Ghostscript wrapper for D:\Digitalizacja\MTS84_t22z1_4_PDF_artyku³y\mts84_t22z1_2.pdf M E C H A N I K A TEORETYCZNA I STOSOWANA 1/2, 22 (1984) SOME CONSIDERATIONS ON THE CONSTITUTIVE LAW IN THERMOPLASTIC1TY Th. L E H M A K N Suhr-Universitdt, Lehrstuhl fur Mechanik 1. Introduction This paper deals with a phenomenological theory of large, non-isothermic deformations of solid bodies which can be considered as classical continua. We suppose that the ther- modynamical state of each material element is uniquely defined by the values of a finite set of state variables even in irreversible processes. Such a phenomenological theory is, of course, restricted to a limited class of materials on the one hand and to processes running not too far from thermodynamical equilibrium on the other hand. A thermo-mechanical process starts in the initial state ££ of the body which is chara- cterized by the initial configuration and by the initial thermodynamical state of each material element. The process is determined by the history of the independent process variables. These are the prescribed thermo-mechanical boundary conditions and the prescribed body forces and energy sources acting inside the body. The course of the process is governed by the material independent field equations (balance equations) and by the constitutive law of the material. We focus our considerations to the constitutive law which governs the local thermo-mechanical process within the thermodynamical state space. Concerning these local thermo-mechanical processes we can distinguish on the first level: 1. strictly reversible processes governed uniquely by thermodynamical state equations, 2. other processes. From the phenomenological point of view we can subdivide the second class into four subclasses: 2.a) plastic deformations characterized by constraint equilibrium states, 2.b) internal processes leading to changes of the internal structure of the material, 2.c) thermal activated processes (without constraint equilibrium states) leading to unlimited creep processes (high temperature creep or long time creep) 2.d) viscous (damping) processes. The internal processes 2.b) may be coupled with processes of the kind 2.a) or 2.c). Ho- wever, they can also occur independently like, for instance, solid phase transformations, recrystallization, or recovery. They may or may not be connected with deformations. Damping processes finally may be correlated to all other kinds of processes including reversible processes as, for instance, in viscoelastic deformations. These considerations suggest a material model as shown in fig. 1. The particular structure T . LE H M AN N and the mutual arrangement of the different elements is determined by the respective constitutive laws. Some particular cases of such constitutive laws will be discussed later. reversible processes - pla st ic deformations - st ruct ura l changes - cre e p processes damping parallel arran ged F ig. 1. M aterial model o The real  thermo- mechanical  process  carries  the body  from  the  initial  state  - Sf into the  actual state $£. All physical  quantities are acting in the respective  current configuration of  the  body.  We  attach to the actual  state if  of the body  an accompanying  fictitious  re- ference  state S? by means  of a  fictitious  reversible  process  which  carries  each  material element from  its actual thermodynamical  state  into an unstressed  state at reference  tem- perature T  (see fig.  2). D uring this fictitious  process the internal variables are kept constant in  order to ensure a unique definition  of reversible energy  [1, 2]. initial  slain fictitious reference  state I incompatible I |   fictitious  reversible  process |   (internal  variables  frozen) actual state F ig..  2.  Thermo- mechanical  process There is no real  or fictitious  process  leading  from  the initial  state Ś ? to the  reference state <£. Therefore  it becomes  unneccessary  to introduce any strain  tensor  defining  the non- reversible  deformations  uniquely.  We need,  however,  a  unique  decomposition of mechanical  work  rate into its reversible  part  and its remaining  parts.  