An anisotropic and nonhomogeneous compressible linear thermo-microstretch elastic cylinder is subject to zero body loads and heat supply and zero lateral specific boundary conditions. The motion is induced by a time-dependent displacement, microrotation, microstretch, and temperature variation specified pointwise over the base. Further, the motion is constrained such that the displacement, microrotation, microstretch and temperature variation and their derivatives with respect to time at points in the cylinder and at a prescribed time are given in proportion to, but not identical with, their respective initial values. Two different cases for these proportional constants are treated. It is shown that certain integrals of the solution spatially evolve with respect to the axial variable. Conditions are derived that show that the integrals exhibit alternative behavior and in particular for the semi-infinite cylinder that there is either at least exponential growth or at most exponential decay.

1. Introduction

The theory of micromorphic bodies was introduced by Eringen [1, 2] in order to adequately describe the behavior of materials that have internal structure, such as liquid crystals, blood flow, polymeric substances, and porous materials. Deformation of particles that compose the material contributes to the macroscopic behavior of the body. Eringen also developed the theory of microstretch elastic solids [3] which is a generalization of the micropolar theory [4]. The particles of the solid with microstretch can expand and contract independent of translations and the rotations which they execute. Later, Eringen developed the theory of thermo-microstretch elastic solids [5]. For this class of materials, De Cicco and Nappa [6] derived the equations of the linear theory of thermo-microstretch elastic solids with the help of an entropy production inequality proposed by Green and Laws [7]. Bofill and Quintanilla [8] studied existence and uniqueness results. In the case of semi-infinite cylinders with the boundary lateral surface at null temperature, Quintanilla [9] established a spatial decay estimate controlled by an exponential of a polynomial of second degree. The spatial and temporal behavior of thermoelastodynamic processes for microstretch continuum materials was studied by Ciarletta and Scalia [10].

The class of the nonstandard problems attracted the attention of many researchers in the last two decades: Ames, Payne, Knops, Song, Ciarletta, Chiriţă, Quintanilla, Straughan, Passarella, and others. Knops and Payne [11] studied spatial behavior for the motion of a (semi-infinite) cylinder composed of a nonhomogeneous anisotropic linear elastic material and subject to zero body force and zero lateral boundary conditions. The initial displacement and velocity are not prescribed, nor is the asymptotic behavior at large axial distance. It is prescribed a proportion between displacement and velocity at a given time and, respectively, their initial values. Similar problems were studied by Chiriţă and Ciarletta [12] for the theory of linear thermoelasticity without energy dissipation and by Bulgariu [13] for the theory of elasticity with voids.

We consider a cylinder occupied by an anisotropic nonhomogeneous compressible linear thermo-microstretch elastic material, which is subject to null supply terms and null lateral boundary conditions. The internal energy density per unit of initial volume is assumed to be positive definite, and the constitutive coefficients are assumed bounded from above. Initial data are not prescribed, neither is the asymptotic behavior at large axial distance. We establish decay and growth exponential estimates with respect to axial variable for an integral of cross-sectional energy.

The problem studied in this paper finds application in geology and structural engineering. In [11] we have the example of a pile driven into a rigid foundation that prevents movement of the lateral boundary. The time-dependent displacement, microrotation, microstretch, and variation of temperature prescribed over the excited end constrains the motion such that the displacement, microrotation, microstretch, and variation of temperature and their derivatives with respect to time at points in the cylinder and at a prescribed time are given in proportion to, but not identical with, their respective initial values. It is desired to predict the deformation at each cross-section of the pile in terms of the base displacement, microrotation, microstretch, and variation of temperature.

2. Notation and Basic Formulation

Consider a prismatic cylinder Ω3 whose bounded uniform cross-section 𝐷2 has piecewise continuously differentiable boundary 𝜕𝐷.

The standard convention of summation over repeated suffixes is adopted, and a subscript comma denotes the spatial partial differentiation with respect to the corresponding cartesian coordinate and a superposed dot denotes differentiation with respect to time. Greek subscripts vary over {1,2}, and Latin subscripts vary over {1,2,3}. The letter 𝑠 is reserved for use as a time integration variable.

With respect to the chosen Cartesian coordinates, a partial volume of the cylinder will be denoted byΩ𝑧1,𝑧2=𝐱Ω𝑧1𝑥3𝑧2,(2.1) and, for semi-infinite cylinder, it is convenient to introduce the abbreviationsΩ(𝑧)=𝐱Ω𝑧𝑥3,Ω0(𝑧)=𝐱Ω𝑧𝑥3attime𝑡=0.(2.2) For the cylinder of finite length 𝐿 (if the cylinder is semi-finite we take 𝐿=), Ω is equivalent with Ω(0,𝐿), while Ω and Ω(0) are equivalent for the semi-infinite cylinder. To be more explicit, we will employ the notation 𝐷(𝑥3,𝑡) to indicate that respective quantities are to be evaluated at time 𝑡 over the cross-section whose distance from the origin is 𝑥3.

