on an analytical determination of initial points of crack propagation

6
FISUEER, K. F., Determination of Initial Points of Crack Propagation 229 ZAMM 619 229 -234 (1981) K.-3’. FISCHER On an Analytical Determination of Initial Points of Crack Propagation In der vorliegenden Arbeit wird eine Miiglichkeit der Berechnung von Initidpunkten der Rijausbreitung von der BiJ- spitze untersucht. Diese besteht darin, bek der Formulierung charakteriatischer GroJen zur J2iJbeurteilung neben dem ersten Glied der Reihenlosung f u r die elastiachen FeldgroJen in Ripspitzenumgebung weitere Olieder zu beriicksichtigen. Es wird vor allem ein derartig erweitertes Hih-Kriterium vorgeatellt. Eina qualitativ new Aussage bildet die Berechnung von hypothetischen Initialpunkten der Rijausbreitung vor der Rgspitze. Ahnliche Ergebnisse erhiilt man bei der Untersu- chung e k e s erweiterten Spannungs-Parameter-Kriteriums und einea erweiterten Kriteriums der Mehrachsigkeitszahl des Spannungszustandes. In the present paper, a possibility of determination of initial points of crack propagation in front of the crack tip is in- vestigated. This possibility consists in the fact that apart from the first term ofthe series solution for the elasticfield variables in crack tip vicinity further terms are considered in the formulation of characteristic quantities for crack judgment. Above all an extended Sih-Criterion is presented. A qualitatively new information results in the determination of hypothetical initial points of crack propagation in front of the crack tip. We obtain similar results if we consider the extended stress- parameter-criterion and the extended multi-axial-value-criterion. B aamoii pa6o~e uaysae~ca BO~MO~HOCT~ onpenenemrr HasanbHMx ToseK pacnpocTpaHeHm T~~UHH~I nepen OCT~H~M TpeWuwI. 3 ~ a BO~MO~HOCT~ COCTOHT B TOM, ¶TO, ~cxnmarr nepsMB sneH pFiga pemenurr AJIH ynpyrmx nepeMemMx noan, B nenocpegcmemoi BJIE~~OCTE~ OT oc~p~lrr TpeuuHu EanbHeZimne sneHM paccMaTpusaIoTcs c noMoubw nocTpoenusI xapamepHcTnsecmx Benmm nna oqeHm Tpeqm. KIpeHcne Bcero, npeAcTasneH pacmHpeHHm8 KpmepHZl CII. ECasecTsenHo HcmaH nH4opMaqurr npencTasneHa onpe- AenemeM rmoTeTmecmx HasanbnbIx ToYeK pacnpocTpaHeHmFi TpewuHbi nepex OCT~H~M Tpexrimm. no- KoBHbIe pesynmami nonysawrcrr, ecm paccnroTpeTb pacmupeHHMii IcpuTepui HanpFixenm EI pacrrru- pelIHbIfi KPllTepHn MHOrOOCeBO~ BeJIH9HEIbI. 1, Int,roduction In literature, various fracture criteria of crack fracture mechanics are described, [ 11, [2], [3]. They are classified, for instance, according to the main physical quantities such as strain energy, stress, deformation. The form of this physical magnitude, that is specific for the particular criterion, is called a “characteristic quantity”. Fracture cri- teria include information on the following two basic hypotheses [Zj, [4]: 1 st hypothesis: Assumption about crack propagation direction in case of crack instability. 2nd hypothesis: The crack becomes unstable, if the characteristic quantity for crack judgment reaches a critical value. In the following, we restrict ourselves to linearly elastic material behaviour and static, in-plane problems. The stress and displacenient field in the crack tip vicinity provides the basis for the determination of the charac- teristic quantities. The stress field takes the following form approximately: p and v are the shear modulus and POISSON’S ratio respectively. The a:, a: denote the generalized intensity factors. With n = 1, the v-1/2-singularitycharacteristic for crack problems is obtained, In most cases only the terms for n = 1 of the stress field in the crack tip vicinity are considered for formulating criteria of crack fracture mechanics at present. For improved clarify such criteria will be refered to as “simple” criteria in the following. Now a survey on the formal extension of fracture criteria will be given in such a way that in addition to the first term further terms are considered in the formulation of characteristic quantities. Such criteria will be refered to as “extended” criteria of crack fracture mechanics, cf. [5]. COTTERELL was the first who underlined the significance of including further terms in formulating charac- teristic quantities, [6]. WILLIMS and Ew~G, for instance, used an extended normal stress criterion with two terms to treat a plane GRIFFITH-crack under tension loading, [“I. In the following, the extended SIR-Criterion, the extended stress-parameter-criterion and the extended multi- axial-value-criterion will be discussed as examples for extended criteria and their significance for the determination of hypothetical initial points of crack propagation. 2. Extension of the Sih-Criterion The strain energy density factor X is used as a characteristic quantity, [l], [8]. I n general, S can be expressed by S=rW*. (2) W* describes the strain energy density. In the following, 6 and r describe the coordinates in a polar coordinate system with the origin in the crack tip.

