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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.

The theory of micromorphic bodies was introduced by Eringen [

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 [

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 [

Consider a prismatic cylinder

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

With respect to the chosen Cartesian coordinates, a partial volume of the cylinder will be denoted by

In this paper, we consider the theory of thermo-microstretch elastic solids. The equations of this theory are [

the evolutive equations:

the constitutive equations:

the geometric relations

The constitutive coefficients and

We assume that

By taking into account that

In what follows we denote with

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

We will use the notations

By dedublation of this quadratic form we have

We introduce the following notations:

From (

By using relations (

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:

By direct differentiation with respect to

Finally, the above equation yields

We will choose the parameter

In this context, we note that

Next, we want to obtain an appropriate estimate for the function

Because we imposed (

In this section we determine the spatial evolution of the solution of the nonstandard problem

Let us first consider the case (a). Because

For the case (b), we have

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

If

If there is

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

In this paper we have discussed only the case when

In the previous sections we have considered the nonstandard problem

We are interested in what conditions we would have to take for the constants

The internal energy density per unit of volume is a positive definite quadratic form and so, for

Combining relations (

Using the inequality

Combining relations (_{1,2,3,5} hold true and

From relations (

In conclusion, if we replace conditions (_{1,2,3,5} and (

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.