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Spatial Behavior in Linear Theory ofThermoviscoelasticity with VoidsAndreea Bucur aa Faculty of Mathematics , “Al. I. Cuza” University of Iaşi , Iaşi , RomaniaPublished online: 21 Jan 2015.

To cite this article: Andreea Bucur (2015) Spatial Behavior in Linear Theory of Thermoviscoelasticity with Voids, Journal ofThermal Stresses, 38:2, 229-249, DOI: 10.1080/01495739.2014.985566

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Journal of Thermal Stresses, 38: 229–249, 2015Copyright © Taylor & Francis Group, LLCISSN: 0149-5739 print/1521-074X onlineDOI: 10.1080/01495739.2014.985566

SPATIAL BEHAVIOR IN LINEAR THEORY OFTHERMOVISCOELASTICITY WITH VOIDS

Andreea BucurFaculty of Mathematics, “Al. I. Cuza” University of Iasi, Iasi, Romania

In the present article we study the spatial behavior of the solutions to the initialboundary value problem associated with the linear theory of thermoviscoelasticmaterials with voids. We prove a set of properties for an appropriate time-weightedsurface power function, which allows us to obtain an idea of the domain ofinfluence in linear thermoviscoelasticity with voids. Some spatial estimates of theSaint–Venant type, for bounded bodies, and Phragmén–Lindelöf type, for unboundedbodies, are obtained. Such estimates are characterized by time-dependent as well astime-independent decay and growth rates.

Keywords: Spatial behavior; Spatial decay and growth; Thermoviscoelasticity with voids

INTRODUCTION

The theory of elastic materials with voids is a recent generalization of theclassical theory of elasticity and has been formulated by Goodman and Cowin[1]. The intended applications of this theory are to the fields of soil mechanics,foundation engineering and to manufactured porous materials. The nonlinearversion of this theory was developed by Nunziato and Cowin [2]. The basic premiseunderlying this theory is the concept of a material for which the bulk density iswritten as the product of two fields, the matrix material density field and the volumefraction field. This representation of the bulk density introduces an additionaldegree of kinematic freedom and is compatible with the theory developed byGoodman and Cowin [1] to describe the flowing granular materials. The lineartheory of elastic materials with voids was formulated by Cowin and Nunziato [3].Later, Iesan [4] extended this theory to the thermoelastic case. For a review of theliterature on thermoelastic materials with voids the reader is referred to [5–13].

The theory presented in [14] includes a rate effect in the volumetric responsewhich may be caused by the inelastic surface effects in the vicinity of void boundary.Iesan [15], extended this theory to the case when the time derivative of the straintensor and the time derivative of the gradient of the volume fraction are includedin the set of independent constitutive variables. The basic equations for the linearand nonlinear theory of thermoviscoelastic materials with voids were established by

Received 12 April 2014; accepted 12 April 2014.Address correspondence to Andreea Bucur, Faculty of Mathematics, “Al. I. Cuza” University

of Iasi, Carol I., No. 11, 700506 Iasi, Romania. E-mail: bucur.andreea.v@gmail.com

229

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Iesan [15]. Here, we study the spatial behavior of solutions to the initial boundaryvalue problem associated with the theory of thermoviscoelastic materials with voids.

To prove the spatial behavior we use the method developed by Chiritaand Ciarletta [19], namely the time-weighted surface power function method.Therefore, following [16–20] we will introduce a time-weighted surface powerfunction associated with the solution of the initial boundary value problem andthen we will study some of its general properties. On this basis we establish spatialdecay estimates of Saint–Venant type (for bounded bodies) and some alternatives ofPhragmén–Lindelöf type (for unbounded bodies). Such estimates are characterizedby independent as well as time-dependent decay and growth rates. Combining thesetwo sorts of estimates one can obtain a complete description for the spatial behaviorof the solutions.

The article is structured in three main parts. First, we mention the basicequations of the linear theory of thermoviscoelasticity with voids, we formulatethe initial boundary value problem and we discuss some positive definitenessassumptions upon the internal energy density and the dissipation energy density.Then, we establish some estimates that are useful in proving the main resultsdescribing the spatial behavior. We introduce an appropriate time-weighted surfacepower function and we prove some general properties. Finally, Theorems 2, 3 and4 give a complete description of the spatial behavior of the solutions to the initialboundary value problem associated with the linear theory of thermoelasticity withvoids.

BASIC EQUATIONS

We consider a body that in the reference configuration (at time t = 0) occupiesa bounded or unbounded regular region B of the Euclidean three-dimensional space.Let �B be the boundary of B and let us assume that is a piecewise smooth surface.The body is referred to a fixed system of rectangular Cartesian axes Oxi �i = 1� 2� 3��

Throughout this article the Latin indices have the range �1� 2� 3�, Greeksubscripts have the range �1� 2� and the usual summation over repeated subscripts isemployed. We use a subscript preceded by a comma for partial differentiation withrespect to the corresponding Cartesian coordinate and a superposed dot denotespartial differentiation with respect to time.

