and tτ2, and ψ7,...,ψ12 are functions of tτ1, tτ2,
+ ψ 9 T1 T2 M3 + ψ10 T12 M3 + ψ11 T1 M2 M3
and tτ3. 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(tτ) 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)
Mα =
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
Tα = 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
dτ
Tαβ = δ ij
T
τα
τβ
-∞
( ).
()
ε ij
ε ij (τα )
t
ε ij τ γ
ε ij τβ
∫ (t - τ)
+2
dτ
(A8)
Tαβγ = 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(tτ)
and ψ2 = (tτ) 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(tτ) in eq A4 in which ψ1 and ψ2
the constitutive equation for the Green-Rivlin
are functions of tτ1, ψ3,...,ψ6 are functions of tτ1
material.
24
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