In this paper, we consider the theory of thermo-microstretch elastic solids. The equations of this theory are [5](i)the evolutive equations: 𝑡𝑗𝑖,𝑗+𝑓𝑖=𝜌̈𝑢𝑖,𝑚𝑗𝑖,𝑗+𝜀𝑖𝑟𝑠𝑡𝑟𝑠+𝑔𝑖=𝐼𝑖𝑗̈𝜑𝑗,𝜋𝑖,𝑖𝜎+=𝐽̈𝜓,𝜌𝑇0̇𝜂=𝑞𝑖,𝑖+𝑠inΩ×(0,𝑇),(2.3)(ii)the constitutive equations: 𝑡𝑖𝑗=𝐴𝑖𝑗𝑟𝑠𝑒𝑟𝑠+𝐵𝑖𝑗𝑟𝑠𝜅𝑟𝑠+𝐷𝑖𝑗𝑟𝛾𝑟+𝐴𝑖𝑗𝜓𝛽𝑖𝑗𝑚𝜃,𝑖𝑗=𝐵𝑟𝑠𝑖𝑗𝑒𝑟𝑠+𝐶𝑖𝑗𝑟𝑠𝜅𝑟𝑠+𝐸𝑖𝑗𝑟𝛾𝑟+𝐵𝑖𝑗𝜓𝐶𝑖𝑗𝜃,3𝜋𝑖=𝐷𝑟𝑠𝑖𝑒𝑟𝑠+𝐸𝑟𝑠𝑖𝜅𝑟𝑠+𝐷𝑖𝑗𝛾𝑗+𝑑𝑖𝜓𝜉𝑖𝜃,3𝜎=𝐴𝑟𝑠𝑒𝑟𝑠+𝐵𝑟𝑠𝜅𝑟𝑠+𝑑𝑖𝛾𝑖+𝑚𝜓𝜁𝜃,𝜌𝜂=𝛽𝑟𝑠𝑒𝑟𝑠+𝐶𝑟𝑠𝜅𝑟𝑠+𝜉𝑖𝛾𝑖𝑞+𝜁𝜓+𝑎𝜃,𝑖=𝑘𝑖𝑗𝜃,𝑗in[Ω×0,𝑇),(2.4)(iii)the geometric relations𝑒𝑖𝑗=𝑢𝑗,𝑖+𝜀𝑗𝑖𝑘𝜑𝑘,𝜅𝑖𝑗=𝜑𝑗,𝑖,𝛾𝑖=𝜓,𝑖,onΩ.(2.5) In the above equations we have used the following notations: 𝑡𝑖𝑗 is the stress tensor, 𝑚𝑖𝑗 is the couple stress tensor, 𝜋𝑖 is the microstress vector, 𝜎 is the scalar microstress function, 𝜂 is the specific entropy, 𝜌 is the mass density (mass in the reference configuration), 𝑓𝑖 is the body force, 𝑔𝑖 is the body couple, is the (scalar) body load, 𝑠 is the heat source density, 𝑞𝑖 is the heat flux vector, 𝐼𝑖𝑗 is the microinertia tensor, 𝐽 is the microstretch inertia, and 𝜀𝑖𝑗𝑘 is the alternating symbol. The variables of this theory are as follows: 𝑢𝑖 the components of the displacement vector, 𝜑𝑖 the components of the microrotation vector, 𝜓 the microstretch function, and 𝜃 the variation of temperature from the uniform reference absolute temperature 𝑇0.

The constitutive coefficients and 𝐼𝑖𝑗 are prescribed functions of the spatial variable with the following symmetries:𝐴𝑖𝑗𝑟𝑠=𝐴𝑟𝑠𝑖𝑗,𝐶𝑖𝑗𝑟𝑠=𝐶𝑟𝑠𝑖𝑗,𝐷𝑖𝑗=𝐷𝑗𝑖,𝑘𝑖𝑗=𝑘𝑗i,𝐼𝑖𝑗=𝐼𝑗𝑖.(2.6) Moreover we have𝑘𝑖𝑗𝜃,𝑖𝜃,𝑗0,𝐼𝑖𝑗𝜑,𝑖𝜑,𝑗0.(2.7)

We assume that 𝜌,𝐼𝑖𝑗,𝐽 and the constitutive coefficients are continuous and bounded fields on the closure Ω. We also assume that the constitutive coefficients are continuous differentiable functions on Ω and𝜌(𝐱)𝜌0>0,𝑎(𝐱)𝑎0>0,𝐽(𝐱)𝐽0>0,𝐼(𝐱)𝐼0>0,(2.8) where 𝐼(𝐱) denote the minimum eigenvalue of 𝐼𝑖𝑗(𝐱) and 𝜌0,𝑎0,𝐽0,𝐼0 are constants.

By taking into account that 𝑘𝑖𝑗 is a positive definite tensor, we have𝑘𝑚𝜃,𝑖𝜃,𝑖𝑘𝑖𝑗𝜃,𝑖𝜃,𝑗𝑘𝑀𝜃,𝑖𝜃,𝑖,(2.9) where 𝑘𝑚 and 𝑘𝑀 are related to the minimum and the maximum eigenvalue (conductivity moduli) for 𝑘𝑖𝑗. By using the Schwarz's inequality we have𝑞𝑖𝑞𝑖𝑘𝑀𝑘𝑖𝑗𝜃,𝑖𝜃,𝑗.(2.10)

In what follows we denote with 𝒫 the nonstandard problem structured by equations (2.3)–(2.5) with null supply terms, supplemented by the lateral boundary conditionṡ𝑢𝑖𝑡𝛼𝑖𝑛𝛼=0,̇𝜑𝑖𝑚𝛼𝑖𝑛𝛼=0,̇𝜓𝜋𝛼𝑛𝛼=0,𝜃𝑞𝛼𝑛𝛼[][],=0,(𝐱,𝑡)(𝜕𝐷×0,𝐿)×0,𝑇(2.11) conditions on the base𝑢𝑖(𝐱,𝑡)=𝑎𝑖𝑥𝛼,𝑡,𝜑𝑖(𝐱,𝑡)=𝑏𝑖𝑥𝛼𝑥,𝑡,𝜓(𝐱,𝑡)=𝑐𝛼,𝑥,𝑡𝜃(𝐱,𝑡)=𝜏𝛼[],,𝑡,(𝐱,𝑡)𝐷(0)×0,𝑇(2.12) and the final values at time 𝑇 of the displacement, microrotation, microstretch, and variation of temperature and their derivatives with respect to time are proportional to their initial values, that is,𝑢𝑖(𝐱,𝑇)=𝜆𝑢𝑖(𝐱,0),𝜑𝑖(𝐱,𝑇)=𝜆𝜑𝑖𝜃(𝐱,0),𝜓(𝐱,𝑇)=𝜆𝜓(𝐱,0),(𝐱,𝑇)=𝜈𝜃(𝐱,0)for𝐱Ω,(2.13)̇𝑢𝑖(𝐱,𝑇)=𝛼̇𝑢𝑖(𝐱,0),̇𝜑𝑖(𝐱,𝑇)=𝜇̇𝜑𝑖(𝐱,0),̇𝜓(𝐱,𝑇)=𝛽̇𝜓(𝐱,0),𝐱Ω,(2.14) where 𝐧 is the unit outward normal on 𝜕𝐷,𝑎𝑖(𝑥𝛼,𝑡),𝑎𝑖(𝑥𝛼,𝑡),𝑐(𝑥𝛼,𝑡), and 𝜏(𝑥𝛼,𝑡) are prescribed differentiable functions compatible with the initial/final data and the lateral boundary data. The constants 𝛼,𝜇,𝜆,𝛽, and 𝜈 are prescribed and satisfy the conditions||𝜇||||𝛽||||𝜆|||𝛼|>1,>1,>1,>1|𝜈|>1.(2.15)

