S. Chirita et al.

parallel. The Kelvin-Voigt material is considered a solid since the deformation is reversible(though not suddenly). More information on the linear theory of viscoelasticity can be foundin [3–6].

This paper deals with the three–dimensional linear theory of the Kelvin-Voigt materials(see for example Eringen [3] p. 325 for the basic equations). In fact, we consider a finite(semi-infinite) prismatic cylinder consisting of a homogeneous and isotropic Kelvin-Voigtmaterial, one of whose plane ends is subject to a displacement which varies harmonicallyin time while the other end (for a finite cylinder) and the lateral surface are clamped, andsubject also to standard initial conditions. Following the research line introduced by Flavinand Knops [7] for the linearly damped wave equation and the linearly elastic damped cylin-der, and developed further for other models in [8–12] and references therein, we investigatethe spatial behavior of solutions describing harmonic vibrations in a Kelvin-Voigt elasticmaterial.

The results obtained in the aforementioned papers suggest exponential decay of activityaway from the excited end, provided the frequency of vibration is lower than a critical valueand the constitutive coefficients satisfy some positive definiteness conditions. In this work,by using some different measures, we prove that the dissipative effects may be used to obtainthe same kind of results but without any restriction condition on the elastic coefficients andfor every value of the frequency of vibrations. Of course, the decay rates depend on theLamé coefficients, the viscosities and the frequency of vibration.

2 Formulation of the Problem

Throughout this paper, we refer the motion of a continuum to a fixed system of rectan-gular Cartesian axes 0xk (k = 1,2,3). We shall employ the usual summation and differ-entiation conventions: Latin subscripts are understood to range over the integers (1,2,3)

whereas Greek subscripts are confined to the range (1,2), summation over repeated sub-scripts is implied, subscripts preceded by a comma denote partial differentiation with re-spect to the corresponding Cartesian coordinate, and a superposed dot denotes time differ-entiation.

Let B denote the interior of a right cylinder of length L > 0 whose cross section isbounded by one or more piecewise smooth curves. Choose the Cartesian coordinates suchthat the origin lies in one end of the cylinder and such that the x3–axis is parallel to thegenerators. Let D(x3) denote the cross section of the cylinder corresponding to the axialdistance x3, and let ∂D(x3) denote the cross-sectional boundary. We denote by π the lateralsurface of the cylinder, that is π = ∂D × [0,L].

Let us assume that an isotropic and homogeneous Kelvin-Voigt material occupies B .Considering that the natural state is unstressed, then in the absence of the body force, thesystem of field equations for the linear theory are (cf. Eringen [3], p. 325)

– the equations of motion

trl,r = ρ0ul , (2.1)

where ρ0 is the density mass at time t = 0, trl are the components of the stress tensor, ul

are the components of displacement vector field;– the constitutive equations

trl = λennδrl + 2μerl + λ∗ennδrl + 2μ∗erl , (2.2)

On Spatial Behavior of the Harmonic Vibrations in Kelvin-Voigt

where δrl is the Kronecker delta, λ and μ are the Lamé coefficients, λ∗ and μ∗ are coeffi-cients of viscosity and erl are defined by

– the geometrical equations

erl = 1

2(ur,l + ul,r ). (2.3)

The dissipation inequality [3]

λ∗err ell + 2μ∗erl erl ≥ 0, (2.4)

implies that

μ∗ ≥ 0, 3λ∗ + 2μ∗ ≥ 0. (2.5)

Let us consider the initial boundary value problem defined by the above field equations,the initial conditions

ul = a(1)l (x1, x2, x3), ul = b

(1)l (x1, x2, x3), in B at t = 0, (2.6)

and the lateral boundary conditions

ul = 0 , on π × [0, t0) (2.7)

together with the end boundary conditions

ul = ul(x1, x2) exp(−iωt), on D(0) × [0, t0) (2.8)

and

ul = 0, on D(L) × [0, t0) (2.9)

where ω is a strictly positive constant and i is the unit complex, that is i = √−1.It is easy to see that

ul = Ul(x1, x2, x3, t) + vl(x1, x2, x3) exp(−iωt), (2.10)

where Ul absorbs the initial conditions and satisfies the null boundary conditions and theequations (2.1), (2.2) and (2.3), while vl satisfies the boundary value problem consisting ofthe field equations

