approached through both empirical and analytic
mation of the strains. As shown in Mellor (1975),
in differential form it is
means (see Lockett 1972). The former involves
fitting relatively simple mathematical expressions
E σ EM EK
E
2σ
to experimental data, which gives results with
E
+
σ
+ M + M + K
t 2 ηM ηK ηK t ηM ηK
limited ranges of application. In contrast, the ana-
lytical approach is more rigorous, originating in
fundamental axioms of physics. An example of
E E ε
2ε
= EM
+ M K
(A2)
.
an empirical relationship follows from the obser-
t 2 ηK t
vation in Lockett (1972) that the equation
in which σ and ε are understood to be functions
σ
σ
ε t = εe sinh + εd tn sinh
,
(A6)
of time. Equation A2 is a special case of the gen-
σe
σd
eral relationship,
Pσ = Qε
where σ is stress, εt is the total strain, εe, εd, σe, and
(A3)
σd are constants, is a good representation of the
where P and Q are the differential operators
Note that if σe is large, then
2
n
P = a0 + a1
+ a2
+ K + an
2
tn
t
t
σ σ
sinh ≈ ,
σe σe
2
m
Q = b0 + b1
+ b2
+ K + bm
.
t2
tm
t
so that, with the substitution (εe/σe) = 1/E, eq A6
describes a Maxwell material with a linear elastic
The forms of the constants a0, a1,..., b0, etc., can be
spring and a dashpot which, for n=1, follows the
determined for the four-parameter model by com-
hyperbolic sine relationship for the relationship
parison with eq A2.
between stress and strain rate. Equation A6 or its
Equation A3 can be derived from the integral
version with the linear spring can be extended to
form of the one-dimensional stress-strain relation-
the four-parameter model by simply adding the
ships for linear viscoelastic fluids,
contribution of the Voigt model shown in Figure
t
ε
A1, with either linear or nonlinear elements de-
∫ G(t - τ) τ dτ
σ=
(A4)
pending upon the data which is to be fitted to the
-∞
equation. Thus, there is some flexibility to the
empirical relationships with respect to fitting data,
t
σ
∫ J(t - τ) τ dτ
ε=
but the resulting equations could become incon-
venient for solving boundary value problems.
-∞
As an illustration of the analytical approach
where G and J are relaxation and creep functions
we use the example of the stressstrain relation-
respectively. However, there is some loss of gen-
ship for a Green-Rivlin material, because it is simi-
erality in making this step (Christensen 1971) since
lar to a relationship used by Brown et al. (1973)
the complete spectrum of possible creep or relax-
and Brown (1976) to describe the nonlinear
ation times in the integrals in eq A4 is replaced by
deformational behavior of snow. After assuming
a discrete number in the differential form (eq A3).
isotropy, homogeneity and deformation under iso-
Equations A4 are the one-dimensional forms
thermal conditions, the relationship can be writ-
of the integrals
ten in matrix form as (Lockett 1972)
t
ε kl
σ(t) = ∫-∞ {Iψ 1 T1 + ψ 1 M1} dτ1
∫ Gijkl (t - τ)
t
dτ
σ ij =
i, j = 1, 2, 3.
(A5)
τ
-∞
+∫-∞ ∫-∞ {Iψ 3 T1 T2 + Iψ 4 T12
t
t
and its equivalent for strain in terms of stress,
which are the three-dimensional forms of the con-
+ ψ 5 T1 M2 + ψ 6 M1 M2 } dτ1 dτ2
stitutive relationships for linear viscoelastic ma-
terials.
{
The problem of defining constitutive relation-
t
t
t
+∫-∞ ∫-∞ ∫-∞ Iψ 7 T123 + Iψ 8 T1 T23
ships for nonlinear viscoelastic materials has been
23
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