Wave splitting in linear viscoelasticity
M. Romeo
D.I.B.E., Università, via Opera Pia, 11/a, 16145 Genova, Italy
(Received 12 November 1997; revised and accepted 30 June 1998)
Abstract– The problem of transient propagation in a homogeneous linear viscoelastic solid is investigated by means of the wave splitting technique. Two mechanical viscoelastic models are considered in particular, corresponding respectively to a single Maxwell element and to the series of a Maxwell element plus a Kelvin element. Analytical solutions for the split field are evaluated in terms of the wave propagator via the Laplace transforms. The reflectivity of a viscoelastic half-space is explicitly for the two mechanical models and the corresponding inverse problem is approached. Finally, the reflection and transmission problem for a transient wave is solved for a viscoelastic layer.Elsevier, Paris
wave propagation in solids / linear viscoelasticity
1. Introduction
The aim of the present paper is to obtain some definite results for the one-dimensional problem of transient propagation in a viscoelastic medium. This problem is of common interest for various investigations ranging from seismic phenomena in geophysics to the mechanical behaviour of viscoelastic polymers. Within the context of a linear theory of viscoelasticity, the direct problem is based on the knowledge of a relaxation functionG(τ ) which characterizes the stress state of the material for any given history of the applied strain. In some cases the available mechanical model is not sufficiently refined to give a relaxation function which works effectively for practical purposes. This fact is also due to the difficulty of measuring specific parameters such as the characteristic relaxation times. In those cases in which a stress–strain law holds in a differential form, a definite expression can be worked out for G(τ ) as a sum of exponentials of τ. The most simple example of such relaxation functions arises from the Maxwell model which require the knowledge of only one relaxation time. A more realistic behaviour of viscoelastic media can be achieved by the Maxwell–Kelvin model, which involves two relaxation times. It has been remarked that this last model generally agrees with experimental results on creep tests (Jaunzemis, 1967). Moreover, as previously shown (Caviglia and Morro, 1990), the analysis of creep tests for such materials allows us to obtain reliable values of the relaxation times and, in turn, a satisfactory expression ofG(τ ). Applications of the Maxwell–Kelvin model or, more generally, of exponential-like relaxation functions are customary in numerical computations for wave scattering problems (Ammicht et al., 1987; Caviglia and Morro, 1992), but no exact or approximate analytical solutions have been derived for the propagation and the reflection problems.
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 functionG(τ ). 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 att=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 regionB of the three-dimensional space. The displacement u(x,t) and the Cauchy stress T(x,t) are taken to be CN functions of timet inR+ (N >1) and piecewise smooth functions of the positionx in an unbounded subset
V ⊆R3. We assume that a linear stress-strain law holds in a differential form. More precisely, denoting by e=12[∇u+(∇u)T]the infinitesimal strain tensor and byee=e−13treandTe=T−13trTthe deviatoric parts
ofeandTrespectively we have
N
X
k=0
e
pk∂t(k)Te= N
X
k=0
e
qk∂t(k)ee, (1-a)
N
X
k=0
b
pk∂t(k)trT= N
X
k=0
b
qk∂t(k)tre, (1-b)
where ∂t(k) denotes the k-order time derivative and pek, pbk, qek, qbk (k = 0,1, . . . , N ) are real numbers.