Trl,r = −ρ0ω2vl,

Trl = (λ − iωλ∗)vn,nδrl + (μ − iωμ∗)(vr,l + vl,r ),(2.11)

subject to

vl = 0 , on π (2.12)

and

vl = ul(x1, x2) , on D(0) (2.13)

vl = 0, on D(L). (2.14)

S. Chirita et al.

From (2.11) we deduce the following partial differential equations for the functions vl :

(μ − iωμ∗)vl,rr + [λ + μ − iω(λ∗ + μ∗)]vr,rl + ρ0ω2vl = 0. (2.15)

It is established in Appendix, under the further assumptions μ > 0, 3λ + 2μ > 0, thatUl tends to zero as t → ∞. So that, Ul represents essentially the transient and vl exp(−iωt)

the forced oscillation. We recall that an element of damping has been considered in theequations of motion associated with the problems investigated by Flavin and Knops [7] inorder to prove that Ul tends to a constant value. In the context of Kelvin-Voigt model theseadditional terms in the equations of motions are unnecessary because of the dissipative termspresent in the constitutive equations.

In what follows we study the spatial behavior of the amplitude of the steady-state vi-bration satisfying (2.15) under the boundary conditions (2.12), (2.13) and (2.14). Since theeffects caused by Ul in the cylinder are transient, the decay estimates for vl are of interestfor large time t .

3 Spatial Behavior for the Amplitude of the Steady-State Vibration

In this section we shall study the spatial behavior of the amplitude of the considered vibra-tions under the assumptions suggested by the dissipation inequality (2.4), that is

μ∗ > 0, 3λ∗ + 2μ∗ ≥ 0. (3.1)

To this end we associate with the amplitude of the steady-state vibration the cross-sectionalfunctional

K(x3) = −∫

D(x3)

[iμ(vl,3vl − vl,3vl) + i(λ + μ)(vr,rv3 − vr,rv3)

+ ωμ∗(vl,3vl + vl,3vl) + ω(λ∗ + μ∗)(vr,rv3 + vr,rv3)]dA, (3.2)

where the superposed bar denotes the complex conjugate. Further, we shall establish thefollowing result describing the spatial behavior of the amplitude of the harmonic vibrationin concern.

Theorem 1 In the context of a finite cylinder made of a Kelvin-Voigt material, the cross-sectional functional K(x3) represents an acceptable measure of the solution vr , in the sensethat it is positive for all vr and it vanishes only when vr = 0. Moreover, it satisfies thefollowing exponential decay estimate

0 ≤ K(x3) ≤ K(0) exp

(−2ωμ∗

ξM

x3

), x3 ∈ [0,L], (3.3)

where the positive constant ξM is given by

ξM = 1√χ1

[|μ| + 4|λ + μ| + ω(4λ∗ + 5μ∗)] (3.4)

and χ1 is the lowest eigenvalue in the two–dimensional clamped membrane problem for thecross section D.

On Spatial Behavior of the Harmonic Vibrations in Kelvin-Voigt

Proof By direct differentiation in (3.2), we obtain

dK

dx3(x3) = −

∫D(x3)

[i(λ + μ)(vr,rv3,3 − vr,rv3,3) + 2ωμ∗vl,3vl,3

+ ω(λ∗ + μ∗)(vr,rv3,3 + vr,rv3,3)]dA

−∫

D(x3)

[iμ(vl,33vl − vl,33vl) + i(λ + μ)(vr,r3v3 − vr,r3v3)

+ ωμ∗(vl,33vl + vl,33vl) + ω(λ∗ + μ∗)(vr,r3v3 + vr,r3v3)]dA. (3.5)

On the basis of (2.15), the square bracket of the second integral may be written in theform

iμ(vl,33vl − vl,33vl) + i(λ + μ)(vr,r3v3 − vr,r3v3) + ωμ∗(vl,33vl + vl,33vl)

+ ω(λ∗ + μ∗)(vr,r3v3 + vr,r3v3)

= −i(λ + μ)(vr,rρvρ − vr,rρvρ)

