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Ovyt80

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lArt

icleCwY=Qi G=wt= ltm x@ Ov=tUB VvD u=O}t OQw;Q@

�Q=}Wv=O� �u=L] u=QO=v wQUN�OWQ= T=vWQ� � VwNrO u=UL=

R=wy= u=QtJ O}yW x=oWv=O 'l}v�t |UOvyt xwQo '|UOvyt |xOmWv=O

wQ u}ty R= "OwW|t sUH Ctw=kt Vy=m Ea=@ |y=o sUH QO Ov=tUB |=yVvD OwHwu}= |=Q@ "CU= Q=OQwNQ@ |Q=}U@ C}ty= R= CavY QO =yVvD u}= OQw;Q@ w X}NWD

xO=iDU= VwQ QO "OQm xO=iDU= ?QNtQ}e |=yV}=tR; QO CwY=Qi G=wt= R= u=wD|t Qw_vtQ=WDv= CaQU C=Q}}eD 'CU= O=wt 1l}DUq=wDUwm ; X=wN Q@ |vD@t xm CwY=Qi G=wt= R=u}= QO "OvwW|t |UQQ@ 'Ov=tUB |=yVvD u=O}t R= Qw@a s=ovy CwY=Qi u=UWm G=wt=

Q=Qk VvD CLD xm OwOLt=v |Oa@xU u=UWm \}Lt l} QO xQP CmQL CqO=at 'j}kLDxS} w Q}O=kt |rm |xr�Ut 'CmQL CqO=at |xOW|]N |xir-wt xU "OvwW|t |UQQ@ 'OQ=O

=Q VvD CLD s=UH= QO Q=WDv= p@=k G=wt= CaQU 'xS} w Q}O=kt u}= "OvyO|t p}mWD =Qw VvD ?ULQ@ CaQU C=Q}}eD 'xOt; CUO x@ CqO=at R= xO=iDU= =@ "OvyO|t p}mWDQO u; Q=WDv= CaQU Q@ Gwt V@=D |x} w=R Q}F-=D u}vJty w 'sw}v}twr; QO |Oa@l} VvQm

"CU= xOW xO=O u=Wv Oqwi w sw}v}twr;

naderan k@scu.ac.irehsan.delkhosh@gmail.com "xD}U}DUq=wDUwm ; 'Ov=tUB VvD 'CwY=Qi G=wt= '?QNtQ}e uwtR; %|O}rm u=oS=w

xtOkt "1u}= u}}aD |=Q@ |}=yVqD 'Ov=sUH QO Gwt CmQL =} R=wQB u=tR |Q}oxR=Ov= Q@|=ylU}O 'xO}tN |=yjQw 'xOW |Q=mWwH C=a]k w K]Ut |=yjQw QO =yVvDxDiQo CQwY [14�7]|Oqwi |=yxr}t w =yxrwr w C=a]k |Q=mWwH pLt 'Q:wOt

"CU=CaQU C=Q}}eD Qou=Wv xm CU= |}x]@=Q uOQw; CUO x@ Q[=L j}kLD hOyCQwYx@\}Lt uDiQo Q_v QO =@ "OW=@ Ov=tUBVvD Qw[L QO 'sUH QO G=wt= Q=WDv=R= xO=iDU= =@ 'VvD CLD sUH QO |tHL G=wt= Q=WDv= CaQU |x]@=Q '|y=vDt=vGwt OwQw |x} w=R Q}F-=D u}vJty [15]"OwW|t G=QNDU= 2OwOLt |v=UWm |x} Q_vu=tR w |@Uv CaQU 'xD}U}DUq=wDUwm ; ?}=Q[ 'sUH uwQO R= u; Qw@a CaQU Q@=@ |}xLiY G=wt= "OQ}o|t Q=Qk xar=]t OQwt VvD u=O}t =@ u; \=@DQ= w Gwt R=wQBOvwW|t QWDvt u; QO xm |]}Lt Q=DN=U QO xm OvwW|t ZQi lJwm |xvt=O

"�Ov=COWsm G=wt=� Ovvm|tv O=H}= |Q}}eD

Ov=tUB VvD |=Q=O sUH QO |tHL G=wt= Q=WDv= "2|tHL |=ywQ}v uDiQo Q_v QO uwO@ 'Xi |O=t C=YDNt =@ xQPl}CmQL CqO=at

[1]%CWwv u}vJ u=wD|t =Q@�ij@xj

= �0@2ui@t2 i; j = 1; 2; 3 (1)

