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In one of his papers on the theory of thermoelasticity, M. A. Biot [2] has presented a solution to the differential equations very similar to the well-known Boussinesq-Papcovitch solution in the theory of elasticity. In this note it is proved that this solution is complete, the proof being based upon Mindlin's theorem of completeness of the Boussinesq-Papcovitch solution in elasticity. 1. Basic equations of thermoelasticity. In a three-dimensional space a bounded open region D is considered. A space point will be denoted by x and the boundary of D will be denoted by B. It is assumed that B + D is a regular region (see Kellogg [1]). The mathematical formulation of the linear thermoelastic problem for an isotropic solid in the absence of body forces, inertia effects and internal heat supply consists of the following differential equations [2], holding for all x £ D, and for all values of the time variable larger than some initial value, which will be taken as zero: (X + aOVV-u + mV2u /3V0 = 0, xED, t > 0, (1) c, dd/dt + I3T0 d(V-n)/dt /cV20 = 0, x £ D, t> 0, (2) where u: displacement vector of the solid material, T0: absolute reference temperature, 6: incremental temperature, t: time, X, ix'Lamp's constants, /3 = (X + |a: coefficient of volumetric thermal expansion, cspecific heat per unit volume in the absence of deformations, k: thermal conductivity. The vector function u and the scalar function 6 will be assumed to be continuous, together with at least their first two partial space derivatives and their first partial time derivatives. It is furthermore assumed that the first partial time derivatives of these functions possess continuous partial space derivatives up to the second order at least. Functions having these properties will be said to be of class C<2,1>. Functions of which only continuity in space and of the first n partial space derivatives is required will be said to be of class C("'. Clearly a function of class Cu'm) (m > 0) is also of class CM. For a complete description of a particular problem a set of appropriate initial and boundary conditions must be given in addition to the differential equations. In this note, however, restriction is made to general solutions of the differential equations, regardless of the boundary and initial conditions. *Reeeived September 15, 1967. 486 A. VERRUIJT [Vol. XXVI, No. 4 As already noted by Biot, the theory of thermoelasticity is mathematically analogous to the theory of deformation of a fluid-saturated porous elastic material, which is often called the theory of consolidation of porous media. The quantity /30 in Eq. (1) is then replaced by the pressure of the fluid in the pores, and in the second equation the coefficients have a different physical meaning. The major difference between the two branches of science arises in applications, because in thermoelastic problems the first term in Eq. (2) usually dominates the second one, whereas in problems of consolidation the usual magnitude of the material constants results in the second term in Eq. (2) dominating the first. For a mathematical treatment this difference is of course irrelevant. 2. Biot's solution. A solution of the differential equations (1) and (2) has been given by Biot [2] in the following form: u = V(<p + r-ifc) + 7<r, (3) 6 = [(X + 2M)/j8] W + [2(X + 2/i)/j8 + 7(X + rf/fiV • tfr, (4) where 7 is a constant related to the material properties by 7 = —2(X + 2n + /32ro/c»)/(X + n + fi2T0/c,), (5) and where <p and are a scalar and a vector function, respectively, which satisfy the following differential equations d(VV)/df = CV2VV, (6) V2tt = 0. (7) In Eq. (6) C represents another constant, which is related to the fundamental material properties by „ _ k H2y c, (X + 2M+ /32T„/c,)' W Apart from some minor differences in notation the equations (3), (5), (6), (7), and (8) were given in Biot's paper [2]. The expression (4) for the incremental temperature was not given explicitly by Biot, but follows immediately from his considerations. It can easily be verified by direct substitution from (3) and (4) into (1) and (2) that any pair of functions <p and it satisfying Eqs. (6) and (7) indeed represents a solution of the problem. The question naturally arises whether this solution is complete, or, in other words, whether any solution of the basic equations (1) and (2) admits a representation in the form of Eqs. (3) and (4). That this is indeed the case will be shown below. 