plastic notched bars

J. Mech. Phys. Solids,1968, Vol. 16,pp. 205to 213. Pcrgmmn Press. Printedin GreatBritain.

CALCULATIONS ON THE BENDING OF RIGID/PLASTIC NOTCHED BARS By D. J. F. EWING Department

of Applied Mathematics

and Theoretical

(Received ZDtk January

Physics,

Cambridge

1068)

STANDARD plane-strainsolutionsfor the bending of notched rigid/plastic

bars apply only if the notch is sufkiently deep (GREEN 1966). Following Green, critical notch depths are calculated for a range of V-notches with drcular roots, under pure bending. In particular, the conventional deep-notch soluton does not apply to the pure bending of the standard Charpy-Izod test specimen. The influence. of the indenter on the slipline field for the Charpy test is also studied. The constraint factor, but not the stress-state around the notch, is found to be sensitive to the indenter conditions assumed. Detailed calculations are presented.

1.

OBJECTIVES

SLIPLINE fields solutions describing the plane-strain bending of V-notched bars, including in particular the standard Charpy-Izod specimen, were first given by GREEN (1958, 1956) and by GREEN and HUNDY (1956). This last paper contains a very full account of the practical and experimental significance of the theory. The analysis concerns quasi-static bending either by pure couples (fourpoint bending) or by an indenter, as in the Charpy test (three-point bending). Variants of these solutions have since been proposed by several writers. LIANI~ and FORD (1958) extended the theoretical and experimental analysis to notches with miscellaneous shapes in pure bending. ALEXANDER and KOMOLY (1962) recalculated the three and four-point bending solutions for the actual Izod specimen, allowing for the finite fillet radius of the notch root, but neglecting effects due to the indenter. In all the slipline fields that were calculated explicitly, yielding was assumed to be localized around the minimum section. The slipline fields then depend only on the local notch shape near its root, and on the loading conditions; they are independent of the notch depth. As Green emphasizes, this can only be true if the notch is sufficiently deep. For shallower notches, the initial deformation spreads to the surface of the bar on either side of the notch. Green has suggested possible forms, without detailed calculations. The calculations presented here are of two kinds. First, following Green, the critical notch-depth/core-width ratios above which the deep-notch solution applies under pure bending are given for a range of notch angles and notch root radii. It is found in particular that the notch in the standard Charpy specimen is not deep enough for the conventional deep-notch pure-bending solution to apply. Next, the effect of the indenter on the slipline fleld for the Charpy test is examined. In this THE

205

D. J. F. EWING

206

with bending taking place about the indenter, the deep-notch type of solution does apply, as was shown by Green; this is also confirmed by the experiments of Green and Hundy. Following the suggestion of Alexander and Komoly, we idealize the actual circular indenter by replacing it by a flat-topped punch of width 2b, and analyse the effect of varying this small dimension b. We again allow for the finite notch root radius.

test,

2.

CRITICALNOTCH DEPTH IN FOUR-POINTBENDING

Let a be the minimum width of bar at the notch. We seek the critical width w,; i.e. the minimum width for which the deep-notch solution applies. Figure 1 illustrates Green’s construction (1956) for a lower bound WC to w,.

W

U

Fxa. 1.

Green’s

constructionfor a lower bound WQ to the pure-bendingeritiealwidth

WC.

We first consider notch halfangles u less than one radian, and sharp notches (Fig. la). We use Green’s lettering in this figure, but not his other notation. Regions LMPO, QRS, together with the circular arc PBQ of radius R, (and reflections in ZZ’) represent his deep-notch slipline field. The plastic triangles OML, QRS, with sides b (OM) and c (QR), are respectively in tension and compression, with p = -k and + k; so OPBQTR is an u-line. We are using the usual notation of slipline field theory (HILL 1959, Chapter VI); thus, k is the plane-strain yield shear stress and p is the hydrostatic pressure. The parameters R, b, c and the angles 4 and 8, which fix the deep-notch field, can be calculated in terms of a and u (GREEN 1953): we find B = $7f + & (1 -

a).