'xDi=}v pmW Q}}eDCr=L QO|r=oJ �0 'x]kv QOVvD QwUv=D |=yxir-wt �ij u; QO xm

?} Q[ ?=NDv= x@ VvD |ak=w u=R}t R= `q]= sOa 'C=a]k |L=Q] w p}rLD QOQO Ov=tUB |=yVvD |xR=Ov= w =DU=Q X}NWD "Ot=Hv=|t C=a]k |q=@ u=v}t]=?=NDv= =Q ?U=vt u=v}t]= ?}=Q[ =D Ovm|t ltm u=L=Q] w u=UOvyt x@ s=UH=OQ}o|t Q=Qk xO=iDU= OQwt Ov=tUB VvD |Q}oxR=Ov= |=Q@ |OOaDt |=yVwQ "Ovvm\=kv 'OOaDt |=yC}r@=k w =yR=}Dt= R= |Q=OQwNQ@ seQ|ra =yu; R= l} Qy xmX=wN Q@ xm CU= CwY=Qi G=wt= R= xO=iDU= '=yVwQ u}= R= |m} "OvQ=O R}v |ia[|=yVvD x@ 'G=wt= u}= Q=WDv=CaQU u; QF= Q@ w xOw@ Q=wDU= O=wt |xD}U}DUq=wDUwm ;

"OQ=O |oDU@ sUH QO OwHwtu=UWm |=yC@=F x@ swUwt?}=Q[ R= xO=iDU= wCmQL CqO=at uDiQo Q_v QO =@h}YwD |=Q@ 'p}Uv=DB |SQv= `@=D QO xm �� �2 w 1 pw=OH� swU w swO |x@DQt�� [1]OvwW|t h} QaD OwOLt |=ypmW Q}}eD p}rLD QO OQov=Uty |xO=t Q=DiQQO =Q |L]U w |WQ@ '|rw] |m}v=mt G=wt= CaQU Q@ VvD Q}F-=D u=R}t u=wD|tCQw=Ht QO sy w sHL QO sy Ov=tUB |=yVvD xm =Hv; R= [4�2]"OQm OQw;Q@ s=UH==D xDiQo CQwY |}=yVqD 'OvwW|t `} RwD CN=wvm} Q}e Qw]x@ s=UH= Kw]UuDW=O [5]"OwW Q}PBu=mt= CN=wvm} Q}e VvD |=yu=O}t |Q}oxR=Ov= w X}NWDR= &CU= |QwQ[ Ov=tUB |=yVvD |@=} RQ= |=Q@ swU |x@DQt |v=UWm C@=F Q}O=kt|UOvyt QO OQ@ Q=mQB O=wt R= |[a@ |=Q@ ?}=Q[ u}= j}kO u}}aD Qw_vtx@ wQ u}='xQ=W= OQwt |=yVwQ |Q}oQ=mx@ =@ [6]"CU= xDiQo s=Hv= |O=} R |@ QHD |=yQ=m|vD@t ,qwtat xm Ov=tUB |=yVvD |Q}oxR=Ov= ?U=vt C=R}yHD R= xO=iDU= R}v w

pw�Ut xOvU} wv �"1391 4 4 VQ}PB '1391 1 19 x}LqY= '1390 10 17 Ci=} QO %M} Q=D

CwY=QiG

=wt=lt

mx@Ov=t

UBVvDu

=O}tOQw

;Q@%OW=@ Q=QkQ@ 7 |w=Ut=v xm u; Q@ \wQWt

uStatici � uDynamici (7)

|xrO=at CQwYx@ =Q 6 |xrO=at u=wD|t 'OW=@l}vwtQ=y GwtCmQL xm ZQi u}= =@%CWwv 8

ui = "iXi +miAei(!t�kjxj) (8)

|}=Hx@=H Q=OQ@ |O=y |=yTwv}Ummi 'Tv=mQi ! '|rY= |=yVvQm "i u; QO xmO=OQ=Qk w CU= Gwt Q=OQ@ |=yxir-wt kj w |}=Hx@=H |xvt=O A '|rY= |=yQwLt =@u=UWm |}=Hx@=H Qou=Wv pw= |xrtH "CU= jO=Y j QQmt T}Ov= |wQ \ki `tH"CU= xDiQo Q=Qk VvD u=O}t CLD xm CU= |OQov=Uty Ot=H sUH QO x}rw=|}xLiY Gwt Q=WDv= QF=Q@ xm CU= |mJwm Q=}U@ pmW Q}}eD Qou=Wv swO |xrtH(kj) Gwt Q=OQ@ |=yxir-wt "CU= xOt; OwHw x@ OQov=Uty=v sUH l} u=}t R=CQwYx@ lj Gwt Q=OQ@ |O=y |=yTwv}Um w �K� Gwt OOa ?ULQ@ u=wD|t =Q'|}=[i w |O=t C=YDNt u}@ |x]@=Q R= xO=iDU= =@ "CWwv kj = Klj = 2�

