Generalization of Eshelby's Formula for a Single Ellipsoidal Elastic Inclusion to Poroelasticity and Thermoelasticity

Eshelby's formula gives the response of a single ellipsoidal elastic inclusion in an elastic whole space to a uniform strain imposed at in nity. Using a linear combination of results from two simple thought experiments, we show how this formula may be generalized to both poroelasticity and thermoelasticity. The resulting new formulas are important for applications to analysis of poroelastic and thermoelastic composites, including but not restricted to rocks. PACS numbers: 81.05.Rm, 81.40.Jj, 91.60.Ba, 62.22.Dc, 03.40.-t, 46.30.Cn 1 Probably the single most referenced work in the extensive and rapidly growing literature on elastic composites is Eshelby's paper1 on the response of a single ellipsoidal elastic inclusion in an elastic whole space to a strain imposed at in nity. Eshelby found that a uniform strain at in nity results in a uniform strain within the ellipsoidal inclusion. This simple fact was then used in more detailed calculations to obtain the fourth-rank tensors relating these two uniform strains. The tensors themselves are not simple in general since they involve elliptic integrals, but Eshelby was able to enumerate and explicitly evaluate all of these integrals for simple shapes like spheres, oblate and prolate spheroids, needles and disks. The results have been found to be immensely useful in the analysis of composite materials, since most inclusion shapes commonly of interest can be approximated by some ellipsoid. E ective medium theories2 8 for elastic constants have very often been based on static or dynamic approximations that make explicit use of Eshelby's formulas. Recent reviews of e ective medium theories and rigorous bounding methods applied to composite analysis are available.9 10 It would be of considerable interest to have an identity analogous to Eshelby's result available in other, more complex, problems in composites analysis (e.g., piezoelectric composites11). Two problems that are themselves relatively straightforward generalizations of elasticity are poroelasticity and thermoelasticity. In poroelasticity, we allow the possibility that the elastic materials contain connected voids or pores and that these pores may be lled with uids under pressure which then couples to the mechanical e ects of an externally applied stress or strain. In thermoelasticity, we include the e ects of temperature on the elastic materials and consider the coupling between thermal expansion and externally applied stresses and strains. In fact, it is known that problems in these two subjects have very similar mathematical structure,12 15 so that solutions found in one generally carry over with only minor modi cations to the other. 2 The main purpose of this paper is to show that the results of two simple thought experiments can be combined to produce a rigorous generalization of Eshelby's formula valid for either poroelasticity or thermoelasticity. Then, the hard part of Eshelby's work in computing the elliptic integrals (needed to evaluate the fourth-rank tensors) can be carried over to these new results with only trivial modi cations. We will rst discuss the problem in terms of poroelasticity and later point out the modi cations necessary to map onto thermoelasticity. In our notation, a superscript i refers to the inclusion phase, while superscripts h and refer to host and composite media, respectively. In this application the composite is a very simple one, being an in nite medium of host material with a single ellipsoidal inclusion of the ith phase. The basic result of Eshelby is then of the form e(i) pq = Tpqrse rs; (1) where e(i) is the uniform induced strain in the inclusion, e is the uniform applied strain of the composite at in nity, and T is the fourth-rank tensor relating these two strains.4 The summation convention for repeated indices is assumed in expressions such as (1). In elasticity, the components of T depend explicitly on the elastic constants of both the host and inclusion. In our rst thought experiment, we consider that in the absence of poreuid e ects in poroelasticity (or thermal e ects in thermoelasticity), the formula (1) must remain unchanged. In poroelasticity the only di erence induced by the generalization from elasticity is an implicit one arising from the interpretation of the elastic constants used in evaluating the fourth-rank tensor T . Two types of bulk and shear moduli must be considered in poroelasticity, frame moduli (of the overall porous medium, often called the \frame") and grain or mineral moduli 3 (of the purely solid constituents). In the absence of any pore uid, only external con ning stresses are operative and the only pertinent moduli are the frame moduli, corresponding to moduli one would measure for a porous sample of the material drained of all uid. We will use the symbols K , K(h), and K(i) for the frame bulk moduli of the composite, the host, and the inclusion, respectively. The frame shear moduli are given similarly by , (h), and (i). The host and inclusion frame moduli16 