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Analysis of a radial crack in a circular ring segment
ANALYSIS OF A RADIAL CRACK IN A CIRCULAR RING SEGMENT Army MWS
PETER G. TRACY and Mecblulics Research Center, wa-,
rbhs. 02lrr U.S.A.
INTRODUCTiON
A TEST specimen frcqucntly encountered in fracture toughness testing of shell materials is the
curved beam with crack as illustrated in Fii. 1. A typical loading system consisting of point loads at A, B, and E is shown. The circular ring segment with a crack emanating from the circular surface with the smaller radius can be considered the equivalent plane problem of elasticity. Simikr problems have been solved by Joncs[4] and[S] using the Gnitc clement approach. ‘Ibis probkm can bc neatly handld by using results obtained by Ricc[7], which arc closely rclatcd to the ‘weight function’ concepts of Bcuckncr[ I]. Rice has shown that if K1 and the displacement field arc known as functions of crack length for a primary symmetrical loading system on a body, the stress intensity factor, & can bc found for any other symmetrical loading system on the body by performing a simple integration. It is clear that the loading system consisting of point loads as illustrated in Fig. I leads only to K1 vale of stress intensity, which can be found if K1 amI the dispkccmcnt field arc known for a primary loading system. This approach obviates the need to introduce singukritks to handle the point loads and a!Iords the ability to analyze the probkm if the cffccts of dilTcrcntloading systems arc to bc analyzed in the CourWofthctcsting. The method used to analyze the primary lu&ing system is basic&y the ‘modifkd . mrrppmpcollocotion’ (MMC) plan combincd with a partitioniq~of the M. This procedure, described by Bowk et ol.[2),is used to compute stress intensity facturs and displacements for the ~loadiagqstaa,whicbisbendingcauplesMappliadatthecndsBCandADinFie.l. This tcchniquc involves the representation of the crack surface by a simpk mappingfunction and
Fig. 1. Circular ring a&ion with crack. 253
254
PETERG.TRACY
the prescription of force and stress conditions oa the boundary in a least squares sense. Since the geometry is subdivided into several regions, each with its own stress function representation, the stress functions in neighboringregions must be ‘stitched’usiag force and displacement conditions at their common boundary. ANAtYSIs
Frequent use will be made of the complex variable foraurlation of MuskhelishviIi[6]aad the required notation is briefly summarized. The functions I+(Z)aad S(z) in the Muskhelishvili forarulatioa are arm&ticfunctions of the complex variable 2 - x + iy. Ia terms of 4(z) aad #V(Z) the stresses and di@acemeats ia Cartesian coordiaates caa be expressed as UX+ a, = 2[9’(2)+ ml a, - uX+ 2ir- - Z[ff$“(z)+ S’(z)]
(1)
sad (2) where 17=3-4v
@ule strata)
E and v beii Young’smodulus aad Poisson’s ratio, respectively. Here and throughout prisms denote difIerea&ioa and the bars complex conjugatioa. The resuitaat force aloag aa arc s caa be expressed as
(3) where X ds aad Y. ds are the horixontal aad vertical compoaeats, respectively, of the force actiag aoraml to the element ds. The solution will be carried out using the modified mapping-collocation (MMC) method introduced by Bowie and NealI3.l.Thus the auxilhuy complex W-planeis introduced along with the mapping function z s u2(w) x -R,e-‘r*-(“ml
(4)
Points lyiag on a horixontal line ia the W-planeare mapped by @z(w)iato poiats lying on an arc ia the Z-plane. More specifically,points lying on the real axis in the W-plane are arapped into points lyiag on an arc of radius RI centered about the origia in the Zplaae. Points in the W-plane lying on a vertical line become poiats lyiag on a radial line from the originin the physical or Z-plane. Thus the physical geometry ia Fii. 1 corresponds to the geometry in the W-planeshown in Fig. 2. The point w corresponding to the point t = r& ia the range of odw) caa be found to be w=(f-&)+iln@-)
(5)
Analysis of a radial crack
ringsegment
a
Fig.ZTbeW-plane. by solving (4)for w. It is easily shown by using (5) that (see Figs. 1 and 2) b=28
(6)
In termsof @z(w)equations(2) and (3) can be written
@l(W)-;-
#(w)+
u:(w)4
q+(w)
_
@2(W)--;-
(w)
+
-
9(w)
= f1+
v2
-
o;oWb-W=2G(u+iu)
0
PI
904 - NB)t(wNaJ 9(w) = SC@&0 an analytic functions of the c.ompIsxvariablew.
