Deformation and fracture of an amorphous PdCuSi alloy in V-notch bending tests—I

DEFORMATION AND FRACTURE OF AN AMORPHOUS Pd-Cu-Si ALLOY IN V-NOTCH BENDING TESTS-I. MODEL MECHANICS OF INHOMOGENEOUS PLASTIC FLOW IN NON-STRAIN HARDENING SOLID H. KIMURAt and T. MASUMOTO The Research Institute for Iron. Steel and Other Metals, Tohoku University Sendai, Japan (Receioeh 26 February 1980; irkrevised form 27 March 1980) Abstract-Model mechanics of the inhomogeneous plastic deformation of amorphous Pd7,,sCu6!&s as a non-strain hardening solid is proposed. The dependence of the yield stress on plate thickness, notch angle, and notch depth is investigated with the V-notch bending test. Plastic deformation under plane stress occurs by an antiplane strain mechanics, consistent with predictions for Dugdale’s plastic zone and also contraction in the present model. In specimens under plane strain, the general yield deformation pattern shows a plastic hinge. Assuming Von Mises’ criterion, a plastic constraint factor of 1.20 is calculated, in good agreement with that of Green’s slip line field. The dependence of serrated flow on the extension rate was also measured over the range of 1 x lo-’ to 1 x 10’ mm/min. Serrated flow is found to disappear at a critical extension rate (Q. i, = Aexp(-H/kT) can be confirmed as the empirical mechanical equation of state for serrated flow. The activation energy (H) for onset of serrations is 0.35 eV. Rhsum&-Nous proposons un modble pour la dkformation plastique hCtCrog&ne d’un alliage Pd7,,JCu6Si18,s considirh comme un solide sans consolidation. Nous avons ttudie la variation de la limite ilastique en fonction de l’kpaisseur de la plaquette, de l’angle et de la profondeur de l’entaille, au wurs d’un essai de flexion avec entaille en V. La d&formation plastique sous contra&e plane se produit par une d&formation antiplanaire whCrente avec les pr&visions dans la zone plastique de Dugdale et avec le modQle propost en ce qui wnceme la contraction. Dans les hhantillons sous dkformation plane, la wurbe de la dhormation prtsente une region plastique. D’apr&s le critkre de Von Mises, nous avons calcult un facteur de contrainte plastique de 1.20, en bon accord avec celui du champ de lignes de glissement de Green. Nous avons Cgalement mesurC la variation de Woulement en hachures en fonction de la vitesse d’allongement entre lo-* et 10’ mm/min. Les hachures de l%wulement plastique disparaissent pour une vitesse d’Clongation critique (Q. Nous pouvons done confirmer l’kquation & = Aexp(-HlkT) wmme &quation empirique d’ttat de I’Ccoulement en hachures. L’knergie d’activation (H) pour le dtbut des hachures est de 0.35 eV. Zusammenfassung-Modelimedhaniken werden fiir die inhomogene plastische Verformung der amorphen Legierung Pd,,,sCu6Si16,s als ein sich nicht verfestigender Festkijrper vorgescblagen. Die Abtingigkeit der FlieDspannung van Plattendicke, Kerbwinkel und Kerbtiefe wird mit dem V-Kerbbiegeversuch studiert. Die plastische Verformung im ebenen Spannungszustand liiuft ilber eine Mechanik ab, die mit den Aussagen der plastischen Zone in Model1 von Dugdale und mit unserer Beschreibung der Kontraktion vertriiglich ist. 1~ Proben mit ebenem Spannungszustand weist das allgemeine Verformungsverhalten ein plastisches Gelenk auf. Aus von Mises Kriterium folgt fdr den Verstiirkungsfaktor ein Wert von 1.20, der gut mit dem Gleitlinienfeld von Green iibexeinstimmt. Die Abhiingigkeit des ruckweisen FlieDens von der Dehngeschwindigkeit wurde ebenfalls iiber einen Bereich von 1 x lo-* bis 1 x 10’ mm/min gemessen. Das ruckweise Flief3e.n verschwindet bei einer kritischen Dehngeschwindigkeit i,. Als empirische mechanische Grundgleichung kann kc = Aexp( -H/kT) Nr das ruckweise FlieBen bestltigt werden. Die Aktivierungsenergie H fiir Einsetzen des ruckweisen FlieDens betr;igt 0,35 eV. 1. INTRODUCTION Amorphous metals are higher strength materials [l-33, especially the Fe-based systems, which exhibit very high fracture stresses, in excess of 300 kg/mm* [4-q. Amorphous metals are capable of

t Present address: Cornell University, Bard Hall, Ithaca, New York, U.S.A.