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Page 1: On an Analytical Determination of Initial Points of Crack Propagation

FISUEER, K. F., Determination of Initial Points of Crack Propagation 229

ZAMM 619 229 -234 (1981)

K.-3’. FISCHER

On an Analytical Determination of Initial Points of Crack Propagation

In der vorliegenden Arbeit wird eine Miiglichkeit der Berechnung von Initidpunkten der Rijausbreitung von der BiJ- spitze untersucht. Diese besteht darin, bek der Formulierung charakteriatischer GroJen zur J2iJbeurteilung neben dem ersten Glied der Reihenlosung fur die elastiachen FeldgroJen in Ripspitzenumgebung weitere Olieder zu beriicksichtigen. Es wird vor allem ein derartig erweitertes Hih-Kriterium vorgeatellt. Eina qualitativ new Aussage bildet die Berechnung von hypothetischen Initialpunkten der Rijausbreitung vor der Rgspitze. Ahnliche Ergebnisse erhiilt man bei der Untersu- chung e k e s erweiterten Spannungs- Parameter-Kriteriums und einea erweiterten Kriteriums der Mehrachsigkeitszahl des Spannungszustandes.

I n the present paper, a possibility of determination of initial points of crack propagation in front of the crack t ip i s in- vestigated. This possibility consists in the fact that apart from the first term ofthe series solution for the elastic field variables in crack t ip vicinity further terms are considered in the formulation of characteristic quantities for crack judgment. Above all an extended Sih-Criterion is presented. A qualitatively new information results in the determination of hypothetical initial points of crack propagation in front of the crack tip. We obtain similar results i f we consider the extended stress- parameter-criterion and the extended multi-axial-value-criterion.

B aamoii pa6o~e uaysae~ca B O ~ M O ~ H O C T ~ onpenenemrr HasanbHMx ToseK pacnpocTpaHeHm T ~ ~ U H H ~ I nepen O C T ~ H ~ M TpeWuwI. 3 ~ a B O ~ M O ~ H O C T ~ COCTOHT B TOM, ¶TO, ~cxnmar r nepsMB sneH pFiga pemenurr AJIH ynpyrmx nepeMemMx noan, B nenocpegcmemoi B J I E ~ ~ O C T E ~ OT oc~p~lrr TpeuuHu EanbHeZimne sneHM paccMaTpusaIoTcs c noMoubw nocTpoenusI xapamepHcTnsecmx Benmm nna oqeHm Tpeqm. KIpeHcne Bcero, npeAcTasneH pacmHpeHHm8 KpmepHZl CII. ECasecTsenHo HcmaH nH4opMaqurr npencTasneHa onpe- AenemeM rmoTeTmecmx HasanbnbIx ToYeK pacnpocTpaHeHmFi TpewuHbi nepex O C T ~ H ~ M Tpexrimm. no- KoBHbIe pesynmami nonysawrcrr, ecm paccnroTpeTb pacmupeHHMii IcpuTepui HanpFixenm EI pacrrru- pelIHbIfi KPllTepHn MHOrOOCeBO~ BeJIH9HEIbI.

1 , Int,roduction

In literature, various fracture criteria of crack fracture mechanics are described, [ 11, [2], [3]. They are classified, for instance, according to the main physical quantities such as strain energy, stress, deformation. The form of this physical magnitude, that is specific for the particular criterion, is called a “characteristic quantity”. Fracture cri- teria include information on the following two basic hypotheses [Zj, [4]: 1 st hypothesis : Assumption about crack propagation direction in case of crack instability. 2nd hypothesis : The crack becomes unstable, if the characteristic quantity for crack judgment reaches a critical

value. In the following, we restrict ourselves to linearly elastic material behaviour and static, in-plane problems. The stress and displacenient field in the crack tip vicinity provides the basis for the determination of the charac- teristic quantities. The stress field takes the following form approximately:

p and v are the shear modulus and POISSON’S ratio respectively. The a:, a: denote the generalized intensity factors. With n = 1, the v-1/2-singularity characteristic for crack problems is obtained,

In most cases only the terms for n = 1 of the stress field in the crack tip vicinity are considered for formulating criteria of crack fracture mechanics at present. For improved clarify such criteria will be refered to as “simple” criteria in the following.

