The Elastic Strain Energy of Coherent Ellipsoidal Precipitates in Anisotropic Crystalline Solids: Applications to the Ar
- PDF / 402,695 Bytes
- 6 Pages / 414.72 x 648 pts Page_size
- 92 Downloads / 295 Views
eyC = Sil Cklmn emT ,
(1)
where Ckj,,, are the elastic stiffnesses of the matrix phase, and in spherical coordinates Sijk
d dO0f,"sin 2X 0-
= 87
#
-1
2
2
2
(Z1 + a Z2 +
0
-1
2~ Zi Zl Mjk + +Zj ZI2 z2)3/2 Mik 3
263 Mat. Res. Soc. Symp. Proc. Vol. 321. ©1994 Materials Research Society
(2
where Mik = Cijkl zj zt, and
z = (sin 0 cos 0, sin 0 sin 0, cos 4),
where a = a2 / al,,3 = a3 / al and al, a2, and a3 are the three 6 semi-axes of an ellipsoid. The elastic strain energy per unit volume of a precipitate is given by T c T 1 (3) W = -L ( e 'a - e c ) C~ita eij . In the presence of an applied load where a precipitate forms in a system with strain eqA due to an external stress P4, the mechanical energy change AE for this system due to the formation of a unit volume of a precipitate would be'
AE = W + W,
(4)
where WW,,represents the interaction energy between the precipitate and the applied load and is given by'
(5)
A eiT . Wnt= - Cijkl eki
In the case of an inhomogeneous system, where the elastic stiffnesses of the precipitate and maT trix are not the same, an 'equivalent' stress-free transformation strain eijcan be obtained by solv. 6 ing (6) eTT ) Cijkl (efkc - ekT)* Cijta (ekc where Cij*k are the elastic stiffnesses of the precipitate and eij is the actual stress-free transformation strain suffered 6 by the precipitate. Therefore, the strain energy per unit volume of such a precipitate is given by
W= -L ( e
7v-e
t)Cyt eT'iy.(7
Analogous to the homo eneous case, if a system is under an externally applied load which produces stress and strain eiq, then the mechanical energy change AE for the system after formation of a unit volume of such a precipitate is also given by Eq. (4), and W is given by Eq. (7). The interaction energy is given by AT*
Wi,,
AT
Cij• e• eij + Cijk eki ei.
(8)
For the case of an applied load, an 'equivalent' stress-free transformation strain ei is obtained by solving' (9) Cijkt (e,ý- eki + eta) Cijkl (et - eld + ek) COMPUTATIONAL PROCEDURE For the given values of a and P, the Sijk4 in Eq. (2) can be calculated by numerical integration if the orientation relationship between the precipitate and matrix is known. Because Siju = Sia in Eq. (1), Eqs. (1) and (6) [or Eqs. (1) and (9) in the case of externally applied load] can be combined as a group of 12 linear equations with the constrained strain eyi and the 'equivalent' stress-free trans-
264
formation strain eqT as the independent variables. The actual stress-free transformation strain T* ei can be evaluated using the lattice parameters of both precipitate and matrix phases and the orientation relationship between the two phases. The aragonite to calcite transformation has been extensively studied1°,11. Although there are several possible orientation relations between the two phases during the transformation, the predominant relation for calcite nucleating inside single crystal aragonite is (000)A = (0001)c and [100]A = [10T0]c ". This orientation has been used in the present study and the Cartesian coordinates take
Data Loading...
