HW5 - EN222 Mechanics of Solids Homework 5 Cracks and...

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EN222: Mechanics of Solids Homework 5: Cracks and Contacts Due Thursday March 16, 2006 Division of Engineering Brown University 1. The figure shows a periodic array of edge dislocations with in an infinite solid. Show that the vertical stress along can be calculated as 12 0, bb == b 2 0 x = 1 22 2 2 2 11 4( 1 ) x Eb 22 1 2( 1 ) n Eb x nL x πν =+ −− σ = L L x 1 x 2 Sum the series analytically (MAPLE will do it). L 2a x 1 x 2 2. Using the results of problem 1, set up an integral equation to calculate the burgers vector distribution that represents the solution to an infinite array of parallel cracks with length 2 a and spacing L , which is loaded by a uniform stress 22 far from the crack. Show that the integral equation can be expressed as 1( ) (,) ) ) bd kx b d gx x ξξ ξ πξ + (= ( ∫∫ where 2 22 ( ) 4(1 ) / ( , ) ( / )cot( ( ) / ) 1/( ( )) E a L xa L x νσ =− = Use the numerical procedure discussed in class to solve the integral equation for 2 () / 1 b φ The stress intensity factors for a mode I crack can be extracted from the solution for
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HW5 - EN222 Mechanics of Solids Homework 5 Cracks and...

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