undergoing extensive plastic flow Cl]. This plastic flow occurs in the form of the highly localized shear bands [l, 31. This raises the interesting question of what kind of flow unit can be constructed topologitally, since dislocations which have been extablished as line defects for correlated atomic motion in a crystal may not be useful descriptions in amorphous metals which have no long range order. Experimentally, plastic flow in amorphous metals is qualitatively

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characterized by non-strain hardening [S] and a structural change [S, 101 inside the shear deformation zones. Several elegant models for micro-mechanism have been proposed, which predict the yield stress [l l-131 or inhomogeneous-homogeneous transition [14, IrJ etc. by making assumptions about the atomic structure. On the premise that the purpose of research on the strength of matter consists in a quantitative comprehension of the actual strength and its related phenomena, we believe that the most promising approach at the present time is to investigate the plasticity in random structure on a macro-mechanical basis. However, it is hard to say that yield stress and nonhardening property can be well-defined by the following facts: 1. lack of uniaxiality accompanied by antiplane strain mode [la] and plane strain necking mode [lfl; 2. apparent strain hardening caused by multiplication and intersection of slip bands [18]; 3. microscopic softening in cumulative slip deformation mode [18]; 4. macroscopic softening by superimposed hydrostatic tension [19]; 5. dynamical softening in serrated flow [S, 181; 6. inverse effect of strain rate on fracture stress [ 1,5j. The primary purpose of this investigation is to establish the quantitative model mechanics which can apply to most amorphous alloy. The secondary purpose is to present the kinetics equation of dynamical flow which depends on the ease with which atoms can be rearranged locally. The bend specimen with a notch will mable us to provide quantitative observation for inhomogeneous flow. 2 ~P~IM~TAL

PRO~DUR~

Pd7&u6Si16 was chosen because its relatively low quenching rate ailows the preparation of bulk specimens [20]. Examination by means of the differential scanning calorimetry and X-ray diffraction showed that the specimens were amorphous. Figure 1 shows the shape and dimension for the ‘standard’ V-notch specimen used. Quenched rods were ground into a rectangular prismatic bar of plate

Pd-Cu-Si

ALLOY

BENDING

thickness (B) and piate width (W’) of 1 mm. Notches with a notch angle (a) of 45”, a depth (u) of 0.2 mm, and a radius of curvature in notch roots (p) of about 0.04 mm were made with a specially dressed diamond wheel. Finally, reference marks spaced 4 mm apart (21) were inscribed onto the specimen. In order to study geometrical effects on plastic flow behavior, the plate thickness, notch angle and notch depth of the ‘standard’ V-notch bend specimen as shown in Fig. 1 was varied, with plate thicknesses ranging from 0.2-1.5 mm, notch depths from 0.1 to 0.5 mm and notch angles from 30-150”. An Instron mechanical testing machine (maximum load of 500 kg) operated at an extension rate of 0.1 mm/min. was used to deform the specimens in three point bending. Strain rate effects were studied with standard 45” V-notch bend specimens deformed at cross head speeds ranging from O.OOOl-lOOmm/min. and at temperatures ranging from 77-473 K. The plastic deformation process was observed in real time with an optical microscope. For detailed observations of slip bands we used a scanning electron microscope and an optical microscope with Nomarski contrast. 3. MODEL MECHANICS OF PLASTIC DEFORfviATION IN NON-STRAIN HARDENING SOLID 3.1 Plane stress We firstly provide the model mechanics for antiplane strain mode by which tensile failure of unnotched amorphous thin sheet occurs typically under the condition of plane stress. 3.1.1. Dugdale plastic zone. Dugdale predicted for a rigid perfectly plastic solid that the two dimensional plastic zone size (R) is related’ to the applied net stress (CT& and the crack length (a) by the following relationship [Zl] :

(1) where CT~is yield stress in uniaxial tension.

B a 0204so a=O.tl-0.4s d=N)-lw

2

mm Fig.

