Bulletin No. 241 n / D, Petrography and Mineralogy, 31 

benes IE, Chemistry and Physics, 43

DEPAETMENT OF THE INTERIOR

UNITED STATES GEOLOGICAL SURVEY

CHARLES D. WALCOTT, DIRECTOR

EXPERIMENTS

ON

SCHISTOSITY AND SLATY CLEAYAGE

BY

F.

WASHINGTON

GOVERNMENT PRINTING OFFICE

190,4

CONTENTS.

Page.

Letter of transmittal................................-.............;........ 7

Abundance of fissile rocks ....-.'.........-...--........-...-.....-....-..- 9

Importance of cleavage ................................................... 10

Theories of cleavage.. L................................................... 11

Distinction between Sharpe's theory and mine .............. r .............. 12

Means of studying the strain ellipsoid....................................... 12

Linear compression....................................................... 13

Boiling........:....................................................-.... 13

Scission engine........................................................... 13

Ceresin and its treatment.................................................. 14

Experiments with clay ................................................... 15

Other materials tried ..................................................... 16

Strain ellipsoids in ceresin .....-............'............................... 16

Construction of strain ellipsoids ........................................'... 17

Cleavage on the two theories.......... r................................... 17

Cleavage actually found .................................................. 18

Absence of slip cleavage .................................................. 19

Significance of bubbles ..................................................... 19

Scission experiments ..................................................... 20

Results for ceresin........................................................ '20

Results for clay .......................................................... 21

Lessons drawn ........................................................... 21

Mathematical notes....................................................... 24

Note on the theory of slaty cleavage ................................... 24

Note on the inscription of an ellipse in a parallelogram.................. 28

Note on the computation of tan GO ..................................... 30

Note on the volume changes in the formation of slate................... 31

Index ......................:...-.-.-.--...-.-----..-.-.-----...-.-....... 33

3

ILLUSTRATIONS.

Paga

PLATE I, fig. 1. Scission engine........................................... 14

II, fig. 2. Strain ellipsoids in ceresin ................................ 16

3. Cleft strain ellipsoid in ceresin ............................. 16

4. Diagram of distortion in rolling clay........................ 16

5. Diagram to show piercing of cylinders..................... 16

6. Cake of ceresin compressed to two-thirds original height.... 16

7. Cake of lead compressed to two-thirds original height...... 16

8. Cake of clay compressed to two-thirds origiiial height, showing peripheral ruptures................................. 16

9. Cake of lead compressed to one-third original height ....... 16

III, fig. 10. Three cakes of ceresin compressed to one-third original height,

enlarged.. ............................................. 18

11. Diagram to show strain ellipsoids in one quadrant of compressed ceresin cake.................................... 18

12. Diagram to show cleavage on Sharpe's theory.............. 18

13. Diagram to show cleavage on Becker's theory .............. 18

IV, fig. 14. Probable cleavages on Becker's theory..................... 20

15 to 22. Photographs of actual cleavages...................... 20

V, fig. 23. Loci of major axes of ellipsoids corresponding to fig. 6...... 22

24. Block of ceresin faulted by scission........................ 22

25. Block of ceresin showing cleavage due to scission .......... 22

26. Block of ceresin showing cleavage due to double scission.... 22

27. Block of clay showing cleavage due to scission ............. 22

VI, fig. 28. Diagram of slide, but not maximum slide.................. 24

29. Diagram of maximum slide............................... . 24

30. Pure shear compounded of two scissions................... 24

31. Scission and angle of rotation f.i-v .......................... 24

VII, fig. 32. Inscription of ellipse in parallelogram ..................... 28

33. Approximate inscription of ellipse in curvilinear quadri- -.

lateral-----------------_-_.---.--..--.......-...--. * 28

5

LETTER OF TRANSMITTAL.

DEPAKTMENT OF THE INTERIOR, 

UNITED STATES GEOLOGICAL, SURVEY,

Washington, D. C., June 4> 1904-

SIR: I have the honor to transmit herewith a paper by myself 

describing experiments made for. the purpose of testing disputed 

theories of schistosity and slaty cleavage.

Very respectfully,

GEORGE F. BECKER,

Geologist in Charge Division of

Chemical and Physical Research. 

Hon. CHARLES D. WALCOTT,

Directs United States Geological Survey.

EXPERIMENTS ON SCHISTOSITY AND SLATY 

CLEAVAGE.

By GEORGE F. BECKER.

Abundance of fissile rocks. A very large part of the rocks exposed 

at the earth's surface exhibit schistosity or cleavability not ascribable 

to sedimentation and often well marked in masses of igneous origin. 

In many cases the surfaces along which rocks cleave intersect one 

another, producing a foliated structure. In other instances the cleavage occurs in nearly parallel planes or along surfaces having large 

radii of curvature. None of these phenomena are confined to highly 

.indurated rocks, as might be inferred from some discussions, but are 

found well developed in soft and fragile unmetamorphosed shales, as 

all who have made observations in disturbed Tertiary areas are aware. 

Specimens of these shales are so fragile, however, that they are scantily 

represented in collections. Slaty cleavage, as the term is used in this 

paper, means simply the most regular and extreme form of cleavability 

or schistosity, in which the laminae are thin and are bounded by substantially parallel surfaces, no matter whether the material exhibiting 

these properties is indurated, as in the valuable roofing slates, or is soft 

and fragile, as in the comparatively worthless shales. Though argillaceous rocks exhibit the most perfect cleavage, this does not apparently 

differ in kind from the schistosity sometimes found in grit beds, limestone, granite, quartzite, and basic eruptives. A structure which is 

at least analogous is found in rolled or drawn metals, in terra-cotta 

articles, such as roofing tiles, and in pastry. a In short, slaty cleavage is 

a structure, or a group of structures not readily distinguishable from 

one another, and has a character of its own independent of the material which may exhibit it, although some substances are better fitted 

than others to displa}^ it in perfection. This relationship is recognized 

in the French term " schiste," which, in Gallic usage, comprehends 

shales, slates, and the crystalline schists of English writers. In 

English usage schist is commonly s}7 nonymous with crystalline schist, 

but the frequent use of this adjective would seem to imply the existence of schists which are not crystalline, and these could hardly be 

defined otherwise than as equivalent to shales.

aCf. Daubr6e, A., G6ol. Exp., pp. 398 and 428.
10 SCHISTOSITY AND SLATY CLEAVAGE. [nuLL.24i.

Slaty cleavage, even in highly indurated rock, passes over by insensible gradations into less simple forms of schistosity, and vast masses 

of the crystalline schists show cleavage planes in systems intersecting 

one another at acute angles. It has always seemed to me that a true 

explanation of cleavage must include the theory of foliated schists as 

well as of roofing slate. The association of mica with cleavable rocks 

has often been 'insisted upon, perhaps with too great emphasis. Some 

shales contain much mica, but others contain only a little. According 

to Professor Rosenbusch, shales (Schieferthone) have the same mineralogical composition as clays, and mica is not an essential constituent. 

Sorby mentions clay slates which carry very little mica, and G. W. 

Hawes a describes a roofing slate from Littleton, N. H., which on 

microscopic examination is found to consist of fragments of quartz 

and feldspar as fine as dust, although in the larger part of the rocks 

called clay slate in New Hampshire he found abundant mica. Now 

clay slates and some shales have as good cleavage as mica-schists. 

Again, quartz-schists and other cleavable rocks contain very little 

mica. The grit beds or sandy strata found in slates do not always 

contain much mica, and yet their cleavage is manifestly of the same 

origin as that of the slate in which they are embedded. It is well 

known that near intrusive masses mica-schist not infrequently passes 

over into ordinary schists, and these into phyllites and clajr slates, as if 

the amount of mica were characteristic of the degree of metamorphism 

rather than an index of the cleavability. "Compression," says Sir 

Archibald Geikie, "may give rise to slaty cleavage. But it has often 

been accompanied or followed by further internal transformations in 

the rocks. Chemical reactions have been set up and new minerals 

have been formed." In the study of slates it is often manifest that a 

portion of the mica is secondary. Thus on blind joints (Ausweichungsclivage of Heim) large continuous sheets of mica are frequently 

found. Muscovite is also well known to be one of the chief decomposition products of the feldspars, an alteration which is readily intelligible from a chemical standpoint. The increase of the mica content 

of phyllites as compared with shales seems to me most reasonably 

accounted for as a concomitant of the genesis of cleavage, though not 

an essential one.

Importance of clewvage. Schistosity as a structure is important, and 

it is a part of the business of geologists to explain its origin. Slaty 

cleavage has further and greater importance as a possible tectonic 

feature. Scarcely a great mountain range exists, or has existed, along 

the course of which belts of slaty rock are not found, the dip of the 

cleavage usually approaching verticality. Are these slate belts equivalent to minutely distributed step faults of great total throw, or do they 

indicate compression perpendicular to the cleavage without attendant

a Hitchcock, C. H., Geology of New Hampshire, vol. 3,1878, pt. 4, pp. 237,238.
BECKER.] THEOBIES OF CLEAVAGE. 11

relative dislocation? Evidently the answer to this question is of first 

importance in the interpretation of orogenic phenomena.

