Consider then a small defect with a radius p situated at the center of an elastically isotropic sphere of radius R. The defect is modeled in four different

ways:

As a center of dilatation (CD): the defect is created by displacing the inner radius r of the surrounding matrix by an amount

u(P) — cp [A2]

Here, the parameter c is referred to as the strength of the defect.

As a cavity (CA): the inner surface is loaded by a pressure p, giving rise to a radial stress component in the surrounding medium, which has the boundary value
partial differentiation. Hooke’s law then connects the stresses with the elastic part of the strains, which for the isotropic case reads as

Oij m(uij A Uj;i) A 1dijuk, k 2pej [A5]

Here, repeated indices imply a summation and a comma before an index, say j, means a partial deriva­tive with regard to xj. The mechanical equilibrium equations are Oj;j — 0, or after substitutions of Hooke’s law

(m A 1)uy; ji A pui; jj — 0 [A6]

The pressure p can also represent radial forces that the defect exerts on its nearest neighbor atoms of the surrounding matrix, and they are then referred to as Kanzaki forces.

As an inclusion (INC): the material within the defect region is subject to a transformation strain ej as if it had endured a phase transformation. We shall treat here only the simple case of an isotropic transformation strain ej — pd, j where dj is the Kronecker matrix.

As an inhomogeneity (IHG): In addition to a transformation strain, the defect region has also acquired different elastic constants from the sur­rounding matrix.

The boundary conditions at the interface between the defect and its surrounding matrix are that the radial displacement and the radial stress component must be continuous.

These different models of defects correspond in essence to the three possible boundary conditions on the defect-matrix interface, namely the Dirichlet boundary condition for the CD, the von Neuman condition for the CA, and the mixed one for the INC and IHG.

One could also consider these three different boundary conditions for the external surface. However, prescribing a value for the displacement of the external surface or a mixed boundary condition implies the presence of yet another surrounding medium, in which the defect-containing sphere would be embed­ded. Here, these cases will not be considered.

We then restrict the following treatment to the boundary condition of a free external surface for which

Or(R) = 0 [A4]

The solution of the elasticity problems associated with all defect models is rather elementary and can be found in many textbooks. However, as a way to intro­duce our notation, we sketch the procedure.

The deformation is described by a displacement field «i(r), from which the strain tensor is obtained by

We assumed here that the transformation strains eij are constant within the defect region.

For a spherically symmetric problem, the dis­placement field possesses only one radial component u, which satisfies the differential equation 1 d / 2 s

7dT(r u)

The solution to this equation is

u(r) — At A B/r2 [A8]

with two unknown constants A and B for each region, for the matrix that surrounds the defect and for the defect region itself. However, for the defect region B — 0. The remaining constants are to be determined by the boundary conditions. For later use we give the radial stress component

sr — (2m a 31)(a — m) — 4mB/r3

— 3K (a — m) — 4mB/r3 [A9]

for the case of an isotropically transformed inclusion, that is, when ej — mdj. Furthermore, instead of the Lame’s constants m and l, we introduced the bulk modulus K and the shear modulus m. The following ratio involving the two elastic constants will appear often, and we reserve the symbol

4m _ 2(1 — 2v) 3K — 1 A v

for it, where v is Poison’s ratio and gE is the Eshelby factor.

The hydrostatic stress is related to the elastic dilatation or the lattice parameter change by the equation

1 . Da

Oh — — ff„ — 3K (a — m) — 3^ [A11]

3 a

We see that for a defect in the center of an isotropic sphere the lattice parameter changes are uniform throughout the matrix.

The density of strain energy, on the other hand, is strongly peaked near the defect, as can be seen from the formula

f (r) = [1] [2] GijUij = 2K(A — rj)[3] + 6pB2/r6 [A12]

Integrating this function over the entire sphere gives the strain energy associated with the defect,

tR

(r)r2dr [A13]

0

Finally, let us state the formulae for volume changes. The external volume change is computed from the equation

A V = 4pR2u(R) = 4pR[4] [5](A + B/R3)

or AV = 3(A + B/R3) [A14]

In a similar fashion, one defines the change of the defect volume due to the constraints imposed by the surrounding matrix, as

AU = 3(A + B/p3) [A15]

A last point must be made regarding the measure­ment of the lattice parameter change.

If the atoms in the defect region contribute to the diffraction pattern just as the matrix atoms, we must calculate the appropriate average lattice parameter change over the entire volume V. We denote this
average by < > brackets. However, if the defect region consists of a very different crystal structure or atoms with much lower form factors, then the defect region does not contribute to the diffraction pattern, and the lattice parameter change is due to the matrix only. When the defect region is excluded, the lattice parameter change is indicated by ( ) parentheses.

We obtain the coefficients A and B by solving the equations for the boundary conditions. For example, for the case of an inhomogeneity, the radial stress components vanish on the external surface,

sr(R) = 3KA — 4pB/R3 = 0 [A16]

On the interface between the inclusion region and the matrix, that is, at r= p, the displacements and the radial stress components must be continuous, or

AiP = Ap + B/p2 [A17]

and

3Ki(Ai — r) = 3KA — 4pB/p3 [A18]

Here, the subscript ‘I’ designates a parameter for the inclusion region, while parameters for the matrix are without a subscript. Solution of the three eqns [A17]-[A19] determines the three parameters A, B, and AI listed in the first three rows and the last column of Table A1. We note that the stress within the inhomogeneous inclusion is purely hydrostatic (because BI = 0), and the results do not depend on

Table A1 The solutions for the integration constants A and B, and volume changes, lattice parameter changes, and defect strain energies that derive from them

INC Г oS 1 + o 3 r p3 1 + o r(1 + oS) 1 + o 3r(1 + oS) 1 + o 3rS rS ro(1 — S)S 1 + o 6 2 1 — S 6r Dll 1 + o

IHG

Table A2 Simplified expressions obtained from the general ones in Table 2 for small defect concentrations, S << 1 Defect type CD CA INC IHG A for matrix coS ps ■qmS VoS 3K 1 + m 1 + ko B for matrix о со pp3 3 VP3 -3 CO 4m 1 + m 1 + ko A for defect NA NA V V — of defect 1 + m 1 + ko 3c 3p 3v 3v V u) 4m 1 + m 1 + ko of solid 3cS(1 + m) 3pS (1 + m) 3vS 3 1 + m 3vS 4m 1 + ko — for solid cS(1 + m) ps vS 1 + o 3K 1 + ko — for matrix cSo ps VimS VoS a 3K 1 + m 1 + ko Defect energy U0 12mc2o 3p2 u 6mv2u 6mv2u 8m 1 + m 1 + ko

the shear modulus of the inhomogeneity, but only on the ratio

к = K/KI [A19]

For the inclusion (INC), its bulk modulus is the same as for the matrix. Therefore, its parameters follow from those for the IHG by setting к = 1.

With the parameters A and B determined, it is straightforward to evaluate volume changes, lattice parameter changes, and the defect strain energy U0. Table A1 contains a tabulation of all these para­meters for the four different defect types, and for all possible values of the defect volume fraction S. For application to defect concentrations in irradiated materials, it may be assumed that S<<1, and the expressions in Table A1 can be simplified to those shown in Table A2.