� lj%OQw; CUO x@ |}=Hx@=H |=yxir-wt ?ULQ@ u=wD|t =Q 8 |xrO=at

ui = "iXi +miAei(!t�K((1+"1)X1l1+(1+"2)X2l2+(1+"3)X3l3))

(9)

xDWwv u=mt Q}}eD |} RH C=kDWt ?ULQ@ xm =Q pY=L |]NQ}e CmQL CqO=at|]N u=wD|t |m}D=DU= pmW Q}}eD C}akwt pwL Qwr}D \U@ R= xO=iDU= =@ 'OvwW|t'OW=@ VvD CLD sUH QO Gwt Q=WDv= CaQU Qou=}@ V = Cp = !

K Qo = "OQm|=yC@=F w |rY= |=yVvQm ?ULQ@ Ci w Bi 'Ai |=yQDt=Q=B |Ov@xDUO =@CqO=at u}= 'Ov=xOW xDWwv CUw}B QO xm |}xvwox@ 'swU w swO |x@DQt |v=UWm%OW Oy=wN xDWwv 10 CqO=at j@=]t X3 w X2 'X1 |=DU=Q xU QO xOW|]N

m1(A1 +A2 +A3) +m2A4 +m3A5 = 0m1B1 +m2(B2 +B3 +B4) +m3B5 = 0m1C1 +m2C2 +m3(C3 + C4 + C5) = 0 (10)

CmQL CqO=at pL "3XNWt |xLiY l} QO Gwt Q=WDv= "1"3

QDxO=U =Q CmQL CqO=at u=wD|t 'OwW QWDvt XNWt |xLiY l} QO Gwt Qo =u=vJ �1 pmW� |DQ=mO C=YDNt x=oDUO QO =Q xar=]t OQwt sUH Qw_vtu}O@ "OQm

"Gwt Q=WDv= C=LiY V}=tv w C=YDNt x=oDUO "1 pmW

"Oqwi w sw}v}twr; swO |x@DQt |v=UWm |=yC@=F "1 pwOHv

E � �

GPa GPa GPa

0 34 71 26 48 59 28 sw}v}twr;0 28 220 85 110 Oqwi

"Oqwi w sw}v}twr; swU |x@DQt |v=UWm |=yC@=F "2 pwOHn m l

GPa GPa GPa

�408 �401 �311 sw}v}twr;�710 �580 �270 Oqwi

`tH O=OQ=Qk "CU= xi |}=[i C=YDNtx@ Xi R= |}=Hx@=H u=O}t |=yxir-wt ui w"xi = xi(X1; X2; X3) w ui = xi �Xi w xOw@ Q=QkQ@ |Q=QmD T}Ov= |wQsUH |v=UWm |WvQm |SQv= |r=oJ `@=D R= |Q}ojDWt =@ VvD QwUv=D |=yxir-wt(E) |Sv=Qo q OwOLt VvQm |=yxir-wt ?ULQ@ '(�) xDi=}v pmW Q}}eD Cr=L QO

%OvwW|t xDWwv 2 |xrO=at CQwYx@[�] = [J]

�@'@E

�(2)

CQwYx@ R}v (I) OL=w QwUv=D w(J) 3u}@ wm =S?ULQ@ |Sv=Qo q OwOLtVvQm QwUv=D%OwW|t xDWwv 3 |xrO=at

E = 12 (JTJ� I) (3)

%CU= x@U=Lt p@=k 4 |x]@=Q R= u}@ wm =S T} QD=t[J] =

�@xi@Xi

�(4)

Q=Okt xU R= |a@=D CQwYx@ OQov=Uty u=UWm Ot=Hl} |=Q@ |WvQm |SQv= |r=oJ[2w1]%OwW|t xDWwv |Sv=Qo q VvQm QwUv=D I3; I2; I1 |=}=B

� = l + 2m3 I31 + �+ 2�

2 I21 � 2mI1I2 � 2�I2 + nI3 (5)

|x@DQt |v=UWm |=yC@=F s=vx@ xO=t O}OH C@=F Q=Okt xU n w m w l u; QO xmw sw}v}twr; |=Q@ Q}O=kt u}= 2 pwOH QO "OvwW u}}aD O}=@ V}=tR; =@ xm OvDUy swU

"Ov=xOW x�=Q= OqwipY=L |=yCQ=@a |v} Ro}=H w '2 |xrO=at QO 5 =D 3 CqO=at uO=OQ=Qk =@X3;X2;X1 |=yCyH QO |]NQ}e CmQL CqO=at '1 |xrO=at QO =yVvD |=Q@