Theories of cleavage. The earliest theory of slaty cleavage assimilated it to mineral cleavage, a view not tenable after microscopic study. 

Mr. John Phillips in 1843 was .the first to interpret slaty cleavage as 

an effect of mechanical strain. Mr. Daniel Sharpe in 1849 offered 

the explanation now most generally accepted by geologists, viz, that a 

fracture perpendicular to the line of pressure would run along the 

flattest faces of the component grains and meet the smallest number 

of them- a theory implying that the mass is heterogeneous and that 

the adhesion between the component particles is smaller than the 

cohesion within the particles. Dr. H. C. Sorby later described a variety 

of cleavage due to the presence of numerous microscopic blind joints. a 

Professor Tyndall in 1856 made exceedingly interesting experiments 

on this subject, obtaining admirable cleavage in wax. He denied that 

heterogeneity aided cleavage. In his first paper he asserted most 

emphatically that the cleavage was perpendicular to the direction of 

pressure, but in a footnote and in a later paper he indicated decided 

doubt as to this perpendicularity. Professor Daubree (1879) also dissented from the view that heterogeneity is essential to slaty cleavage, 

and ascribed this structure to gliding (" glissement," slide) in the mass, 

which is equivalent to a denial of .the perpendicularity of the causative 

force to the consequent cleavage. b In 1893 1 published a theory 

founded on experiment and analysis. It is in agreement with Daubree's 

idea, but more precise. According to this theory, cleavage is due to 

a weakening of cohesion c along planes of maximum tangential strain (or 

maximum slide). It is susceptible of proof d that deformation due to 

pressure is actually effected by relative motion of the mass in opposite 

directions parallel to these planes. When this movement exceeds the 

elastic limit and falls short of the breaking strain, it would seem inevi-

« Sorby's theory of slate was that the'preliminary effect of pressure on argillaceous strata is to give 

the mica an irregular distribution, and the final effect to rearrange the mica in new parallel planes. 

This hypothesis is still accepted in some text-books. To me it appears loo fanciful for serious discussion. Jf mica scales, in all possible orientations, were to be mingled with mud, their average inclination to any plane would be 32° 42' (or the well-known "average latitude of all places north of the 

equator"). If such a mass were to be compressed until the average angle were only 2° to a given 

plane, the thickness of the mass must be reduced to one-eighteenth. Jf a sediment containing mica 

were to be treated according to Sorby's theory, it would seemingly be needful to press it at first in 

such a way ns to increase its thickness some eighteen times and then to compress it again in another 

direction to about its original thickness.

The attempt has been made to account for the cleavage surfaces on Sorby's hypothesis as maximum 

cleavability. I can not concur in this view. The cleavage in slate is confined within very narrow 

limits, perhaps one degree. Slate may be broken indeed at greater inclinations, but the ruptured 

surfaces do not then show imperfect cleavage; they are conchoidal or irregular and destitute of 

cleavage.

t> The reader will find a digest of the literature in Mr. Alfred Barker's memoir on slaty cleavage, 

Brit. Assoc., 1885, p. 813, and some further notes in my paper on finite homogeneous strain, flow, and 

rupture of rocks, Bull. Geol. Soc. America, vol. 4,1893, pp. 75-87.

cAs H. Rogers put it, " the cohesive force is obviously at a minimum of intensity in the direction 

perpendicular to these planes" of cleavage. Trans. Royal Soc., Edinburgh, vol. 21, 1857, p. 450.

ft See note on the theory of slaty cleavage appended to this paper.
12 SCHISTOSITY AND SLATY CLEAVAGE. [BULL.241.

table that the cohesion should be diminished and that cleavage should 

result. a Only in ideally brittle substances is there no interval between 

the elastic limit and the breaking strain. It is of course certain that, 

the material constituting slate has been strained far beyond its elastic 

limit; and that it has a breaking strain is often manifested by blind 

'joints. These planes stand at an angle of 45° or more to the direction 

,of greatest local linear compression. There are at least two sets of 

them, and maybe four, symmetrically disposed with reference to this 

direction. In cases of double cleavage so usual in disturbed areas, 

both in distinct development and in the more or less irregular form of 

ordinary schistosity and foliation cleavage is produced on both sets 

of planes, so that a cross section shows acute-angled rhombs somewhat 

like those indicated in fig. 13, PI. Ill, and in fig. 30, PI. VI. In the 

case of forces acting on a supported mass at an acute angle to the 

plane of support, it was shown in my paper that the effect of viscositj7 

would be to suppress all but one set of cleavages and to accentuate 

this remaining one. Tyndall's experiment, property considered, was 

shown to be a case of this kind, and it was maintained that the cleavage of roofing slate is thus to be explained. If this be true, a belt of 

slate is equivalent to a great fault distributed over an infinite number 

of infinitesimal steps. 6

Distinction between Sharpens theory and mine. The distinction 

between Sharpe's theory and mine is well defined. If in any portion 

of the mass before strain a small sphere is supposed to be marked out, 

this sphere after strain will have become an ellipsoid, called the strain 

ellipsoid. If Sharpe's theory is correct, the cleavage due to pressure 

will be in surfaces perpendicular to the smallest axis of the strain ellipsoid. If my theory is correct, the cleavage will make with this smallest axis an acute angle equal to or greater than 45°, and increasing as

the strain grows greater.

Means of studying the strain ellipsoid. The general nature of the 

experiments needed to compare the theories is made plain by this contrast. It amounts to a study of the strain ellipsoid. One means to this 

end is to incorporate into a mass to be experimented upon small spheres 

of the same material as the remainder, but of a distinguishable color. 

After straining is effected dissection or rupture, by exposing the distorted sphere, will show the local character of the strain and the posi-

«A pertinent illustration of weakening of cohesion is afforded .by bars of mild steel ruptured by 

tension. The most plausible a priori idea of rupture by tension is that it would occur in a plane 

perpendicular to the line of force; and in hard steels this mode of rupture is often observed. In 

mild steels, on the other hand, the surface of rupture is rough and granular, the grains approaching 

pyramidal forms. The aggregate surface of such a fracture is far in excess of that of the mean 

plane. Now, since the rupture must follow a surface of least resistance, the resistance per unit area 

on the pyramidal faces must have been much smaller than that on the plane surface perpendicular 

to the tension. This can be due only to a weakening of cohesion along the pyramidal faces.

bThis theory embraces rupture as well as simple and double or multiple cleavage. So far as jointing and cleavage due to blind joints are concerned, it has been accepted by some geologists, who 

regard true cleavage as a distinct phenomenon.
BECKER.] STUDY OF STRAIN ELLIPSOID. 13

tions of the axes of the strain ellipsoid. Numerous experiments of 

this description have been made for this paper, and some of them are 

illustrated in fig. 2, PI. II. They are instructive, but not sufficiently 

so. If it were practicable to build up an adequate block of small 

spheres and to fill in the.interstices uniformly with the same material 

differently colored, the mass after strain would show the character of 

the strain ellipsoid at uniformly spaced intervals. This is impracticable, but the result sought can be attained very approximately in 

another wa}r. If a block of material be pierced with fine holes at regular intervals, forming in one plane a network of small squares such as 

is shown in fig. 5, PI. II, and the holes be filled with coloring matter, 

then strain, followed by dissection, will show the figures into which 

the small squares have been distorted. If the squares were very small, 

the sides of the distorted figures would be nearly straight and parallel. 

It would then be easy by a geometrical construction to inscribe ellipses 

tangent .to the four sides at their middle points, and these ellipses

would accurately represent the.section of the strain ellipsoid. Even 

if the sides Avere somewhat curved, the strain at the center of the 

curvilinear parallelogram would be very closely represented by an 

ellipse found by a rational system of interpolation, as will be described 

in a note appended to this paper. Experiments have been made in 

this way also, the distorted figures being photographed and photographically enlarged to a convenient scale for constructing the ellipses.

Linear compression. Experiments by Tyndall's method are very 

simply and easily executed, but the resulting strain is highly complex 

and indeed could not be discussed as a problem of pure mechanics in 

the present state of knowledge of the transmission of energy in plastic 

bodies. By the means noted in the last paragraph such experiments 

can be sufficiently elucidated for an investigation of slaty cleavage.

jRolling. Another known means of producing slaty cleavage is by 

rolling out a cake of suitable material as a cook prepares pastry, or by 

passing the mass between rolls. It is easj' to prick holes perpendicular to 

the surface of the cake before rolling, fill them with pigment, and, after 

distortion, to dissect the mass. The. nature of the effect is seen in 

fig. 4 of PI. II, the originally vertical lines being drawn out into parabola-like curves which are vertical only at the apex. Cleavage may 

be developed especially near the upper and lower surfaces of the rolled 

mass, to which it is nearly parallel. Such experiments, however, give 

results which are less definite and less easilj7 discussed than those given 

by Tyndall's method. The manner of applying the force and the degree 

of rolling affect the result, as a matter of course, and it seems difficult 

to establish a standard of conditions.

