and *t*τ2, and ψ7,...,ψ12 are functions of *t*τ1, *t*τ2,

+ ψ 9 T1 T2 M3 + ψ10 T12 M3 + ψ11 T1 M2 M3

and *t*τ3. Depending upon their form, each ψa

+ ψ12 M1M2 M3} dτ1 dτ2 dτ3 + K

could include several constants which, if the phys-

(A7)

ics were known completely, might be determined

analytically. However, in general, they are found

by fitting the equation to experimental data. Thus,

100

the approach is really semiempirical, rather than

where **I **= 0 1 0

completely analytical. The equation could be writ-

001

ten in inverted form, with strain as a function of

stress. In that case, the functions ψa would be

creep functions analogous to *J*(*t*τ) above. Note

σ = σ ij

that the equation is written in three dimensions

although it is often reduced to one dimension for

ε ij (τα )

application as was done in Brown et al. (1973)

**M**α =

and Brown (1976).

τα

Finally, to demonstrate the connection between

the relatively simple relationships in eq A3 and

ε ij (τα )

A4 as compared to A7, we note that, for isotropic

*T*α = δ ij

τα

materials, eq A5 can be simplified to (Lockett 1972)

()

ε ij

*t*

ε ij (τα )

ε ij τβ

∫ λ(t - τ)

σ ij = δ ij

*d*τ

*T*αβ = δ ij

T

τα

τβ

-∞

( ).

()

ε ij

ε ij (τα )

*t*

ε ij τ γ

ε ij τβ

∫ (t - τ)

+2

*d*τ

(A8)

*T*αβγ = δ ij

T

τα

τβ

τγ

-∞

which, for constant λ and , reduces to the famil-

= 0 *for i *≠ *j*

iar relationship for a linear elastic material. How-

δij =

= 1 *for i *- *j*

ever, introducing the substitutions ψ1 = λ(*t*τ)

and ψ2 = (*t*τ) puts eq A8 into the notation of

The functions ψa are relaxation functions

eq A7 and shows that it is the linearized form of

(analogous to *G*(*t*τ) in eq A4 in which ψ1 and ψ2

the constitutive equation for the Green-Rivlin

are functions of *t*τ1, ψ3,...,ψ6 are functions of *t*τ1

material.

24