"O};|t CUO x@"1 %OwW xDiQo Q_v QO VN@ wO R= |@}mQD \}Lt R= x]kv Qy |}=Hx@=H Qo =Ov=tUB VvD u=O}t l} OwHw QF= Q@ CU= umtt xm (uStatici ) |m}D=DU= VN@Q=WDv= QF= Q@ xm (uDynamici ) |m}t=v}O VN@ "2 &OW=@ xOW O=H}= x}rw= VvD =}u} RDQ=m C=YDNt QO |}=Hx@=H |=yxir-wt 'O};|t OwHw x@ xvt=Osm |}xLiY G=wt=

%CWwv u}vJ u=wD|t =QuTotali = uStatici + uDynamici ; i = 1; 2; 3 (6)

76

VwNrO

u=UL=

wu=L]

u=QO=vwQ

UN

=yQwLt R=|m} O=ODt= QO Gwt|Dkw '|[Qa w|rw] G=wt= Q=WDv= |=DU=Q "2 pmW"O@=D@

QWDvt VvD O=ODt= QO Gwt Qo = sw}v}twr; QO |rw] G=wt= CaQU C=Q}}eD "3 pmW"OwW

QWDvt VvD O=ODt= QO Gwt Qo = sw}v}twr; QO |[Qa G=wt= CaQU C=Q}}eD "4 pmW"OwW

w l1 = 1 'OwW QWDvt �VvD =@ =DU=Qsy� x QwLt O=ODt= QO ,qFt Gwt Qo ='xS} w Q}O=kt |R=UxO=U wCmQL CqO=at QO Q}O=kt u}= |v} Ro}=H =@ "l2 = l3 = 0Q=WDv= |=DU=Q Qou=Wv CaQU pw= T}Ov= =yu; QO xm O};|t CUO x@ 16 \@=wQ|=yC@=F� CU= sUH C=QP CmQL =} V=aDQ= |=DU=Q Qou=Wv swO T}Ov= w 'Gwt

%�Ov=xOW h} QaD CUw}B QO ai�0V 2

ii = a1 + a2"i + a3 ("j + "k)

�0V 2ij = �+ a5"i + a4"j + a6"k

�0V 2ik = �+ a5"i + a6"j + a4"k (16)

OwtVW w|rw] Owt xU |=Q@ G=wt=CaQU C=Q}}eD '4 w 3 |=ypmW |=yQ=Owtv QOuwQO VvD O=ODt= QO Gwt xm |DQwY QO '|Oa@l} VvQm =} VvD x@ C@Uv |[Qa

"OW=@ z; y; x |=yQwLt Q@ j@]vt ?}DQDx@ 3 w 2 '1 |=yCyH xm s} Q}o|t Q_v QO%Ovm|t jOY 11 |x]@=Q QO Gwt Q=OQ@ |O=y |=yTwv}Um

l21 + l22 + l23 = 1 (11)

u}vJ xQP CmQL CqO=at w l2 = 0 'OwW QWDvt xz |xLiY QO Gwt xJv=vJ%OvwW|t xDWwv2664 G11 � �0V 2 0 G13

0 G22 � �0V 2 0G31 0 G33 � �0V 2

37752664 m1m2m3

3775 = 0

(12)

"Ov=xOW xDWwv CUw}B QO Gij Q}O=ktu=v}tQDO O}=@ OW=@ xDW=O |y}O@ Q}e ?=wH 12 CqO=at x=oDUO xm u}= |=Q@QO Q=WDv= p@=k G=wt= |xYNWt |xrO=at ?}DQD u}= x@ "OW=@ QiY Q@=Q@ u; ?}=Q[12 CqO=at x=oDUO |xS} w Q}O=kt 'xrO=at u}= |=yxW} Q "O};|t CUO x@ sUH|Owta |[Qa Owt l} '�SH� |ki= |[Qa Owt l} Q=WDv= CaQU w OvDUy

%Ov};|t CUO x@ �L� |rw] Owt l} w �SV �

�1 = �0V 21 = 12�G11 +G33 +

q(G11 �G33)2 + 4G213

��2 = �0V 22 = G22

�3 = �0V 23 = 12�G11 +G33 �

q(G11 �G33)2 + 4G213

�(13)

"OQm |]N w O=O \U@ Qwr}D |QU R= xO=iDU= =@ =Q =yCQ=@a u}= R= l} Qy u=wD|t%p=Ft u=wvax@

�(xz)3 = �0V 23 = f (")