To guaranteetraction-freeconditionson the arc AB in the physicalregion(see Fii. 1) the extukon aqgimentsof Muskhelishviliarcutilized.If S’ denotesthe regionabovetherealaxis in the W-pbm andS’ Benotesthe regionbelow,the function 4(w) is definedby
where
Thus S(w) em be written 8s 9(w) = and 0 and (7’) become
-d(w)-~~~~w), WCS’
(10)
PETER
236
0.TRACY
(11’) If #(w ) is analytic in the region of the W-plane extended regions ABCD and CDEF respectively added advantage of eliminating funCtiO&
reducing the probkm to
physical region plus the provides the only one cracks,
the physical rquires the intraductioa along with the
auxiUary
function.
+r-t-9.
FiikIlxtmddngioar.
(12)
Analysis of a radial crackina circularringsegment Z-ploll0
Do,
Dec
A
E
B
Fig. 4. Partitiowd zones.
PXXWXW&I~ as inf2] the representations for &&) and &&v) are ‘stitched’by requiring along the segment EF (See Fig. 4) that
where the nota& (fi + i&j, and (u + io), denote the forces and disph~~ments, respectively, as represented in r&on j. The representations of &(w) and Cp&) must be chosen consistent with the comlitkms outlived thus far. Taking into account a sing&&y in MC), for g = -i, din(t) is written (bII(0
= “‘4 5 f+iaJ” ’
of9
Obviously because of the stress sylnmctry about the iIxq&wy axis only the irnq&my ~~f~~v~sof II ~~~~~ci~tsforev~v~of n ~~~.We further assume dxlF@)o; mg Ga(w- WI)”
(17)
where wo is real and the midpoint of the line segment EB in Fig. 4. To carry out the MMC method antes of &(l) and #t&w) are used. Stati arc chosen along (see Fii. 4) BC, CG, and the crack surface and either $n(f) is inserted in (14)or #m(w) is inserted into (1I), whichever is appropriate, to obtain conditions between the Q and between the cl. To obtain conditions relating the at to the G, stations arc chosen along (we Fii. 4) EF and conditions written according to (ll), (1I’), (14),(14’)and (15).In general the number of conditions exceeds the munber of unknown coe&icnts of #&) and dew) and a k@ squares minimization of error is employed to obtain a War system of shnultaneous equations in the aI and cr. This system ii then solved to obtain #n(g) and 4&w). BOUNDARYCONDITIONS The bou&%y conditions in the problem are those normal stresses along the cads (see Fig. 1) AI) and BC which give rise to pyre bend&g. The form of these stresses is given by ~~~0 and Goodier[91to be
EFM Vol. 7, No. 2-E
pm
zs8
0. TRACY
where N=(R:r, = ir I,and M is the moment. Elsewhere on the boundary and crack the normal and shear stresses are zero. It can be shown that for stresses of this form -4M fzR12ln N
(19)
fi + ifi = -
NUMERICAL REsuLm
Once the coefBcientsfor &U(W)and &I({) are known, it is a simple matter to calculate K, using +&). Sii et a1.[8]have shown that K = KI - iIG = 2(2n)‘” lim (2 - &)‘n#(z) z-4 where zc is the location of the crack tip in the physical region. Equation (20) leads to
(20)
&I(l) E (+4)2(w)- 23)I/2 o;(w)o;(l)
K = 2(2#
= 2 h+ (
)
‘n+h(i).