− iμ(vl,ρρvl − vl,ρρvl) − ωμ∗(vl,ρρvl + vl,ρρvl)

− ω(λ∗ + μ∗)(vr,rρvρ + vr,rρvρ). (3.6)

By using the relation (3.6), the divergence theorem and the boundary conditions (2.12)in the second integral of (3.5), we deduce

dK

dx3(x3) = −

∫D(x3)

[2ωμ∗vl,rvl,r + 2ω(λ∗ + μ∗)vr,rvl,l

]dA. (3.7)

Since λ∗ + μ∗ is nonnegative, we have

−dK

dx3(x3) ≥

∫D(x3)

2ωμ∗vl,rvl,rdA. (3.8)

On the other hand, by using the arithmetic–geometric inequality, the Schwarz’s inequalityand ∫

D(x3)

vl,ρ vl,ρdA ≥ χ1

∫D(x3)

vl vldA, (3.9)

where χ1 is the lowest eigenvalue in the two–dimensional clamped membrane problem forthe cross section D, we deduce for the terms in the right hand of (3.2) the following estimates

∣∣∣∣∫

D(x3)

iμ(vl,3vl − vl,3vl)dA

∣∣∣∣ ≤ 2|μ|(∫

D(x3)

vl,3vl,3dA

)1/2( 1

χ1

∫D(x3)

vl,ρ vl,ρdA

)1/2

≤ |μ|√χ1

∫D(x3)

vl,r vl,rdA; (3.10)

∣∣∣∣∫

D(x3)

i(λ + μ)(vr,r v3 − vr,rv3)dA

∣∣∣∣ ≤ |λ + μ|√χ1

∫D(x3)

(vr,r vl,l + v3,ρ v3,ρ)dA; (3.11)

∣∣∣∣∫

D(x3)

ωμ∗(vl,3vl + vl,3vl)dA

∣∣∣∣ ≤ ωμ∗√

χ1

∫D(x3)

vl,r vl,rdA; (3.12)

S. Chirita et al.

∣∣∣∣∫

D(x3)

ω(λ∗ + μ∗)(vr,r v3 + vr,rv3)dA

∣∣∣∣

≤ ω(λ∗ + μ∗)√χ1

∫D(x3)

(vr,r vl,l + v3,ρ v3,ρ)dA. (3.13)

So we can write

|K(x3)| ≤∫

D(x3)

(D1vl,r vl,r +D2vl,l vr,r

)dA, (3.14)

where

D1 = 1√χ1

[|μ| + |λ + μ| + ω(λ∗ + 2μ∗)],(3.15)

D2 = 1√χ1

[|λ + μ| + ω(λ∗ + μ∗)].

The distinct eigenvalues of the linear transformation associated with the positive definitequadratic form

σ = D1vl,r vl,r +D2vl,l vr,r , (3.16)

are

ξm = D1, ξM = D1 + 3D2. (3.17)

From (3.14) we deduce

|K(x3)| ≤∫

D(x3)

ξMvl,r vl,rdA. (3.18)

Hence, it follows immediately from (3.8) and (3.18) that

|K(x3)| ≤ − ξM

2ωμ∗dK

dx3(x3) , x3 ∈ [0,L] . (3.19)

Since K(·) is a non–increasing function and K(L) = 0 (see (2.14) and (3.2)), it followsthat K(x3) ≥ 0 for every x3 ∈ [0,L]. Moreover, integrating (3.8) with respect x3 on [x3,L],it follows that

K(x3) ≥∫

B(x3,L)

2ωμ∗vl,rvl,rdV ,

where B(x3,L) = D(x3) × [x3,L]. This relation together with the boundary conditionsprove that K(x3) = 0 implies vr = 0 on B(x3,L), so that K(x3) is an acceptable measure ofthe amplitude of the steady state vibration. Therefore, (3.19) became

dK

dx3(x3) + 2ωμ∗

ξM

K(x3) ≤ 0, x3 ∈ [0,L]. (3.20)

By integration one obtains the estimate (3.3) and the proof of Theorem 1 is complete. �

On Spatial Behavior of the Harmonic Vibrations in Kelvin-Voigt

Let us discuss further the case of a semi-infinite cylinder (that is the case when L → ∞).We have the only two possibilities: (a) K(x3) ≥ 0 for all x3 ∈ [0,∞), or (b) there existsx∗

3 ∈ [0,∞) so that K(x∗3 ) < 0.