= 12�G11 +G33 �

q(G11 �G33)2 + 4G213

�= f (0) +

��@f@G11

@G11"i

+ @f@G13

@G13"i

+ @f@G33

@G33"i

��"i

(14)

[16]'CU= xOW s=Hv= R}v Qo}O |xLiY wO QO G=wt= Q=WDv= |=Q@ jwi C=}rta x@=Wt"OwW|t |Q=OOwN C=@U=Lt C=}�RH uOQw; R= Q=YDN= C}=aQ |=Q@ |rw

|Oa@l} VvD Cr=L "2"3=Q CmQL CqO=at u=wD|t 'sUH QO (�) |Oa@l} VvD u=O}t OwHw CQwYQO?} Q[ uDiQo Q_v QO =@ 'CU= ?U=vDt VvQm =@ VvD xm Cr=L u}= |=Q@ "OQm QDxO=U

%OwW|t xDWwv u}vJ =yVvQm '(E) nv=} pwOt w (�) uwU=wB"1 = " = �

E"2 = "3 = ��" (15)

|rY= |=yCyH O=ODt= QO CU= umtt OwW OQ=w sUH K]U Q@ Owta Gwt Qo =|rY= CyH xU QO |[Qa w |rw] G=wt= Q=WDv= |=DU=Q 2 pmW QO "O@=} Q=WDv=

"CU= xOW xO=O u=Wv |Oa@l} VvD x@ C@Uv z; y; x

77

CwY=QiG

=wt=lt

mx@Ov=t

UBVvDu

=O}tOQw

;Q@

"sw}v}twr; |=Q@ |Oa@l}VvDCr=L QO G=wt=CaQU|@Uv C=Q}}eD Q=Owtv "5 pmW

C=Q}}eD u=wD|t 'OwW Q=QmD `]kt hrDNt \=kv |=Q@ Ov}=Qi u}= Qo = "OQw; CUO x@ =Q"OQm s}UQD =Q `HQt K]U l} =D hrDNt \=kv |xrY=i ?ULQ@ VvD

Gwt CaQU Q@ Q=WDv= |x}w=R Q}F-=D "4"3\@=wQ R= u=wD|t |rw] w |WQ@ G=wt= CaQU Q@ Q=WDv= |x} w=R Q}F-=D |xar=]t |=Q@Qo = p=Ft u=wvax@ "OQm xO=iDU= 'OvW=@ xOW ?DQt |Oa@l} VvD ?ULQ@ xm 13'6 '|rw] Gwt |=Q@ w xHQO 25 w 20 '15 'z QwLt =@ |WQ@ Gwt Q=WDv= |x} w=Rj@] Oqwi w s}v}twr; QO |WQ@ G=wt= CaQU C=Q}}eD Q=Owtv 'OW=@ xHQO 12 w 9

"OW Oy=wN 7 pmW j@] |rw] G=wt= |=Q@ w 6 pmW

"Q=WDv= |x}w=R Q}}eD QF= Q@ Oqwi w sw}v}twr; QO|WQ@ GwtCaQU C=Q}}eD "6 pmW

"Q=WDv= |x}w=R Q}}eD QF= Q@ Oqwi w sw}v}twr; |rw] Gwt CaQU C=Q}}eD "7 pmW

�0 = 2730 kg=m3 |r=oJ =@|tw}v}twr; sUH |=Q@ '�2 pmW� O@=} Q=WDv= sUHVvD |=DU=Q QO |rw] Gwt CaQU '3 pmW Q=Owtv x@ xHwD =@ "CU= xOW sUQGwt |=Q@ xm |r=L QO O@=}|t Vy=m VvQm =} VvD V}=Ri= QF= Q@ '|WWm |QwLt4 pmW |=yQ=Owtv "O@=}|t V}=Ri= VvD |=DU=Q Q@ Owta C=yH QO xQWDvt |rw]w Q=WDv= O=ODt= xm u}= x@ xDU@ '|WQ@ =} |[Qa G=wt= CaQU xm OvyO|t u=Wvu}= "Ovvm|t Q}}eD 'OvW=@VvD O=ODt= Q@ j@]vt =} Owta =yu; |=Q@\}Lt C=QPCmQLj@]vt Qo}Om} Q@ uQ=kD Crax@ =yOwt R= |[a@ CaQU xm OvyO|t u=Wv =yQ=Owtv

"CU= xOW |UQQ@ "4"3 VN@ QO G=wt= CaQU Q@ Q=WDv= |x} w=R Q}F-=D "OwW|t

|@Uv CaQU C=Q}}eD "3"3'CwY=Qi G=wt= ltmx@ VvD |Q}oxR=Ov= VwQ QO xHwD OQwt |=yQDt=Q=B R= |m}|x]@=Q j@=]t "CU= |QwLt VvQm =} VvD ?ULQ@ Gwt CaQU |@Uv C=Q}}eD