The initial displacement, microrotation, microstretch, and variation of temperature and their derivatives with respect to time at points in the cylinder are not prescribed. The conditions specified on the end 𝐷(𝐿) for a finite cylinder, or at asymptotically large axial distance for the semi-infinite cylinder, are also not prescribed.

We will use the notations 𝜘={𝜓𝑖𝑗,𝜒𝑖𝑗,𝜈𝑖,𝜙} and 𝜘(𝛼)={𝜓(𝛼)𝑖𝑗,𝜒(𝛼)𝑖𝑗,𝜈𝑖(𝛼),𝜙(𝛼)}, 𝛼=1,2. We assume that the internal energy density per unit of volume is a positive definite quadratic form, therefore we can write𝑚𝜓𝑖𝑗𝜓𝑖𝑗+𝑚𝜒𝑖𝑗𝜒𝑖𝑗+𝑚𝜈𝑖𝜈𝑖+𝑚𝜙22𝑊(𝜘)𝑀𝜓𝑖𝑗𝜓𝑖𝑗+𝑀𝜒𝑖𝑗𝜒𝑖𝑗+𝑀𝜈𝑖𝜈𝑖+𝑀𝜙2,forall𝜘,(2.16) where 𝑚,𝑀,𝑚,𝑀,𝑚,𝑀,𝑚,𝑀 are positive constants related to the minimum and, respectively, maximum eigenvalues of the positive definite quadratic form 2𝑊(𝜘)=𝐴𝑖𝑗𝑟𝑠𝜓𝑖𝑗𝜓𝑟𝑠+𝐶𝑖𝑗𝑟𝑠𝜒𝑖𝑗𝜒𝑟𝑠+𝐷𝑖𝑗𝜈𝑖𝜈𝑗+𝑚𝜙2+2𝐵𝑖𝑗𝑟𝑠𝜓𝑖𝑗𝜒𝑟𝑠+2𝐷𝑖𝑗𝑟𝜓𝑖𝑗𝜈𝑟+2𝐸𝑖𝑗𝑟𝜒𝑖𝑗𝜈𝑟+2𝐴𝑖𝑗𝜓𝑖𝑗𝜙+2𝐵𝑖𝑗𝜒𝑖𝑗𝜙+2𝑑𝑖𝜈𝑖𝜙.(2.17)

By dedublation of this quadratic form we have𝜘2(1),𝜘(2)=𝐴𝑖𝑗𝑟𝑠𝜓(1)𝑖𝑗𝜓(2)𝑟𝑠+𝐶𝑖𝑗𝑟𝑠𝜒(1)𝑖𝑗𝜒(2)𝑟𝑠+𝐷𝑖𝑗𝜈𝑖(1)𝜈𝑗(2)+𝑚𝜙(1)𝜙(2)+𝐵𝑖𝑗𝑟𝑠𝜓(1)𝑖𝑗𝜒(2)𝑟𝑠+𝜓(2)𝑖𝑗𝜒(1)𝑟𝑠+𝐷𝑖𝑗𝑟𝜓(1)𝑖𝑗𝜈𝑟(2)+𝜓(2)𝑖𝑗𝜈𝑟(1)+𝐸𝑖𝑗𝑟𝜒(1)𝑖𝑗𝜈𝑟(2)+𝜒(2)𝑖𝑗𝜈𝑟(1)+𝐴𝑖𝑗𝜓(1)𝑖𝑗𝜙(2)+𝜓(2)𝑖𝑗𝜙(1)+𝐵𝑖𝑗𝜒(1)𝑖𝑗𝜙(2)+𝜒(2)𝑖𝑗𝜙(1)+𝑑𝑖𝜈𝑖(1)𝜙(2)+𝜈𝑖(2)𝜙(1).(2.18) We can remark that (𝜘,𝜘)=𝑊(𝜘). By the Cauchy-Schwarz inequality, we have𝜘(1),𝜘(2)𝑊𝜘(1)1/2𝑊𝜘(2)1/2.(2.19)

We introduce the following notations: 𝑇𝑖𝑗=𝑡𝑖𝑗+𝛽𝑖𝑗𝜃, 𝑀𝑖𝑗=𝑚𝑖𝑗+𝐶𝑖𝑗𝜃, Π𝑖=3𝜋𝑖+𝜉𝑖𝜃, Σ=3𝜎+𝜁𝜃. If, in (2.16), we take 𝜘={(1/𝜌0)𝑇𝑖𝑗,(1/𝐼0)𝑀𝑖𝑗,(1/𝐽0)Π𝑖,(1/𝜌0)Σ}, we obtain2𝑊𝜘𝑀𝜌20𝑇𝑖𝑗𝑇𝑖𝑗+𝑀𝐼20𝑀𝑖𝑗𝑀𝑖𝑗+𝑀𝐽20Π𝑖Π𝑖+𝑀𝜌20Σ21𝜛𝜌0𝑇𝑖𝑗𝑇𝑖𝑗+1𝐼0𝑀𝑖𝑗𝑀𝑖𝑗+1𝐽0Π𝑖Π𝑖+1𝜌0Σ2,(2.20) where 𝜛=max𝑀𝜌0,𝑀𝐼0,𝑀𝐽0,𝑀𝜌0.(2.21)