/Scission engine. It is evidently most desirable to make experiments 

by. producing simple well-known strains which will or may lead to 

cleavage. For this purpose I designed what may be called a "scission
14 SCHJSTOSITY AND SLATY CLEAVAGE. [BULL. 241.

engine," a simple mechanism, shown without its cover in fig. 1, PI. I. 

A block of material to be experimented upon is placed in the space B, 

the cover (provided with grooves .to receive PP) is put on, screwed 

down onto pillars, and the driving screw set in motion. The bars F H 

revolve round the fixed pivots F, and the hinges H bend while the 

plates PP slide in their own planes. The area and volume of the space 

B remain constant and the strain produced is approximately a scission 

(or shearing motion). On four of the six faces, however, there is 

friction, which interferes somewhat, but not seriously, with the perfection of the strain. 05 If a circle is inscribed on the upper surface of 

the block before strain, it becomes approximately an ellipse showing 

the orientation of the strain ellipsoid. When the space B is thoroughly 

filled, no rupture within the mass is possible, the finite movement being 

distributed over an infinite number of planes; but if the space is not 

filled, complex strains result and rupture ma}^ occur. The engine 

gives about the unit strain, or produces a block with two angles of 45°.

Ceresin and its treatment. The experiments to be described have 

been made principally with ceresin and a smooth clay such as is used by 

sculptors for modeling. I have tried white wax, which was the material employed by Tyndall, but white ceresin is preferable. This substance is refined ozocerite and consists of a mixture of paraffins. The 

material at my disposal melts at about 60° C., but some of the component paraffins solidify at a higher temperature, so that at 60° the cooling melt is a pasty mass, like thick oatmeal gruel. It contracts greatly 

in solidifying and should be cast at as low a temperature as practicable to prevent radial crystallization. Shavings of the solidified mass 

under the microscope show brilliant polarization colors, so that the 

mass is a crystalline solid. A particle fused on a slide and allowed to 

cool also, shows high double refraction and exhibits a hypidiomorphic 

structure, analogous to that of semiporphyritic granites of excessively 

fine grain. Castings chilled in ice and salt and then broken with a 

hammer display a very fine-grained granular structure and somewhat 

conchoidal fracture, with no apparent radial crystallization.

The strong analogy between this solid and a rock is deserving of 

special emphasis. I can not see how deformations in masses of ceresin 

can possibly differ in character from those in the vastly less tractable 

crystalline rocks.

In carrying^ out Tyndall's experiment with ceresin I cast'cylinders 

about li inches in diameter and nearly 2 inches in length. After the 

cylinders were cold the ends were planed down until all trace of the 

pit due to contraction was removed. Before compressing them they 

remained in a thermostat for some hours at 35°, because at considerably lower temperatures they rupture or crumble too easily. Com-

«To,obviate in some measure the effect of friction toward the center of the block, I made the space 

J5 6 cm. long and only 4 cm. wide1.
U. S. GEOLOGICAL SURVEY

SCISSION ENGINE.
BECKER.] MATERIALS FOE EXPERIMENT. . 15

pression w,as efl'ected in a copying press between two heavy glass 

plates mounted on boards. To avoid superficial chilling of the ceresin 

cylinders the glass plates were warmed to approximately the same 

temperature as the ceresin. Good cleavage is not obtained unless the 

cylinders are reduced to one-third of the original length, and more 

nearly perfect cleavage results from still further reduction. For 

accurate study the compressive force should be applied to the cylinders so slowly as not to rupture the edges of the cakes by tension. 

The compressed cakes were placed in a mixture of ice and salt for an hour 

or two, and then, being held edgewise on a small anvil, were broken by 

striking smart blows on the edge with a light hammer.

It is best to carry out such experiments in series. Some cylinders 

may be pierced with a network of fine holes (fig. 5, PI. II) and, after 

compression, cut across in the plane of the perforations. To get the 

perforations sensibly in the same plane after compression, both piercing and squeezing must be carefully done, Other cylinders, of the

same dimensions but not pierced, may be compressed to the same 

extent as the perforated cylinders and then, after chilling, split to 

show cleavage. -If the C37 linders are squeezed too rapidly the edges 

will split a contingency to be avoided because the distribution of 

strain then becomes irregular and eludes systematic discussion.

For experiments on scission, blocks were planed to fit the opening 

B as accurately as possible and the blocks were given a temperature 

of about 20° before straining. They were cooled and broken as in 

the experiments on linear compression.

Experiments ivith day. The clay used burns to a pleasing terra-cotta 

color, without much shrinkage. Slides of the burnt mass show that 

the clay contains a large amount of finely divided quartz and a small 

amount of black mica in minute scales. There is also a trace of 

organic matter in this clay, for when first heated it blackens. For 

use, the clay should be moistened as little as is compatible with convenient modeling and kneading.

From a well-kneaded lump of clay it is easy to cut cylinders similar 

to those of ceresin described above. . Excellent cleavage can be produced by compressing them in a press and burning the cakes lightly 

in an assay muffle furnace. Indeed, to get cleavage, it is sufficient to 

press a pellet, say a centimeter in diameter, under a spatula and toast 

it over a Bunsen lamp! Yet in some respects clay is inferior to ceresin 

for this experiment, because c}7 linders, after reduction to about fiveeighths of their original diameter, crack at the edges, so that the 

strain can not be followed systematically by the method given above. 

Some of these cracks are meridional and due to tension, while others 

occur at about 45° to the line of force and are true joints on planes of 

maximum tangential strain. Such a cake of clay is shown in fig. 8, PI. 

II. Precisely similar cracks are produced in steel cylinders subjected to
16 SCHISTOSTTY AND SLATY CLEAVAGE. [BULL. 241.

end pressure, and it is clear from these phenomena that the moist clay 

behaves mechanically like a true solid, relative motion of the particles 

taking place at an acute angle to the line of pressure. I have tried 

subdividing clay cylinders by a network of pin holes in one plane, 

and then compressing and dissecting them as described on a previous 

page. Up to the point where the edges begin to rupture, the deformation is exactly the same as in the cylinders of ceresin of similar 

dimensions and degree of compression.

On the other hand, clay behaves better than ceresin in the scission 

engine, apparently because I did not succeed in casting ceresin without 

small bubbles of included air, while it is easy to knead the clay until 

air is expelled. In the scission engine there is no tendency to cubical 

compression provided the space B is homogeneously filled, but this 

proviso is essential. In experiments on ceresin one acute angle of the 

space B is usually found empt}7 , and though this space is never large, 

it is a disturbing condition. With well-kneaded clay the space B 

remains full after strain, and such blocks after burning show.excellent 

cleavage.

Clay is a very instructive material to experiment upon for two reasons: Most or much of the natural slate is of argillaceous origin; and, 

again, the burning of the strained clay cakes may properly be considered as a true metamorphism, which nevertheless does not obliterate 

the cleavage mechanically induced.

Other materials tried. Plaster of Paris pellets compressed under a 

spatula at just the right moment during the "setting" process and 

then dried out exhibit cleavage, a fact which is interesting because of 

the accompanying formation of selenite crystals. 1 have tried plaster 

paste in the scission engine repeatedly, but have not been able to hit 

exactly the right conditions. Air bubbles get into the liquid plaster 

while it is being poured into the space B, and if stress is applied too

soon the plaster naturally sets without any development of cleavage. 

Moreover, plaster seems to begin to set from the outside, so that the 

space B was probably never homogeneously filled with a substance 

fitted to display the properties of homogeneous strain.

Lead cylinders pierced with holes and afterwards filled with tin wire 

and then compressed show by dissection just the same strains as do 

ceresin blocks. Such cases are shown in figs. 7 and 9, PI.. 11. Similar 

results have been obtained with aluminium. I had hoped that these 

metals, cooled to the temperature of liquid air, would become brittle 

enough to exhibit cleavage, but was disappointed. The lead cakes 

seemed as tough as sole leather and I could not produce the least indication of a crack in the aluminium b}T the most vigorous use of the 

hammer.

Strain ellipsoids in ceresin. In fig. 2, PI. II, is shown a series of 

cakes of ceresin into which spherical pellets of ceresin tinged with a
U. S. GEOLOGICAL SURVEY BULLETIN NO. 24-1 PL. II

Fig. 2.

Fig. o.

FIG. 2. STRAIN ELLIPSOIDS IN CERESIN.

FIG. 3. CLEFT STRAIN ELLIPSOID IN CERESIN

FIG. 4. DIAGRAM OF DISTORTION IN ROLLING CLAY

FIG. 5. DIAGRAM TO SHOW PIERCING OF CYLINDERS

FIG. 6. CAKE OF CERESIN COMPRESSED TO TWO-THIRDS ORIGINAL HEIGHT

FIG. 7. CAKE OF LEAD COMPRESSED TO TWO-THIRDS ORIGINAL HEIGHT.

FIG. 8. CAKE OF CLAY COMPRESSED TO TWO-THIRDS ORIGINAL HEIGHT, SHOWING

	PERIPHERAL RUPTURES. 

