North holland series in applied mathematics and mechanics 21 an introduction to thermomechanics

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 or otherwise, wi

AN INTRODUCTION TO THERMOMECHANICS

Hans ZIEGLER

Swiss Federal Institute of Technology, Zurich

and University of Colorado, Boulder

Second, revised edition

1983

N O R T H - H O L L A N D PUBLISHING C O M P A N Y - A M S T E R D A M · NEW YORK · O X F O R D

All rights reserved 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 or otherwise, without the prior

permission of the Copyright owner

First printing 1977 Second, revised edition 1983

PUBLISHERS:

NORTH-HOLLAND PUBLISHING COMPANY

AMSTERDAM OXFORD NEW YORK

SOLE DISTRIBUTORS FOR THE U.S.A A N D C A N A D A :

ELSEVIER SCIENCE PUBLISHING COMPANY, Inc

52 VANDERBILT A V E N U E NEW YORK, N.Y 10017, U.S.A

Library of Congress Cataloging in Publication Data

Ziegler, Hans, 1910 - An introduction to thermomechanics

{North-Holland series in applied mathematics and mechanics, 21)

PRINTED IN THE N E T H E R L A N D S

o b j e c t of t h e r m o d y n a m i c s , o n t h e o t h e r h a n d , h a s a l w a y s b e e n a finite

v o l u m e , e g , a m o l e , a n d t h e s t a t e w i t h i n t h e b o d y h a s b e e n tacitly

a s s u m e d t o b e t h e s a m e t h r o u g h o u t t h e e n t i r e v o l u m e It is s u r p r i s i n g t h a t this p h i l o s o p h y h a s b e e n m a i n t a i n e d even at t h e a g e o f statistical a n d

q u a n t u m m e c h a n i c s , a l t h o u g h it is clearly i n c o n s i s t e n t w i t h t h e first

f u n d a m e n t a l law in its c o m m o n f o r m : A t least p a r t of t h e h e a t s u p p l y

a p p e a r i n g in this law is d u e t o h e a t flow t h r o u g h t h e s u r f a c e of t h e b o d y

t h e r m o d y n a m i c s as a field t h e o r y in m u c h t h e s a m e w a y as c o n t i n u u m

m e c h a n i c s h a s b e e n t r e a t e d for m o r e t h a n 2 0 0 y e a r s In s u c h a field t h e o r y ,

r e a s o n a b l y fast p r o c e s s e s c a n b e t r e a t e d w i t h t h e s a m e e a s e as slow o n e s ,

vii

a n d r e s t r i c t i o n t o r e v e r s i b l e p r o c e s s e s b e c o m e s u n n e c e s s a r y F i n a l l y , this field t h e o r y is t h e p r o p e r f o r m in w h i c h t h e r m o d y n a m i c s a n d c o n t i n u u m

T h e first is a s t a t e f u n c t i o n , d e p e n d e n t o n t h e free e n e r g y , t h e s e c o n d is

c o n n e c t e d w i t h t h e d i s s i p a t i o n f u n c t i o n I n view of later d e v e l o p m e n t s ( C h a p t e r 14) t h e r o l e o f t h e t w o f u n c t i o n s is e m p h a s i z e d T h e d e f o r m a t i o n

h i s t o r y is r e p r e s e n t e d in t h e s i m p l e s t p o s s i b l e m a n n e r , n a m e l y b y i n t e r n a l

p a r a m e t e r s

C h a p t e r 5 d e a l s w i t h t h e c h a r a c t e r i s t i c p r o p e r t i e s o f v a r i o u s m a t e r i a l s A

c o n t r o v e r s y c a n n o t b e a v o i d e d ; in this r e s p e c t I a s s u m e full r e s p o n s i b i l i t y for t h e final c h a p t e r s

C h a p t e r 14 r e t u r n s t o t h e b a s i s of t h e r m o d y n a m i c s T h e classical t h e o r y ,

r e s t r i c t e d t o r e v e r s i b l e p r o c e s s e s , tacitly e x c l u d e s g y r o s c o p i c f o r c e s W i t h exactly t h e s a m e r i g h t t h e y m a y b e e x c l u d e d in t h e i r r e v e r s i b l e c a s e T h e

g e n e r a l i t y A special w o r d of t h a n k s is d u e t o m y s o n , P r o f e s s o r