From (2.17)–(2.20), we have1𝜌0𝑇𝑖𝑗𝑇𝑖𝑗+1𝐼0𝑀𝑖𝑗𝑀𝑖𝑗+1𝐽0Π𝑖Π𝑖+1𝜌0Σ2𝑒=2𝑖𝑗,𝜅𝑖𝑗,𝛾𝑖,1,𝜓𝜌0𝑇𝑖𝑗,1𝐼0𝑀𝑖𝑗,1𝐽0Π𝑖,1𝜌0Σ[]2𝑊1/22𝑊𝜘1/212𝜛𝑊𝜌0𝑇𝑖𝑗𝑇𝑖𝑗+1𝐼0𝑀𝑖𝑗𝑀𝑖𝑗+1𝐽0Π𝑖Π𝑖+1𝜌0Σ21/2,(2.22) where 𝑊=𝑊({𝑒𝑖𝑗,𝜅𝑖𝑗,𝛾𝑖,𝜓}), and consequently we obtain1𝜌0𝑇𝑖𝑗𝑇𝑖𝑗+1𝐼0𝑀𝑖𝑗𝑀𝑖𝑗+1𝐽0Π𝑖Π𝑖2𝜛𝑊.(2.23)

By using relations (2.4), (2.5), and (2.17), we obtain𝑡𝑖𝑗̇𝑒𝑖𝑗+𝑚𝑖𝑗̇𝜅𝑖𝑗+3𝜋𝑖̇𝛾𝑖1+3𝜎̇𝜓+𝜌̇𝜂𝜃+𝑇0𝑞𝑖𝜃,𝑖=̇1𝑊+𝑇0𝑘𝑖𝑗𝜃,𝑖𝜃,𝑗̇+𝑎𝜃𝜃.(2.24)

3. A Differential Inequality

The aim of this section is to obtain a differential inequality for an appropriate function related to the cross-sectional energy flux.

We introduce the following function:𝐼𝑥3=𝑇0𝐷(𝑥3,𝑠)𝑒𝜔𝑠𝑡3𝑖̇𝑢𝑖+𝑚3𝑖̇𝜑𝑖+3𝜋31̇𝜓+𝑇0𝑞3𝜃𝑑𝑎𝑑𝑠,0𝑥3𝐿,(3.1) where 𝜔 is a positive parameter at our disposal whose values will be defined later.

By direct differentiation with respect to 𝑥3 in (3.1) and by using the evolutive equations (2.3) with null supply terms, we obtain 𝑑𝐼𝑑𝑥3𝑥3=𝑇0𝐷(𝑥3,𝑠)𝑒𝜔𝑠𝜌̇𝑢𝑖̈𝑢𝑖+𝐼𝑖𝑗̇𝜑𝑖̈𝜑𝑗+3𝐽̇𝜓̈𝜓+𝜌̇𝜂𝜃𝑡𝛼𝑖,𝛼̇𝑢𝑖+𝑡3𝑖̇𝑢𝑖,3𝑚𝛼𝑖,𝛼̇𝜑𝑖𝜀𝑗𝑖𝑘𝑡𝑗𝑖̇𝜑𝑘+𝑚3𝑖̇𝜑𝑖,33𝜋𝛼,𝛼̇𝜓+3𝜎̇𝜓+3𝜋3̇𝜓,31𝑇0𝑞𝛼,𝛼1𝜃+𝑇0𝑞3𝜃,3𝑑𝑎𝑑𝑠,(3.2) and by using the geometric relations (2.5), we have 𝑑𝐼𝑑𝑥3𝑥3=𝑇0𝐷(𝑥3,𝑠)𝑒𝜔𝑠𝜌̇𝑢𝑖̈𝑢𝑖+𝐼𝑖𝑗̇𝜑𝑖̈𝜑𝑗+3𝐽̇𝜓̈𝜓+𝜌̇𝜂𝜃+𝑡𝑖𝑗̇𝑒𝑖𝑗+𝑚𝑖𝑗̇𝜅𝑖𝑗+3𝜋𝑖̇𝛾𝑖1+3𝜎̇𝜓+𝑇0𝑞𝑖𝜃,𝑖𝑑𝑎𝑑𝑠𝑇0𝐷(𝑥3,𝑠)𝑒𝜔𝑠𝑡𝛼𝑖̇𝑢𝑖+𝑚𝛼𝑖̇𝜑𝑖+3𝜋𝛼1̇𝜓+𝑇0𝑞𝛼𝜃,𝛼𝑑𝑎𝑑𝑠.(3.3) The divergence theorem and the lateral boundary conditions (2.11) ensure us that the final integral in the right-hand term vanishes. Using relation (2.24), we deduce 𝑑𝐼𝑑𝑥3𝑥3=12𝑇0𝐷(𝑥3,𝑠)𝑒𝜔𝑠𝜕𝜕𝑠𝜌̇𝑢𝑖̇𝑢𝑖+𝐼𝑖𝑗̇𝜑𝑖̇𝜑𝑗+3𝐽̇𝜓̇𝜓+2𝑊+𝑎𝜃2+𝑑𝑎𝑑𝑠𝑇0𝐷(𝑥3,𝑠)1𝑇0𝑒𝜔𝑠𝑘𝑖𝑗𝜃,𝑖𝜃,𝑗𝑑𝑎𝑑𝑠.(3.4)