Now a survey on the formal extension of fracture criteria will be given in such a way that in addition to the first term further terms are considered in the formulation of characteristic quantities. Such criteria will be refered to as “extended” criteria of crack fracture mechanics, cf. [5].

COTTERELL was the first who underlined the significance of including further terms in formulating charac- teristic quantities, [6]. WILLIMS and E w ~ G , for instance, used an extended normal stress criterion with two terms to treat a plane GRIFFITH-crack under tension loading, [“I.

In the following, the extended SIR-Criterion, the extended stress-parameter-criterion and the extended multi- axial-value-criterion will be discussed as examples for extended criteria and their significance for the determination of hypothetical initial points of crack propagation.

2. Extension of the Sih-Criterion

The strain energy density factor X is used as a characteristic quantity, [l], [8]. In general, S can be expressed by

S = r W * . (2) W* describes the strain energy density. In the following, 6 and r describe the coordinates in a polar coordinate system with the origin in the crack tip.

Page 2: On an Analytical Determination of Initial Points of Crack Propagation

230 BISCIIER, K. F., Determination of Initial Points of Crack Propagation

The criterion is based upon the fact that the fracture proceeds froin an failed “interior element” located in the vicinity of the crack tip. It is expedient to use a representation of W* by meanR of the principal normal RtreRscs:

where (3 - 4 v ) J plane strain, (3 - v ) / ( l + Y) , plane stress.

.-( The following sum and difference of principal normal stresses will result! :

k k

n = 1 n-1 (ar + o*)’ = 2 bnr(n-3Y2 , (.; - 02)2 = 2 enr(n-3)/2 .

The coefficients bn, cn depend upon 6 and the generalized intensity factors. Now we write the strain energy density fact,or (or function, respectively) 8:

k E X = r W * .

The superscript index indicates the number of the terms considered in determining the corresponding quantity. Using equations (3), (4) we obtain the decomposition:

1 For r /a -+ 0 and lim (r/a)o A 1, equation (6) changes into the wellknown 8:

?/4+0

The coefficients ail depend upon 6 and POISSON’S ratio V . With rla -A 5, the following dependence resiilts:

(7) According to the 1st hypothesis of the simple criterion, the locus of the local minimum of the strain energy density factor governs the crack propagation direction. Due to the dependence given in equation (7), the following relations

are available for t,he determination of the point $,,($o, ;b) in which 8 = ~!5’~i,,:

8, = XnlS, 5 ) *

k k

(8) l k b = B

k 1: (asjaa = 0 , a s p 6 = o} a=$ ,

k If there exists such a point, an extended Sm-Criterion can be stated: 1 s t hypothesis : Crack propagation in the elastic range proceeds from the point

At this point the strain energy density factor X becomes B local minimum. [J radially towards Go,

k

E 2nd hypothesis : The crack becomes unstable when the minimum 8-factor reaches a critical value Sc.

P,, is refered to as the Hypothetical Initial Point of the Crack Propagation, [9], [lo], (see Fig. 1). E

E Now at first we base our considerations on the GRImsrn-crack under Mode I-loading, 6, = 00.

1 P

‘>/

Fig. 1. Positions of Iiiitial Points nf (’rack Propagation near tile Orark Tip

Page 3: On an Analytical Determination of Initial Points of Crack Propagation

FISCHER, K. F., Determination of Initial Points of Crack Propagation 231

If we take k = 4 into account from equations (8) result:

For comparison one obtains: 3 1 x - 1 2 f o = 2 ( - Z z T ) '

where

Figure 2 Bhotvs the dependence of the distance between crack tip and initial point upon POISSON'S ratio.

Fig. 2. Dependence of the distance between initial points crack tip upon Porssox's ratio Y, Mode I

Now we get the following values for the critical load (v = 0.3):

and

00

Thus the simple criterion can be understood as the lower limit for the critical load of the crack. The value pc, which was determined according to the exact solution of the stress field at the crack tip (cf. equations (13)), shows that, with k = 4, a good approximation has been obtained.

Now again, we base our considerations on the angled GRIFFITH-Crack. Thus, the relations (8) lead to a non linear set of equations for the determination of the coordinates of the initial point :

This set of coupled equations is solved numerically for k = 4 and Y = 0.3 (plane strain) for various load applica- tion angles.