(22)
The points { = i and w = iL correspond to the crack tip in the l- and W-planes, respectively. Because of the symmetries it follows that d&(i) and thus K are real. At this point it is convenient to introduce the function (23) where we= u,~,,_~,.The well-knownresult that for I 4 RI, HI + 1.12was used to check solutions for small cracks. Using the method descril by I&em, K, can be calculated for the body in F@ 1 subjected to the folhiqJ two point load systems:
p, L
ux =
-P,
s
2 =
z2
=
23 =
R,e’“n”-r’
(240
-if2
-P,z=z*=iR2
a? =
4P, 2’22
(249
4P, 2 = 23
As in (18)the normal and shear stresses elsewhere on the boundary and crack surface are zero. SummarMg1711.lL can be found for any symmetrical loading system on a body, if K, and the displacement field are known as functions of crack length for one symmetricalloading system on the body. This fact is shown in the following relation: K,=
Is t.hdS
(25)
Analysis of a radialcrackin a dmlar ringsegment
259
where t is the stress vector on the boundary S and h is referred to as the weight function defined by E au(l) (26) WVY, 0 = 2&w(1) y+where X1”‘(1)is the stress intensity value known for a part&x&uloading system as a function of 1 and u(l) is the displacement field for that particular loading system. For the body subjected to the conditions of (18) one can find &“‘(I) and u(l) if 4&g) and 44~) are known for a sufficient number of crack lengths. Adjusting displacements such that u = 0 at z2and z3and using the fact that the loads are point loads, it is clear that the stress intensities for a given 1 for the conditions of (24a)and (24b)are given by: PE au K1 =Kl”‘;jr I+-3
(2W
PE av Ks = 2K1”‘z I r-4)
respectively. For this case, we define
In order to obtain HI values for the loading of (24)the displacements u and u as a function of 1 were &ted to piece-wise polynomiais to facilitate the cakulation of the derivatives in (26).
-0 Fig. 5. v&es
1 /R,
ofH, = K,/u.(d)‘” for bending.
260
PETER 0. TRACY
Results for vwIona I/(& - R,) vdwa are preseti in Fii 3 for the RdR, ratha of 2,1*75,1.5 and l-23 for the loading of (18).It is clear that the direction and magnitude of the point loads at zz and 2%of (24) can be h&d by using combinations of the loading systems (24a) and (24b). Obviously, from (zs) and (26)the value of KSfor such a comb~ti~ loading is the sum of the condom from each of the loadings (24a) and (24b). In Fii 6, data is presented for the loadiqs (%a) and (24b)and a qombmatioo.The combination loadingis chosen such that the point loads at zz and zs of (24)are normal to the circular arc with greatest curvature, which implies the ma@ude is l/2@ since the value of 8 (see Fig. 1) remains tied at n/4. Acknowitdgrmartr-% auhr ti8hasto acknowledge the interest and encoutage.ment of C. E. Frame and 0.
L. B&c in
thiSWOk
[l] pi_)&“““,
A novel principle for tb8 come
of stl@a UlteMity factm. z &$cw. bffztk beck. 9). 529-546
[2] 0. L ‘i&vie, C. B. Reese aud D. M. Neal+Solutiom of pbuwproblem of claaticity utihiq qppL ius& %, 767-772 {Em).
puti-
cmcqts.
J.
[fl A. T. Jonea, Fmture tou&ma testing with sections of cyliadhl shells. SL-7MO50, Saadia Labotamh, Livmnore, Calif., Nov. 1973. IuHIMW, Some Bark Pmblems ofthe Mathmtktd IItatwy of Elastkity. NooM, Gmdngcn, Holhd @] s31M * m Jam&. Rice, Soas remarka on &tic crack-tip stress fields, ht. J. Soiids, Stfucturcs, 8,751-738 (1972). @I 0. C. Sib. P. C. Paris and F. Erdogprr,Crack-tip stress-Wty fsccors for plam ext55skmaad pltttc beeding probka~s. I. tqqe He&. 29,306 (1%2). [9] S. Tii and J. N. Good&, 7hmy of Elwicity. McGraw-Hin. New York (1951). (k?aceiocd June 1974)
PETER G. TRACY and Mecblulics Research Center, wa-,
rbhs. 02lrr U.S.A.