Finally, the above equation yields𝑑𝐼𝑑𝑥3𝑥3𝑥=𝐸3𝑥,𝑇𝐸3+,0𝑇0𝑥𝜔𝐸3+,𝑠𝑑𝑠𝑇0𝐷(𝑥3,𝑠)1𝑇0𝑒𝜔𝑠𝑘𝑖𝑗𝜃,𝑖𝜃,𝑗𝑑𝑎𝑑𝑠,(3.5) where𝐸𝑥3=1,𝑡2𝐷(𝑥3,𝑠)𝑒𝜔𝑠𝜌̇𝑢𝑖̇𝑢𝑖+𝐼𝑖𝑗̇𝜑𝑖̇𝜑𝑗+3𝐽̇𝜓2+2𝑊+𝑎𝜃2[].𝑑𝑎,𝑡0,𝑇(3.6) Therefore, by means of relations (2.13) and (2.14), we have𝑑𝐼𝑑𝑥3𝑥3=12𝑒𝜔𝑇𝛼21𝐷(𝑥3,0)𝜌̇𝑢𝑖̇𝑢𝑖+1𝑑𝑎2𝑒𝜔𝑇𝜇21𝐷(𝑥3,0)𝐼𝑖𝑗̇𝜑𝑖̇𝜑𝑗+1𝑑𝑎2𝑒𝜔𝑇𝛽21𝐷(𝑥3,0)3𝐽̇𝜓2+1𝑑𝑎2𝑒𝜔𝑇𝜆21𝐷(𝑥3,0)+12𝑊𝑑𝑎2𝑒𝜔𝑇𝜈21𝐷(𝑥3,0)𝑎𝜃2+𝑑𝑎𝑇0𝑥𝜔𝐸3,𝑠𝑑𝑠+𝑇0𝐷(𝑥3,𝑠)1𝑇0𝑒𝜔𝑠𝑘𝑖𝑗𝜃,𝑖𝜃,𝑗𝑑𝑎𝑑𝑠.(3.7)

We will choose the parameter 𝜔 so that we have0<𝜒𝜔=12𝑒min𝜔𝑇𝛼21,𝑒𝜔𝑇𝜇21,𝑒𝜔𝑇𝛽21,𝑒𝜔𝑇𝜆21,𝑒𝜔𝑇𝜈2,1(3.8) assumming that 𝜔 ranges in the set20<𝜔<𝑇||𝜇||||𝛽||||𝜆||minln|𝛼|,ln,ln,ln,ln|𝜈|(3.9) if we suppose that conditions (2.15) hold true.

In this context, we note that𝑑𝐼𝑑𝑥3𝑥3𝜒𝜔𝐷(𝑥3,0)𝜌̇𝑢𝑖̇𝑢𝑖+𝐼𝑖𝑗̇𝜑𝑖̇𝜑𝑗+3𝐽̇𝜓2+2𝑊+𝑎𝜃2+𝑑𝑎𝑇0𝑥𝜔𝐸3,𝑠𝑑𝑠+𝑇0𝐷(𝑥3,𝑠)1𝑇0𝑒𝜔𝑠𝑘𝑖𝑗𝜃,𝑖𝜃,𝑗𝑑𝑎𝑑𝑠0,(3.10) in view of assumptions of the positive definitiveness of 𝑊 and 𝑘𝑖𝑗. Therefore, we can conclude that 𝐼(𝑥3) is a nondecreasing function with respect to 𝑥3 on [0,𝐿].

Next, we want to obtain an appropriate estimate for the function 𝐼(𝑥3). Using the constitutive equations, the Schwarz's inequality, the arithmetic-mean inequality, and relation (2.23), we obtain the inequality1𝜌0𝑡𝑖𝑗𝑡𝑖𝑗+1𝐼0𝑚𝑖𝑗𝑚𝑖𝑗+1𝐽03𝜋𝑖𝜋𝑖=T𝑖𝑗𝜌0𝛽𝑖𝑗𝜌0𝜃𝑇𝑖𝑗𝜌0𝛽𝑖𝑗𝜌0𝜃+𝑀𝑖𝑗𝐼0𝐶𝑖𝑗𝐼0𝜃𝑀𝑖𝑗𝐼0𝐶𝑖𝑗𝐼0𝜃+Π𝑖3𝐽0𝜉𝑖3𝐽0𝜃Π𝑖3𝐽0𝜉𝑖3𝐽0𝜃1+𝜀11𝜌0𝑇𝑖𝑗𝑇𝑖𝑗+1𝐼0𝑀𝑖𝑗𝑀𝑖𝑗+13𝐽0Π𝑖Π𝑖+11+𝜀11𝜌0𝛽𝑖𝑗𝛽𝑖𝑗+1𝐼0𝐶𝑖𝑗𝐶𝑖𝑗+13𝐽0𝜉𝑖𝜉𝑖𝜃21+𝜀112𝜛𝑊+1+𝜀1𝑀2𝜃2,𝜀1>0,(3.11) where 𝑀2=maxΩ1𝜌0𝛽𝑖𝑗𝛽𝑖𝑗+1𝐼0𝐶𝑖𝑗𝐶𝑖𝑗+13𝐽0𝜉𝑖𝜉𝑖.(3.12) Using Schwarz's inequality, arithmetic-mean inequality, and relations (2.10) and (3.11), we obtain the estimate||𝐼𝑥3||𝑇0𝐷(𝑥3,𝑠)𝑒𝜔𝑠𝜀221𝜌0𝑡3𝑖𝑡3𝑖+1𝐼0𝑚3𝑖𝑚3𝑖+3𝐽0𝜋3𝜋3+𝜀32𝑇0𝑎0𝑞3𝑞3+12𝜀2𝜌̇𝑢𝑖̇𝑢𝑖+𝐼𝑖𝑗̇𝜑𝑖̇𝜑𝑗+3𝐽̇𝜓2+12𝑇0𝜀3𝑎𝜃2𝑑𝑎𝑑𝑠𝑇0𝐷(𝑥3,𝑠)𝑒𝜔𝑠𝜀221𝜌0𝑡𝑖𝑗𝑡𝑖𝑗+1𝐼0𝑚𝑖𝑗𝑚𝑖𝑗+3𝐽0𝜋𝑖𝜋𝑖+𝜀32𝑇0𝑎0𝑞𝑖𝑞𝑖+12𝜀2𝜌̇𝑢𝑖̇𝑢𝑖+𝐼𝑖𝑗̇𝜑𝑖̇𝜑𝑗+3𝐽̇𝜓2+12𝑇0𝜀3𝑎𝜃2𝑑𝑎𝑑𝑠𝑇0𝐷(𝑥3,𝑠)𝑒𝜔𝑠1𝜔𝜀2𝜔2𝜌̇𝑢𝑖̇𝑢𝑖+𝐼𝑖𝑗̇𝜑𝑖̇𝜑𝑗+3𝐽̇𝜓2+𝜀21+𝜀1𝜛𝜔𝜔2𝜀2𝑊+2𝑀21+𝜀1𝜔𝑎0𝜀1+1𝜔𝑇0𝜀3𝜔2𝑎𝜃2+𝜀3𝑘𝑀2𝑎01𝑇0𝑘𝑖𝑗𝜃,𝑖𝜃,𝑗𝑑𝑎𝑑𝑠,𝜀1,𝜀2,𝜀3>0.(3.13) Further, we equate the coefficients of all energetic terms in the last integral imposing that1𝜔𝜀2=𝜀21+𝜀1𝜛𝜔=𝑀2𝜀21+𝜀1𝜔𝑎0𝜀1+1𝜔𝑇0𝜀3=𝜀3𝑘𝑀2𝑎0,(3.14) and hence we have𝜀2=11+𝜀1𝜛,𝜀3=2𝑎0𝑘𝑀𝜔1+𝜀1𝜛,(3.15) where𝜀1=1212𝑇0𝑀2+𝜔𝑘𝑀2𝑎0𝑇0𝜛+12𝑇0𝑀2+𝜔𝑘𝑀2𝑎0𝑇0𝜛2𝑀+42𝑎0𝜛.(3.16)

