rheology -principles, measurements, and applications

Also from Exercise 3.4.3 4.6.1 Relaxation After a Step Strain for the Lodge Equation The shear stress is given by eq... 4.6.2 Stress Growth After Start-up of Steady Shearing for the Lod

This book is printed on acid-free paper

Copyright 0 1994 by Wiley-VCH, Inc All rights reserved

Originally published as ISBN 1-56081-579-5

Published simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA

01923, (978) 750-8400, fax (978) 750-4744 Requests to the Publisher for permission should be

addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York,

NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ@IWILEY.COM

Library of Congress Cataloging-in-Publication Data:

Macosko, Christopher W

Rheology : principles, measurements, and applications / by

Christopher W Macosko : with contributions by

Ronald G Larson [et al.]

p cm.-(Advances in interfacial engineering series)

Includes bibliographical references and index

ISBN 0-471-18575-2 (alk paper)

1 Rheology I Larson, RonaldG 11 Title III Series

QC189.5.M33 1993

CIP Printed in the United States of America

20 19 18 17 16 15 14 13

Even the mountains flowed

before the Lord

From the song of Deborah

after her victory over the

Isaac Newton

“Think what God has determined to do to all those who submit themselves to His righteousness and are willing to receive His ggt ”

James C Maxwell June 23, 1864

“Zn the distance tower still higher peaks, which will yield to those who ascend them still wider prospects, and deepen the feeling whose truth is emphasized by every advance in science, that ‘Great are the works of the Lord’ ”

J.J Thomson,

Nature, 81, 257 (1909)

20 years with over 800 attendees Many of the examples, the top- ics, and the comparisons of rheological methods included here were motivated by questions from short course students Video tapes of this course which follows this text closely are available

My consulting work, particularly with Rheometrics, Inc., has pro- vided me the opportunity to evaluate many rheometer designs, test techniques, and data analysis methods, and fortunately my con- tacts have not been shy about sharing some of their most difficult rheological problems I hope that the book’s approach and content have benefited from this combination of academic and industrial applications of rheology

As indicated in the Contents, two of the chapters were writ- ten by my colleagues at the University of Minnesota, Tim Lodge and Matt Tirrell With Skip Scriven, we have taught the Rheolog- ical Measurements short course at Minnesota together for several years Their contributions of these chapters and their encourage- ment and suggestions on the rest of the book have been a great help Ron Larson, a Minnesota alumnus and distinguished member

of the technical staff at ATT Bell Labs, contributed Chapter 4 on

nonlinear viscoelasticity We are fortunate to have this expert con- tribution, a distillation of key ideas from his recent book in this area I collaborated with Jan Mewis of the Katholieke Universiteit Leuven in Belgium on Chapter 10 on suspensions Jan’s expertise and experience in concentrated suspensions is greatly appreciated Robert Secor, now of 3M, prepared Appendix A to Chapter 3, con-

cerned with fitting linear viscoelastic spectra, during his graduate studies here Mahesh Padmanabhan was very helpful in prepara- tion of much of the final version, particularly in writing and editing parts of Chapters 6 and 7 as well as in preparing the index This manuscript has evolved over a number of years, and so many people have read and contributed that it would be impossible

to acknowledge them all My present and past students have been particularly helpful in proofreading and making up examples In

addition, my colleagues Gordon Beavers and Roger Fosdick read early versions of Chapters 1 and 2 carefully and made helpful sug- gestions

A major part of the research and writing of the second sec- tion on rheometry was accomplished while I was a guest of Martin

xvii

Laun in the Polymer Physics Laboratory, Central Research of BASF in Ludwigshafen, West Germany The opportunity to dis- cuss and present this work with Laun and his co-workers greatly benefited the writing Extensive use of their data throughout this book is a small acknowledgment of their large contribution to the field of rheology

A grant from the Center for Interfacial Engineering has been very helpful in preparing the manuscript Julie Murphy supervised this challenging activity and was ably assisted by Bev Hochradel, Yoav Dori, Brynne Macosko, and Sang Le The VCH editorial and production staff, particularly Camille Pecoul, did a fine job I apol- ogize in advance for any errors which we all missed and welcome corrections from careful readers

Chris Macosko August 1993

xViii / ACKNOWLEDGMENTS

PREFACE

Today a number of industrial and academic researchers would like

to use rheology to help solve particular problems They really don’t want to become full-time rheologists, but they need rheolog- ical measurements to help them characterize a new material, ana- lyze a non-Newtonian flow problem, or design a plastic part l hope this book will meet that need A number of sophisticated in- struments are available now for making rheological measurements

