Nonhomogeneous Straining and Fracture Mechanism in a Filled ElastomerMaria Ronay Citation: Journal of Applied Physics 35, 629 (1964); doi: 10.1063/1.1713427 View online: http://dx.doi.org/10.1063/1.1713427 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/35/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Molecular-dynamics simulations of mechanical reinforcement in filled elastomers AIP Conf. Proc. 1599, 62 (2014); 10.1063/1.4876778 Masstocharge ratio and kinetic energy of positive ion emission accompanying fracture of a filledelastomer Appl. Phys. Lett. 41, 827 (1982); 10.1063/1.93709 The Fracture Energy and Some Mechanical Properties of a Polyurethane Elastomer Trans. Soc. Rheol. 15, 217 (1971); 10.1122/1.549209 The Mechanical Behavior of a Filled Elastomer at High Strain Rates Trans. Soc. Rheol. 13, 323 (1969); 10.1122/1.549135 The Character of the StressStrain Function for Highly Filled Elastomers Trans. Soc. Rheol. 12, 303 (1968); 10.1122/1.549110

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.I () (' R ~ ,\ L () F ,\ [> I'L I E I) [> [f \' SIC S \' () L \' ,\1 r-: 3 S. :\ (), 3 (T \\' () l' ,\ R T " l' ,\ R T I) ,vI ,\ R C][ I') 6 4

Nonhomogeneous Straining and Fracture Mechanism in a Filled Elastomer

:'IARIA RONAY

Coi1lmbi([ l'niversity, New York, New York 10027 rReceivecl 30 August 1963)

This paper presents an experimental study of the strain concentrations arising near the edges of elliptical cutouts in uniformly strained, tlat strips of polyurethane rubher filled with a high volume percentage of an inorganic salt. The strain con cent rations are st udied under various condit ions of uniform st raining, for strains up to fracture initiation.

The fracture mechanism at the root of the cutout is studied micrographically and it is found that the local fracture process is similar to that in unfilled elastomers, with consecutive rupture of highly deformed st rands of the elastomeric material.

1. INTRODUCTION

ELASTOMERS filled with a high volume concentra­tion of inorganic slats are used for various purposes

in modern technology, such as clamping materials or solid propellants, The high volume concentration of the lIller produces a heterogeneous material, the deforma­tional behavior of which is not easily reproducible by cons tit ut ive equations of conventional continuum mechanics while its failure cannot be dealt with on the basis of concepts of crack propagation and fracture mechanics in a continuous medium (modified Griffith condition). Although the mechanical response at very small strains depends mainly on t he response of the elastomer and the volume concentration of the filler, the response at large strains is, in addition, significantly affected by part ide size and dist ribution. The pro­nounced heterogeneity of the material is illustrated by micrographs of tIle surface of the material even at moderate magnification (X4-t). Figure 1 (a) shows the high volume concentration of the salt (potassium chloride) and the resulting heterogeneous struet ure of the combined material, while Fig. 1 (b) demonstrates t he heterogeneity of the elastomeric structure itself, with small residues of an insoluable constituent (alumi­num powder), after the salt has been dissolved and the swelled elast orner rest ored to its original over-all geometry by drying.

\rhile the basic viscoelasticity of the combined material has been well established, the complexity and essential nonlinearity of its deformational response forces coupled with the necessity to consider finite strains makes any theoretical analysis of boundary value problems involving such materials quite difficult, since the usual simplifications which produce an ana­lytically manageable theory (homogeneity, linearity, small strains) cannot be justified, even in first approxi­mation, when considering operational conditions of solid propellant grains.

This applies particularly to fracture problems in nonhomogeneous strain fields, which are of considerable practical significance in view of the current use of highly filled elastomers in various geomet ric configura­tions. Since such configurations frequently involve st rain

6.2<)

concentrations and strain gradients of considerable severity, their successful use will depend on the pos­sibility of completely avoiding cracks or at least on close control of their propagat ion. Since the study of strain concentrations with the aid of photoelastic materials of properties that differ widely from those of highly filled elastomers is hardly relevant to the problem of failure of the latter, it seeme(l expedient to design a

(a)

(b)

FIG. 1. (a~ Microstructure of filled polyurethane (X46); (b) MICrostructure after removal of soluble filler.

