Diffusion of a perfect fluid in a linear elastic stress field

MECH. RES. COMM. Vol.3, 245-250, 1976. Pergamon Press. Printed in USA. D IFFUS ION OF A PERFECT FLU ID IN A L INEAR ELAST IC STRESS F IELD E.C. A i fant i s Depar tment of Chemica l Eng ineer ing and Mater ia l s Sc ience , Un ivers i ty of M innesota , M inneapo l i s , M innesota 55455 (Received 12 January 1976; accepted as ready for print 19 March 1976) Introduction and S,,..~ry The current knowledge for the diffusion of a perfect fluid (may be an elastic gas) in a linear isothermal elastic stress field (may be an elastic solid) is sum,mrized by the following two results [i, 2, 3, 4]. (i) The diffusion process can be described mathematically by a linear partial differential equation of a parabolic type which is basically derived by applying a simple modification to the first Fick's law of diffusion. (il) Independently of the above macroscopic arguments there is available a microscopic theory which invokes a diffusion mechanism based on the formation of vacancy - interstitial pairs and finally proposes an exponen. tial dependence of the diffusivity D on the trace of the stress tensor a. On the other hand some experimental results have shown the above de- pendence to be linear. In all these studies it is tacitly assumed that the fluid appears as a dilute solution such that its density is small compared to the density of the solid. It is further assumed that the solid suffers a static deformation which is in- dependent of the presence of the gas. The present work retains the above assumptions but it further develops a ra- tional mechanical model based only on the principles of continuum mechanics. The model is constructed by fully exploiting'the balance laws of mass and momentum for the fluid constituent. Constitutive assumptions are made for the stress of the fluid and a "diffusive force" which is appropriately introduced to account for the diffusion effects. These constitutive assumptions are re- stricted to obey the objectivity condition (or principle of frame indiffer- ence). Thus within a unified phenomenologlcal theory, free from the restric- tive assumptions of Fick type theories and the uncertainties of microscopic arguments, it was possible to establish theoretically the following results: (i) The above mentioned classical diffusion equation is derived as a special case of our theory. (ii) The linear dependence of the diffusion coefficient D on the trace of Scientific Communication - abbreviated 245 246 E.C. AIFANTIS Vol.3, No.4 the stress tensor ~ is proven to be a natural consequence of the pro- posed theory. The development of the theory There is a strong, current interest in discovering and interpreting the me- chanical interaction effects between the stress field of elastic solids and fluids diffusing through them. In the classical treatment, mainly due to Cottrell, a diffusion equation is derived by adding stress dependent terms to the first empirical Fick's law of diffusion. In particular, and on no mechanical basis or rational evidence, classical treatments give p~ = -Dr 0 + MpVq , (i) where p and v are the density and the velocity of the fluid, ~ the trace N of the stress tensor and D, M phenomenological coefficients. Substitution of (i) into a differential equation expressing the conservation of mass of the fluid, yields DV2p ~t = - MVo ° V p (2) For the derivation of equation (2) a constant temperature field was assumed throughout such that the coefficients D, M are constants. Also, the fact that the trace of the stress tensor in a linear elastic isotropic solid with a vanishing external body force field is harmonic, i.e. v2 = 0 , (3) was used. Some work exists in the literature dealing with the solution of the diffusion equation (2), mainly due to Leeuven [i], for a limited number of boundary conditions. Also, a considerable number of experimental scientists are interpreting their results based on the differential equation (2) and its solution. However, there has not been much interest in a rational theoreti- cal derivation of the diffusion equation, independent of the restrictive Vol.3, No.4 DIFFUSION OF A PERFECT FLUID 247 assumptions of Fick's laws. Thus I attempt here to re-examlne the problem based on the general principles of continuum mechanics, namely, the balance equations of mass, linear and angular momentum. I further assume that, because of the small concentration of the fluid, the mechanical response of the solid is unaffected by the pres- ence of the fluid, while the presence of the solid is recognized by the fluid through its strain tensor, e . For conformity with the classical theories, attention is confined to the case in which the strain tensor e enters only through its first invarlant, tre. To account for diffusion phenomena I pos- tulate the existence of a "diffusive force" vector acting on each component of the fluid-solid system, as an external force field~ Thus it enters into the equation of motion to describe the exchange of momentum between the fluid and the solid. The functional forms of the "diffusive force" vector and the stress tensor for the fluid are deduced from general constitutive assumptions restricted by the principle of frame indifference [5], (i.e., the independence of the mate- rial response on the change of frame or observer). The deduced constitutive forms are inserted in the equation of motion and subsequent use of the con- tinuity equation gives rise to a diffusion law, derived exclusively from a purely dynamical theory. I proceed now to illustrate in mathematical terms the proposed theory. The field equations of continuity and motion of the fluid are ~0 + dlv(p~) = 0 (4) 5t d ivE+ p~ = p~ ; ~=S t (5) where p is the density of the fluid, v its velocity, T the stress tensor N of the gas, ~ the "diffusive force" vector and the symbol "t" denotes transposition. 