The dotted lines represent a slipline extension of this deep-notch field, isolating non-plastic shoulders GIFUL; all stresses vanish beyond WGK. IF is a circular arc of radius R’; FD is a straight-line segment of length b’, and GIK is a plastic triangle of side GI = a’. These three parameters are fixed by the three conditions of overall static equilibrium of the shoulder. All angles arising may be determined in terms of u from Hencky’s theorem and direct geometry. Define /l = 8~ - u. Then the angular spans of PB and FI both equal g/l, and the angle X of arc BQ is always equal to $7r + 9.

Calculations on the bending of rigid/plastic notched bars

207

The point of this construction is as follows. Suppose the unstressed material above WGK is removed, together with the rigid core of material; replace the latter by the equivalent distribution of surface tractions along OPBQT. Shoulder GIFULW is now able to rotate rigidly relative to the rest of the bar. Using this deformation mode and the plastic stress-field in a familiar virtual-work argument, we see that for all widths zp,< WC every equilibrium stress-field compatible with the tractions assumed along OPB is necessarily overstressed somewhere. For details see GREEN (1956). Thus, wC is certainly not less than WC. w, will be equal to WC if this field can be completed; i.e. if some field of stress can be found in the rigid regions that does not violate the yield criterion. This cannot as yet be proved but seems plausible; Green shows that this field is the limiting case of a possible family of (incomplete) slipline fields at subcritical width. The deep-notch field (and so Green’s construction) may be modified in the usual way to allow for a fillet of radius T at the notch root. See, for example, Alexander and Komoly (op. cit.) or JOHNSONand SOWERBY(1967). The modification needed to Green’s construction will be evident. Values of WC and DG, together with the parameters describing the deep-notch field and the constraint factorf, are given in Table 1 at the end of this section. DG is the dimension ZK. The whole field is essentially similar to the rotating-shoulder fields providing lower bounds to the critical widths of V-notched bars (EWING and HILL 1967; EWING 1968) and is calculated here by exactly the same method. For angles a greater than one radian, the deep-notch core OPBQT is replaced by a singular point N, as in Fig. lb. The angles 8 and p = 87 - u are now equal, while PB and FI again have angular span $8. The modification to Green’s construction is as shown: the field defined by the slipline NPMU and the singularity at N may be continued as far as UF. The construction of the shoulder follows as before. Exactly the same virtual-work argument shows that the assumed stresses along ON cannot sustain an admissible equilibrium stress-field if rp,< WG. For sharp notches there is an exact correspondence with the shoulder geometry of the tension-bar problem; but this is no longer so when a circular fillet is introduced. The deep-notch field parameters b (= OM) and d (= NS) may be expressed algebraically in terms of r, a, and p as b@

= [a + r -

(1 + B) r@l/@ +

41/2 = [(a + 4 (1 + B>- 4/(2

8), + k%

and the deep-notch constraint factor f may be written (normalizing a = 1 unit) f =

(1 + r - =8)2 +

-_I!--(1 2

+

B

+ r + re@)2 -

r2 (e2p -

1).

When a < 1, corresponding expressions would be very complicated. The transitional case CC= 1 provides a useful numerical check on the computer programme used to generate Table 1. For the larger values of r, and the smaller values of a, the parameter b’ becomes negative. This means that UF lies to the other side of MD, so that the third parameter fixing the shoulder dimensions is the angular span of UM rather than the straight-line length b’ of MU. Strictly speaking, the calculation method should be modified to allow for this, since UM is no longer straight. This was not thought

D. J. F. EWING

208

worthwhile in view of the modest values of Y and b’ arising in the cases involved, since the inaccuracies involved are very small. The corresponding values of WC appear bracketed in Table 5. In his paper Green suggested an alternative approximate construction [given by BISHOP (19%) in the tension-bar context] for W, and Do. The plastic stressfield is again continued into the rigid regions (the continuation being defined uniquely, as before, either by sliplines PML and PB or by NPML and the singularity at N), this time up to a stress-free surface beginning at 1;. This surface LK* is continued from OL until it becomes parallel to the flat bar side at K*. Figure 2 illustrates the procedure when the notch is a circular arc, with N a singular point of stress. LN and L’N are equiangular spiral segments. The angle of the fan LNE is # and the angle turned through along NL or LK* is /3. Such a deep-notch