H a n s h e i n r i c h Ziegler, for his v a l u a b l e linguistic a s s i s t a n c e in t h e

p r e p a r a t i o n of t h e t e x t I finally e x p r e s s m y g r a t i t u d e t o D r C a r l o S p i n e d i for his h e l p , p a r t i c u l a r l y in p r o o f r e a d i n g , a n d t o t h e D a n i e l J e n n y

F o u n d a t i o n for s u p p o r t in t h e p r e p a r a t i o n of t h e d r a w i n g s

C H A P T E R 1

MATHEMATICAL PRELIMINARIES

I n o r d e r t o d e s c r i b e t h e configuration of a n a r b i t r a r y b o d y , w e n e e d a

reference system, e g , a rigid b o d y o r f r a m e s e r v i n g a s a b a s i s f o r t h e

o b s e r v e r A n y q u a n t i t a t i v e t r e a t m e n t r e q u i r e s a coordinate system fixed t o

I n this s e c t i o n w e will briefly d i s c u s s t h e p r i n c i p a l r u l e s o f t e n s o r a l g e b r a

I n s o m e cases w e will restrict o u r s e l v e s t o t y p i c a l e x a m p l e s w h i c h a r e easily

g e n e r a l i z e d , a n d w e will l e a v e p a r t of t h e p r o o f s t o t h e p r o b l e m s e c t i o n

T h i s is in fact t h e t r a n s f o r m a t i o n (1.16)i for η = 3 A n o t h e r f o r m o f t h e

q u o t i e n t l a w s t a t e s t h a t t h e set t(i j k) d e f i n e s a t e n s o r t if t(i j k)^ is a

T h e y a r e i d e n t i c a l w i t h t h e c o m p o n e n t s o f t h e a n t i m e t r i c p a r t o f t h e t e n s o r

Sj k a n d h e n c e d o n o t d e p e n d o n its s y m m e t r i c p a r t O n a c c o u n t o f (1.30)

a n d ( 1 2 9 ) ! ,

^ijk^k = i^ijk^kpq^pq = \^kij^kpq s pq

= \ (Sipdjq ~ diqdjp)Spq = ! ( % ~ ty) = % ] · (1.32)

T h u s , t h e r e l a t i o n

4 P r o v e t h a t t h e set t(ij, k) d e f i n e s a t e n s o r tijk if t(i,j, k)ry is a v e c t o r

Xi A s s u m e t h a t t h e b o d y is a g y r o w i t h fixed p o i n t Ο a n d a n g u l a r velocity

ω , , a n d find its a n g u l a r m o m e n t u m Z), a n d its k i n e t i c e n e r g y Γ

real s o l u t i o n μ) satisfying ( 1 4 2 ) T h i s s o l u t i o n d e f i n e s t h e first p r i n c i p a l

axis o f ty L e t u s i n t r o d u c e a n e w c o o r d i n a t e s y s t e m x[ t h e first axis o f

f u n c t i o n is itself a s c a l a r , w e call it a scalar-valued function o f t h e given

t e n s o r s I n a s i m i l a r m a n n e r , w e d e f i n e vector- o r tensor-valued tensor

b u t m a k e it a r u l e t o t r e a t i n d i c e s f o l l o w i n g a c o m m a as if t h e y w e r e t h e first o n e s

T h e c o n c e p t s d e f i n e d a b o v e a r e called differential operators T h e i r i n d e x

theorem of Gauss It is easily g e n e r a l i z e d for r e g u l a r , i e , for piecewise

s m o o t h s u r f a c e s a n d a l s o for n o n - c o n v e x b o d i e s since a n y b o d y o f this t y p e

called Green's first identity E x c h a n g i n g t h e r o l e s o f φ a n d ψ a n d

s u b t r a c t i n g t h e r e s u l t f r o m ( 1 1 0 2 ) , w e o b t a i n Green's second identity

of a s i n g l e - v a l u e d p o t e n t i a l φ satisfying Laplace's equation Δφ = 0 in K,

a n d t h e last c o n d i t i o n yields v * g r a d ^ = d ^ / d v = 0 o n A G r e e n ' s first

2 Verify t h e i d e n t i t i e s

d i v c u r l u i = 0,

c u r l c u r l ν = g r a d div ν - A v

3 T h e i n s t a n t a n e o u s v e l o c i t y field Vi(x k ) o f a rigid b o d y m a y b e w r i t t e n

as Vj = i>,-0) + e ijk a>jX ky w h e r e i>/0) a n d ω ] a r e c o n s t a n t v e c t o r s S h o w t h a t t h e

in O

5 R e c o n s i d e r t h e p r o o f c o n t a i n e d in t h e last a l i n e a o f t h i s s e c t i o n W h y

is t h e c o n d i t i o n t h a t V b e s i m p l y c o n n e c t e d e s s e n t i a l ?

