State Griffith theory of brittle fracture. On the basis derive an expression for fracture stress? State Orwan$'$s modification

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In practice it is observed that fracture occurs at much lower values of stresses as compared to theoretical cohesive strength of the metal. The first explanation of this discrepancy was proposed by Griffith. Theory proposed by him though applicable to perfectly brittle materials like glass it has helped to understand fractures in metals to great extent

According to Griffith, submicroscopic fine cracks present in brittle material produce a localized stress concentration of sufficient magnitude which reaches horizontal cohesive strength value for that location. Nominal stress value can be much lower than localized value.

When crack grows into fracture more surface area is created. This requires energy that overpowers cohesive force between the atoms. This energy is supplied by elastic strain released because of spreading of crack. Griffiths criterion for crack propagation states “A crack will propagate when decrease in elastic strain energy is at least equal to the energy required to create new crack surface”.

Consider a thin plate of negligible thickness having elliptical crack of interior length 2c as shown in Fig

The maximum stress at the tip of crack is

$\sigma_{max} = \sigma \Bigg[ 1 + 2\bigg(\frac{c}{\rho}\bigg)^{1/2}\Bigg] \approx 2 \sigma \bigg(\frac{c}{\rho}\bigg)^{1/2}$

Before fracture starts, maximum stress at the crack tip shall rise to theoretical value of cohesive strength. Once both are equal, crack propagates. The stress is then can be called nominal fracture stress, σf, is given by(Where σ is tensile energy acting normal to crack surface. Negative sign indicates released elastic strain energy)

$\sigma_f = \bigg(\frac{E \gamma \rho}{4ac}\bigg)^{1/2}$

The sharpest crack (i.e. maximum stress concentration) can be represented by ρ = a. Thus,

$\sigma_f = \bigg(\frac{E \gamma}{4c}\bigg)^{1/2}$

Surface energy due to presence of crack length = 2c

$U_z = 4 \gamma c$

Strain energy released in a thin plate of unit thickness is given by Inglis as follows:

$U_e = \frac{\pi \sigma^2 c^2}{E}$

The total change in potential energy resulting from creation of crack

ΔU=Us+Ue

Griffiths criterion states that the crack will propagate under a constant applied stress σ, if an increase in crack length conserves the total energy of the system or increment an surface energy is balanced by decrease in elastic strain energy.

Griffith’s criterion can be expressed as follows for incremental change in systems energy with crack length:

$\frac{\delta U_e}{\delta C} = \frac{\delta U_s}{\delta c} \Rightarrow \frac{2\pi \sigma^2c}{E} = 4\gamma \Rightarrow \sigma = \bigg(\frac{2E\gamma}{c \pi} \bigg)^{1/2}$

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