The governing equations of the linear theory of anisotropic and inhomogeneousthermoviscoelastic materials with voids are given by (see, Iesan [15])

• the equations of motion

tji�j + �fi = �ui (1)

Hj�j + g + �l = �� (2)

• the energy equation

�T0 = Qj�j + �S (3)

in B × �0� ��,

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LINEAR THEORY OF THERMOVISCOELASTICITY 231

• the strain-displacement relation

ers = 12

(ur�s + us�r

)(4)

and• the constitutive equations

tij = Cijrsers + Bij + Dijk�k − �ij + S∗ij

Hi = Aij�j + Drsiers + di − ai + H∗i

g = −Bijeij − � − di�i + m + g∗ (5)

� = �ijeij + a + m + ai�i

Qi = kij �j + firsers + bi + aij�j

with S∗ij , H∗

i and g∗ given by

S∗ij = C∗

ijrsers + B∗ij + D∗

ijk�k + M∗ijk �k

H∗i = A∗

ij �j + G∗rsiers + d∗

i + P∗ij �j (6)

g∗ = −F ∗ij eij − �∗ − �∗

k�k − R∗j �j

in B × �0� ��.

Here, tij are the components of stress tensor, Hi are the components ofthe equilibrated stress vector, g is the intrinsic equilibrated force, ui are thecomponents of the displacement vector, is the change in volume fraction from thereference state, � is the equilibrated inertia in the reference state, is the entropydensity per unit mass, � is the density mass, fi are the components of the bodyforce vector, l is the extrinsic equilibrated body force, S is the heat supply andT0 is the absolute temperature of the body in the reference configuration. Theconstitutive coefficients are prescribed functions depending on the spatial variablex, continuously differentiable on B and satisfying the following symmetries

Cijrs = Cjirs = Crsij� Bij = Bji� Dijk = Djik� �ij = �ji� Aij = Aji (7)

C∗ijrs = C∗

jirs = C∗rsij� B∗

ij = B∗ji� D∗

ijk = D∗jik� A∗

ij = A∗ji� kij = kji

(8)M∗

ijk = M∗jik� G∗

rsi = G∗sri� F ∗

ij = F ∗ji� P∗

ij = P∗ji� firs = fisr � aij = aji

According to Iesan [15], the constitutive coefficients are satisfying thefollowing dissipation inequality

� ≥ 0 (9)

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with

� = C∗ijrseij ers + �∗2 + A∗

ij �i�j + 1T0

kij �i �j + �B∗ij + F ∗

ij�eij

+ �D∗ijk + G∗

ijk�eij�k +(

M∗ijk + 1

T0

fkij

)eij �k + �d∗

i + �∗i ��i

+(

R∗j + 1

T0

bj

) �j +

(P∗

ij + 1T0

aji

)�i �j (10)

We also assume that �, � and a are strictly positive fields on B, so we will have

��x� ≥ �0 > 0

��x� ≥ �0 > 0 (11)

a�x� ≥ a0 > 0

where �0, �0 and a0 are positive constants.Further, we will assume that the dissipation density � is a positive definite

quadratic form in terms of eij� � �i and �i, so there exist �∗m� �∗

M� �∗m� �∗

M� a∗m� a∗

M� km

and kM such that

� ≤ �∗Meij eij + �∗

M2 + a∗M�0�i�i + 1

T0

kM �i �i (12)

and

� ≥ �∗meij eij + �∗

m2 + a∗m�0�i�i + 1

T0

km �i �i (13)

Throughout this article, we will consider that the quadratic form W , definedby

W = 12

Cijrseijers + 12

�2 + 12

Aij�i�j + Bijeij + Dijkeij�k + di�i (14)

is positive definite in terms of eij� and �i. This means that there exist positiveconstants �m and �M such that

�m�eijeij + 2 + �0�i�i� ≤ 2W ≤ �M�eijeij + 2 + �0�i�i� (15)

To the field equations (1)–(6) we adjoin the initial and boundary conditions.The initial conditions are

ui�x� 0� = u0i �x�� �x� 0� = 0�x�� �x� 0� = 0�x�

(16)ui�x� 0� = v0

i �x�� �x� 0� = �0�x�� for x ∈ B

with u0i , 0, 0, v0

i and �0 prescribed and continuous functions.

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LINEAR THEORY OF THERMOVISCOELASTICITY 233

The boundary conditions related to this theory of thermoviscoelastic materialswith voids are the following:

ui�x� t� = ui�x� t� on �1 × �0� ��� tji�x� t�nj = ti�x� t� on �2 × �0� ��

�x� t� = �x� t� on �3 × �0� ��� Hj�x� t�nj = H�x� t� on �4 × �0� �� (17)

�x� t� = �x� t� on �5 × �0� ��� Qj�x� t�nj = Q�x� t� on �6 × �0� ��

where we denoted by �k, k = 1� 2� � � � � 6, the subsets of �B with the followingproperties �1 ∪ �2 = �3 ∪ �4 = �5 ∪ �6 = �B and �1 ∩ �2 = �3 ∩ �4 = �5 ∩ �6 =∅. Also, we will consider that ui, , , ti, H and Q are given functions and that ni

are the components of the outward unit normal vector to the surface of �B. Wedenote by ��� the initial boundary value problem defined by Eqs. (1)–(6), the initialconditions (16) and the boundary conditions (17).

Some Auxiliary Results

Here we establish some estimates that we will use further to prove the spatialbehavior of the solutions of the initial boundary value problem defined in theprevious section.

