A PROOF OF COMPLETENESS OF THE GREEN-LAMI TYPE SOLUTION IN THERMOELASTICITY

A proof of completeness of the Green-Lain6 type solution for the unified governing field equations of conventional and generalized thermoelasticity theories is given. ,KEYWORDSAND PHRASES. Thermoelasticity, Generalized Thermoelasticity, Green-Lam6 solution, completeness of solution.


INTRODUCTION
In ], the author presented three complete solutions for the following system of coupled partial differential equations which may be interpreted as a unified system of governing field equations of the conventional and generalized models of the linear thermoelasticity theory of homogeneous and isotropic materials: (1.1 a,b) The notation employed in these equations and those to follow are as explained in 1].One of the three solutions of the system (1.1) presented in 1] is analogous to the Green-Lam6 solution in classical elastodynamics [2]; this solution is described by the following relations: That is, if the known function F is represented by the relation (1.6) (by virtue of the Helmholtz resolution D g g (.5) of a vector field), then a solution {u_, 0} for the system (1.1) is given by the representations (1.2) and (1.3) where and are arbitrary scalar and vector functions (respectively) obeying the partial differential equations (1.4) and (1.5).Here D, D 2, /93 and D 5 are partial differential operators defined by ]" "' fl '-t   (1.9) It was also shown in [1 that the solution described above is complete in the sense that if the known function_,F is represented as in (1.6), then every solution {u_, 0} of the system (1.1) admits a representation given by the relations (1.2) and (1.3) with # and obeying the equations (1.4) and (1.5).
The proof of completeness suggested in [1 was an extension of the proof given in [2] in the context of classical elastodynamics.This proof makes the hypothesis that in the representation (1.6) for F the function g is divergence-free (that is, div g 0) and infers that also has to be divergence-free.
The object of the present Note is to give a proof of the completeness of the Green-Lain6 type solution that does not make the hypothesis that div g 0 and consequently does not infer that div g O.This proof is motivated by the work of Long [3] in classical elastodynamics and is analogous to that given in [4] in the context of the theory of elastic materials with voids.

PROOF OF COMPLETENESS
Consider solution {u_, 0} of the system (1.1).By virtue of the Helmholtz representation of a vector field, u may be expressed as _u= l+ot +curl) (2.1) for some scalar field p and a vector field q.
This condition may be taken to be valid when u and 0 obey homogeneous initial conditions.
Taking the divergence of both sides of (2.3) and noting that div V 7 2 and div curl is the zero (2.7) Taking the curl curl of both sides of (2.3) and noting that curl V is the zero operator and curl curl Vdiv-V 2 we obtain the equation V 2 curl(D2-g,_) =. (2.8) where This equation implies that [4, Appendix] q= r0+ 1 (2.9) V2(curl 0) 0_..
Since this holds for any t, we should have div r2 0 and div 3 0 from which it follows that Ilr2 curl 2, for some 2, dj3.
We now define the function qt(P, t) by vf(Q,t-R/C)dv D (2.15) where =t02 +3 (2.16) and R is the distance from the field point P to a point Q, the integration (over D) being w.r.t.Q.
With the aid of the identity V o 3 )f q(Q, R[c) dV -4 n q c20t 2 R D and the relations (1.8), (2.14) and (2.16), expression (2.15) yields DN= D_l.Using the relation (2.11), we now find that N obeys the equation/ lg g, which is the desired governing equation (1.5)   for .
Thus, we have shown that, given any solution {, 0} for the system (1.1), one can construct functions 0 and such that and 0 can be represented by the relations (1.2) and (1.3) with 0 and obeying the equations (1.4) and (1.5).
This completes the prf of completeness of the Green-Larn6 type solution for the system (1.1).
Note that no where in the proof it has been assumed that div g 0 and inferred that div has to be zero.
ACKNOWLEDGEMENT.This work is supported by the U. S. Government Fulbright Grant #15068 under the Indo-American Fellowship Program.The author is thankful to Bangalore University, Bangalore and the University Grants Commission, New Delhi for nominating him for the Fellowship and the Indo- U.S. subcommission on education and culture for awarding the Fellowship.His thanks are also due to Professor Lokenath Debnath for the facilities.