INTRODUCTiON
A TEST specimen frcqucntly encountered in fracture toughness testing of shell materials is the
curved beam with crack as illustrated in Fii. 1. A typical loading system consisting of point loads at A, B, and E is shown. The circular ring segment with a crack emanating from the circular surface with the smaller radius can be considered the equivalent plane problem of elasticity. Simikr problems have been solved by Joncs[4] and[S] using the Gnitc clement approach. ‘Ibis probkm can bc neatly handld by using results obtained by Ricc[7], which arc closely rclatcd to the ‘weight function’ concepts of Bcuckncr[ I]. Rice has shown that if K1 and the displacement field arc known as functions of crack length for a primary symmetrical loading system on a body, the stress intensity factor, & can bc found for any other symmetrical loading system on the body by performing a simple integration. It is clear that the loading system consisting of point loads as illustrated in Fig. I leads only to K1 vale of stress intensity, which can be found if K1 amI the dispkccmcnt field arc known for a primary loading system. This approach obviates the need to introduce singukritks to handle the point loads and a!Iords the ability to analyze the probkm if the cffccts of dilTcrcntloading systems arc to bc analyzed in the CourWofthctcsting. The method used to analyze the primary lu&ing system is basic&y the ‘modifkd . mrrppmpcollocotion’ (MMC) plan combincd with a partitioniq~of the M. This procedure, described by Bowk et ol.[2),is used to compute stress intensity facturs and displacements for the ~loadiagqstaa,whicbisbendingcauplesMappliadatthecndsBCandADinFie.l. This tcchniquc involves the representation of the crack surface by a simpk mappingfunction and
Fig. 1. Circular ring a&ion with crack. 253
254
PETERG.TRACY
the prescription of force and stress conditions oa the boundary in a least squares sense. Since the geometry is subdivided into several regions, each with its own stress function representation, the stress functions in neighboringregions must be ‘stitched’usiag force and displacement conditions at their common boundary. ANAtYSIs
Frequent use will be made of the complex variable foraurlation of MuskhelishviIi[6]aad the required notation is briefly summarized. The functions I+(Z)aad S(z) in the Muskhelishvili forarulatioa are arm&ticfunctions of the complex variable 2 - x + iy. Ia terms of 4(z) aad #V(Z) the stresses and di@acemeats ia Cartesian coordiaates caa be expressed as UX+ a, = 2[9’(2)+ ml a, - uX+ 2ir- - Z[ff$“(z)+ S’(z)]
(1)
sad (2) where 17=3-4v
@ule strata)
E and v beii Young’smodulus aad Poisson’s ratio, respectively. Here and throughout prisms denote difIerea&ioa and the bars complex conjugatioa. The resuitaat force aloag aa arc s caa be expressed as
(3) where X ds aad Y. ds are the horixontal aad vertical compoaeats, respectively, of the force actiag aoraml to the element ds. The solution will be carried out using the modified mapping-collocation (MMC) method introduced by Bowie and NealI3.l.Thus the auxilhuy complex W-planeis introduced along with the mapping function z s u2(w) x -R,e-‘r*-(“ml
(4)
Points lyiag on a horixontal line ia the W-planeare mapped by @z(w)iato poiats lying on an arc ia the Z-plane. More specifically,points lying on the real axis in the W-plane are arapped into points lyiag on an arc of radius RI centered about the origia in the Zplaae. Points in the W-plane lying on a vertical line become poiats lyiag on a radial line from the originin the physical or Z-plane. Thus the physical geometry ia Fii. 1 corresponds to the geometry in the W-planeshown in Fig. 2. The point w corresponding to the point t = r& ia the range of odw) caa be found to be w=(f-&)+iln@-)
(5)
Analysis of a radial crack
ringsegment
a
Fig.ZTbeW-plane. by solving (4)for w. It is easily shown by using (5) that (see Figs. 1 and 2) b=28
(6)
In termsof @z(w)equations(2) and (3) can be written
@l(W)-;-
#(w)+
u:(w)4
q+(w)
_
@2(W)--;-
(w)
+
-
9(w)
= f1+
v2
-
o;oWb-W=2G(u+iu)
0
PI
904 - NB)t(wNaJ 9(w) = SC@&0 an analytic functions of the c.ompIsxvariablew.