Following [8], we introduce the linear space �4 as the set of all four-dimensional displacements fields U defined by

U �= �ui�√

�0� (18)

Corresponding to U ∈ �4, we will introduce the state of strain E�U� defined by

E�U� �= �eij�U�� �√

�0�i�U�� (19)

and we will denote by � the vector space of all objects of the form (19). We alsodefine the magnitude of E ∈ � by

�E� �= �E · E�1/2 = (eijeij + 2 + �0�i�i

)1/2(20)

For every state E ∈ � we define the field

S�E� ={

Sij�E�� G�E��1√�0

hi�E�

}(21)

where

Sij�E� = Cijrsers + Bij + Dijk�k (22)

G�E� = −Bijeij − � − di�i (23)

hi�E� = Aij�j + Drsiers + di (24)

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We can observe that S�E� belongs to �. In what follows, we consider the followingbilinear form

� �E�1�� E�2�� = 12

[Cijrse

�1�ij e�2�

rs + ��1��2� + Aij�1��i

�2��j + Bij

(e

�1�ij �2� + e

�2�ij �1�

)+ Dijk

(e

�1�ij

�2��k + e

�2�ij

�1��k

)+ di

(�1�

�2��i + �2�

�1��i

)](25)

for every E����U� ∈ �� � = 1� 2, where E����U� ={e

���ij �U�� ����U��

√�0

����i �U�

}�

Clearly, with the aid of the symmetry relations (7) and (8), we obtain that

� �E�1�� E�2�� = � �E�2�� E�1��� ∀ E�1�� E�2� ∈ � (26)

We can also observe that

� �E� E� = W�E�� ∀ E ∈ � (27)

Taking into account that W�E� is a positive definite quadratic form for every E ∈ �,and by using the Cauchy–Schwarz inequality, we deduce

� �E�1�� E�2�� ≤ �W�E�1���1/2�W�E�2���1/2� ∀ E�1�� E�2� ∈ � (28)

On the other hand, according to (20), the magnitude of S�E� is given by

�S�E�� =[Sij�E�Sij�E� + G2�E� + 1

�0

hi�E�hi�E�]1/2

(29)

Therefore, in view of the relations (22)–(24) and (29), we have

�S�E��2 = Sij�E�Sij�E� + G2�E� + 1�0

hi�E�hi�E�

= CijrsersSij + BijSij + Dijk�kSij − BijeijG − �G

− di�iG + 1�0

Aij�jhi + 1�0

Drsiershi + 1�0

dihi (30)

If we denote by

S�E� ={

Sij�E�� −G�E��1√�0

hi�E�

}(31)

then we have

�S�E��2 = 2� �E� S�E�� (32)

According to the assumption that W�E� is a positive definite quadratic form, forE = S�E�, we obtain

2W�S�E�� ≤ �M

(SijSij + G2 + 1

�0

hihi

)(33)

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LINEAR THEORY OF THERMOVISCOELASTICITY 235

By means of the Schwarz inequality and by using the relation (33), we get

�S�E��2 ≤ 2�MW�E� (34)

Consequently, in view of (34) we have

Sij�E�Sij�E� + 1�0

hi�E�hi�E� ≤ 2�MW�E�� ∀ E ∈ � (35)

Using the constitutive equations (5), (6) and relations (22)–(24) we obtain

tijtij + 1�0

HiHi = �Sij − �ij �tij + 1�0

�hi − ai �Hi + S∗ijtij + 1

�0

H∗i Hi

≤[�Sij − �ij ��Sij − �ij � + 1

�0

�hi − ai ��hi − ai �

]1/2

×[trstrs + 1

�0

HsHs

]1/2

+(

S∗ijS

∗ij + 1

�0

H∗i H∗

i

)1/2

×[trstrs + 1

�0

HiHi

]1/2

(36)

Hence, we get

tijtij + 1�0

HiHi ≤ 2[�Sij − �ij ��Sij − �ij � + 1

�0

�hi − ai ��hi − ai �

+ S∗ijS

∗ij + 1

�0

H∗i H∗

i

](37)

We recall that for every second-order tensors Mij� Rij and every positive number �

the following inequality holds

�Mij + Rij��Mij + Rij� ≤ �1 + ��MijMij +(

1 + 1�

)RijRij (38)

Therefore, using (38) in (37) we get

tijtij + 1�0

HiHi ≤ 2[�1 + ��

(SijSij + 1

�0

hihi

)+

(1 + 1

�

)M2 2

+ S∗ijS

∗ij + 1

�0

H∗i H∗

i

](39)

where we denoted by

M2 = maxB

(�ij�ij + 1

�0

aiai

)(40)

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Moreover, with the aid of the inequality established in (35) we obtain

tijtij + 1�0

HiHi ≤ 4�M�1 + ��W�E� + 2(

1 + 1�

)M2 2

+ 2(

S∗ijS

∗ij + 1

�0

H∗i H∗

i

)(41)

for all � > 0�Now, we proceed to estimate S∗

ijS∗ij + 1

�0H∗

i H∗i and QiQi. Taking into account

the constitutive equations (6), we get

S∗ijS

∗ij + 1

�0

H∗i H∗

i = C∗ijrsersS

∗ij + B∗

ij S∗ij + D∗

ijk�kS∗ij

+ M∗ijk �kS

∗ij + 1

�0

A∗ij �jH

∗i + 1

�0

G∗rsiersH

∗i

+ 1�0

d∗i H∗

i + 1�0

P∗ij �jH

∗i (42)