This  means  at the same  time  a  unique  decomposition  of  the  deformation  rate  into  corresponding  parts. Furthermore we require a unique measure for the  reversible  strain  serving  as thermody- namical state  variable. In  the following we shall  at first  discuss  the mechanical  and  thermodynamical  frame for  the formulation  of the  constitutive  law in our sense  of a  phenomenological  theory. Then we shall  compare some  different  constitutive  laws. We shall  also  discuss by which experiments  the  material  parameters  and functionals  entering  the constitutive  laws  can be determined. Finally we shall point to certain coupling effects  occuring in some thermo- mechanical  processes. CONSTITUTIVE LAW IN THERMOPLASTICITY 5 2. Mechanical and thermodynamical frame for the formulation of the constitutive law A thermo-mechanical process in a body can be described with respect to an independent Euklidian space of observation endowed with a space-fixed coordinate system xa. However, we can also relate the process description to a comoving and codeforming body-fixed coordinate system |*. Many authors prefer the first possibility. Concerning the formula- tion of constitutive laws, however, the second way offers many advantages. This cannot be discussed in all details. Only some aspects may be emphasized. The initial position of a material point at time t may be denoted by x­ = x\l). (1) The base vectors and the matric of the space-fixed coordinate system in this position are fa = ga(.X e), #a/3 = gaf>(x e). (2) At time t the actual position of the same material point is x« = x*(xe,t). (3) (3) describes the motion of body in the space of observation. The base vectors and the metric of the space-fixed coordinate system in the actual position are £« - S«(*e), gaP = £«„(*")• (4) The velocity of the material point in the space of observation is va = i"*{iga&- (10) I n these formulas m ean F T ; tran sposed of  F, {  } s : symmetric part, {  }A : antimetric  (skew- symmetric)  part, Ip:  covariant  derivation  in  the  actual  position. F r o m  (8) we  can  derive  different  strain  ten sors. T h is is  well  kn own .  We  refer  to  [3, 4]. The true stress m easured  in the actual configuration,  the so- called  C auchy  stress  ten sor, is f f. (11) With  respect  to  energy  considerations  we  in troduce  th e  weighted  C auch y  stress  ten sor S  =  1-  J £   =  y  <%§*?  -   Ą gagf-  .  '  (12) Then  we  can  write  th e  specific  work  rate  in  th e  form * - - 4 - SM 2 - 4 - Jfdg.   03) Q. S U sing  a  body- fixed  coordin ate  system  a  m aterial  po in t  keeps  its  coordin ate  f'  durin g th e  whole  process.  Th e  base  vectors  an d  t h e  m etric  in  th e  initial  con figuration  of  th e body  are  denoted  by The  corresponding  quantities  in  th e  actual  configuration  are g,  -   gj(F,  t),  g tk   -   g tk (i r ,  t).  (15) The  deformation  of  th e  body  can  be m easured  by  th e  quan tities  [1, 3] ql- Pg*,  .(a- y- Mn-  ( »6> Relating  q' k  to  the  actual  con figuration ,  we  obtain  th e  ten sor F ro m  (17)  we  can  derive  arbitrary  strain  ten sors  by  m ean s  of  isotropic  ten sor  functions [1,  3].  F urth erm ore  th e  deformation  rate  is  expressible  in  the  form 2 (rVM* Ź5 ( ' )  =   - 5-   denotes  th e  m aterial  derivative  with  respect  to  tim e  (£ '  held  fixed)  which  is at different  from  th e  substan tial  tim e  derivative  in  th e  space- fixed  coordin ate  system.  I t corresponds  t o  one  of  th e  Oldroyd- derivations  [4  t o  7,  1].  