NORTH-HOLLAND SERIES IN APPLIED MATHEMATICS AND MECHANICS EDITORS: E BECKER Institut fur Technische Mechanik Hochschule, Darmstadt B B U D I A N S K Y Division of Applied Harvard Sciences University W.T KOITER Laboratory of Applied University H.A Institute Mechanics of Technology, Delft LAUWERIER of Applied University of Mathematics Amsterdam V O L U M E 21 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 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 P U B L I S H I N G C O M P A N Y - A M S T E R D A M · N E W YORK · O X F O R D © N O R T H - H O L L A N D PUBLISHING COMPANY—1983 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: 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 OXFORD NEW YORK SOLE DISTRIBUTORS FOR T H E U S A A N D C A N A D A : ELSEVIER SCIENCE PUBLISHING C O M P A N Y , 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 {North-Holland series in applied mathematics Bibliography: pp Includes index Thermodynamics Continuum I Title thermomechanics and mechanics, 21) mechanics QC311.ZE 531 76-973 ISBN 0-444-86503-9 P R I N T E D IN T H E N E T H E R L A N D S PREFACE C o n t i n u u m m e c h a n i c s d e a l s w i t h d e f o r m a b l e b o d i e s I n its e a r l y s t a g e s it w a s c o n f i n e d t o a few s p e c i a l m a t e r i a l s a n d t o p a r t i c u l a r s i t u a t i o n s , n a m e l y t o ideal liquids or t o elastic solids u n d e r i s o t h e r m a l or a d i a b a t i c c o n d i t i o n s I n t h e s e s p e c i a l c a s e s it is p o s s i b l e t o s o l v e t h e b a s i c p r o b l e m , i e , t o d e t e r m i n e t h e flow a n d p r e s s u r e distributions or the d e f o r m a t i o n a n d stress fields in p u r e l y m e c h a n i c a l t e r m s T h i s is d u e t o t h e fact t h a t t h e s o l u t i o n c a n b e d e v e l o p e d f r o m a set o f d i f f e r e n t i a l e q u a t i o n s w h i c h d o e s n o t contain the energy balance From the viewpoint of general continuum mechanics, however, p r o b l e m s o f t h i s t y p e a r e s i n g u l a r A n y o n e w o r k i n g in t h i s field k n o w s t h a t s o o n e r o r l a t e r h e g e t s i n v o l v e d in t h e r m o d y n a m i c s T h e r e a s o n f o r t h i s is t h a t in g e n e r a l a c o m p l e t e set o f d i f f e r e n t i a l e q u a t i o n s c o n t a i n s t h e e n e r g y balance Since p a r t of the energy exchange takes place as heat flow, the a p p r o p r i a t e f o r m o f t h e e n e r g y b a l a n c e is t h e first f u n d a m e n t a l l a w o f thermodynamics, and it b e c o m e s clear therefore that it is impossible to separate the mechanical aspect of a p r o b l e m generally from the t h e r m o d y n a m i c processes accompanying the motion T o obtain a solution, the f u n d a m e n t a l laws of b o t h mechanics a n d t h e r m o d y n a m i c s must be applied In gas dynamics and in t h e r m o e l a s t i c i t y this has long been recognized T h i s s i t u a t i o n h a s its c o u n t e r p a r t in t h e r m o d y n a m i c s U n t i l r e c e n t l y t h e i n t e r e s t in t h i s field w a s a l m o s t e x c l u s i v e l y f o c u s e d o n p a r t i c u l a r l y s i m p l e b o d i e s , mainly o n inviscid gases, characterized by certain state variables as, e g , v o l u m e , p r e s s u r e a n d t e m p e r a t u r e I n o t h e r b o d i e s , h o w e v e r , o r if v i s c o s i t y is t o b e t a k e n i n t o a c c o u n t , o n e is c o m p e l l e d t o u s e c o n c e p t s f r o m continuum mechanics, replacing the volume by the strain tensor and the p r e s s u r e b y t h e s t r e s s t e n s o r It m a y e v e n b e n e c e s s a r y t o h a v e r e c o u r s e t o the m o m e n t u m theorems, and to account for the kinetic energy in f o r m u l a t i n g t h e first f u n d a m e n t a l l a w I n s h o r t , t h e r m o d y n a m i c s c a n n o t be separated from continuum mechanics I n v i e w o f t h e s e s t a t e m e n t s it b e c o m e s c l e a r t h a t c o n t i n u u m m e c h a n i c s and thermodynamics are inseparable: a general theory of continuum vi m e c h a n i c s a l w a y s i n c l u d e s t h e r m o d y n a m i c s a n d vice v e r s a T h e e n t i r e field is t r u l y i n t e r d i s c i p l i n a r y a n d r e q u i r e s a u n i f i e d t r e a t m e n t , w h i c h p r o p e r l y b e d e n o t e d a s thermomechanics may S u c h a u n i f i e d t r e a t m e n t is t h e topic of this b o o k In order to a m a l g a m a t e t w o b r a n c h e s of science, o n e needs a c o m m o n l a n g u a g e C o n t i n u u m m e c h a n i c s h a s a l w a y s b e e n a field t h e o r y , e v e n in its r u d i m e n t a r y f o r m s like h y d r a u l i c s o r s t r e n g t h o f m a t e r i a l s T o t r e a t e v e n such a simple p r o b l e m as b e n d i n g of a b e a m , o n e m u s t recognize t h a t the states of strain a n d stress d e p e n d o n position a n d possibly o n t i m e T h e o b j e c t o f 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 