My goal is to help readers select the proper type of test for their applications, to interpret the results, and even to determine whether or not rheological measurements can help to solve a par- ticular problem

One of the difficult barriers between much of the rheology literature and those who would at least like to make its acquain- tance, if not embrace it, is the tensor That monster of the double subscript has turned back many a curious seeker of rheological wisdom To avoid tensors, several applied rheology books have been written in only one dimension This can make the barrier seem even higher by avoiding even a glimpse of it Furthermore, the one-dimensional approach precludes presentation of a number

of useful, simplifying concepts

1 have tried to expose the tensor monster as really quite a friendly and useful little man-made invention for transforming vec- tors It greatly simplifies notation and makes the three-dimensional approach to rheology practical I have tried to make the incorpo- ration of tensors as simple and physical as possible Second-order tensors, Cartesian coordinates, and a minimum of tensor manipu- lations are adequate to explain the basic principles of rheology and

to give a number of useful constitutive equations With what is presented in the first four chapters, students will be able to read and use the current rheological literature For curvilinear coordi- nates and detailed development of constitutive equations, several good texts are available and are cited where appropriate

Who should read this book, and how should it be used? For the seasoned rheologist or mechanicist, the table of contents should serve as a helpful guide These investigators may wish to skim over the first section but perhaps will find its discussion of

constitutive relations and material functions with the inclusion of both solids and liquids helpful and concise I have found these four chapters on constitutive relations a very useful introduction to rheology for first- and second-year engineering graduate students

1 have also used portions in a senior course in polymer processing The rubbery solid examples are particularly helpful for later de- velopment of such processes as thermoforming and blow molding There are a number of worked examples which students report are helpful, especially if they attempt to do them before reading the solutions There are additional exercises at the end of each chap- ter Solutions to many of these are found at the end of the text

xv

In Part I of the book we only use the simplest deformations, primarily simple shear and uniaxial elongation, to develop the im- portant constitutive equations In Part I1 the text describes rheo-

meters, which can measure the material functions described in Chapters 1 through 4 How can the assumed kinematics actually

be achieved in the laboratory'? This rheometry material can serve the experienced rheologist as a useful reference to the techniques presently available Each of the major test geometries is described with the working equations, assumptions, corrections, and limita- tions summarized in convenient tables Both shear and extensional rheometers are described Design principles for measuring stress and strain in the various rheometers should prove helpful to the new user as well as to those trying to build or modify instruments The important and growing application of optical methods in rheol- ogy is also described

The reader who is primarily interested in using rheology to

help solve a specific and immediate problem can go directly to a chapter of interest in Part I11 of the book on applications of rheol-

ogy These chapters are fairly self-contained The reader can go

back to the constitutive equation chapters as necessary for more background or to the appropriate rheometer section to learn more about a particular test method These chapters are not complete discussions of the application of rheology to suspensions and poly- meric liquids; indeed an entire book could be, and some cases has been, written on each one However, useful principles and many relevant examples are given in each area

