《复合材料 Composites》课程教学资源（学习资料）第五章 陶瓷基复合材料_martensitic transformation ic ceramics

P.M. Kelly, L.R. Francis Rose/ Progress in Materials Science 47(2002)463-557 orientations that are twin-related about(100)m. Similarly, variants ABC and ABC obeying orientation relationship C-1, will appear to be twin-related about(001)m. It is also possible to combine variants, such as ABC and BaC, to produce adjacent monoclinic variants that appear as(110)m rotation twins. The formation of apparently twin-related variants is, in essence, anaccident of the crystallography of the t-m transformation. The early transmission microscope observations of the martensitic transformation conia reported by Bailey in 1964 revealed extensive twinning on(100)m and(110)m [31 and numerous sub- sequent investigations of the t->m martensitic transformation in a variety of zirco- niaalloys' invariably indicated twinning on(100)m,(001)m and, to a lesser extent, on(110)m. Initially, it was suggested via an erroneous argument - that the twinning was a manifestation of the lattice invariant shear [27]. The similarity between the subsequent observations of twins in transformed ellipsoidal particles in Mg-partially stabilised zirconia(Mg-PSZ)[32, 33] and the well-established internal twinning in certain ferrous martensites appeared to support the LIS/twinning sug gestion. However, in their theoretical analysis of the transformation Choudhry and Crocker [34] concluded that (100)m twinning was not an allowable lattice invariant shear system for any of the three basic types of correspondence A, B or C and that (001)m twinning only gave real solutions in the case of correspondence B. Similarly Kelly and Ball [35] showed that neither(100)m nor(001)m twinning could be a lattice variant shear system in the tetragonal to monoclinic transformation in zirconia with correspondence C, because neither of these planes intersected the Bain cone The only two remaining explanations for this omnipresent twinning, particularly n the transformation of isolated tetragonal particles, was deformation twinning esulting from external stresses or that the 'twins'are really individual monoclinic martensite plates, which just happen to be twin-related [35. If the latter is true and it is entirely consistent with the predictions of the phenomenological theory then it is very likely that in any one variant there is a significant component of the shape strain that opposes the shape strain in the adjacent 'twin variant. Hence, the formation of twin-related variants would provide a mechanism for reducing the overall shear strain of the transformed region. It is the twin-related variant explan- ation that is now universally accepted and, long before the reason for the formation of these variants was fully understood, the presence of these was taken to mean that the shear strain in the transformed region was completely accommodated leaving only a pure dilatation [36-39]. The formation of variants therefore has extremely important consequences in transformation toughening and these will be considered in more detail in the next section. It is worth noting that the importance of forming variants depends on the number of variants that can form and the rela- tive orientation of the shape strain in adjacent variants. The orientation relationship between the variants and the fact that it mimics a twin relationship turns out to be of little direct relevance to transformation toughening. If the two twin orientations could form via a mechanism that did not involve a significant component of the shape strain being in opposite directions in the two variants, there would be rela tively little reduction in the overall"strain energy arising from the transformation strains”[32

     ' (   3 ""4 0     ?  *  * ?        $ ?(  '  $$   ' (   3"" 4 ,  $           ?   ?*   $  P         $$  3 "4    '  
    $$ ' (       H G     $   
    
       $              )    $   F   851  +  '    3 ""4  3 "4 M% N    ( /       
             )  (   H  G     '    3 ""4 3"" 4      +  V "W ,    '              '   '             M!2N
   '   /     '       $  $   ($     )    3(0C4 M%!%%N   '(    '          $$  $$   ,09'   (    - '              =   =  M%1N    3 ""4 '   '    '                $    $     ?   3"" 4 '              $   0     F M%7N '    3 ""4   3"" 4 '                             )    '    $  ?      $    F   
 '    +$      $  '   $            $   '    '        +     H'  G             $  '  P     ' ( M%7N ,            '   $     $                      .  $    $    $$  $     P H' G   -      ' (   '  $                      ,  ' (   +$(     '    $                   '      $    H'  G '                '  $        $    M%56%8N
         + $    /           '             +   , '       $        $             (        $    P   
     $ '             '    $                   ,  ' '                       .  $    $      $$       '     '   (          HH               GG M%!N 12? ,-- . #-"- &! "  / ,    $!!  01 2)344