%R= CU= CQ=@a |@Uv CaQU 17

�Vij =Vij � V 0

ij

V 0ij

(17)

C@Uv 17 |x]@=Q R= xJv=vJ "CU= VvD uwO@ \}Lt QO Gwt CaQU V 0ij u; QO xm

%CW=O s}y=wN OwW xDiQo jDWt (") VvQm x@d�V11d"

= a2 � 2a3�2a1

= L11

d�V13d"

= a5 � (a6 + a4) �a5 � a4

= L13

d�V31d"

= a4 � (a5 + a6)�a5 � a4

= L31

d�V32d"

= a6 � (a6 + 2a5 � a4)�a5 � a4

= L32

d�V33d"

= a3 � (a2 + a3)�2a1

= L33 (18)

h} QaD CUw}B QO ai |=yC@=F w OvwW|t xO}t=v xD}U}DUq=wDUwm ; ?}=Q[ Lij|x@DQt |v=UWm |=yC@=F R= xO=iDU= =@ sw}v}twr; |=Q@ ?}=Q[ u}= Q}O=kt "Ov=xOW�2w 1 pw=OH QO GQOvt� 7900kg/m3 |r=oJ =@ Oqwi w sw}v}twr; swU w swO

"Ov=xOW GQO 3 pwOH QO w x@U=LtQO w sw}v}twr; |=Q@hrDNt C=yH QO |@UvCaQU C=Q}}eD '\@=wQ u}=ltm x@u} QDW}@ 'pmW u}= |=yQ=Owtv QO "Ov=xOW sUQ 5 pmW QO |Oa@l} VvD Cr=LQO "CU= �0 6 R= QDsm 'sw}v}twr; QO 0 001 VvQm |=R=x@ |@Uv CaQU C=Q}}eD'Gwt Q=WDv= CaQU |=Hx@ 'OQ=O Q=Qk |Oa@l} VvD CLD sUH xm |}=yV}=tR;QO xvwtv QO Gwt Q=WDv= u=tR COt "OwW|t |Q}oxR=Ov= Gwt CmQL =} R=wQB u=tRu=tR �t 'xa]k Ct=N[ h u; QO xm 'V3 = 2h

�t %=@ CU= Q@=Q@ 4wm =� Tr=B VwQu=tR |Q}oxR=Ov= =@ "CU= xz |xLiY QO |[Qa Gwt CaQU V3 w 'Gwt R=wQBj} Q] R= "OwW|t x@U=Lt Gwt CaQU 'Ovm|t CmQL Gwt xm |}xrY=i w Gwt R=wQBu; Q}_v VvD u=wD|t '4 w 3 |=ypmW QO =yCaQU QwLt |wQ CaQU u}= p=kDv=

"sw}v}twr; |xD}U}DUq=wDUwm ; ?}=Q[ "3 pwOHL33 L32 L31 L13 L11

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UN OW=@ |WWmVvD xm |Dkw Gwt Q=WDv=CaQU 'CU=VvD O=ODt= Q@ Owta =yu; |=Q@"OvwW|t O}}-=D R}v [15w2]OwHwt \@=wQ |UQQ@ =@ '=y|Q}oxH}Dv u}= "O@=}|t V}=Ri="Ow@ Oy=wN TmaQ@ xH}Dv OvwW QWDvt sUH K]U x@ C@Uv x} w=R CLD G=wt= Qo =CwY=Qi Gwt CaQU |x@U=Lt w xD}U}DUq=wDUwm ; |xO}OB R= xO=iDU= =@|Q}oxR=Ov= =Q xO=t QO OwHwt |m}v=mt |=yVvD Q=Okt u=wD|t 'sUH R= |Qw@a|=yQ=Owtv R= xm u=vJ "OW=@ Ov=tUB VvD =} |r=ta= VvD Ov=wD|t VvD u}= "OQm|]N |a@=D CwY=Qi G=wt=CaQU C=Q}}eD 'CU=O}BVvD w VvQm?ULQ@CaQU|@Uv CaQU C=Q}}eD ?}W |xU}=kt R= "CU= sUH QO OwHwt VvD w VvQm R=VvD O=ODt= QO xm |rw] G=wt= |=Q@ |@Uv CaQU C=Q}}eD xm OwW|t x_Lqt=y|UQR=@ QO OwW|t x}YwD =Pr 'CU= QDW}@ =yOwt Q}=U x@ C@Uv OvwW|t QWDvthrDNt |=yVvQm QO Gwt R=wQB u=tR uwJ "OQ}o Q=Qk C} wrw= QO Owt u}= R= xO=iDU=

