Cockcroft-Latham Fracture Criterion and Bulk Formability of Copper Base Alloys
Cockroft-Lathamov kriterij loma in masivna preoblikovalnost bakrovih zlitin
B. Ule*, V. Leskovšek, B. Breskvar, Institute of Metals and Technology, Ljubljana, Slovenia
K. Kuzman, D. Švetak, Faculty for Mechanical Engineering, Ljubljana, Slovenia F. Kofol, Kolektor, Idrija, Slovenia
The ductility of metalic materials is generally defined as the ability to deform plastically vvithout fracture. It is usually expressed as a measure of the strain at fracture in a simple tension test1. Hovvever, the percentage elongation in a tensile test is often dominated by the uniform elongation, vvhich is dependent on the slope of the stress/strain curve. The end of uniform elongation coincides vvith the onset of plastic instability accompanied by voids nucleation, their grovvth and coalescence. It appears that the elongation value is too complex to be regarded as a fundamental property of a material and it seems reasonable to assume that any criterion of fracture vvill be based on some combination of stress and strain rather than on either of these quantities separately. Tvvo grades of copper base alloys used for the production of commutators for electrical motors vvere tested in compressing and stretching. The bulk formability of these alloys vvere projected using the Cockcroft-Latham criterion23, based on the tensile strain energy density at fracture. This criterion emphasizes the importance of tensile stresses in fracture and can be applied to a variety of cold vvorking processes.
Key vvords: ductility, formability, fracture, Cockcroft-Latham fracture criterion, copper base alloys
Duktilnost kovinskih materialov je v splošnem definirana kot sposobnost, da se plastično deformirajo brez pojavljanja razpok. Običajno jo izrazimo z lomno deformacijo pri enostavnem nateznem preiskusu1. I/ odstotkih izmerjen celokupni raztezek pri nateznem preiskusu je dokaj odvisen od enakomernega raztezka, to je od strmine krivulje napetost-deformacija. Konec enakomernega raztezanja sovpada s pojavljanjem plastične nestabilnosti, ki jo spremlja nastajanje por, njihova rast in združevanje. Zdi se, da je raztezek preveč kompleksen in ga ne moremo smatrati kot osnovno materialno lastnost. Zato je smiseln privzetek, po katerem bo moral lomni kriterij temeljiti prej na neki kombinaciji napetosti in deformacij, kot le na eni posamični količini. Za tlačno in natezno preiskušanje smo izbrali dve bakrovi zlitini za izdelavo kolektorjev pri elektromotorjih. Masivno preoblikovalnost teh zlitin smo opredelili s Cockcroft-Lathamovim kriterijem23, ki temelji na gostoti natezne deformacijske energije. Ta kriterij poudarja pomen nateznih napetosti pri lomu in ga lahko uporabitmo pri različnih procesih preoblikovanja v hladnem.
Ključne besede: duktilnost, preoblikovalnost. lom, Cockcroft-Lathamov kriterij loma, bakrove zlitine.
1. Introduction
The evolution of ductile damage within the deforming body is considerably influenced bv the stress state in the material. It was suggested bv Siebel4 that the cracking in metalworking is as-sociated with induced tensile stresses. even in processes such as forging that are predominantly compressive. The importance of tensile stresses is indirectly confirmed by the large inerease in ductilitv vvhen the materials are deformed under hydrostatic pressure1. Pugh and Green'' demonstrated that superimposing a hydrostatic pressure in the extrusion process greatly enhanced
* Dr. Boris ULE. dipl. inž., IM'I Ljubljana. Lepi pol 1 1, 61000 Ljubljana
reductions could bc achieved in ductile materials, and that even some brittle materials could be extruded vvithout difficulty. In a tensile test of a eylindrical test specimen the stresses at the minimum section of the neck can be calculated in different vvavs" 1 and may be considered to be the sum of tvvo parts. One part. the equivalent stress. is equal to the current yield stress and is constant aeross the cross-section. The other part, a hydrostatic tension, varies from zero at the periphery to a peak value at the centerline. As stated by Cockcroft and Latham '. that the use of the criterion based on total plastic vvork per unit volume at the fracture point, vvhich vvould take into account only the equiva-lent stress i.e. the current yicld stress, is not a proper solution.
The current y i c ki slrcss, unlike the pcak stress, is not intluenced by the shape of the neeked region. Consequently, the neck shape should have no effect 011 the fracture strain. a conclusion which is eontrary 10 cxperimental facts'. Thercfore Cockcroft' and Cockcroft and Latliam1 proposed a criterion based 011 the tensile strain energv densitv wherc the magnitude of the highest normal stress is taken into account. At tensile testing this would bc the stress aeting in the ccnterline where fracture is initiated.