a sufficiently l a r g e n u m b e r o f a t o m s W e will see, h o w e v e r , t h a t for t h e

e x p l a n a t i o n o f c e r t a i n p h e n o m e n a (crystal elasticity, t h e r m a l effects, etc.)

t h e m o l e c u l a r s t r u c t u r e h a s t o b e t a k e n , a t least t e m p o r a r i l y , i n t o a c c o u n t

W e will n o t specify a t p r e s e n t w h e t h e r t h e continuum c o n s i d e r e d is a g a s ,

a l i q u i d , o r a solid; in fact, t h e s e t e r m s will n o t b e d e f i n e d u n t i l C h a p t e r 5

W e will a s s u m e , h o w e v e r , t h a t t h e b o d y is d e f o r m a b l e , in c o n t r a s t t o t h e rigid b o d y t r e a t e d in e l e m e n t a r y m e c h a n i c s R e f e r r i n g a c o n t i n u u m t o a

c a r t e s i a n c o o r d i n a t e s y s t e m , w e d i s t i n g u i s h b e t w e e n spatial points, fixed in

t h e r e f e r e n c e s y s t e m , a n d material points o r particles, c o n s i d e r e d t o b e

e l e m e n t s o f t h e c o n t i n u u m a n d t h u s t a k i n g p a r t in its m o t i o n I n a s i m i l a r

m a n n e r w e d i s t i n g u i s h b e t w e e n s p a t i a l a n d m a t e r i a l c u r v e s , s u r f a c e s a n d

v o l u m e s

F o r a n a r b i t r a r y t i m e t t h e s t a t e of m o t i o n of a c o n t i n u u m is d e s c r i b e d b y

a velocity field vk (Xj) It specifies t h e velocities of all m a t e r i a l p o i n t s at t i m e

t a n d will b e a s s u m e d t o b e c o n t i n u o u s a n d d i f f e r e n t i a b l e T h e field lines of

vicinity of Ρ w i t h a n g u l a r velocity w a b o u t P T h i s a n g u l a r velocity,

r a t e plane; if t w o p r i n c i p a l e x t e n s i o n r a t e s v a n i s h , w e call it uniaxial

A velocity field is called plane if all v e l o c i t y v e c t o r s a r e p a r a l l e l t o a given

T h e instantaneous distribution o f t h e t e n s o r t k l n in t h e vicinity o f Ρ is

is called t h e circulation o f t h e flow a r o u n d t h e c l o s e d c u r v e C , d e n o t e d in

41

for every c l o s e d r e g u l a r s u r f a c e T h e v o r t e x lines h a v e b e e n d e f i n e d in

S e c t i o n 2.1 as t h e field lines o f w k(xj, t) A vortex tube is f o r m e d b y t h e

v o r t e x lines p a s s i n g t h r o u g h t h e p o i n t s o f a closed c u r v e A vortex filament

KINETICS

K i n e t i c s d e a l s w i t h t h e forces a c t i n g o n a b o d y a n d w i t h t h e m a n n e r t h e s e forces i n f l u e n c e its m o t i o n W e k n o w f r o m p a r t i c l e m e c h a n i c s t h a t , t o a

c e r t a i n e x t e n t , t h e c o n n e c t i o n b e t w e e n forces a n d m o t i o n d e p e n d s o n t h e

c h o i c e ( o r , t o b e m o r e p r e c i s e , o n t h e s t a t e of m o t i o n ) of t h e r e f e r e n c e

s y s t e m T h i s a m b i g u i t y , h o w e v e r , d i s a p p e a r s if w e restrict o u r s e l v e s t o inertial s y s t e m s as r e f e r e n c e f r a m e s

I n o r d e r t o f o r m u l a t e t h e s e p r i n c i p l e s , let u s recall a few c o n c e p t s u s e d

t h r o u g h o u t m e c h a n i c s (cf [4]) T h e power o r rate of work of a f o r c e is t h e

r e m a i n i n g faces a r e dAj = dAvj W e d e n o t e t h e stress v e c t o r a c t i n g o n dA

b y σ ( ν ) a n d t h e stress v e c t o r s o n t h e o t h e r faces b y - a y I n this w a y , t h e

stress v e c t o r o f a s u r f a c e e l e m e n t t h e e x t e r n a l u n i t v e c t o r o f w h i c h p o i n t s in

t h e p o s i t i v e d i r e c t i o n Xj is aj, a n d its c o m p o n e n t s <7,y a r e t h e normal stresses