are the only ones that can appear in the expressions for the tensors T in poroelasticity. The general relations between strains and stresses in an isotropic poroelastic medium take the form e(h) pq = S(h) pqrs rs + (h) 3K(h) pf pq; (2) where is the applied external stress, S(h) is the compliance tensor of the host frame material and (h) is the Biot-Willis parameter17 of the host medium. When a uniform saturating uid is present in the pores of a microhomogeneous poroelastic medium, the resulting uniform strains in an isotropic medium are related to applied uniform (hydrostatic) stresses by e(h) pq = "pc pf 3K(h) + pf 3K(h) m # pq = pc (h)pf 3K(h) pq; (3) where pc = 1 3 ss is a uniform external (at in nity in these inclusion problems) con ning pressure (positive under compression) and K(h) m is the grain or mineral bulk modulus of the host material. From (2) and (3), the Biot-Willis parameter17 is seen to be given by (h) = 1 K(h)=K(h) m . Expressions similar to (3) apply to e pq and e(i) pq , with the corresponding changes in the bulk moduli and other parameters. Now for our second thought experiment, we consider under what circumstances the host medium and the ellipsoidal inclusion will expand or contract at the same rate. See Figure 1. 4 This scenario is possible in poroelasticity because there are two adjustable elds present. (No such possibility exists in the purely elastic problem of Eshelby.) Analogous problems were rst discussed originally in thermoelasticity18 20 and more recently in poroelasticity.13 The trick is that, if a ratio of pc and pf can be found so that e(h) and e(i) change at the same rate, then so must e and, furthermore, no local concentrations of stress develop. The resulting macroscopic strains are uniform; the macroscopic stresses are uniform; and, therefore, stress equilibrium conditions are trivially satis ed. Thus, the entire analysis of these stress states reduces to simple algebra. We know from earlier work13 that the uniform expansion/contraction ratio can be found for any two-phase poroelastic composite, and the single ellipsoidal inclusion example considered here is just an especially simple two-phase problem. In the uniform expansion/contraction scenario, once the pore pressure pf (which is uniform throughout host and inclusion because of assumed open-pore boundary conditions) has been speci ed then we know that the con ning pressure pc needed to produce a uniformly expanded or contracted state is given by pc=pf = ( (h)=K(h) (i)=K(i)) 1=K(h) 1=K(i) R; (4) depending only on the (assumed known) physical properties of the host and inclusion. This result was obtained by setting e(h) = e(i) and solving for the pc=pf ratio. Then, the strains of the reference states of the composite, host, and inclusion materials are given by " pq(pf ) = pf 3K (R ) pq; (5) "(h) pq (pf ) = pf 3K(h) R (h) pq; (6) 5 and "(i) pq (pf ) = pf 3K(i) R (i) pq; (7) all of which are equal " pq = "(h) pq = "(i) pq = pq(pf=3) (h) (i) = K(h) K(i) , by construction. The ratio R is the one de ned in (4), which is easily veri ed by equating (6) and (7) and solving for R. [The remaining equality among (5)-(7) determines the value of for the composite.13] Thus, the nal form of the generalization of Eshelby's formula to poroelasticity is given by e(i) pq "(i) pq = Tpqrs (e rs " rs) : (8) We see that, if the poreuid pressure vanishes (e.g., pf = 0 in the absence of a pore uid), then the uniform strains " disappear from (8) and it reduces exactly to (1) as it should. For the other limiting case, when the pore pressure has been speci ed to be pf 6= 0, then the uniform strains " in (8) can be computed from (5) and (7). Now, if the strain at in nity happens to be chosen to be equal to this uniform strain, (8) shows that the inclusion strain takes the value at in nity as it should. Since the equation for e(i) is linear, these two cases are enough to determine the behavior for arbitrary values of e and pf . The deceptively simple equation (8) is the main result of this paper. The same formula with slightly di erent interpretations of the symbols also applies to the thermoelastic problem as we will now show. For thermoelasticity, (2) is replaced by e(h) pq = S(h) pqrs rs + (h) pq; (9) where is again the applied external stress in the host, S(h) is the compliance tensor of the host material, (h) is the linear thermal expansion coe cient of the host material, and is the 6 temperature change. Equation (3) is replaced by e(h) pq = pc 3K(h) (h) pq; (10) where K(h) is the bulk modulus of the host material and pc is again the uniform con ning pressure at in nity. (We do not need to distinguish types of pressure in the thermoelastic problem, so the subscript c is super uous in this case.) Again similar expressions obtain for the inclusion phase and for the composite medium as a whole. To ensure uniform expansion or contraction in the thermoelastic problem, we see that the ratio of pressure to temperature change must be pc= = 3 (h) (i) 1=K(h) 1=K(i) X; (11) again depending only on the physical properties of the host and inclusion phases. The uniform strains in the