TESTS-l

I, V-notch bend specimen, dimensions in mm.

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312.1General J&?id slip line jield by Green. Figure 3 the plane strain slip line field for general yield-

showk

ing of 4.5” V-notch rigid-perfectly

bend specimen as derived

plastic solid [25].

tral angle (y) can be a characteristic value

of 67”

is predicted

for a

A plastic hinge ten-

under

quantity full

plane

and its strain

state [25,263 p = 67’

3.2.2 General yield stress and plastic constraint factor. The bending moment (M) at general yielding is

Fig. 2. Antiplane strain mode. 3.1.2

Contraction in antiplane strain mode. The

contraction, accommodated by local necking in antiplane strain mode [22] as shown in Fig. 2 can be a macroscopic variable in the fully three dimensional elastic plastic treatment of notch tips. With a modified Hahn and Rosenfield model, the relation between in plane crack opening displacement (4) and out of plane contraction (D) can be obtained with a simple form (see Fig. 2): notch root

2 D sin fi = @

(2)

The angle (fi) which the siip plane forms to the thickness direction is related to the width of the plastic zone (R,) at the surface and plate thickness (B): B = tan-’ B/R,

(3)

The relationship between the J integral (J) and the crack opening displacement (4) for a non-strain hardening rigid-perfectly plastic solid is given by (see Appendix): 3 = cr,@

(4)

The crack opening displacement after general yielding can be related to the displacement (u) for a rigidperfectly plastic solid by the following equation (see Appendix): # = (l/ur)J* *11

(5)

The bend angle 8 is reiated to the pin d~spIa~ment (u) [23,24] by: e= tan-‘u/l

(6a)

when the bend angle is small, equation (6a) reduces to: e = u/t Equation of(S), upon substitution (6b) becomes:

0’3)

(6b) of equation (2) and

given by the following equation which provides a lower bound solution for the notch case (assuming Von Mises criterion) [27] : M,, = f$

x (~ndingarm)

= rrouB(W - a)*/2JS

(9)

where oGY is the general yield stress on the net section. Green predicted the following equation which provides the upper bound solution [25,263 for the general yielding pattern as shown in Fig. 3. MGr = 126arB(W - a12/2*Jj

wt

The ratio of the general yield stress (a& to the tensile yield stress (au) of the unnotched round bar will be called the plastic constraint factor L [2fl : L = QYh

f.11)

The plastic constraint factor of general yielding pattern in Fig. 3 is given by division of equation (1) to equation (2) as follows: L = 1.26 [25,26]

(12)

In this investigation we will use the general yield stress, as defined by equation (9) as a variable rather than the general yield load. It should be pointed out that at this stage equation (9) is strictly formal. 4. RJSXJLTS 4.1 Plastic deformation in Vlnotch bending Figure 4 shows the load-extension (cross-head displacement) curve for standard 45” V-notch specimen at room temperature. Figure 5 shows the corresponding sequence of plastic defo~ation patterns observed

(7) where

3.2 Plane strain Next,

we

deformation strain.

provide the mechanics of plastic hinge in V-notch bend specimen in plane

Fig. 3. Green’s slip line field for the V-notch with a circular fillet in plane strain. (25).

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General yie!dinp

Fig. 4. Load-extension curve for a standard V-notch bend specimen at room temperature.

on the side surface of the specimen. The load at which slip band formation at the notch root could first be observed agreed closely with the initial deviation from Iinearity (elastic limit: Ps) in Fig. 4. Further Ioading

the elastic slope right up to the previous load level prior to unloading (see Fig. 4). effhcts in Y-notch Gail

beyond the elastic limit initiated new slip bands at the

4.2 G~~tric~l

notch which grew in the piastic hinge pattern shown in Fig. 5, as well as sets of straight slip bands, inclined z 54” to the surface on the compression side (Fig. Sa). Upon the further loading, plastic deformation proceeded in an irregular, jerky way (jerky flow) which finally gave way to the regular way (serrated flow) at high extensions shown in Fig. 4. Serrated Row sets in when the plastic hinges propagate completely across the cross-section of the side surface: general yielding which occurs at a plastic-hinge central-angle (y) angle of 65” (Fig. 5b). Experimentally, general yielding is a