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630 MARIA RONAY

T I-f----- - -.-

201 P~1o.2!l ~1~~:1h \

1.0" a

-1------- -

lL--__ ~ ~---~---_J

-~------r__ --------~

p=0.0625 S:-:lO.l88 b P~10.309 c

_~ __ ~9 __ r- -c---~ ~8 __

FIG. 2. Dimensions of specimen and cutouts used.

simple experiment for the direct study of the interrela­tion in highly filled elastomers, between the process of uniform over-all straining and the distribution of strain, and the initiation and propagating of cracks in strain fields in which the initial inhomogeneity could be arbitrarily varied.

2. DESCRIPTION OF EXPERIMENT

The filled elastomer, polyurethane rubber filled mainly with potassium chloride (about 50% by volume, particle size between 50 and 300 p.), prepared in sheets 2.5 in, wide, 2.0 in. long, and ts in. thick, was uniformly straJrr:ed in a simple frame; the free length between the frame edges was 1.0 in.

Inhomogeneous strain fields were produced by cutting in the center of the sheet (Fig. 2) : (a) a circle of diame­ter d=0.25 in.; (b) an ellipse with diameters a=0.309 in., b=0.188 in., the direction of the major axis being normal to the direction of the straining; (c) the same ellipse as in (b) but with the major axis parallel to the direction of the straining; (d) a straight sharp incision of length a=0.309 in. normal to the direction of the straining. The relevant radii of curvature at the root of the strain concentration in the direction of the straining are p=fi in. (ellipse b), p=j in. (circle), and P=/s in. (ellipse c); therefore, the ratios tlp=2.5, 1, and 0.5, respectively, where t denotes the axis of the ellipse normal to the direction of the straining or the radius of the circle. The "radius of curvature" at the incision p"'O produces a ratio tip -t 00 ; it is, however, uncertain to what extent the edges of the incision are damaged in the cutting process, fa,cilitating premature cracking.

In order to avoid premature failure of the sheets along the clamped edges, strips of the same material were inserted between the metal clamps and the sheet. The strain was applied by careful parallel motion of the clamps with the aid of the screws, and was measured by observing, under a microscope, the deformation of the vertical and horizontal diameters of several small

circles (or half-circles) traced on each sheet. The radius of these circles was kept sufficiently below the smallest radius at the edge of the cutout to ensure that the strain averaging over the diameter, implied in measur­ing strain at a point by observing the deformation of a finite circle around the point into an ellipse, should not entail a significant underestimate of the strain at this point· in view of the inhomogeneity of the strain field of the cutout.

The "over-all" strain was determined from the deformation of a large Ci-in.-diam) circle surrounding the cut, and its value checked against the deformation of the small circles midway between the cut and the edges, as this measure is believed to represent a more relevant measure of the "over-all" strain along the center line of the sheet than the average strain related to the change of distance between the edges of the frame, considering the nonhomogeneous distribution of the strain across the frame.

The following straining processes were applied:

(1) Straining to failure initiation in consecutive steps of (logarithmic) strain of roughly 2.5%. With observa­tion of the intermediate strain distributions the dura­tion of this process was approximately 3 h to fracture initiation. Subsequently, the growth of the crack with time was observed.

.36

.32

.28

.24

.20

.16 I: of • .12

o strain at fracture L 01.3"10 overall)

• at 4.88% overoll stroin I

J I I

II .08

.04

I

/ V I(

12 o

!l I.c:o .~!l .~ .~!l ~ t---_ ~) ~ ..... .......

-.04 -- longitudinal strain ---- transverse stroin , i ,-( -.08

I· 2Jj()"

FIG. 3. Distribution of strain' at 4.88% over-all strain and at fracture initia­tion for ellipse and step­wise straining.

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NON HOM 0 G ENE 0 U SST R A I N I N G AND F RAe T U REM E C HAN ISM 631

(2) Straining to failure initiation in a single step with observation of maximum local and over-all strain; failure with rapidly spreading crack was attained in about 2 min.