T h is  m aterial  tim e  derivative is  objective  in con tradiction t o th e substan tial  time derivative,  since  th e rigid body  rotation s T. LEHMANN Q, however, can n ot be uniquely determined since the reference state 3! is a geometrically óó incompatible state. F rom (25) we derive an additive decomposition of the total deform ation rate D =  {F- F~x} s =  [F- F- n  +\ F- F- F-X- F_~l\   . (27) I w co  U uo co w w J.v U sing the polar decomposition the first therm on the right h an d side reads iF- F- H  ={VQQ T -   V- 1 ]  +iVQQ T - F " 1 } W) 00  is \ (0 (~} (;, (r) J, \ «  fa  fa (r)  j  s 4 H ^ ^ F F } + „ {F -   ^ -  F - i - F "1 ^ -   FJ. 2  l(r)  (~)  ( 0  (Pj)  *  1(0  (r)  (r)  (/• )  (r)  (r)J We  see  this expression  depends  on th e arbitrary  (local)  spin of th e  reference  configuration. The  reason is that th e  substantial  time  derivative  entering  this  expression  is n ot  objective. If,  however,  th e  reversible  behaviour  of  th e  material  is  isotropic  th e expression  for  th e work  rate reduces to (29) Q  6  "   l   w)  0)  (O  (o I  6  loo (0  (O  (o  h =  ~ (5- v- ty.v+i- S :\ QI FF~1]  QĄ . Q  ~  (o  w  Q  [fi  1(0  (O  I s W J This  means t h at th e first  term  on the right  h an d  side  an d  therefore  also  th e  second  term become independent of the rotation of th e reference  configuration. I n  this case we  can  define ^ J F F   +  F F ) ,  (30) (?)  l  lOO  00  00  00) as  reversible deformation  rate  an d S  = S •   V' 1   and  V  (31) "  (o  (o as  conjugated  pair  of  stress  and strain  with  respect  to the reversible  deformations. H o- wever, it must  be emphasized  once m ore  th at  this  is  only  possible in the  case  of  isotropy of  the reversible  deformations [4]. We  avoid  these  restrictions  when  we base  our considerations  on a  multiplicative de- composition  of the tensor q  writing qt  =  gimL r t"g sk   -   qi r q r k 0 )  ( r >  (32) with  qr k  =   *grsg st   an d  q\ T  =   gim*g mr . (0  (0 This  leads  again  to an additive  decomposition  of the  total  deformation  rate  according to 4  =   Uq- yw*  =  ̂ ( r 1 )  (q)r  + j ( r 1 ) i ( ? | r  (si, T, bji), (42) S fr) as thermodynamic state function. From (42) we derive thermic state equation: .1 T,b, pij (43) (r) vSl (r) caloric state equation: J = —-ryfr — s(.si> T, b, /?*). (44) Concerning the changes of the specific free enthalpy we obtain from the equations (42) and (40) the two expressions ip = w + w q' \i+r—— ske/—^-sksi — Ts—Ts (45a) (r) (0 Q Q (>•) Q (r) From the equations (43), (44), (45a) and (45b) we finally derive: balance equation for specific reversible work: - ( 4 6 ) balance equation for remaining specific energy supply: balance equation for specific entropy (Gibbs equation): -oir*—^ETPk- A48) C O N ST I T U T I VE LAW I N TH ERMOFJLASTICITY 11 In the balance equation (48) we have to decompose the evolution of the specific entropy in to its reversible part s and its irreversible, dissipative part 's (entropy production)- W W) T ś  = T s+T s.  (49) (r)  w Concerning  this  decomposition  within  the  frame  of  a  phenomenological  theory  we  have to  distinguish  four  different  classes  of  processes: a)  strictly  reversible,  non- dissipative  processes  governed  by  state  equations  and re- presenting a sequence of equilibrium states; b)  irreversible,  dissipative  processes  characterized essentially  by non- equilibrium states; c)  dissipative  processes  appearing as  a sequence of equilibrium states; d)  non- dissipative  processes  appearing  as  a  sequence  of  equilibrium  states  but not- governed  by  state equations. On  the  micro- level  only  th e  classes  a)  and  b)  occur  which  can  be  treated  within  the frame  of  the  classical  theory  of  reversible  or  irreversible  processes,  respectively.  The existence  of  class  c) is  due  to  the fact  that some  irreversible  processes  on the micro- level may  have  very  short  relaxation  times.  Thus  these  dissipative  processes  appear  on  the macro- level  as  a  sequence  of  equilibrium  states  as,  for  instance,  plastic  deformations. The  occurence  of  processes  of  class  d)  is  a  consequence  of  the  fact  that  on  the macro- level  we  are  dealing  in  a  so- called  small  (incomplete) state  space.  Therefore  certain non- dissipative  processes  become  dependent  on  th e  history  of  the  processes  as,  for  instance, anisotropic  hardening  (and  softening)  due  to  inelastic  deformations  and  connected with storing  an d restoring  of  mechanical  energy. F rom  these  facts  it  follows  th at  the  contributions to  the  entropy  production have  to  be defined  within  th e  constitutive  law.  