On the third-and fourth-order constants of incompressible isotropic elasticity

On the third- and fourth-order constants of incompressible isotropic elasticity Michel Destradea) School of Electrical, Electronic, and Mechanical Engineering, University College Dublin, Belfield, Dublin 4, Ireland Raymond W. Ogden Department of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, Scotland, United Kingdom (Received 15 March 2010; revised 25 August 2010; accepted 28 September 2010) Consider the constitutive law for an isotropic elastic solid with the strain-energy function expanded up to the fourth order in the strain and the stress up to the third order in the strain. The stress–strain relation can then be inverted to give the strain in terms of the stress with a view to considering the incompressible limit. For this purpose, use of the logarithmic strain tensor is of particular value. It enables the limiting values of all nine fourth-order elastic constants in the incompressible limit to be evaluated precisely and rigorously. In particular, it is explained why the three constants of fourth-order incom! and D! are of the same order of magnitude. Several examples of application of pressible elasticity l, A, the results follow, including determination of the acoustoelastic coefficients in incompressible solids and the limiting values of the coefficients of nonlinearity for elastic wave propagation. C 2010 Acoustical Society of America. [DOI: 10.1121/1.3505102] V PACS number(s): 43.25.Dc, 43.25.Ed, 43.25.Fe, 43.25.Ba [ROC] I. INTRODUCTION Extracting the condition for incompressibility from a stress–strain relation can be an ambiguous process because it leads to an infinite limit for one (or more) of the elastic stiffnesses and, eventually, to the appearance of a hydrostatic stress term proportional to an arbitrary Lagrange multiplier (to be determined from boundary and/or initial conditions).1,2 In linear isotropic elasticity, the relation between the infinitesimal stress r and the infinitesimal strain ! reads 1 m! ; 2 Pages: 3334–3343 !¼% 1 3 trðrÞd þ r; 2E 2E (4) (1) and there is no arbitrary quantity. Turning now to nonlinear isotropic elasticity, we must first of all make a choice of the measures of stress and strain. Physicists and acousticians seem to favor the pair consisting of the Green–Lagrange strain tensor e! and the second Piola– Kirchhoff stress !t , and they expand, in the so-called weakly nonlinear theory, the strain-energy density W in terms of three isotropic invariants of the strain. Hence, the Landau and Lifshitz4 expansion can be conducted to fourth order as5 where k and l are the Lamé coefficients and d is the identity. The incompressible limit is equivalent to the condition tr ! ¼ 0, which leads to the limiting case A! C! k W ¼ I!12 þ lI!2 þ I!3 þ B!I!1 I!2 þ I!13 þ E!I!1 I!3 þ F!I!12 I!2 3 2 3 2 4 ! ! ! ! (5) þ GI2 þ H I1 ; r ¼ k trð!Þd þ 2l!; k ! 1; r ¼ %pd þ 2l!; (2) where p is an arbitrary scalar. By way of contrast, a strain– stress relation is more amenable to the imposition of incompressibility, because it leads to unambiguous, finite limit(s) for one (or several) compliance(s).3 Hence, in linear isotropic elasticity, we go from !