As an outgrowth of experimental evidence of the influence of stress stale on duetile fracture, several other eriteria vvere also suggcsled for the predietion of fracture in complex stress states. A modilication of the Cockcroft-Latham criterion vvhich in-cludcs a hydrostatic-prcssure term vvas suggested bv Bro/o et al.1 . Other eriteria of importance vverc proposed bv Oyane Clift11 Hoffmanner1' and Osakada"'. Sueh eriteria vvere success-lullv applicd bv a number of investigators to a varietv of eold vvorking operations1 '.
2. Fracture criterion
With the Cockcroft and Latliam tensile ductility approach, the fracture is predieted vvlien
J(7 (o" /a)df = C
(1)
Load
Figure I: The geomelry al the neeked region and the distribution of axial stress <r„. Slika I: Geometrija vratu in porazdelitev aksialne napetosti it,,.
factor representing the effect of the highest tensile stress. cr*. and C is a material constant. If there is no tensile stress operating but only a compressive stress. it* = 0 and fracture docs not occur'.
The expression (1) has the dimensions of vvork per unit vol-ume (N/nr = Nm/m' = J/m1). The reduced form
a ile = C
.2)
vvherc t' is equivalent i.e. effective strain, č, is cquivalcnt strain al fracture, cr is cquivalent i.e. effective stress, o is the highest tensile stress, (it*/i~t) is a non-dimcnsional stress-conccntration
Load
is used for calculation. The correction for neeking and pro-jections for .1 could bc obtained bv different approaches Wright and coworkersIH. for instance, successfulv used the equa-tions of Davidenkov and Spiridonova . vvhereas the equations of Bridgmair vvere used in our experiments.
Fig. 1 illustrates the geometrv at the neeked region and the distribution of axial stress bv this localized deformation. I11 ae-cordance to Bridgman\ the variation of the stresses in a minimum scction of a neeked bar in tension is given as
= ct(i + 2 p/R) In (\ + R / 2p)
<T, = <7„„ = a In
R:+ 2Rp - ; 2 Rp
a
o + cs„
(3a)
(3b)
(3c)
vvhere (r„. is the average stress on the minimum scction of the neck (load divided bv minimum neck area). R is the radius of the minimum cross scction at the neck, r' is the so called "strained length" (r'= r R/R„), R,, is the radius of initial cross scction (R„ = 5 mm) and p is the radius of curvature of the neck profile. In the ccnterline of the neeked region, vvhere r = r' = 0. z = 0 the stress component it,, reaches the highest value ir....... = <r
- - , R + 2p a = <7 + o In-—
(4)
Considering the strain hardening. the llovv curve can bc ap-proximated by several constitutive equations. The most common are proposed by Hollomon", Ludvv ik" . Svvift24 and Vocc . The most simple one is (lic Hollomon povver lovv relation:
a = KI"	(5)
vvhere 11 is the strain-hardening exponent and K is the strength coefficient of the material. Both constants could bc determined simply bv the values of 0.2'v offset yield strength and fracture stressls or bv computer least-squares fits by plotting hi o against hi i. vvhere 11 and K are the slope and intercept respectivcly. Hovvever, the strain-hardening exponent n. and the strength coefficient, K. could bc also evaluated front the tensile test dala bv applving the criterion of instabilitv at the onset of neeking. da/de = it. From this. it can bc shovvn that at neeking. vvhere the ultimate tensile strength is measured. the strain-hardening exponent is given as
f„ = 11	(M
Once the strain-hardening cxponent is knovvn. the strength coefficient K in the Eq. (31 can be eusilv determined bv means of ultimate tensile strenath au„, as
K = cruls (2 ■ 71S28/n)
\ n
(7)
From the results of Eq. (5), a relationship of it 10 t' can be determined bv using Eqs. (3a) and (4) and assuming that the

necking is initiated at the ultimate tensile strength o\lls when ti",, = <r„ and vvhen E achieves n (Hq. 6). assuming also that cr increases linearlv vv ith equivalent strain (Fig.2). Considering the hatched areas on Fig. 2. the integral (2) can be separated as follovv s
C = J<7 de = J rr de + | a de
(S)
B> substituting the knovvn function <r r.v. t' and using a Irape-zoidal formula for the second integral, vve finallv get
r *	i, r
/1+1 "
(9)
			« Of
		0* ----	Of
		—---/ ' , o vs e / / / / / / / / /	
			
			
Ey	E,
True Strain
Figure 2: Effective stress. a. and peak stress. a , versus effective strain at tensile testing (schematically).