FIG. 9. CAKE OF LEAD COMPRESSED TO ONE-THIRD ORIGINAL HEIGHT.
DECKER.] DETEKMINING STEAIN ELLIPSOIDS. 17

mere trace of vermilion were introduced during the process of casting. 

The cylinders after compression were cut radially to show the strain 

ellipsoids thus produced. If a composite photograph were to be taken 

of these and similar cases the position of the strain ellipsoid, in tnost 

portions of the cross section would appear.

Many cakes containing pellets were cooled below the freezing point 

and split. These specimens show that the surfaces of cleavage are not 

parallel to the major axes of the strain ellipsoids. In some cases the 

cleavage developed by the hammer intersected the pellets, leaving no 

doubt whatever on this point (fig. 3, PI. II). It is relatively seldom, 

however, that such a crack forms without splitting away a large part 

of the pellet and leaving some doubt' as to the exact position of the 

major axis. Hence the observer is driven to a comparison between 

the dissected cakes without cracks and the split cakes. It seemed 

desirable, therefore, to devise a means of determining once for all the 

position of the strain ellipsoid in any and every part of the cake. This 

will now be described.

Construction of strain ellipsoids. Fig. 5, PI. II, is a diagram indicating the way in which cylinders were pierced with holes, forming a 

rectangular network covering a quarter of the cross section. A thread 

smeared with dry vermilion powder was drawn, after the piercing 

needle and thus the interior of the perforations was coated with pigment.

Photographs of. three cakes compressed after perforation and cut 

to show the network are shown on an enlarged scale in -fig. 10, PL 

III. They are as nearly alike as the imperfection of the appliances 

used would permit. The central vertical lines in the cylinders were 

not absolutely central, nor were the plates between which the compression was effected accurately parallel planes. Hence, after compression the central line in each case is somewhat buckled.

From the middle one of these photographs a tracing was made, 

slightly modified by comparison with the other two, and then very 

much enlarged. In the diagram so procured ellipses were drawn by 

the methods explained at the end of this paper. The areas of the 

ellipses were next checked by a simple computation and found sensibly 

correct, showing that no considerable error had occurred in copying 

or construction. The value of the axes of the ellipses being found 

and the volume of the ellipsoids known, the planes of maximum tangential strain are also immediately deducible. The major axes were 

drawn through the ellipses and the positions of the planes of maximum 

tangential strain were shown by broken lines. The diagram was then 

reduced photographically to the same scale as the photographs from 

which it was derived. The result is shown in fig. 11, PI. III.

Cleavage on the two theoi^ies. It is now eas^ to draw through this 

quadrant of the figure representing the cross section of the cake a set of 

10026 No. 241 04 2
18 '. SCHISTOS1TY AND SLATY .CLEAVAGE. [BULL.241.

lines which are sensibly tangent to the directions of the major axes of 

the ellipses, and this set of lines represents the cleavages which the 

cake should have according to Sharpe's theory. This result is illustrated by fig. 12, PI. Ill, where, for the'sake of completeness, all four 

quadrants are filled out. Similarty a diagram can be prepared illustrating my theoiy, and this is given in fig. .13, PI. III. It will be 

observed that the two diagrams differ very radically and that the choice 

between them must be an easy one.

The system of cracks likely to occur on Sharpe's theory is made 

sufficiently evident by the figure just mentioned. The other theory 

shows an interlacing of possible fractures near the center, which is 

more complex and could not be entirely developed in a single cake 

without comminution. For this reason the most probable and impor 

tant fractures are given separately "in fig. 14, PI. IV.

The surfaces which at the end of the straining process are surfaces 

of maximum tangential strain were never at any step of the process 

perpendicular to the direction of greatest linear contraction. To make 

sure of this, I have constructed the strain ellipses for a case in which 

a cylinder of ceresin pierced with a network of holes was compressed 

to two-thirds its original length. Fig. 6, PL II, shows the cross section of the strained mass, and fig. 23, PL V, shows lines coinciding 

with the direction of the major axes of the strain ellipsoids for this 

case. Comparison of the last diagram with that previously discussed 

for a strain twice as great (fig. 12, PL III) shows their analogy, and 

proves the statement made in the first sentence of this paragraph.

Cleavage actually found. It is exceedingly difficult to give satisfactory illustrations of the cleavages actually obtained by Tyndall's 

experiment. In the first place, the most instructive cakes are those 

which go to pieces under the hammer; but then the residual flakes are 

too delicate to be cut across radially for photographing, and were 

this accomplished only a single section could be figured, whereas the 

observer may examine them in three dimensions. Again, when the 

cakes are so gently hammered as merely to crack from the edges and 

are subsequent^ separated into two or more pieces, tension ruptures 

are produced as well as true cleavage; but these are not distinguishable in a photograph. Especially characteristic in radial section is the 

way in which the cleavage meets the outer edge of the cake. Seen in 

cross section the thinner part of the split cake at the outer edge is 

shaped like one horn of a crescent moon. This characteristic is shown 

by every cake, and yet it is not strikingly apparent in eveiy section 

illustrated in figs. 15 to 22, PL IV, although it is well shown in several of them. The cakes break last at the axis, and here there is.niost 

danger of tension rupture in forcing the opposite portions asunder. 

In two or three of the specimens illustrated, however, there is evidence
U. S. GEOLOGICAL SURVEY BULLETIN NO. 241 PL. Ill

FIG. 10. THREE CAKES OF CERESIN COMPRESSED TO ONE-THIRD ORIGINAL HEIGHT, ENLARGED.

FIG. 11. DIAGRAM TO SHOW STRAIN ELLIPSOIDS IN ONE QUADRANT OF COMPRESSED CERESIN CAKE.

FIG. 12. DIAGRAM TO SHOW CLEAVAGE ON SHARPE'S THEORY.

FIG. 13. DIAGRAM TO SHOW CLEAVAGE ON BECKER'S THEORY.
BKCKBB.] CLEAVAGES FOUND. 19

of double fracture of opposite inclination at the axis (figs. 19 and 20). 

In figs. 18 and 21, PI. IV, the cleavage is seen passing through the 

axis at an angle to the median line.

On the whole, even the photographs show a very reasonable agreement with the inference from analysis indicated as probable cleavages 

in fig. 14, PL IV. I may add that the cleavability about midway 

between the edge and the axis is often so perfect as to defy illustration; the flakes are frequently so thin that print can be read through 

them.

I must have made. Tyndall's experiment a hundred times, using wax 

on some occasions and ceresin on others. In no case have I broken a 

cake which behaved as it should on Sharpe's theory, illustrated in fig. 

12, PL III. The section of a cake most nearly in accord with that 

theory which I have seen is shown in fig. 22a, PL IV, and especially 

for that reason. Even this exceptional instance exhibits features not 

in agreement with that theory, while another section of the same cake 

(fig. 225) does not at all resemble the diagram constructed for the loci 

of the major axes of the strain ellipsoid. The general features of the 

cleavage along surfaces of maximum tangential strain seem always to 

be recognizable when the terminations of the cylinders are plane ana 

the compression is sufficient to produce good cleavage in the squeezed 

cakes, but not so excessive that the minor axes of the strain ellipsoids 

are reduced to almost insensible length. Evidently, in order to make 

any instructive comparison between the two theories, these conditions 

must be fulfilled.

Absence of slip cleavage. The semitranslucency of ceresin is very 

advantageous for these experiments in some respects. If a cake of 

ceresin is examined in a strong light and at the same time partly 

shaded, any internal cracks can readily be detected. When cylinders 

are being compressed for Tyndall's experiment, between glass plates 

of the proper temperature, the first cracks to form seem always to be 

at the edge of the cake; and when the pressure is applied so gradually 

as to avoid this peripheral splitting I find no internal cracks unless 

the glass plates were too cold.

Significance of bubbles. On the other hand, my cakes all contain 

numerous minute bubbles of air, carried into the mold in casting. During compression these are flattened, and are then equivalent to minute 

blind joints. The flattened bubbles are, of course, oriented exactly 

as are the strain ellipsoids. They are perfectly visible in the photographs of the dissected cakes, and comparison shows that the orientation of the bubbles is indistinguishable by the eye from that of the 

strain ellipsoids obtained by construction as shown in my diagram, 

fig. 11, PL III. Were the bubbles smooth internally, as they would 

be in a glass, it might be possible to dispense with the construction.
20 SCHI8TOSITY AND SLATY CLEAVAGE. [BULL.241.

They are not smooth, however, and seern to be lined with minute 

paraffin crystals, so that their evidence, though confirmatory, is not 

sufficient.

I feel fully justified in asserting that there is in my experiments just 

described no blind jointing (Ausweichungsclivage) or slip cleavage in 

the directions in which cleavage actually takes place, or in directions 

called for by my theory of cleavage. But the bubbles tend to weaken 

the mass in the directions in which cleavage should occur on Sharpe's 

theory. Hence the weakening of cohesion, to which I attribute cleavage 

(along the surfaces of maximum tangential strain), must be so great 

as more than to counterbalance the effect of the bubbles.

