A thin steel plate contains a straight crack running from its centre out to the rim

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Partial differential equations and boundary-value problems

Conditions of Validity, Ch The three equations, their characteristics and what makes a problem well posed

Problem

A thin steel plate contains a straight crack running from its centre out to the rim. Taking plane polar coordinates (r,θ)(r,\theta) centred on the crack tip, the plate occupies 0<r<a0<r<a, 0<θ<2π0<\theta<2\pi (the domain angle at the tip is a full 2π2\pi, since the two faces of the zero-width crack lie at θ=0\theta=0 and θ=2π\theta=2\pi), with a=0.10 ma=0.10\,\mathrm{m}. Both crack faces are held at the same fixed temperature, taken as the zero of the temperature deviation uu. Near the tip, the leading term of the steady temperature deviation u(r)u(\mathbf{r}) satisfying ∇2u=0\nabla^2u=0 is known to be u(r,θ)=A r1/2sin⁡(θ/2),u(r,\theta)=A\,r^{1/2}\sin(\theta/2), where AA is a constant fixed by the data on the rim r=ar=a, and ∇2=1r∂r(r∂r)+1r2∂θ2\nabla^2=\dfrac{1}{r}\partial_r(r\partial_r)+\dfrac{1}{r^2}\partial_\theta^2 in plane polar coordinates. ((a)) Verify that u=Ar1/2sin⁡(θ/2)u=Ar^{1/2}\sin(\theta/2) satisfies ∇2u=0\nabla^2u=0 for r>0r>0 and that it vanishes on both crack faces θ=0\theta=0 and θ=2π\theta=2\pi. ((b)) Show that ∣∇u∣=∣A∣/(2r)|\nabla u|=|A|/(2\sqrt r) for every θ\theta, so that ∣∇u∣→∞|\nabla u|\to\infty as r→0r\to0 along every direction. State which hypothesis of the energy uniqueness proof this violates, and which hypothesis the maximum-principle proof needs instead; confirm that the latter holds at the tip. ((c)) Two temperature fields u1u_1 and u2u_2 both solve the same Dirichlet problem on this cracked plate (identical data on the rim r=ar=a and identical, matching data on both crack faces). Using the maximum principle, show u1=u2u_1=u_2 throughout the plate.

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