The plastic deformation of a notched bar, i.e. the shapes of the plastic zones, is generally a&ted by the notch geometry which determines the stress distribution below the notch root. 4.2.1 Plate tfiickness eficts, Figure 6 shows the variations of the general yield stress and the plastic con-

well defined process which can easily be followed in situ by observing

the formation of slip bands on the specimen surface with an optical microscope. Plastic deformation beyond general yielding proceeds initially coupled with a slight (2-3x) load increase {section b in Fig. 4) which can be referred to as apparent strain-hardening in the defined flow load ‘PFy’ followed by a plateau (section c). Fig (5~) shows that the characteristic feature of flow events after general yielding is initially a rotation of the end sections of the bar around the slip plane of two plastic hinges; both the material inside and outside of the hinge stays rigid and this so called primary hinge broadens by the formation of new shear bands spreading from the notch. Finally, slip bands characterized as plastic ‘wing’ spread occasionally from top of face to plastic hinge. Figure (5d) shows a sheared off plastic hinge, in the serrated flow occurs during shearing of a single plastic hinge. Resides, when a specimen is unloaded and then reloaded, the toad increases with

straint factor with plate thickness. At the critical plate thickness ratio to net section of B/G - a = 0.6 a transition between plane stress and plane strain occurs in the deformation mode. The general yield stress and the plastic constraint factor in plane strain specimen increase with plate thickness up to a plate thickness ratio of about 1.25,and then become constant. Figure 7 gives examples of the deformation pattern at general yielding in specimens with the pIate thickness ratio of (a) 0.74; (b) with notch angle of 150”;and (c) with the notch depth ratio to plate width of a/W = 0.49. The specimen with the small constraint factor of 1.1 at the plate thickness ratio of 0.74 has the smaller central angle of 52” obtained from Fig. 7a compared to 65” for a standard specimen. Figure 8 shows the load-extension curves for V-notch bending specimens in plane stress with plate thickness ratios of (a) 0.49 and (b) 0.26. Figure 9 are micrographs of the plastic deformation patterns observed at the side surface of the specimens as well as looking down into the notch. The sequence shows (a) local yielding, (b) general yielding, (c) after general yielding, (d) well past general yielding, corresponding to the labels in the Ioad-extension curve of Fig. 8(a). The loading in the vicinity of the elastic limit produces slip bands which

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E

F-

Flol;~:hiiknes;,~ 0

$ &00

I -Teaikg

mode

B I W-o 1.6

I

I

0

Q&v

d-c ,““_/ i ,’

-.I

3 2

s -L3 2

L

-1.2 8 ” 1.1*=

..~---“-I 0.2

I

z

/

4 s”lOOIn

2.0

Plastic hinge mode-

-

$ E CM0 5

5

net se;t$n I

I 0.4

I 0.6

Plate

t 0.0

I 1.0

thickness B,

I 1.2

I 1.4

- 1.08 t 1.6

mm

Fig. 6. Variation of net stress at generaI yielding and constraint factor with plate thickness.

curve out from the corners of the notch and, at the compression side, the slip bands are relatively straight and form an angle of about 55” with the compression axis [(Fig. 9(a)]. Upon further loading, the plastic deformation at the notch root occurs via simultaneous shear slip on a plane inclined to the thickness direction, that is, in the antiplane strain mode, and also spread from the corners of notch root perpendicular to the tensile axis [(Fig. 9(b)]. When the load approaches the plateau in Fig. 8, the two slip bands in the tensile and compressive side oom~ect,and the state of general yielding is reacht& The net stress at general yield&g is calculated via equation (9). After general yM&ng local necking proceeds by Shea&g along the two iatffaacted and strongly developed deformation ~VSS& rwldconsequently the notch side surfaces

Fig. 7(a)

move in a contractional motion that preserves their flat faces as expected in the model proposed [Fig. 9(a-d)]. Note that the net stress at general yielding in plane stress increases with thickness. 42.2 Notch angle and notch depth ejkcts. Figure 10 shows the variation of the general yield stress and the plastic constraint factor with notch angle. Figure 11 shows the variations of the general yield stress and the plastic constraint factor with notch depth ratio. All specimen showed a plastic hinge pattern at general yielding and no marked change of the deformation mode was observed as shown in Fig. 7, even though in slip line field theory pure bending occllrs only at a critical angle[2gJ and below the notch depth ratio of 0.18 [26j. The general yield stress and the plastic constraint factor remain nearly constant

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1.2 mm -1

Fig. 7. General yield deformation patterns for specimens (a) with plate thickness ratio of 0.74; (b) with notch angle of lx)“, and (c) with a/W = 0.49. within the experiment range of the notch angle and below the notch depth ratio of 0.3. For higher ratios, both quantities increase and attain a value of 213 kg/mm2 and 1.42 respectively at notch depth ratio of 0.49 and its plastic hinge has a central angle (a) of 75” obtained in Fig 7(c), which is somewhat larger than that of a standard speoimen (65”).