These  contributions  comprehend: 1.  the immediately dissipated  specific  work ft m  ft- ft,  (50) (d)  ( 0  <*) where w  denotes  the  specific  mechanical work  stored  in  changes  of  the internal CO structure  of  the material, 2.  the irreversible  part  of  heat flux 3.  the entropy production T rj due to other dissipative  processes  which may be involved in  internal  processes,  in  energy  supply  by  sources,  and  (as  far  as  not  negligible) in energy fluxes  different  from heat. According to th e second law  of thermodynamics the entropy production cannot become negative.  This means l O . ( 51) The  dissipative  (rate dependent an d rate  independent) processes  can be treated by means of  so- called  dissipative  potentials.  H ow  this  can be  done shall  not be  discussed  here.  We refer  to  [21]. 12 T. LEHMANN Within the thermodynamical frame which is given by the relations (42) and (46) to (51) the constitutive law has to be defined. It consists of a) state function for the specific free enthalpy governing also immediately the reversible processes, b) evolution laws for the non-reversible deformations, c) evolution laws for the internal variables, d) flux laws for energy (heat flux and possibly other fluxes) e) laws of entropy production (w, Trj). (d) As already mentioned we shall disregard energy fluxes different from heat. In this case the evolution, laws for the internal variables degenerate to first order ordinary differential equations in time of the form b = b(si,T,b,ftcJi,t), (52a) }lk = }l(sl,T,b,^, Vsl,T). (52b) Otherwise they represent first order partial differential equations containing also the gradients of the state variables (for more details see [1]). 3. Some different models concerning the constitutive law in thermoplasticity Many different models of constitutive laws are introduced in order to describe the inelastic behaviour of solid bodies, particularly of polycrystalline metals. Some of them are more directed to small deformations occuring in creep and relaxation processes. Others aim at large deformations in general processes. Another group deals with special problems connected with solid phase transformations occuring in quenching processes [22 to 24] or in deformations of so-called memory-alloys [25, 26]. All these models fit the frame of the general material model given in fig. 1. They emphasize special features respectively. In this paper we cannot give a comprehensive survey of all existing theories. Only four of them are selected to demonstrate some different points of view. 3.1 Krempl's and Cernocky's theory of thermo-viscoplasticity. Krempl's and Cernocky's theory of thermo-viscoplasticity [27], [28] relates primilarily to creep and relaxation processes. Therefore it takes into account only small strains. The central constitutive equation reads m[ik-k[ffik, etk, T]aik = (ttk-Gik[ers, T\, (53) with v[£iu> T]'- Poisson's ratio, T ], k[ct lk ,e lk ,T ], G lk [e rs ,T ], v[s ik ,T ], an d a[T \ have to be determ in ed experimentally. This theory which does n ot con tain any yield con dition is presented in functional form avoiding th e in troduction of in tern al variables. Th e disadvantage of such a theory, however, is t h at it can n o t include such ph en om en a like recrystallization or solid phase  trans- for- m ation s,  since  in  t h a t  case nt  an d k  d o  n ot  depen d  any  m ore  on  the  total  strain  uniquely. 3.2  H art's  theory  of  thermo- vlscoplasticity.  H art 's theory  of thermo- viscoplasticity  [29 to  31] in ten ds t o cover th e whole field  of non- elastic deform ations, i.e. as well viscoplastic  processes as  therm ally  activated  creep  an d  relaxation  processes.  The  theory  is  based  on  a  material m odel  whis  is  scetched  in  fig.  3  using  the  custom ary  Theological  diagram  representations. _ii v  i ( a) k ^ A A M A r (of  t i 1 ( b)" - —I d'  =   dk  m"»r  nr F ig.  3.  