The double or schistose, cleavage which is called for by theory and 

is illustrated in fig. 13, PI. Ill, is not often displayed except by the 

diversity in the directions of the surfaces of fracture near the axis of 

the cakes; but the fact that near the axis the cake may split in either 

of two directions shows that there are two intersecting cleavages.

Scission experiments. The purpose of experimenting with scission 

was, as has been explained, to produce a simple strain which, if it 

could be made to lead to cleavage at all, would indicate beyond doubt 

whether the surfaces coincided with one of those of maximum tangential strain. In a scission one of these directions is parallel to two 

faces of a' distorted rectangular mass, those, viz, which undergo no 

change of area. In this direction the same set of particles is subject 

to maximum tangential strain from the inception of the process to its 

completion, however long a time that may be. There is a second set 

of planes of maximum tangential strain, but as the strain increases in 

amplitude fresh sets of material particles continually replace one 

another in these latter planes, so that any one set of particles undergoes 

maximum tangential strain along these planes only for an infinitesimal 

time. Hence, either a smaller effect, or at least a different effect, will 

be produced on this second set of planes. If my theory is correct, 

cleavage is to be looked for only parallel to the planes of constant 

area, for reasons indicated in a note on the theory appended to this 

paper. With the unit shear, or when the acute angle of the distorted 

mass is 45°, the major axis of the strain ellipsoid makes an angle of 

about 32° (i tan -1 2) with the undistorted planes. This strain is shown 

in fig. 31, PL VI.

Results for ceresin. I never expected to get a perfectly smooth 

slat}r cleavage by scission, for reasons stated in the appended note on 

my theory. Experiments on the scission of ceresin blocks are not very 

satisfactory. If the blocks have as high a temperature as I found best 

for Tyndall's experiment (where the strains for the most part are of 

much greater amplitude than my scission engine will produce), the 

blocks show no cleavage at all. At 20° C. and 0° C. cleavage sometimes results and is sometimes absent or insensible. When the cleav-
U. S. GEOLOGICAL SURVEY BULLETIN NO. 241 PL. IV

FIG. 14. PROBABLE CLEAVAGES ON BECKER'S THEORY. 

FIG. 15-22. PHOTOGRAPHS OF ACTUAL CLEAVAGES.
BECKEH.] ' SCISSION EXPERIMENTS. 21

age is a vanishing quantity or absent the mass breaks with a more or 

less conchoidal fracture. When there is any distinct cleavage it is 

parallel to the undistorted planes.

One difficulty with ceresin is that, on account of air bubbles, a the 

block does not completely and homogeneous!37 fill the space. If it did, 

rupture would be impossible. In fig. 24, PL V, is shown an instance 

of rupture. The circle stamped on the block before strain is distorted 

to an oval which is dislocated by three tiny faults. The joints formed 

are exactly parallel to the undistorted plane. The truncated upper 

left-hand corner shows the failure to fill out the space.

In fig. 25, PI. V, is shown a block which did not entirely fill out the 

space, but was not jointed like fig. 24, PI. V. When struck on the back 

with a hammer it developed cracks having a tendency to parallelism 

with the undistorted planes. In one case, after subjecting a cake to 

scission, I cut.off one sharp corner perpendicularly to the plane of no 

distortion and filled out the opposite face with a wedge, so as to reduce 

the block once, more to a rectangle. This was again strained in the 

engine. After chilling it broke with some regular^. It is shown in 

fig. 26, PI. V, where X is the mass which was added before the second 

strain.

Results for day. For scission clay is a far better material than 

ceresin, as was mentioned above, and in all the cases I have tried clay 

blocks subjected to unit strain and lightly burned show cleavage parallel to the planes of no distortion. A veiy highly instructive specimen was produced accidentally. The distorted block was dried on the 

water bath and then heated in an assay muffle furnace, which, however, 

grew hot too rapidly. The escaping water vapor burst the block into 

many pieces. These, fortunately, were of such sizes that it proved 

practicable to fit them together and restore the outlines of the block. 

This specimen is illustrated 'by a photograph (fig. 2Y, PI. V). The 

ellipse is visible and not faulted, and the cleavage is manifestly parallel 

to the planes of no distortion.

Lessons drawn. The experiments described above, which constitute 

a study in plasticity, appear to me to demonstrate that true cleavage 

(wholl}' free from blind joints, or Ausweichungsclivage) can be produced both in ceresin and in clay. Burning the cla}^ does not obliterate the cleavage. The cleavage does not coincide even approximately 

with the direction of the major axes of the strain ellipsoids. Neither 

does the cleavage correspond to the position of the major axes of the 

strain ellipsoids at any previous stage of the strain. On the other 

hand, the orientation of the cleavage does correspond to the position

nit might be better to prepare in another way blocks of ceresin for scission. The melted mass 

might be Arery gradually cooled, with very gentle stirring, in a flat-bottomed pan, and there allowed 

to solidify without pouring. Then blocks could be sawn out of the mass and planed to fit the engine. 

Such blocks would be free from bubbles. The experiments with clay are so satisfactory that I did 

riot try this method.
22 scHtSTostT? AtfD SLATY CLEAVAGE. [BULL. 241.

of the surfaces of maximum tangential strain. Cakes of ceresin linearly compressed exhibit and elucidate slat}^ cleavage near their edges. 

Toward their axes of symmetry they show and explain the double or 

multiple cleavage so characteristic of the crystalline schists. This last 

important phenomenon is almost unintelligible on Sharpe's theor}^ for 

if the greatest linear contraction were .perpendicular to one set of 

cleavages in the schists, it could not also be perpendicular to the 

other. If it be suggested that the two (and sometimes four) cleavages 

were successively impressed on the schist, the answer is that observation is inconsistent with this explanation, the distribution of minerals 

and their mutual relations contradicting the idea. The evidence presented shows that rupture and cleavage follow the same surfaces, 

cleavage being due, so far as can be told, to weakened cohesion a 

state of things in absolute accord with DaubreVs experiments and 

Hei-m's observation that Ausweichungsclivage and cleavage without 

rupture are sometimes visible in the same slide and are parallel to each 

ot-her. a The effect of pressure in the direction of greatest linear contraction, on the other hand, is only to force molecules closer together in 

the line joining their centers. How this approach of molecules to one 

another might increase their cohesive attraction I can understand; but 

how it could weaken it is, to me, a mystery. So far as I know, no 

theory of molecular attraction has been formulated which would 

account for such a weakening.

The deformation of ceresin implies only a trifling expenditure of 

energy or a minute evolution of heat. When firm rocks are deformed, 

especially igneous rocks, the amount of work done or of heat evolved 

must be very great. That such deformation should be accompanied 

by the genesis of "secondary minerals is in accordance with all the 

results of the study of metaraorphism. Now, experiments of my own, 

not yet published, show conclusively that ciystals tend to grow in the 

direction of least resistance, as might indeed be assumed, although they 

exert a linear force in any direction . Secondary minerals in a slate originating in a firm rock will thus tend to develop chiefly in the direction 

of cleavage. It is not improbable that the secondary development of 

mica on cleavage planes may further facilitate the cleavage to which 

it owes its existence, much as slickensides on a faulted surface facilitate 

further faulting. If the mica were assumed as the origin of the 

cleavage, it would be necessary to show how it could be generated and 

oriented in planes independent of the stratification without obliterating

aMechanismus der Gebirgsbildung, vol. 2, 1878, pp. 56 and 59. Heim attributes cleavage to movements (Ausweichungeu) of the mass perpendicularly to the direction of pressure. It is evident that 

he refers to relative movements of adjacent portions of the mass, or what I call tangential movements. 

These necessarily imply forces locally inclined to the direction of the relative motion, for, in general, 

in any solid or in any hyperviscous liquid strained at a finite rate, a tangential displacement implies 

a force containing a tangential component. Heim insists that rupture and cleavage, when due to 

one force, are parallel, a point in which I agree with him. On the other hand, master joints and 

false cleavage occur characteristically at angles of more than 45° to the slaty cleavage.
U. S. GEOLOGICAL SURVEY BULLETIN NO. 241 PL. V

Fig. -;;.

FIG. 23. LOCI OF MAJOR AXES OF ELLIPSOIDS CORRESPONDING TO FIG 6.

FIG. 24. BLOCK OF CERESIN FAULTED BY SCISSION.

FIG. 25. BLOCK OF CERESIN SHOWING CLEAVAGE DUE TO SCISSION.

FIG. 26. BLOCK OF CERESIN SHOWING CLEAVAGE DUE TO DOUBLE SCISSION.

FIG. 27. BLOCK OF CLAY SHOWING CLEAVAGE DUE TO SCISSION.
CONCLUSIONS REACHED. 23

the stratification, as well as why grit beds in slate show cleavage. This, 

it seems to me, has never been successfully done.

The experiments made would indicate that the order of linear compression in slates, supposing their volume inalterable, is about in the 

ratio of 1 to 3 or 4, and that it is accompanied by large slides or tangential strains; but the degree of strain needful to produce cleavage 

probabty depends both on the nature of the material and on the rate of 

straining. .From field observations I. suspect that a compression of 

one-half sometimes suffices.