¼% m 1þm trðrÞd þ r; E E (3) where v is Poisson’s ratio and E is Young’s modulus, to the limiting case a) Author to whom correspondence should be addressed. Electronic mail: michel.destrade@ucd.ie 3334 J. Acoust. Soc. Am. 128 (6), December 2010 ! B, ! and C! are the three third-order constants, E, ! F, ! where A, ! ! G, and H are the four fourth-order constants, and eÞ; I!1 ¼ trð! I!2 ¼ trð! e2 Þ; I!3 ¼ trð! e3 Þ; (6) are, respectively, of first, second, and third order in the strain. (Note that we have placed overbars on the elastic constants so as to avoid conflict with the standard notation E for Young’s modulus, which is used extensively in what follows.) Thus, the terms in I!12 and I!2 are “second-order” terms, the terms in I!13 , I!1 I!2 , and I!3 are “third-order” terms, and so on. However, the incompressibility limit is not easily implemented with this choice, because it is a combination of the invariants,6 specifically 2 4 I!1 % I!2 þ I!12 þ I!13 % 2I!1 I!2 þ I!3 ¼ 0: 3 3 0001-4966/2010/128(6)/3334/10/$25.00 (7) C 2010 Acoustical Society of America V Downloaded 10 Jan 2011 to 79.97.235.126. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.jsp This, in particular, means that I!1 is now a second-order quantity. This makes it complicated to arrive at the fourth-order expansion of incompressible nonlinear elasticity, for which the strain-energy function has the form A! W ¼ lI!2 þ I!3 þ D!I!22 ; 3 (8) where D! is another constant. However, it is advantageous to use, instead of Green– Lagrange strain, an alternative measure of strain, namely the logarithmic strain, since this leads to a very simple expression of the incompressibility constraint. The logarithmic strain e is defined by 1 eÞ; e ¼ lnðd þ 2! 2 (9) and we consider the three independent invariants I1 ¼ trðeÞ; I2 ¼ trðe2 Þ; I3 ¼ trðe3 Þ; (10) respectively, of orders one, two, and three, analogously to I!1 , I!2 , and I!3 . Then incompressibility is expressed exactly as I1 ¼ 0; (11) which must hold identically for all stresses and strains. These considerations suggest the following protocol for finding the limiting values of the elastic constants in incompressible weakly nonlinear elasticity (Secs. II and III). First, write down the relation between stress and strain. Since we are considering isotropic materials, the relevant stress measure t conjugate to logarithmic strain e is the Kirchhoff stress tensor on rotated coordinates related to t! by t ¼ ðd þ 2! eÞ!t (for a general discussion of conjugate stress and strain tensors see Ogden7). Then invert this relation to find the strain in terms of the stress and, more particularly, I1 in terms of invariants of t. Finally, note that I1 must be zero for all t. In what follows, we conduct this process explicitly for thirdorder elasticity. Then we present the results of the fourthorder case, omitting many of the (cumbersome and lengthy) details of the calculations. In Sec. IV, we give a few examples of the applications, and we see that the results established in compressible elasticity can be taken to their incompressible limit, without having to re-write and re-solve the equations of motion and boundary conditions. Previously, Hamilton et al.8 have shown that there should remain only three elastic constants in the incompressible limit of Eq. (5) (Ogden9 had in fact proved this result 30 years earlier). They also found some limited information on the behavior of the other six constants, by extending the work of Kostek et al.10 from third-order to fourth-order elasticity. Based on a comparison with the equation of state of inviscid fluids, it was found that l and A! remain, a new fourth-order constant D! emerges, and k ! 