the state within the b o d y has been 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 been m a i n t a i n e d even at t h e age of statistical a n d quantum mechanics, although it is c l e a r l y i n c o n s i s t e n t with the first f u n d a m e n t a l l a w in its c o m m o n f o r m : A t least p a r t o f t h e h e a t s u p p l y a p p e a r i n g in t h i s l a w 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 o f t h e b o d y A s long as this process goes o n , t h e t e m p e r a t u r e of t h e elements n e a r t h e surface differs from the o n e of the elements further inside t h e b o d y ; the s t a t e o f t h e b o d y is t h e r e f o r e n o t h o m o g e n e o u s There are t w o ways out of this dilemma T h e h i s t o r i c a l w a y , still d o m i n a t i n g v a s t a r e a s o f t e a c h i n g in t h e r m o d y ­ n a m i c s , c o n s i s t s in t h e r e s t r i c t i o n t o i n f i n i t e l y s l o w p r o c e s s e s I n p l a c e o f actual processes o n e considers sequences of (homogeneous) equilibrium s t a t e s E x c e p t f o r a few s p e c i a l c a s e s , s u c h i d e a l i z e d p r o c e s s e s a r e p r a c t i ­ cally r e v e r s i b l e , a n d t h i s e x p l a i n s w h y in classical t h e r m o d y n a m i c s (or rather thermostatics) the limiting case of reversibility plays such a d o m i ­ n a n t r o l e H o w e v e r , t h e e n g i n e e r e n g a g e d in t h e c o n s t r u c t i o n o f t h e r m o m e c h a n i c a l m a c h i n e r y c a n n o t limit h i m s e l f t o i n f i n i t e l y s l o w p r o c e s s e s a n d hence has never t a k e n this restriction seriously T h e situation strongly re­ s e m b l e s t h e o n e in p r e - N e w t o n i a n m e c h a n i c s w i t h its a t t e m p t s t o d e v e l o p dynamics from purely static concepts T h e m o d e r n w a y o u t o f t h e d i l e m m a is d i f f e r e n t b u t s u r p r i s i n g l y s i m p l e : instead of infinitely slow processes o n e considers infinitesimal elements of t h e b o d y in w h i c h a p r o c e s s t a k e s p l a c e , a d m i t t i n g t h a t t h e s t a t e v a r i a b l e s differ from element to element In other words: one conceives t h e r m o d y n a m i c s a s a field t h e o r y in m u c h t h e s a m e w a y a s 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 f o r m o r e t h a n 0 y e a r s I n 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 a s s l o w o n e s , vii a n d restriction t o reversible processes b e c o m e s unnecessary Finally, 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 m e c h a n i c s a r e easily a m a l g a m a t e d The strong interdependence thermodynamics was generally of continuum recognized about mechanics three and decades ago V a r i o u s s c h o o l s h a v e s i n c e c o n t r i b u t e d t o t h e r m o m e c h a n i c s , e a c h f r o m its p o i n t o f v i e w a n d in its o w n l a n g u a g e o r f o r m a l i s m It is n o t t h e a i m o f t h i s book to report on the various approaches nor to compare them The book is i n t e n d e d a s a n introduction to this fascinating field, b a s e d o n the simplest possible a p p r o a c h E x c e p t f o r a n i n t r o d u c t i o n t o t h e t h e o r y o f c a r t e s i a n t e n s o r s t h e first three chapters are concerned with the mechanical laws governing motion of a continuum They are based on considerations of the mass geometry, o n the principle of virtual power a n d o n a general form of the r e a c t i o n p r i n c i p l e It is well k n o w n t h a t t h e m o s t g e n e r a l a p p r o a c h to c o n t i n u u m m e c h a n i c s m a k e s u s e o f t h e d i s p l a c e m e n t field a n d o f m a t e r i a l , a n d hence curvilinear, c o o r d i n a t e s F o r a beginner, however, this a p p r o a c h presents c o n s i d e r a b l e m a t h e m a t i c a l difficulties t h a t a r e a p t t o o b s c u r e the p h y s i c a l c o n t e n t s S i n c e p h y s i c s d e s e r v e s p r i o r i t y in a n i n t r o d u c t i o n o f t h i s t y p e , a t r e a t m e n t b a s e d o n t h e v e l o c i t y field h a s m a n y a d v a n t a g e s a n d h a s t h e r e f o r e b e e n p r e f e r r e d T h i s k i n d o f a p p r o a c h h a s b e e n p r e s e n t e d in a masterly fashion by Prager in his "Introduction to Mechanics of C o n t i n u a " , a n d s i n c e t h e r e is n o t m u c h p o i n t in m a k i n g c h a n g e s j u s t f o r t h e s a k e o f o r i g i n a l i t y , t h e first t h r e e c h a p t e r s a n d c e r t a i n p o r t i o n s o f t h e subsequent applications are similar to the corresponding parts of P r a g e r ' s book C h a p t e r deals with thermodynamics representation, familiar from textbooks It s t a r t s from in t h i s field, the classical introduces and discusses the concept of (independent a n d d e p e n d e n t ) state variables, a n d s h o w s h o w t h e f u n d a m e n t a l l a w s c a n b e f o r m u l a t e d in t e r m s o