Principal Stresses and Invariants 20

Finite Deformation Tensors 24

2.4 General Viscous Fluid 83

2.2.1 Rate of Deformation Tensor 72

2.3.1 Uniaxial Extension 79

2.4.1 Power Law 84

2.4.2 Cross Model 86

vii

2.4.3 Other Viscous Models 86

2.4.4 The Importance of ZZm 89

2.4.5 Extensional Thickening Models 91

2.5.1 Other Viscoplastic Models 95

4.3 Simple Nonlinear Constitutive Equations I46

4.3.1 Second-Order Fluid I46

4.3.2 Upper-Convected Maxwell Equation 149

4.3.3 Lodge Integral Equation I53

4.4 More Accurate Constitutive Equations I58

4.4.1 Integral Constitutive Equations I58

4.4.2 Maxwell-Type Differential Constitutive

Equations 166

4.5 Summary I70

4.6 Exercises I71

References I72

Part II MEASUREMENTS: RHEOMETRY 175

5.3.2 Shear Strain and Rate I91

5.3.3 Normal Stresses in Couette Flow I95

5.3.4 Rod Climbing I98

5.3.5 End Effects 200

5.3.6 Secondary Flows 202

5.3.7 Shear Heating in Couette Flow 203

5.4 Cone and Plate Rheometer 205

5.4.1 Shear Stress 206

5.4.2 Shear Strain Rate 207

5.4.3 Normal Stresses 208

5.4.4 Inertia and Secondary Flow 209

5.4.5 Edge Effects with Cone and Plate 213

5.7.1 Rotating Cantiliver Rod 227

5.3 Concentric Cylinder Rheometer 188

5.5 Parallel Disks 217

5.6 Drag Flow Indexers 222

5.7 Eccentric Rotating Geometries 226

CONTENTS 1 h

5.7.2 Eccentric Rotating Disks 227

5.7.3 Other Eccentric Geometries 231

6.2.2 Wall Slip, Melt Fracture 244

6.2.3 True Shear Stress 247

6.4 Other Pressure Rheometers 266

6.4.1 Axial Annular Flow 266

6.4.2 Tangential Annular Flow 267

6.4.3 Tilted Open Channel 268

9.2 Review of Optical Phenomena 381

9.2.1 Absorption and Emission

9.4 Flow Birefringence: Principles and Practice 393

9.4.1 The Stress-Optical Relation 393

9.5 Flow Birefringence: Applications 408

9.5.1 Stress Field Visualization 408

9.5.5 Dynamics of Block Copolymer Melts 415

9.5.6 Dynamics of a Binary Blend 415

9.5.7 Birefringence in Transient Flows 416

10.5.1 Monodisperse Hard Spheres 455

10.5.2 Particle Size Distribution 458

10.5.3 Nonspherical Particles 459

10.5.4 Non-Newtonian Media 460

10.5.5 Extensional Flow of Ellipsoids 460 10.6.1 Electrostatic Stabilization 462

10.6.2 Polymeric (Steric) Stabilization 464

10.7.1 Structure in Flocculated Dispersions 465

10.7.2 Static Properties 467

10.7.3 Flow Behavior 468

10.5 Brownian Hard Particles 455

10.6 Stable Colloidal Suspensions 461

10.7 Flocculated Systems 465

xii / CONTENTS

11.2 Polymer Chain Conformation 476

11.3 Zero Shear Viscosity 479

11.5 Concentrated Solutions and Melts 497

Effect of Molecular Weight Distribution 506

= [ 2 2 yl] * [ i] = (3,2, -1) (a vector)

APPENDIX / 515

IT = trT = sum of the diagonal components of T = 3 + 2 + 0 = 5

Thus the state of stress or stress tensor at the test point is

(b) What is the net force on the 1 mm2 surface whose normal is

ii=21+22?

t n = f i * T = ( l - 1 O ) [ O 1 0 0 -2 0 ] = - ( l , O , - 2 ) 1

1 f,, = a,tn = -(l21 - 223) in newtons

Jz

516 / APPENDIX

(c) The normal component off, normal to li is

since ii’ = dx’/Jdx’I

1.10.5 Inverse Deformation Tensors

Substituting

I (da’ F-I) (da’ F-I) - - (da’ F-I) ((F-l)T da’) = fi’.c-’.fi’

- -

P 2 - da’ a da’ v a t 2

1.10.6 Planar Extension of a Mooney-Rivlin Rubber

(a) (b) The boundary deformations will be the same as in Example

1.8.2 Thus, by eq 1.8.8 B will be

TI I = (2CI + 2CZ)(ar2 - a-2)

This result has exactly the same functional dependence as the neo- Hookean model Thus measurements of T I I in planar extension could not differentiate between the two However

T22 = 2C1(1 - a-*) + 2C2(a2 - 1)

which has a dependence on a that differs from the neo-Hookean

1.10.7 Eccentric Rotating Disks

Note that in the literature this geometry&is called the Maxwell or- thogonal rheometer or eccentric rotating disks, ERD (Macosko and

Davis, 1974; Bird, et al., 1987, also see Chapter 5 ) Usually, the

coordinates 22 = y and 23 = E are used

note that this is the same deformation as simple shear of eq 1.4.24 with slightly different notation

(b) Using eq 1.5.2., we can readily evaluate the stresses

The stress components acting on the disks will be T % 3 = t 3

The force components can be calculated by integrating these stresses over the area of the disk

(a) From a right triangle formed with the bubble radius, R, as the

hypotenuse and the initial sheet radius, Ro, as the base, we obtain R2 = Ri + ( R - h)2 and thus R = ( R i + h 2 / 2 h )