Equation (1) arises in connection with empirical models of a viscoelastic solid, wherepk and qk 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 withpeN,pbN6=0, a couple of relaxation functionsG(τ )e andG(τ )b ∈C∞(R+)
exists such that
T(x,t)=G(e 0)ee(x,t)+1
3G(b 0)tre(x,t)I+
Z t
0
f
G′(τ )ee(x,t−τ )+1
3Gc′(τ )tre(x,t−τ )I
dτ, (2)
whereGe andGb satisfy the following linear differential equations, appropriate to quantities with a superimposed
tilde or hat
N
X
k=0
pk∂τ(k)G(τ )=q0 (3)
with the initial conditions
G(0)= qN pN
, ∂τ(r)G|0= 1
pN
qN−r − r−1
X
k=0
pN−r+k∂τ(k)G|0
The solution to Eqs (3) and (4) can be easily obtained by the Laplace transform. We have
Laplace transform of Eq. (5) we get
G(τ )=
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 caseN =1 and, denoting by κandµrespectively the elastic bulk modulus and the elastic shear modulus, we have
e
p0=pb0= 1
τM
, pe1=pb1=1, qe0=qb0=0, qe1=2µ, qb1=3κ,
whereτM is the Maxwell relaxation time. Substitution into (6) yields
e
G(τ )=2µG(τ ), G(τ )b =3κG(τ ), (7)
where
G(τ )=exp(−τ/τM). (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
whereτK is the Kelvin relaxation time and whereα is the ratio between the Young’s moduli of the Maxwell
element and the Kelvin element. Substitution into (6) yields (7) with
G(τ )= τ1τ2
3. Transient propagation and wave splitting
We assume here that the subsetV consists of the half-space{(x, y, z)∈R3|x>0}and denote bye1the unit
normal to the plane boundary ofV. Let us suppose that a uniform displacementu0(t)and a uniform traction w0(t)=T(0, t)e1are given at the surfacex=0 fort >0, while no deformation occurs fort60 throughoutV. We look for solutions to the one-dimensional problem
ρ∂t2u(x, t)=∂xw(x, t), x∈R+, t∈R+, (10)
whereρis the mass density of the medium andw=Te1. Equation (10) splits into a couple of scalar equations
which, accounting for (2) and (7), read
ρ∂t2uk=∂xwk, (11-a)
ρ∂t2u⊥=∂xw⊥, (11-b)
whereuk=u1andu⊥represents one of the two componentsu2andu3, and where
wk(x, t)=
4
3µ+κ
∂xuk(x, t)+
Z t
0
4
3µ+κ
G′(τ )∂xuk(x, t−τ )dτ, (12-a)
w⊥(x, t)=µ∂xu⊥(x, t)+
Z t
0 µG ′(τ )∂
xu⊥(x, t−τ )dτ. (12-b)
In the following we shall consider one of the two scalar problems given by (11-a), (12-a) and by (11-b) and (12-b) for the quantitiesuandwwhich will be taken as the displacement and the traction of pure compressive or pure shear motions. Posingv=∂tuand taking into account that∂xu(x,0)=0 forx∈R+, we can rewrite
our problem in the following form
ρ∂tv(x, t)=∂xw(x, t), (13-a)
∂tw(x, t)=σ ∂x
v(x, t)+
Z t
0
G′(τ )v(x, t−τ )dτ
(13-b)
forx∈R+andt∈R+, together with the boundary conditions
v(0, t)=v0(t), (14-a)
w(0, t)=w0(t) (14-b)
and the initial conditions
v(x,0)=0, (15-a)
w(x,0)=0. (15-b)
In Eq. (13)σ =43µ+κ for compressive motions andσ =µfor shear motions. According to the viscoelastic model outlined in the previous section, we assume thatG(τ ) has a bounded first derivative inR+. Hence the
integral operatorK defined as
(Kf )(t)=f (t)+
Z t
0 G
for anyf ∈L1(R+), is invertible and the system (13) can be written as
The operatorK−1can be easily derived from (16) on applying the Laplace transform. We obtain
K−1f(t)=
In order to obtain a diagonalized form of the matrix at the right-hand side of Eq. (17), we introduce the quantities (v+, v−)T as follows
whereP (t)andS(t)are suitable functions inL1(R+), to be determined according to the condition
(S·P ·f )(t)=f (t) (21)
for anyf ∈L1(R+). We observe that, in view of Eq. (15) we have
v+(x,0)=v−(x,0)=0, ∀x∈R+. (22)
Substitution of (19) into (17) and the use of (20) allows us to obtain
∂x
consequence of the convolution’s properties and of Eq. (22). We finally impose the diagonalization condition
P (t)= ±1
and Eq. (23) becomes
∂x
and where the choice of the sign in (25) has been performed according to the meaning ofv+andv−as forward
and backward propagating disturbances. Explicit expressions forP (t)and Q(t)=S(t)/ccan be derived for the Maxwell model and the Maxwell–Kelvin model described in Section 2. With the help of a table of Laplace transforms (Roberts and Kaufman, 1966), we have
P (t)=δ(t)exp(−t/2τM)+E
respectively for the Maxwell model and the Maxwell–Kelvin model witht>0, whereδ(t)is the usual Dirac delta distribution and the functionsEandF are given by
E(t, β)=βI0(βt)+I1(βt)exp(−βt),
F (t, β)=βI0(βt)+I1(βt)exp(−βt),
whereI0andI1are modified Bessel functions.