composite, host, and inclusion phases when pc= = X are given by " pq( ) = 3K (X 3 K ) pq; (12) "(h) pq ( ) = 3K(h) X 3 (h)K(h) pq; (13) and "(i) pq ( ) = 3K(i) X 3 (i)K(i) pq: (14) Again, all three of these strains (12)-(14) are equal by construction. Now equation (8) can be reinterpreted for the thermoelastic single ellipsoidal inclusion problem by simply using (12) and (14) in place of (5) and (7). If a change of temperature 7 occurs, then a strain e pq imposed at in nity will result in the strain e(i)pq in the inclusion. If theimposed strain happens to equal the one that would produce the uniform strain determinedby (12)-(14), then (8) guarantees that the uniform strain outside is the same one that resultsinside the inclusion. If there is no change in temperature = 0, then the terms in " drop outof (8), and the problem reduces correctly to Eshelby's original problem.It is worthwhile to note that (8) could have been derived in an equally rigorous, but per-haps less intuitive manner, without the use of our two thought experiments | just as Levin'sderivation18 of the thermoelastic composites' formula was obtained in a less intuitive fashionthan the one of Cribb.19 For example, the book of Mura21 makes extensive use of the rathertechnical concepts of \eigenstress," \eigenstrain," and \stress-free strain," special cases of whichcould have been designed to permit an alternative derivation of (8) for the case of thermoelas-ticity. We believe however that the derivation presented here is much simpler, more intuitive,and easier to grasp.The result (8) is of great practical value for many applications as mentioned earlier in thepaper. For example, the result can be used in a very direct way to rederive the results ofBerryman and Milton13 and then generalize these results approximately to multicomponentporous composites using e ective medium theory.8 Another important application is the com-putation of long-wavelength scattering from an ellipsoidal inclusion in an in nite medium. Suchresults have been shown to be very useful in e ective medium theories6 8 for elastic composites.Scattering from a spherical inclusion of one poroelastic material imbedded in another has beencomputed previously by Berryman22 and by Zimmerman and Stern,23 but to date no resultsare known for scatterers having more general shapes (such as ellipsoids) in poroelastic applica-tions. Earlier work of Mal and Knopo 24 writing elastic scattering formulas in terms of integral8 equations valid for long wavelengths has been used previously in scattering-based formulationsof e ective medium theory for elasticity.5;7 By generalizing the methods of Mal and Knopoto poroelasticity and thermoelasticity, the formulas presented here will make it possible to ob-tain scattering formulas for arbitrary ellipsoidal-shaped inclusions with much less e ort thanhas been expended previously just for the spherical case, and also permit the generalization ofe ective medium theory25 27 to proceed more easily into the complex realms of poroelasticityand thermoelasticity.Work performed under the auspices of the U. S. Department of Energy by the LawrenceLivermore National Laboratory under contract No. W-7405-ENG-48.References1. J. D. Eshelby, Proc. R. Soc. London Ser. A 241, 376 (1957); J. D. Eshelby, in Progress inSolid Mechanics, R. Hill (ed.), North Holland, Chapter 3 (1961).2. R. Hill, J. Mech. Phys. Solids 13, 213 (1965).3. B. Budiansky, J. Mech. Phys. Solids 13, 223 (1965).4. T. T. Wu, Int. J. Solids Struct. 3, 1 (1966).5. G. T. Kuster and M. N. Toksoz, Geophysics 39, 587 (1974).6. J. Korringa, R. J. S. Brown, D. D. Thompson, and R. J. Runge, J. Geophys. Res. 84, 5591(1979).7. J. G. Berryman, J. Acoust. Soc. Am. 68, 1820 (1980).8. J. G. Berryman and P. A Berge, Mech. Mat. 22, 149 (1996).9 9. S. Torquato, Appl. Mech. Rev. 44, 37 (1991).10. J. G. Berryman, in American Geophysical Union Handbook of Physical Constants, editedby T. J. Ahrens (AGU, New York, 1995), pp. 205{228.11. J. H. Huang and J. S. Yu, Composites Engng. 4, 1169 (1994).12. M. A. Biot, J. Appl. Phys. 27, 240 (1956).13. J. G. Berryman and G. W. Milton, Geophysics 56, 1950 (1991).14. A. N. Norris, J. Appl. Phys. 71, 1138 (1992).15. J. G. Berryman and G. W. Milton, Appl. Phys. Lett. 61, 2030 (1992).16. Note that in order to introduce well-de ned frame moduli into the analysis at this stage,we have assumed that the porosity of both the host and inclusion phases occurs on a muchner length scale than the size of the inclusion itself.17. M. A. Biot and D. G. Willis, J. Appl. Mech. 24, 594 (1957).18. V. M. Levin, Mech. Solids 2, 58 (1967).19. J. L. Cribb, Nature 220, 576 (1968).20. K. Schulgasser, J. Mater. Sci. Lett. 8, 228 (1989).21. T. Mura, Micromechanics of Defects in Solids, Kluwer, Dordrecht, 1987.22. J. G. Berryman, J. Math. Phys. 26, 1408 (1985).23. C. Zimmerman and M. Stern, J. Acoust. Soc. Am. 94, 527 (1993).10 24. A. K. Mal and L. Knopo , J. Inst. Math. Its Applic. 3, 376 (1967).25. J. G. Berryman, J. Acoust. Soc. Am. 91, 551 (1992).26. B. Budiansky, J. Composite Materials 4, 286 (1970).27. N. Laws, J. Mech. Phys. Solids 21, 9 (1973).