H art's  material  model Th e  constitutive  law  consists  of  the  following  set  of  equations  (omitting  the  reversible processes) d dl-   Ą - - ffl- tf,  (57) (0   (/) a (f) d M  _  C«)  ^ (a) ff (a) Ą   =  M ai, ( o) G (58) (59) (60) (61) 14 T. LEHMANN b* ~ = dr{o*,a). (62) O («) (a) In these formulas denote a* i i 1 '* = s ^ d = 1/ 4$ : ! (hardness): -  s r , d l k : stress deviator stress invariant ; deformation rate invariant scalar- valued  internal variabk a' k  (stored anelastic  strain): tensor- valued  internal variable G: R: Q- M,  m,  n,  %,f: shear  modulus gas  constant activation  energy constants  1 to determine material  functions  |  experimentally tjc  or  tl  respectively  are  n ot  additional  internal  variables.  They  are  determined  (by  th e relations 4- tŁ +tt  (63) (f)  («) and 4  -   4  m dl+4  =  4 + \   (64a) ( 0  V)  (<*)  (a)  («) or 1 ^ ^  *  ri,  ,  (64b) (/ ) (a) which  result  from  the material  model  (fig.  3). The system of  constitutive  equations  (57) to  (64) is  derived  from  the evaluation  of  experi- mental results  rather than  from  fundamental  thermodynamical considerations. The  scalar- valued  constants  and  material  functions  can  be  determined  from  uniaxial  experiments with  different  loading  histories.  The  integration  of  th e  constitutive  equations  becomes rather  complicated  in  arbitrary  processes.  The  theory  simplifies  when  t h e  viscous  overst- resses  tl  can be assumed  remaining  small. (f) 3.3  Raniedd's theory of  thermoplasticity.  Raniecki's  theory  of  thermoplasticity  [21] is  based on the thermodynamical frame  given in chapter 2. I t is  restricted t o non- isothermic elastic- plastic  deformations.  Changes  of  the  internal  structure  independent  of  plastic  deforma- tions  and  creep:  or  relaxation  processes  are  n ot  considered.  The  frame  of  the  theory, however, allows for  such  extensions. CON STITUTIVE LAW IN THERMOPLASTICITY 15 I n the particular case which is fully treated in [21] the specific free enthalpy is assumed in the form where a, b, §{ represent in tern al variables. The corresponding G ibbs equation (48) reads 1 , ,, 1 ,, dw** . dw**  • dm**  v . Q  co e da 3J (66) The  14  quantities r . l ,« A  - W* J f r >  • ~o~ k̂i  ^*   —  " *    ̂ ł (67) are  considered  as  dissipative  therm odynam ic forces.  The conjugated  rates  (fluxes)  are 4 (0 The  existence  of  a  dissipation  poten tial  $(z(r>) with  the  property (68) (69) is  assumed  defining  the  entropy  production .  F o r  rate  independent  plastic  behaviour 0 m ust  be  a  homogenious  function  of  order  one  with  respect  to  z ( r ) . N ow  the  existence  of  addition al  yield  conditions  in  the  space  of  the  dissipative  ther- modynamic  forces  is  assumed  which  may  also  depend  explicitly  on  the  state  variables. They  are  chosen  in  the  special  form : (70a) (70b) =   l(.tt- Bl)(tf~B  b, T ) =  0, F<«  =   [Bl k BW 2 - YV\ B,  T ) =   0. The  resulting  rate  equation s  (evolution  laws)  read if 0 ) 0  if =  0, <  0, v 4 (0 - x2 dpi a  = if if if =  0, =  0, < o, if if x>  > 0 , A  >  0, 0,  f'2'  <  0, (71a) (71b) >  0, <  0, <  0, =  0, <  0, >  0, (71c) 16 T. LEHMANN if , F < 2 > = 0 A<2> > 0 ( 7 1 d )6 = 0 if F<2> < 0. The quantities A( 1), A( 2) can be calculated from th e consistency con dition s i r ( 1 > = 0 or F < 2 ) = 0, respectively. Th e inelastic behaviour is completely governed by th e two yield con dition s (70a) an d (70b). I t should be emphasized t h at F ( 2 ) does n ot depen d o n the stresses. Therefore it does n o t represent a yield con dition in th e stress space. If th e general form of th e yield con dition s is given as proposed by (70a) an d (70b) then th e constitutive law can be determined im m ediately from experimental investigations of simple cyclic processes. Therefore this th eory m ay prove its ability particularly for such cyclic processes . 