ON THE KIENZLER-DUAN FORMULA FOR THE HOOP STRESS AROUND A CIRCULAR VOID

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Ευπραξία Δασκαλόπουλος
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1 13 ЦРНОГОРСКА АКАДЕМИЈА НАУКА И УМЈЕТНОСТИ ГЛАСНИК ОДЈЕЉЕЊА ПРИРОДНИХ НАУКА, 21, 216. ЧЕРНОГОРСКАЯ АКАДЕМИЯ НАУК И ИСКУССТВ ГЛАСНИК ОТДЕЛЕНИЯ ЕСТЕСТВЕННЫХ НАУК, 21, 216 THE MNTENEGRIN AADEMY F SIENES AND ARTS PREEDINGS F THE SETIN F NATURAL SIENES, 21, 216. UDK 621.8: Vldo Lubd * N THE KIENZLER-DUAN FRMULA FR THE HP STRESS ARUND A IRULAR VID Abstt The deivtion of the Kienzle-Dun fomul fo the hoop stess ound iul void used b eithe emote loding o neb intenl soue of stess is pesented bsed on the Fouie seies nlsis without efel to the Poisson oeffiient of ltel onttion, s in the oiginl deivtion b Kienzle nd Dun (1987). The fomul fo the longitudinl stess below the fee sufe of hlf-spe due to neb intenl soue of stess is lso deived b mens of the limiting poess fom the solution to the poblem of n intenl soue of stess ne n infinitel lge iul void, nd b n independent nlsis without efel to the fome poblem. The use of the deived fomuls in the dislotion nd inlusion poblems of mehnis of solids nd mteils siene is disussed. KIENZLER-DUANVJ FRMULI ZA BRUČNI NAPN K KRUŽNG TVRA Sžetk Kienzle-Dunov fomul z obučni npon oko kužnog otvo usljed spoljšnjeg opteećenj ili unutšnjeg izvo npon je izveden n bzi Fouieove nlize, bez uvođenj u nlizu Poissonovog koefiijent elstičnosti koji je koišćen u du Kienzle i Dun (1987). Fomul z uzdužni npon ispod slobodne povšine poluposto usljed obližnjeg unutšnjeg izvo npon je izveden ko gnični slučj unutšnjeg izvo npon u blizini beskončno velikog otvo, * Pof. D. V. A. Lubd, The Montenegin Adem of Sienes nd Ats, Montenego, nd Univesit of lifoni, Sn Diego, A , USA.

2 14 Vldo Lubd ztim nezvisnom nlizom ne pozivjući se n ješenje poblem beskončnog posto oslbljenog kužnim otvoom. Diskutovn je znčj ovih fomul z nlizu dislokionih poblem i inkluzij u mehnii čvstih tijel i nui o mteijlim. σ θ (, θ) = 2[σ θ(, θ) σ (, θ) [σ θ(,θ)+σ (,θ). (, θ) J M σ zθ (, θ) =2σ zθ(, θ).

3 n the Kienzle-Dun Fomul fo the Hoop Stess ound iul Void 15 = σ (,) = 2[σ (,) σ (,). d Φ = Φ(, θ) σ = 1 Φ Φ 2 θ 2, σ θ = 2 Φ 2, σ θ = 1 2 Φ θ 1 2 Φ θ. σ(, θ) σθ (, θ) σ θ (, θ) = ˆσ (, θ) = σ(, θ) ˆσ θ (, θ) = σθ (, θ) σ (, θ) =σ (, θ)+ˆσ (, θ) σ θ (, θ) σ θ (, θ)

4 16 Vldo Lubd q q d d h h q q d d h h d (, θ) d (, ) (ξ,η) (, θ) (, ) (ξ,η) σθ σθ (, θ) (, θ) = = σ(, θ) σ(, θ) = = = = Φ = A f 1 (θ)+ n f n (θ)+ n+2 B n g n (θ). Φ = A [ f 1 (θ)+ n f n (θ)+ n+2 B n g n (θ). f n (θ) =A n os nθ + n sin nθ, g n (θ) =B n os nθ + D n sin nθ, f n (θ) =A n os nθ + n sin nθ, g n (θ) =B n os nθ + D n sin nθ, = = = (, [ θ) =2A +2f (θ) n(n 1) n 2 (θ)+(n 1)(n 2) σ (θ) (, θ) =2A +2f 1 (θ) n(n 1) n 2 f n (θ)+(n + 1)(n 2) n g n (θ), θ(, [ θ) =2A +6f (θ)+ n(n 1) n 2 (θ)+(n 1)(n 2) σ (θ) θ(, θ) =2A +6f 1 (θ)+ n(n 1) n 2 f n (θ)+(n + 1)(n + 2) n g n (θ), θ(, θ) = 2f 1(θ) [ σ (n 1) n 2 n(θ)+(n 1) n(θ) θ(, θ) = 2f 1(θ) (n 1) n 2 f n(θ)+(n + 1) n g n(θ),

