Strees Analysis in Elastic Half Space Due To a Thermoelastic Strain

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1 IOSR Junal f Mathematics (IOSRJM) ISSN: Vlume, Issue (July-Aug 0), PP Stees Analysis in Elastic Half Space Due T a Themelastic Stain Aya Ahmad Depatment f Mathematics NIT Patna Biha India Abstact: The stess distibutin n elastic space due t nuclei f them elastic stain distibuted unifmly n the cicumfeence f a cicle f adius R situated in the place = λ f the elastic semi space f Hkean mdel has been discussed by Nwacki: The Fce stess and cuple stess have been detemined. The fe stess educes t the ne btained by Nwacki f classical elasticity. I. Intductin: Analysis f stess distibutin in elastic space due t nuclei f themelastic stain distibuted unifmly n the cicumfeence f a cicle f adius situated in the plane Z = h f the elastic semi space f Hkean mdel has been discussed by Nwacki. This nte is an extensin f the analysis f abve pblem f micpla elastic semi-space. Fce stess ji and cuple stess ji have been detemined due t pesence f nuclei f themelastic stain situated in the place Z = h inside the semi space. The fce stess educes t the ne btained by Nwascki f classical elasticity. II. Basic Equatins: We cnside a hmgenus istpic elastic mateial ccupying the sami infinite egin Z O in cylindical pla cdinate system (,, Z). It has been shwn by Nwacki [64] that is in the case when the macdisplacement vect and mictatin depend nly n and the basic equatins f equilibium u w f mic-pla they f elasticity ae decmpsed int tw mutually independent sets. Hee we shall be cncened with the set = (u, O, u ) and the tatin vect = (O,,O): u w e T ( )( ) u ( ) e T ( )( u ( ). ( ) u u ( )( ( ) 4 0 u Whee e = ( ) = = (3λ + ) t u, u = displacement cmpnents = Cmpnent f tatin vect λ,,,, = elastic cnstants T (, ) = tempeatue distibutin t = cefficient f themal expansin...(6.) 46 Page

2 T the displacement vect Stees Analysis In An Elastic Half Space Due T A Themelastic Stain (u, O, u ) and the tatin vect u state f fce stess ij and cuple stess ij 0 ij = = (O,, O) is ascibed the fllwing w ij = III. Stess-Stain elatins : The elatin between stess tens ij, ij and displacement and tatin in the cylindical u w cdinates ae given by 3 u = et u = e T u = et u u u u = ( ) ( ) u u u u = ( ) ( ) = ( ) ( ) = ( ) ( ).(6.) = ( - ), = ( - ) 4 Fllwing Nwacki [08], we intduce displacement ptentials, and tatin ptential V such that = = ( ). (6.3) 47 Page

3 = v Substituting ( 6.3) in (6.) we get Stees Analysis In An Elastic Half Space Due T A Themelastic Stain T ( ) ( ) ( ) v (6.4) T ( ) ( ) ( ) ( ) v ( ) 4v 0 The abve equatins ae satisfied if = m T (( -) V = 0 Whee = elated by 5 ( ) ), 4 = - 4 ( ) m =,.(6.5) V. (6.6) and V and ae T slve (6.5) we wite = +.. (6.7) V = V + V Whee and V ae paticula integals f nn-hmgeneus pat and, V ae geneal slutins f hmgeneus pat. Nw f paticula integal we have = mt (6.8) and V = 0 and f geneal slutin we have = 0 ( -) V = 0 (6.9) 6 IV. Slutin f the title pblem : We cnside nuclei f them elastic stain distibuted unifmly n the cicumfeence f a cicle f adius and situated in the plane = h inside the elastic half space. The stess distibutin ij can be cnsideed as sum f tw stess systems S and S. The system S cnstitute stess distibutin ij f infinite elastic space cntaining tw nuclei f themelastic stains situated in the planes = h and = -h distibuted unifmly alng the cicumfeences f the cicles, each f adius. The secnd system S cnstitutes stess distibutin ij cespnding t elastic semi-space in the isthemal state. The stess ij is s chsen that the bunday cnditins n the plane = O. = 0, = 0, = 0 ae satisfied. The themelastic displacement ptential cespnding t ij satisfies the equatin 7 = m (R - R) [(-h) - ( + h)].(6.0) Whee = x + y and (x) epesents Diac delta functin. Repesenting the ight hand side f the equatins (6.0) by the Fuie Integal 48 Page