Slika*2: Ekvivalentna napetost o in maksimalna napetost o v odvisnosti od ekv ivalentne deformacije pri natezanju (shematsko).
3. Experimental methods
Tvvo copper base alloys declared as CuAg().2 (OF) and CuA«0.2 respectivelv vvere selected for the present study. Both allovs, commercially available as one-half inch diameter vvires, contain 0.2 vvt. 7< Ag. This type of copper base allov s has good creep strength at elevated temperatures and high softening temperature and is used for instance for the production of commuta-tors for electrical motors.
The cxperimental CuAg().2 allov contained 0.01 vvt. 7c O and 0.005 vvt. 7, P vvhereas the CuAg().2 (OF) alloy i.e. oxygen free allov contained < 0.005 vvt. 7, O and 0.002 vvt. 7, P. Hovvever, the bulk vvorkabilitv of CuAg().2 allov vvith higher oxygen content vvas essentiall) wor.se than that of oxygen free allov. As shown on Fig. 3, a duetile damage occured at eold bending of commutator segments from such oxygen containing CuAg().2 allov. On the contrary, the bulk formability of experimental oxy-gen free alloy was excellent as it didrft present any problems in metal vvorking processes.
The microstructure of both allovs is shown in Fig. 4 and 5. The microstructure of oxygen containing CuAg().2 allov (Fig. 4) consisled of relatively small grains with tvvinned areas. vvhereas the microstructure of oxygen free CuAg0.2 alloy (<0.005 7< O) (Fig. 5) is characterized by somewhat larger. equiaxed grains

V-i*-

r S®«**;
■ ./j ■ -rmg^f-..
50jjm Ž
h
H


Figure 4: Microstructure of oxygen containing CuAg0.2 allov . shovv ing relatively small grains vvith some tvvinned areas.
Slika 4: Mikrostruktura kisik vsebujoče zlitine CuAg0.2. Razmeroma drobna kristalna zrna s posameznimi področji dvojeenja.
Figure 3: Cracks on commutator segments occured at eold bending. Slika 3: Razpoke, nastale na krakih kolektorja pri upogibanju v hladnem.
Figure 5: Microstructure of oxygen free CuAg().2 allov. shovv ing equiaxed grains vvith irregular boundaries. Some grains contain tvvinned areas.
Slika 5: Mikrostruktura kisika proste zlitine CuAg0.2. Enakoosna kristalna zrna z iregularnimi mejami in dvojčki v posameznih zrnih.
with irregular boundaries and twinned areas. The microstructure of both alloys is typical for the as annealed state where only a slight cold reduction was applied at the calibration drawing.
Tensile flovv curves were obtained using an Instron machine by pulling standard tensile specimens vvith gauge sections of 10 mm diameter and 100 mm length at a cross-head speed of 1 mm/min. The neck profile radius vvas established from the photographs of the necked region using an appropriate geomet-rical aproximation. A discontinuous compression testing vvas carried out on a instrumented hydraulic press on specimens vvith 10 mm in diameter and initial height of 12 mm. In order to min-imize the frictional constraints, teflon vvas used on contact surfaces and constant frietion coefficient of 0.024 vvas achieved. Cumulative height reductions reached 86 %. The strain rate at compression testing vvas comparable vvith that at the uniform tension and remained nearlv constant over most of the strain range. Load-displacement data obtained vvere fed into a com-puter by points for stress-strain analysis. A correction factor for the adequate compensation of frietion vvas also incorporated into the eomputer program.
4. Results
Fig. 6 are the plots of load vs. elongation at tensile test of both experimental alloys. The values obtained at testing are listed in Table I. The yield stress at tensile testing vvas determined as 0.2 'h Offset Yield Stress. The yield stress measured at compression testing is also included in the table (values in paren-thesis). As alreadv mentioned, the radius of the minimum cross seetion of the neck R. and the radius of curvature of the neck profile r. of the tensile specimens vvere measured too. By the CuAg0.2 alloy (0.01 % O) the neck profile radius vvas 2.20 mm. the minimum cross seetion radius vvas 2.85 mm, vvhereas at oxygen free CuAg0.2 alloy (<0.005 '/i O), vvhere more a rupture than a fracture vvas observed, the neck profile radius vvas only 0.15 mm and the minimum cross seetion radius vvas 1.8(1 mm.
