Near the tip of a crack in a thin elastic plate

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Foundations of mathematical language · Calculus of change and accumulation

Conditions of Validity, Ch Multiple integrals: Fubini, the change-of-variables theorem and integrals that do not exist

Problem

Near the tip of a crack in a thin elastic plate, the local strain-energy density is modelled as

u(r) = u_0 (r/a)^{-p}

, where

r

is distance from the tip measured in the plane of the plate, so the relevant number of dimensions is

n=2

, and

a

is a fixed length scale. The total stored energy within radius

a

of the tip is

\int_{r\le a} u(r)\,\mathrm{d}A

, and the total energy from radius

a

out to infinity (an idealised infinite plate) is

\int_{r\ge a} u(r)\,\mathrm{d}A

. ((a)) Take the realistic exponent

p=1

(stress fields near a crack tip scale as

r^{-1/2}

, so the energy density scales as

r^{-1}

). Decide whether the energy stored within radius

a

of the tip is finite. ((b)) With the same

p=1

, decide whether the energy stored from radius

a

out to infinity is finite. ((c)) Suppose instead a different defect gives

p=3

far from the core. Decide whether the energy from radius

a

to infinity is now finite.

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