3. Proof of completeness of Biot's solution. The starting point of the proof is the following theorem: Theorem 1. For any solution v0 of the homogeneous linear problem (X + m)V V -v + mV2v = 0, x E D, (9) where D is the interior 0/ a regular, bounded region and where v is to be of class C{n) in D (n > 2), there exists at least one pair of functions f0 , F0 of class C(n) in D, such that v„ = V(/0 + r-F„) 2[(X + 2/x)/(X + At)]F„ , x G D, (10) 1969] BIOT'S SOLUTION OF THE COUPLED THERMOELASTIC PROBLEM 487 V2/0 = 0, x ED, (11) V2F0 = 0, x E D, (12) where r is the -position vector of the point x. Equation (9) is the differential equation for the displacement vector in an isotropic linear elastic material in the absence of body forces. The solution (10) was given by Papcovitch [3], and independently, but somewhat later, by Neuber [4]. It is usually referred to, however, as the Boussinesq-Papcovitch solution because the fundamental parts of the solution were already presented by Boussinesq in his memoir on the theory of elasticity [5]. The presentation of the solution by Papcovitch already implied that it is a complete solution. An explicit proof of the completeness theorem stated above has been given by Mindlin [6] (see also Eubanks and Sternberg [7]). The proof is based upon Helmholtz' decomposition theorem which asserts that any vector function of class C<n> (n > 0) in a regular region can be represented as the sum of the divergence of a scalar function and the curl of a vector function, both functions being of class C(n+1' in that region (see Phillips [8]). The pair of functions f0, F0 may be replaced by a pair of functions h0, t^o of the same class according to the following relations h0 = fo + [1 + (2/7) (X + 2 M)/(X + M)]r-F„ , x ED, (13) *0 = — (2/7) (X + 2m)/(X + n)F0 , x ED, (14) where 7 is an arbitrary nonzero constant. If it is furthermore assumed that 7 is finite, then there exists a unique inverse relationship between h0 , iI-0 and / , F0 , fo = ho + [I + (7/2) (X + m)/(X + 2p)]r • t(r„ , X ED, (15) Fo = —(7/2) (X + M)/(X + 2MHo , x ED. (16) Substitution of Eqs. (15) and (16) into Eqs. (10), (11) and (12) at once leads to the following generalization of Theorem 1. Theohem 2. For any solution v0 of the homogeneous linear problem, (X + M)VV -v + V2v = 0, x E D, where D is the interior of a regular, bounded region and where v is to be of class C(n> in D (n > 2), and for any constant value of 7 other than zero and infinite, there exists at least one pair of functions h0 , tl'o of class Cln) in D, such that v0 = V (h0 + T'tto) + 7*to > x E D, (17) V2h0 + [2 + 7(X + M)/(X + 2/OJV -ife0 = 0, x ED, (IS) V2t£o = 0, x ED. (19) Equation (17) constitutes a somewhat more flexible representation than Eq. (10) because of the arbitrariness of 7. This is compensated, however, by the fact that the governing equation for h0 , Eq. (18), is more complicated than the one for /0 , Eq. (11). Equation (1) is the inhomogeneous counterpart of Eq. (9), and therefore, in accordance with a fundamental theorem from the theory of linear differential equations (see e.g. Ford [9]), the complete solution of the inhomogeneous equation (1) consists 488 A. VERRUIJT [Vol. XXVI, No. 4 of the sum of a particular solution of that equation and the complete solution of the corresponding homogeneous equation (9). A particular solution of the inhomogeneous equation (X + (i)VV'U + mV2u pve0 = 0, X e D, (20) where 80 is an arbitrary function of class C(m) in D (•m > 1), and where u is to be of class C<n> in D (n > 2), will be sought in the form u* = Vn, x G D, (21) with <pn a scalar function of class C<n+1>. Substitution of (21) into (20) shows that this amounts to determining a function <p^ such that V[(X + 2ft)vV* pe0] =0, xE D, which will surely be satisfied if the expression between brackets vanishes, (X + 2m)VV* P0o = 0, i£ D. (22) This is Poisson's equation, a solution of which is (see e.g. Phillips [8])

The completeness of Biot’s solution of the coupled thermoelastic problem