/ , ·· ·* \ \
1 ** '1 ν ···· / /
±1 /*··. ···*!
0
F ^
r
. Other two axes
G Any one axis η tn - ι τ ι tn η (general) , ι 0 to 1 - 1 to 0
with sum = ^ 1
FIG. 8. T a b l e of elongations and contractions occurring in t y p i c a l pure shearing m o v e m e n t s .
G O N I O M E T R Y O F F L O W A N D R U P T U R E 529 serving the deformation of a reference grid imprinted on the surface. W i t h this arrangement, it is possible t o produce in the central area the complete range of t w o and three-dimensional irrotational pure shears. This is better illustrated b y a consideration of Fig. 8, which lists the elongations and conti actions which occur in the elastic boundary material in a series of typical m o v e m e n t s . T h e table is compiled for a small m a x i m u m strain of
1 arbitrary unit (positive for an expansion, negative for a contraction) in any one direction. T h e strain in the second direction is prescribed and that in the third direction is calculated from the other t w o on the assumption that the b o u n d a r y material is incompressible. T h e a c c o m p a n y i n g diagrams illustrate the deformation that a circle marked on the material in its initial state (dashed line) w o u l d suffer in each of the m o v e m e n t s . In all but F, it can be seen that there are certain lines in the strain geometry which d o not change their length but only their direction during the m o v e m e n t ("lines of zero e l o n g a t i o n " ) and these are shown dotted. Cases A, B, and C, and D, E, and F, respectively, correspond t o the same t w o deformations viewed from three mutually perpendicular directions. E v e n in the intermediate cases G, it is possible t o arrange that the boundary member is situated in any one of the three main planes of the three-dimensional m o v e m e n t . T h e changes from A to C and from D to Ε are, in practice, trivial as they cor-respond merely t o a rotation of Fig. 7 through 90 deg. but they are included for clarity of discussion.
If n o w a thin film of the material t o be examined is spread on the bound-ary member, the m o v e m e n t of the latter in the c o m m o n surface is trans-mitted t o it provided that the t w o adhere throughout the experiment. T h e transmission of the action is governed b y the laws of the equilibrium of forces and of the preservation of continuity. It can be shown that, when the boundary lies in a main plane of the deformation, only stresses normal t o and strains in the boundary plane are transmitted from one material to the other.
b. Rotational Movements
T h e most c o m m o n rotational action is " s i m p l e " or laminar shear. It has been shown in Section I I that the stress-strain geometry of laminar shear is complicated b y the fact that not only d o the main directions of stress and strain rotate in the material as the strain becomes large but they m a y also deviate from the main directions of their time derivatives. T h e discussion is therefore limited t o the case of small strains where the directions of maxi-m u maxi-m contraction and extension and zero elongation maxi-m a y be taken as con-stant since this condition is likely t o favor the formation of clear perturba-tion patterns. W i t h this restricperturba-tion, the laminar shearing m o v e m e n t then becomes identical with the t y p e Β m o v e m e n t of Fig. 7.
Some more complicated types of rotational shear are the subject of a continuing investigation.
3. EX P E R I M E N T A L TE C H N I Q U E
T h e m e t h o d is u n d o u b t e d l y capable of a large number of variations, de-velopments, and applications and it is proposed t o give only a brief a c c o u n t of the examples which have been investigated so far.
Whereas the strains are measured b y the actual m o v e m e n t of a grid printed on the substrate, the stresses can be measured b y the distortion of this grid from the position it would have taken had there been n o test ma-terial applied. T h e stress normal t o the substrate m a y be measured in principle b y transferring this stress t o a fluid.
A form of the general instrument has already been described but because of their experimental simplicity, it is frequently convenient t o use other forms of b o u n d a r y member, each of which can cover a certain limited range of deformations.
T y p e F m o v e m e n t (Fig. 7) can be p r o d u c e d b y inflation and deflation of a balloon of sufficiently large radius. Greater variability is obtained b y using a cylindrical balloon with an additional tension along its long axis.