FIG. 2. The Bishop-type construction for an approximation WB to WC, applied to a circular notch.

field is valid only if the ’ effective ’ notch half angle 4~ - fl is not less than one radian ; tensionbar problem, the dimensions Wg, Dg of this field may be expected to be very dose to W, and &!, Values of WB and WC as calculated by computer are compared in Table 1; they are seen to be very nearly equal. In all these cases DB and Do were found to agree to four decimal places. But, unlike the tension-bar case, no method of proving this field to be complete is known to the writer. An advantage of this approximation is that it can be done graphically, if desired. when u > 1, WB and Ds may be evaluated analytically. Consider first the fully circular notch of Fig. 2. The radius of curvature p (t) of LE after turning through an angle t from L may be written, i.e. if ~$(a + r) < O-64 (GREEN 19.58). By analogy with the co~esponding

p(t) ==q!!

3 u~t~~~! n=O

The coefficients an are given by

a, = e@ 5

(-)”

B716+~+i

m-0

(writing & = p/ +nf.) and so satisfy the recurrence-relationships a, + an+1 = e@ t%+n

*=e@

---I.

So, by algebra exactly similar to the corresponding tension-bar problem [described in detail in EWXNG (1968, Appendix)], we And

Calculations on the bending of rigid/plastic notched bars WB = a + &(e@ -

209

1)2,

where More generaily, when the deep-notch slipline field extends up to the straight sides of the notch as in Fig. lb, we have WB = 0 + *r (es -

BB =

re@ (P) + AO

If2 + b 42

(es -

I),

b v“J (Ao + JO).

This follows by combining the sharp-notch and round-notch Bishop fields, using the slipline field superposition principle (HILL 1967). The sharp-notch field dimensions are available from the tension-bar problem. For a < 1, no such simple analytical expressions are available, because the angles B and B are unequal. As before, the transitional case a = 1 checks the computer programme used to generate Ws and DE. TABLE I. cc.

da

8.21" 7*5O 11.32”

0

15O lY*60° 22.5'