I infer that a slate belt is equivalent to a distributed fault or a step 

fault with infinitesimal steps, whose total displacement is of the same 

order as the thickness of the slate. The direction of this faulting 

(according to the results reached in my former discussion) is given bjT 

the intersection of the cleavage plane with a plane perpendicular to 

the grain of the slate, and is therefore ordinarily not greatly inclined 

to the horizon. Were the grain vertical it would indicate horizontal 

faulting. The major axis of the strain ellipsoid lies in the plane which 

is perpendicular to the grain of the slate but at a considerable angle 

to the cleavage.

The force to which cleavage is due lay in this same plane at an angle 

to the cleavage which would be zero if the strain were unalloyed scission, barely conceivable in nature, and finite for all other strains. 

There is no simple relation between the direction of the force producing strain and the directions of the axes of the strain ellipsoid for any 

case of rotational strain. In the case of slate the direction of the force 

lay within the acute angle between the direction of greatest linear compression (or the smallest axis of the strain ellipsoid) and the cleavage. 

In the case of symmetrically developed double or multiple schistosity, 

not infrequent in the crystalline schists, rotation was absent and the 

direction of force coincided with that of the smallest axis of the 

strain ellipsoid bisecting the obtuse angle between the cleavages. It 

appears probable, from the experiments, that the angle between the 

slaty cleavage and the local direction of the force to which it is due 

may vary within Avide limits.

For other consequences of the theory confirmed by the experiments 

here described 1 must refer to my former memoir. a

"It is easier to test the experimental results reached in this paper, now that it is written, than to 

examine microscopically even a very small suite of rocks. The following method of verification is 

suggested: Any young student or handy janitor can prepare a set of cylinders of ceresin cast at the 

lowest practicable temperature, with flat ends, and of a diameter equal to the length. Keep ten or 

twelve such cylinders in a thermostat over night at 35° C. Compress them, three at a time, between 

heavy, somewhat warm glass plates in a copying press to one-third of their original length, so slowly 

as not to burst the edges. Put the cakes in ice and salt for an hour or more. Then cut one or more of 

the cakes in two on a plane which includes the axis. Examination of the minute bubbles should 

show whether my diagram of the distribution of strain ellipsoids adequately expresses the facts. If 

it does so, the figures exhibiting the alternative cleavage, according to Sharpe's theory or mine, must 

also be correct. Split the rest of the cakes by striking them edgewise with a hammer, and compare 

the cleavage with the diagrams.
MATHEMATICAL NOTES. 

NOTE ON THE THEOttY OF SLATY CLEAVAGE.

The theory of rupture of rock masses under pressure which I have 

propounded a is that fracture occurs along planes of maximum tangential strain, or, as it is also called, of maximum slide. Cleavage I regard 

as due to a weakening of cohesion, antecedent to rupture, on these 

same planes of maximum slide, the effects being influenced by viscosity, 

although the direction is independent of viscosity. For the full development of this theory the reader must be referred to the former 

memoir, just cited, but some essential features should be included here.

The planes of maximum tangential strain in a homogeneously strained 

mass are readily found. Their position relative to the major axis of 

the strain ellipsoid is independent of the cubical compression to which 

the mass may have been subjected, and of any rotation which the axes 

may have undergone relatively to the elements of mass. These positions are therefore dependent only on the two pure undilational shears, 

at right angles to each other, which determine the relative magnitude 

of the axes of the strain ellipsoid. These two shears may be separately 

considered.

The first problem is, then, to find the planes of maximum tangential 

strain in an irrotational shear ellipsoid. In this ellipsoid the section 

containing the greatest and least axes is an ellipse of the same area as 

the corresponding great circle of the original sphere. If the radius of 

the circle-is taken as unity, the major axis of the ellipse may be called 

a (the "ratio of shear")-and the minor axis will be l/«, so that all 

lines in the ellipse parallel to the major axis of the ellipse exceed their 

original length in the ratio <*, and all lines parallel to the minor axis 

have been reduced in length in the ratio l/<*.

In the circle draw any parallelogram, for instance one with its center at the center of figure, and from the center draw two radii parallel 

to the sides of the parallelogram, making angles £ and ^ with the 

major axis, and meeting the circle at points a? y and xl y^ as shown in 

fig. 28, PI. VI. Then in the ellipse there will be a corresponding 

parallelogram and set of points which may be indicated by the same 

letters primed.

a Finite homogeneous strain, flow, and rupture of rocks: Bull. Geol. Soc. America, vol. 4,1893, pp. 

13-90.

24
U. S. GEOLOGICAL SURVEY BULLETCN NO. 241 PL. VI

Fig. 31.

FIG. 28. DIAGRAM OF SLIDE, BUT NOT MAXIMUM SLIDE. 

FIG. 29. DIAGRAM OF MAXIMUM SLIDE.

FIG. 30. PURE SHEAR COMPOUNDED OF TWO SCISSIONS. 

FIG. 31. SCISSION AND ANGLE OF ROTATION p-v.
BECKER.] MATHEMATICAL NOTES. 25

Now slide will change the angles of the original parallelogram, or 

the inclination of the radii in the circle, and the greater this change 

the greater the slide. If, therefore, the maximum slide is sought, the 

deflection of each of the two radii must be a maximum, and symmetry 

shows that $ and 3^ must be equal. Hence the problem reduces to 

finding the radius in the circle which experiences the greatest change 

of direction through the strain.- This is very easy, for by definition 

(see fig. 29, PI. VI),

tan 3=y/«; tan 3'= J x

so that

tBT1 /S 5n_ tan (3-3)-

which has its maximum value when

tan 3=«; tan 3' = I/ a. 

Thus the maximum value of

Of OC~l

tan(3-3') = 2 

The total change of the angle of the original rhomb is measured by 

twice this quantity, and a a~~ l is known as the "amount of shear."" 

In polar coordinates the equation of the ellipse is

and when or=cot 3', evidently r=l, so that the directions in which 

slide is a maximum for a single shear are those in which the radii 

have preserved their original length. In other words, if the circle 

is superposed upon the ellipse, the intersections of the two curves 

are the extremities of the radii in question. Hence, also, the planes 

of maximum tangential strain in the shear ellipsoid are the circular 

sections.

This subject can also be profitably considered from another point of 

view, that of the stresses involved. If a rod is subjected to a finite 

tensile load Q, I have shown 6 that the resultant stress (force per unit 

area), R, the normal stress, N, and the tangential stress, jT, in one 

component shear, may be written as follows:

Q . acw 8 COS2 '

=5!(a+I) 2 sin2 3'cos2 3'.

. 

"It would be much better to measure shear by the quantity § and to call this the amplitude

of shear. 

6 The finite elastic stress-strain function: Am. Jour. Sci., 3d ser., vol. 44, 1893, pp. 337-356.
26 SCHISTOSITY AND SLATY CLEAVAGE. [BOLL. 241.

Comparing this value of It? with the polar equation of the ellipse, it 

appears that for any value of S'

so that the resultant load or initial stress, or stress into final area, is the 

same on any section. Now, for the circular section, or tan $' = I/ #, 

1V=0, so that the entire load is tangential, and although the tangential 

stress (as is 'Well known) is greatest for 3' =45°, the tangential load, 

Ttr T, is greatest for tan S7 =!/<*, and then becomes 7rT=rtQl3=7rjR. 

Thus, according to the theory here set forth, rupture and cleavage are 

determined by maximum tangential load, not by maximum tangential 

stress.

If a second shear is applied to the mass in a plane at right angles to 

the first, the effect in the plane of the first shear is only to reduce the 

height without altering the breadth. Consequently no further slide 

is produced in the plane of the first shear and no further tendency to 

the impairment of cohesion or to its dissolution exists. On the other 

hand, the angle of the planes of maximum tangential strain to the 

major axis of the ellipse is modified. If the final angle made by these 

planes with the major axis A, is &?, and if B and ^are the other axes 

of the ellipsoid (A> C> B) it is e&sy to see a that in the plane AB

or if the mass is incompressible, so that 'ABC=\,

tan oo=B.

From the point of view of cleavage and rupture, the inner mechanism of a pure shear is important. Suppose the rhomb shown in fig. 

30«, PL VI, to be divided into an infinite number of equal strips, 

each of length equal to a side, and that these be slid over one another 

as the cards of a pack can be slipped. Then the resulting figure may 

be a square, shown at 1). Divide this square anew into strips at right 

angles to the former divisions and shift these new strips as shown 

in fig. 30c, PI. VI. The result of the double process will be a rhomb 

identical with that due to pure shear, shown in the preceding diagram, 

fig. 29, PI. VI.*

Now, not only is shear produced by this mechanism, but it seems 

impossible to devise any other mechanism by means of which it can 

be produced. The significant point is that action is almost confined 

to planes parallel to the sides of the rhomb; elsewhere the only relative movement wjricli occurs is mere approach or separation of molecules on lines joining their centers. It is conceivable that the lengthenaBull. Geol.'Soc. America, vol. 4, 1893, pp. 34 and 22.

