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## Computational & Applied Mathematics

*On-line version* ISSN 1807-0302

### Comput. Appl. Math. vol.31 no.1 São Carlos 2012

#### http://dx.doi.org/10.1590/S1807-03022012000100009

**Fundamental solution in the theory of micropolar thermoelastic diffusion with voids**

**Rajneesh Kumar; Tarun Kansal ^{*}**

Department of Mathematics, Kurukshetra University, Kurukshetra-136 119, India E-mails: rajneesh_kuk@rediffmail.com / tarun1_kansal@yahoo.co.in

**ABSTRACT**

In the present article, we construct the fundamental solution of system of differential equations in the theory of micropolar thermoelastic diffusion with voids in case of steady oscillations in terms of elementary functions. Some basic properties of the fundamental solution are also established. Some special cases are also discussed.

**Mathematical subject classification: **74Bxx, 74Fxx, 74Hxx.

**Key words: **fundamental solution, micropolar thermoelastic diffusion with voids, steady oscillations.

**1 Introduction **

The linear theory of elasticity is of paramount importance in the stress analysis of steel, which is the most common engineering structural material. To a lesser extent linear elasticity describes the mechanical behavior of other common solid materials, e.g., concrete, wood and coal. However, this theory does not apply to the behavior of many new synthetic materials of the elastomer and polymer type, e.g., polymethyl-methacrylate, polythylene, polyvinyl chloride.

Modern engineering structures are often made up of materials possessing an internal structure. Polycrystalline materials, materials with fibrous or coarse grain structure come in this category. Classical theory of elasticity is inadequate to represent the behavior of such materials. The micropolar elasticity theory takes into consideration the granular character of the medium, and is intended to be applied to materials for which the ordinary classical theory of elasticity fails owing to the microstructure of the material. Within such a theory, solids can undergo macro-deformations and micro-rotations. The motion in this kind of solids is completely characterized by the displacement vector and the microrotation vector, whereas in case of classical elasticity, the motion is characterized by the displacement vector only. The micropolar theory have been extended to include thermal effects by Eringen (1970, 1999) and Nowacki (1966a,b,c). Boschi and Iesan (1973) extended a generalized theory of micropolar thermoelasticity.

Iesan (1986) established a linear theory of thermoelastic materials with voids. He presented the basic field equations and discussed the conditions of propagation of acceleration waves in a homogeneous isotropic thermoelastic material with voids. He showed that transverse wave propagates without effecting the temperature and the porosity of the material. Iesan (1987) extended the thermoelastic theory of elastic material with voids to include initial stress and the initial heat-flux effects. Dhaliwal and Wang (1995) also formulated a thermoelasticity theory for elastic material with voids to include heat flux among the consecutive variables and assumed an evolution equation for the heat-flux. Chirita and Scalia (2001) and Pompei and Scalia (2002) studied the spatial and temporial behavior of the transient solutions for the initial-boundary value problems associated with the linear theory of the thermoelastic materials with voids by using the time-weighted surface power function method. Scalia, Pompei and Chirita (2004) considered the steady time harmonic oscillations within the context of linear thermoelasticity for materials with voids and derived the spatial decay results for the amplitude of harmonic variations in a cylinder.

Scalia (1992) considered a grade consistent micropolar theory of thermoelasticity for materials with voids. Passarella (1996) introduced a theory of micropolar thermoelasticity for materials with voids based on the Lebon (1982) law for heat conduction.

Diffusion is defined as the spontaneous movement of the particles from a high concentration region to the low concentration region and it occurs in response to a concentration gradient expressed as the change in the concentration due to change in position. Thermal diffusion utilizes the transfer of heat across a thin liquid or gas to accomplish isotope separation. Today, thermal diffusion remains a practical process to separate isotopes of noble gases (e.g. xexon) and other light isotopes (e.g. carbon) for research purposes. In most of the applications, the concentration is calculated using what is known as Fick's law. This is a simple law which does not take into consideration the mutual interaction between the introduced substance and the medium into which it is introduced or the effect of temperature on this interaction. However, there is a certain degree of coupling with temperature and temperature gradients as temperature speeds up the diffusion process. The thermodiffusion in elastic solids is due to coupling of fields of temperature, mass diffusion and that of strain in addition to heat and mass exchange with the environment.

Nowacki (1974a,b,c, 1976) developed the theory of thermoelastic diffusion by using coupled thermoelastic model. Uniqueness and reciprocity theorems for the equations of generalized thermoelastic diffusion problem, in isotropic media, was proved by Sherief et al. (2004) on the basis of the variational principle equations, under restrictive assumptions on the elastic coefficients. Due to the inherit complexity of the derivation of the variational principle equations, Aouadi (2007) proved this theorem in the Laplace transform domain, under the assumption that the functions of the problem are continuous and the inverse Laplace transform of each is also unique. Aouadi (2008) derived the uniqueness and reciprocity theorems for the generalized problem in anisotropic media, under the restriction that the elastic, thermal conductivity and diffusion tensors are positive definite. Recently, Aouadi (2009) derived the uniqueness and reciprocity theorems for the generalized micropolar thermoelastic diffusion problem in anisotropic media. Also, Aouadi (2010) derived the uniqueness, reciprocity and existence theorems for the thermoelastic diffusion problem with voids in anisotropic media.