P.M. Kelly, L.R. Francis Rose/ Progress in Materials Science 47(2002)463-557 479 2.3. The shape strain, stress-induced transformation and self-accommodation The shape strain in a martensitic transformation is an invariant plane strain that consists of a shear parallel to the habit plane, plus a dilatational strain normal to the habit plane. The latter is an expansion or a contraction that corresponds to the volume change(An, which, in the case of the t->m transformation in zirconia, is ar expansion of between 0.04 and 0.05. As will be shown later, the shear component(y) of the shape strain(s)in this transformation is 0. 15-0. 16, which is 3 or 4 times larger than the dilatational component. Clearly, significant strain energy will be associated with a martensitic transformation that involves such large strains. To appreciate the significance of this large strain energy it is necessary to consider the various energy terms involved in the transformation In order to nucleate a martensite plate it is necessary to overcome a free energy barrier AWn[7, 40, 41]. The overall change in energy A w that accompanies the for- change AG per unit volume associated with the transformation from the unstable parent phase to the stable product phase and the energy required to form the nucleus or embryo 140, 41]. In simple terms this can be expressed as △W=△GF+△ USTRAIN+△ USURFACE (2.2) where AUSTRAIN is the strain energy associated with the shape strain of the mar tensite plate(proportional to the volume of the nucleus), AUSURFACE is the surface energy of the interface between the product martensite the parent phase (propor- tional to the surface area of the nucleus)and v is the volume of the transformed region. In cases where the transformed region has some form of internal structure it may be necessary to include an additional term A UINTERNAL on the right hand de of Eq. (2.2). Note that for the transformation to proceed at all AGe must be Using Eshelby's analysis of the elastic field of an ellipsoidal inclusion [42]. Chris- tian [7, 43] derived an expression for the elastic strain energy SE per unit volume of a thin, oblate spheroid of radius R and semi-thickness t(I<<R) Se- A where u is the shear modulus and v is poissons ratio for both the matrix and the martensite plate. SE is the product of a term that depends on the shape of the spheroid( /R)and a shape independent term that involves the elastic constants and he transformation strains. Hence, Eq.(2.3)can be can simplified to SE=(/R)y, where y is the right hand side of (2.3)divided by(/R). Using this expression the value of UstRain is given by the product of the volume of the prolate spheroid (V=4R-t/3)and the strain energy per unit volume SE-ie (/R)y

-)-    *!      *!!   
$               $          $     $ $             $
  +$          $       3
(4 '       
      )     +$   ' ""1  ""7 * '   '     $  3
4   $   34       " 76" 5 '  %  1          $  =  .    '      '                 
$$    .                           ,       $           
9 M21"1 N
   
9   $   (                  
8 $        '            $ $    $  $    /         M1"1 N , $     +$   <
9 
8( 
  
    '
0
*,I        '   $     (   $ 3$ $          4
0L*=;       '  $     $ $ 3$ $ (          4  (           ,  '                         
,I
;I*        ;/ 3!!4 I           $   
8      L  ; G       .    $    M1!N = (   M21%N    +$           $ $          $     "   (   3UU"4< $      1  "

(!  !  ! 
 !    '         G        +     $ $  $      $   $   $  39"4   $ $                     - ;/ 3!%4    $ .  $J39"4 '       3!%4     39"4 L   +$     0
*,I     $        $  $  3(J1"! 9%4      $     $   39"4 ,-- . #-"- &! "  / ,    $!!  01 2)344 128