5 n the Kienzle-Dun Fomul fo the Hoop Stess ound iul Void 17 () θ = () () = () σ(,θ)=2a = () 2f 2 (θ), σθ(,θ)=2a +2f 2 (θ), σθ(,θ)= f θ = 2(θ). = (,θ)=2a 2f 2 (θ) θ(,θ)=2a +2f 2 (θ) θ(,θ)= f 2(θ) (,θ)=2a 2f 2 (θ) θ(,θ)=2a +2f 2 (θ) θ(,θ)= f 2(θ) (,θ)=2a 2f 2 (θ) θ(,θ)=2a +2f 2 (θ) 2(θ) σ 2A (,θ)=2a 2A /2 (,θ)+σ 1 2f 2 (θ), σ θ(,θ)=2a +2f 2 (θ), σθ(,θ)= f 2(θ). /2 1 2A (,θ)+σ θ (,θ) 2A 1 /2 1 2A (,θ)+σ θ (,θ) 2A 2A 2A 1 /2 1 (,θ)+σ θ 2A (,θ) 2A 2A 1 /2 1 2A θ (,θ) (, θ)dθ = I1 /2 I1 = σ(,θ)+σ θ (,θ) θ(, θ)dθ =2A (, θ)dθ 2A 2A θ(, θ)dθ =2A (, θ)dθ 2A θ(, θ)dθ =2A (, θ)dθ θ(, θ)dθ =2A (, θ)dθ 1 θ(, θ)dθ =2A σ (, θ)dθ = 1 σ θ(, θ)dθ =2A. θ(, θ) 2σ(, θ) =2f (θ)+ 3n(n 1) n 2 (θ)+(3n 2)(n 1) (θ) θ(, θ) 2σ(, θ) =2f 1 (θ)+ 3n(n 1) n 2 n (θ)+(3n 2)(n 1) n n (θ) θ(, θ) 2σ(, θ) =2f 1 (θ)+ 3n(n 1) n 2 n (θ)+(3n 2)(n 1) n n (θ) θ(, θ) 2σ(, θ) =2f 1 (θ)+ 3n(n 1) n 2 n (θ)+(3n 2)(n 1) n n (θ) θ) 2σ(, θ) =2f 1 (θ)+ [ 3n(n 1) n 2 n (θ)+(3n 2)(n 1) n n (θ) σθ(, θ) 2σ(, θ) =2f 1 (θ)+ 3n(n 1) n 2 f n (θ)+(3n 2)(n + 1) n g n (θ). σ σθ σθ σθ θ σ σθ θ (, (, θ) θ) (, θ) (, θ) (, θ) (, θ) ˆσ ˆσ (, θ) = σ(, θ) ˆσ θ (, θ) (, θ) = σ(, θ) ˆσ θ (, θ) ˆσ (, θ) = σ(, θ) ˆσ θ (, θ) ˆσ (, θ) = σ(, θ) ˆσ θ (, θ) ˆσ (, θ) = σ(, θ) ˆσ θ (, θ) ˆσ (, θ) = σ(, θ) ˆσ θ (, θ) = ˆΦ =Â ln 1 ˆf1 (θ)+ n ˆfn ˆΦ (θ)+ n+2 (θ) =Â ln 1 ˆf1 (θ)+ n ˆfn (θ)+ n+2 n (θ) ˆΦ =Â ln 1 ˆf1 (θ)+ n ˆfn (θ)+ n+2 n (θ) ˆΦ =Â ln 1 ˆf1 (θ)+ n ˆfn (θ)+ n+2 n (θ) ˆΦ =Â ln 1 ˆf1 (θ)+ [ n ˆfn (θ)+ n+2 n (θ) ˆΦ =Â ln + 1 ˆf1 (θ)+ n ˆfn (θ)+ n+2 ĝ n (θ), ˆf n (θ) =Ân os nθ Ĉn sin nθ n (θ) ˆB n os nθ ˆD n sin nθ ˆf n (θ) =Ân os nθ Ĉn sin nθ n (θ) ˆB n os nθ ˆD n sin nθ ˆf n (θ) =Ân os nθ Ĉn sin nθ n (θ) ˆB n os nθ ˆD n sin nθ ˆf n (θ) =Ân os nθ Ĉn sin nθ n (θ) ˆB n os nθ ˆD n sin nθ ˆf n (θ) =Ân os nθ + Ĉn sin nθ, ĝ n (θ) = ˆB n os nθ + ˆD n sin nθ. ˆσ (, θ) =Â 2 ˆf 3 1 (θ) n(n+1) n 2 ˆfn (θ)+(n 1)(n+2) n n (θ) ˆσ (, θ) =Â 2 ˆf 3 1 (θ) n(n+1) n 2 ˆfn (θ)+(n 1)(n+2) n n (θ) ˆσ (, θ) =Â 2 ˆf 3 1 (θ) n(n+1) n 2 ˆfn (θ)+(n 1)(n+2) n n (θ) ˆσ (, θ) =Â 2 ˆf 3 1 (θ) n(n+1) n 2 ˆfn (θ)+(n 1)(n+2) n ˆσ θ (, θ) = Â 2 n (θ) ˆσ (, θ) =Â 2 2 ˆf [ 3 1 (θ) n(n+1) ˆf n 2 ˆfn (θ)+(n 1)(n+2) n ĝ n (θ), 3 1 (θ)+ n(n+1) n 2 ˆfn (θ)+(n 1)(n 2) n n (θ) ˆσ θ (, θ) = Â 2 ˆf 3 1 (θ)+ n(n+1) n 2 ˆfn (θ)+(n 1)(n 2) n n (θ) ˆσ θ (, θ) = Â 2 ˆf 3 1 (θ)+ n(n+1) n 2 ˆfn (θ)+(n 1)(n 2) n n (θ) ˆσ θ (, θ) = Â 2 ˆσ θ (, θ) ˆf 3 1(θ)+ (n 1) n 2 ˆf n (θ)+(n 1) n n(θ) ˆσ θ (, θ) ˆf 3 1 (θ)+ n(n+1) n 2 ˆfn (θ)+(n 1)(n 2) n n (θ) ˆσ θ (, θ) = Â ˆf 3 1(θ)+ (n 1) n 2 ˆf n (θ)+(n 1) n n(θ) ˆσ θ (, θ) ˆf [ 3 1 (θ)+ n(n+1) n 2 ˆfn (θ)+(n 1)(n 2) n ĝ n (θ), ˆf 3 1(θ)+ (n 1) n 2 ˆf n (θ)+(n 1) n n(θ) ˆσ θ (, θ) ˆf 3 1(θ)+ (n 1) n 2 ˆf n (θ)+(n 1) n n(θ) ˆσ θ (, θ) = 2 ˆf [ 3 1(θ)+ (n + 1) n 2 ˆf n (θ)+(n 1) n ĝ n(θ).