(3) Creep straining in equal (logarithmic)' strain increments to failure applied in intervals of one-half to one day.

(4) Repeated slow strain cycling to predetermined strain followed by straining to fracture.

The first straining process was imposed on specimens with different cutouts, while the other processes were imposed on specimens with circular cutout only.

3. RESULTS OF EXPERIMENTS AND DISCUSSION

The results of the investigation are presented in Figs. 3 to 8. Figures 3 to 5 (a) present the observed longitudinal and transversal strain distributions along the center line of specimens with different cutouts, under straining in consecutive steps. The strain distri­butions at 5% strain and at failure are shown (straining process 1). Figure 5 (b) shows the strain distribution for the circular cutout shown in Fig. 5 (a) but under slow

o strain at fracture

I (9.5% overall) I - at 5.4% overall ~ iT strain i

i I

I

I

I /: !

/ I ,

,j II I V

I

25 I.~O .~5 J

.~o .~5 ~i'\ :'-V

----. , ---- 1_

2.50"

(a)

.36

.32

.28

.24

.20

.16

.08

.04

I

o

-.04

-.08

o strain at fractuni I ! (7.3% overall) I

-at 4.98"10 overall strain-I

i I I

/

I, II /

.25 I.?<, '"7 .5~ .25 :Lincisia n --.., --longitudinal strain ----transverse strain

I II I

I· 2.50"

(b)

FIG. 4. (a) Distribution of strain at 5.4% over-all strain and fracture initiation for ellipse (b) and step-wise straining; (b) Distribution of strain at 4.89% over-all strain and fracture initiation for incision and step-wise straining.

o atraln at fractu,.,

+-(l0.7%overalll I

- at 528"10 overall strain

.36 o atrain at fracture 1

(7.4% overall) i • at 5.08% overall strain-

I

.32

I

I i I : '

i III 'I

I

II , ,

i I /1

V

I

I

I i I ! I

I

I III V

.28

.24

.20

.16 0::

] III

.12

.08

.04

25 I.cr' .7[5 i.:r .~J v"'\ :: ' ~ ...I " ,::::A __

I I

.25 I.~O .~5 .510 .~5

~ " V

",'

~/

o

i " " -.04 -- longitudinal strain ---- transverse strain

I II I

2.50" I -.08

2.50"

FIG. 5. (a) Distribution of strain at 5.28% over-all strain and fracture initiation for circle and step-wise straining. (b) Distribu­tion of strain at 5.08% over-all strain and fracture initiation for circle and creep straining.

stepwise creep straining (straining process 3). In Fig. 6 the strain distribution in a specimen with circular cutout strained in a single step to 8% over-all strain and unstrained the following day is presented.

Although the classical equations of elasticity are applicable only in first rough approximation, the two­dimensional stress-strain relations

E~1=0"1-V0"2,

E~2= -VO"l+0"2, (3.1)

where 0"1, ~l denote the longitudinal and 0"2, ~2 the lateral principal components of stress and strain, respectively, would indicate that for a lateral stress 0"2 ~ VO"l, the lateral strain ~2 ~ 0, while for 0"2 ~ 0, ~2/ ~l ~ - v. In the considered longitudinally strained wide sheets with circular hole, the latter condition arises at the edge of the cutout, while the former, with v«!, is characteristic of a location close to the edge at which the lateral stress 0"2 attains a maximum value. For the circular cutout the observed distributions of the lateral strain clearly show the existence of this point at a distance of somewhat less than /6 in. inside of the edge of the hole:

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632 MARIA RONAY

at this point the observed transverse strain is zero or close enough to zero to suggest a relatively high in­tensity of transverse stress. For the elliptic cutout (b) the zero point of lateral strain is too close to the edge; for the elliptic cutout (c) the transverse stress intensity is too low to produce the clear effect observable for the circular cutout. The distance of about 0.4 of the radius of the circular cutout is, in fact, the distance at which 0'2rnux occurs according to classical elastic theory.l It is obvious, moreover, that because of the small absolute values of the lateral displacements the accuracy of optical observation of the lateral deformation of the diameters of small circles traced on a relatively rough rubber surface cannot be very high; for the circular cutouts the accuracy is, however, sufficient to demon-

.28

.24

.20

. /6

.12 c .~

1ii .08

.04

o strained to 8% overall strain • unstrained (foliowinO day)

-Ionoitudinol strain ---- transverse strain

I

FIG. 6. Distribution of strain for circle at 8% rapidly applied strain and after-strain release.

strate the location of the high lateral stress intensity which plays a major role in the early stages of cracking (see Sec. 4).

