A centre crack of length 2a = 0.2 mm is perpendicular to a tensile stress σ in a plateof soda glass. Young’s modulus for soda glass is 70 GPa and the surface energy perunit area is 0.5 J m−2.(a) Determine the strength of the plate, i.e. the maximum stress it can support,according to Griffith’s theory.(b) What is the fracture toughness, Kc, of the soda glass?
Question
A centre crack of length 2a = 0.2 mm is perpendicular to a tensile stress σ in a plateof soda glass. Young’s modulus for soda glass is 70 GPa and the surface energy perunit area is 0.5 J m−2.(a) Determine the strength of the plate, i.e. the maximum stress it can support,according to Griffith’s theory.(b) What is the fracture toughness, Kc, of the soda glass?
Solution
(a) According to Griffith's theory, the strength of the plate (σ) can be calculated using the formula:
σ = sqrt((2Eγ)/πa)
where: E = Young's modulus = 70 GPa = 70 * 10^9 Pa γ = Surface energy per unit area = 0.5 J/m^2 a = half the length of the crack = 0.2 mm / 2 = 0.1 mm = 0.1 * 10^-3 m
Substituting these values into the formula gives:
σ = sqrt((27010^90.5)/(π0.110^-3)) σ = sqrt((7010^9)/(π0.110^-3)) σ = sqrt(7*10^14) σ = 2.65 * 10^7 Pa
So, the strength of the plate according to Griffith's theory is approximately 26.5 MPa.
(b) The fracture toughness (Kc) can be calculated using the formula:
Kc = σ * sqrt(πa)
Substituting the values we have:
Kc = 2.65 * 10^7 * sqrt(π0.110^-3) Kc = 2.65 * 10^7 * sqrt(3.1410^-4) Kc = 2.65 * 10^7 * 0.0177 Kc = 4.69 * 10^5 Pam^0.5
So, the fracture toughness of the soda glass is approximately 0.47 MPa*m^0.5.
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