"O}; pta x@ CkO C}=yv u; |Q}oxR=Ov= QO O}=@ 'CU= lJwm |OOa

s}v}@|t =yQ=Owtv u}= QO 'OwW|t \=@vDU= 4 w 3 |=ypmW R= xJv; Tmax@CaQU 'OvwW QWDvt xvwtv K]U x@ C@Uv QiY R= Q}e |}x} w=R CLD G=wt= Qo = xm

"O@=}|t V}=Ri= |rw] G=wt= CaQU w Vy=m |WWm |=yVvD =@ |WQ@ G=wt=

|Q}oxH}Dv "4CwY=Qi G=wt= CaQU xm CU= u; R= |m =L VywSB u}= QO xOt; CUO x@ G}=Dvw VvD |=DU=Q x@ C@Uv Gwt Q=WDv= O=ODt= 'VvD |xR=Ov= w CyH 'xO=t TvH x@O=ODt= xJv=vJ "OQ=O |oDU@ �Gwt uOw@ |[Qa =}|rw]�\}Lt C=QPCmQL O=ODt=|=Q@ w Vy=m |rw] Gwt CaQU '|WWm VvD |=Q@ OW=@ u=Um} VvD w Gwt Q=WDv=C=QP CmQL O=ODt= xm |WQ@ G=wt= |=Q@ "O@=}|t V}=Ri= u; CaQU '|Q=Wi VvD

=yCWwv=B1. Acoustoelasticity2. �nite deformation theory3. Jacobian4. Pulse-Eco

�References� `@=vt1. Murnaghan, F.D., Finite Deformation of an Elastic

Solid, Applied Mathematics Series, Wiley, New York(1951).

2. Hughes, D.S. and Kelly, J.L. \Second-order elastic defor-mation of solids", Physical Review, 92(5), pp. 1145-1149(1953).

3. Husson, D. and Kino, G.S. \A perturbation theory foracoustoelastic e�ects", J. Appl. Phys., 53(11), pp. 7250-7258 (1982).

4. Husson, D. \A perturbation theory for acoustoelastic ef-fect of surface waves", J. Appl. Phys., 57(5), pp. 1562-1568 (1985).

5. Delsanto, P.P., Olivero, G. and Scalerandi, M. \Acousto-elastic e�ects in elastic media with nonuniform initialstress", Res Nondestr Eval, 12, pp. 105-118 (2000).

6. Crecraft, D.I. \The measurement of applied and residualstresses in metals using ultrasonic waves", Journal ofSound and Vibration, 5(1), pp. 173-192 (1967).

7. Kudryavtsev, Y. and Kleiman, J., Ultrasonic Techniqueand Device for Residual Stress Measurment, IntegrityTesting Laboratory Inc., 80 Esna Park Drive, Units 7-9,Markham, Ontario, L3R 2R7 Canada.

8. Kumar, R. and Partap, G. \Wave propagation incircumferen-tial direction in micropolar cylindricalcurved plate", International Journal of Mechanics andSolids, ISSN 0973-1881, 4(1), pp. 117-125 (2009).

9. Blessing, G.V., Hsu, N.N. and Proctor, T.M. \Ultrasonicshear wave measurement of known residual stress in alu-minum", Experimental Mechanics, 24(3), pp. 218-222(1984).

10. Qu, J., He, Y., Yang, Y., Wang, Y. and Tian, C. \Re-search on ultrasonic measurement for the welding resid-ual stress in the ring welded seam for a RPV plate",Transactions of the 14th International Conference onStructural Mechanics in Reactor Technology (SMiRT14), pp. 417-424 (1997).

11. Tanala, E., Bourse, G., Fremiot, M. and De Belleval,J.F. \Determination of near surface residual stresses onwelded joints using ultrasonic methods", NDT & EInternational, 28(2), pp. 83-88 (1995).

12. Akhshik, S. and Moharrami, R. \Improvement in ac-curacy of the measurements of residual stresses dueto circumferential welds in thin-walled pipe usingRayleigh wave method", Nuclear Engineering and De-sign, 239(10), pp. 2201-2208 (2009).

13. Rizzo, P. and Lanza di Scalea, F. \E�ect of frequency onthe acoustoelastic response of steel bars", ExperimentalTechniques, pp. 40-43 (November-December 2003).

14. Schmerr, L.W., and Song, S.-J., Ultrasonic Nondestruc-tive Evaluation Systems: Models and Measurements,Springer, New York (2007).

