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The influence of the stress state on the plastic zone size
THE INFLUENCE OF THE STRESS STATE ON THE PLASTIC ZONE SIZE D. AURICH Bundesanstaltfti MaterialprWmg,1Berlin45. Germany Abbrt-“Linear eiastic” fracturemechanicsis based on the “planestrain”conditionwhich meansdr, = 0. Really stress states can occur in stn~cturesor components where dcz2>O. Two examples are discussed. Using the v. Mises criterionand the Sneddonequationsthe influenceof an externalstress o:, act& parallel to the crack front on the plastic zone size is calculatedand demonstratedby exampics.
INTRODUCTION L-VALUESare related in fracture mechanics “per detithem” which is defined by the relation
to the stress state “plane strain”,
04 = Y(U,+ u2)
(1)
u12) arc the three principal stresses, and o3 = a, in the coordinate system of a crack @ii l)[l].Inthe~ofacrrsckthe”planestrain”conditionisonlycoPrelatedtoa~stress state, but in practice there can be effective different stress states in the swnmding of cracks. This paper deals with the iunucnce of (zn. On principle oa is variable from 0 to a~.Therefore it can be also
where
uzz> v(ur.x+ UYY) as whlbe discussedbelow. Besides “plane strain” the condition “plane stress” is a sig&k.ant one
too, where a, = 0. If a cracked plate is loaded as shown in Fii. 1, the material close to the crack front causes a constraint. The intensity of this constraint depends on the material properties and the thickness of the plate. The constraint (‘) causes the stress 0, u,c = a; = k(u, + uz) = k(u, + u=) where k PO... Y depending on the matuial and the plate thichess as mwltioncd above. ,a
/’
i_---------_.--
Ia Fii. 1. Coordi~te system of a crack. 761
D. AURICH
762
Theoretically one can superimpose an additional externally (‘) produced stress at to the stress ah produced by the constraint. The stress at may vary from -Q)to +a, and if there is no further constraint, a:, does not induce any change.of the stresses a, and arv. Therefore uzz= a&+ u&.
(2)
Of course uII has 00 intluence 00 the stress intensity factor IG too, as Paris and Sih[Z] have stated, if the same condition is performed (00 further constraint). Common knowledgeis that the Mereaces of the plastic zooe sizes and of the critical fracture mcchanic~reierrsdtouplanestress”aad”piencstrain”areoriOiDltsdfromastrosO influenceofu~ffthishtruefor
ooe has to suppose that the tiueoce
of au is not limited to a, 5; ~(0;. + Us,,).Really larger vaiwsofu,shouldteduceth!p~uwes~and~criticplfrachvtmechanicvrluMlturtBsr 00. The intensity of the influence of an 00 the plastic zooe size and 00 the critical v&es will depend on the flow and fracture hypothesis to be applied. It may be a questioo whether stress states with
may occurintech&al structures iocoooectioo with cracks or not. ReaUythere are some simple cases where this may happeo. For example: (1) A“thickwaUed”opene&dtubeuaderinternalpreamnwithasqWnpa&forceF’ arbending~~bothcausineateaaionu‘Linthccrackedretioaofthe~pPnaeltothe front of a crack in one of the positioos demonstrated in Fig 2. (2) A “thick walled” pressure vessel with a crack in one of the positions in Fii 2. The hoop stress induced by the inter&l preseure p causes in both cases the stress
u:,= Y(Un + u&7)
(3)
by the co&mint at the crack ftmnt. A lon#udinaI force oo the tube respectively the intuarl pressure of the vessel induces the additional stress
a:, = UL.
(4)
Therefore in these examples UZZ =
v(u,
+uyy)+uL
(5)
IOpractice there may be of course many other examples of structures where stress states of this kind may be induced. The consequences of these considerations for the plastic zooe size will be discussed below. The consequences for the critical fracture mechanic values will be discussed in another paper[3].