4. Evaluation of the wave propagator
be obtained by (20) and the evolution of the transient forx >0 can be determined by the equation
where∂(·)stands for derivative with respect to the argument involved in the convolution. A convenient form of
the solution to Eq. (28) can be achieved in terms of a wave propagatorP+(x, t)which is defined as
v+(x, t)=P+(x,·)·v+(0,·)(t), (29)
where v+(x, t) satisfies Eq. (28). A comprehensive treatment of wave propagators in the context of
one-dimensional problems is given in Karlsson (1996) and applied to the more general case of inhomogeneous media. According to Eq. (29) the propagatorP+(x, t)can be viewed as the solution to Eq. (28) corresponding
to aδ-like pulse atx=0, that is, under the boundary condition v+(0, t)=δ(t). Substitution of (29) into (28)
where the condition (22) has been exploited. The use of Eq. (27) and the arbitrariness of v+(0, t) yield the following equation forP+(x, t)
∂xP+(x, t)= −
In the case of the Maxwell model,G(s)is given by (8) and an exact solution (31) is obtained in the form
PM+(x, t)=exp(−t/2τM)
where H (t) is the Heaviside unit step function. Equation (32) corresponds to a superposition of a directly transmitted wave and a transient wave propagating with speed c and whose amplitude decays exponentially in time. In the case of a Maxwell–Kelvin model we are unable to work out the inverse transform in Eq. (31). However, a realistic approximation of the argument of the exponential in (31) allows us to bypass this difficulty. For the Maxwell–Kelvin model it results in
Figure 1.Wave propagators for shear waves in a copper half-space for the Maxwell model(P+
M)and the Maxwell–Kelvin model(PMK+ )as functions
ofxfort=100 s,τM=6·105s,τK=2s,α=1/30.
As frequently occurs (Caviglia and Morro, 1990), we have α ≪1 and τM ≫τK. Then an approximated
expression of the second root on the right-hand side of (33) holds for anys >0, such that
s
s LG(s) ≃
s
s
s+ 1 τM
+2α τK
s s+τ1
K
. (34)
In view of (34) we can work out the inverse transform in (31) to obtain
P+
MK(x, t)≃exp
−x c
α 2τK
P+
M(x, t)+
A(x,·)∗P+
M(x,·)
(t) , (35)
wherePM+ is given by (32) and
A(x, t)=exp(−t/τK)
1 τK
√ t
r
αx 2cI1
2
τK
s
αx τK
t
.
5. Exact solutions for the reflectivity
Here we assume that the planex=0 coincides with the interfaceSbetween a homogeneous elastic medium (x <0) and a homogeneous viscoelastic medium modelled as in the previous sections (x > 0). We are interested in the reflectivity at this interface with respect to a transient wave impinging onS from the elastic medium. To this aim we impose the continuity of the field(v, w)T atx=0, i.e.
for anyt>0, where 0+ and 0− refer respectively to the limiting values from the right and from the left of
x=0. In view of Eq. (3.19), Eq. (36) reads
and whereρe,σeare quantities pertaining to the elastic medium. After the multiplication of Eq. (37) to the left
byD−1we arrive at
We define the reflectivityR(t)atSas follows
v−(0−, t)=R(·)·v+(0−,·)(t). (39)
The definition (39) differs from that given in He and Ström (1996) since it implicitly contains the reflection coefficient. As will be clear in the results of this section, this choice allows us to identify the reflection coefficients with the instantaneous reflectivity at the surfaceS. Substituting into (38), we obtain
v+(0+, t)=1
Invoking the causality principle we assume that no backward waves occur forx>0, whencev−(0+, t)=0
and Eq. (41) yields the following Volterra-type integral equation forR(t) √ρ
Substitution into (40) yields
which, according to the definition (39) and Eq. (19), is a consequence of the continuity ofv(x, t)atS. Once Eq. (42) has been solved, the result (43) gives the boundary condition for the propagation problem stated in the previous section. Incidentally, we observe that Eq. (42) can be conversely regarded as a Volterra-type integral equation for S(t)if the reflectivity R(t) is known fort ∈R++. This inverse problem has been investigated
numerically in Ammicht et al. (1987). An exact solution of Eq. (42) can be obtained for the viscoelastic models here considered. The use of the Laplace transform allows us to write
R(t)=L−1
and, taking into account Eqs (25) and (27), we obtain
R(t)=L−1
respectively for the Maxwell and the Maxwell–Kelvin models, and where
N (ξ )=2χexp(−ξ )
J1 being the Bessel function of order one. The quantity ν accounts for that part of the reflected amplitude
which is present irrespective of the viscoelastic relaxation. Hence it represents the instantaneous reflectivity and corresponds to the reflection coefficient of the purely elastic case. The results (45) and (46) are illustrated infigure 2for shear waves on a aluminium(elastic)–copper(viscoelastic) interface.