Further, to obtain an estimation for S∗ijS

∗ij + 1

�0H∗

i H∗i , we need to evaluate every term

of (42). For instance, we have

C∗ijrsersS

∗ij + 1

�0

G∗rsiersH

∗i ≤ (

C∗mnpqC

∗mnpq

)1/2(ersS

∗ij ersS

∗ij

)1/2

+(

1�0

G∗mnpG

∗mnp

)1/2(ersers

1�0

H∗i H∗

i

)1/2

≤ (C∗

mnpqC∗mnpq

)1/2(ersers

)1/2(S∗

ijS∗ij

)1/2

+(

1�0

G∗mnpG

∗mnp

)1/2(ersers

)1/2(

1�0

H∗i H∗

i

)1/2

(43)

and hence we get

C∗ijrsersS

∗ij + 1

�0

G∗rsiersH

∗i ≤ �1

(ersers

)1/2(

S∗ijS

∗ij + 1

�0

H∗i H∗

i

)1/2

with

�1 = (C∗

mnpqC∗mnpq

)1/2 +(

1�0

G∗mnpG

∗mnp

)1/2

(44)

Therefore, applying the same procedure for the other terms of (42), we deduce

S∗ijS

∗ij + 1

�0

H∗i H∗

i ≤[�1

(ersers

)1/2 + �2�� + �3

(�0�j�j

)1/2

+ �4

(1T0

�j �j

)1/2](S∗

ijS∗ij + 1

�0

H∗i H∗

i

)1/2

(45)

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LINEAR THEORY OF THERMOVISCOELASTICITY 237

with

�2 = (B∗

mnB∗mn

)1/2 +(

1�0

d∗md∗

m

)1/2

�3 =(

1�0

D∗mnpD

∗mnp

)1/2

+(

1

�20

A∗mnA

∗mn

)1/2

(46)

�4 = (T0M

∗mnpM

∗mnp

)1/2 +(

T0

�0

P∗mnP

∗mn

)1/2

and �1 given by (44). Thus, the inequality (45) can be written in the following form(S∗

ijS∗ij + 1

�0

H∗i H∗

i

)1/2

≤ �1

(ersers

)1/2 + �2�� + �3

(�0�j�j

)1/2

+ �4

(1T0

�j �j

)1/2

(47)

Moreover, we remark that

S∗ijS

∗ij + 1

�0

H∗i H∗

i ≤ 4[�2

1eij eij + �22

2 + �23�0�i�i + �2

4

1T0

�i �i

]≤ 4

[�2

1

�∗m

�∗meij eij + �2

2

�∗m

�∗m2 + �2

3

a∗m

a∗m�0�i�i

+ �24

km

1T0

km �i �i

](48)

In conclusion, we obtain the following estimation

S∗ijS

∗ij + 1

�0

H∗i H∗

i ≤ �M�1 (49)

with

�1 = �∗m

2eij eij + �∗

m

22 + a∗

m

2�0�i�i + 1

2T0

km �i �i (50)

and

�M = maxB

{8�2

1

�∗m

�8�2

2

�∗m

�8�2

3

a∗m

�8�2

4

km

}Therefore, a useful consequence of the relations (49) and (41) is

tijtij + 1�0

HiHi ≤ 4�M�1 + ��W�E� + 2(

1 + 1�

)M2 2 + 2�M�1 (51)

with �1 defined by (50).

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To estimate QiQi we introduce the following notations

qi = kij �j

qi = firsers + bi + aij�j (52)

Then, we have

qiqi = (kij �j

)qi

≤ (krskrs

)1/2( �j �j

)1/2(qiqi

)1/2(53)

so that we deduce

qiqi ≤ 2T0KM

(1

2T0

km �i �i

)(54)

with

KM = krskrs

km

In a similar way, we shall estimate qiqi. From (52) it follows

qiqi = (firsers + bi + aij�j

)qi

≤ [(fmnpfmnp

)1/2(ersers

)1/2 + (bmbm

)1/2��

+(

1�0

amnamn

)1/2(�0�j�j

)1/2](qiqi

)1/2(55)

Hence, relation (55) implies

qiqi ≤ 3(�2

1eij eij + �22

2 + �23�i�i

)≤ �M

(�∗

m

2eij eij + �∗

m

22 + a∗

m

2�0�i�i

)(56)

with

�1 = (fmnpfmnp

)1/2� �2 = (

bmbm

)1/2� �3 =

(1�0

amnamn

)1/2

(57)

and

�M = maxB

{6�2

1

�∗m

�6�2

2

�∗m

�6�2

3

a∗m

}In view of the relations (38) and (52) we obtain

QiQi = (qi + qi

)(qi + qi

)≤ �1 + �′�qiqi +

(1 + 1

�′

)qiqi (58)

for every �′ > 0.

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LINEAR THEORY OF THERMOVISCOELASTICITY 239

Therefore, by means of (54), (56) and (58), we deduce

QiQi ≤ 2T0KM�1 + �′��2 + �M

(1 + 1

�′

)�3 (59)

with

�2 = km

2T0

�i �i

�3 = �∗m

2eij eij + �∗

m

22 + a∗

m

2�0�i�i (60)

Next we establish a preliminary integral identity that is necessary inproving the main results describing the spatial behavior of the solutions of theinitial boundary value problem ���. The next lemma presents a time-weightedconservation law of total energy.