If a disk, prepared b y stretching rubber sheets in even tension across a circular wire frame, is given a slight extension or compression across a diam-eter, the deformation in the centre is of t y p e B, with the sheet in the xy-plane. If the circle is first deformed into an ellipse and then given a similar m o v e m e n t , the m o v e m e n t s along the x- and 7/-axes will not n o w be equal and opposite and their ratio will depend on the eccentricity of the initial ellipse.
If a long strip of elastic sheet is pulled along its greatest length (x-axis), the resulting deformation near the centre of the strip will correspond fairly closely t o the t y p e D. If the rigid grip holding the strip is orientated along the i/-axis, then in the neighborhood of the grip, the deformation resulting from a pull along the x-axis corresponds t o t y p e A since there can be n o m o v e m e n t in the ?/-direction. T h u s there is a continuous transition from t y p e A t o t y p e D as one proceeds from the grip toward the centre.
T h e ideal requirements of the b o u n d a r y material are
(i) It should not react chemically with the material under test.
(it) There should be n o diffusion into or swelling of one material b y the other.
(Hi) T h e t w o materials should adhere.
(iv) T h e b o u n d a r y material should deform uniformly without perturba-tion throughout the transiperturba-tion region where disturbances are ap-pearing in the test sample.
(υ) W h e n it is required t o measure forces as well as m o v e m e n t s , the resistance t o flow in the b o u n d a r y m e m b e r should be small c o m -pared with that in the material under test.
T h e swelling of natural rubber b y oils and m a n y organic liquids renders
G O N I O M E T R Y O F F L O W A N D R U P T U R E 531 it unsuitable for the testing of m a n y systems. M a n y synthetic rubbers have a better chemical resistance but their elastic extensibility is n o t always adequate. In quantitative tests, the possible production b y the mechanical action of changes in temperature in the rubber has t o be considered and counteracted, if necessary, b y isothermal devices. These temperature changes remain negligible in vibrational experiments with small amplitudes over a w i d e range of frequencies since they are caused b y the v e r y small frictional heat of the rubber but in unidirectional deformations, considerable amounts of free energy are added t o the heat content of the rubber in ex-tension and subtracted in contraction. T h e associated temperature changes are then v e r y noticeable.
In a few instances, other b o u n d a r y materials have been used with success (for example, cellophane film as a substrate for vacuum-deposited alu-m i n u alu-m ) and one can envisage the possibilities of a nualu-mber of others, such as a second, more ductile metal as a substrate for a thin film of a brittle metal.
Ideally, the material under test should b e sandwiched between t w o parallel deformable boundaries t o i m p r o v e the h o m o g e n e i t y of the imposed action in the direction perpendicular t o the sheet. In the preliminary ex-periments the second sheet has been omitted t o facilitate observation. I t is possible that its reintroduction might affect the present results.
4 . PR E L I M I N A R Y RE S U L T S A N D DI S C U S S I O N
T h e m o s t marked feature of the early results is the great clarity of b o t h the direction and the periodicity of the rupture pattern which results from the m u c h i m p r o v e d h o m o g e n e i t y of the imposed mechanical action and the similarities observed in the behavior of substances widely different in their chemical and physical properties. T h i s latter point is well illustrated b y Plate I V . These patterns were obtained b y simple stretching of the m a -terials deposited on rubber sheet (type D). F r o m left t o right, these are
PLATE I V . A c o m p a r i s o n of the rupture patterns p r o d u c e d in three different ma-terials b y a t y p e D a c t i o n .
corn flour powder, a paste of corn flour in decyl alcohol, and aluminum vacuum-deposited on rubber (for which w e are indebted t o Messrs. W . E d w a r d s and C o . , Crawley, E n g l a n d ) . T h e different angle appearing in the last picture m a y be due t o distortion of the rubber b y the metal.