15. Rose, J.L., Ultrasonic Waves in Solid Media, CambridgeUniversity Press, Cambridge, U.K., New York (1999).

16. Delkhosh, E. \Investigation of residual stresses in ma-terials using ultrasonic waves", M.Sc. Thesis, Faculty ofEngineering, Shahid Chamran University of Ahwaz, Iran(2011).

79

CwY=QiG

=wt=lt

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UBVvDu

=O}tOQw

;Q@

A4 = l1l2(a0 + a7("1 + "2) + a8"3)

A5 = l1l3(a0 + a7("1 + "3) + a8"2)

B1 = A4

B2 = ��0V 2 + l21(�+ a5"1 + a4"2 + a6"3)

B3 = l22(a1 + a2"2 + a3("1 + "3))

B4 = l23(�+ a6"1 + a4"2 + a5"3)

B5 = l1l3(a0 + a8"1 + a7("3 + "2))

C1 = A5

C2 = l2l3(a0 + a7("2 + "3) + a8"1)

C3 = ��0V 2 + l21(�+ a5"1 + a6"2 + a4"3)

C4 = l22(�+ a6"1 + a5"2 + a4"3)

C5 = l23(a1 + a2"3 + a3("1 + "2))

G11 = A1 +A3 + �0V 2

G13 = A5

G22 = B2 +B4 + �0V 2

G33 = C3 + C5 + �0V 2

G31 = G13 = C1 = A5

CUw}BCqO=at ?}=Q[ w �=y�i� O=wt swU w swO |x@DQt |v=UWm |=yC@=F |Ov@xDUO

%�Gij w Ci 'Bi 'Ai� CmQL

a0 = �+ �

a1 = �+ 2�a2 = 5�+ 10�+ 2l + 4ma3 = �+ 2la4 = �+ 2�+m

a5 = �+ 4�+m

a6 = �+m� 12n

a7 = 2 (�+ �) + 2l +m

a8 = �m+ 12n+ 2l

A1 = ��0V 2 + l21(a1 + a2"1 + a3("2 + "3))

A2 = l22(�+ a4"1 + a5"2 + a6"3)

A3 = l23(�+ a4"1 + a6"2 + a5"3)

80

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h RESIDUAL STRESS FIELDESTIMATION BY ULTRASONICWAVESK. Naderan Tahan(corresponding author)naderan k@scu.ac.irE. Delkhoshehsan.delkhosh@gmail.comDept. of Enginnering College of Mechanics,Shahid Chamran University of Ahvaz

Sharif Mechanical Engineering JournalVolume 29, Issue 2, Page 75-80, Original Article

c Sharif University of Technology

� Received 7 January 2012; received in revised form 7 April2012; accepted 24 June 2012.

AbstractIn certain cases, the presence of residual stress reducesthe strength of parts, hence, its detection and estima-tion is of great importance. Ultrasonic waves have thepotential to be employed in nondestructive tests to dothis task. The analysis is based on the acoustoelasticityproperties of the material that studies changes in wavevelocity when passing through a stressed medium. Toderive the equations of motion for a particle in a stressedelastic solid, it is supposed that the displacement vectorconsists of two parts: a static part, which may be devel-oped by a residual or applied stress �eld, and a dynamicpart, which is produced due to wave propagation andis considered a small amplitude harmonic motion. The�nite displacement theory is employed to derive the gen-eral form of the non linear equations of motion in threedimensions for an incident wave at various angles, whichare made linear by the Taylor expansion method. Vari-ous components of the wave velocity are computed fromlinear equations as an eigenvalue problem. Relative ve-locity changes versus one dimensional stress and strainin an aluminum block, and the e�ect of wave incidenceangle on the propagation velocity are established andplotted "It is shown that the speed of wave in a stressedelastic solid depends not only on the material proper-ties of the solid, but, also, on the wave incidence angleand the propagation and particle displacement directionrelative to the stress direction. From the presented di-agrams, it is revealed that if the direction of incidence

113

and propagation of the wave coincides with the directionof normal stress, then, the velocity of the longitudinalmode will decrease in tension and increase in compres-sion. On the other hand, the velocity of shear wave willincrease in tension and decrease in compression. Theseconclusions agree with the results in other references.Comparing the slope of relative velocity diagrams in astressed body, we conclude that the velocity of longitudi-nal waves that are incident along the normal stress com-ponent, is greater than other wave modes, although themagnitude of this change is very small for either modes;say, about 0.6 percent for longitudinal waves at 70 MPatension in aluminum. Therefore, precise measurementof the ight time of the wave is necessary.

Key Words: residual stress, ultrasonic wave, non de-structive test, acoustoelasticity.