The reflectivity functions ReM(t) and ReMK(t) respectively show, a typical one- and two-step behaviour.
The relaxation times τM and τK are placed within the time intervals at which the transitions occur. Hence
if we are interested in the inverse problem, any functionR(t)e given by the reflection data yields only roughly
Figure 2.ReflectivityR(t)e as given by Eqs (45) and (46) for shear waves impinging on a aluminium(elastic)–copper(viscoelastic) interface for the
Maxwell model (—) and the Maxwell–Kelvin model (• —).◦ τM,τKandαare same as infigure 1.
pulses, we expect to obtain accurate values ofR(t)e only for short time intervals (0,t )¯ , (¯t≪τM, τK). In the
sequel we show how Eqs (45) and (46) can be exploited to deriveτM andτK from the reflection data att=0.
In the Maxwell case we can obtainτM directly fromReM(0). SinceLM(ξ ,0)=ξ /τM, by (45) we get
τM=
ν1
e
RM(0)
, (47-a)
where
ν1=
Z ∞
0 N (ξ )ξdξ . (47-b)
In the Maxwell–Kelvin case we deriveτMandτKfrom the quantitiesReMK(0)andReMK′ (0)=(dReMK/dt)t=0.
SinceLMK(ξ ,0)=αξ /τK, we easily obtain from (46)
e
RMK(0)=
1
τM +
α τK
We also have
Substitution into the time derivative of (46) yields
e
The system (48) and (49) admits the following solution for the couple 1/τM and 1/τK
1
6. Transmission through a layer
As a final step we formulate the problem of transient transmission for a homogeneous viscoelastic layer of thicknessd. Here we assume that the domainV is given by{x∈R3|06x6d}and that the half-spaces x <0 and x > d correspond to homogeneous elastic media with mass densities and elastic moduli denoted respectively byρe,σeandρe¯,σe¯. Similarly to Eq. (36) we impose the continuity ofvandwacrossx=d and
Eq. (29) we define the backward wave propagatorP−(x, t)by the following relation
v−(x, t)=P−(x,·)∗v−(d,·)(t). (51)
It is worth remarking that our definition of P− differs from that given in Karlsson (1996) since we are
essentially interested in homogeneous media where forward and backward modes propagate independently. Accordingly, if aδ(t)pulse is given at x=d it backward propagates with amplitudeP−(x, t) (06x 6d),
positiond−x. In other words, the forward and backward propagators must comply with the property
P−(x, t)=P+(d−x, t) (52)
for anyx∈ [0, d]and anyt∈R+. According to (52), Eq. (51) gives, in particular
v−(0+, t)=P+(d,·)∗v−(d,·)(t).
An inversion procedure, analogous to that used to work out Eq. (18), yields
v−(d−, t)=
Owing to Eqs (37) and (53), the continuity condition (50) can be cast in the form
and accounting for the definitions (19) and (20) we obtain
where the entries of the matrixA(t)are given by
A11= Denoting byR(t)the reflectivity function for the layer, from Eq. (54) we get
A21(t)+A22∗R(t)=0. (57)
We can also define a transmission functionT(t)such that
fort∈R+. Substitution into (6.5) allows us to obtain
T(t)=A11(t)+A12∗R(t). (58)
Equation (57) represents the Volterra integral equation forR(t)which is appropriate for the layer’s problem. Once it has been solved, the transmission function follows from Eq. (58).