T h e breakup pattern m a y be regarded as consisting of groups of waves with a statistically well-determined orientation, wavelength, and ampli-tude. In some instances, there is only a single group of waves with a w a v e front perpendicular t o the direction of m a x i m u m contraction (e.g., p o l y -styrene film) a n d / o r elongation.
In general, there occur t w o or more intersecting groups of waves with w a v e fronts orientated either perpendicular t o the directions of m a x i m u m elongation or contraction, a n d / o r perpendicular t o the directions of maxi-m u maxi-m shear, a n d / o r parallel t o the lines of zero elongation. T h e appearance of any one, t w o , or more groups, simultaneously or in different experiments with the same material, depends on the material and on the mechanical action imposed and can be varied deliberately. T h e final rupture often occurs in a complex fashion b y the opening up of cracks not only along the troughs of the waves of the original perturbation pattern but also along the troughs of a new group of w a v e s appearing at the instant of rupture.
In some materials, the result depends on the rate of deformation. In a sufficiently slow deformation, silicone bouncing p u t t y is able t o flow fast enough t o maintain continuity and rupture does not occur even at very large strains. A t somewhat faster rates of strain, the w a v e front of dis-turbances extends perpendicular t o the direction in which the major change of length occurs.
T o avoid the complications introduced b y this time-dependent behavior, most of the investigations so far have been made with dry powders for which the rate of deformation seems t o play little part. Plate V illustrates the behavior of corn flour p o w d e r in a reciprocating t y p e Β m o v e m e n t on the flexible disc. Here, as in Plate I V , the disturbances seem t o be oriented parallel t o the lines of zero elongation in the respective m o v e m e n t s .
A l t h o u g h the same perturbation pattern is obtained whether the action is a reciprocating one about the initial position or a unidirectional m o v e -ment, it has been established, particularly in experiments with powders, that m u c h clearer patterns are obtained b y the former technique. In a recipro-cating shear, the pattern appears gradually over a period of time which increases as the amplitude of the m o v e m e n t is reduced. It is possible t o de-v e l o p the pattern slowly with a de-vibrational strain of de-v e r y small amplitude, a strain m a n y times smaller than that required t o p r o d u c e breakup in a single unidirectional action and the analogy between this behavior and the phenomenon of "fatigue" in metals is a striking one.
T h e new technique is particularly suited t o the examination of dilatant
G O N I O M E T R Y O F F L O W A N D R U P T U R E 533
PLATE V . Corn-flour p o w d e r in a r e c i p r o c a t i n g action of t y p e Β
materials. T h u s a highly dilatant corn flour-water paste remains v e r y mobile and deforms freely with the b o u n d a r y u p t o v e r y large amounts of strain if the action is sufficiently slow. A t a sufficiently high rate of strain, the surface of the material goes dry u p o n the imposition of a v e r y small strain. If the action is stopped at this point, v e r y rapid relaxation t o a m o -bile fluid takes place but, if the apparently dry material is given a further small strain, rupture occurs, usually normal t o the direction of m a x i m u m extension if the sample is sufficiently thick and along the lines of zero elongation if the sample is thinly spread. Qualitatively similar results h a v e been obtained with a wide range of dilatant pastes (e.g., metal powders in mineral oils; glass beads in water; corn flour in glycol, glycerol, and decyl alcohol) but in certain of these, relaxation t o a mobile fluid on cessation of the shear does n o t occur immediately as in the corn flour-water system ( t y p e I ) . Sometimes it occurs spontaneously but m u c h more slowly ( t y p e I I ) and this m a y depend only on the increased viscosity of the liquid phase.