Extension

4.3 Dynamicd &kts on phk

&fkmcdon

Figure 12 shows the load-extension curves at various temperatures and extension rates. It can be seen that serration heights gradually decrease with increasing rates, and in the case of a constant temperature of 223 K serrated flow disappears above the

mm

Fig. 8. Load-extension curves for thin bend specimens with plate thickness ratio of (a) 0.49 and (b) 0.26.

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c

P

-I

z

c5

bmo-

;;

(4%

3

-.-_r-.-.

-.-.-

L

Ti

21% ‘5

TjlOO-

---_~-9-~-

-13

L

-ee--

‘0

" -s2u

it Y

2 s! z

- 1.1‘= I:

0

I

20

I

I

1

40 60 80 Notch angle ff

I

1

100 120 , degree

I

l&O

-lOa’ ,

160

Fig. 10. Variation of the general yield stress and constraint factor with notch angle. extension rate of I mm/min within the resolution of the load recording. The ‘critical’ extension rate for onset of serrations can be found and is a temperature dependent quantity. The appearance and disappear-

ante in temperature range of serrated flow in the plastic hinge deformation mode is similar to that in compression under a different stress state[8]. Figure 13 shows the general yield stress; tensile yield stress for an unnotch~ round bar and plastic constraint factor as a function of temperature. The general yield stress is aimost constant between 300 and 473 K and below room temperature, rapidly increases with decreasing temperature. Plastic constraint factors and central angles are independent of temperature. 5. DiSCUSION 5.1 The mechanics of V-notch bending and inhomogeneity of plastic flow

b&h

bepthi

&ate

width

al W

Fig. 11. Variations of general yield stress constraint factor with notch depth ratio.

5.1.1 Plane stress. Figure 14 shows the plastic zone size, as a function of the ratio of net stress to yield stress (a,&$. The applied loads (Papp) below the general yield load (Per) in Fig. 8 will be approximateIy proportional to stress applied gross, and hence we assume Papp/Pcr *(W/W - a) = ‘~.~~/a~ We

40L 77K

195

i

35

1

22 I i

291

, 352 aCxxn{o.ool+o.or-+0.2+0.5+l+2+530

0.1 rnmlmin

ol X

:~ 0.1 mm

Extension Fig. 12. Load-extension curves for standard specimens at various temperatures between 77 and 352 K at a constant extension rate of 0.1 mm/min. and at extension rates ranging from 0.0001-S mm/min.

KIMURA

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Pd-Cu-Si ALLOY BENDING TESTS-1 I 0.10 l

:0oS-

f$

r50

_

0

(a) B/0=0.49 (b) = 0.26

i’

l

/

5

“\<+iZZZ

bar

’

o----O-

E c

E x

-1k

ijioo-

- 1.3 ” 5

z

L & E (3

l-

1.2 .u 5 -1.1 ,o

-*--* I 0

I

I 300

I 400

I 500

200 Testing temperature ,

100

’

600

0

Bend &le

LOP.

K

Fig. 13. General yield stress and constraint factor as a function of temperature.

Fig. 15. Contraction at a notch root as a function of measured bend angle.

@o = @GpGyjB

I

i_

1.0 0.6 0.6 02 Ok Net stress I yield Stress ON,&

Measured plastic zone size as a function of the ratio of net stress to yield stress. The solid line denotes prediction ol the Dubdale model (u,,,/uy = P,,,/PGu.