Because we imposed (3.14), we can multiply (3.13) by 𝑐=𝜔𝜀2 and obtain 𝑐||𝐼𝑥3||𝑇0𝑥𝜔𝐸3,𝑠𝑑s+𝑇0𝐷(𝑥3,𝑠)1𝑇0𝑒𝜔𝑠𝑘𝑖𝑗𝜃,𝑖𝜃,𝑗𝑑𝑎𝑑𝑠,(3.17) and from (3.10) we obtain the first-order differential inequality𝑐||𝐼𝑥3||𝑑𝐼𝑑𝑥3𝑥3,𝑥3[]0,𝐿.(3.18)

4. Spatial Behaviour of 𝐼(𝑥3)

In this section we determine the spatial evolution of the solution of the nonstandard problem 𝒫 by integrating (3.18). We first consider that the cylinder has a finite length (i.e., 𝐿<). We have only two possibilities: (a) 𝐼(𝑥3)0 for all 𝑥3[0,𝐿] or (b) there exists 𝑧[0,𝐿] so that 𝐼(𝑧)>0. It is easy to see that, if 𝐼(𝐿)0, we are in the case (a).

Let us first consider the case (a). Because 𝐼(𝑥3) it is a nondecreasing function with respect to 𝑥3 on [0,𝐿], we have𝑑𝐼𝑑𝑥3𝑥3𝑥+𝑐𝐼30,0𝑥3𝐿,(4.1) which after integration leads to the following Saint-Venant-type decay estimate:𝑥0𝐼3𝐼(0)𝑒𝑐𝑥3,0𝑥3𝐿.(4.2) For a cylinder of finite length 𝐿, we have to prescribe such boundary conditions on the end 𝑥3=𝐿 that implies 𝐼(𝐿)=0, and then we will predict a spatial exponential decay as described in (4.2).

For the case (b), we have 0𝐼(𝑧)𝐼(𝑥3) for 𝑧𝑥3𝐿, and hence (3.18) implies the inequality0𝑑𝐼𝑑𝑥3𝑥3𝑥𝑐𝐼3,𝑧𝑥3𝐿,(4.3) which, by integrating, yields the following growth estimate𝐼𝑥3𝐼(𝑧)𝑒𝑐(𝑥3𝑧),𝑧𝑥3𝐿.(4.4)

Let us discuss further the case of a semi-infinite cylinder (i.e., the case when 𝐿).

If 𝐼(𝑥3)0 for all 𝑥3[0,), we obtain that 𝐼(𝑥3)0 as 𝑥3, and hence relation (3.10) gives the following decay estimate for the weighted total energy:𝑥3𝐼(0)𝑒𝑐𝑥3,0𝑥3<,(4.5) where𝑥3=𝜒𝜔Ω0(𝑥3)𝜌̇𝑢𝑖̇𝑢𝑖+𝐼𝑖𝑗̇𝜑𝑖̇𝜑𝑗+3𝐽̇𝜓2+2𝑊+𝑎𝜃2𝑑𝑣+lim𝑥3𝑇0𝑥3𝑥3𝜔𝐸(𝜗,𝑠)𝑑𝜗𝑑𝑠+𝑇0Ω(𝑥3)1𝑇0𝑒𝜔𝑠𝑘𝑖𝑗𝜃,𝑖𝜃,𝑗𝑑𝑣𝑑𝑠,0𝑥3<.(4.6)

If there is 𝑧[0,) so that 𝐼(𝑧)>0, for the semi-infinite cylinder, 𝐼(𝑥3) becomes unbounded for 𝑥3 and hence (𝑥3) is infinite.

We have established a Phragmén-Lindelöf alternative type for the semi-infinite cylinder.

5. Further Comments

In this paper we have discussed only the case when |𝛼|>1,|𝜇|>1,|𝛽|>1,|𝜆|>1, and |𝜈|>1. When we take the conditions |𝛼|<1,|𝜇|<1,|𝛽|<1,|𝜆|<1 and |𝜈|<1 we cannot find a suitable bound because there is nonuniqueness in this case (see e.g., Quintanilla and Straughan [14] for an argument concerning this subject).

