The theory that fracture proceeds when the elastic energy released as a crack extends exceeds the energy of the new surfaces. The fracture stress is given by √(2Eγs/(πa)).
Japanese name: グリフィスの理論
Griffith theory appears in 1 practice question across 1 topic, including Fracture mechanics and fatigue.
In 1920 Griffith proposed judging whether a crack grows not from the stress at its tip but from an energy balance. When a crack grows slightly in a stressed plate, the elastic strain energy stored around it is released. On the other hand, creating new fracture surfaces requires surface energy. While the crack is short, the energy needed to create surface is larger and the crack does not grow, but the released energy increases faster the longer the crack is, so beyond a certain length the released energy wins and the crack runs through all at once, accelerating as it goes.
When a stress σ acts perpendicular to a crack of length 2a in a wide plate, in plane stress an elastic energy of πσ²a²/E per unit thickness is released, and the surface energy is 4aγs. Differentiating the total energy with respect to a and setting it to zero gives the fracture stress σf = √(2Eγs/(πa)). In plane strain, E is replaced by E/(1 − ν²). For example, for a glass with E = 70 GPa and γs = 1.0 J/m², a = 10 µm gives σf of about 67 MPa, roughly a hundredth of the theoretical strength of about 7 GPa. The strength is inversely proportional to √a.
Griffith's equation works well for glass, in which almost no plastic deformation occurs, but applied to metals it greatly underestimates the strength. Orowan and Irwin proposed adding the work consumed in the plastic zone at the crack tip to the surface energy. In ductile steels this plastic work is thousands of times the surface energy or more and accounts for most of the energy needed for fracture. Irwin further expressed the energy balance by the energy release rate G and the stress field at the crack tip by the stress intensity factor K, showed the relation between the two and laid the foundation of linear elastic fracture mechanics.
Griffith's theory shows that the strength of a brittle material is set less by the material itself than by the largest crack it contains. This is why the strength of the same glass differs greatly depending on how it has been scratched, and why specimens made by the same process show scattered strengths. Larger parts are more likely to contain large cracks, so their average strength is lower. The Weibull distribution treats this scatter and size effect statistically. Griffith himself showed by experiment that thinner glass fibers are stronger. This is why controlling crack size and proof testing are emphasized in the design of ceramics.
Answer: C. About 211 MPa
2Eγs/(πa) = 2×70×10⁹×1.0/(3.14×1.0×10⁻⁶) ≈ 4.46×10¹⁶ Pa², and its square root is about 2.11×10⁸ Pa, or about 211 MPa. The Griffith equation comes from the condition that a crack grows unstably when the elastic strain energy released as it extends exceeds the energy 2γs of the two new surfaces it creates. About 149 MPa drops the factor of 2 that comes from forming two surfaces, and about 374 MPa forgets to divide by π. Because σf is inversely proportional to √a, making the crack 4 times longer halves the fracture stress.
The released elastic energy grows with the square of the crack length, while the surface energy grows in proportion to the length. Solving the condition at which their rates of increase balance gives the square of the stress inversely proportional to crack length, so the strength is inversely proportional to √a.
Griffith's equation can be rewritten as σf√(πa) = √(2Eγs). The left side has the form of the stress intensity factor K and the right side is a constant of the material, so the fracture condition is the same as K reaching a critical value. For brittle materials such as glass, K_IC ≈ √(2Eγs).
It can be used directly for materials such as glass and ceramics, in which almost no plastic deformation occurs before fracture. For metals and polymers, the energy consumed by plastic deformation and other processes at the crack tip must be included, and the theory extended in that form is the basis of present-day fracture mechanics.
The Griffith equation comes from the condition that a crack grows unstably when the elastic strain energy released as it extends exceeds the energy 2γs of the two new surfaces it creates.
A glass (Young's modulus E = 70 GPa, surface energy γs = 1.…Young's modulus Energy release rate Fatigue striations Miner's rule Modified Goodman diagram Paris law Stress concentration factor Stress intensity factor