f a field t h e o r y A c h a r a c t e r i s t i c p o i n t o f t h e p r e s e n t t r e a t m e n t is t h e f a c t t h a t t h e stress a p p e a r s as t h e s u m of a quasiconservative a n d a dissipative stress 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 with t h e dissipation function In view of later developments ( 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 parameters C h a p t e r deals with the characteristic properties of various materials A viii r o u g h c l a s s i f i c a t i o n o f b o d i e s is p r e s e n t e d , a n d t h e c o n s t i t u t i v e e q u a t i o n s o f s o m e c o n t i n u a a r e d i s c u s s e d T h e g e n e r a l t h e o r e m s e s t a b l i s h e d in t h e preceding chapters, supplemented by the proper constitutive determine the thermomechanical behavior of a given relations, body This is i l l u s t r a t e d in C h a p t e r s t h r o u g h 1 , w h i c h d e a l w i t h t h e a p p l i c a t i o n o f t h e theory to various types of c o n t i n u a C h a p t e r s 12 a n d 13 c o n t a i n a s h o r t o u t l i n e o f g e n e r a l t e n s o r s a n d t h e i r a p p l i c a t i o n in t h e s t u d y o f l a r g e d i s p l a c e m e n t s T h e r e p r e s e n t a t i o n f o l l o w s t h e lines o f G r e e n a n d Z e r n a in t h e i r e x c e l l e n t b o o k o n "Theoretical E l a s t i c i t y \ T h e i n c l u s i o n o f t h i s m a t e r i a l m a k e s it p o s s i b l e , in p a r t i c u l a r , , t o p o i n t o u t (a) t h e i m p o r t a n c e o f a p r o p e r c h o i c e o f t h e s t r a i n m e a s u r e a n d o f t h e c o r r e s p o n d i n g s t r e s s , a n d (b) t h e d i f f e r e n c e b e t w e e n c o v a r i a n t a n d contravariant c o m p o n e n t s of a tensor, essential for t h e p r o o f of the o r t h o g o n a l i t y c o n d i t i o n in C h a p t e r 14 U p t o a n d i n c l u d i n g C h a p t e r 13 t h e s u b j e c t m a t t e r , in s p i t e o f a p e r s o n a l t i n g e in t h e p r e s e n t a t i o n , remains within confines that appear to be generally accepted by n o w T h e remainder of the b o o k transgresses these t r a d i t i o n a l l i m i t s It m a y b e c o n s i d e r e d , t o g e t h e r w i t h C h a p t e r , a s a synopsis of the a u t h o r ' s c o n t r i b u t i o n s to t h e r m o m e c h a n i c s , published f r o m 1957 o n w a r d s , o c c a s i o n a l l y w i t h t h e a s s i s t a n c e o f D r J u r g N a n n i a n d P r o f e s s o r C h r i s t o p h W e h r l i It is c l e a r t h a t in a s y n o p s i s o f t h i s t y p e m a n y p o i n t s w h i c h o n c e s e e m e d e s s e n t i a l b u t h a v e lost t h e i r i m p o r t a n c e c a n b e dropped, and it is e q u a l l y obvious that many thoughts which once a p p e a r e d v a g u e h a v e s i n c e a s s u m e d a m o r e c o n c i s e f o r m I n c i d e n t a l l y , in a field which is still in a state of development a certain amount of 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 t h i s 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 o f t h e r m o d y n a m i c s T h e c l a s s i c a l t h e o r y , restricted to reversible processes, tacitly excludes gyroscopic forces W i t h e x a c t l y 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 o b v i o u s w a y o f d o i n g t h i s is t o a s s u m e t h a t t h e d i s s i p a t i v e s t r e s s e s a r e d e t e r m i n e d b y t h e d i s s i p a t i o n f u n c t i o n a l o n e m u c h in t h e s a m e w a y a s t h e q u a s i c o n s e r v a t i v e f o r c e s d e p e n d o n t h e free e n e r g y F o r c e r t a i n s y s t e m s , t o be called e l e m e n t a r y , the connection between dissipative stresses dissipation function then turns out to have the form of a n condition, and orthogonality a n d it f o l l o w s t h a t t w o s c a l a r f u n c t i o n s , t h e free e n e r g y a n d t h e dissipation function (or the rate of e n t r o p y p r o d u c t i o n ) completely govern a n y kind of process ix C h a p t e r 15 s h o w s t h a t t h e o r t h o g o n a l i t y c o n d i t i o n is e q u i v a l e n t t o a n u m b e r of e x t r e m u m principles, a m o n g t h e m a principle of m a x i m a l rate of e n t r o p y p r o d u c t i o n T h i s last principle suggests a generalization of the o r t h o g o n a l i t y c o n d i t i o n for systems of the so-called c o m p l e x type This g e n e r a l i z a t i o n will b e r e f e r r e d t o a s t h e orthogonality principle, a n d it is e a s y t o see t h a t it r e d u c e s t o O n s a g e r ' s s y m m e t r y r e l a t i o n s in t h e l i n e a r c a s e F i n a l l y , C h a p t e r s 16 t h r o u g h 18 a r e c o n c e r n e d w i t h a p p l i c a t i o n s o f the orthogonality condition a n d the orthogonality principle to various types of continua A s already m e n t i o n e d , I have tried to keep the m a t h e m a t i c a l formalism a s s i m p l e a s p o s s i b l e I a s s u m e , h o w e v e r , t h