The Basis of Linear Elastic Fracture Mechanics

Abstract

In this chapter we will discuss the use of elastic solutions to predict the strength of parts containing sharp cracks. Immediately, we are faced with the difficulty that, whether we consider the line crack or the elliptical crack with semiminor axis shrunk to zero, the stresses at the crack tip are infinite for any value of load. Thus, the elastic solutions by themselves are not adequate for fracture prediction. Basically, two approaches have been taken in fracture calculations to overcome this dilemma without becoming involved in a detailed analysis of the “process zone” at the crack tip. One is based on energy considerations and takes advantage of the fact that the strain energy associated with the stress field around a crack is finite because the stress becomes infinite in an infinitesimally small region. The other makes use of the stress intensity factor to describe the state of stress in the crack tip region. At first sight it is perhaps confusing to have two different approaches but they can be shown to be equivalent.

Problems

• 1. Mica, a transparent almost ideally brittle material, can be cleaved, as shown below, by a smooth rigid wedge. One cannot measure the force P in such an experiment, but the deflection y of the cleaved strip near the crack front can be measured optically by counting interference fringes. Obreimoff [32] made such an experiment and for h = 0.011 cm found

  $$ \begin{array}{ll}y=0.04{x}^2\hfill & \mathrm{where}\ x\ \mathrm{and}\ y\ \mathrm{are}\ \mathrm{in}\ \mathrm{centimeters}.\hfill \end{array} $$

  Treating the cleaved strip as a cantilever of length a, and ignoring shear deflections, you are asked to estimate G _c (dyn/cm) for Obreimoff’s muscovite mica. We recall that the deflection curve of a cantilever of length a with end load P is given by

  $$ \begin{array}{lll}y=\frac{P}{EI}\left(\frac{a{x}^2}{2}-\frac{x^3}{6}\right)=\frac{Pa{x}^2}{2EI}\hfill & \mathrm{if}\hfill & x\ll a\hfill \end{array} $$

  Take E for mica as 2 × 10¹² dyn/cm^2.

  

  (assume unit width perpendicular to paper).

• 2. It has been proposed that fracture tests be conducted on double cantilever beam specimens loaded by end moments as shown. Recalling that the end rotation ϕ of a cantilever beam of length a is given by $\phi = Ma/EI$, you are asked to obtain an approximate value for K _I for this specimen for plane strain conditions. Express the result in terms of M and the specimen dimensions.

  

• 3. To measure the fracture toughness G _c of an adhesive joint, the test sketched below has been proposed. The strain energy of a plate clamped at its rim and centrally loaded is given by

  $$ \begin{array}{lllll}U=\frac{8\pi D{x}^2}{a^2}\hfill & \mathrm{where},\hfill & D=E{h}^3/12\left(1-{v}^2\right)\hfill & \mathrm{and}\hfill & x=\frac{P{a}^2}{16\pi D}\hfill \end{array} $$

  You are asked to obtain an expression for G in terms of P, D, and any other quantities involved.

  

• 4. A plate, as shown below, is clamped at the ends but no external loads are applied. If the ambient temperature decreases but the grips do not move, then an axial tensile stress will be developed in the plate (transverse stresses can be neglected well away from the grips). For a high-strength steel plate 1 in. thick with the properties given below, you are asked to estimate the temperature change that will cause catastrophic propagation of a 75 mm long through crack:

  • Young’s modulus, E = 200 GPa

  • Coefficient of thermal expansion, α = 10.8 × 10⁻⁶/°C

  • Critical crack energy release rate, G _Ic = 5250 J/m²

  • Yield strength, S _y  = 1400 MPa

  • Poisson’s ratio, ν = 0.3

  Assume plane strain and no change of G _Ic with temperature which may be a reasonable assumption if we are considering moderate changes in temperature. Also neglect any corrections.