Deformation Of Adaptive Heterophase Crystal
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EO 0
-1 00
0 ; -1)
2 =•
X 0 0
-1)
•
3=Eo0
-1
j
0
299 Mat. Res. Soc. Symp. Proc. Vol. 360 01995 Materials Research Society
(1)
For X =2 these self-strains describe the transformation of the BCC to the FCC lattice ("Bainstrain"). This case, which corresponds to some martensitic transformations in Cu-base shape memory alloys 1,2 , is considered as an example. As shown before3 -4 the equilibrium mesostructure in general consists of plane parallel layers of a product phase separated by layers of an initial phase. The product phase itself consists of two or more different domains forming plane, parallel alternations (Fig.2). In the case of cubic-tetragonal transformation with X > 1, under extension along [100] direction, the polydomain product consists of two domains, 1 and 2 (Fig. I) or I and 3, where 1 is always the largest fraction 3 . The average self-strain of the product phase is:
'=(1-0a)
1 +0"
2
(E.(a)
0
0
0
e2 (a)
0
0
0
e3(a)
=
E' =[1 (Z+1]e E; 22 =1-l+a(X+l)]o 0
(2)
e3 =-e
where a is the volume fraction of domain 2. Under contraction along [100] the polydomain product consists of domain 2 and 3, then the average self-strain is (63 ()
- a')
+
3-
0 0
0 El (aW)
0
0 0 e2 (a')
(3)
where c' is the volume fraction of domain 3.
X3 (0011I~
X2 [1010]F~
Xi [100] Fig. 13 variants of the self-strain for a cubic tetragonal transformation. 1-J3
' AO001 j Fig.2 The evolution of the mesostructure under elongation.
300
The problem is to find the equilibrium mesostructure which corresponds to the minimum in free energy at a fixed temperature and a fixed external strain E. It is assumed that there is no plastic deformation or fracture (i.e., the strains are compatible and the crystalline system is coherent). The difference of the elastic moduli of phases is neglected for simplicity. EQUILIBRIUM MESOSTRUCTURE AND STRESS-STRAIN RELATION FOR THE DISPLACEMENT CONTROLLED DEFORMATION For the two-phase crystal, containing a volume fraction 03of the product phase, the average self-strain is ý = /3k(a). The free energy of the two-phase crystal per unit volume the product phase under the strain e along [100] is: F=Afr!+(1/2) E( E- /3 e,(a F=Afr!+(l/2)E(1-fre 3 )
))2
+•(n, a, /3)
for elongation for contraction
(4)
where e(n,a, p3)=/3(1-/3)l/2Ee22 (a) is an energy of the internal stresses arising due to crystalline coherency. For simplicity it is assumed that the crystal is elastically isotropic. For contraction e2(o')--0 at ox'=l/X+l and the coherent energy is absent. Thus the free energy of two-phase state is f = opri + [e - /3(Z - a(X + 1))]2 + p3(1 2 f=i-4i+(e-r!) a'=1l(X+l)
p3)[-1 + a(z + 1)]2
for elongation for contraction
(5)
where dimensionless parameters introduced are f = F / I/2Ee; = / E0; = Af/I/ 2Ec 02. 4 is the dimensionless temperature with the phase equilibrium temperature To as a reference temperature. The equations of equilibrium: af/ao•=af/aP=0
(6)
determine the equilibrium mesostructure, i.e. the equilibrium fractions of the product phase (g3o) and of domain 2 in it
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