&Th'c two component strains are scissions, and it is_ susceptible of easy algebraic proof that if two 

scissions of equal amplitude are superposed in such a manner as to produce pure shear the two scission planes must be at right angles to each other. If the ratio of the resultant shear is cr, the ratio of 

shear in each of the scissions must be ^/a.
BECKER.] MATHEMATICAL NOTES. 27

ing of elements parallel to the major axis should cause rupture 

perpendicular to this direction, or in planes parallel to the plane of 

the mean and minor axis. It is also true that when short cylinders are 

subjected to linear compression, their edges sometimes split by tension 

meridionally or in planes perpendicular to the plane of the two greater 

axes of the strain ellipsoid, and that the blocks sometimes yield along 

planes of maximum tangential strain. But that the approach of molecules alorig the smallest axis, on lines joining their centers should tend 

to weaken their qohesion and produce cleavage perpendicular to the 

smallest axis seems to me most ..improbable from a mechanical point 

of view, and I have found no experimental evidence of such an effect. 

Such an effect, however, is implied in Sharpe's theory of cleavage.

In a pure shear the lines of maximum tangential strain do not coincide throughout the strain with the same material particles. The first 

particles to undergo this strain stood originally at 45° to the line of 

pressure, while those which ultimately underwent maximum tangential 

strain originally lay at angles of less than 45° to the line of pressure 

i. e., tan ~i\la. These lines of strain thus rotate through wedges of 

the strained mass. The axes of the ellipsoid, however, in any so-called 

pure strain coincide with the same sets of particles throughout, and 

maintain an invariable direction.

In rotational strains, on the other hand, the axes of the ellipsoid 

wander, so that successive sets of particles become axial. This rotation of the axes affects also the rotation of the lines of maximum slide. 

The axial rotation adds to the rotation of. one set of lines of maximum 

slide and diminishes the rotation of the other set. Hence, in rotational 

strain one set of sliding planes wanders through the mass faster than 

the other. Consequently, also, on one side of the minor axis a given 

radial layer of particles is subjected to maximum tangential strain for 

a longer time than the corresponding layer on the other side. The 

angle of rotation of a strain is the angle between either axis of the 

strain, ellipsoid and the line which passed through the same set of 

particles before strain began.

The extreme case of rotational strain, and the most important one, 

is scission (fig. 31, PI. VI). In scission the rotation of the axes of the 

strain ellipsoid exactly compensates for the rotation which one set of 

planes of maximum strain would have in an irrotational strain of equal 

amount. Hence, in scission this latter set of planes does not rotate at 

all, or, in other words, the same set of material particles is subject to 

maximum tangential strain from the inception of strain to its conclusion. The other set of planes rotates through the mass just twice 

as quickty as it would if the strain were "pure."

Now, all real matter is viscous, and a solid displays its viscosity by 

yielding gradually to force up to a certain limit. A mass to which force 

is applied for a brief time interval resists deformation not only in virtue
28 SCHISTOSITY AlsTD SLATY CLEAVAGE. [BULL. 241.

of its ; ' rigidity," a but in virtue of its viscosity also. Hence, in the case 

of rotational strains in viscous solids, those layers of particles through, 

which the planes of maximum tangential strain move more slowly will 

experience greater permanent deformation than the layers on. the 

other side of the minor axis. These last, when the difference of'rotation is considerable, will either recover elastically or rupture like a 

brittle body, thus giving rise to master joints and false cleavage.

Hence, according to the theory here propounded, s\&ty cleavage 

will result, in suitable material, from, rotational strains, and will be 

found on the side of the least axis of the strain ellipsoid from which 

rotation takes place. In other words, it will occur at an angle to the 

major axis, the tangent of which is the third root of BZIA C, but at 

only one of the angles so denned. It does not seem probable, from a 

theoretical point of view, that scission loy itself will produce relatively 

perfect or smooth cleavage. If the cleavage due to scission alone had 

a certain roughness, and if the mass were further subjected to a linear 

compression that would reduce it to a fourth of its original thickness, 

this roughness would also be reduced to a fourth, or the cleavage 

would be four times as smooth as in simple scission. Furthermore, 

the conditions under which scission alone is produced must be 

extremely exceptional among rocks. All deformations, however, can 

be resolved into scissions; so that, if it were to be said that cleavage 

is due to strains compounded of scissions, this would merery be equivalent to asserting that cleavage is due to deformation.

NOTE ON INSCRIPTION OF AN ELLIPSE IN A PARALLELOGRAM.

This problem is easily solved algebraically if, for example, the parallelogram is derived from a square by displacements. It is thus 

treated in my former memoir on homogeneous strain. For such discussions as have been offered in this paper it is convenient to use a 

graphical method.

Let the sides of the parallelogram in fig. 32. PL VII, be 2a' and 25', 

the value of a' being greater than that of 5', and let the acute angle 

between the sides be $. Draw through the center of the figure lines 

parallel to the sides, then these lines will be conjugate diameters of the 

inscribed ellipse. By the properties of conjugate diameters these are 

connected with the axes a and 5 by the two relations,

(adb5)2 =a'W2 ±2a' J' sin ^.

a Rigidity is resistance to change of shape when {he force is so gradually applied that viscosity does 

not come into play. Rubber is a rigid body with a low modulus of rigidity. Rigidity in this technical sense is a property common to all solid bodies.
BULLETIN NO. 241 PL. VII

Fig. 33.

FIG. 32. INSCRIPTION OF ELLIPSE IN PARALLELOGRAM.

FIG. 33. APPROXIMATE INSCRIPTION OF ELLIPSE IN CURVILINEAR QUADRILATERAL.
BECKBH.] MATHEMATICAL NOTES. 29

From the center, o., draw a line od perpendicular to a' and of length 

a'. Connect the foot of J' (or the point e) with d. Then in the triangle ode the angle doe is 90° ?/>, and therefore

d£=a'*+l>'z -Za'V sin ^=(a-l}z .

Again draw from d to the top of the shorter conjugate diameter the 

line df. Here the angle fod is 90°+^, and hence

d[f=an+V*+W 1)' sin if> = (a+V)\

With de as radius, describe a circle about d cutting fd at k and prolong /$ to. ^. Then, evidently,

so that the magnitude of the axes is known; To find their position, 

draw through o lines parallel to the chords ek and eg. The line/$ will 

then be divided as in the ordinary construction of the ellipse, founded 

on the theorem that if two fixed points, on a right line, are constrained to move on rectangular axes, the curve generated by any 

other point on the line will be an ellipse whose greatest and least 

diameters lie in the given rectangular axes. The construction 

shows that f is a point on the ellipse. There are two positions of the 

axes found which -are compatible with this condition, but only one 

which is compatible with the further condition that ft shall be tangent 

to the curve. Hence a is parallel to ek and 5 is parallel to eg.

I find that a closely analogous construction was given by Mannheim, 

in 1857." That here presented has certain advantages.

To find the mean radius of the ellipse, draw the line fh tangent to 

the circle. Then

or twice the radius required. Also the angle fdh is the acute angle 

between mean radii of the ellipse. If the length \/aJ is set oft' on a, 

and lines are drawn from this point to the extremities of the minor 

axis, they will be parallel to the mean radii and correspond to the sides 

of the rhomb in fig. 29.

If only the position of the axes in the rhomb is wanted, it can conveniently be found as follows without determining the magnitude of 

the axes. Let 5 be the angle between a and a1 '/ then

COS

To prove this equation, write the equation of the ellipse referred to its 

conjugates as axes in the well-known form w'zlafZ -\-y'zlb'z = l. If a? and

a See Williamscm's Diffl. Calc., 9th ed., p. 374.
30 SCHISTOSITY AND SLATY CLEAVAGE. [BULL. 241.

y are abscissa and ordinate of a point on the ellipse referred to rectangular axes, one of them containing «',

Substitution gives an equation of the ellipse in rectangular coordinates 

equivalent to

a

and it is well known that

which gives the value stated above. This value may be old, but I do 

not recall having seen it. . When it is desirable to draw the strain ellipse characteristic of the 

central point of a curvilinear quadrangle representing a small portion 

of a strained mass, a rational method which cannot be very erroneous 

is as follows: In fig. 33, PL VII, at the central points, m, of two opposite curved sides draw tangents and from their intersection a line, /?, 

bisecting the angle between the tangents. Proceeding in the same 

way with the other two curvilinear sides, find a second bisectrix. 

Next draw through the middle points, m, of the curvilinear sides lines 

parallel to the bisectrices. These parallels form a parallelogram in 

which, as shown above, it is easy to inscribe an ellipse, which, how- 

 ever, may not make an exact contact with the curved sides, though it 

usually approaches .closely to contact.

To examine the legitimacy of this construction, suppose that not 

merely the sides were given but a large number of intermediate 

curves. Then at the center there would be a small parallelogram 

whose sides would have directions intermediate between the tangents 

at the points m. Evidently the simplest hypothesis is that the directions of these sides would, each be the mean of those of the two opposite tangents, and this assumption is made in the construction. Though 

the spacing of the intermediate curves would in general vary, it is perfectly legitimate to assume that for small distances the spacing would 

vary linearly, so that in the central parallelogram points corresponding to m would lie at distances apart which are simply proportional to 

those of these points on the curvilinear quadrangle. No further 

assumption is made in the construction, and I can see no doubt that it 

is sufficiently accurate for any such purpose as that to which it has 

been applied in the foregoing paper.