Let us first consider case (a). Then the relation (3.19) leads to the differential inequality(3.20) and hence we deduce the spatial decay estimate (3.3). We have to outline that in thiscase the volume energetic measure

E(x3) =∫

B(x3)

[μ∗vl,rvl,r + (λ∗ + μ∗)vr,rvl,l

]dV, (3.21)

with B(x3) = D(x3) × [x3,∞), exists and is finite.Let us now discuss case (b). Since K(·) is a non–increasing function on [0,∞), it follows

that we have

K(x3) < 0 for all x∗3 ≤ x3 < ∞ (3.22)

and hence the relation (3.19) leads to the following differential inequality

dK

dx3(x3) − 2ωμ∗

ξM

K(x3) ≤ 0, x∗3 ≤ x3 < ∞. (3.23)

By integrating this last differential inequality we obtain the following estimate

−K(x3) ≥ −K(x∗

3

)exp

[2ωμ∗

ξM

(x3 − x∗

3

)]> 0, x∗

3 ≤ x3 < ∞. (3.24)

We have to observe that in this case the volume energetic measure E(x3) is infinite.These results are embodied in the following Phragmèn-Lindelöf alternative result.

Theorem 2 In the context of a semi-infinite cylinder we have the following alternative:(a) for the amplitudes having a finite volume energetic measure E(x3), the K(x3), as givenby (3.2), is an acceptable measure (in the sense that it is positive for all vr and it vanishesonly when vr = 0) which decays spatially faster than the exponential exp(− 2ωμ∗

ξMx3), or

(b) for the amplitudes having an infinite volume energetic measure E(x3), the −K(x3) growsspatially faster than the exponential exp[ 2ωμ∗

ξM(x3 − x∗

3 )].

4 Further Measures

In the present section we prove how we can relax the assumption (3.1), as was suggestedby the dissipation inequality (2.4). In fact, we shall establish spatial decay estimates for therelaxed assumptions upon the viscosities λ∗ and μ∗ as follows

μ∗ > 0, λ∗ + 2μ∗ > 0. (4.1)

To this end we introduce the function

Jκ(x3) = −

∫D(x3)

[iμ(vl,3vl − vl,3vl) + i(λ + μ)(vr,rv3 − vr,rv3)

+ ωμ∗(vρ,3vρ + vρ,3vρ) + ω(λ∗ − μ∗ + 2κ)(vρ,ρv3 + vρ,ρv3)

+ ω(λ∗ + 2μ∗)(v3,3v3 + v3,3v3)]dA, x3 ∈ [0,L], (4.2)

where κ is a positive parameter at our disposal. Further, we establish the following result.

S. Chirita et al.

Theorem 3 In the context of a finite cylinder, for all κ ∈ (max(−2λ∗ − 2μ∗,0),2μ∗), thefunction J

κ(x3) defined by (4.2) represents a measure of the amplitude of harmonic vibration

vr , in the sense that it is positive for all vr and it vanishes only when vr = 0. Moreover, itsatisfies the following spatial estimate

0 ≤ Jκ(x3) ≤ J

κ(0) exp

(−2ω

√χ1

ζM

x3

), x3 ∈ [0,L], (4.3)

where ζM is an appropriate positive constant depending on λ, μ, λ∗, μ∗, ω and κ.

Proof We first note that the direct differentiation and the use of (2.15) and the boundaryconditions (2.12) in (4.2) furnish

dJκ

dx3(x3) = −2ω

∫D(x3)

μ∗(v1,2 − v2,1)(v1,2 − v2,1)dA

− 2ω

∫D(x3)

[(λ∗ + 2μ∗)v3,3v3,3 + (λ∗ + κ)(vρ,ρ v3,3 + vρ,ρv3,3)

+ (λ∗ + 2μ∗)vρ,ρ vα,α]dA

− 2ω

∫D(x3)

[μ∗(v3,αv3,α + vα,3vα,3)

+ (μ∗ − κ)(vρ,3v3,ρ + vρ,3v3,ρ)]dA. (4.4)

