A closed-ended thick-walled steel cylinder has inner radius

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids · Solid continuum mechanics and complex fluids

Bending, Ringing, Breaking and Flowing, Ch Dislocations and the onset of plastic flow

Problem

A closed-ended thick-walled steel cylinder has inner radius a=50 mma=50\,\mathrm{mm}, outer radius b=100 mmb=100\,\mathrm{mm}, tensile yield stress σY=350 MPa\sigma_Y = 350\,\mathrm{MPa}, and Burgers vector bB=0.248 nmb_{\rm B}=0.248\,\mathrm{nm} for its bcc iron lattice. It is pressurised internally by a pressure pp. At the bore (r=ar=a) the stresses are, quoted from the thick-cylinder (Lamé) analysis, σθ=p b2+a2b2−a2,σr=−p,σz=p a2b2−a2.\sigma_\theta = p\,\frac{b^2+a^2}{b^2-a^2},\qquad \sigma_r = -p,\qquad \sigma_z = p\,\frac{a^2}{b^2-a^2}. ((a)) Find the internal pressure at which the bore first yields, according to the Tresca criterion and according to the von Mises criterion. ((b)) At the Tresca first-yield pressure, an edge dislocation at the bore lies on the 45∘45^\circ plane, with its Burgers vector along the direction of maximum shear in that plane, on which the shear stress is the maximum shear stress τmax⁡=12(σθ−σr)\tau_{\max} = \tfrac12(\sigma_\theta-\sigma_r). Find the Peach-Koehler force per unit length this shear exerts on the dislocation.

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