a t t h e r e a d e r is f a m i l i a r w i t h vector algebra and analysis, with the basic laws of mechanics and t h e r m o d y n a m i c s , w i t h t h e e l e m e n t s o f g e o m e t r y in ^ - d i m e n s i o n a l s p a c e a n d of the theory of functions, a n d with the n o t i o n of convexity p r o v i d e t h e r e a d e r w i t h a m e a n s o f t e s t i n g his g r a s p o f t h e To matter, p r o b l e m s have been a d d e d at the end of each section wherever this was possible In the second edition of this b o o k the thermodynamic aspect of c o n t i n u u m m e c h a n i c s h a s been stressed wherever this seemed desirable; besides, s o m e weak points have been strengthened In C h a p t e r a section d e a l i n g m a i n l y w i t h i n v a r i a n t s h a s b e e n a d d e d , a n d in t h i s c o n t e x t t h e b a s i c i n v a r i a n t s o f s e c o n d - o r d e r t e n s o r s h a v e b e e n r e d e f i n e d C h a p t e r s 11 a n d 18, d e a l i n g w i t h v i s c o e l a s t i c i t y , h a v e b e e n e x t e n d e d t o i n c l u d e t h e r m a l e f f e c t s T h e first o n e a p p e a r s s u p p l e m e n t e d b y a s e c t i o n , t h e s e c o n d o n e h a s b e e n c o m p l e t e l y r e w r i t t e n S e c t i o n 14.4 a p p e a r s in a n e w f o r m , a s d o C h a p t e r 16, o n n o n - N e w t o n i a n l i q u i d s , a n d C h a p t e r 17, o n p l a s t i c i t y I n C h a p t e r 15 a s e c t i o n d e a l i n g w i t h t h e d e r i v a t i o n o f t h e s e c o n d f u n d a m e n t a l law from the orthogonality condition has been a d d e d O n the whole, the terminology has been simplified, particularly in c o n n e c t i o n with the classification of materials (fluids, solids a n d viscoelastic bodies) M a n y m i n o r c h a n g e s h a v e b e e n m a d e , a n d m i s p r i n t s o f t h e first e d i t i o n h a v e b e e n e l i m i n a t e d M o s t o f t h e p r o b l e m s h a v e b e e n r e f o r m u l a t e d in s u c h a w a y that they n o w show the m a i n results I a m greatly indebted to Professors William Prager and W a r n e r T K o i t e r , w h o h a v e b o t h c r i t i c a l l y r e a d t h e m a n u s c r i p t o f t h e first e d i t i o n a n d p r o v i d e d n u m e r o u s suggestions for i m p r o v e m e n t I a m also grateful to Professors R a l p h C Koeller a n d William L W a i n w r i g h t for pointing o u t t h a t s o m e o f t h e a p p l i c a t i o n s in S e c t i o n 15.3 a n d C h a p t e r 16 l a c k e d χ generality A Hansheinrich special word of thanks Ziegler, for his valuable is d u e to my linguistic son, Professor assistance in the p r e p a r a t i o n o f 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 , particularly in p r o o f r e a d i n g , and to the Daniel Jenny 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 o f t h e d r a w i n g s Z u r i c h , J u l y 1982 H a n s Ziegler CHAPTER MATHEMATICAL PRELIMINARIES I n o r d e r t o d e s c r i b e t h e configuration reference system, of a n arbitrary b o d y , we need a e.g., a rigid b o d y o r f r a m e serving as a basis for the 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 t h i s r e f e r e n c e f r a m e O u r first t a s k is t o d e v e l o p t h e m a t h e m a t i c a l t o o l s needed for the description of the m o t i o n or, m o r e generally, of a n y process in which framework the body must in consideration be consistent with takes the fact part that The mathematical the choice of the c o o r d i n a t e s y s t e m is a r b i t r a r y I n c o n s e q u e n c e , o u r s t a r t i n g p o i n t m u s t b e the study of coordinate t r a n s f o r m a t i o n s R e s t r i c t i n g o u r s e l v e s in this c h a p t e r t o 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 s , w e will d e v e l o p t h e c o n c e p t o f t h e cartesian tensor Cartesian t e n s o r s L e t u s r e f e r ( F i g 1.1) t h e t h r e e - d i m e n s i o n a l p h y s i c a l s p a c e t o a g i v e n Fig 1.1 Cartesian coordinate systems 341 + (18.65) A c c o r d i n g t o ( ) t h e m a t e r i a l is c o m p o u n d , a n d t h e o r t h o g o n a l i t y c o n d i t i o n is t o b e a p p l i e d t o t h e v a r i o u s Φ separately W e thus have (Γ) $d') = v (r) dan = άΡ + δα+ ν (18.66) oa 0) where = Φ= ( ) λ'ά (1) c o r r e s p o n d i n g t o l i n e a r e l e m e n t s o n t h e l e f t - h a n d s i d e o f F i g 18.2 a n d a nonlinear spring on the right, lead to the nonlinear differential equation G i J + < t ( )^- ^(D // A i J = A ( ) ( (i) £ + ]^ (ΐ)) ( ΐ)^· έ £ 343 18.3 Hereditary integrals In Section 11.2 t h e r e s p o n s e of linear m o d e l s h a s been described by i n t e g r a l e q u a t i o n s A s i m i l a r d e s c r i p t i o n is p o s s i b l e f o r l i n e a r v i s c o e l a s t i c m a t e r i a l s W e will d i s c u s s it f o r t h e i s o t r o p i c c a s e C o m p a r i n g the differential e q u a t i o n (18.53) with (11.24) a n d (11.35), we n o t i c e t h a t t h