In the previous sections we have considered the nonstandard problem 𝒫 in which the proportionality coefficients of displacement, microrotation, microstretch, and variation of temperature with their derivatives with respect to time at the time 𝑇 and their respective initial values are given by (2.13) and (2.14). We consider a similar nonstandard problem 𝒫 given by (2.3)–(2.5) with null supply terms, and instead of conditions (2.13), we have𝑢𝑖(𝐱,𝑇)=𝜆1𝑢𝑖(𝐱,0),𝜑𝑖(𝐱,𝑇)=𝜆2𝜑𝑖(𝐱,0),𝜓(𝐱,𝑇)=𝜆3𝜃𝜓(𝐱,0),(𝐱,𝑇)=𝜈𝜃(𝐱,0),𝐱Ω,(5.1) where 𝜆𝑖 and 𝜆𝑗 can be different for 𝑖𝑗, with conditions (2.14) remaining valid. The problem 𝒫 has the lateral boundary conditionṡ𝑢𝑖𝑡𝛼𝑖𝑛𝛼=0,𝜑=0,̇𝜓𝜋𝛼𝑛𝛼=0,𝜃𝑞𝛼𝑛𝛼[][]=0,(𝐱,𝑡)(𝜕𝐷×0,𝐿)×0,𝑇(5.2) and the conditions on the base𝑢𝑖(𝐱,𝑡)=𝑎𝑖𝑥𝛼,𝑡,𝜑𝑖(𝐱,𝑡)=𝑏𝑖𝑥𝛼𝑥,𝑡,𝜓(𝐱,𝑡)=𝑐𝛼,𝑥,𝑡𝜃(𝐱,𝑡)=𝜏𝛼[].,𝑡,(𝐱,𝑡)𝐷(0)×0,T(5.3)

We are interested in what conditions we would have to take for the constants 𝜆1,𝜆2, and 𝜆3 so that our study given in the previous sections may follow the same path.

The internal energy density per unit of volume is a positive definite quadratic form and so, for 𝜘={𝑒𝑖𝑗,𝜅𝑖𝑗,𝛾𝑖,𝜓} in (2.16), at the moment 𝑡=𝑇, we have2𝐷(𝑥3,𝑇)𝑊𝑑𝑎𝑚𝐷(𝑥3,𝑇)𝑒𝑖𝑗𝑒𝑖𝑗𝑑𝑎+𝑚𝐷(𝑥3,𝑇)𝜅𝑖𝑗𝜅𝑖𝑗𝑑𝑎+𝑚𝐷(𝑥3,𝑇)𝛾𝑖𝛾𝑖𝑑𝑎+𝑚𝐷(𝑥3,𝑇)𝜓2𝑑𝑎=𝑚𝐷(𝑥3,0)𝜆21𝑢𝑗,𝑖𝑢𝑗,𝑖+2𝜆1𝜆2𝜀𝑗𝑖𝑘𝑢𝑗,𝑖𝜑𝑘+2𝜆22𝜑𝑘𝜑𝑘2+𝑑𝑎𝑚𝜆222𝐷(𝑥3,0)𝜑𝑗,𝑖𝜑𝑗,𝑖𝑑𝑎+𝑚𝜆222𝐷(𝑥3,0)𝜑𝑗,𝑖𝜑𝑗,𝑖𝑑𝑎+𝑚𝜆23𝐷(𝑥3,0)𝜓,𝑖𝜓,𝑖𝑑𝑎+𝑚𝜆23𝐷(𝑥3,0)𝜓2𝑑𝑎.(5.4) The condition required in (5.2) that 𝜑=0 on 𝜕𝐷(𝑥3) gives us the possibility to apply the Poincaré inequality𝐷(𝑥3,0)𝜑𝑗,𝑖𝜑𝑗,𝑖𝑑𝑎𝛿𝐷(𝑥3,0)𝜑𝑖𝜑𝑖𝑑𝑎,(5.5) with 𝛿 a positive constant. Using the arithmetic-mean inequality, we can deduce2𝜆1𝜆2𝜀𝑗𝑖𝑘𝑢𝑗,𝑖𝜑𝑘||𝜆21𝑢𝑗,𝑖||||𝜆2𝜀𝑗𝑖𝑘𝜑𝑘||𝜀𝜆21𝑢𝑗,𝑖𝑢𝑗,𝑖2𝜀𝜆22𝜑𝑘𝜑𝑘,𝜀>0.(5.6)

Combining relations (5.5), (5.6), and (5.4), we have2𝐷(𝑥3,𝑇)𝑊𝑑𝑎(1𝜀)𝑚𝜆21𝐷(𝑥3,0)𝑢𝑗,𝑖𝑢𝑗,𝑖+𝑑𝑎2𝑚2𝑚𝜀+𝑚𝛿2𝜆22𝐷(𝑥3,0)𝜑𝑖𝜑𝑖+𝑑𝑎𝑚𝜆222𝐷(𝑥3,0)𝜑𝑗,𝑖𝜑𝑗,𝑖𝑑𝑎+𝑚𝜆23𝐷(𝑥3,0)𝜓,𝑖𝜓,𝑖𝑑𝑎+𝑚𝜆23𝐷(𝑥3,0)𝜓2𝑑𝑎.(5.7) Requiring that 1+𝑚𝛿4𝑚1<𝜀<1,(5.8) we ensured that the brackets from the right-hand terms in (5.7) are positive.