763
The inliuence of the stress state on the plsstic zonesize
~~~~OF~~ON~P~C~~~ The plastic zone size of sufEcicnt ductile metals can be calculated by the v. Mises
hypothesis[4]. The principal stresses 01 and uz are derived from the Sneddon-quations[Sl: u,
(a)
=-&os;(l+sin;)
where IG is the stress intensity factor. The princii stress a3 depends on r, if it is caused by the constraint of the crack front; it does not depend on r, if it is caused otherwise. (1) If as = uzzis constant in the surrounding of the crack, the plastic zone size r, is given by
and with cp= 0 (on the ligament)
where a, is the yield point. If o, = 0, qn (9) changes into the “pIane stress” case given by Irwin and McClintock[4]. (2) If
wheretheflrsttermdescribesthatpartofa, whichiscausedbyamaximumoanstraintdc,=O at the crack aud derived from qns (31,(6) and (7) and the second term mpreeents an externally produced stress which is constant in the surroundingofthccrack,theplasticzonesizer,isgiven by r, _
(1-2aw2 -&&2cos2f(2-2~+-~((l-2~)2+3i~
>I ”
and with cp= 0 (on the ligament)
Ifu’ zx- 0, qn (12)changes into the “plane strain” case given by Irwin and l&Clintock too[4], if
a:, > 0, the deformation increment becomes den > 0. Equation (9) becomes identical with qn (12) when at the elasti~tic
interface (13)
Equation (12)becomes identical with qn (9) for the “plane stress” case when the superimposed StreSS
This means that a compression stress of this magnitude has to be applied parallel to the crack
764
D. AURICH
front of a “plane strain” sample to change the stress from the “plane strain” to the “plane stress” condition. This fallows from the fact, that at some tie of Kr the princ@alstrcsees u1 and a2 reach the ma@tude of uYat the same distance from the crack tip r - K?/2wu~ (ploae stress plastic zone radius) as well under ‘*pIanestrain” as under “piane stress” caditia as Fii 3 shows. However under ‘plane stmiC condition the point (P) referred to is placed within the elastic stress field whereas under “plane stress” condition it is placed at the eiasti@astic interface. From eqn (1) follows for this point u3 = uu = Y(UI+ a*) * 2W,.
(1s)
Tbereforeu,=O(plaaestress)canbereschsdinathickplatebyu’,appmprirrteecpl(14). F~4demon&Wthcinflucnceofa:,ontheplesticzancsizecakulaW by eqn (12)for some data combhlatim: Y= O-3, (1) K, = HI0 N/mm)”
u, = SO0N/mm’
(2) K, = 2!WO N/mm=
a, = 750N/mm’
(3) KI - lOOON/mm*
a, - loo0 N/mm*.
-600
-400 AMNlONAl
-2&l EXIERNAL
sram
a,: N/mm’
Fig. 4. Dependence of the plastic umesizeontheadditiwIaI extend
stress
ff&.
The iniblenceofthe stressstateon the plasticzone size
765
Example 1 demonstratesthis rathersigaiikantly: CT',=0
r,,= 2.55mm(planestrain)
a’,=2OON/mm*
r, = l-30 mm
The plastic zone size reducesby about one half when a’, increaacsfrom0 to200N/mm2.
REFERENCES [l] 0. J. Irwin. Hmd6uch der PhgsuS (Ed. S. Fliigge), Vol. 6, pp. W-590. Sptingcr, Berlin (1958). [2] P. C. Paris aed C. C. Sih. ASTM Spa. Techn. Publ. No. 381, pp. 30-81 (1965). P] D.AlUieh,tObCpUblirbcd. [4] P. A. McClintock and G. R. w. UTM Spec. Tcchn. Publ. No. 381, pp. 84-113 (MS). [a I. N. S&don, Proc. Roy. Sot. Am, 229-m (1946). (Receioed March 1974)
INTRODUCTION L-VALUESare related in fracture mechanics “per detithem” which is defined by the relation
to the stress state “plane strain”,
04 = Y(U,+ u2)
(1)
u12) arc the three principal stresses, and o3 = a, in the coordinate system of a crack @ii l)[l].Inthe~ofacrrsckthe”planestrain”conditionisonlycoPrelatedtoa~stress state, but in practice there can be effective different stress states in the swnmding of cracks. This paper deals with the iunucnce of (zn. On principle oa is variable from 0 to a~.Therefore it can be also
where
uzz> v(ur.x+ UYY) as whlbe discussedbelow. Besides “plane strain” the condition “plane stress” is a sig&k.ant one
too, where a, = 0. If a cracked plate is loaded as shown in Fii. 1, the material close to the crack front causes a constraint. The intensity of this constraint depends on the material properties and the thickness of the plate. The constraint (‘) causes the stress 0, u,c = a; = k(u, + uz) = k(u, + u=) where k PO... Y depending on the matuial and the plate thichess as mwltioncd above. ,a
/’
i_---------_.--
Ia Fii. 1. Coordi~te system of a crack. 761
D. AURICH
762
Theoretically one can superimpose an additional externally (‘) produced stress at to the stress ah produced by the constraint. The stress at may vary from -Q)to +a, and if there is no further constraint, a:, does not induce any change.of the stresses a, and arv. Therefore uzz= a&+ u&.