e s t r e s s d e v i a t o r is a f u n c t i o n a l o f t h e d e v i a t o r i c s t r a i n a n d t h a t it m u s t h a v e t h e p a r t i c u l a r f o r m α/* ο w h e r e / ' is a r e l a x a t i o n f u n c t i o n d e p e n d e n t o n t h e c o e f f i c i e n t s o f ( ) T h e relaxation function c a n be d e t e r m i n e d , as described in Section 11.2, from the response to a strain pulse For the material corresponding to the m o d e l s o f F i g 18.1 o r , w e h a v e , a c c o r d i n g t o P r o b l e m o f S e c t i o n 11.2, 7'(0 = < < ° > ' + ( ^ -*')exp[- ± t ] w h e r e , as we h a v e seen in P r o b l e m of Section , (18.78) > , 0 and A g l a n c e a t ( ) s h o w s t h a t t h e i s o t r o p i c s t r e s s is a f u n c t i o n a l o f t h e i s o t r o p i c s t r a i n a n d t h e t e m p e r a t u r e T h e first c o n t r i b u t i o n h a s t h e f o r m \J(t-t*) J — dt*, dt* (18.79) w h e r e is a r e l a x a t i o n f u n c t i o n d e p e n d e n t o n t h e c o e f f i c i e n t s p (s) ( = 1, , and # ( ) n) T h e s e c o n d c o n t r i b u t i o n is g i v e n b y J"(t-t*)^-dt*, ο di9* dt* (18.80) w h e r e / " is a n o t h e r r e l a x a t i o n f u n c t i o n , d e p e n d e n t o n t h e c o e f f i c i e n t s p (s) a n d r* W e t h u s h a v e s) < " > , 0, tfp0 a n d r > > / ? M ° > (1 Since the (1 relaxation functions depend on the coefficients in the differential e q u a t i o n s (18.53) a n d (18.54), they are ultimately d e t e r m i n e d b y t h e free e n e r g y a n d t h e d i s s i p a t i o n f u n c t i o n o f t h e m a t e r i a l ; h o w e v e r , t h e i r c o n n e c t i o n w i t h t h e s e g o v e r n i n g f u n c t i o n s is b y n o m e a n s m a n i f e s t T h e i n e q u a l i t i e s t o w h i c h t h e c o e f f i c i e n t s in ( ) a n d ( ) a r e s u b j e c t obviously ensure that the relaxation functions are monotonically decreasing and tending t o w a r d s positive values, as suggested by the model of Fig 18.4 In the case of t h e relaxation function argument of condition / ? ( ) / the exponential function > ; the inequality ^ ( ) / (18.78), e.g., is n e g a t i v e o n >p the exponential function positive, and # ( ) / ( ) ^ account of the the ' renders the coefficient of ) ' > ensures that / ' tends to a positive limit Similar s t a t e m e n t s hold for the relaxation functions (18.82) a n d (18.83) I n c e r t a i n m a t e r i a l s t h e r e s p o n s e , a s far a s d i l a t a t i o n is c o n c e r n e d , is practically elastic H e r e , the internal p a r a m e t e r s zero, and the governing functions (18.46) (r=l,2, and (18.47) are degenerate accordingly With the notations η+1 Σ fr =K a = r) and η+ Σ# r) a = Κα (r) (18.84) (18.51) reduces to kk 3K[e -3a(i9-i9 )], kk (18.85) as w a s t o be expected o n a c c o u n t of (7.77) a n d ( ) W h i l e t h e r e s p o n s e 345 t o d e s t o r t i o n s is still g o v e r n e d b y ( 7 ) , E q ( ) r e p l a c e s t h e i n t e g r a l e q u a t i o n ( ) F i n a l l y , w i t h e -*0, kk incompressible, and o k k a->0, K-+00 the material becomes turns into a reaction T h e t r a n s i t i o n t o a n i s o t r o p i c m a t e r i a l s is s t r a i g h t f o r w a r d H o w e v e r , t h e t r e a t m e n t o f n o n l i n e a r v i s c o e l a s t i c m a t e r i a l s is far m o r e c o m p l i c a t e d [ ] CONCLUSION I n t h e P r e f a c e o f t h i s b o o k it h a s b e e n s t r e s s e d t h a t C h a p t e r s t h r o u g h 13 d e a l w i t h t h e g e n e r a l l y a c c e p t e d p a r t s o f t h e r m o m e c h a n i c s , w h e r e a s t h e last five c h a p t e r s d e v e l o p t h e a u t h o r ' s i d e a s c o n c e r n i n g m a x i m a l r a t e o f entropy production S t a r t i n g p o i n t o f t h i s l a t t e r d e v e l o p m e n t is t h e o b s e r v a t i o n ( S e c t i o n ) t h a t i n classical t h e r m o d y n a m i c s t h e n o t i o n o f g y r o s c o p i c f o r c e s h a s n e v e r occurred; they have been tacitly excluded despite their conservative c h a r a c t e r [43] Since t h e classical t h e o r y h a s been s o very successful in t h e t r e a t m e n t o f r e v e r s i b l e p r o c e s s e s , it s e e m s r e a s o n a b l e t o e x c l u d e g y r o s c o p i c forces also in t h e t r a n s i t i o n t o dissipative systems T h e o b v i o u s w a y t o d o t h i s is t o p o s t u l a t e t h a t t h e d i s s i p a t i v e f o r c e s a r e d e t e r m i n e d b y t h e dissipation function a l o n e F o r elementary systems (Section 14.3), where t h e velocities define a single v e c t o r o r t e n s o r , this p o s t u l a t e leads t o a n orthogonality condition in velocity space: t h e dissipative force is o r t h o g o n a l t o t h e dissipation surface in t h e e n d point of t h e corresponding velocity vector a k It is r e m a r k a b l e t h a t t h e o r t h o g o n a l i t y c o n d i t i o n is e q u i v a l e n t w i t h a p r i n c i p l e o f m a x i m u m r a t e o f e n t r o p y p r o d