Lemma 1. Suppose that P ⊂ B is a regular region with regular boundary �P� Then,for every solution �U� � of the initial boundary value problem ���, and for each t ∈�0� ��, we have

∫P

e−�t

{12

�[ui�t�ui�t� + ��t�2

] + 12

a �t�2 + W�E�t��

}dv

+∫ t

0

∫P

e−�s� dv ds +∫ t

0

∫P

e−�s{�

2�[ui�s�ui�s� + ��s�2

]+�

2a �t�2 + �W�E�s��

}dv ds

=∫ t

0

∫P

{12

��ui�0�ui�0� + ��0�2� + 12

a �0�2 + W�E�0��

}dv ds

+∫ t

0

∫P

e−�s

[�fi�s�ui�s� + �l�s��s� + 1

T0

�S�s� �s�

]dv ds

+∫ t

0

∫�P

e−�s

[tji�s�njui�s� + Hj�s�nj�s� + 1

T0

Qj�s�nj �s�

]da ds (61)

where � is a prescribed positive parameter.

Proof. To prove (61), we multiply the first equation of motion with ui, the secondone with , and we add them to obtain

12

�

�s

[�ui�s�ui�s� + ���s�2

]= �fi�s�ui�s� + �l�s��s� − tij�s�eij�s� − Hj�s��j�s� + g�s��s�

+ [tji�s�ui�s� + Hj�s��s�

]�j

(62)

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240 A. BUCUR

By means of the relations (3), (5), (6) and (62), we deduce

12

�

�s

[�ui�s�ui�s� + ���s�2 + 2W�E�s�� + a �s�2

]+ �

= �fi�s�ui�s� + �l�s��s� + 1T0

�S�s� �s�

+[tji�s�ui�s� + Hj�s��s� + 1

T0

Qj�s� �s�

]�j

(63)

Then, by multiplying (63) by e−�s we deduce the following:

�

�s

{12

e−�s[�ui�s�ui�s� + ���s�2 + 2W�E�s�� + a �s�2

]}+ �

2e−�s

[�ui�s�ui�s� + ���s�2 + 2W�E�s�� + a �s�2

]+ e−�s�

= e−�s

[�fi�s�ui�s� + �l�s��s� + 1

T0

�S�s� �s�

]+ e−�s

[tji�s�ui�s� + Hj�s��s� + 1

T0

Qj�s� �s�

]�j

(64)

If we integrate the above relation over P × �0� t�, then we obtain

12

∫P

e−�t[�ui�t�ui�t� + ���t�2 + 2W�E�t�� + a �t�2

]dv +

∫ t

0

∫P

e−�s�dv ds

+∫ t

0

∫P

�

2e−�s

[�ui�s�ui�s� + ���s�2 + 2W�E�s�� + a �s�2

]= 1

2

∫P

[�ui�0�ui�0� + ���0�2 + 2W�E�0�� + 1

2a �0�2

]dv

+∫ t

0

∫P

e−�s

[�fi�s�ui�s� + �l�s��s� + 1

T0

�S�s� �s�

]dv ds

+∫ t

0

∫P

e−�s

[tji�s�ui�s� + Hj�s��s� + 1

T0

Qj�s� �s�

]�j

dv ds

The divergence theorem leads to the conclusion, and the proof is complete. �

Spatial Behavior

Let �U� � be a solution of the initial boundary value problem (�)corresponding to the given data � = �fi� l� S� u0

i � v0i � 0� �0� 0� ui� � � ti� H� Q�. As

in [8], we consider a fixed time T > 0 and we will denote by DT the support of theinitial and boundary data, the body force, the extrinsic equilibrated body force andthe heat supply on �0� T�� i.e., the set of all points in B so that:

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LINEAR THEORY OF THERMOVISCOELASTICITY 241

i) if x ∈ B, then

u0i �x� �= 0 or v0

i �x� �= 0 or 0�x� �= 0 or �0�x� �= 0 or 0�x� �= 0 (65)

or

fi�x� s� �= 0 or l�x� s� �= 0 or S�x� s� �= 0 for some s ∈ �0� T� (66)

ii) if x ∈ �B, then

ui�x� s� �= 0 or ti�x� s� �= 0 or �x� s� �= 0 or H�x� s� �= 0 or(67)

�x� s� �= 0 or Q�x� s� �= 0

for some s ∈ �0� T��

In what follows we will assume that DT is a bounded properly regular region.Otherwise we will substitute DT by the smallest regular region including DT andwhich is contained in B.

On this basis we introduce the set Dr� r ≥ 0, by

Dr ={

x ∈ B � DT ∩ ��x� r� �= ∅}

(68)

where ��x� r� is the closed ball with radius r and center at x� Let L be the diameterof the region B\DT � We will use the notation Br for the part of B contained in B\Dr

and for r1 > r2 we set B�r1� r2� = Br2\Br1

� Furthermore, we will denote by Sr thesubsurface of �Br contained inside of B and whose outward unit normal is orientedto the exterior of Dr�

For the solution �U� � of the initial boundary value problem (�), we associatea time-weighted surface power function, I�r� t�, defined by

I�r� t� = −∫ t

0

∫Sr

e−�s

[ti�s�ui�s� + H�s��s� + 1

T0

Q�s� �s�

]da ds (69)

∀r ≥ 0� t ∈ �0� T�, with ti�s�, H�s� and Q�s� given by

ti�s� = tji�s�nj H�s� = Hi�s�ni Q�s� = Qi�s�ni (70)

and � a prescribed positive parameter.Furthermore, we set

J�r� t� =∫ t

0I�r� s�ds (71)

for all r ≥ 0 and t ∈ �0� T�� In the next theorem we prove some properties of thetime-weighted surface power function.