At this stage we choose the parameter κ in such way that

max(−2λ∗ − 2μ∗,0) < κ < 2μ∗, (4.5)

so that we have

|λ∗ + κ| < λ∗ + 2μ∗, (4.6)

|μ∗ − κ| < μ∗. (4.7)

With these choices we have

−dJκ

dx3(x3) ≥ 2ωμ∗

∫D(x3)

(v1,2 − v2,1)(v1,2 − v2,1)dA

+ 2ωτ1

∫D(x3)

(vρ,ρ vα,α + v3,3v3,3)dA

+ 2ωτ2

∫D(x3)

(v3,αv3,α + vα,3vα,3)dA, (4.8)

where

τ1 = λ∗ + 2μ∗ − |λ∗ + κ| > 0,(4.9)

τ2 = min(κ,2μ∗ − κ) > 0.

Moreover, we deduce

On Spatial Behavior of the Harmonic Vibrations in Kelvin-Voigt

−dJκ

dx3(x3) ≥ 2ωτ3

∫D(x3)

vα,ρ vα,ρdA + 2ωτ1

∫D(x3)

v3,3v3,3dA

+ 2ωτ2

∫D(x3)

(v3,αv3,α + vα,3vα,3)dA, (4.10)

where

τ3 = min(μ∗, τ1). (4.11)

Thus, from (4.10) we deduce that Jκ(x3) is a non–increasing function on [0,L]. Since we

have Jκ(L) = 0, it follows that J

κ(x3) is a positive quantity and hence, by a procedure

similar with that used in theorem 1, we can conclude that Jκ(x3) appears as an acceptable

measure of the amplitude in concern.Furthermore, we write (4.2) in the form

Jκ(x3) = −

∫D(x3)

[iμ(vρ,3vρ − vρ,3vρ) + i(λ + μ)(vρv3,ρ − vρv3,ρ)

+ i(λ + 2μ)(v3,3v3 − v3,3v3)]dA − ω

∫D(x3)

[μ∗(vρ,3vρ + vρ,3vρ)

− (λ∗ − μ∗ + 2κ)(vρv3,ρ + vρv3,ρ)

+ (λ∗ + 2μ∗)(v3,3v3 + v3,3v3)]dA (4.12)

and note that the arithmetic–geometric and Schwarz inequalities and the relation (3.9) give

Jκ(x3) ≤ 1√

χ1{|μ| + |λ + μ| + ω[μ∗ + |λ∗ − μ∗ + 2κ|]}

∫D(x3)

vα,ρ vα,ρdA

+ 1√χ1

{|λ + μ| + |λ + 2μ| + ω[λ∗ + 2μ∗ + |λ∗ − μ∗ + 2κ|]}∫

D(x3)

v3,αv3,αdA

+ 1√χ1

(|μ| + ωμ∗)∫

D(x3)

vρ,3vρ,3dA

+ 1√χ1

[|λ + 2μ| + ω(λ∗ + 2μ∗)]∫

D(x3)

v3,3v3,3dA. (4.13)

From the relations (4.10) and (4.13) we obtain the following first–order differential in-equality

2ω√

χ1Jκ(x3) + ζM

dJκ

dx3(x3) ≤ 0, x3 ∈ [0,L], (4.14)

where

ζM = max

{1

τ2(|μ| + ωμ∗),

1

τ3{|μ| + |λ + μ| + ω[μ∗ + |λ∗ − μ∗ + 2κ|]},

1

τ2{|λ + μ| + |λ + 2μ| + ω[λ∗ + 2μ∗ + |λ∗ − μ∗ + 2κ|]},

1

τ1[|λ + 2μ| + ω(λ∗ + 2μ∗)]

}. (4.15)

S. Chirita et al.

Thus, by a direct integration in (4.14) we obtain the decay estimate (4.3) and the proofof theorem is complete. �

Following a procedure similar with that in the above section, the results may be easyextended to a semi-infinite cylinder. Thus, we have

Theorem 4 In the context of a semi-infinite cylinder we have the following alternative:(a) for the amplitudes having a finite volume energetic measure F(x3), with

F(x3) =∫

B(x3)

vl,rvl,rdV ,

the function Jκ(x3), as given by (4.2) with κ ∈ (max(−2λ∗ − 2μ∗,0),2μ∗), is a measure

which decays spatially faster than the exponential exp(− 2ω√

χ1ζM

x3), or (b) for the amplitudeshaving an infinite volume energetic measure F(x3), the −J

κ(x3) grows spatially faster than

the exponential exp[ 2ω√

χ1ζM

(x3 − x∗3 )].