In Table II the values of Hollomon eonstants calculated from the compression- and tensile lests data are listed. The eonstants. obtained from tensile test data vvere determined bv various methods of analysis, because the high percent error in £u makes sometimes the Eq. (6) unacceptable. Hovvever, n is the exponent of an empirical equation (5) and it is not surprising that this equation cannot accurately deseribe the stress-strain curves in the vvhole strain range. Man et al.-" deseribed the tensile curves
Figure 6: Load vs. elongation at tensile lest of both experimental allovs.
Slika 6: Odvisnost med obremenitvijo in ra/.tezkom pri nate/nem preiskušanju obeh eksperimentalnih /litin.
Table I: Tensile test results
Tabela I: Rezultati nate/nega preizkušanja
	Vield stress' (MPa)	Tensile strength <rul, (MPa)	Uniform elongation eux 100 (%)	Total elongation e,n X 111(1 (<7r)	Reduction in area Z\ 11)1) (%)
CuAg0.2 allov (0.01 O)	306 (321)	323	->	9	68
CuAg0.2 (OF) allov(<0.005r i O)	2^2 (248)	258	142	27	87
* The values in parenthesis vvere obtained by compression testing
Table II: Comparison of Hollomon eonstants bv various test methods
Tabela II: Primerjava Hollomonovih konstant dobljenih pri različnih metodah preizkušanja
	Strength coefficient K (MPa)	Strain-hardening e.iponent n	Correlation coefficient r
Compression Test			
CuAg0.2 allov (0.01 'i O)	397	0.090	0.977
CuAg0.2 (OF) allovir «0.005 <k O)	379	0.139	0.993
Tension test			
CuAg0.2 allov (O.of% Oi	431	0.087	
CuAg0.2 (OF) alloy «0.005 O)	385	0.133	
Calculated from the true stress-true strain tensile test data "Calculated from the uniform elongation /Eq. (6)/
Table III: Values of the material eonstants Tabela III: Vrednosti materialnih konstant
	(MPa)	CT„ u (MPa)	eu	o (MPa)	e,	(T , (MPa)	C/Eq(9)/ (MJ/m1)
CuAg0.2 allov (0.01 % O)	306	330	0.019	435	1.124	652	548
CuAgO,2(OF) allov «0.005 %0)	212	288	0.129	423	2.041)	124d	1499
of copper using tvvo equations of Hollomon tv pe". This kind of analysis is based on the assumption that a change in deformation mechanism occurs during the deformation.
The collection of final results is given in Table III vvhere 0.2 °/( offset yield stress. a . average ultimate stress. <r,, „. and fracture stress. <rr, vvith corresponding strains, f„ and 8, respec-tively, the highest tensile stress <r as vvell as the material constant C are shovvn. The value of C, i.e. the value of tensile strain energy density of oxygen free CuAg0.2 allov (<0.005 c< O) is extremely large. This allov vvith lovv oxygen content exhibited
more a rupture than a fracture at tensile testing resulting in adequate small radius of curvature ofthe neck profile while the tensile stress in necked region strongly increased. In spite ofthe fact that the original Bridgman5 analysis and the similar analysis of Davidenkov and Spiridonova7 give the best approximate procedure for the calculation of stress distribution in the necked region, novvadavs, more accurate numerieal solutions can be obtained with FEM.
5. Discussion
Using the fracture criterion based on the model of Cockcroft and Lathanr '. the tensile strain energy density at fracture fortwo copper base allovs was calculated. The calculation utilized the results of tensile and compression tests. When the constant cross-head-speed is used at tensile test, the strain rate decreases slighty during the homogenous deformation and then rises rapidly as neeking occurs. The rise in strain rate during the neeking results in an anomalous rise in flow stress and restriets the usefulness of the data obtained by strains smaller than those prevailing during the onset ofthe neeking27. Probably some discrepancies betwccn the tensile and the compression test results (see Table II) could be explained also vvith Bauschinger effect and vvith the assump-tion that a change in deformation mechanism occurs during tensile deformation. Zanki2* and Schvvink and Vorbrugg2" found ihesc kind of stages also in the tensile curves of annealed nickel and copper at lovv strains. Furthermore. Mishra at al.1" proposed that in the range of uniform strain the Hollomon lavv overesti-mates the flovv stress in the initial stages and underestimates it in the final stages. It seems therefore, that the tensile test data of experimental copper base allovs should fit better vvith double-n method vvhich uses the tvvo Hollomon equations. Kleemola and Nieminen-1 found out that the use ofthe Hollomon equation for pure copper in an annealed state and after 40 pet deformation gives a misleading pieture of the strain-hardening properties of the material because the strain-hardening exponent n is not equal to the correct eu value. The same vvas also observed in our CuAg0.2 experimental alloy vvith 0.01 % oxygen vvhere computer least-squares fits by plotting tensile test data had to be used instead of F.q. (6). In opposition to tensile test data, the compression test data are obtained from a much larger strain range and it is therefore assumed that the use of the simple Hollomon povver lavv relation is justified. The regression analy-sis of compression test data also gives satisfactory high linear correlation coefficients (0.98 and 0.99 respectively).
