation energy

The crack initiates at K_Iscc and thereafter propagates at a rate governed by the slowest process, which most of the time is the rate at which corrosive ions can diffuse to the crack tip. As the crack advances so K rises (because crack length appears in the calculation of stress intensity). Finally it reaches K_Ic , whereupon fast fracture ensues and the component fails. One of the practical difficulties with SCC is its unexpected nature. Stainless steels, for example, are employed because under most conditions they are 'passive', i.e. effectively inert. Very often one finds a single crack has propagated while the rest of the metal surface stays apparently unaffected.

References

1. Knott, Fundamentals of Fracture Mechanics (1973).
2. Foroulis (ed.), Environmentally-Sensitive Fracture of Engineering Materials (1979).
3. West JM, Basic Corrosion & Oxidation (Horwood 1986, 2nd edn), chap.12.