In some other instances (type I I I ) , spontaneous relaxation does n o t occur at all, at least within a reasonable time, but the application of a small-amplitude reciprocating shear brings a b o u t a rapid return t o the m o b i l e state. E v e n the relaxation of t y p e I I materials is greatly speeded b y this means. If the amplitude of the strain is increased b e y o n d a certain a m o u n t ,
relaxation does not take place. It has further been observed that, although a shiny liquid surface m a y be reformed b y the relaxation process, healing of the cracks is not complete and a repetition of the shearing action results in the reappearance of the former cracks. A similar difficulty in annealing stress crazing in p o l y m e t h y l methacrylate has been reported b y Russell.46
In materials of sufficient tensile strength, the imposition of the mechani-cal action through rigid grips frequently produces a distribution of stress and strain which is sufficiently h o m o g e n e o u s over a wide enough region in the material for a clear pattern of disturbances t o appear, though the present m e t h o d is superior in that it offers a m u c h wider choice of the mechanical actions t o be applied.
There occur in the literature a number of reports of the formation of perturbation patterns covering a wide range of substances. M o s t of the in-vestigations refer t o single materials or single groups of materials and the generality of these effects and the similarities in the behavior of materials differing greatly in their physical and chemical properties d o n o t appear t o have been c o m m e n t e d upon previously. T h e occurrence of regular flow figures ("Luders' lines") has been observed in metals, marble, and w a x ,47 and in tracing p a p e r .48 T h e occurrence of periodic patterns in p o w d e r films over which an air stream is passing in a vibratory laminar shearing m o v e -m e n t4 9' 50 m a y be another example of the same p h e n o m e n o n though the periodicity in this case has been attributed t o vortex formation in the air flow. Stress-crazing in high polymers has been shown t o be the result of the formation of similar regular p a t t e r n s ,4 6 , 51 '5 2 the cracks in this instance o c -curring perpendicular t o the direction of m a x i m u m tension and n o t at all in the direction of m a x i m u m compression. In physical metallurgy a number of examples are k n o w n of rupture occurring either in the direction of maxi-m u maxi-m tension or along shear planes.
All the materials which have been investigated b y the present m e t h o d are isotropic in their initial state. If the material under test is initially anisotropic, then some directions in the material will be weaker than others and these directional properties must b e superimposed on those of the stress-strain g e o m e t r y . Either set m a y predominate so that the direction in which rupture takes place in these materials m a y depend either on the geometry of the deformation or on the molecular or crystalline structure of the substance. In crystalline materials, it is probable that the latter set of
4 6 E . W . Russell, Nature 165, 91 (1950).
4 7 A . N a d a i , " T h e o r y of F l o w and Fracture of S o l i d s . " M c G r a w - H i l l , N e w Y o r k , 1950.
4 8 A . H . Nissan, private c o m m u n i c a t i o n .
4 9 M . Waller, Nature 166, 961 (1950).
5 0 M . Waller, J. Sei. Instr. 31, 410 (1954).
6 1 C . C . H s a i o and J. A . Sauer, J. Appl. Phys. 21, 1071 (1950).
5 2 B . Maxwell and L . F. R a h m , Ind. Eng. Chem. 41, 1988 (1949).
G O N I O M E T R Y O F F L O W A N D R U P T U R E 535 properties is m o s t important and deformation occurs along certain crystal-lographically important directions.
I n addition t o the o b v i o u s utility of the m e t h o d for determining the mechanical properties of a material at rupture, it is likely t o p r o v e a useful tool in a n u m b e r of other p r o b l e m s . T h e reproduction in coarse-grained systems, such as pastes and p o w d e r s , of m a n y of the features of the b e -havior in deformation of metals should lead t o a better understanding of the latter. T h e paste and p o w d e r systems h a v e the great advantage that the determination of the mechanical properties can here b e supplemented b y microscopic observation of the m o v e m e n t of each unit in the flow process.
T h e study of the changes in mechanical properties which occur when a liquid phase is a d d e d t o a p o w d e r or when the properties of the t w o phases and the interface in a paste are varied should b e fruitful from the v i e w p o i n t of lubrication. E v e n from qualitative experiments, it is easy t o distinguish marked differences in the behavior of pastes c o m p o u n d e d of the same