The data presented in Table lil (C-values) shovvs that at oxygen free CuAg().2 alloy the integral of the maximal tensile stress over the plastic strain path /Eq. (9)/ reaches a value of ap-prox. 1500 MJ/m". vvhich is ncarl v three times larger than that of CuAg0.2 allov vvith 0.01 % oxygen (C approx. 550 MJ/m1). Hovvever. the reduction in area at tensile test of oxygen free CuAg0.2 allov is only for one quarter larger than that of CuAg0.2 allov vvith 0.01 % oxygen. It seems that a considerable inerease in ductilitv may result bv an on!y relatively lovv improvement in the reduction-in-area value. so that such way of expressing the duetiIity shovvs little diserimination in very duetile metals. Namely. once the neeking develops, a "negative feedback" effect occurs. vvhich tends to prevent the exhibition of really large ductilitv values'. At the onset of neeking, the stress system changes and a component of hydrostatic tensile stress is gener-ated vvhich is superimposed on the axial stress. The hydrostatic tensile stress inereases as the neck becomes deeper, vvith the result that the fracture is more likely to occur. This situation leads
to a stabilizing effect and an inerease in ductility causes a deeper neck to be formed, hence higher local stresses are developed and a greater possibility of fracture occurs. The Cockcroft-Latham criterion of duetile fracture is therefore a much more reasonable criterion than the criterion based simply on reduction-in-area value at tensile test. Hovvever. it should be noticed that at the neck profile vvhich is characteristic for rupture instead of fracture, the extremely high value of tensile stress <r . and high tensile strain energy density C. are somevvhat doubtful. The equations of Bridgman5 /(3a,b and c) and (4)/ and equations of Davidenkov and Spiridonova becomes questionable at very lovv radius of curvature of the neck profile typical for the rupture and vvhen asymptotic continuum mechanics analyses could not be applied for the deseription of stress state due to local singularitv.
The excellent bulk formability ofthe experimental CuAg().2 oxygen free alloy vvith regard to the verification based on the Cockcroft-Latham criterion is not surprising. This criterion has the desirable feature for homogenous compression, in vvhich čase the tensile stress o' is zero and no fracture limit is predieted. This coincides vvell vvith the experimental results at homogeneous (frictionless) compression. Hovvever, follovving the theoretical analysis of Kuhn et al.1, it is possible to relate the C-value calculated from tensile test data /Eq. (9)/ and the principal tensile and compressive surface strains, i.e. the circumferencial and axial strains measured on the barreled equatorial free surface at fracture in upset lest. It vvas also experimentally confirmed that there is a linear relationship betvveen the tensile and compressive surface strains at fracture by upsctting by rolling and by bending. Consequently, the representation of fracture data as a plot of tensile versus compressive strains at fracture became a useful form for the analysis ofthe fracture in eold forming processes resulting in a "forming limit diagram concept". The Cockcroft-Latham tensile strain energy fracture criterion is consistent vvith this concept'.
6. Conclusions
The Cockcroft-Latham tensile strain energy density criterion vvas used for a reliable evaluation of the ability of tvvo copper alloys to metalvvorking operations. This criterion implies that the fracture depends both on the stresses imposed and on the strains developed and it could be deseribed as "fracture vvill occur vvhen the plastic vvork of the largest tensile stress, per unit volume, reaches a characteristic critical value".
It vvas demonstrated that there is a good qualitative agree-ment betvveen the predietion of formability under bulk metalvvorking conditions based on this criterion and the observed behaviour in the production of commutators forelectical motors. The plastic vvork done per unit volume by oxygen free CuAg().2 alloy vvith excellent bulk formability is nearly three times larger than that done by the oxygen containing CuAg0.2 alloy (0.01 % O), vvhile, the difference in the tensile reduction of area of both alloys vvas much smaller. Consequently, the reduction-in-area value at tensile test cannot be appreciated in general.
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