(b) Deformation in Membrane

a1 = a2 near the pole because the bubble is symmetric

aI(Y2a3 = 1 for an incompressible solid

Thus a3 = l/a: or 6/S0 = ( A x , / A x ) ~ We can determine the thickness of the bubble by measuring the stretch near the pole (c) Stresses in the Membrane Applying the neo-Hookean model

TI I and T22 can be treated as surface tensions where I' is the stress in the membrane times unit thickness r = TI I 6 Using the membrane balance equation, eq 1.8.5

since R I = R2 = R for a sphere Substituting gives

1.10.9 Film Tenter

(a) Equate the volumetric flow rate at the entrance and exit

V i n A i n = U o u t A o u t (1 m/s)(0.5m)(150 x 10-6m) = (3m/s)(l m)h

h = 2 5 ~ m (b) Find the stress on the last pair of clamps

The extensions are fixed by the tenter

0 0

B i j = [! 0 ]

0 0 a;

For the neo-Hookean solid

Substituting for B gives

but T33 = 0, no external forces acting in 23 direction, perpendicular

to the film Therefore

Thus the force exerted per unit area in the last pair of clamps is

(c) Assume that the torque needed to turn the roller is due only to

the force required to stretch the film in the 21 direction The force

is the stress component tl times the film cross-sectional area a1

2.8.1 B and D for Steady Extension

An extensional flow is steady if the instantaneous rate of change of length per unit length is constant

1 dl

= k = constant

1 dt

APPENDIX / 521

For a general extension

Therefore, for a general steady extensional flow

The rate of deformation tensor is just the first time derivative

while for an incompressible material (conservation of volume) it gives (recall eq 1.4.6)

522 / APPENDIX

Thus

and for steady uniaxial

Now we can solve for the invariants

I I I B = 1 (for all incompressible materials)

For the rate of deformation we can take the time derivatives

of Bij or reason directly Again by symmetry €2 = €3 and for an incompressible material 120 = tr 2D = 0 Thus

€ 1 + €2 + €3 = 0

which gives

€ 1 = -2€7

APPENDIX I 5 2 3

Thus

and the invariants are

(b) Steady Equal Biaxial Extension This is the reverse of uniaxial

extension a b = a: and a2 = l/ab

Thus

and for steady equal biaxial

The first invariant is

(2.8.9)

1 1 1 2 D = 21:

We note that although equal biaxial extension is just the reverse of uniaxial, the invariants of B are different Therefore we would ex-

pect material functions measured in each deformation to be different

in general Another common approach to equibiaxial extension is

to let ( r b = cry2 and = 241, basing ~e~lgth change on the sides rather than the thickness of the samples

(c) Steady Planar Extension In this case, as we saw in Example

1.8.1, a2 = 1 Then from conservation of volume a1 = 1 /a3, and thus

a; 0 0

B i j = [ : 0 1 l / a o:]

and for steady planar extension

and

2.8.2 Stresses in Steady Extension

(a) Power Law Fluid Apply eq 2.4.12 to the kinematics found

in Exercise 2.8.1 The results are:

Uniaxial extension

Biaxial

APPENDIX I 5 2 5

Planar

(b) Bingham Plastic We can use the constitutive equation to

rewrite the yield stress criteria in terms of B Since r = GB

Bingham Plastic results are summarized in the Table 2.8.1,

2.8.3 Pipe Flow of a Power Law Fluid

You need to increase the pipe diameter Recall eq 2.4.21

Q =

Let Ql = Q2; Ap1 = Am, 2Ll = L2; m , n =constants and solve for R2/ R I

From eq 2.4.22 the ratio of shear rates in the two pipes will be

TABLE 2.8.1 / Bingham Plastic Results

Hookean t , Criteria Newtonian for 11, < 7; I I , for I I , > 7;

Equal biaxial TII - T22 = G ( l / a i -all L G2(a/ + 2/41) 2-11 - T22 = 3q0i + 437,

< 3 q 0 2

526 I APPENDIX

2.8.4 Yield Stress in Tension

Using the results of Example 2.8.2 we obtain

3.4.2 Two-Constant Maxwell Model

The two-constant integral linear viscoelastic model is

APPENDIX / 527

3.2.10 and solving the definite integrals of the exponentials (check

any standard integral table) For example, with eq 3.2.8

Using eq 3.2.10 gives

and that the energy dissipated over a length of time t is

energy dissipated = 4 = s 0 t : D dt For small amplitude sinusoidal oscillations, this expression be- comes