248 E.C. AIFANTIS Vol.3, No.4 The assumed constitutive equations for ~ and ~ are: Z = ~(0, tre, V0, ~) (6) % = ~(0, tre, vp, ~) (7) The principle of frame indifference restricts NT and ~ to be isotropic functions of their arguments. Thus, they have the representations [6], NT = Ionl+ I i~ ~ + I f0 m qp + 13(v ~ V 0 + V 0 m v) (8) ~!r = y IE+ Y f0 (9) where I ° , , , ' II 12' 13 YI Y2 are scalar functions of tr~ and the special invari~nts ~ .~, q0-v0, ~ .vp and the symbol "m" denotes tensor product. I assume the fluid to be at rest, initially, with a constant equilibrium density Po During diffusion I consider small deviations from the equili- brium such that the density becomes D = D o + c~ while the velocity is v=¢~'. Here ¢ is a small dimensionless parameter, such that I can neglect powers in ¢ above the first compared with unity. In the light of the above approx- imation I expand the equations (8) and (9) around the equilibrium in a Taylor's series. Inserting the resultant expressions for T and @ into the equa- tion of motion (5) and returning back to the variables P and ~ I obtain ~v - c£0 -clVtr ~- c2DVtr~e - c2tr~eVD + 0o(C3+c4tr~e)~+Do(C5+c6tr~e)VD = D o ~-~ (i0) where cF; ~ = 0,1,2,3,4,5,6 are constants. Taking the divergence of both sides of equation (I0) and using the continuity equation (4) to eliminate the velocity, it follows that ~2p +c2.~ ) ~P c3. V2 p -c4*Vp • v~ c5.~ v2p = 0 (ii) At 2 (Cl* ~-~ - where c F¢: ; ~ = 1,2,3,4,5 are new constants related to those in (i0), as well as to the Poisson's ratio ~ and the modulus of elasticity E. Here, is the trace of the stress tensor of the solid. For the derivation of Vol.3, No.4 DIFFUSION OF A PERFECT FLUID (II) the well-known relations of linear-elasticity were used, i.e. 72 tre = 0 (12) N I - 2~ (13) and tr~ - ~ Equation (II) is a hyperbolic type (telegrapher's equation), which is advan- tageous compared to the classical equation (2) which is a parabolic type. That is, it is well known that hyperbolic, not parabolic, equations are appropriate for the description of physical phenomena exhibiting motion. For conformity with the classical theories, the inertia term ?2p ~t 2 is neglected and thus equation (II) is reduced to ~--~ DV2p - MV~ " vp + NaY 2 ~t = p (14) where D, M, N are constants related to those in Eq. (II). Equation (14) can be written in the form ~0 D, V20 ~t = - MV~ • Vp ; D* = D + N~ (15) Examining equation (15), we observe that this is identical to the classical result of Eq. (2) except that the "effective diffusion coefficient" D* is proven to be a linear function of the trace of the stress tensor, ~ . This fact, i.e. the linear dependence of the diffusivity on the hydrostatic pres- sure, is also experimentally verified [3,4]. The result indicated by (15)2 is important, particularly in cases where the stresses are large. In these cases the effective diffusivity D* deviates considerably from the ordinary diffusion coefficient D. This deviation re- suits in a significant change of the diffusion pattern obtained by the solu- tion of Eq. (15)1. The proposed theory may be successfully applied in prob- lems of diffusion of gases in dislocation and crack fields. Phenomena such as 249 250 .E.C. AIFANTIS Vol.3, N©.~ precipitation hardening, hydrogen embrittlement and stress corrosion cracking may use the above model to explain the mechanisms involved. For the diffusion of hydrogen in an infinite Nickel plate, containing a cen- tral crack of length 0.I in. and loaded under plane strain conditions with an applied stress of I0.000 psi, it was calculated, [7], that in the neigh- borhood of the crack D*/D varies from i0 to infinity. The infinite value occurs because the linear elastic solution for the above configuration pre- dicts infinite stresses right at the tip of the discontinuity. Another interesting point is that the classical equation (2) does not pre- dict any stress effect on the diffusion process in the case of a uniform stress field. This effect is illustrated in the derived equation (15), by the first term of its right hand side. Re ferences I. H. P. Van Leeuven, J. of Corrosion, NACA 29, 197 (1973) 2. C. G. Homan, Acta Met, 12, 1071 (1964) 3. I. Onishi, Y. Kikuta, T. Araki and T. Fujii, Yosetsu Gakkaishi 32, No 2, 147 (1969) 4. J. F. Cox and C. G. Homan, Phys. Rev. B5, No 12, 4755 (1972) 5. C. Truesdell and W. Noll, The Non-Linear Field Theories, In Flugge's Handburch der Physik, Band III/I. Berlin-Heidelberg-New York: Spriuger (1965). 6. A. J. M. Spencer, Theory of Invariants, In Continuum Physics I, Edited by Eringen, Academic Press (1971) 7. E. C. Aifantis, Ph.D. Thesis; University of Minnesota (1975) Abbreviated Paper -- For further information, please see the author.