u c t i o n S*® ( S e c t i o n ) : t h e velocity c o r r e s p o n d i n g t o a given dissipative force maximizes t h e rate of e n t r o p y p r o d u c t i o n s u b j e c t t o t h e s i d e c o n d i t i o n US*® = Afa , w h e r e ϋ is k the temperature This principle evokes the second fundamental law, a n d since t h e latter c a n b e s h o w n (Section 15.3) t o follow f r o m t h e f o r m e r , t h e extension of the principle of maximal rate of entropy production t o m o r e g e n e r a l p r o c e s s e s s u g g e s t s itself T h e o r t h o g o n a l i t y principle t h u s o b t a i n e d (Section 14.4) provides t h e possibility t o treat c o m p l e x processes, consisting of several elementary processes coupled with each other T h e principle represents a n extension of O n s a g e r ' s t h e o r y i n t o t h e n o n l i n e a r field It is clear, however, that these principles are of essentially t h e r m o d y n a m i c n a t u r e T h e y p r e s u p p o s e a n entirely statistical system a n d are n o t valid, e.g., in systems o f t h e t y p e considered in rigid-body m e c h a n i c s T h i s is t h e r e a s o n w h y it is i m p o s s i b l e t o t e s t t h e m b y m e a n s o f 347 m a c r o s c o p i c m e c h a n i c a l m o d e l s T h e y e v e n fail in c a s e s w h e r e m a c r o ­ s c o p i c e l e m e n t s a r e p r e s e n t in a n o t h e r w i s e s t a t i s t i c a l s y s t e m , a n d t h e y a r e i n v a l i d in p a r t i c u l a r in c o m p o u n d s y s t e m s ( S e c t i o n ) , w h e r e s e v e r a l uncoupled elementary processes take place simultaneously Contrary to o u r n o t i o n s a c q u i r e d in p o t e n t i a l t h e o r y , it c a n in f a c t n o t b e e x p e c t e d t h a t t w o entirely separated (and hence independent) processes m a x i m i z e the total rate of e n t r o p y p r o d u c t i o n a l t h o u g h this h a p p e n s for t h e individual processes O n e o f t h e c o n s e q u e n c e s o f t h e l a s t s t a t e m e n t s e e m s o n first g l a n c e paradoxical: a complex process does not approach a c o m p o u n d one when c o u p l i n g is c o n s i d e r e d t o d e c r e a s e b e y o n d a n y l i m i t S i m i l a r p h e n o m e n a , h o w e v e r , h a v e b e e n o b s e r v e d i n o t h e r f i e l d s , e g , in t h e p r o b l e m o f t r a c e effects in s t a b i l i t y [ 4 ] , in P r a n d t P s b o u n d a r y l a y e r t h e o r y ( S e c t i o n ) o r in G i b b s ' p a r a d o x [ ] T h e e x p l a n a t i o n is p r o b a b l y s i m i l a r t o t h e o n e in t h e last e x a m p l e : p r o c e s s e s a r e e i t h e r c o u p l e d o r u n c o u p l e d ; a c o n t i n u o u s transition b e t w e e n t h e t w o d o e s n o t really exist T h e m a i n i d e a s m e n t i o n e d a b o v e a r e e x p o u n d e d in C h a p t e r s 14 a n d T h e y i m p l y t h a t t h e r e s p o n s e o f a n y m a t e r i a l is d e t e r m i n e d by two g o v e r n i n g f u n c t i o n s , t h e free e n e r g y a n d t h e d i s s i p a t i o n f u n c t i o n The r e m a i n i n g t h r e e c h a p t e r s , dealing with a p p l i c a t i o n s , are i n t e n d e d t o test these ideas Far from uncovering any inconsistencies, they confirm and i n t e g r a t e t h e k n o w n r e s u l t s a n d e v e n s i m p l i f y s o m e o f t h e m T o b e s u r e , in v i s c o e l a s t i c i t y , t h e r e still r e m a i n s t h e t a s k t o e s t a b l i s h t h e connection between the constitutive relations and the governing functions I n c i d e n t a l l y , t h e i d e a s u n d e r l y i n g o r t h o g o n a l i t y h a v e b e e n t a c i t l y u s e d in t h e first p a r t o f t h i s b o o k ( u p t o C h a p t e r 13) a n d e v e n in m u c h o f t h e l i t e r a t u r e n o t a t all c o n c e r n e d w i t h o r t h o g o n a l i t y W e have noted, e.g., that the equation (4.42) for the internal forces, where (4.39) is v a l i d o n l y if i n t e r n a l f o r c e s of the gyroscopic type are excluded A c c o r d i n g t o S e c t i o n , t h i s c o n d i t i o n is p r e c i s e l y t h e n u c l e u s o f t h e t h e o r y o f o r t h o g o n a l i t y , a n d it is i n t e r e s t i n g t o n o t e t h a t it is e v e n u s e d b y authors averse to orthogonality 348 A s a n o t h e r e x a m p l e , let u s cite t h e e q u a t i o n i ^ ^ - = a++Arf 2))^ (9.74) ( r e p r e s e n t i n g t h e i s o t r o p i c p a r t o f t h e d i s s i p a t i v e s t r e s s in a n o n - N e w t o n i a n l i q u i d I n S e c t i o n w e a r g u e d s o m e w h a t a m b i g u o u s l y t h a t σ{?), s i n c e its p o w e r is z e r o , m a y b e c o n s i d e r e d a s a q u a s i c o n s e r v a t i v e s t r e s s a n d t h a t t h e r e f o r e t h e f a c t o r o f dy o n t h e r i g h t - h a n d s i d e m u s t b e z e r o T h i s r e s u l t , r e a c h e d b y d e t o u r , c a n b e o b t a i n e d d i r e c t l y a n d u n e q u i v o c a l l y (as in c o n n e c t i o n