To guaranteetraction-freeconditionson the arc AB in the physicalregion(see Fii. 1) the extukon aqgimentsof Muskhelishviliarcutilized.If S’ denotesthe regionabovetherealaxis in the W-pbm andS’ Benotesthe regionbelow,the function 4(w) is definedby
where
Thus S(w) em be written 8s 9(w) = and 0 and (7’) become
-d(w)-~~~~w), WCS’
(10)
PETER
236
0.TRACY
(11’) If #(w ) is analytic in the region of the W-plane extended regions ABCD and CDEF respectively added advantage of eliminating funCtiO&
reducing the probkm to
physical region plus the provides the only one cracks,
the physical rquires the intraductioa along with the
auxiUary
function.
+r-t-9.
FiikIlxtmddngioar.
(12)
Analysis of a radial crackina circularringsegment Z-ploll0
Do,
Dec
A
E
B
Fig. 4. Partitiowd zones.
PXXWXW&I~ as inf2] the representations for &&) and &&v) are ‘stitched’by requiring along the segment EF (See Fig. 4) that
where the nota& (fi + i&j, and (u + io), denote the forces and disph~~ments, respectively, as represented in r&on j. The representations of &(w) and Cp&) must be chosen consistent with the comlitkms outlived thus far. Taking into account a sing&&y in MC), for g = -i, din(t) is written (bII(0
= “‘4 5 f+iaJ” ’
of9
Obviously because of the stress sylnmctry about the iIxq&wy axis only the irnq&my ~~f~~v~sof II ~~~~~ci~tsforev~v~of n ~~~.We further assume dxlF@)o; mg Ga(w- WI)”
(17)
where wo is real and the midpoint of the line segment EB in Fig. 4. To carry out the MMC method antes of &(l) and #t&w) are used. Stati arc chosen along (see Fii. 4) BC, CG, and the crack surface and either $n(f) is inserted in (14)or #m(w) is inserted into (1I), whichever is appropriate, to obtain conditions between the Q and between the cl. To obtain conditions relating the at to the G, stations arc chosen along (we Fii. 4) EF and conditions written according to (ll), (1I’), (14),(14’)and (15).In general the number of conditions exceeds the munber of unknown coe&icnts of #&) and dew) and a k@ squares minimization of error is employed to obtain a War system of shnultaneous equations in the aI and cr. This system ii then solved to obtain #n(g) and 4&w). BOUNDARYCONDITIONS The bou&%y conditions in the problem are those normal stresses along the cads (see Fig. 1) AI) and BC which give rise to pyre bend&g. The form of these stresses is given by ~~~0 and Goodier[91to be
EFM Vol. 7, No. 2-E
pm
zs8
0. TRACY
where N=(R:r, = ir I,and M is the moment. Elsewhere on the boundary and crack the normal and shear stresses are zero. It can be shown that for stresses of this form -4M fzR12ln N
(19)
fi + ifi = -
NUMERICAL REsuLm
Once the coefBcientsfor &U(W)and &I({) are known, it is a simple matter to calculate K, using +&). Sii et a1.[8]have shown that K = KI - iIG = 2(2n)‘” lim (2 - &)‘n#(z) z-4 where zc is the location of the crack tip in the physical region. Equation (20) leads to
(20)
&I(l) E (+4)2(w)- 23)I/2 o;(w)o;(l)
K = 2(2#
= 2 h+ (
)
‘n+h(i).