(2)
Of course uII has 00 intluence 00 the stress intensity factor IG too, as Paris and Sih[Z] have stated, if the same condition is performed (00 further constraint). Common knowledgeis that the Mereaces of the plastic zooe sizes and of the critical fracture mcchanic~reierrsdtouplanestress”aad”piencstrain”areoriOiDltsdfromastrosO influenceofu~ffthishtruefor
ooe has to suppose that the tiueoce
of au is not limited to a, 5; ~(0;. + Us,,).Really larger vaiwsofu,shouldteduceth!p~uwes~and~criticplfrachvtmechanicvrluMlturtBsr 00. The intensity of the influence of an 00 the plastic zooe size and 00 the critical v&es will depend on the flow and fracture hypothesis to be applied. It may be a questioo whether stress states with
may occurintech&al structures iocoooectioo with cracks or not. ReaUythere are some simple cases where this may happeo. For example: (1) A“thickwaUed”opene&dtubeuaderinternalpreamnwithasqWnpa&forceF’ arbending~~bothcausineateaaionu‘Linthccrackedretioaofthe~pPnaeltothe front of a crack in one of the positioos demonstrated in Fig 2. (2) A “thick walled” pressure vessel with a crack in one of the positions in Fii 2. The hoop stress induced by the inter&l preseure p causes in both cases the stress
u:,= Y(Un + u&7)
(3)
by the co&mint at the crack ftmnt. A lon#udinaI force oo the tube respectively the intuarl pressure of the vessel induces the additional stress
a:, = UL.
(4)
Therefore in these examples UZZ =
v(u,
+uyy)+uL
(5)
IOpractice there may be of course many other examples of structures where stress states of this kind may be induced. The consequences of these considerations for the plastic zooe size will be discussed below. The consequences for the critical fracture mechanic values will be discussed in another paper[3].
763
The inliuence of the stress state on the plsstic zonesize
~~~~OF~~ON~P~C~~~ The plastic zone size of sufEcicnt ductile metals can be calculated by the v. Mises
hypothesis[4]. The principal stresses 01 and uz are derived from the Sneddon-quations[Sl: u,
(a)
=-&os;(l+sin;)
where IG is the stress intensity factor. The princii stress a3 depends on r, if it is caused by the constraint of the crack front; it does not depend on r, if it is caused otherwise. (1) If as = uzzis constant in the surrounding of the crack, the plastic zone size r, is given by
and with cp= 0 (on the ligament)
where a, is the yield point. If o, = 0, qn (9) changes into the “pIane stress” case given by Irwin and McClintock[4]. (2) If
wheretheflrsttermdescribesthatpartofa, whichiscausedbyamaximumoanstraintdc,=O at the crack aud derived from qns (31,(6) and (7) and the second term mpreeents an externally produced stress which is constant in the surroundingofthccrack,theplasticzonesizer,isgiven by r, _
(1-2aw2 -&&2cos2f(2-2~+-~((l-2~)2+3i~
>I ”
and with cp= 0 (on the ligament)
Ifu’ zx- 0, qn (12)changes into the “plane strain” case given by Irwin and l&Clintock too[4], if
a:, > 0, the deformation increment becomes den > 0. Equation (9) becomes identical with qn (12) when at the elasti~tic
interface (13)
Equation (12)becomes identical with qn (9) for the “plane stress” case when the superimposed StreSS
This means that a compression stress of this magnitude has to be applied parallel to the crack
764
D. AURICH
front of a “plane strain” sample to change the stress from the “plane strain” to the “plane stress” condition. This fallows from the fact, that at some tie of Kr the princ@alstrcsees u1 and a2 reach the ma@tude of uYat the same distance from the crack tip r - K?/2wu~ (ploae stress plastic zone radius) as well under ‘*pIanestrain” as under “piane stress” caditia as Fii 3 shows. However under ‘plane stmiC condition the point (P) referred to is placed within the elastic stress field whereas under “plane stress” condition it is placed at the eiasti@astic interface. From eqn (1) follows for this point u3 = uu = Y(UI+ a*) * 2W,.
(1s)
Tbereforeu,=O(plaaestress)canbereschsdinathickplatebyu’,appmprirrteecpl(14). F~4demon&Wthcinflucnceofa:,ontheplesticzancsizecakulaW by eqn (12)for some data combhlatim: Y= O-3, (1) K, = HI0 N/mm)”
u, = SO0N/mm’
(2) K, = 2!WO N/mm=
a, = 750N/mm’
(3) KI - lOOON/mm*
a, - loo0 N/mm*.
-600
-400 AMNlONAl
-2&l EXIERNAL
sram
a,: N/mm’
Fig. 4. Dependence of the plastic umesizeontheadditiwIaI extend
stress
ff&.
The iniblenceofthe stressstateon the plasticzone size
765
Example 1 demonstratesthis rathersigaiikantly: CT',=0
r,,= 2.55mm(planestrain)
a’,=2OON/mm*
r, = l-30 mm
The plastic zone size reducesby about one half when a’, increaacsfrom0 to200N/mm2.
REFERENCES [l] 0. J. Irwin. Hmd6uch der PhgsuS (Ed. S. Fliigge), Vol. 6, pp. W-590. Sptingcr, Berlin (1958). [2] P. C. Paris aed C. C. Sih. ASTM Spa. Techn. Publ. No. 381, pp. 30-81 (1965). P] D.AlUieh,tObCpUblirbcd. [4] P. A. McClintock and G. R. w. UTM Spec. Tcchn. Publ. No. 381, pp. 84-113 (MS). [a I. N. S&don, Proc. Roy. Sot. Am, 229-m (1946). (Receioed March 1974)