with (16.3), (16.88), (17.4) a n d (17.7)) f r o m t h e principle of a b s e n t d i s s i p a t i v e f o r c e s , s t a t e d a t t h e e n d o f S e c t i o n 14.3 a n d b a s e d o n t h e notion that the thermodynamics dissipation function is the key to irreversible BIBLIOGRAPHY [1] W Fliigge, Tensor Analysis and Continuum Mechanics (Springer, Berlin, 1972) [2] A.J Spencer, Theory of invariants, in: E C Eringen, ed., Continuum Physics Vol (Academic Press, New York, 1971) p 239 [3] E and F Cosserat, Sur la thoorie de l'olasticito, Ann [4] H Ziegler, Vorlesungen uber Mechanik [5] W Traupel, Die Grundlagen [6] J.W Gibbs, Collected Toulouse 10 (1896) (Birkhauser, Basel, 2nd ed., 1977) der Thermodynamik (Braun, Karlsruhe, 1971) p Works Vol (Yale Univ Press, New Haven, CT, 1948) p 44 [7] H Ziegler, Systems with internal parameters obeying the orthogonality 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Continuum-thermo-mechanics, in: J.F Besseling and A M A van der Heijden, eds., Trends in Solid Mechanics (Delft Univ Press, Delft, 1979) p 39 [42] R.M Christensen, Theory of Viscoelasticity, An Introduction (Academic Press, New York, 1971) [43] H Ziegler, Principles of Structural Stability (Birkhauser, Basel, 2nd ed., 1977) [44] H Ziegler, Trace effects in stability, in: H Leipholz, ed., Instability Systems, IUTAM Symp Herrenalb [45] A Sommerfeld, Thermodynamik of Continuous (Springer, Berlin, 1971) p 96 und Statistik (Deutsch, Thun and Frankfurt, 2nd ed., 1977) p 66 [46] H Ziegler, Chemical reactions and the principle of maximal rate of entropy production, Z Angew Math Phys 34 (1983) in press SUBJECT INDEX acceleration 34 accompanying coordinate system 26 acoustics 134 adiabatic process 59 alternating tensor analytic function antimetric tensor 96 axis of symmetry 114 conservation condition conservation of mass 36 38 constitutive relations 75 continuity equation 39 continuum 25 contraction contravariant basis 219 contravariant components 219 convective change 34 34 barotropic process 87 convective derivative basic invariants Bauschinger effect 15 189 convex surface Couette flow Becker's rule 330 coupled processes 259 Beltrami-Michell equation 103 covariant basis 219 180, 265 163, 294 Bernoulli equation 92 covariant components 219 body force 42 covariant derivative 228 boundary layer equations 155 creep 197 boundary layer theory 155 creep compliance 201 creep phase creep theory of metals 201 cross effect 163 115 bulk modulus bulk viscosity caloric equation of state 81, 103 82 79 crystal classes cartesian coordinate system curl cartesian tensor curvilinear coordinate system Cauchy-Riemann equations 96 characteristic equation 11 characteristic system 11 Christoffel symbol circulation Clausius-Duhem inequality 228 38 73 coefficient of thermal expansion 121 coherent velocities 258 complex plane complex potential 96 96 complex process complex velocity compound process configuration 259 97 259 212 19 217 95 d'Alembert's paradox deformation 28 deformation rate 28 density dependent state variables 34 58 15 deviator differentiable function 96 differential equation of heat conduction differential operators diffusion equation dilatation dilatation rate 80, 123 19 151 31,33 30 352 dilatation ratio 240 gas Dirac function 141 gas constant 131 gas dynamics 137 displacement field dissipation function dissipation rate dissipation surface 32 63, 76 Gauss' theorem 81 20 62 generalized strains 316 254 generalized stresses 316 dissipative force 60 Gibbs' equation dissipative work 60 global yield surface distortion 30 gradient divergence 19 gradual relaxation 144 Green's identities 22 gyroscopic forces 251 gyroscopic systems 252 Doppler effect dual tensor dual vector Duhamel's differential equation dummy index 76 elastic-perfectly-plastic material 191 elastic-plastic material 191 elementary process 258 elliptic differential equation 138 energy theorem 53 entropy 59 entropy flow 70 entropy production 59 entropy supply 59 equilibrium 45 Euler's approach 233 Euler's differential equation 88 extension 33 extension rate 29 extension ratio 239 Hamilton-Cayley equation flow flow rule 199 58, 70 14 189 heat capacity 80, 130 heat conduction 79, 280 heat flow heat supply Heaviside function Helmholtz' vortex theorems homogeneous dissipation function 54, 70 58 196 90 261 homogeneous material 79 Hookean solid 86 Hooke's law 86 hydrostatic pressure hydrostatic stress 80 103 hyperbolic differential equation 138 hyperplane 267 hypersurface 267 ideal gas 130 ideal liquid first fundamental law 19,228 123 hardening material elastic body 62 316 87 impact response 328 95 incoherent velocities 259 178 incompressible liquid 81 fluid 77 incompressible material 39 forces 58 independent state variables 58 Fourier's law 79 index notation free energy 61 inertia force functional fundamental equation of gas dynamics fundamental equation of hydrostatics fundamental laws 65 137 89 58, 70 43 inhomogeneous material 79 instantaneous distribution 34 instant relaxation integrity basis 198 15 353 internal energy 53,58 internal forces Maxwell grid 331 internal