(22)
The points { = i and w = iL correspond to the crack tip in the l- and W-planes, respectively. Because of the symmetries it follows that d&(i) and thus K are real. At this point it is convenient to introduce the function (23) where we= u,~,,_~,.The well-knownresult that for I 4 RI, HI + 1.12was used to check solutions for small cracks. Using the method descril by I&em, K, can be calculated for the body in F@ 1 subjected to the folhiqJ two point load systems:
p, L
ux =
-P,
s
2 =
z2
=
23 =
R,e’“n”-r’
(240
-if2
-P,z=z*=iR2
a? =
4P, 2’22
(249
4P, 2 = 23
As in (18)the normal and shear stresses elsewhere on the boundary and crack surface are zero. SummarMg1711.lL can be found for any symmetrical loading system on a body, if K, and the displacement field are known as functions of crack length for one symmetricalloading system on the body. This fact is shown in the following relation: K,=
Is t.hdS
(25)
Analysis of a radialcrackin a dmlar ringsegment
259
where t is the stress vector on the boundary S and h is referred to as the weight function defined by E au(l) (26) WVY, 0 = 2&w(1) y+where X1”‘(1)is the stress intensity value known for a part&x&uloading system as a function of 1 and u(l) is the displacement field for that particular loading system. For the body subjected to the conditions of (18) one can find &“‘(I) and u(l) if 4&g) and 44~) are known for a sufficient number of crack lengths. Adjusting displacements such that u = 0 at z2and z3and using the fact that the loads are point loads, it is clear that the stress intensities for a given 1 for the conditions of (24a)and (24b)are given by: PE au K1 =Kl”‘;jr I+-3
(2W
PE av Ks = 2K1”‘z I r-4)
respectively. For this case, we define
In order to obtain HI values for the loading of (24)the displacements u and u as a function of 1 were &ted to piece-wise polynomiais to facilitate the cakulation of the derivatives in (26).
-0 Fig. 5. v&es
1 /R,
ofH, = K,/u.(d)‘” for bending.
260
PETER 0. TRACY
Results for vwIona I/(& - R,) vdwa are preseti in Fii 3 for the RdR, ratha of 2,1*75,1.5 and l-23 for the loading of (18).It is clear that the direction and magnitude of the point loads at zz and 2%of (24) can be h&d by using combinations of the loading systems (24a) and (24b). Obviously, from (zs) and (26)the value of KSfor such a comb~ti~ loading is the sum of the condom from each of the loadings (24a) and (24b). In Fii 6, data is presented for the loadiqs (%a) and (24b)and a qombmatioo.The combination loadingis chosen such that the point loads at zz and zs of (24)are normal to the circular arc with greatest curvature, which implies the ma@ude is l/2@ since the value of 8 (see Fig. 1) remains tied at n/4. Acknowitdgrmartr-% auhr ti8hasto acknowledge the interest and encoutage.ment of C. E. Frame and 0.
L. B&c in
thiSWOk
[l] pi_)&“““,
A novel principle for tb8 come
of stl@a UlteMity factm. z &$cw. bffztk beck. 9). 529-546
[2] 0. L ‘i&vie, C. B. Reese aud D. M. Neal+Solutiom of pbuwproblem of claaticity utihiq qppL ius& %, 767-772 {Em).
puti-
cmcqts.
J.
[fl A. T. Jonea, Fmture tou&ma testing with sections of cyliadhl shells. SL-7MO50, Saadia Labotamh, Livmnore, Calif., Nov. 1973. IuHIMW, Some Bark Pmblems ofthe Mathmtktd IItatwy of Elastkity. NooM, Gmdngcn, Holhd @] s31M * m Jam&. Rice, Soas remarka on &tic crack-tip stress fields, ht. J. Soiids, Stfucturcs, 8,751-738 (1972). @I 0. C. Sib. P. C. Paris and F. Erdogprr,Crack-tip stress-Wty fsccors for plam ext55skmaad pltttc beeding probka~s. I. tqqe He&. 29,306 (1%2). [9] S. Tii and J. N. Good&, 7hmy of Elwicity. McGraw-Hin. New York (1951). (k?aceiocd June 1974)