3.4 Another proposal for the constitutive law of thermoplasticity. The auth or h as proposed a generalized constitutive law as well for  elastic- plastic  as for  elastic- viscoplastic  beha- viour  [1, 2]. Changes  of th e internal structure  of  th e m aterial as, for  in stan ce, by  recrystalli- zation  or  solid  phase  tran sform ation s  can  be  in cluded.  Lon g  tim e  creep  an d  relaxation processes  represent  a  separate  m echanism which  can  be  added as  indicated  in th e  m aterial model  fig.  1. This  will n o t be  treated here. The theory  is em bedded in  the frame  developed  in ch apter 2.  Th e  specific  free  en th alpy is  assumed  to  be  given in  the  form  (42)  which  can  be  specialized  in  m an y  cases  to y f ó , T , b,  # ,  <4) = ę *(s{, T ) + W **(T , b, B, A) .  with B - plfi  an d A =  a £ a j.  (  } b, (S' k ,  an d x' k   represent  in tern al  variables.  F r o m  (72)  we  derive  by  m ean s of th e  th erm ic state  equation  (43)  the increm ental law  for  th e  reversible  deform ation s 4 = 4(s'k,TJl,T).  (73) (0  00 I n  many  cases  it  can  be  approxim ated  by  a lin ear  hypoelastic  law  [1, 32] 4 = ~\   +  Up  V + xi\ 61.  (74) (O  2G (qK } Concerning  th e balance  equation  for  the  rem ain in g  specific  energy  supply  (47)  we  obtain from  (72) w q'\ t +r  = c p T +Bfsi+hb'+gB+clA w (0 ,B,A) = ̂ - ,  g(T ,b,B,A)= &. 8B  ' W CoN STmmvr; LAW IN THERMOPLASTICITV 17 With respect to the inelastic deformations apart from thermally activated creep and re- laxation  processes  we  assume  th at  two different  mechanisms  contribute  to these  defor- m ations.  Therefore  we put 4 = 4+4  (76) ( 0 ) (s) an d  accordingly w  -   - l- Ą df  =   ~sl k d k t +Ą -  s l k d\  =  w + w.  (77) ( 0  Q  ( 0  Q  GO  S  W  (p)  (*) dl  represents  the  plastic  or viscoplastic  deformations  resulting  from  slip processes  which 00 are  governed  by th e actual  stress  state,  dl is  connected  with  certain  rearrangements  of m the  distribution  of  lattice  defects  due to stress  increments.  In principle  we assume  that these two  different  mechanism depend on different  yield surfaces  F(p), F(s)  which are defined in the space of the thermodynamic state variables  and can be interpreted as yield conditions in  the stress  space  (depending  on the remaining  state  variables).  Restricting  ourselves for  the  present  to plastic  deformations  we can  write T ,   b>   pQ  =   o ,  (78) F^   = F^ (4,T ,b,4)  = 0.  (79) The  introduction  of an additional  internal  variable  «' k  into  (79)  is  necessary  in order to allow for independent  changes  of  (78)  an d  (79). Concerning th e corresponding  deformation  rates we assume L L r  (80) dl  =  R(tL - «l),  R(Ą ,T ,b,A).  (81) W T h is  m ean s  t h a t  d' k   is  govern ed  by t h e  n o rm alit y  rule  a n d t h a t  dl depen ds  prim arily on (P)  w t h e  ch an ges  of t h e  effective stress .8- 'J- 4.  (82) The  internal  variables  oc{,  fil k  represent  the  so- called  back- stresses.  Concerning the  evolu- tion  laws  of  the  internal  variables  we suppose  th at  they  reflect  an interaction  between hardening or softening  processes  due to inelastic  deformations  d k   and dl  on the one hand and  certain  annealing  processes  (recrystallization,  recovery)  on the  other hand.  Therefore we  write b -  — (tl- xM+4-  (tL - PM- bH4, T , b, B, A),  (83a) Q  W  Q  ( P )  (!>) h  = "£- PkW 4,T ,b,B),  (83b) (p)  c o V 4^ Cdl- oi{&(s l k ,T ,b,A).  (83) (5)  («i) 2  Mech.  Teoret.  i  Stos.  1—2/ 84 18 T. LEHMANN Decomposing the specific work rate w into the immediately dissipated part w and the stored part w according to (A) w + w = w = w+w (84) (p)  (j) If) (d)  (A) (see eq. (50)) and defining the entropy production T r\ due to internal processes we obtain from the balance equation (75) certain restrictions for the evolution laws (83a) to (83c) for the internal variables. Putting the resulting expressions into the consistency condition for the plastic deformations F