45

Fig 3.2 Stresses acting on an infinitesimal tetrahedron

(i=j) a n d t h e shear stresses (/=£/) a c t i n g o n t h e e l e m e n t

T h e stresses give rise t o s e c o n d - o r d e r f o r c e s , w h e r e a s t h e b o d y a n d t h e

Fig 3.3 Stresses acting on an infinitesimal cuboid

ie uk Xj(Qv k \ 0 dV=\e iJk XjQf k dV+^e iJk Xj(a k/ -gv k Vi)vi dA

3 2 T h e state of stress

A c c o r d i n g t o S e c t i o n 3.1 t h e s t a t e o f stress w i t h i n a c o n t i n u u m is

d e s c r i b e d b y t h e field o f t h e stress t e n s o r O n c e t h i s field is k n o w n , t h e stress v e c t o r a c t i n g o n a n y s u r f a c e e l e m e n t in a n a r b i t r a r y p o i n t is g i v e n b y ( 3 6 ) S i n c e t h e stress t e n s o r is s y m m e t r i c , it is s u b j e c t t o t h e r e s u l t s

o b t a i n e d in S e c t i o n 1.3 T h u s , t h e r e exists a t least o n e s y s t e m o f o r t h o g o n a l

principal axes a t e a c h p o i n t P T h e s u r f a c e e l e m e n t s n o r m a l t o t h e s e a x e s

a r e called principal elements T h e y a r e free o f s h e a r s t r e s s e s , a n d t h e

c o r r e s p o n d i n g n o r m a l stresses σ λ , , r e f e r r e d t o a s principal stresses, a r e

r e f e r r e d t o as t h e mean normal stress

A s t a t e of stress is called plane in a given p o i n t Ρ if o n e of t h e p r i n c i p a l

stresses, e g , o m is z e r o T h e s u r f a c e e l e m e n t p e r p e n d i c u l a r t o t h e

c o r r e s p o n d i n g p r i n c i p a l axis x3 is t h e n stress-free A c c o r d i n g t o ( 3 1 8 ) , t h e

stress v e c t o r a c t i n g o n a n a r b i t r a r y s u r f a c e e l e m e n t p a s s i n g t h r o u g h Ρ h a s

t h e c o m p o n e n t s σ λ v {, σπν2, 0, a n d h e n c e is p a r a l l e l t o t h e e l e m e n t w h i c h is

free of stresses If t w o p r i n c i p a l stresses, e g , σ η a n d σΙ Π v a n i s h in P , t h e

s t a t e of stress is called uniaxial H e r e t h e c o m p o n e n t s of t h e stress v e c t o r

57

vicinity o f e q u i l i b r i u m U n d e r t h i s c o n d i t i o n , m o s t p r o c e s s e s a r e r e v e r s i b l e ,

a n d as a c o n s e q u e n c e c e r t a i n t e x t s o n t h e r m o d y n a m i c s c l a i m t h a t every sufficiently s l o w p r o c e s s is r e v e r s i b l e E v e n t h i s s t a t e m e n t is i n c o r r e c t : t h e

b e d e s c r i b e d b y a set o f m u t u a l l y i n d e p e n d e n t kinematical coordinates o r

parameters ak ( £ = 1 , 2 , , n) a n d b y t h e a b s o l u t e temperature, w h i c h will

s y s t e m ; w e call t h e s e q u a n t i t i e s independent state variables A n y f u n c t i o n

of t h e m will b e r e f e r r e d t o as a dependent state variable o r a state function

T h e first fundamental law o f t h e r m o d y n a m i c s s t a t e s t h a t t h e r e exists a

s t a t e f u n c t i o n U(a k, t9), called t h e internal energy, s u c h t h a t

A c c o r d i n g t o t h i s t h e o r e m t h e t o t a l d i f f e r e n t i a l o f t h e i n t e r n a l e n e r g y is t h e

s u m of t h e e l e m e n t a r y w o r k d o n e o n t h e s y s t e m a n d t h e e l e m e n t a r y h e a t