(13b)

Substitu~on of equation (2) in equation (4) yields: J = (2sinfi*ar)D

(14)

Equation (13b), which states that the contraction at general $elding increases with thickness, agrees with the experimental results shown in Fig. 15. Substitution of the equation (13b) into the equation (14), therefore, gives the following expression for the value of the J integral at general yielding (.I&: JGY =

(13

@r@o~

Since the J integral and the net stress at general yielding are related (see Appendix),the net stress at general yielding is the applied stress required to attain the crack opening displacement at general yielding. The linear relation between the J integral and the plate thickness (see equation (15) erplains the rapid increase in the net stress at general yielding shown in Fig. 9. This conclusion is important in an engineering sense where one is interested in the fracture strength of an unnotched ribbon.

- t.4

.’

03a)

= 2Doysinfi/I#

5:: 1.5

Fig. 14.

30

, d&e

and the contraction (Dcy) at general yielding can be written as:

define the farthest front of slip bands extending from the notch tip in Fig. 9 as the two dimensional plastic zone size perpendicular to the tensile axis. As the plate thickness decreases, the shape of the plastic zone will approach a straight line [see equation (3) /? = 55” in this study]. The experimental results are in good accordance with Dugdale’s relationship. Figure 15 shows the measured contraction (D) as a function of the bend angle (Q). The linear relation between the bend angle and the contraction predicted by equation (7) is in excellent agreement with the experimental results obtained with specimens of different thickness. This experimental varification justifies the various assumptions made in constructing equation (8). We may therefore conclude that the plastic flow in two and three dimensions is that of a non-strain hardcning rigid-perfectly plastic solid. We further consider the physical meaning of J integral at general yielding qualitatively. Since the dimensionlesscrack opening displacement (@ofbelow general yielding can be characterized by the two dimensional Dugdale model, the crack opening displacement in the three dimensional flow pattern (Qi~r)

,//:

Cl

5 1.3 ‘, c 1.2 s .u 1.1 ‘; 0 1.0 a’ _i 1.6:

0 Prcdiion 0 Standard 0 aiW=0.49 0 BkzO.75 0 Wa=l.88 0 #rlZO e o( =150

by Green spceimen d

6 o/O

/“’

I LO Central

I 50 angle

I 60

I 70

I, 80

tl , degree

Fig. 16. Relation between measured central angles of pb tic hinge and pIas& constraint factor.

KlMURA

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10'3 Extension

Fig. 17. Relationship

AMORPHOUS

10-z

10"

ALLOY

BENDING

b

195K

0

223

I

TESTS-1

IO

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102

mm I min

rate

between the logarithm of serration temperatures.

5.1.2 Plane strain. Figure 16 shows the relation between the plastic constraint factor and central angles of the plastic hinge as measured from Fig. 5 and Fig. 7. The central angle of 65” for the standard specimen with 45” notch agrees with equation (8) presented in Green’s slip line field. The variation of plastic constraint factor in a systematic manner with the central angle reflects a change of stress state with changing notch geometry. We may offer the central angle of the plastic hinge as a parameter which plays an important role in the calculation of the plastic constraint factor, since no theoretical data are available. An angle of 55” on the compressive side is predicted by theory for slip under plane stress [29]. The experimental shear field which extrapolates well to a singularity point, located approximately at the center point of notch curvature [see Fig. S(b)], still is regarded as that of an acute notch. According to Ewing, following equation (12), the average constraint factor for acute notches in three point bending is about 1.22 [30]. This value accords with the plastic constraint factor of 1.20 obtained by using welldefined general yield stress based on the microscopic inhomogeneity in Figs 6, 10 and 11. The good quantitative identity between the theoretical and experimental yield patterns and the plastic constraint factor under ‘full’ plane strain conditions validates the assumptions underlying the theory; i.e., an amorphous Pdis-Cue,-Sii6 alloy behaves as a non-strain hardening material which obeys the Von Mises yield criterion and not a pressure dependence yield criteria.

5.2 Kinetics of plastic deformation 5.2.1 Serrated flow.Figure 17 shows the relationship between the logarithm of the serration heights observed on the record charts (L\a) and the extension rate 6) for various testing temperatures. The nearly linear relationship in two regions of the extension rate can be represented by a logarithmic time law Afl=g