Theorem 2. Let �U� � be the solution of the initial boundary value problem ���corresponding to the given data � = �fi� l� S� u0

i � v0i � 0� �0� 0� ui� � � ti� H� Q� and

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242 A. BUCUR

suppose that DT is a bounded set. Then, for each t ∈ �0� T�, the time-weighted surfacepower function, I�r� t�, has the following properties:

(i) For 0 ≤ r2 < r1�

I�r1� t� − I�r2� t� = −∫

B�r1�r2�e−�t

{12

�[ui�t�ui�t�+��t�2

] + 12

a �t�2 + W�E�t��

}dv

−∫ t

0

∫B�r1�r2�

�

2e−�s

{�[ui�s�ui�s� + ��s�2

] + a �s�2

+ 2W�E�s��}dv ds −

∫ t

0

∫B�r1�r2�

e−�s�dv ds (72)

(ii) I�r� t� is a continuous differentiable function on r ≥ 0, and

�I

�r�r� t� = −

∫Sr

e−�t

{12

�[ui�t�ui�t� + ��t�2

] + 12

a �t�2 + W�E�

}da

−∫ t

0

∫Sr

e−�s

{�

2�[ui�t�ui�t� + ��t�2

] + �

2a �t�2

+ �W�E�

}da ds −

∫ t

0

∫Sr

e−�s�da ds (73)

(iii) For each fixed t ∈ �0� T�� I�r� t� and J�r� t� are non-increasing functions withrespect to r;

(iv) I�r� t� satisfies the following first-order differential inequality

�

��I�r� t�� + �I

�r�r� t� ≤ 0 r ≤ 0 (74)

where

� = maxB

{c�

��M�1

�0

}c =

√2�M�1 + �0�

�0

(75)

and �0 is the positive root of the following algebraic equation

�2 +(

1 − M2

a0�M

− ��0KM

2a0T0�M

− ��0�M

4a0T20 �M

)� − M2

a0�M

= 0 (76)

(v) J�r� t� satisfies the following first-order differential equation

√t��t�

�J

�r�r� t� + �J�r� t�� ≤ 0 r ≥ 0 t ∈ �0� T� (77)

where

��t� = maxB

{��t��

�M�3

�0

}��t� =

√2t�M �1 + ��t��

�0

(78)

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LINEAR THEORY OF THERMOVISCOELASTICITY 243

and � is the positive root of the following algebraic equation

�2 +(

1 − M2

a0�M

− �0KM

2a0T0�Mt− �0�M

4a0T20 �Mt

)� − M2

a0�M

= 0 (79)

(vi) If B is a bounded body, then I�r� t� and J�r� t� are positive functions.

Proof. To prove (i) we will first take in Lemma 1, P = B�r1� r2�� with r1 ≥ r2 ≥ 0�

Then, using Eqs. (69), (70) and the definition for B�r1� r2�, we will get

I�r1� t� − I�r2� t� = −∫ t

0

∫�B�r1�r2�

e−�s

[tji�s�ui�s� + Hj�s��s� + 1

T0

Qj�s� �s�

]njda ds

Moreover, with the aid of the divergence theorem, the previous equation can bewritten in the form

I�r1� t� − I�r2� t� = −∫ t

0

∫B�r1�r2�

e−�s

{tji�j�s�ui�s� + tji�s�ui�j�s� + Hj�j�s��s�

+ Hj�s��j�s� + 1T0

Qj�j�s� �s� + 1T0

Qj�s� �j�s�

}dv ds (80)

If we substitute the Eqs. (1)–(4) into Eq. (80) then we obtain

I�r1� t� − I�r2� t� = −∫ t

0

∫B�r1�r2�

e−�s

{12

�

�s

[�ui�s�ui�s� + ���s�2

+ a �s�2 + 2W�E�s��

]+ �

}dv ds (81)

which integrated by parts leads to the conclusion. Thus, we have (i).Parts (ii) and (iii) follow from the dissipation inequality, relation (72) and from

the assumption that W is a positive definite quadratic form.We now prove (iv). By means of the Schwarz’s inequality and by using the

arithmetic-geometric mean inequality into Eq. (69) we obtain

�I�r� t�� ≤∫ t

0

∫Sr

e−�s�tji�s�ui�s� + Hj�s��s� + 1T0

Qj�s� �s��da ds

≤∫ t

0

∫Sr

e−�s

{�1

2�0

[tij�s�tij�s� + 1

�0

Hi�s�Hi�s�

]+ 1

2�1

[�ui�s�ui�s� + ���s�2

] + 12�2T0

a �s�2

+ �2

2a0T0

Qj�s�Qj�s�

}da ds (82)