P. M. Kelly, L.R. Francis Rose/Progress in Materials Science 47(2002 )463-557 The surface energy term A USURFACE is given by the surface area of the spheroid (2TR for a thin spheroid) times the surface energy r [41]. Hence, for a trans- formed spheroid of radius R and thickness t, Eq (2.2) becomes [41] △W=(4R21/3)△G+(4xR21/3)(/R)+(2rR2)r When Aw is plotted as a function of t and R, Eq(2. 4)defines a surface with a addle point determined by aa w/at=dAW/aR=0 [41]. from which it follows that the critical saddle point energy -the energy of nucleation AWn-is given by 32y2r3 (2.5) nd the critical dimensions of the nucleus tn and rn by 2T (26) =F= cψ (2.7) For a subcritical spheroidal nucleus or embryo to reach a critical size, pass over the free energy barrier and grow into a full fledged martensite plate, the most favourable path to follow is one that minimises Aw with respect to t and R at all stages [41]. This optimal relationship is given by tt R This leads to the 'classic relationship between the free energy change Aw and the The difference in free energy between the parent and product phases(AG) is essentially linear with absolute temperature. At some temperature To the energies are the same (AG=0). In order to nucleate the martensite transformation, the material must be cooled down to its Ms temperature when the value of A Gc is suffi ciently large to drive the transformation(Fig. 5). This critical value of AG obviously depends on AUSURFACE and AUSTRAIN, or more importantly on the values of the surface energy r and the shape independent part (y) of the strain energy. The surface energy is not amenable to any significant externally imposed hanges. However, if the strains AV and y that make up the shape independent part of the strain energy y could be accommodated or compensated in some way, then the critical value of AG would be reduced and the transformation could occur at a higher Ms. For example, if the constraint imposed by the surrounding matrix could be removed - say by extracting the transformable particles from the material then the effective Ms temperature would be raised

  
0L*=;         $  3 !"!      $ 4      M1 N -     (   $     "     ;/ 3!!4   M1 N<
9  1"! %
8  1"! %"  !"!    A
9 $         " ;/ 3!14 .   '    $     
99J
99"J" M1 N   '     '       $        
9    <
9  %!!% %
81                 " <   !
8   "  1
8!   !!          $           ) $         '    T   $      $    '     
9 '   $    "    M1 N
$    $   <  ! "   
   H G   $ '    
9      "
 O    '  $  $  $ 3
84     '    $ *  $       3
8J"4 ,                    '    $ '   
8 S(            3  74
    
8    $ 
0L*=; 
0
*,I    $           $ $ $ 34     
        . + $    - '    
( 
  $  $ $ $              $    '       
8 '                    +$       $        +         +      $          O   $ '     1?" ,-- . #-"- &! "  / ,    $!!  01 2)344

P. M. Kelly, L.R. Francis Rose/Progress in Materials Science 47(2002 )463-557 Increasing I Decreasing constraint Decreasing I Increasing assistance I assistance Tetragonal nergy 0 U for the Monoclinic Tetragonal STRAIN Trans'n Temperatu △Gv Fig. 5. The relationship between the free energy change AGeV for the tetragonal e monoclinic trans- formation in zirconia, showing the variation with temperature and the two energy components AUsTRAIN and△ USURFACE. At the Ms temperature△GF=△ USTRAIN+△ USURFACE. Decreasing△ StrAin b ncreasing the stress assistance or reducing the surrounding constraint raises the Ms temperature, while Another possibility is via the application of an external stress. In the presence of n applied stress oa, the shape change can do work in the direction of the stress and this effectively reduces AUSTRAIN. As Patel and Cohen showed [44], the work done by the shape strain(AUwoRk) in a simple two-dimensional situation is given by △ WorK=1/2 yasin26±1/25oa(1+cos26) (2.9) where y is the shear component of the shape strain, s is the dilatational component (E=An and 8 is the angle between the applied stress and the normal to the habit plane of the martensite plate. In the more general three-dimensional situation △ WorK=aaEz where E, is the total strain(shear and dilatational components) resolved in the direction of the applied stress. The effective overall strain energy is now reduced to (AUSTRAIN-AUWORK). This means that the chemical free energy required to nucleate the transformation is less and the martensite transformation can be made to occur at a temperature higher than the Ms- the transformation has been stress induced(see Fig. 5)

*  $       $$     +   ,  $    $$     $    '            O  
0
*,I *   =  ' M11N  '      $   3
AK4   $ ' (       <
  !
!! !  "!  # '
   $    $        $  3J
(4     '  $$           $     $ ,     (    <
  $  % ' )      3       $  4          $$   
O       '   3
0
*,I
AK4
       /                         $              (  3   74   7
  $ '    
8(             (     )    '      '  $   '   $ 
0
*,I 
0L*=; *   $
8( J
0
*,I>
0L*=; E 
0
*,I                      $ '   
0
*,I  '   $ ,-- . #-"- &! "  / ,    $!!  01 2)344 1?