30°

8Y+i0

45O

1 radian

Rla

bla

c/a

WBlU

WC/a

Dola

f

aml2k 2.515 2.449 2*388

l/32

0.3886 0*8882 O-8555

@015o 0

0.5027 0*5081 o-5298

1.4229 (1~4208) pm5387 1.4222 (1.4206) 0.5839 l-8931 (1-3918) 0*5397

1.2606 I*2696 1CM.W

0 l/82 l/l6

oea85a 0~8851 0.8251

MM32 o-0155 0

O%%%@ 0*5297 oma2

1.4174 (1*4164) 0.5852 i-3925 (1~3915) omoo 1.3679 (1.8671) 0*5w5

l-2693 1.2430 1.2274

0 l/82 l/l0

0*3780 0.8516 osa34a

0.0744 0~0492 0*0249

0.5116 o-5828 0.5540

0.5834 1.4082 14Q76 1.8875 1.8869 0.5917 1.8667 (1.3661) 0.5940

1.2594 1.2427 1.2273

2,178

0 l/32 l/l6

oG684 0*3438 W8198

9-1091 0+362 o-0633

9.5206 1*3946 0*5384r l-3774 O-5582 1+8663

1.3942 1‘3771 l-3699

0~5941 0*5958 OG976

l-2572 l-2415 1.2268

2.047

0 l/32 l/l6

0.8527 O-8398 O-8089

O-1479 O-1272 O-1065

O-5882 0%5490 0+5666

1.3762 1.8622 I*3431

1-8760 1.3620 1~8430

0~6930 o+o32 0.6085

l-2529 I.2885 1.2249

1,916

0 l/82 l/l6

0.8304 0.81x1 0~2913

0*1913 9.1727 0.1541

0.5502 0*5649 0.5796

I.8525 l+Ml2 1*8290

1.3524 l-3412 l-8299

0.6160 0.6147 0.6185

1.2458 1.2826 1.2294

I*785

0 l/32 l/l6

0+2751 0*2598 os?445

O-2751 O-2598 O-2445

0*5391 0*6009 O-6128

1*2994 1.2929 1.2346

l-2994 1.2920 l-2346

0*6495 l-2220 O-6458 l-2125 0*6421. l-2033

l-571

0

0

2.809 2.265

The case r = 0, GL= 30’ was calculated by Green. The values of the auxiliary parameters R’, b’, a’, as calculated here by computer are (with Green’s values in brackets) R’/a = 06127 (O-59), V/a = 0.0718 (O-077) and a’/a = 0.0379 (0.042), Green used the from which Wofa = 1.89425 (l-39) and Z?o/a = 0.5941 (0.59).

D. J. F. EWXNG

2x0

step-by-step calculating method; the main numerical errors then come in computing the couple resultant along OMD, For record purposes the auxiliary dimensions in the case u = 22.5’, r/a = 0.03125 are R’/a = 0.56122, b’/a = 0.00293and a’ = 0.04643, with &/a = O-59166 and Woja = 1.38687. Since this last is greater than 1.25, the deep-notch solution does not apply to the standard Charpy specimen in pure bending. The maximum tensile stress uplaat the notch root is given by 2k (1 + &r - z) as long as yielding extends at least as far as the straight portions of the notch. For an ideally sharp notch, am has a maximum 2.51 x 2k corresponding to u = 3.21”, b = 0. For still smaller notch angles the deep-notch slipline field is unaltered and so am does not change. The corresponding value 1.261for the constraint factor is a universal upper bound for all notch shapes in pure bending, as was noted by GREEN (1953). Rounding the notch root slightly decreases the maximum attainable values; thus, when T = 0.25 mm u= has a maximum of 2.38 x 2k co~esponding to M = 11.32O. 3.

ANALYSIS OF THE CHARPY TEST

We here calculate the slipline field, constraint factor and tensile stress at the notch root for the standard Charpy test piece in three-point bending. We wish in particular to analyse the corrections to be expected due to the indenter. The indenter surface in practice is a circular arc of about 24 mm radius (details are given by ALEXANDERand KOIMOLY). We replace this, as suggested by them, by a flattopped punch of width 2b, and analyse the effects on the slipline field of varying this small dimension b. We shall allow for the finite fillet radius r (= 4 mm) at the notch root. One half of the plane strain slipline field is drawn in Fig. 3. The indenter is supposed not to sink in; the condition for this is that zkb (1 + &r) > kF where kF is the applied shearing force per unit breadth.

(b) FIG. a. (a) Slipline field for the Charpy test. The indenter is idealized as a flat punch of breadth 2b. (b) Enhwgement showing fictitious sliplines CB and CD, for convenient calculation of the traction resultants along BD.

Calculationson the bending of rigid/plasticnotched bars

21x

In the standard Charpy test the lever arm I is 20 mm. In the standard Izod test, the specimen is loaded as a cantilever, with 1 = 22 mm. The same slipline field applies, but is restricted to one half of the bar, with b = 0. Green and Hundy originally calculated this field with r = b = 0 for the practical cases I = 20 mm and 22 mm. Alexander and Komoly confirmed their values and calculated the corresponding fields for r = & mm, b = 0. Green used the graphical construction already described to verify that the standard specimen was wide enough for his deep-notch solution to apply, His value for the critical-width ratio was 1.22, less than the actual ratio l-25. Green and Hundy’s experiments confirmed that deformation does not spread to the upper surface. In calculating this field we have to find values of R, d, #, y, and applied force kF so that kF balances the tractions acting on sliplines ABD and DE. Setting up the equations is helped by a simple artifice*: continuing the fully-plastic field into the rigid region by a fictitious fan BCD, as in Fig. 3b. This allows the traction resultants and moments on the circular slipline BD to be written down by inspection. The derivation and numerical solution of these equations is outlined at the end of this section. The results are shown in Table 2. o, is the maximum tensile stress across the root (Alexander and Komoly give instead the ma.ximum hydrostatic tension) and the constraint factor f is defined with reference to the pure bending of an unnotched bar; i.e. f = 2Fl/ka2.? TABLE

