540 M. Romeo
especially in connection with transmission lines He, 1993; Åberg et al., 1995; Kristensson, 1995. The viscoelastic model is outlined in Section 2 where a differential form of the stress–strain law is used to obtain the
relaxation function Gτ . The wave splitting formulation is applied in Section 3 to both pure compressive and pure shear motions. Then, in Section 4, we analyze the split field in terms of a wave propagator which, via a time
convolution, relates the wave field at a fixed point of the space to the wave field at any other point. The integro- differential equation for the wave propagator is analytically solved for the Maxwell model and for a suitable
approximation of the Maxwell–Kelvin model. An exact solution for the reflectivity is derived in Section 5 at an elastic–viscoelastic interface for each viscoelastic model and it is shown how the reflectivity data at t = 0 can
be exploited to derive the characteristic relaxation times. Finally, the problem of transient propagation through a viscoelastic layer is solved in Section 6 arriving at a suitable expression for the reflection and the transmission
functions in terms of a series expansion.
2. Linear viscoelastic model
Let us consider a homogeneous and isotropic viscoelastic medium which occupies an unbounded region B
of the three-dimensional space. The displacement ux, t and the Cauchy stress Tx, t are taken to be C
N
functions of time t in R
+
N
1 and piecewise smooth functions of the position x in an unbounded subset
V ⊆ R
3
. We assume that a linear stress-strain law holds in a differential form. More precisely, denoting by
e =
1 2
[∇u + ∇u
T
] the infinitesimal strain tensor and by
e
e = e −
1 3
tr e and
e
T = T −
1 3
tr T the deviatoric parts of e and T respectively we have
N
X
k=0
e
p
k
∂
k t
e
T =
N
X
k=0
e
q
k
∂
k t
e
e ,
1-a
N
X
k=0
b
p
k
∂
k t
tr T =
N
X
k=0
b
q
k
∂
k t
tr e, 1-b
where ∂
k t
denotes the k-order time derivative and
e
p
k
,
b
p
k
,
e
q
k
,
b
q
k
k = 0, 1, . . . , N are real numbers. Equation 1 arises in connection with empirical models of a viscoelastic solid, where p
k
and q
k
are given in terms of the constitutive parameters of a virtual network of springs and dashpots. Gurtin and Stemberg 1962,
have shown that if Eq. 1 holds with
e
p
N
,
b
p
N
6= 0, a couple of relaxation functions
e
Gτ and
b
Gτ ∈ C
∞
R
+
exists such that
T x, t =
e
G
e
e x, t +
1 3
b
G
0tr ex, tI +
Z
t
f
G
′
τ
e
e x, t − τ +
1 3
c
G
′
τ
tr ex, t − τI dτ,
2 where
e
G and
b
G satisfy the following linear differential equations, appropriate to quantities with a superimposed
tilde or hat
N
X
k=0
p
k
∂
k τ
Gτ = q 3
with the initial conditions G
0 = q
N
p
N
, ∂
r τ
G| =
1 p
N
q
N−r
−
r−1
X
k=0
p
N−r+k
∂
k τ
G| r = 1, . . . , N − 1.
4
Wave splitting in linear viscoelasticity 541
The solution to Eqs 3 and 4 can be easily obtained by the Laplace transform. We have LGs =
q +
P
N j =1
P
N k=j
p
k
g
k−j
s
j
P
N k=0
p
k
s
k+1
, 5
where g
r
= ∂
r τ
G| . Denoting by s
h
h = 0, . . . , N the roots of the equation
P
N k=0
p
k
s
k+1
and taking the inverse Laplace transform of Eq. 5 we get
Gτ =
N
X
h=0
q +
P
N j =1
P
N k=j
p
k
g
k−j
s
j h
p
N
Q
06l6N l6=h
s
h
− s
l
exps
h
τ .
6 In the next sections we shall consider two particular models for viscoelastic media. The first one is the well
known Maxwell model which deserves special mention for its simplicity. In this case N = 1 and, denoting by κ
and µ respectively the elastic bulk modulus and the elastic shear modulus, we have
e
p =
b
p =
1 τ
M
,
e
p
1
=
b
p
1
= 1,
e
q =
b
q = 0,
e
q
1
= 2µ,
b
q
1
= 3κ, where τ
M
is the Maxwell relaxation time. Substitution into 6 yields
e
Gτ = 2µGτ ,
b
Gτ = 3κGτ , 7
where Gτ = exp−ττ
M
. 8
The second model is obtained by joining in series a Maxwell element and a Kelvin element. The effectiveness of such a model has been remarked by Caviglia and Morro 1990 in connection with creep tests. This is a
second-order model N = 2 with
e
p =
b
p =
1 τ
M
τ
K
,
e
p
1
=
b
p
1
= 1
τ
M
+ 1
τ
K
1 + α,
e
p
2
=
b
p
2
= 1,
e
q =
b
q = 0,
e
q
1
= 2µ
τ
K
,
b
q
1
= 3κ
τ
K
,
e
q
2
= 2µ,
b
q
2
= 3κ, where τ
K
is the Kelvin relaxation time and where α is the ratio between the Young’s moduli of the Maxwell element and the Kelvin element. Substitution into 6 yields 7 with
Gτ = τ
1
τ
2
τ
1
− τ
2
1 τ
K
− 1
τ
1
exp−ττ
1
− 1
τ
K
− 1
τ
2
exp−ττ
2
, 9-a
where τ
1
τ
2
= τ
M
τ
K
, 1
τ
1
+ 1
τ
2
= 1
τ
M
+ 1
τ
K
1 + α. 9-b
Equation 9 has been exploited in particular for the problem of reflection of plane harmonic waves at a fluid–viscoelastic interface Romeo, 1990.
542 M. Romeo
3. Transient propagation and wave splitting