6 18 Vldo Lubd s q s ^ s q ^ s q q = ˆσ (, θ) = σ(, θ) ˆσ θ (, θ) = σθ (, θ) σ (, θ) σθ (, θ) = = σ(, θ) σθ (, θ) = ˆσ (, θ) = σ(, θ) ˆσ θ (, θ) = σθ (, θ) Â ˆf 1 (θ) = 2A 2f 1 (θ), = [ n(n + 1) n 2 ˆfn (θ)+(n 1)(n + 2) n ĝ n (θ) [ n(n 1) n 2 f n (θ)+(n + 1)(n 2) n g n (θ). 2 3 ˆf 1(θ) =2f 1(θ), [ (n+1) n 2 ˆf n (θ)+(n 1) n ĝ n(θ) = [ (n 1) n 2 f n(θ)+(n+1) n g n(θ).

7 n the Kienzle-Dun Fomul fo the Hoop Stess ound iul Void 19 Â = 2 2 A, Â 1 = 4 A 1, Ĉ 1 = 4 1, n(n + 1) n 2 Â n (n 1)(n + 2) n ˆBn = n(n 1) n 2 A n +(n + 1)(n 2) n B n, n(n + 1) n 2 Â n + n(n 1) n ˆBn = n(n 1) n 2 A n + n(n + 1) n B n. (Ân, ˆB n ) (Ĉn, ˆD n ) ( n,d n ) n 2 Â n =(n 1) n 2 A n + n n B n, n ˆBn = n n 2 A n (n + 1) n B n, (Ĉn, ˆD n ) ( n,d n ) [ ˆσ θ (, θ) =2A +2f 1 (θ)+ 3n(n 1) n 2 f n (θ)+(3n 2)(n+1) n g n (θ). ˆσ θ (, θ) =2A + σ θ(, θ) 2σ (, θ), 2A = 1 2 [σ θ(,θ)+σ (,θ). σ θ (, θ) =σ θ(, θ)+ˆσ θ (, θ), σ θ (, θ) = 2[σ θ(, θ) σ (, θ) [σ θ(,θ)+σ (,θ).

8 2 Vldo Lubd b = σ (,) = 2[σ (,) σ (,) σ (,) = = = σ (,) σ (,) σ τ σ (,)=σ σ (,)+σ τ (,).

9 n the Kienzle-Dun Fomul fo the Hoop Stess ound iul Void 21 s s s s = σ σ = ˆσ (,)= σ σ (,)+σ τ (,). ˆσ (,) σ (,)= 2σ σ (,), σ = σ (,). σ σ (,)=σ (,). ˆσ (,)=σ (,) 2σ (,). σ (,) σ (,)=σ (,)+ˆσ (,),

10 22 Vldo Lubd s s s =s s = = σ(,)=σ (,) σ (,) 1 s (-,).8 σ/σ / µ(b /)/[π(1 ν) σ (, ) / σ =

11 σ (, ) / σ = µ(b /)/[π(1 ν) n the Kienzle-Dun Fomul fo the Hoop Stess ound iul Void 23 σ (,) = 2[σ (,) σ (,). b σ (, ) = µb ( 2 2 ) (1 ν) ( ) 2, µb ( ) σ (, ) = (1 ν) ( ) 2, σ (, ) = σ 4η 2 (1 + η 2 ) 2, σ = µ(b /) π(1 ν), η = / = ± σ σ (, ) = σ /2 σ (, ) =

12 24 Vldo Lubd J 1 J 2 L M

13 n the Kienzle-Dun Fomul fo the Hoop Stess ound iul Void 25

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