In view of Eqs (27), (31), (55), (56) and (44), it is possible to write the solution of Eq. (57) in the form
R(t)=L−1
LR(s)
−LP+(2d, s)LR(s)
1−LP+(2d, s)LR(s)LR(s)
(t),
whereR(t)has the form (44) withχin the place ofχ. Now we observe that, in view of (44) and (31),
LR(s) <1, LR(s) <1, LP+(2d, s) <1
for anys∈R+. Hence the following expansion holds
1−LP+(2d, s)LR(s)LR(s)−1=1+ ∞
X
n=1
LP+(2nd, s)LR(s)nLR(s)n.
Accordingly, we obtain
R(t)=R0(t)+
∞
X
n=1
R0(·)∗P+(2nd,·)∗R(·)∗n·R(·)∗n (t), (59)
where
R0(t)=R(t)−P+(2d,·)∗R(·)(t) and where the exponent∗ndenotesn-times convolution.
Equation (59) expresses the reflectivity of a layer in terms of the reflectivities R and R at the interfaces between an elastic medium and a viscoelastic half-space. It accounts for successive reflections within the layer by means of a series expansion. The proper reflectivities for a Maxwell layer or a Maxwell–Kelvin layer can be obtained from (59) after substitution of Eqs (32), (45) and (35), (46) respectively. We finally note that, as is expected
R(0)=R(0), R′(0)=R′(0)
hence the solution to the inverse problem described in the previous section remains unchanged in the layer’s case.
7. Conclusion
solutions for the field in terms of a wave propagator. The results show that the amplitude ofvdecays according to the typical relaxation times of the model considered; a comparison between the Maxwell and the Maxwell– Kelvin cases displays a quite noticeable difference of amplitudes in a wide range of travel distances and times. An exact solution is also derived for the reflectivityR(t) across an elastic–viscoelastic interface for both the Maxwell and the Maxwell–Kelvin models. The results reveal typical jumps of R(t) at the relaxation times pertinent to each model. In spite of the wide use of the wave-splitting technique to solve propagation and reflection problems in electromagnetic dispersive media, analytical results for the equivalent problem in linear viscoelasticity are absent for exponential-type models which allow for more than one relaxation time. The results of the present paper are addressed to this gap, with a particular emphasis on the reflection direct and inverse problems. In this respect we have also shown that the relaxation times can be derived by the knowledge of the reflection coefficientR(0)and of the derivativeR′(0). This approach should be considered as a method to investigate the relaxation properties of a viscoelastic material in alternative to creep tests. Finally, in Section 6 we have formulated the problem of transient transmission through a homogeneous layer arriving at a solution in terms of a series expansion which describes a multiple scattering between the edges of the layer.
References
Åberg I., Kristensson G., Wall D.J.N., 1995. Propagation of transient electromagnetic waves in time-varing media—direct and inverse scattering problems. Inverse Problems 11, 29–49.
Ammicht E., Corones J.P., Krueger R.J., 1987. Direct and inverse scattering for viscoelastic media. J. Acoust. Soc. Am. 81, 827–839. Caviglia G., Morro A., 1990. On the modelling of dissipative solids. Meccanica 25, 124–127.
Caviglia G., Morro A., 1992. Inhomogeneous Waves in Solids and Fluids. World Scientific, Singapore.
Gurtin M.E., Sternberg E., 1962. On the linear theory of viscoelasticity. Arch. Rational Mech. Anal. 11, 291–356.
He S., 1993. A ‘compact Green function’ approach to the time-domain direct and inverse problems for a stratified dissipative slab. J. Math. Phys. 34, 4628–4645.
He S., Ström S., 1996. Time-domain wave splitting and propagation in dispersive media. J. Opt. Soc. Am. A 13, 2200–2207. Jaunzemis W., 1967. Continuum Mechanics. Macmillan, New York.
Karlsson A., 1996. Wave propagation for transient waves in one-dimensional media. Wave Motion 24, 85–99.
Kristensson G., 1995. Transient electromagnetic wave propagation in waveguides. J. Electtrom. Waves Appl. 9, 645–671. Roberts G.E., Kaufman H., 1966. Table of Laplace Transforms. W.B. Saunders Co., Philadelphia.