P. M. Kelly, L.R. Francis Rose/Progress in Materials Science 47(2002 )463-557 Two points are worth noting. For the t-m transformation in zirconia, AV=0.045 and y=0.15. Hence, the larger shear component of the shape strain will have a far greater influence than the dilatational strain. More importantly, there is a crucial difference between a dilatational strain and a shear strain. a shear strain can chang sign, while a dilatational strain cannot. This means that for a positive volume change, uniaxial tension will provide positive stress-assistance, while uniaxial com- pression will oppose transformation(negative stress-assistance) However, provided a suitably oriented variant is available, both uniaxial tension and compression will generate the shear stress necessary to interact with the shear component of the shape strain and lead to positive stress-assistance c The other important point is that stress-induced martensitic transformation is not mited to situations involving externally applied stress. Internal stresses--such as those resulting from anisotropic expansion or contraction -can also stress-induce the transformation. In fact, the initial transformation itself will often generate local nternal stresses that influence subsequent transformation in surrounding regions There are two examples of this behaviour that are particularly relevant to zirconia and to the resulting transformation toughening The first is the formation of self-accommodating martensite plates. As discussed in Section 2.2, for a single basic correspondence (A, B or C) the t-m transfor mation in zirconia can result in 4 sets of crystallographically equivalent pairs The application of the phenomenological theory(discussed in the Section 3)shows that, not only are the members of these pairs twin-related to each other, but they have shape strain directions that are essentially equal and opposite. When the first martensite plate forms, the shape strain automatically generates opposing stresses in he surrounding matrix [see Fig. 6(a) and(b)]. These stresses increase as the initial plate grows and they contribute to the eventual halt in transformation. The surrounding matrix is now subject to a local internal stress that could lead to further stress-induced martensite plate formation- provided the crystallography of the system satisfies certain conditions. The stress-induced subsequent variant must have a habit plane that is close to that of the original martensite plate generating the internal stress- within say 300, and preferably less. In addition, the shape strain of the second variant must be essentially equal and opposite to that of the initial variant. The dilatational component of the shape strain AV is always of the same sign, so it is only the shear components of the shape strain that will be equal and opposite. Provided the shear components are equal and opposite, the forma- tion of these self-accommodating martensite variants (in pairs)can result in a transformed volume, where the overall shear strain of the pair is effectively zero This is illustrated in Fig. 6(c). This is known as the formation of self-accommodating martensite variants. Two points are worth noting. First, the favoured sites for nucleation of the second (and subsequent)self-accommodating variant(s) are the stress concentration at the ends of the initial variant marked X in Second, the dilatational component of the shape strain can never be accommodated and the dilatational strain associated with the pair of plates is he sum of the dilatational strain components of the two individual plates in the

' $   '        
      )   
(J""17 
J" 7 -     $    $   '      T          $       O '            *       '         
     $       +    '  $   $    (  '   +   ( $  '  $$     3   ( 4 - ' $               +      $  '          '     $    $      $    ( 
 $  $    (                    + $$    ,             $  +$          (       ,            '         T  /           
  ' +$       $    )                
.       (      $  *     0  !!         $  3    ?4  
   (    )        1     $  /  $  
$$     $       3     0  %4 '          $  ' (       $          /  $$  A  .    $     $       $$         + M   534  34N
       $  '              
    +  ' P              (    $     $      $      .       
 (   /         $            $       '    %"  $   ,     $              /  $$          
    $    $  
( '          $    $    '   /  $$        $   /  $$    (     (        3  $  4           '        $  O  ) 
     534
 '       (        
' $   '                    3  /4 (     3 4                      HXG    534 0       $    $    !               '   $   $           $    '    $   $  1?! ,-- . #-"- &! "  / ,    $!!  01 2)344