1; k G! ¼ ; 2 B! ¼ %k; 4k E! ¼ ; 3 k ! D! ¼ þ B! þ G; 2 J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010 ! F! ¼ %C; see Zabolotskaya et al.11 and Jacob et al.6 for the latter iden! F, ! and H! remains undetertity. However, the behavior of C, mined. Moreover, the full comparison with fluids ultimately leads to the identities: l ¼ 0, A! ¼ 0, and D! ¼ 0, which, in considering an elastic solid, are not satisfactory because they imply that some information might be missing from the list ! and D! could play a role. One can in Eq. (12), where l, A, only go so far in comparing the behavior of a solid to that of a fluid; a particular disconnect emerges when comparing the behavior of a solid in the incompressible limit, where the speed of a longitudinal wave should tend to infinity, to that of an isentropic liquid, where the speed of a longitudinal wave is finite (Sec. IV D). The differences existing between a compressible solid and its incompressible counterpart must be tackled within the framework of solid mechanics. In effect, many questions have remained open in the literature, as attested by the following comments. Domański12 remarks that “the details of the derivation [of Eq. (8)] are not quite clear from the mathematical point of view,” and that “surprisingly, the experiments confirm that, in spite of being a combination of the higher order constants, the fourth-order constant D! is of a similar order of magnitude as the secondorder shear Lamé constant l and the third-order Landau con! Catheline et al.13 measure A! and B! for agar–gelatin stant A.” based phantoms and note that “the huge difference between these third-order moduli is striking since in more conventional media such as metal, rocks, or crystals they are of the same order.” They then propose an “intuitive justification” for this difference, which “does not hold for a theoretical explanation.” Jacob et al.6 record that “surprisingly, the experiments confirm that, although expressed as function of ! the moduli D! is of the order of ! G, compression moduli k, B, ! and add magnitude of the so-called shear moduli l! and A,” that “no explanation has been given for this order of magnitude.” We address each of these points in the course of this paper. II. THIRD-ORDER INCOMPRESSIBILITY In this section, we focus on third-order elasticity, so that the strain-energy function (5) reduces to C! k A! W ¼ I!12 þ lI!2 þ I!3 þ B!I!1 I!2 þ I!13 : 2 3 3 (13) At this order, the inversion of Eq. (9) gives e! ¼ e þ e2 þ 2e3 =3, so that 2 I!1 ¼ I1 þ I2 þ I3 ; 3 I!2 ¼ I2 þ 2I3 ; I!3 ¼ I3 : (14) It follows that the third-order expansion of W in terms of the invariants of the logarithmic strain e reads !! " A C! k 2 þ 2l I3 þ ðB! þ kÞI1 I2 þ I13 ; W ¼ I1 þ lI2 þ 3 2 3 (12) M. Destrade and R. W. Ogden: Fourth-order incompressible elasticity (15) 3335 Downloaded 10 Jan 2011 to 79.97.235.126. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.jsp Second, note that the stress must remain finite in the incompressible limit. Hence, the last term in the expression for t in Eq. (18) remains finite, so that A remains finite: which is also written as W¼ Em E A I2 þ I2 þ I3 2ð1 þ mÞð1 % 2mÞ 1 2ð1 þ mÞ 3 C þ BI1 I2 þ I13 ; 3 (16) where E and v are second-order constants (Young’s modulus and Poisson’s ratio) and A, B, and C are third-order constants with respect to the logarithmic strain, with the connections, A=l ¼ Oð1Þ: With these two conditions for incompressibility, the expression for b in Eq. (21) reduces to b¼% E¼ 3k þ 2l l; kþl A ¼ A! þ 6l; m¼ k ; 2ðk þ lÞ B ¼ B! þ k; ! C ¼ C; (17) to the Lamé and Landau coefficients. The (conjugate) stress t ¼ @W=@e then expands as t¼ ! " Em I1 þ BI2 þ CI12 d ð1 þ mÞð1 % 2mÞ ! " E þ 2BI1 e þ Ae2 : þ 1þm (18) ð1 % 2mÞ I2 ; 2 1 corrected to the second order in the strain. These can be inverted to give I12 , I2, and then I1 ¼ aT1 þ bT2 þ W¼ mð2 % mÞð1 % 2mÞ ð1 % 2mÞ3 : % ð2B þ 3CÞ 3 E3 E (21) For the exact incompressibility condition [Eq. (11)] to hold for all stresses and strains, we must have a ¼ b ¼ c ¼ 0. First, a ¼ 0 is clearly equivalent to either of k ! 