Strain Energy Release Rate (G)

 

Theoretical Stress Approach

    Atomic Physicists have allowed us to understand that two atoms (or ions) can attract or repel each other whilst remaining as two separate entities. Their attractive forces remain at short and long ranges but repulsive forces are only significant at short range. We could hence measure the force between two atoms as the distance between them was varied.

    Results have shown a net attractive force at large atomic distances and a net repulsive force at small distances, with no force at some equilibrium distance r₀. When we consider the potential energy in the bond:

 

    Since the aim is to find out the magnitude of the force and stress required to cause a simple crystalline solid to fracture across an arbitrary plane, we need to consider each atomic bond as a spring element initially at a separation of r₀ in order to carry out the calculation. Consider the simplest case when we try to pull apart 2 atoms so that the inter-atomic spacing increases from r₀ to r₀+δr:

 

    Assuming that if the bond is stretched past the maximum force F_max, the bond will become unstable and eventually break if the strain continues to increase. An estimation of the tensile strength could hence be obtained by calculating F_max. Similarly, we can use the same idea but converting the force now to a stress (force per unit area) so that the value can be used in more general situations. The value of stress can be arrived by realising that each atom effectively occupies an area of r₀², the stress is then given by F/r₀². By computing the stress-strain relationship, one would see:

 

    which is given by the equation:

 

    where λ is the wavelength such that σ = σ_max at x=λ/4.

    The shaded area of the stress-strain curve is the work of fracture - the energy required to separate two atoms from their equilibrium position to infinity which is equivalent to the enery required to create two new surfaces (of the crack).  By defining the surface energy of the free surfaces created by fracture to be γ, the shaded area must then be equivalent to 2γ.

    By realising that the work fracture is also equivalent to the minimum energy of the U-r curve at r = r₀, one can equate that:

 

    which yields the theoretical fracture stress (σ_max). However, in reality this approach is somehow inaccurate when the obtained values are compared to those in practice.

 

Energy Balance Approach (Griffith Cracks)

    According to a solution due to Inglis, the stress at the ends of the cracks is given by:

 

    As the crack tip radius becomes infinitely sharp, it can be assumed that ρ -> the lattice spacing r₀, hence:

 

    Assuming that fracture occurs when σ = σ_max:

 

    where in terms of the fracture stress σ_F:

 

    However, this approach has serious limitations as it assumes too much by taking a totally elastic solution to predict a stress which in reality the solution is far from pure elastic.

    Griffith had then overcame the objections by modifying the approach in terms of thermodynamic in order to avoid having to consider the processes of the crack tip.

    Consider the Load-Displacement graph below for an infinite plate containing a central, through thickness crack with crack length 2a and is subjected to a remotely applied uniform tensile stress σ:

 

    It can be seen that under a fixed grip conditions, the extension of the crack from 'a' to 'a+da' results in the release of elastic strain energy from the plate equivalent to 1/2 (P₁-P₂)u₁. Intuitively, it would be reasonable to make the assupmption that this energy is consumed in the work of fracture and hence the energy required to create the two new crack surfaces. The same principles can be applied to the fixed load condition to give 1/2 P₁(u₂-u₁), thus by letting u₂-u₁=δu and P₁-P₂=δP:

• Fixed grip = -1/2 δPu (Strain energy release)
• Fixed load = -1/2 Pδu (Potential energy release)

    where the displacement, u and the load, P are related via a simple linear equation

u = CP

    where C is the compliance of the system which has the inverse units to stiffness

    By rewriting the linear equation when the values are tend to zero:

δu = CδP

    and by substituting into the previous equations under fixed grip/ load conditions:

•  -1/2 δPu =-1/2 CPδP
•  -1/2 Pδu =  -1/2 CPδP

    which means that there is no difference in the energy released when an infinitesimally small increment of crack growth occurs between the two different loading conditions.

    Then here is where Griffith's theory came into play. His thermodynamic description of the fracture process made an important connection in recognising that the driving force for crack extension is the energy that this is used up as the energy required to create the two new surfaces created by the fracture. By exploiting his theory, he had arrived at the following relations for the change in energy of a body with crack length:

 

    These equations have now allowed us to relate energy, applied stress and the crack length together. The task now is to create a method that allows us to predict the onset of fracture as a function of crack length and applied stress.

    Consider a body containing a crack of length 'a' subjected to an external stress σ. The total energy of the system is:

 

    Thus the instability condition for crack growth can be expressed as:

 

    by re-arranging the above equation in terms of stress σ, where  σ = σ_F:

 

    In reality, these relations are only the basis for further extrapolation since these two equations are only valid for truly brittle materials like glass, but not for quasi-brittle materials such as metals. Although they are not applicable to fracture conditions in the more ductile materials, the inherent inverse square relationship between 'a' and 'σ_F' is found to be applicable.

    The development of Fracture Mechanics has therefore taken the fundamental thermodynamic argument of Griffith and brought further by modifying the definition of the way that the energy in the system is calculated.

 

Computing the Strain Energy Release Rate (G)

    So afterall how can we determine G either analytically or by experiment such that a value can be input to predict the critical crack length with a known applied load?

    Recall the diagram:

 

     Since the strain or potential energy release for an increment of crack growth δa is given by Gδa per unit thickness, by defining B as the thickness of the plate:

 

    However, this can not be easily measured experimentally. By letting δa => 0 so invoking the compliance relationship between the load and displacement u=CP, the relationship can be re-arranged to:

 

    and as δa => 0:

 

    which holds for both fixed grip and constant load conditions.

    Hence,  having successfully determined the compliance function for an object with a particular geometry (B) experimentally, the energy release rate (G) can be derived as a function of crack length. A further step can also be taken to calculate the critical strain energy release rate (G_C) or simply, fracture toughness.

 

G for Quasi-Brittle Materials

    As mentioned at the end of the section "Energy Balance Approach (Griffith Cracks)", when applying Griffith fracture stress equation to 'tought' materials like a thin sheet of aluminium showed that the inverse square root relationship was maintained but that the constant product of E was much larger than the classical surface energy term (2γ).

    In fact, for quasi-brittle materials a great deal of plastic deformation energy was consumed prior to and during the fracture event at and near the crack tip. Orowan and Irwin independently suggested a modification to the thermodynamic balance of Griffith by including a pastic work term γ_P :

 

 

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