= 0 (85) we can calculate the factor X in (80). The requirement A > 0 leads to the so- called  loading  condition.  F or details  see [1], The theory  simplifies  when we  assume  coinciding yield conditions pm  =   jpw  _  JF   (86) Then the additional internal variable Ą   is  dispensible.  This  case is  treated fully  in  [1]. There  also  the extension  of  this  approach  to  elastic- viscoplastic  behaviour  can be  found. The  material  functions  and  parameters  entering  this  approach  can  not  be  determined only from simple monotonie or cyclic experimental tests since the theoretical frame  supposes two  independent  yield  mechanisms.  Therefore  additional  experiments  with  non- propor- tional loading pathes are needed as  shown  in [33]. 4.  Some  additional  remarks The  approach  described  in  3.4  introduces  in  common  with  Raniecki's  theory  an  ad- ditional yield condition. In contradiction to Raniecki's  theory  this  second yield condition contains  also  the  stresses.  Furthermore in  the  approach  3.4  the  corresponding  yield me- chanism  is  not governed  by  the normality rule.  This  leads  to  the appearance  of  an  addi- tional term (81) in the evolution law  for  the inelastic deformations. A similar  term appears also in H art's theory  as  equation  (64b) shows. Therefore  the approach 3.4  combines  some features  of  H art's and Raniecki's theory, whereas  Krempl's  and Vernocki's  theory  follows another concept. Experimental  investigations  with  regard  to  the  constitutive  law  concern  the determi- nation  of  subsequent  yield  conditions  after  different  pre- loading  histories  on  the  one hand  and  inquiries  on  stress- strain- temperature  relations  (yield  mechanisms)  in  different loading  processes  on  the  other  hand.  The  investigation  of  subsequent  yield  conditions leads  to different  results  depending on the method of determination  [34 to 40]. Experiments  with  partial  unloading  [35  to  39]  may  result  in  definitions  of  yield con- ditions which  don't enclose the stress  origin as fig.  4 shows. On the other hand experiments with total unloading  [39 to 40] may lead to  concave yield conditions  (see fig.  5) Yield  conditions  which  do  not  enclose  the  stress  origin  cannot  be  associated  with deformation  rates  governed  by  the  normality  rule  if  the  corresponding  yield  mechanism CON STITUTIVE LAW IN THERMOPLASTICITY 19 T=lb/ in x10 F ig. 4. Effect of proof strain on subsequent yield surface after partial unloading (IKEG AMI [39]) - 300 F ig. 5. Effect of proof strain on subsequent yield surface after total unloading (G U P TA, LAU ERT [40]) is considered t o be essentially dissipative. This would con tradict the second law of ther- m odyn am ics. Therefore  th e  experim en tal  facts  suggest  an  approach  with  two  different  yield  mecha- n ism s  as  discussed  in  3.4.  One  m ain  aspect  is  scetched  in  fig  6.  Th e  in n er yield  condition J F ( S )  belongs  t o  (small)  deform ation  rates Ą   which  are  n o t  governed  by  th e  norm ality w rule.  These  deform ation s  are  con n ected  with  certain  rearran gem en ts  of  th e  distribution of  lattice  defects  a n d  represen t  essentially  n on dissipative  processes.  T h e  outer  yield  con- dition Fm  correspon ds  t o  th e  usual  definition  of  plastic  yielding. With in  th e  stress  space between Fw  an d Fw  we  obtain  ap art  from  t h e  strictly  reversible  deform ation s  repre- sen ted  by di  only  sm all  addition al  deform ation s  represen ted  by d{  un til  also  th e  yield fr)  w con dition Fm  is  fulfilled.  This cooperation between Fw  an d  J F ( P )  can explain  th e  hysteresis lo o p  in  u n lo ad in g- —relo ad ing  an d  th e  difference  in  th e  behaviour  a t  reloadin g  an d  at loadin g  in  th e  opposite  direction  after  un loadin g. H owever,  it  m ay  be  em phasized  once  m ore  t h a t  in depen den t  of  th e  respective  corre- lat io n  of  th e  two  yield  con dition s,  even  if  they  coincide,  an  influence  of  th e  existence 2* 2 0 T. LEHMAN N 4   •   dj,   .   