The extended criterion, in contrast to the simple criterion, leads - both for the crack propagation angle and the minimum energy density factor - to quantitatively differing results. Pig, 3 shows, for instance, the dependence of the crack propagation angle Upon the load application angle.

It is necessary to mention that nearly the game results are obtained if we use the exact solution for the stress field in crack tip vicinity.

The qualitative difference between the simple and extended SIH-Criterion is expressed by the prediction of the existence of an initial point of crack propagation in front of the crack tip. The significance of the prediction of initial points will be discussed in brief.

First it should be said that the determination of such points is based solely on the formulation of an extended strain energy density factor (or, better: -function) and the minimization of the parameter. Test results are known from literature, where crack nucleus formation (or now: initial points) in front of the crack tip was observed in polycarbonete samples, [I l l . In order to enable a comparison with the results obtained by means of analytic cal-

Page 4: On an Analytical Determination of Initial Points of Crack Propagation

232 FISCHER, K. F., Determination of Initial Points of Crack Propagation

1

k Fig. 3. Dependence of the crack propagation angle 6, upon the load application angle a, Y = 0.3 (plane strain)

Table 2. Positions of initial points according the extended SIR-Criterion

crack sharpiicss &, according to s =@/a GARDE/WEISS [11]

00

t,, (solution for ellipse) v = 0.3 v = 0.4

0.0032 0.00697 0.016 0.031 1 0.032 0.0577

0.0718 0.0295 0.0868 0.0349 0.0972 0.0385

culations, it is necessary to consider the influence of the V-notch radius in formulating 8. From this the values given in Table 1 result for the distance of the initial points from the roots of the notches for various “crack sharpness degress” 6 = pla where e denotes the notch radius.

Because of the difference bet,ween of the crack models (tension bar with double outer notch [ll] and here ellipse in the infinite plane) only a qualitative comparison can be made. The theoretical prediction stating that lo increases with growing 6, is substantiated.

3. Extension of the Stress-Parameter-Criterion

W m a Tzu C H I A N ~ proposed in 1975 a so called stress parameter criterion for combined mode fracture, [12]. The simple criterion includes the following basic hypotheses: 1 s t hypothesis: Crack propagation proceeds in radial direction vertically to the maximuin normal stress on

an iso-W*-line, i.e.

{ m p a 1 = 0, aw/w 1 < o } , ~ = ; ~ (16)

1 where @ is t,he st,ress parameter

KI 6 @ = -=-- [l +cos 1.9 - 3M sin 61 cos-. Ij,B 2

2nd hypothesis: The crack becomes unstable when the maximum normal stress on an iso-W*-line reaches a critical value:

1 1 *81$=,;, =i, max ‘gl;l*=const. = OC (18)

Magnitude Jf is defined by iM A K,,/KI. K I and KII are the stress intensity factors for symmetrical and anti- metrical crack loading. Since magnitude M can be understood as a measure of deviation from the symmetrical case it is refered to as the symmetry factor, cf. [2], [5].

If we extend the criterion in the sense described above, and use the exact stress field around the crack tip of

a GRIFFITH-crack under Mode I-loading, the stress parameter reads as follows, 6, = oo: k

Page 5: On an Analytical Determination of Initial Points of Crack Propagation

F I S C H ~ , K. F., Determination of Initial Points of Crack Propagation 233

! I

m

Fig. 4. Positions of initial points versus POISSON’S ratio according 7 m 75 the extended stress-parameter-criterion

&=/;fn - m

We get maximum value of @ according to equations (20) :

m

lo is the distance between the initial points of crack propagation and the crack tip. Fig. 4 shows the dependence of the disttance upon POISSON’S ratio v.

4. Extension of the Multi- Axial-Value-Criterion

KOCHENDORFER and SCROLL found in 1957 an inverse proportional connection between a so called value of niulti- axiality of stress stat: and the fracture stress of steel samples by means of experimental methods, [13]. The multi- axial value is defined as follows:

Jz is the 2nd invariant of stress deviator.

hypotheses of the simple criterion are the following :

1 s t hypothesis:

Now we interpret this parameter as a characteristic value of a crack fracture criterion, cf. [14]. The basic

1 Crack propawtion proceeds in radial direction where SZ reaches a maximum, i.e.

2nd hypothesis: The crack becomes unstable when K,(6,) reaches a critical value Klc where

I ; , ( ~ ~ ~ K I ( S ; m R a / g ~ R ~ , e = ~ ~ * (23) If we extend this criterion and use the exact stress field solution around the crack tip of a GItIlrFITH-crack under

Mode I-loading (tl0 = O O ) , initial points are calculated200. k

The distance between initial points and the crack tip reads as follows:

We take plane strain into account only. Fig. 5 shows the dependence of this distance upon POISSON’S ratio.