0 0.5 1.0

39629 4*0881 4.2173

103.543 103.755 104.119

9431 11.532

2.4779 22456 2.0322

30.2 293 29.1

l-7804 13196 13714

l-224 l-251 1.287

2944 2.051 2.064

20

0 o-5 1.0

4.0219 4*1475 4.2771

102.375 102.590 102.957

7.376 9510 Xl.625

24078 2.1757 1.9624

32*5 32-l 31.4

1.9448 1.9884 29460

1.216 1.243 l-279

2903 2.010 2-023

22

0 05 I-0

33441 3.9772 4.1132

101439 101.763 102.221

7394 9.736 11+758

2.5333 23625 20904

34-4 33.8 32.9

17714 13114 13636

l-218 1.245 l-281

l-970 i-982 1,997

20

0 0.5 1.0

3.9085 4.0419 4.1781

loo*370 100696 101.153

7.755 9.809 ll*844

2.4597 2.2291 2.0171

36-6 35.9 35.0

1.9356 1.9802 2.0383

l-210 1.238 1.274

1.933 l-944 1.960

22

0

0.25

2.

7,309

(a) We fzst consider the tractions acting along the ~~~~r spirai segment DE. Ox, Oy be local coordinate axes directed along aud perpendicular to OE (Fig. lb), so that

Let

LZ?B = Teeeos 8, yD = Teesin 8. Then the resultant traction resolved along Ox due to material on the rJg& of ED is given *This

was pointed

out to me by Dr. FL HILL

tThe expression given by Alexander discrepancies with their resulta for

in another

and Komoly

forf

thecaseb = 0 lie within

Thus,

they report an xmliquidated

residual

of fin theiF

context.

The device ~eema surprisingly

by

little known.

is inoorrect; they use in effect, 2E1 (I + t)/?&. The small the uncertainties of their trial-and-error numerical method.

notation)(El - E&)/k=

0*0078unitson p. 269 of their work.

D. J. F. EWING

212 e

e

s

k dx - (1 + 24 k dy =

kd ( - 2Zy) = - 2k0yn s 0

0

Similarly, the traction resolved along Ox is 2k&D, and the anti-clockwise moment about 0 is

Thus, U&etractions acting on ED due to natem’al on its bight are entirely equivalentto a hydrostatic tension 2kB acting along OD, together with an anti-clockwise puTe couple of #rs (exp (20) - 1). This remark is very useful in setting up traction-equations involving equiangular spiral seg ments. (h) We now set up the equilibrium equations. It is very convenient to resolve along and perpendicular to AB. Accordingly, define A and B as the projections of ABCD parallel to AB and CB respectively; thus A = d + R sin I/J, B = R (1 - cos $). Let k FA and k& be components of the resultant traction on ABD resolved parallel to AB and CB respectively: then we see from Fig. 2b that FA = A + (1 + By) B - 2Rsin # + 214Rcos $, F~=(1+2y)A+B-2$1Rsin(b. These tractions have a clockwise couple kGD about D, where GD = & (A2 + 232)(1 + 2~) -

#Be

+ Bd + Rs sin 4

due to the pressure terms due to the shear-force terms,

i.e. on substituting d = A - R sin # and B - R = - R cos #, Gr~=$(As+B2)(1+2y)+AB-Rs($-sin#cos$). Now let #* be the angle DO makes with AB, so that I,&+ = 4 - &r, and define y* = &r - y, We then have a set of six equations: B = R (1 - cos I/), A +- Tee COB$+ = (a -j- T) cos y* - b sin Y*, B + Tee sin I/S*= (a +