4 m (-R) [ (-h) - ( + h ] Stees Analysis In An Elastic Half Space Due T A Themelastic Stain mr J( ) J( R) Cs( h) Cs( hdd 00 The slutin f (6.0) is epesented by the integal mr J ( R ) J ( ) e ( h ) e ( h ) d., h0 mr J ( R ) J ( ) e ( h ) e ( h ) d, h 0 The stess distibutin f the system ( S ) is btained = = mr = = mr 8 _..6.)..6.) ( ) ( ( h) ( h) J( R) J( R) J ( ) e e d ( ) = - ( ) ( ( h) ( h) J ( R) J ( ) J "( ) e e d V. Geneal Slutin f Hmgeneus Equatins: Applying Kankel tansfm t equatin (6.9), the geneal slutin f half space is given by = ( A B ) e J( ) d (6.4) and V = ( L Me ) J ( ) d. (6.5) e 9 whee = and L,M,A, B ae sme functins f, t be detemined by bunday cnditins. Equatins (6.4) give L = - B. (6.6) Knwing the functins, and V the fce stesses and cuple stesses ae calculated by the elatins = u e " ( " ) " 49 Page

5 = = = + = (+) = (-) = (+) = (-) Stees Analysis In An Elastic Half Space Due T A Themelastic Stain " ( " ) " " ( ) " " " ( ) " " V " V " V " V " V " 0 - (-) - (+) V " V " Since the bunding suface = 0 is fee fm tactins, we have n = O, S + S = O Thus = + = O = + = O = + = O Since = O, we get = O fm (6.8) This gives L = -- M (.6.9) L = -- ( ) B Als, fm (6.6) we get M = -- L = ( )( ) B The slutin f equatin " ( ) 4 V " Is btained as 3 " B e a e J( ) d ( )( Whee a O 4 ( ) 50 Page

6 Stees Analysis In An Elastic Half Space Due T A Themelastic Stain Bunday cnditins (6.8), yield A = 4 a P () ( B = P () (6.0) ( ) Whee P () = h mrj ( R) e a ( / ) Substituting expessins f, and V with values f A and B in (6.0), we btain ij and ij with the help f the elatins (6.7) 3 4 a0 ( P( ) e J ( ) d 3 ( )( ) e a e P( ) J( ) d 4 3 e p( ) J ( ) d = 3 + = ( ) 4 a P( ) e J ( ) d (-) e a (/ ) e P( ) J( ) d( ) 3 +4 ( ) e a e P( ) J( ) d 4 ( 3 ( e e ) P( ) J( ) d 3 " ( ) " 3 ( e e ) ( ) J( ) ( ). J( ) x P( ) " ( )( ) ( ) 4 ( ) e e P J ( ) d Stess distibutin in the elastic half space is btained by adding (6.3) and (6.) Thus ". (6.) ( h) ( h) mr J( ) J ( ) e e J( R) d 3 4 a P J ( ) J( ) P( ) e d 3 ( )( ) e a e J ( ) J( ) P( ) d 4 e p( ) j0( ) d Page

7 " Stees Analysis In An Elastic Half Space Due T A Themelastic Stain 4 ( h) ( h) m R e e. J ( ) J ( R ) d +4 a P( ) e J ( ) d -- 4 ( )( ) e a e P( ) J ( ) d 4 3 e P( ) J ( ) d " ( h) ( h) m R e e J ( R) J ( ) d a ( ) e P( ) J( ) d 3 ( )( ) e a e p( ) J( ) d 4 3 e P( ) J ( ) d " 5 ( h) ( h) m R e e J ( R) J ( ) d 3 ( ) 4 a P( ) e J ( ) d 3 3 4( ) ( ) e a ( ) e P( ) J ( ) d 3 4 ( )( ) e a e P( ) J ( ) d 4 ( ) 3 ( e e ) P( ) J ( ) d " ( ) ( e e ) J ( ) J ( ) xp( ) d ( )( ) 4 " ( e e ) P( ) J ( ) d 5 Page

8 ( ") ( ") " ( ) ( ") Stees Analysis In An Elastic Half Space Due T A Themelastic Stain ( h) ( h) m R e e J ( R) J ( ) d +R xj ( ) d... (4 a ) e ( )( ) e a e p( ) 6. (6.) F = O, the micpla cuple stess vanishes and in that case = = O, a = O, = 0. Thus we get fm (6.) ( h) ( h) mur J ( ) j ( ) e e J ( ) d ( ) J 3 P( ) e ( ) J ( ) ( ) d ( h) ( h) mur J( ) J ( ) e e J( R) d J ( ) 3 ( J ( ) ( ) P( ) e d ( h) ( h) mur e e J ( R) J ( ) d 4 P( ) e J ( ) d = ( h) ( h) = - 7 mur e e J ( R) J ( ) d 3 ( ) P( ) e J ( ) d.. (6.3) u = u = 0 h whee P () educes t (- ) e J (R). Results in (6,3) have been btained in f Hkean them elasticity. 53 Page

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