Pd-Cu-Si

(16)

where m is a stress exponent. The stress exponent is close to zero in the lower extension rate region and -4.5 in the higher extension rate region, independent of temperature. As a logarithmic quantity, the appear-

heights and extension

rate at various

ante and disappearance of serrated flow is not a critical phenomena occurring at a well defined extension rate at a certain test temperature. We define, therefore, a critical extension rate for onset of serrations by extrapolating the experimental data to the load fluctuations below 1 kg/mm’ where the extension curves become practically smooth. Figure 18 is an Arrhenius plot of the thus defined critical extension rate. It can be seen that the plot of the logarithmic critical extension rate versus the reciprocal temperature shows good linearity and that the data can therefore be represented by

where H is the activation energy for onset of serrations, k the Boltzman constant, and A is preexponential factor. The activation energy is 0.35 f 0.02 eV. The activation energy of serrated flow will be an important material parameter for the discussion of the flow micromechanism. 5.2.2 Temperature dependence of general yield stress. The increase at low temperatures, shown in Fig. 13 in the genera1 yield stress, defined by inhomogeneity of plastic flow under a stress state of hydrostatic tension, unlike the fracture stress [4] and compressive flow stress [8] may provide macroscopic character of thermal activation process. The plastic constraint factor and central angle of plastic hinge independent of temperature in Fig. 13, however, indicates no specification of activation process: little effects of hydrostatic tension and degree of inhomogeneity for plastic flow in activation region as well as plateau region. The deformation in the plateau region where serrated flow dominates also cannot be understood in terms of athermal behavior. Tentatively, we offer the following model for the occurrence of a plateau in the temperature dependence of the general yield stress: the stress in the thermally activated region continues to decrease with increasing temperature, but the increase in the serration heights with increasing temperature balances this decline. If this model would indeed apply, which still has to be proven, yielding under inhomogeneous plastic

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t

\

expf-+

&A

t

H =0.3? eV

\ +

\ t

\

I 3

I 4 1000 I T

I 5

I 6

K-1

Fig. 18. Arrhenius plot of the critical extension rate for onset of serrations.

deformation would be dominated vation.

by thermal acti-

6. CONCLUSIONS 1. The net stress at general yielding in plane stress for an amorphous Pd7sCu6Si16 alloy decreases with decreasing thickness. The relation between the measured two dimensional plastic zone size and the applied net stress is in good agreement with the prediction of the Dugdale model. The linear relation between the three dimensional contraction and the bend angle predicted in antiplane strain m&e for a non-strain hardening rigid-perfectly plastic solid, is consistent with the experimental ones in different thickness. 2. The standard 45” V-notch specimen in plane strain exhibits a plastic hinge; the general yield stress of 178 kg/mm2 and a value of 1.20 for the plastic constrain factor. Good agreement between the experimental deformation pattern and constraint factor and the results from Green’s

slip line field theory

does

quantitatively point out that amorphous PdIs-CuGSi16 obeys the Von.Mises Criterion. 3. The general yield stress increases with decreasing temperature below about room temperature. Above room temperature the flow stress is almost constant. The plastic constraint factor is independent of the testing temperature. 4. Serrated flow disappears above a certain critical extension rate (2,). An Arrhenius plot of the critical extension rates is found to be linear and the mechanical equation of state for serrated flow is given by: ’ = Aexp EC The activation energy for onset of serrations is 0.35 eV for an amorphous Pd18-Cu6-Si,h alloy and should be considered as an important material parameter. Acknowledgemenrs-Thanks are due to Professor H. Suto and Professor H. Shimada at Tohqku University for many helpful comments, and also Professor D. G. Ast at Cornell

KIMURA University

for critical

AND

reading

MASUMOTO

AMORPHOUS

of the manuscript

prior

to

publication.

REFERENCES I. T. Masumoto

and

R. Maddin.

Acru mcrtrll. 19, 725

(1971).

2. H. S. Chen and T. T. Wang,

J. crppl, Phw. 41, 5338

(1970). 3. H. J. Leamy. 4. 5.

H. S. Chen and T. T. Wang, h.‘ero//. Trans. 699 (1972). C. A. Pampillo and D. E. Polk. Acta meroll. 22, 741 (1974). T. Masumoto and H. Kimura, Nippon Kink. Gakk. 39,

133 (1975). 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

H. S. Chen and D. E. Polk, J. Non-Cryst. Solids IS, 174 (1974). L. A. Davis, R. Ray, C. P. Chou and R. C. O’Handley, Scripta metall. 10, 541 (1976). C. A. Pampillo and H. S. Chen, Mater. Sci. Engng. 13, 181 (1974). H. S. Chen, Scripta metall. 7, 931 (1973). T. Masumoto and R. Maddin, Muter. Sci. Engng. 19, 1 (1975). J. J. Gilman, J. appi. Phys. 44, 675 (1973). J. C. M. Li, In Distinguished Lectures in Materials Science, Marcel Dekker, New York (1973). M. F. Ashby and J. Logan, Scripra metall. 7,513 (1973). F. Spaepen, Act4 metall.25,407 (1977). A. S. Argon, Acta metall. 27, 47 (1979). H. Kimura and T. Masumoto, Script4 metal/. 9, 211 (1975).