The value of v (Poisson's ratio) obtainable from the negative ratio (~2! El) at the edge of the cutouts are rather erratic. From Fig. 5 a value of v"-0.12-0.15 are obtained at both 5% strain and fracture initiation, sug­gesting high inhomogeneity of the material, while from Fig. 6 a value of v",O.28 is obtained. It is interesting to note that the residual lateral strains in this case are higher than the residual longitudinal strains, another sign of inhomogeneity. For very slowly applied strain

1 A. J. Durelli, A. E. Phillips, and C. H. Tsao, Introduction to Theoretical and Experimental Analysis of Stress and Strain (Mc­Graw-Hill Book Company, Inc., New York, 1958), p. 318.

0,6

0.5

.. 0.4

i £ {0.3

"" g 02

0.1

~1

41 ( I

I

I I

20 40 60 80 time, hours

FIG. 7. Crack length at constant over-aU strain as a function of time .

100

[Fig. 5(b)] the order of magnitude v",0.37 at about 16%, while at or close to the fracture strain of 26% the values of v for creep straining drop to about v",O.2S [Fig. S(b)]. Thus the inhomogeneity of the response appears to increase with increasing rate of straining.

In interpreting the meaning of such values of Pois­son's ratio (which are much lower than those usually reported), it is necessary to consider that (a) the values are associated with relatively large longitudinal strains for which even for an incompressible material the ratio2

(3.2)

for €I = 0.2, therefore, v",0.43 for an incompressible material and v<O.43 for a compressible material; (b) because of the high volume concentration of the filler, the imposition of large strains is associated with the

0.6

, -·max. fracture strain i--- overall fracture strain

J-+++~~-I-4+-~ .. I I I I , I, I

.5 I I Ii' , eQ4~4--+-+-+~~--+-~~-~-~~~H-~--~-U

1ii I Q2~~~~~+~~--+-}~~~-+-L~~~+-~

10-1 I 10 time to fracture, hours

FIG. 8. Fracture strain as a function of time to fracture.

2 A. M. Freudenthal, Acta Tech. 41, 417 (1962).

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NON HOM 0 G ENE 0 U SST R A I N I N G AND F R ACT U REM E C HAN ISM 633

4.0,.----r-1--1.------,..---,---_

-- at fracture ••••••••••• at 5"10 overall .traln

~ , ~ k:"~9!i"l~ .E /10.7"10 ..... g 3.2 /' ,.

1 overall .train rf3"1o ••• / .....

g 8 2.8f-----j---+j-- -~-+---.---.j

/~~

: .. c 'f u; .j

2.4

2.00 4 8 12 16 20

lip

FIG. 9. Strain concentration at edge of cutout as function of local curvature lip.

opening of holes as a result of loss of adhesion between filler and elastomer; the medium thus becomes in­creasingly inhomogeneous and the apparent Poisson ratio tends towards low values, which indicates that longitudinal deformation proceeds by the opening of holes: (decrease of~ density) without significant further later contraction.

The fact that v is higher under very slowly applied ("creep") strain. than for more rapid straining demon­strates the time sensitivity of the deformation. Such time sensitivity necessarily produces a time-dependent crack-propagation process which has been studied by measuring the crack length as a function of time, counted from the first sign of crack initiation (under straining process 1). As soon as crack initiation was noticed, straining was stopped and the crack length measured at several time intervals. Because cracking proceeds under constant over-all strain, the total stored strain energy in the specimen and therefore also the energy stored around the tip of the crack (which pro­vides the driving force for crack propagation) decreases with increasing crack length, resulting in a rather

FIG. 10. Companson ~ of observed strain con- u centrations at 5% over- .2

5

all strain and at fracture 15 4 with stress concentra- '; tion according to clas- ~ sical elastic theory, as 3 functions of parameter 15 3 tip. U

c

E ..

