Fracture Mechanics and Design

Abstract

Fracture mechanics plays an important role in design against failure of a component with an existing crack. This chapter first reviews historical engineering failures and their causes leading to the evolution of fracture mechanics field. Griffith crack theory has been stated and the relevant mathematical relationship has been derived. The stress concentration factor (K_s) has been explained with the aid of mathematical models and the real-life example of aircrafts’ cabin windows with particular reference of the corner radii. A distinction is made between the stress intensity factor (K), the critical stress intensity factor (K_c)_, and the plain strain fracture toughness (K_IC). The design philosophy of fracture mechanics has been explained by reference to its applications in deciding safety of design, material selection, and the design of NDT method. This chapter contains 16 worked examples/calculations, 6 figures, 19 mathematical models/formulae, 6 exercise problems, and 9 MCQs with their answers at the end of the book.

Questions and Problems

• 12.1. (MCQs). Encircle the correct answer for each of the following questions.

  • (a) Which loading mode occurs the most often and results in severe damage to the component?

    • (i) Mode I, (ii) Mode II, (iii) Mode III, (iv) Mode IV.

  • (b) Which loading mode is called the tearing mode?

    • (i) Mode I, (ii) Mode II, (iii) Mode III, (iv) Mode IV.

  • (c) Which loading mode is called the sliding mode?

    • (i) Mode I, (ii) Mode II, (iii) Mode III, (iv) Mode IV.

  • (d) What is the value of geometric factor for a plate with embedded circular crack?

    • (i) Y = 1, (ii) Y = 1.12, (iii) $\mathrm{Y}=\frac{2}{\pi }$, (iv) $\mathrm{Y}=1.1\left(\frac{2}{\pi}\right)$.

  • (e) What is the value of geometric factor for a plate of infinite width with a central crack?

    • (i) Y = 1, (ii) Y = 1.12, (iii) $\mathrm{Y}=\frac{2}{\pi }$, (iv) $\mathrm{Y}=1.1\left(\frac{2}{\pi}\right)$.

  • (f) What is true for a plate with semi-finite width containing a semi-circular surface flaw?

    • (i) Y = 1, (ii) Y = 1.12, (iii) $\mathrm{Y}=\frac{2}{\pi }$, (iv) $\mathrm{Y}=1.1\left(\frac{2}{\pi}\right)$.

  • (g) What is the value of geometric factor for a plate of semi-infinite width with an edge crack?

    • (i) Y = 1, (ii) Y = 1.12, (iii) $\mathrm{Y}=\frac{2}{\pi }$, (iv) $\mathrm{Y}=1.1\left(\frac{2}{\pi}\right)$.

  • (h) Which term is used to express K required for crack propagation?

    • (i) stress intensity factor, (ii) critical stress intensity factor, (iii) toughness.

  • (i) Which term is used to express a material’s resistance to fast fracture in presence of crack?

    • (i) stress intensity factor, (ii) critical stress intensity factor, (iii) fracture toughness.

• 12.2. (a) Explain the role of stress concentration in design of machine components.

  (b) Why are the cabin windows of modern aircrafts designed with a large corner radii?

• 12.3. Derive the mathematical relationship representing the Griffith’s crack theory.

• 12.4. A tensile stress of 70 MPa is applied normal to the major axis of a surface flaw in a large plate of a glass ceramic. The Young’s modulus and the specific surface energy for the glass are 70 GPa and 0.27 Jm⁻², respectively. Compute the maximum length of the crack that is allowable without causing fracture.

• 12.5. A large ceramic plate contains a 0.2-mm-long central crack with a radius of curvature of 0.6 μm at the Crack Tip. A Tensile Stress of 38 MPa Is Applied Normal to the Major Axis of the crack. Calculate the stress concentration factor for the static loading.

• 12.6. A 12-μm long through-thickness central crack is introduced in a 7-mm-thick large plate made of glass having a specific surface energy of 0.18 J/m², and Young’s modulus of 67 GPa. A tensile stress of 33 MPa is applied normal to the major axis of the crack. Calculate the net change in the potential energy of the plate due to the introduction of the crack.

• 12.7. A plate of infinite width contains a central flaw of length 3.2 mm. The plate’s material has a fracture toughness of $43\; MPa\sqrt{m}$. What design (working) stresses do you recommend for this application under mode I loading?

• 12.8. A metal plate of semi-finite width has a plane-strain fracture toughness of $30\; MPa\sqrt{m}$; the plate is subjected to a stress of 220 MPa during service. Design an NDT method capable of detecting an edge crack before the likely growth of the crack.

• 12.9. A plate of semi-finite width contains an embedded circular flaw of length 3 mm. The plain strain fracture toughness of the pate’s material is $50\; MPa\sqrt{m}$. Is the plate’s design safe for an application with a design stress of 200 MPa?