dj, (r)  (si  (pi grip! F ig. 6. Yield surfaces F a n d F<"> of a second yield mechanism rem ain s with respect to th e  stress- strain  relation s  in  com plex loadin g  histories  loadin g  t o  certain  (small)  deviations  from  th e  n orm ality  rule. The  existence  of  a  second  yield  m echanism  is  also  suggested  by  th e  experim ents  of F eigen  [41]  repeated  an d  extended  by  M azihi  an d  D am m  [42].  F ig.  7  shows  t h a t  in  t h e immediate  tran sition  from  pure  tension  t o  an  additive  torsion  in  stress- controlled  expe- rim ents  with  thin- walled  tubes  the  shear  m odulus  appears  reduced  (see  also  [43]).  Th is tran sition  represents  a  n eutral  loadin g  in depen den t  of  the  special  shape  of  the  plastic yield  con dition supposed  it  is  regular.  Accordin g  to  th e  classical  theory  of  plasticity  t h e response  of  th e material  should  be  purely  elastic. 40 30- 20 - 10- - T- »O J t- 100  150  200  250 OF  300  350  400   m m Ck15 G?  0 *  equivalent  Initial G  shear  modulus E- IO"" • valuation:  least  error squares /   ,  .p  o ' 0  ą s  i  15  2  2,5  3  3,5  4 Fig,  7.  Stress- controlled  tension- torsion  experiment  (see  also  F BIG BN   [41D CONSTITUTIVE LAW IN THERMOPLASTICITY 21 The difference between the material response in neutral loading and in unloading which is obvious from the experimental facts shown in fig. 7 suggests immediately to suppose the existence of an additional yield mechanism as assumed in the approach 3.4. Other physical facts which support this approach are the better agreement between theoretical and experimental results in bifurcation problems and other problems with complex load- ing histories [33]. In bifurcation problems also the coupling between thermal and mechanical processes becomes important. The beginning of localization of inelastic deformations leads also to a certain concentration of heat production. This can influence the further development of localization very strongly. References 1. Th. LEHMANN, General frame for the definition of constitutive laws for large non-isothermic elastic plastic and elastic-viscoplastic deformations; in: The constitutive law in thermoplasticity (ed. Th. Leh- mann), CISM Courses and Lectures No. Springer-Verlag Wien/New York (1983). 2. Th. LEHMANN, Einige Aspekte der Thermoplastizitat; ZAMM 63 (1983), T. 3/13. 3. K. THERMANN, Foundations of large deformations; in: The constitutive law in thermoplasticity (ed. Th. 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PHILLIPS, The effect of repeated loading on the yield surface, Acta Mech. 6 (1968) 217/31. 37. A. PHILIPS, The foundations of thermoplasticity — experiments and theory, in: Topics in applied con- tinuum mechanise (J. L. Zeman and F. Ziegler, ed.) Springer-Verlag Wien) New York (1974). 38. A. PHILLIPS and CHONG-WON LEE, Yield surfaces and loading surfaces; experiments and recommenda- tions, Int. J. Sol. Struct. 15 (1979), 715/29. 39. K. IKEGAMI, Experimental plasticity on the anisotropy of metals, in: Mechanical behaviour of ani- sotropic solids, Proc. Euromech. Coll. 115 (J. P. Boehler, ed.) Ed. CNRS Paris (1982), 201/42. 40. N. K. GUPTA and H.-A. LAUERT, A study of yield surface upon reversal of loading under biaxial stress, ZAMM 63 (1983). 41. M. FEIGEN, Inelastic behaviour under combined tension and torsion, Proc. 2 US Nat. Cong. Appl. Mech. 1954. 469/76. 42. P. MAZILU and U. DAMM, Some studies to Feigen's experiments, Internal Report of the Institute of Mechanics, Ruhr-Universitat Bochuni, unpublished as yet. 43. P. MAZILU, Verringerung des Anfang-Schoubmoduls mit zunehmender axialer Dehnung erkla'rt mit Hilfe einesplastischhypoelastischen Modells, ZAMM 64 (1984). CON STITU TIVE LAW IN THERMOPLASTICITY 23 O 3 AK 0 H AX K O H C T H T YT H BH K I X T E P M O r i J I AC T i M H O C T H B pa6oTe o6cy>Kfleno (beH oiweH OJionraecKyio Teopm o 6OJII>U IH X HeH3OfepMHraeciciix T en,  KoropŁ ie  MOJKH O  pacoviaTpHBaTS  KHK  loiaccjprecKH e  cp eflt i.  M b i  npmiHMaeM,  IITO TepMO- cocroflH iie  Kaac^oro  DJieMeHia  MaTepHajia  oflHosHa^HO  on peflejiaeica  aH aieimaM H   KO- H «H o r o  MHTOKecisa  nepeMeH H brx  C OC TOH H I M ,  flaH