---- elastic strell concentrof 100

-- rubber at fracture I •••••• - rubber at 5'Yo overall stroi n

-'" ",-

I" ,. ~.-

.,/" ~ .... " .. ,. " " , "

~.,

/'

2 3 tIp

rapidly decreasing rate of crack propagation. The func­tion of crack length vs time presented in Fig. 7 has been obtained for a specimen with circular cutout; the crack initiation could be observed at a local strain at the edge of the hole of ~l =0.22, f2= -0.049 associated with over-all strain of roughly 8%.

Considering as "time to fracture" the time from the first sign of crack initiation to complete cracking or termination of crack propagation by complete release of the stored energy in the specimen, an approximate relation could be established between localized or over­all strain at crack initiation and time to failure for specimens with circular cutout (Fig. 8). The points associated with times exceeding 1 h have been obtained from the tests presented in Figs. 5 (a) and (b). The test point at 1/30 h was obtained by performing a rapid straining test at which only the longitudinal strain at the edge of the hole and the over-all strain were ob-

I I I _ slape at fracture

.~ & _ .2

.......... slope at 5"10 overall strain f- 0----<> max. fracture stroin

_.-... overall fracture .troin .. .. ....:;

~" //

// " // /; .. :"

It-

---- ----I ! l

0.8 1.6

0.612

0.4 o.s

0.20.4 01 e-._. -._._.- _._.-.-

4 8 12 16 lip

FIG. 11. Strain and slope at fracture and slope at 5% strain as functions of curvature lip.

served; a local strain EI = 0.7 6 which propagated rapidly could be reached before initiation of the crack; the associated over-all strain was 18%. Presentation of the 3 test points in semilogarithmic scale shows a roughly straight-line increase of the logarithmic time to fracture with decrease of strain, both local and over-all.

The effect of repeated strain cycling on the limiting over-all failure strain does not seem to be very sig­nificant provided the straining is relatively rapid. Thus applying 10 cycles of 6% over .. all strain to a specimen with circular cutout (duration of cycle 1 h) followed by rapid straining to failure, the same over-all failure strain of 18% could be attained that was observed under rapid straining without previous cycling. This observation suggests that whatever apparent "fatigue effects" have been reported seem to be the result of time ("creep") sensitivity, not "cycle sensitivity," as

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634 .\1 .\ R 1.\ R 0 :\ ,\ Y

FtG. 12. Local concentrations of strain close to edge of cutout (X18).

might be expected a priori in an essentially viscoelastic material in which the principal deformation processes leading to fatigue (slip and fragmentation of micro­st ructure) are absent. The well-known fact that the lirst prestraining cycle of sufficient intensity to break the adhesive bonds bet ween a significant number of filler particles and binder reduces the shear modulus of the material for subsequent loading cycles can only be of little effect on the (much higher) total strain to failure subsequent to sUl·h cycling. Only very slow cycling accompanied by the characteristic phenomena of local­ized long-time fracture might produce the appearance of a "fatigue eHect." The micrographic study of local cracking presented in Sec. 4 is intended to (lemonstrate the phenomena that produce the time sensitivity of the fracture process.

The test results presented in Figs. 3-5 are summarized in the diagrams in Figs. 9, 10, and 11. Figure 9 illus­trates the effect of curvature at the root of the cutout on t he strain-concentration fact or [tl, tover-all] at 5S~) over-all strain ami at the edge of the cutout at fracture.

hG. 13. Appearance of elongated holes within region of concentrated strain.

FIG. 14. Opcninl!; of large elliptic holes with regions of concentrated strain.