T)

sin y* + b ~0s y*

by direct geometry; FA -

26 re@sin $* = F cos y*,

FB - 26 Tee cos $* = - F sin y* from the linear equilibrium of the tractions ; and finally

from the total moment about D. The symmetries displayed by the six equations when written in this sort of form provide a useful check against algebraic blunders. These six equations may be set up and solved for the six unknowns A, B, R, #, y, F using the Hencky relationship 0=$--y-l. A necessary condition for this solution to be valid is that yielded material should be conilned to the notch fillet; i.e. that OLO > 2!2)". The best way of solving these equations is by the generalized Newton’s rule. Thus, let a be the current approximation to the root x of a system of n non-linear equations 1~(x) = 0 inn unknowns g. Then (in general) a better approximation is a - h, where the ht are found by solving the 1~linear equations, ih, 5-1

(V+%)x-.

- 1~(a)

(i = 1,2,. . . ) n)

Calculations

on the bending of rigid/plastic

Convergence, once it starts, is quadratic. provides starting-values.

21s

notched bars

Here, Green and Hundy’s solution for the case

4.

b = 0

T =

DISCUSSION

For the Charpy test-piece, a = 8 mm and cc = 22.5’. The deep-notch slipfield dimensions in pure bending are, from Table 1, R = 2.818 mm, b = OS94 mm, and c = 0.262 mm*. But the critical width is 11.1 mm; and therefore, as mentioned already, the deep-notch solution does not apply. This explains the observation by KNOTT (1967) that the maximum tensile stress at the notch root (as predicted experimentally from fracture considerations) lay below the value u, =2k (1 ++T --a) for the deep-notch solution. The analysis of the Charpy test shows that the minimum width b necessary to avoid yielding of material around the (idealized) indenter is about 04 mm. The dimensions of the slipline field are fairly sensitive to the dimension b, and the calculated constraint factor is then increased by about S%. But, as would be expected, the state of stress at the notch root is hardly affected; the maximum tensile stress em increases only by about f %. KNOTT (1965, 1967) has found that slip-initiated cleavage fracture in notched specimens occurs when urn reaches a critical (temperature-dependent) value. It follows that his three-point bend calculations (based on Alexander and Komoly’s values for em) are, as expected, hardly affected by indenter effects. ACKNOWLEDOMENTS This work wassupportedby a grant from the Science Research Council. The writer is indebted to Dr. R. HILL, F.R.S. experimental work.

for his guidance and to Dr. J. F. KNOTTfor several discussions about

his

REFERENCES ALEXANDER, J. M. and KOMOLY, T. J. BISHOP, J. F. W. EWINO, D. J. F. EWINO, D. J. F. and HILL, R. GREEN, A. P.

1962 1958 1068

J. Mech. Phys. Solids 10, 265. J. Mech. Phys. Solids 2,48: J. Mech. Phys. Solids 16, 81.

1067

1068 1058

J. Mech. Phys. Solids 15, 115. Q. J. Mech. Appl. Maths. 6,228. J. Mech. Phy8. Solida 6, 250.

1050 1050

J. Me& Phys. Solids I, 128. The Mathematical Theory of Plasticity

GREEN, A. P. and HUNDY, B. B. HILL, R.

(Clarendon

Press,

Oxford). JOHNSON, W. and SOWERBY, Ft. KNOTT, J. F. LIANIS, G. and FORD, H. *There

are small

0967). The diaepandu Sowerby forconfirming

differencea dluppear thin point.

1067

J. Mech. Phys. Solids 15, 255.

1907

Int. J. Mech. Sci., 9,4aa. Proc. Roy. sot. A205,150. J. Mech. Phya. Solide 15, 97. J. Mech. Phys. Solids 7, 1.

1965 1907 1958 between

these and

when

8 numerical

values

caloulated

slip SE corrected

independently In their equation

by

JOEN~ON

and

SOWBBBY

(SO): I ana prnteful

to Mr.