NOTE ON COMPUTATION OF TAN GO.

Let $, 5, c be the axes of the strain ellipsoid, c being vertical to the 

plane of the diagram, and let r* = abc. If x is the original distance of
BECKER.] MATHEMATICAL NOTES. 31

a point from the axis of the cylinder and x' the final distance of the 

same point from the axis of the compressed cake-,

c_2 it x' _%' 

r 2 7f x x

 > 

x

the area of the ellipse. In preparing fig. 11, PL III, I found that the 

ellipses obtained by construction < agreed very fairly with the last 

formula. The differences seemed due to slight inaccuracy in reproducing the network of curved lines in the dissected cakes, fig. 10, PL III. 

The angle co is given b}^

tan2 co=-2= - > 

r x a

or if A and E are the axes found by construction, and if it is assumed 

that £lA=lla,

tan GO ,./_- =_, 

V x A r

lx_ A_a^

In fig. 11, PL III, the ellipses are plotted from the computed axes, 

the ellipticity and orientation being derived from the construction. 

Only a very minute examination would distinguish this from a diagram 

depending solely on construction.

NOTE ON VOLUME CHANGES IN THE FORMATION OF SLATE.

In some theories of slaty cleavage, cubical compressibility of the 

mass is invoked to explain certain .phenomena; but it seems very 

doubtful to me whether this can p\&y any notable part in the process 

of slate making. In plastic deformation a mass must first be strained 

to the elastic limit, at which there is a cubical compression corresponding to the intensity of the force (or to the stress) needed to produce this 

strain. When the stress is increased above this limit plastic deformation sets in and is attended by no further change of volume. Thus
32 SCHISTOSITY AND SLATY CLEAVAGE. [BULL. 241.

the change of volume is a function of the stress at the elastic limit, 

and the cubical compression is one-third of this stress divided by the 

modulus of compressibility.

Now, slates are in part derived from structureless shales which, at 

any considerable distance beneath the surface, are moist. Their compressibility must be intermediate between that of water and that of 

their mineral components. Other slates, again, are produced from firm 

rocks like granite. The moduluses of compression of shale and granite are not known; that of water is 21X106 grams per square centimeter; that of quartz, 387X106 (Voigt), and that of glass from 

347 XlO6 to 437 XlO6 (Everett). Again, the breaking strain of concrete under compressive loads is from 80,000 grams per square centimeter upward, and that of granite is 1,006,000 (v. Bach). Of course 

the elastic limit is lower than the breaking strain. Now, if shale is as 

strong as an inferior concrete and as compressible as water, the cubical 

compression at the elastic limit will be

' 08X1°6

and if granite is as compressible as quartz, the cubical compression 

When it begins to flow will be

1 -006xi°l ,=.00087.

3X38TX106 "

In each case the cubical compression (which is three times the linear 

compression) turns out nearly one-tenth of 1 per cent, a quantity the 

evidences of which it would be very difficult to trace in the field. It 

seems to me that the change in volume of shales and granites must 

be of this order, and that, for most purposes, they may be regarded as 

incompressible.

Doubtless the nietamorphism and hydration of slate is attended by 

changes in density; but densities of 2.60 to 2.80 appear to include 

some clays, all the clay slates and phyllites of which I have a record, 

and the more typical granites. These densities thus afford no evidence 

that considerable increase in density attends the development of 

cleavage.
INDEX.

Pag& 

Aluminium, experiments with............. 16

Angle of rotation, scission and, diagram

showing.......................... 24

Axes of ellipsoids, major, illustration showing location of.................... 22

Becker, G. F., cited on finite elastic stress.. 25 

cited on theory of rupture in rock

masses............................ '24

theory of cleavage held by............. 11

diagram showing................... 18,20

distinction between Sharpe's theory 

and............................... 12

Bubbles, significance of.................... 19-20

Ceresin, characteristics of and experiments

with.............................. 14-15

cleavage obtained in, by scission....... 20-21

by scission, illustration showing.... 22

by Tyndall's experiment........... 18-19

compression of cake of, illustrations

showing effect of................. 10,18

faulting of, by scission, illustration

showing.......................... 22

strain ellipsoids in....................... 10-17

illustrations showing............... 16,18

Clay, cleavage obtained in, by scission .... 21

cleavage obtained in, by scission, illustration showing .................. 22

compression of ca-ke of, illustration

showing effect of................. 16

distortion in rolling, diagram showing. 10 

experiments with ...................... 15-16

Cleavage, association of rupture with...... '22

diagrams showing...................... 18,20

importance of.......................... 10

obtained by experiment................ 18-21

occurrence of........................... 10

of ceresin by scission, illustration showing ............................... 22

of clay by scission, illustration showing ............................... 22

photographs of......................... 20

theories of.............................. 11-12

Tyndall's experiment for, results of.... 18-19

Cleavage, slaty, definition of............... 9

occurrence of........................... 10

mathematical notes on theory of....... 2-1-28

Cleavage, slip, absence of.................. 19

Cleft strain ellipsoid in ceresin, illustration

showing.......................... 16

Compression, illustrationsshowingeffectsof 16 

Compression, linear, experiments in ....... 13

Cylinders, piercing of, diagram showing... 16 

Daubree, A., cited on slaty cleavage ....... 9

theory of cleavage held by ............. 11

10026 No. 241 04 3

Distortion in rolling clay, diagram showing.' 16 

Ellipse, equation of ........................ 25

inscription of, in a parallelogram, notes

on ................................ 28-30

in parallelogram and curvilinear

quadrilateral, diagram showing.. 28' 

Ellipsoids, major axes of, illustration showing location of............ '........ 22

Ellipsoids, strain, construction of.......... 17-18

in ceresin, discussion of................ 10-17

means of studying...................... 12-13

Ellipsoids.strainand cleft strain, in ceresin,

illustrations showing............. 16,18

Faulting of ceresin by scission, illustration

showing.......................... 22

Fissile rocks, abundance of................. 9-10

Geikie, A., quoted on compression ......... 10

Hawes, G. W., cited on composition of roofing slate.......................... 10

Heat, evolution of, by deformation ........ 22

Heim, A., cited on cleavage withqut rupture. 22 

Lead, compression of cakes of, illustrations

showing effect of................. 16

experiments with ...................... 16

Linear compression, experiments in ....... '13

Mathematical note on theory of slaty cleavage ...............................

on volume changes in the formation of 

slate..............................

Mica, association of, with cleavage ........

Minerals, secondary, development of, along

cleavage planes .................. 22

Parallelogram, inscription of ellipse in, diagram showing.................... 28

inscription of an ellipse in, notes on.... 28-30

Phillips, John, theory of cleavage held by.. 11 

Piercing of cylinders, diagram showing.... 16

Plaster of Paris, experiments with......... 16

Quadrilateral, curvilinear, inscription of

ellipse in ......................... 28

Rigidity, definition of...................... 28

Rogers, H., quoted on cohesive force....... 11

Rolling, study of cleavage by .............. 13

Rolling clay,diagram of distortion in...... 16

Rosenbusch, H., cited onmineralogical composition of shales................. 10

Rotation, angle of, diagram showing scission and.......................... 24

Rotational strains, mathematical notes on. 27 

Rupture,association of cleavage with...... 22

Schistosity, importance of.................. 10

occurrence of........................... 9

Scission, angleof rotation in, diagram showing ............................... 24

33
34 INDEX.

Page.

Scission, cleavage of ceresin by, illustration

showing.

cleavage of clay by, illustration showing

faulting of ceresin by", illustration showing

22

22

22

Scission, double, cleavage of clay by, illustration showing .................. 22

Scission engine, illustration showing ...... 14

use of...................... 1.............. 13-14

Scission experiments, result of............. 29-21

Secondary minerals, development of, along

cleavage planes................... 22

Shales, mica in ............................. 10

Sharpe, Daniel, theory of cleavage held by. 11 

theory of cleavage held by, diagram

showing.......................... 18

distinction between Becker's theory 

and................................ 12

Shear, diagram showing.................... 24

Shear, mathematical notes on.............. 25-26

Slate, mica in............................... 10

volume changes in formation of, notes

on\............................... 31-32

Slaty cleavage, definition of................ 9

mathematical notes on theory of....... 24-28

occurrence of........................... 10

Slide, diagrams showing.................... 24

Slip cleavage, absence of................... 19

Sorby, H. C., cited on mica in clay slates... 10

theory of cleavage held by ............. 11

Strain, rotation, mathematical notes on.... 27

Strain ellipsoids, construction of............ 17-18

in ceresin, discussion of ................ 16-17

illustrations showing............... 16,18

means of studying...................... 12-13

Tyndall, John, theory of cleavage held by. 11 

Tyndall's cleavage experiment, results of.. 18-19 

Volume changes in formation of slate, notes

on ................................ 31-32

0