\rith increasingly sharp ntrvature (decreasing root radius) the strain l'oncentration increases much faster at relatively lower strains than at fracture; on the other hand, the strain concentrations at fracture exceed those at 5S~ over-all strain, except for the sharpest curvature. The fact that the strain concentration at fracture ex­ceeds that at lower strains the more the less sharp the curvature, may, at least partly, be due to the fact that fract ure occurs at over-all strains that increase wit h decreasing curv<tt ure or increasing radius (see Fig. 9). (jnly within the range of small root radii}\ in.<p<() does the strain-concentration factor become practically independent of radius and of st rain, at taining a maxi· mum value of roughly 3.5.

Figure 10 shows that such a limiting value is sub­stantially below that obtained from classical elastic theory \rith infmitesimal strain. It also illustrates the deviation between this theory and the observed strain concentration factor at 5% over-all strain. This devia­tion is mainly due to the effect of finite deformations; since the shape of the cutout changes in the course of

FIG. (S. Rupture of strand of elastomer fibers inrming edge of elliptic hole.

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NON H 0 1\[ 0 G E l\ E 0 1] SST R A [ " [ N G :\ N [) F R :\ C T U R E :\1 E C II .\ N ISM 635

FlG 16. Creep extension of elastomer st rands bet ween holes.

the straining so that the curvature at the root of the cutout decreases with increasing strain, the observed strain concentration must necessarily be lower than that obtained from classical theory based on undeformed cutout and infinitesimal strain. It is interesting to note that for the ellipse with major axis in the direction of the direction of the straining, for which the maximum local strain at 5S{ over-all strain is only 12%, the theoretical elastic and observed strain concentrations coincide. For the circular cutout with the well-known classical concentration factor of 3, the observed value is already significantly lower. It is to be expected that such deviation increases with increasing local strain and thus with increasing value of the "notch parame­ter" I p; as Fig. 10 shows, this is borne out by the observations.

Figure 11 illustrates the observed relation bet ween the slope of the strain distribution (strain gradient) at the critical location as well as the fracture strain, local and over-all, and the curvature 1/ p. It appears clearly that the sharply increasing slope of the strain distribu-

FlG. 17. Creep extension of strands between holes close to fracture.

tion does not signiticantly atTert the fracture strain, both local and over-all, which decreases only moderately over the range of curvatures investigated. The very low value of the local fracture strain at the sharp in­cision (1 p= xc) may not be reliable, because of the damage that is easily produced in cutting the incision; t he associated over-all fracture st rain value is, however, consistent with the other results. The local fracture strain is completely independent of the strain gradient.

The above results suggest that for the material investigated the local strain at which fracture occurs depends significantly on rate or duration of straining, but only very moderately on the geometry of the local strain field. This conclusion would justify the assump­tion that the fracture process in the close vicinity of the developing crack is rather localized and not significantly atTected by the wider surroundings of this location. In order to investigate this point, a micrographic study of the developing fracture close to the edge of a specimen with circular cutout was undertaken.

FlG. 18. "Extension of crack" by coakscence of holes.

4. FAILURE MECHANISM IN FILLED ELASTOMERS

In the micrographic study a Leitz Ortholux micro­scope \,"ith Ultrapak light illuminator and Polaroid camera was used. A magnification of X 18 to X 35 was sufficient to bring out the characteristic features of the localized fracture process in highly filled elast omers. This process is shown in the micrographs Figs. 12 to 18.

I t is clearly seen that failure starts by local concen­t rat ions of large strains in the interior of t he specimen, llot only at the surface where the maximum value of fJ

occurs; the localization of strain is accompanied by the formation of numerous holes (Fig. 12). Probably as a result of the transverse stress fJz some of these holes open up while the material is longitudinally stretched (Fig. 13), forming long elliptical cracks at ditTerent locations (Fig. 14). By fracture of a highly strained strip of material separating a hole from the surface,

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636 MARIA RONAY

cracking spreads to the surface (Fig. 15). It is the time­dependent deformation and failure of such individual strands of essentially elastomeric material separating the holes which produces the time sensitivity of the failure process as a whole. Figures 16, 17, and 18 present a clear illustration of the "crack growth" by the coales­cence of elliptical holes through gradual creep fracture, at constant applied strain, of the elastomeric strands (to which small filler crystals are attached), originally separating the individual holes from each other. The forces acting in the strands stabilize a certain geometric configuration of elliptic holes; this configuration gradu­ally changes as the essentially viscoelastic strands de­form extensively. When they fail in finite times in the characteristic manner of viscoelastic or viscoplastic

materials3 with large tensile strain and rapidly decreas­ing cross section, the configuration changes into one in which smaller holes have combined into larger ones with part of the strain energy stored around the previous configuration being released in the process, which thus produces geometric configurations of higher stability.