According to eq 3.3.15, y = yo sin w t , so i, = wyocos w t

Then from eq 3.3.17, t = t h sin wt + ~ C O S w t

Also from Exercise 3.4.3

4.6.1 Relaxation After a Step Strain for the Lodge Equation

The shear stress is given by eq 4.3.19, and y ( r , t’) is given for a

step shear in eq 4.3.20 From these two equations we find

The portion of the integral from zero to t is zero because y (t , t’) = 0 when t’ > 0

APPENDIX / 531

To obtain the first normal stress difference, N I = TI I - 522,

from eq 4.3.18, we must obtain the components B I I ( r , t') and

B22(t, t ' ) for the strain tensor B We find from eq 1.4.24 that

and therefore

B I I (t, r ' ) - B d t , r ' ) = y 2 0 , r' )

As before, y ( t , t ' ) is given by eq 4.3.20 Carrying out the

same manipulations as we did for the shear stress, therefore, yields

The ratio NI / q 2 is then

- Yo

N1

- -

TI 2

which is the Lodge-Meissner relationship given by eq 4.2.8

4.6.2 Stress Growth After Start-up of Steady Shearing for the Lodge Equation

For steady shearing that began at time zero, the history of the strain tensor is given by eq 4.3.21 Therefore, according to eq 4.3.19 the

shear stress is

532 / APPENDIX

Most of the terms above cancel out, leaving

Some of the experiments

Hooke used to establish his

law of extension Note the

marks (0, p, q, r, etc.) he

used to indicate that displace-

ment goes linearly with force

From Hooke, 1678

ELASTICSOLID I 5

He was certainly on the right track to the constitutive equation for the ideal elastic solid, but if he used a different length wire or

a different diameter of the same material, he found a new constant

of proportionality Thus his constant was not uniquely a material property but also depended on the particular geometry of the sam- ple To find the true material constant-the elastic modulus-of his wires, Hooke needed to develop the concepts of stress, force per unit area, and strain Stress and strain are key concepts for rheology and are the main subjects of this chapter

If crosslinked rubber had been available in 1678, Hooke might well have also tried rubber bands in his experiments If

so he would have drawn different conclusions Figure 1.1.2 shows results for a rubber sample tested in tension and in compression

We see that for small deformations near zero the stress is linear with deformation, but at large deformation the stress is larger than

is predicted by Hooke's law A relation that fits the data reasonably

tio for a rubber sample (b)

Schematic diagram of the de-

formation, Data from Treloar

(1975) on sulfur-vulcanized

natural rubber Solid line is

Pa

( 1.1.3) The extension ratio (Y is defined as the length of the deformed

sample divided by the length of the undeformed one:

(1.1.4)

L L'

Figure 1.13

(a) Shear and normal stresses

versus shear strain for a sili-

cone rubber sample subject to

simple shear shown schemati-

cally in (b) The open points

indicate the normal stress

difference T I I - T22 neces-

sary to keep the block at con-

stant thickness x2, while the

solid points are for the shear

stress Notice that the nor-

mal stress stays positive when

the shear changes sign Data

are for torsion of a cylinder

(DeGroot, 1990; see also

Example 1.7.1)

Figure 1.1.3 shows the results of a different kind of experi- ment on a similar rubber sample Here the sample is sheared be- tween two parallel plates maintained at the same separation x2 We

see that the shear stress is linear with the strain over quite a wide

range; however, additional stress components, normal stresses T I I

and T22, act on the block at large strain In the introduction to this part of the text, we saw that elastic liquids can also generate nor- mal stresses (Figure 1.3) In rubber, the normal stress difference depends on the shear strain squared

T I I - T22 = G y 2 (1.1.5) where the shear strain is defined as displacement of the top surface

of the block over its thickness

in the next several sections of this chapter, calling on a few ideas from vector algebra, mainly the vector summation and the dot or scalar products For a good review of vector algebra Bird et al (1987a, Appendix A), Malvern (1969) or Spiegel (1968) is help- ful In the following sections we develop the idea of a tensor and some basic notions of continuum mechanics It is a very simple

y=0.4

y = 0 T22

- 0.4 - 0.2 0.0 0.2 0.4

y = -0.4

ELASTICSOLID I 7

development, yet adequate for the rest of this text and for start- ing to read other rheological literature More detailed studies of continuum mechanics can be found in the references above and in books by Astarita and Marmcci (1974), Billington and Tate (198 I), Chadwick (1976), and Lodge (1964, 1974)