To investigate the boundary value problems of the theory of elasticity and thermoelasticity by potential method, it is necessary to construct a fundamental solution of systems of partial differential equations and to establish their basic properties respectively. Hetnarski (1964a,b) was the first to study the fundamental solutions in the classical theory of coupled thermoelasticity. The fundamental solutions in the theory of micropolar elasticity and thermoelasticity for materials with voids are presented by Scarpetta (1990) and Svanadze et al. (2007) respectively. The fundamental solutions in the microcontinuum fields theories have been constructed by Svanadze (1988, 1996, 2004) and Svanadze et al. (2006). The information related to fundamental solutions of differential equations is contained in the books of Hörmander (1963, 1983).

In this article, the fundamental solution of system of equations in the case of steady oscillations is considered in terms of elementary functions and basic properties of the fundamental solution are established. Some special cases of interest are also discussed.

**2 Basic equations **

Let **x** = (*x*_{1}, *x*_{2}, *x*_{3}) be the point of the Euclidean three-dimensional space E^{3},

and let *t* denote the time variable.

Following Aouadi (2009, 2010), the basic equations for homogeneous isotropic generalized micropolar thermoelastic diffusion with voids in the absence of body forces, body couples, heat and mass diffusive sources are:

where

β_{1} = (3λ + 2µ + *K**)α* _{t}*, β

_{2}= (3λ + 2µ +

*K**)α

*.*

_{c}Here α_{t}, α_{c} are the coefficients of linear thermal expansion and diffusion expansion respectively; is the displacement vector; is the microrotation vector; ϕ* is the volume fraction function; ρ, *C _{E}* are, respectively, the density and specific heat at constant strain; λ, µ,

*K, D, a, b, a*, d*, f**, ξ*, ζ*, α*, β*,

*K**, γ* are constitutive coefficients;

*j*is microintertia density; χ is equilibrated inertia; = Θ -

*T*

_{0}is small temperature increment; Θ is the absolute temperature of the medium;

*T*

_{0}is the reference temperature of the body chosen such that 1; is the concentration of the diffusive material in the elastic body; τ

^{0}is diffusion relaxation time and τ

_{0}is thermal relaxation time; Δ is the Laplacian operator. If τ

_{0}= τ

^{0}= 0, then from (1)-(5), we obtain the basic equations for micropolar thermoelastic diffusion with voids based upon the Fourier classical law of heat conduction.

We define the dimensionless quantities:

where

Upon introducing the quantities (6) in the basic equations (1)-(5), after suppressing the primes, we obtain

where

We assume the displacement vector, microrotation, volume fraction, temperature change and concentration functions as

Using equation (12) in the equations (7)-(11), we obtain the system of equations of steady oscillations as

where

We introduce the matrix differential operator

**F** (**D*** _{x}*) = ||F

*(*

_{gh}**D**

_{x})||

_{9×9}

where

Here ε* _{mrn}* is alternating tensor and δ

_{mn}is the Kronecker delta.

The system of equations (13)-(17) can be written as

**F(D**_{x}**)U(x ) = 0 ,**

where **U** = (**u**, **φ**, ϕ*, *T, C*) is a nine-component vector function on E^{3}.

We assume that

If the condition (18) is satisfied, then **F** is an elliptic differential operator (Hörmander, 1963).

**Definition. ** The fundamental solution of the system of equations (13)-(17) (the fundamental matrix of operator **F**) is the matrix **G** (**x**) = ||*G _{gh}*(

**x**)||

_{9×9}satisfying condition (Hörmander, 1963)

where δ is the Dirac delta, **I** = ||δ_{gh}||_{9×9} is the unit matrix and **x** є E^{3}.

Now we construct **G** (**x**) in terms of elementary functions.

**3 Fundamental solution of system of equations of steady oscillations **

We consider the system of equations

where **H**' and **H**" are three-component vector functions on E^{3}; *Z, L* and *M* are scalar functions on E^{3}.

The system of equations (20)-(24) may be written in the form

where **F*** ^{tr}* is the transpose of matrix

**F**,

**Q**= (

**H**',

**H**",

*Z, L, M*) and

**x**є E

^{3}.

Applying the operator div to the equations (20) and (21), we obtain

where ν* = δ_{5} + δ_{6}.

The equations (26)_{1}, (26)_{3}, (26)_{4} and (26)_{5} may be expressed in the following form

where **S** = (div **u**, ϕ*, *T, C*), ['(**Q** )] = (*d*_{1}, *d*_{2}, *d*_{3}, *d*_{4}) = (div **H**', *Z, L, M*) and

The equations (26)_{1}, (26)_{3}, (26)_{4} and (26)_{5} can be also written as

where

and is the cofactor of the elements *N _{mn}* of the matrix

**N**.

From equations (28) and (30), we see that

where , *m* = 1,2,3,4 are the roots of the equation Γ_{1}(-κ) = 0 (with respect to κ).