17. 18. 19.

L. A. Davis, Scriptu metall. 9, 339 (1975). H. Kimura, Doctorial Thesis, Tohoku University (1978). L. A. Davis and S. Kavesh, J. Muter. Sci. 10, 453 (1975).

20. 21. 22. 23. 24. 25. 26. 2-l. 28. 29. 30.

H. S. Chen and D. Turnbull. Acta metal/. 17, 1020 (1969). D. S. Dugdale, J. Mech. Phys. Solid, 8, 100 (1960). G. T. Hahn and A. R. Rosefield. Inc. J. Fract. Mech. 4, 79 (1968). H. A. Lequear and J. D. Lubahn, Weld. J. Land 33,585 (1954). H. Kimura and T. Masumoto, Act4 metall. 28, 1677 (1980). A. P. Green, Q. Ji Mech. appl. Math. 6,223 (1953). A. P. Green and B. B. Hundy, J. Mech. Phys. So/ids 4, 128 (1956). For example, J. F. Knott, Fundumentuls of Fracture Mechanics Butterworths, London (1974). A. P. Green, J. Mech. Phys. Solids.6, 259 (1956). G. Lianis and H. Ford, PMM 8, 360 (1957). D. J. F. Ewing, J. Mech. Phys. Solids 16, 205 (1968). APPENDIX

The J integral has been developed to deal with a two dimensional crack in a non-linear elastic body, and is defined as [i] : J=

wddy - Ta;ds

> where x - y are rectangular coordinates in the plane of the notch, with x = 0 describing the notch tip, f’ is a closed

Pd- Cu-Si ALLOY BENDING

TESTS-I

1675

path taken counterclockwise around the notch tip, w is the strain energy density function, T is the tension vector perpendicular to F in an outside direction, II is the displacement vector, and ds is a differential element of f. Under the assumption that the integral path f surrounds the Dugdale plastic zone developing of a notch tip, and that Row follows a rigid-plastic solid model, the J integral has the following

J=

-

Lf-&js_ ax

r

=-

value [ii]:

~(4)

-

_ R20,

i0

2d.x

=

2d.v du

g(d) dd

(2)

where 4 is the crack opening displacement, @t at a certain plastic zone size R, v is a distance between two notch planes. In our case the plastic deformation proceeds by an antiplane strain mode as shown in Fig. 2 and the plastic strain in the y direction (6,) is given by lp = 4/B tan /I

(3)

upon substitution of the equation of (3). the equation (2) becomes J = B tan j?

I

41

0

o(e,)

The stress-strain relation under the uniaxial tensile loading is assumed as the following equation u = cc;

(5)

where cre is a constant, n the strain hardening exponent. Substitution of the equation of (5) into the equation of (4) followed by integration gives finally the J integral as J=

(6)

for a rigid-perfectly plastic solid n = 0 and co = cr:

J = QY~,

(7)

the J integral is formally equivalent to the change in potential energy unit area with notch length. The crack opening displacement after general yielding can be related to the displacement (u) [iii]. When specimen behaves as a rigidperfectly-plastic solid and deforms in such a manner that a general yield load can be defined, the J integral is given by J(u) = - (u/E)(aPcr/c%4)

(8)

where L is a constant dependent of geometry of the notch specimen. The equation of (7), upon substitution of the equation of (8) becomes for a rigid-perfectly plastic solid 4t = (l’/~)J*u

J*=

-$ 2

(9)

APPENDIX REFERENCES i. J. R. Rice, J. appl. Mech. 35, 379 (1968). ii. J. R. Rice, Fracture Edited by H. Liebowitz Vol. II. Academic Press New York, (1968). iii. A. A. Wells, J. Br. Welding 10, 563 (1963).