It is obvious that this crack propagation process is sufficiently different from that characteristic for brittle materials or metals to make the application to filled elastomers of concepts of fracture mechanics developed for brittle materials or metals of rather problematic value.

3 M. Reiner and A. M. Freudenthal, Proceedings Fifth Inter­national Congress on Applied Mechanics (John Wiley & Sons, Inc., New York, 1938), p. 232.

JOURNAL OF APPLIED PHYSICS VOLUME 35, NO.3 (TWO PARTS-PART 1) MARCH 1964

Incomplete Liquid Mixing in Crystal Growth from the Melt*

W. R. WILCOX

Aerospace Corporation, El Segundo, California

(Received 5 July 1963; and in final form 14 October 1963)

The boundary-layer treatment of Burton, Prim, and Slichter for prediction of segregation in crystal growth from the melt is extended to arbitrary binary solid/liquid equilibrium phase behavior. The validity of the quasi steady-state approach is examined by derivation of the initial transient. In many cases this transient is appreciable and must be considered for accurate predictions. Eigenvalues for calculation of the transient are given, along with a sample calculation.

I. INTRODUCTION

CRYSTAL growth from the melt encompasses such diverse processes as zone refining, alloy freezing,

and growth of single crystals by the Czochralski, Bridgman, and related techniques. In some cases, it is desired that impurities be separated from the original material (charge); in other cases, a uniform concentra­tion in the resultant crystal is sought. The controlling phenomenon in either case is the segregation of im­purities at the freezing solid/liquid interface. A quan­titative knowledge of this segregation of impurities is essential for all of these processes.

In the most elementary treatments of segregation in a freezing process, it is assumed that no diffusion occurs in the solid and that the liquid is completely mixed.! Solid-state diffusion can generally be safely neglected. Complete mixing in the liquid, however, is an idealiza­tion that is attained only in the limit. Under actual operating conditions, the liquid is usually far from being completely mixed. In the other limit, convection may even be absent, in which case the mass transfer is by

* Research reported in this paper was conducted under U. S. Air Force Contract No. AF 04(695)-169.

1 W. G. Pfann, Zone Melting (John Wiley & Sons, Inc., New York, 1958).

diffusion only. Purely diffusional mass transfer has been treated analytically for zone melting2.3 and for normal freezing,4-S The intermediate condition, that between complete mixing and no mixing, is the most common operating condition in practice. It is this condition of incomplete liquid mixing that is treated in detail here. The quasisteady-state and transient regions are con­sidered for a constant distribution coefficient, for a eutectic forming system, and for a system with a variable distribution coefficient.

II. NOMENCLATURE

aI, a2=arbitrary constants in Eq. (7) aj= jth eigenvalue in Eq. (41), as given by Eq. (42) A = cross-sectional area of zone refined crys tal, cm2 A = fraction of cf>8 due to transient

2 W. R. Wilcox, "Fractional Crystallization from Melts," Ph.D. dissertation, University of California at Berkeley, UCRL-9213 (1960).

3 W. R. Wilcox and C. R. Wilke, Am. Inst. Chern. Eng. J. (to he published).

4 K. R. Hulme, Proc. Phys. Soc. (London) B68,393 (1955). 6 V. G. Smith, W. A. Tiller, and J. W. Rutter, Can. J. Phys.

33, 723 (1955). 6 O. W. Memelink, Philips Res. Rept. 11, 183 (1956). 7 W. A. Tiller, K. A. Jackson, J. W. Rutter, anrl B. Chalmers,

Acta Met. 1, 428 (1953). 8 R, G. Pohl, J. App\. Phys. 25, 1170 (1954).

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