From equation (26)_{2}, it follows that

where .

Applying the operators δ_{5}Δ + µ* and δ_{3} curl to the equations (20) and (21), respectively, we obtain

and

Now

Using equations (33) and (34) in equation (32), we obtain

The above equation can also be written as

Applying the operator Γ_{1}(Δ) to the equation (36) and using equation (29),we get

The above equation may also be written in the following form

where

and

It can be seen that

where are the roots of the equation Γ_{2}(-κ) = 0 (with respect to κ).

Applying the operators δ_{3} curl and δ_{1}Δ +ω^{2} to the equations (20) and (21), respectively, we obtain

and

Now

Using equations (40) and (42) in equation (41), we obtain

The above equation may also be rewritten as

Applying the operator Δ + to the equation (44) and using equation (31), we get

The above equation may also be rewritten in the form

where

From equations (29), (38) and (45), we obtain

where = (**Ψ****'**, **Ψ****"**, Ψ_{3}, Ψ_{4}) and

The equations (30), (39) and (46) can be rewritten in the form

where **J** = || δ_{gh}||_{3×3} is the unit matrix.

In the equations (48)-(52), we have used the following notations:

Now from equations (48)-(52), we have

where

From equations (25), (47) and (53), we obtain

**Θ****U = R**^{tr}**F**^{tr}**U**

The above relation implies

**R**^{tr}**F**^{tr}** = ****Θ**

Therefore, we obtain

We assume that

Let

where

We will prove the following Lemma:

**Lemma. ** The matrix **Y** defined above is the fundamental matrix of operator **Θ** (Δ), that is

**Proof. ** To prove the Lemma, it is sufficient to prove that

Consider

where

On simplifying the right hand side of above relation, we obtain

Similarly, we find that

Also,

Now consider

Using equation (58) in the above relation, we obtain

Similarly, the equations (57)_{1} and (57)_{2} can be proved.

We introduce the matrix

From equations (55), (56) and (63), we obtain

**F**(**D*** _{x}*)

**G**(

**x**) =

**F**(

**D**

*)*

_{x}**R**(

**D**

*)*

_{x}**Y**(

**x**) =

**Θ**(Δ)

**Y**(

**x**) = δ(

**x**)

**I**(

**x**)

Hence, **G**(**x**) is a solution to equation (19).

Therefore we have proved the following Theorem:

**Theorem. ** The matrix **G**(**x**) defined by equation (63) is the fundamental solution of system of equations (13)-(17).

**4 Basic properties of the matrix G(x) **

**Property 1. ** Each column of the matrix **G**(**x**) is the solution of the system of equations (13)-(17) at every point **x** є E^{3} except the origin.

**Property 2. ** The matrix **G**(**x**) can be written in the form

**G** = ||*G _{gh}*||

_{9×9}

** G**_{mn}**(x) = R**_{mn}(**D**_{x})*Y*_{11}**(x),**

** G**_{m,n+3}**(x) = R**_{m,n+3}(**D**_{x})*Y*_{44}**(x),**

** G**_{mp}**(x) = R**_{mp}(**D**_{x})*Y*_{77}**(x)** *m* = 1, 2....., 9 *n* = 1, 2, 3 *p* = 7, 8, 9.

**5 Special cases **

(i) Neglecting the diffusion effect in the equations (13)-(17), we obtain the system of equations of steady oscillations for homogeneous isotropic generalized micropolar thermoelasticity with voids as:

The fundamental solution of the system of equations (64)-(67) is similar as obtained by Svanadze et al. (2007) by changing the dimensionless quantities into physical quantities.

(ii) If we neglect the void effect in the equations (13)-(17), we obtain the system of equations of steady oscillations for homogeneous isotropic generalized micropolar thermoelastic diffusion as:

The fundamental solution of the system of equations (68)-(71) is similar as obtained by Kumar and Kansal (2012).

(iii) If we neglect the micropolar effect in the equations (13)-(17), we obtainthe system of equations of steady oscillations for homogeneous isotropic generalized thermoelastic diffusion with voids as:

The fundamental solution of the system of equations (72)-(75) is similar as obtained by Kumar and Kansal (2012) based upon Lord-Shulman theory of thermoelastic diffusion with voids.

(iv) If we neglect the micropolar and void effects in the equations (13)-(17), we obtain the system of equations of steady oscillations for homogeneous isotropic generalized thermoelastic diffusion as:

The fundamental solution of the system of equations (74)-(76) is similar as obtained by Kumar and Kansal (2012) based upon Lord-Shulman theory ofthermoelastic diffusion.

**6 Conclusions **

The fundamental solution **G**(**x**) of the system of equations (13)-(17) makes it possible to investigate three-dimensional boundary value problems of generalized theory of micropolar thermoelastic diffusion with voids by potential method (Kupradze et al., 1979).

**Acknowledgments. ** One of the authors Mr. Tarun Kansal is thankful toCouncil of Scientific and Industrial Research (CSIR) for the financial support.

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Received: 15/III/11.

Accepted: 11/IX/11.

#CAM-346/11.

* Corresponding author.