Table 2. A comparison of positions of initial points, calculated according to the three extended criteria, is shown in

Table 2. Comparison of Calculated positions of initial points (GRIFFITH-crack, Mode I-loading, plane strain, 6, = Oo).

V r Extended SIR-Criterion Extended Stress- Extended M’ulti-

Parameter-Criterion Axial-Value-Criterion k = 4 k = m k = c a k = m

0.0 0.137 0.122 (a) 0.1 0.116 0.106 1.18 0.2 0.0901 0.0853 0.512 0.3 0.0613 0.0596 0.219 0.4 0.0279 0.0276 0.0606

( O A (0) (0) (0)

0.155 0.0684 0.0237 0.00517 0.000346 (0)

Page 6: On an Analytical Determination of Initial Points of Crack Propagation

234 FISCHER, K. F., Determination of Initial Points of Crack Propagation

Fig. 5. Positions of initial points versus POISSON'S ratio accord- rg ing the extended multi-axial-value-criterion

$5 -- Rcfercnces

1 SIH, G. G., Mechanics of Fracture 1 - Methods of analysis and solutions of crack problems, Noordhoff Int. Publ., Leyden 1973. 2 FISOHER, K.-F., GUNTHER, W., Gegenwartiger Stand der RiDbruchmechanik im Hinblick anf eine praktische Nutzung,

Maschinenbautechnik, 27 (1978) 2, S. 73-76. 3 FISOHER, K.-F., GUNTHER, W., SCHXIDSBERGER, W., Zur Formulierung des Normalspannungskriteriums in der RiDbruch-

mcrhanik. \Visa. Beitrage der IH Zwickau, 4 (1978) 2, S. 119-125. 4 FISCHER, K.-F., GUNTHER, W., Einbeziehung der RiBbruchmechanik in die theoretisrhe Erkliirnng des Trennvorganges, Masrhi-

nenbautechnik, 28 (1979) 5, S. 220-223. 6 FISCHER, K.-F., Zvlr Ermittlung qulitativ neuer Bruchaussagen, die sich aus einer Erweiterung bekannter Bruchkriterien er-

geben, Vortriige zum Problemseminar Bruchmechanik, Technische Universitat Dresden 1979, Heft 1, 8. 151 -168. 6 COTTERELL, B., Notes on the path and stability of cracks, Int. Journal of Fracture Mechanics, 2 (1966), S. 626-533. 7 WILLIAMS, J. G., EWING, P. D., Fracture under complex stress - The angled crack problem, Int. Jour. of Fracture Mechanics,

8 SIH, G. C., Strain-energy-density-factor applied to mixed mode crack problems, Int. Journal of Fracture, 10 (1974) 3 , s . 305-321. 9 FISCHER, B.-F., Zur Erweiterung des Bruchkriteriums von SIR, ZAMM, 68 (1978) 8, S. 331-336.

10 FISCHER, K.-F., On an extension of SIH'S fracture criterion, Int. Journal of Fracture, 15 (1979) 1, S. Rll-R14. 11 GARDE, A. M., WEBS, V., Brittle crack initiation a t the elastic-plastic interface, Metallurgical Transactions, 3 (1972) 11, S. 2811

12 CHIANG, W. T., Fracture criteria for combined mode cracks, ICF 4, Fracture 1977 Vol. 4, Waterloo (Canada) 1977, S. 135-154. 13 KOCHEND~RBER, A., SCHOLL, W., Die Sprodbruchneigung von Stiihlen in Abhanigkeit von Spannnngszustand nnd Temperatnr,

14 FISCHER, K.-F., Zum Verhalten der Mehrachsigkeitsmlil in RiDspitzenumgebnng (to be pnhlished).

8 (1972), S. 441-446.

to 2817.

Stahl und Eisen, 77 (1957), S. 1006-1018.

Eingereicht am 20. 10. 1980

Anschrift : Dr.-Ing. K.-F. FISCHER, Ingenieurhochschulc Zwickau, Abteilung Mathematik-Naturwissenschaften, Wissenschafts- bereich Technisrhe Mechanik, DDR-9541 Znickau, PSF 35