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DoITPoMS Teaching & Learning Packages Brittle Fracture Estimate of the strain energy in a bent beam
Brittle Fracture AimsBefore you startIntroductionWhen do atomic bonds break?Why do cracks weaken a material?Inglis and the crack tip stress ideaCan we calculate the energy changes?What about tension?Another way of expressing the energiesAnother way of calculating the energiesWhy bother if they are the same?Coping with a scatter in strengthSimulation of Weibull modulus experimentWhen does the sample fail completely?Sub-critical crack growth and R-curvesSummaryQuestionsGoing further
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Estimate of the strain energy in a bent beam

In pure bending, where the moment, M, is constant along a beam of length L, the curvature of the beam, κ, and the angle, θ, through which the beam is bent is given by

$$\kappa = {M \over {EI}}{\rm{, so }}\theta = {M \over {EI}} \cdot L$$

The strain energy, Ustrain, is given by

$${U_{{\rm{strain}}}} = {1 \over 2}M\theta = {{{M^2}L} \over {2EI}}$$

Here the moment therefore varies along the length so that Ustrain becomes

$${U_{{\rm{strain}}}} = \int\limits_0^L {{{{M^2}} \over {2EI}}} {\rm{ d}}x$$

where x is the distance along the beam from the point of application of force at the free end. The bending moment is given at any point x by M = Fx  so,

$${U_{{\rm{strain}}}} = {{{F^2}{L^3}} \over {6EI}}$$

The work done, Wforce, by the applied force when the beam is bent is

$${W_{{\rm{force}}}} = {1 \over 2}F\delta $$

This is equal to the strain energy in the beam, Ustrain, so that

$$F = {{3EI\delta } \over {{L^3}}}$$

Substituting for F in the expression for Ustrain and remembering that I = wd3/12 for a beam of thickness d and width w  gives

$${U_{{\rm{strain}}}} = \left[ {{{Ew{d^3}{\delta ^2}} \over {8{L^3}}}} \right]$$

 

For our peeling layer of unit width, i.e. w = 1, then L = c and d = h, so the strain energy in the peeling ligament, UE, is

$${U_{\rm{E}}} = {{E{d^3}{h^2}} \over {8{c^3}}}$$

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