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244 A. BUCUR

According to (51) and (59), we deduce that

�I�r� t�� ≤∫ t

0

∫Sr

e−�s

{1

2�1

[�ui�s�ui�s� + ���s�2

] + 12�2T0

a �s�2

+ 2�1�M�1 + ��

�0

W�E� + �1

�0

(1 + 1

�

)M2 �s�2 + �2KM

a0

�1 + �′��2

+ �2�M

2a0T0

(1 + 1

�′

)�3

}da ds +

∫ t

0

∫Sr

e−�s �M�1

�0

�1da ds (83)

for all r ≥ 0� t ∈ �0� T� and for every �� �′� �1� �2 > 0.Therefore, the relation (83) can be written in the following form

�I�r� t�� ≤∫ t

0

∫Sr

e−�s

{1

�1�

�

2

[�ui�s�ui�s� + ���s�2

] + 2�1�M�1 + ��

�0��W�E�

+[(

1 + 1�

)2�1M

2

�0a0�+ 1

�2T0�

]�

2a �t�2 + �2�M

2a0T0

(1 + 1

�′

)�3

+ �2KM

a0

�1 + �′��2

}da ds +

∫ t

0

∫Sr

e−�s �M�1

�0

�1da ds (84)

As in [20], we will choose �� �′� �1� �2 such that

1�1�

= 2�1�M�1 + ��

�0�= 2�1M

2

�0a0�

(1 + 1

�

)+ 1

�2T0�= �2�M

2a0T0

(1+ 1

�′

)= �2KM

a0

�1+�′�

Thus, we set

�1 = 1c

�2 = 2a0T0c

�(�M + 2T0KM

) �′ = �M

2T0KM

(85)

where c is defined by

c =√

2�M�1 + �0�

�0

and �0 is the positive root of the following algebraic equation

�2 +(

1 − M2

a0�M

− ��0KM

2a0T0�M

− ��0�M

4a0T20 �M

)� − M2

a0�M

= 0 (86)

With these choices the relation (84) can be written

�I�r� t�� ≤∫ t

0

∫Sr

c

�e−�s

{�

2

[�ui�s�ui�s� + ���s�2

] + �

2a �t�2

+ �W�E� + �1� da ds +∫ t

0

∫Sr

e−�s �M�1

�0

�1da ds

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LINEAR THEORY OF THERMOVISCOELASTICITY 245

≤ �

�

∫ t

0

∫Sr

e−�s

{�

2

[�ui�s�ui�s� + ���s�2

] + �

2a �t�2

+ �W�E� + 2�1

}da ds (87)

with

� = maxB

{c�

��M�1

�0

}Therefore, in view of the relations (73) and (87) we obtain

�I�r� t�� ≤ − �

�

�I�r� t�

�r∀r ≥ 0 (88)

so, (iv) is proved.Now, we will prove (v). Using Schwarz’s inequality and the following relation

∫ t

0

∫ s

0f 2���d� ds ≤ t

∫ t

0f 2�s�ds (89)

from Eq. (71) we obtain

�J�r� t�� ≤∣∣∣∣∫ t

0

∫ s

0

∫Sr

e−��

[tji���ui��� + Hj������ + 1

T0

Qj��� ���

]da d� ds

∣∣∣∣≤

{√t∫ t

0

∫ s

0

∫Sr

e−�� 1�0

[tij���tij��� + 1

�0

Hi���Hi���

]da d� ds

}1/2

×{√

t∫ t

0

∫Sr

e−�s[�ui�s�ui�s� + ���s�2

]da ds

}1/2

+{√

t∫ t

0

∫ s

0

∫Sr

e−�� 1a0T0

Qj���Qj���da d� ds

}1/2

×{√

t∫ t

0

∫Sr

e−�s 1T0

a �s�2da ds

}1/2

(90)

If we use now the arithmetic-geometric mean inequality then we obtain

�J�r� t�� ≤√

t

2

∫ t

0

∫Sr

e−�s

{1�3

[�ui�s�ui�s� + ���s�2

] + 1�4T0

a �s�2

}da ds

+√

t

2

∫ t

0

∫ s

0

∫Sr

e−��

{�3

�0

[tij���tij��� + 1

�0

Hi���Hi���

]+ �4

a0T0

Qj���Qj���

}da d� ds (91)

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246 A. BUCUR

with �3 and �4 positive parameters. Thus, by means of (59), (51) and (89) we deduce

�J�r� t�� ≤ √t∫ t

0

∫Sr

e−�s

{1

2�3

[�ui�s�ui�s� + ���s�2

] + 12�4T0

a �s�2

+ 2�M�3�1 + ��t

�0

W�E� + �3M2t

�0

(1 + 1

�

) �s�2

}da ds

+ √t∫ t

0

∫ s

0

∫Sr

e−��

[�4KM

a0

(1 + �′)�2 + �4�M

2a0T0

(1 + 1

�′

)�3

]da d� ds

+ √t∫ t

0

∫ s

0

∫Sr

e−�� �M�3

�0

�1da d� ds (92)

Consequently, relation (92) can be written in the following form:

�J�r� t�� ≤ √t∫ t

0

∫Sr

e−�s

{1

2�3

[�ui�s�ui�s� + ���s�2

] + 2�M�3�1 + ��t

�0

W�E�

+[

1�4T0

+ 2�3M2t

�0a0

(1 + 1

�

)]12

a �s�2

}da ds

+ √t∫ t

0

∫ s

0

∫Sr

e−��

[�4KM

a0

�1 + �′��2 + �4�M

2a0T0

(1 + 1

�′

)�3

]da d� ds

+ √t∫ t

0

∫ s

0

∫Sr

e−�� �M�3

�0

�1da d� ds (93)