Using the inequality 𝑒𝑖𝑗𝑒𝑖𝑗=𝑢𝑗,𝑖+𝜀𝑗𝑖𝑘𝜑𝑘2<2𝑢𝑗,𝑖𝑢𝑗,𝑖+4𝜑𝑘𝜑𝑘,(5.9) from (2.16), at the moment 𝑡=0, we have 2𝐷(𝑥3,0)𝑊𝑑𝑎𝑀𝐷(𝑥3,0)𝑒𝑖𝑗𝑒𝑖𝑗𝑑𝑎+𝑀𝐷(𝑥3,0)𝜅𝑖𝑗𝜅𝑖𝑗𝑑𝑎+𝑀𝐷(𝑥3,0)𝛾𝑖𝛾𝑖𝑑𝑎+𝑀𝐷(𝑥3,0)𝜓2𝑑𝑎2𝑀𝐷(𝑥3,0)𝑢𝑗,𝑖𝑢𝑗,𝑖𝑑𝑎+4𝑀𝐷(𝑥3,0)𝜑𝑘𝜑𝑘𝑑𝑎+𝑀𝐷(𝑥3,0)𝜑𝑗,𝑖𝜑𝑗,𝑖𝑑𝑎+𝑀𝐷(𝑥3,0)𝜓,𝑖𝜓,𝑖𝑑𝑎+𝑀𝐷(𝑥3,0)𝜓2𝑑𝑎.(5.10)

Combining relations (5.7) and (5.10), we have12𝑒𝜔𝑇𝐷(𝑥3,𝑇)12𝑊𝑑𝑎2𝐷(𝑥3,0)12𝑊2(1𝜀)𝑒𝜔𝑇𝑚𝜆212𝑀12𝑀𝐷(𝑥3,0)𝑢𝑗,𝑖𝑢𝑗,𝑖+1𝑑𝑎24𝑚4𝑚𝜀1+𝑚𝛿𝑒𝜔𝑇𝜆228𝑀14𝑀𝐷(𝑥3,0)𝜑𝑖𝜑𝑖+1𝑑𝑎2𝑒𝜔𝑇𝑚𝜆222𝑀1𝑀𝐷(𝑥3,0)𝜑𝑗,𝑖𝜑𝑗,𝑖+1𝑑𝑎2𝑒𝜔𝑇𝑚𝜆23𝑀1𝑀D(𝑥3,0)𝜓,𝑖𝜓,𝑖+1𝑑𝑎2𝑒𝜔𝑇𝑚𝜆23𝑀1𝑚𝐷(𝑥3,0)𝜓2𝑑𝑎,(5.11) and so, instead of relation (3.7), in the case of the problem 𝒫 we obtain 𝑑𝐼𝑑𝑥3𝑥312𝑒𝜔𝑇𝛼21𝐷(𝑥3,0)𝜌̇𝑢𝑖̇𝑢𝑖1𝑑𝑎+2𝑒𝜔𝑇𝜇21𝐷(𝑥3,0)𝐼𝑖𝑗̇𝜑𝑖̇𝜑𝑗+1𝑑𝑎2𝑒𝜔𝑇𝛽21𝐷(𝑥3,0)3𝐽̇𝜓21𝑑𝑎+2𝑒𝜔𝑇𝜈21𝐷(𝑥3,0)𝑎𝜃2+1𝑑𝑎2(1𝜀)𝑒𝜔𝑇𝑚𝜆212𝑀12𝑀𝐷(𝑥3,0)𝑢𝑗,𝑖𝑢𝑗,𝑖+1𝑑𝑎24𝑚4𝑚𝜀1+𝑚𝛿𝑒𝜔𝑇𝜆228𝑀14𝑀𝐷(𝑥3,0)𝜑𝑖𝜑𝑖+1𝑑𝑎2𝑒𝜔𝑇𝑚𝜆222𝑀1𝑀𝐷(𝑥3,0)𝜑𝑗,𝑖𝜑𝑗,𝑖+1𝑑𝑎2𝑒𝜔𝑇𝑚𝜆23𝑀1𝑀𝐷(𝑥3,0)𝜓,𝑖𝜓,𝑖+1𝑑𝑎2𝑒𝜔𝑇𝑚𝜆23𝑀1𝑚𝐷(𝑥3,0)𝜓2𝑑𝑎+𝑇0𝑥𝜔𝔈3+,s𝑑𝑠𝑇0𝐷(𝑥3,𝑠)1𝑇0𝑒𝜔𝑠𝑘𝑖𝑗𝜃,𝑖𝜃,𝑗𝑑𝑎𝑑𝑠.(5.12) We choose 𝜔 to ensure that0<𝜒𝜔=12𝑒min𝜔𝑇𝛼21,𝑒𝜔𝑇𝜇21,𝑒𝜔𝑇𝛽21,𝑒𝜔𝑇𝜈21,(1𝜀)𝑒𝜔𝑇𝑚𝜆212𝑀1,4𝑚4𝑚𝜀1+𝑚𝛿𝑒𝜔𝑇𝜆228𝑀𝑒1,𝜔𝑇𝑚𝜆222𝑀𝑒1,𝜔𝑇𝑚𝜆23𝑀𝑒1,𝜔𝑇𝑚𝜆23𝑀,1(5.13) and so, the range from which we can take 𝜔 is10<𝜔<𝑇𝛼lnmin2,𝜇2,𝛽2,𝜈2,(1𝜀)𝑚𝜆212𝑀,4𝑚4𝑚𝜀1+𝑚𝛿𝜆228𝑀,𝑚𝜆222𝑀,𝑚𝜆23𝑀,𝑚𝜆23𝑀(5.14) if the conditions (2.15)1,2,3,5 hold true and||𝜆1||>2𝑀𝑚||𝜆>1,2||>max8𝑀4𝑚+𝑚𝛿,2𝑀𝑚||𝜆>1,3||>max𝑀𝑚,𝑀𝑚>1.(5.15)

From relations (5.12) and (5.13), we obtain an inequality similar to (3.10) and so, we can continue like in Sections 3 and 4 to obtain a Saint-Venant type estimate or a Phragmén-Lindelöf-type estimate.

In conclusion, if we replace conditions (2.13) with those given in (5.1), the results obtained in Sections 3 and 4 hold true if in (2.11) we change the lateral boundary condition ̇𝜑𝑚𝛼𝑖𝑛𝛼=0 with 𝜑=0 for (𝐱,𝑡)(𝜕𝐷×[0,𝐿])×[0,𝑇] and the constants 𝜆1,𝜆2,𝜆3,𝜈,𝛼,𝛽 and 𝜇 must satisfy conditions (2.15)1,2,3,5 and (5.15).


The author acknowledges support from the Romanian Ministry of Education and Research through CNCSIS-UEFISCSU, Project PN II-RU TE code 184, no. 86/30.07.2010.