To help us see how both shear and normal stresses can act in a material, consider the body shown in Figure 1.2.la Let us cut through a point P in the body with a plane We identify the direction

of a plane by the vector acting normal to it, in this case the unit vector

fi If there are forces acting on the body, a force component f,, will act on the cutting plane at point P In general f,, and fi will have

different directions If we divide the force by a small area d a of the

cut surface around point P, then we have the stress or traction vector

tn per unit area acting on the surface at point P Figure 1.2.1 b shows

a cut that leacjs to a normal stress, while Figure 1.2 l c shows another that gives a shear stress tm Note that Figure 1.2.1 shows two stress vectors of the same magnitude acting in opposite directions This is required by Newton’s law of motion to keep the body at rest Both vectors are manifestations of the same stress component In the discussion that follows we usually show the positive vector only

As we have seen in Figures 1.1.2 and 1.1.3, materials may respond differently in shear and tension, so it is useful to break the stress vector tn into components that act normal (tensile) to the plane fi and those that act tangent or shear to the plane If we pick

a Cartesian coordinate system with one direction fi, the other two directions m and 6 will lie in the plane Thus, t , is the vector sum

of three stress components

So what we have now is a logical notation for describing the normal and shear stresses acting on any surface But will it

be necessary to pass an infinite number of planes through P to

*Many students with engineering or physics backgrounds are already familiar with the stress tensor They may skip ahead to the next section The key concepts in this section are understanding ( 1 ) that tensors can operate on vectors (eq 1.2.10), ( 2 ) standard index notation (eq 1.2.21), (3) symmetry of the stress tensor (eq 1.2.37), ( 4 ) the concept ofpressure (eq 1.2.44), and ( 5 ) normal stress differences (eq 1.2.45)

8 / RHEOLOGY

Figure 1.2.1

(a) A force acting on a body

(b) A cut through point P

nearly perpendicular to the

direction of force The nor-

mal to the plane of this cut

is ii The stress on this plane

is t,, = flu, where a equals

the area of the cut (c) An-

other cut nearly parallel to

the force direction The equal

and opposite forces acting at

point P are represented by

the single component t,,,

Figure 1.2.2

(a) Three mutually perpen-

dicular planes intersecting at

the point P with their associ-

ated stress vectors (b) Stress

components acting on each of

these planes (c) A plane ii

is cut across the three planes

to form a tetrahedron As in

Figure 1.2.1, t,, is the stress

vector acting on this plane

with area a,, For any plane

6, tIz can be determined from

the components on the three

perpendicular planes

X

characterize the state of stress at this point? No, because in fact, the stresses acting on all the different planes are related The stress

on any plane through P can be determined from a quantity called the

stress tensor The stress tensor is a special mathematical operator that can be used to describe the state of stress at any point in the body

To help visualize the stress tensor, let us set up three mutually perpendicular planes in the body near point P, as shown in Figure 1.2.2a Let the normals to each plane be f, f , and 2 respectively

On each plane there will be a stress vector These planes will form the Cartesian coordinate system f, 9 , i As shown in Figure 1.2.2b, three stress components will act on each of the three perpendicular planes Now, if we cut a plane ii across these three planes, we will form a small tetrahedron around P (Figure 1.2.2~) The stress t,

t

‘ i

2

ELASTICSOLID I 9

on plane ii can be determined by a force balance on the tetrahedron The force on ii is the vector sum of the forces on the other planes

-fn = fx + fy + fz (1.2.2) Because force equals stress times area, the balance becomes

-fn = ant, = axt, +arty + a,tz (1.2.3)

where a, is the area of the triangle MOP as indicated in Figure 1.2.2~ From geometry we know that the area a, can be calculated

by taking the projection of a,, on the f plane The projection is given by the dot or scalar product of the two unit normal vectors to each plane

where n, is the magnitude of the projection of f onto 2 Figure 1.2.2b indicates the three components of each stress These com- ponents with their directions can be substituted into the balance above to give

tn = 6 [jET,x + f f T , y +f%T,,

+ 9fTy.x + f i T y y + i i T y , (1.2.8)

+ f f T , + 2iTzy + %T,,]

which, when we take the dot products, reduces to

tn = f h T , , + nyTyx + n,T,,) + f(n,Txy + nyTyy + n,TZy)

10 / RHEOLOGY