We proceed now as in the part (iv), equating the coefficients. So, we will obtain

1�3

= 2�M�3�1 + ��t

�0

= 1�4T0

+ 2�3M2t

�0a0

(1 + 1

�

)= �4KM

a0

�1 + �′� = �4�M

2a0T0

(1 + 1

�′

)Therefore, we set

�3 = 1��t�

�4 = 2a0T0��t�

�M + 2T0KM

�′ = �M

2T0KM

(94)

with

��t� =√

2t�M�1 + ��t��

�0

(95)

and � a positive root of the following algebraic equation

�2 +(

1 − M2

a0�M

− �0KM

2a0T0�Mt− �0�M

4a0T20 �Mt

)� − M2

�Ma0

= 0 (96)

In conclusion, from the definition of DT , the relation (69) and from (iii) we obtain

�J�r� t�� ≤ −√t��t�

�J�r� t�

�r(97)

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LINEAR THEORY OF THERMOVISCOELASTICITY 247

with

��t� = maxB

{��t��

�M�3

�0

}and ��t� given by (95), so the proof is complete. �

The following theorem describes the spatial behavior of the solutions of theinitial boundary value problem ��� for bounded bodies.

Theorem 3 (Spatial behavior for bounded bodies). If �U� � is a solution ofthe initial boundary value problem ��� corresponding to the given data � =�fi� l� S� u0

i � v0i � 0� �0� 0� ui� � � ti� H� Q� and DT is bounded, then, for each

fixed t ∈ �0� T�, the solution �U� �, as measured by I�r� t� or J�r� t� decays spatially inthe following form:

I�r� t� ≤ I�0� t� exp(

−�

�r

)0 ≤ r ≤ L (98)

J�r� t� ≤ J�0� t� exp(

− 1√t��t�

r

)0 ≤ r ≤ L (99)

where � and ��t� are given by (75) and (78).

Proof. Using the part (vi) of Theorem 2, the relations (74) and (77) can be writtenin the following form:

�

�r

[I�r� t� exp

(�

�r

)]≤ 0 0 ≤ r ≤ L (100)

�

�r

[J�r� t� exp

(1√

t��t�r

)]≤ 0 0 ≤ r ≤ L (101)

By integrating with respect to r the relations (100) and (101), we obtain the desiredconclusion and the proof is complete. �

For unbounded bodies, since we do not have sufficient information regardingthe positiveness of the functions I�r� t� and J�r� t�, we derive some results ofPhragmén–Lindelöf type as described in the next theorem.

Theorem 4 (Spatial behavior for unbounded bodies). If {U� � is a solution ofthe initial boundary value problem ���, corresponding to the given data � =�fi� l� S� u0

i � v0i � 0� �0� 0� ui� � � ti� H� Q� and DT is bounded, then for each fixed t ∈

�0� T�� the spatial behavior of the solution �U� � as measured by I�r� t� or J�r� t�, isdescribed by the following alternatives:

(i) If I�r� t� ≥ 0 for all r ≥ 0, then

I�r� t� ≤ I�0� t� exp(

−�

�r

)(102)

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248 A. BUCUR

J�r� t� ≤ J�0� t� exp(

− 1√t��t�

r

)(103)

(ii) If there exists a value rt ≤ 0 so that I�r� t� < 0 and then I�r� t� < 0� J�r� t� <0� ∀r ≥ rt and

−I�r� t� ≥ −I�rt� t� exp[

�

��r − rt�

]∀r ≥ rt (104)

−J�r� t� ≥ −J�rt� t� exp[

1√t��t�

�r − rt�

]∀r ≥ rt (105)

Proof. For the proof of this theorem we follow the method presented in [19] andthe method used in the previous proof. According to Theorem 2, (iii), for each fixedt ∈ �0� T�� I�r� t� and J�r� t� are non-increasing functions with respect to r, so, wehave the following two possibilities:

a) I�r� t� ≥ 0 for all r ≥ 0;

b) there exists rt ≥ 0 such that I�rt� t� < 0.

Part a) proves that the inequalities (74) and (77) can be written in the form (100)and (101), respectively, so, we obtain the inequalities (102) and (103).

In the second case, we have I�r� t� ≤ I�rt� t� < 0� ∀r ≤ rt. Thus, the differentialinequalities (74) and (77) become

�

�r

[I�r� t� exp

(− �

�r

)]≤ 0 ∀r ≥ rt (106)

�

�r

[J�r� t� exp

(− 1√

t��t�r

)]≤ 0 ∀r ≥ rt (107)

By integrating with respect to r the relations (106) and (107) we obtain the desiredconclusion. �

Remark. It is important to remark that the estimates given by Theorem 3 andTheorem 4 are essential to obtain a complete description for the spatial behavior ofthe solutions of the problem ���. Moreover, one can observe that the decay estimates(98), (102) and (104) give a good description for large values of time, while the decayestimates (99), (103) and (105) are suitable for appropriate short values of time.

FUNDING

This work was supported by the strategic grant POSDRU/159/1.5/S/137750.

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