5 Concluding Remarks

In this paper we have obtained some spatial decay estimates for the amplitude of harmonicvibration in a right cylindrical region of finite extent, filled by a Kelvin-Voigt elastic mate-rial, one plane end of which is subject to an excitation which is harmonic in time. We haveemphasized that the dissipative mechanism guarantees the exponential decay of the ampli-tude of the steady-state vibrations without any restriction upon the elastic coefficients λ andμ and for every value of the frequency of vibrations.

Moreover, if μ > 0 and 3λ + 2μ > 0 then the viscoelastic effects give rise ultimately toa steady state vibration (see the Appendix). So, the whole “activity” decays exponentiallyfrom the excited end when t is large enough. When the Lamé coefficients are arbitrary thenthe same conclusion may be obtained provided some special initial conditions, appropriateto the forced oscillation, are adopted (see [7]).

The results are extended to a semi-infinite cylinder to obtain appropriate alternative ofPhragmèn-Lindelöf type.

Acknowledgements The authors are grateful to the reviewers for useful observations leading to improve-ments of the paper. They also acknowledge support from the Romanian Ministry of Education and Research,CNCSIS Grant code ID-401, Contract no. 15/28.09.2007.

Appendix

In this section we mimic the algorithm developed by Flavin and Knops [7] in order to provethat Ul satisfying

Srl,r = ρ0Ul,

Srl = λEnnδrl + 2μErl + λ∗Ennδrl + 2μ∗Erl, in B (A.1)

Erl = 1

2(Ur,l + Ul,r ),

On Spatial Behavior of the Harmonic Vibrations in Kelvin-Voigt

subject to boundary conditions

Ul = 0 , on ∂B (A.2)

and initial conditions of the type

Ul = Al(x1, x2, x3), Ul = Al(x1, x2, x3), at t = 0, (A.3)

decays to zero as t → ∞. The sufficient conditions assuring this result are μ > 0 and 3λ +2μ > 0.

By usual procedures, from (A.1) we deduce

d

dt

[∫B

ρ0

2UlUldV +

∫B

(λErrEll + 2μErlErl)dV

]

= −∫

B

(λ∗Err Ell + 2μ∗ErlErl)dV . (A.4)

Integrating, we deduce that the function

f (t) =∫

B

ρ0

2UlUldV +

∫B

(λErrEll + 2μErlErl)dV,

satisfies

f (0) − f (t) =∫ t

0

∫B

(λ∗Err Ell + 2μ∗ErlErl)dV dt ≥ 0. (A.5)

f (t) being non–negative on [0,∞), we deduce by letting t → ∞ in the above relationthat ∫ ∞

0

∫B

(λ∗Err Ell + 2μ∗ErlErl)dV dt < ∞ (A.6)

and moreover, since∫

B(λ∗Err Ell + 2μ∗ErlErl)dV ≥ 0 for all t in [0,∞), we have

limt→∞(λ∗Err Ell + 2μ∗ErlErl) = 0, x ∈ B. (A.7)

From (3.1), (A.2) and (A.7) it follows that

limt→∞ Ul = 0. (A.8)

Differentiating (A.1)–(A.3) with respect to t and proceeding as above we obtain

limt→∞ Ul = 0. (A.9)

Letting t → ∞ in (A.1) and (A.2) and taking into account (A.8) and (A.9) we deduce thatthe limit (as t → ∞) of Ul , denoted by U 0

l satisfies

srl,r = 0,

srl = λεnnδrl + 2μεrl, in B (A.10)

εrl = 1

2(U 0

r,l + U 0l,r ),

S. Chirita et al.

subject to boundary conditions

U 0l = 0 , on ∂B. (A.11)

Since μ > 0 and 3λ + 2μ > 0, the uniqueness result of elastostatics assures that

limt→∞Ul = U 0

l = 0. (A.12)

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