parameters inviscid fluid 66 65 81 Maxwell material 210 irreversible process 59 Maxwell's yield condition irrotational flow 40 mean extension rate isentropic process 59 mean normal stress isothermal process 59 metric tensor microsystem isotropic fluid isotropic hardening isotropic material isotropic tensor isotropic tensor function Kelvin chain 82 190 maximal dissipation rate Maxwell model 15 50 222 54 25 Navier's differential equation Navier-Stokes differential equation Navier-Stokes fluid Kelvin model 195 nongyroscopic systems Kelvin solid 210 57 normal stress kinematic hardening 190 ordered surfaces kinematic viscosity 149 orthogonality condition Kirchhoff's uniqueness proof Kronecker symbol 106 Lagrange's approach 234 Lame's constants laminar flow 156 Laplace equation 23 Laplace transforms 30 motion Newtonian fluid Laplace operator 195 173, 178 78 331 kinematical parameters 182,271 86 19 200 orthogonality principle 106 149 82 82 252, 253 44 265 viii, 255 ix, 264 orthotropic material 119 parallel flow 150 particle permutation tensor physical components plane deformation rate plane of symmetry 25 231, 244 31 114 liquid 81 plane stress 50 local change 33 plane velocity field 31 local derivative plastic potential 180 plastified element 182 local velocity of sound 34 137 Mach angle 145 Poisson's differential equation 110 Mach cone Mach number 145 Poisson's ratio potential potential flow power 103 22 40 43 Poiseuille flow macrosystem material change material coordinates material material material material curves derivative points surfaces material volumes 145 54 34 236 25 26, 34 25 153, 174,295 power of dissipation 62 powers of a tensor pressure function 88 principal axis 10 25 principal elements 48 25 principal extension rates 30 354 principal shear stress 50 simple body 75 principal stresses 48 12 simple shear 162 simple shearing stress solid 103 77 source 140 principal values principle of absent dissipative forces 260 24 principle of Archimedes principle of least dissipative force 269 spatial curves principle of least velocity principle of maximal dissipation 270 spatial points 25 spatial surfaces 25 25 rate 182, 271 principle of maximal rate of entropy production principle of virtual power pseudo-scalar pseudo-tensor pure heating purely dissipative system purely viscous body quasiconservative force quasilinear liquid quotient law spatial volumes stagnation point 271 43 25 93 standard test 196 star-shaped surface 265 8,223 state functions 58 state of motion 25 59 state variables 58 stationary flow 39 253 76 steady flow 61 Stieltjes integral Stokes' theorem 162 strain energy strain invariants strain rate 39 203 22 85 239 240 28 strain tensor rate of dilatation 30 stream filament 39 rate of extension 29 stream function 95 rate of shear 29 43 streamlines stream tube 26 reaction principle rate of deformation reference system Reiner-Rivlin liquid 161,290 strength of a source stress function relaxation 198 stress tensor relaxation modulus 201 stress vector response restricted creep reversible process 75 197 strong convexity Reynolds stress rigid-perfectly plastic material 158 supersonic flow 176 27 supporting line supporting plane rotation 59 scalar scalar-valued function 4, 221 second fundamental law 59,71 15 shear rate shear strain 103 29 33 shear stress 45 shear modulus subsonic flow summation convention 33, 238 39 140 109 45, 242 43 265 138 138 185 symbolic notation symmetric tensor 265 symmetry operations 114 temperature 57 tensor field tensor function 15 tensor-valued function 15 18 355 theorem of angular momentum 47 velocities 62 theorem o f Gauss 20 velocity field 26 theorem of linear momentum 44 velocity gradient 27 theorem of Stokes 22 velocity of sound 135 theory of the plastic potential thermal conductivity thermal equation of state thermal expansion coefficient 180 79 79 121 velocity potential 40 virtual power 43 viscous fluid 81 viscosity coefficients 82 thermomechanics vi v Mises flow rule 178 Thomson's vortex theorem 90 v Mises material 310 time 25 v Mises yield condition 172 trajectory 26 vortex filament translation 26 vortex lines 28 vortex strength 41 41 Tresca material 311 Tresca's yield condition 179 vortex surface 41 turbulence 157 vortex tube 41 vorticity 28 uncoupled processes 259 uniaxial deformation rate 31 wave equation 135 uniaxial stress 50 weak convexity 265 uniaxial velocity field 31 Weissenberg effect uniform flow 92 unit tensor unrestricted creep 189 variance 221 vector vector gradient vector-valued function 4,219 19,228 15 167, 295 yield condition 172 yield function 178 yield limit 172 yield locus 183,310 yield stress 171 yield surface 172 Young's modulus 103 ... Ziegler, Hans, 1910 - An introduction to {North- Holland series in applied mathematics Bibliography: pp Includes index Thermodynamics Continuum I Title thermomechanics and mechanics, 21) mechanics. .. PUBLISHING COMPANY—1983 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,.. .AN INTRODUCTION TO THERMOMECHANICS Hans ZIEGLER Swiss Federal Institute of Technology, Zurich and University of Colorado, Boulder Second, revised