1; while both E and l remain finite, with l ! E=3. 3336 J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010 (25) # $ C=l ¼ O k2 =l2 : (26) E A I2 þ I3 ; 3 3 (27) or equivalently, in terms of the invariants of the Green strain tensor, ð1 þ mÞ2 ð1 % 2mÞ b ¼ %ðA þ 3BÞ ; E3 m ! 1=2; 3Cð1 % 2mÞ3 : E3 In summary, the conditions given in Eqs. (22)–(24) and (26) are necessary and sufficient for incompressibility. For an incompressible material (with I1 : 0), the strain energy W, therefore, reduces to (20) 1 % 2m ; E c ¼ ðA þ 3BÞ c¼% W¼ where the constants a, b, and c are given by a¼ (24) With the three conditions for incompressibility [Eqs. (22)–(24)], the expression for c in Eq. (21) reduces to ð1 % 2mÞ3 C ! 0; (19) cT12 ; Notice that the first term in the expression for t in Eq. (18) must also remain finite, which means that (1 % 2m)%1I1 remains finite. Clearly, the conditions “b ¼ 0” and “(1 % 2m)%1b is finite” are fulfilled simultaneously when B remains finite: Recall that (1 % 2m)%1I1 and, hence, (1 % 2m)%1c remains finite. It follows that E I1 þ ð2B þ 3CÞI12 þ ðA þ 3BÞI2 ; 1 % 2m " # E2 mð2 % mÞ 2 T2 ¼ I1 þ I2 ; ð1 þ mÞ2 ð1 % 2mÞ2 T1 ¼ E2 27Bð1 % 2mÞ : 4E3 B=l ¼ Oð1Þ: Now we introduce the stress invariants T1 ¼ tr(t) and T2 ¼ tr(t2), and it follows by taking in turn the trace of Eq. (18) and then of its square that T12 ¼ (23) (22) A! E ! A ðI2 % 2I!3 Þ þ I!3 ¼ lI!2 þ I!3 ; 3 3 3 (28) where we have used Eqs. (14)–(17). We use Eqs. (17) to find that, in terms of Lamé and Landau constants, the incompressible limits are Eqs. (22) together with A! ¼ A % 6l ¼ lOð1Þ; # $ C! ¼ C ¼ lO k2 =l2 ; ð1 % 2mÞ3 C! ! 0: B! ¼ B % k ¼ kOð1Þ; ð1 % 2mÞB! ! %E=3; (29) Notice that C! varies in a quadratic manner with respect to k as it goes to infinity in incompressible solids, not in a linear manner, as is incorrectly reported in Refs. 8, 14, and 15; see M. Destrade and R. W. Ogden: Fourth-order incompressible elasticity Downloaded 10 Jan 2011 to 79.97.235.126. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.jsp also Sec. IV D, where it is shown that if C! were linear in k, then longitudinal waves would propagate with finite speed in the incompressible limit. Bearing in mind the advantage of the strain–stress relation (3) in the linear theory, we now take note of its counterpart for the present theory. On use of the definitions of T1 and T2 and Eq. (19) we may, after lengthy algebra, invert the stress–strain equation [Eq. (18)] to give e¼ 1þm m Að1 þ mÞ3 2 t t % T1 d % E3 E E 2 ð1 þ mÞ ½mA % ð1 % 2mÞB'ð2T1 t þ T2 dÞ þ E3 1 þ 3 ½3Bmð2 % mÞð1 % 2mÞ % 3Am2 % Cð1 % 2mÞ3 'T12 d: E (30) In the incompressible limit, as embodied in Eqs. (22)–(26), this reduces to ! " ! " $ 3 # 1 27A 1 e ¼ 3 2E2 þ 3AT1 t % T1 d % 3 t 2 % T2 d ; 4E 3 8E 3 (31) from which it follows immediately that tr(e) ¼ 0. Conversely, the stress–strain relation must accommodate the internal constraint of incompressibility by introducing a Lagrange multiplier, denoted by p, in the following expansions: 2E e þ Ae2 ; 3 !e2 ; ! e þ A! t ¼ %pðd þ 2! eÞ%1 þ 2l! t ¼ %pd þ (32) for the stresses, where we recall that for an isotropic material, t ¼ ðd þ 2! e Þ! t. k A C W ¼ I12 þ lI2 þ I3 þ BI1 I2 þ I13 þ EI1 I3 þ F I12 I2 2 3 3 (34) þ GI22 þ HI14 ; where, in addition to the connections (17), the elastic constants are given by 4A! 28l 2k þ ; E ¼ E! þ 2B! þ þ 3 9 3 k A! 7l G ¼ G! þ B! þ þ þ ; 2 2 6 7l F ¼ F! þ C! % A! % ; 3 H ¼ H! þ A! 7l þ : 6 18 (35) Note that it is a simple matter to invert these relations using Eqs. (17) to give 4A 44l 4k þ þ ; E! ¼ E % 2B % 3 9 3 A 11l k þ ; G! ¼ G % B % þ 2 6 2 11l F! ¼ F % C þ A % ; 3 A 11l H! ¼ H % þ : (36) 6 18 The stress t, to third order in e, is then given by t ¼ kI1 d þ 2le þ BðI2 d þ 2I1 eÞ þ Ae2 þ CI12 d þ EðI3 d þ 3I1 e2 Þ þ 2F ðI1 I2 d þ I12 eÞ þ 4GI2 e þ 4HI13 d: (37) By defining T1 ¼ tr(t), T2 ¼ tr(t2), and T3 ¼ tr(t3), similarly to Sec. II, by manipulating the equations to third order, and after some considerable algebra (which is omitted), we obtain an extension of the expansion (20) to give I1 ¼ aT1 þ bT2 þ cT12 þ a0 T3 þ b0 T1 T2 þ c0 T13 ; (38) where a, b, and c are as given by Eq. (21), while a0 , b0 , and c0 are defined via E5 a0 ¼ ð1 þ mÞ3 ð1 % 2mÞ½2ð1 þ mÞAðA þ 3BÞ % 3EE '; III. FOURTH-ORDER INCOMPRESSIBILITY We now extend the above analysis to include fourthorder terms in the strain-energy function, as in Eq. (5), and we work in terms of the logarithmic strain tensor and its invariants. We use the fourth-order expansion e! ¼ e þ e2 þ 2e3 =3 þ e4 =3 and the identity8 trðe4 Þ ¼ ð1=6ÞI14 % I12 I2 þ ð1=2ÞI22 þ ð4=3ÞI1 I3 to establish the following connections between the invariants of e! and those of e: % E½3ð1 % 5mÞE þ 2ð1 % 2mÞð3F þ 2GÞ'g; E5 c0 ¼ 2m2 ð2 % mÞð1 % 2mÞ2 ðA þ 3BÞ2 % 6mð1 % mÞ ( ð1 % 2mÞ3 ðA þ 3BÞð2B þ 3CÞ ( ½2ð1 þ mÞAðA þ 3BÞ % 3EE' % mð2 % mÞð1 % 2mÞ2 ( ½6ð1 þ mÞðA þ 3BÞB % Eð3E þ 6F þ 4GÞ' 7 28 7 7 I!2 ¼ I2 þ 2I3 % I12 I2 þ I1 I3 þ I22 þ I14 ; 3 9 6 18 þ 2ð1 % 2mÞ4 ½ð1 þ mÞðA þ 3BÞC % EðF þ 6HÞ': (39) (33) The strain energy may then be expressed as a function of the invariants of e, in the form J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010 ( ½ð1 % 2mÞð5 % 4mÞB þ 3ð1 % 2mÞ2 C % mð4 þ mÞA' þ 2ð1 % 2mÞ5 ð2B þ 3CÞ2 þ 3m2 ð1 % 2mÞ 2 4 1 1 1 I!1 ¼ I1 þ I2 þ I3 þ I1 I3 % I12 I2 þ I22 þ I14 ; 3 9 3 6 18 3 1 I!3 ¼ I3 þ 4I1 I3 % 3I12 I2 þ I22 þ I14 : 2 2 E5 b0 ¼ ð1 þ mÞ2 ð1 % 2mÞf2ðA þ 3BÞ The exact incompressibility constraint [Eq. (11)] is enforced for all stresses when a ¼ b ¼ c ¼ 0 and a0 ¼ b0 ¼ c0 ¼ 0. The first three conditions lead to the limits, Eqs. (22)–(26), M. Destrade and R. W. Ogden: Fourth-order incompressible elasticity 3337 Downloaded 10 Jan 2011 to 79.97.235.126. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.jsp thereby bringing great simplifications in the expressions above. Hence in the incompressible limit, we have 81 E a ¼ % ð1 % 2mÞE; 8 4 0 (40) which must tend to zero as v ! 1=2. However, on inspection of Eq. (37), it can be seen immediately that for the stress to remain finite in the incompressible limit, we must have that E is finite, E=l ¼ Oð1Þ; (41) and a0 above does indeed tend to zero as v ! 1=2. As in Sec. II, we see that the first term in Eq. (37) remains finite in the incompressible limit when (1 % 2v)%1I1 remains finite. Clearly here, (1 % 2v)%1a0 remains finite. Equally, for the term in G in Eq. (37) to remain finite, G must itself be finite in the limit: G=l ¼ Oð1Þ: 27 ð1 % 2mÞ2 F : 2 (43) For b0 to tend to zero, and (1 % 2m)%1b0 to remain finite, we must enforce the following behavior for F : ð1 % 2mÞ2 F ! 0; F =l ¼ Oðk=lÞ: (44) In these limits, the I1 I2 F -term in the expression [Eq. (37)] for the stress remains finite, while the I12 F -term vanishes. It remains to consider H. Using the limits above, we see that c0 behaves as E4 c0 ¼ %12ð1 % 2mÞ4 H: (45) For this to tend to zero and (1 % 2m)%1c0 to remain finite, we require ð1 % 2mÞ4 H ! 0; # $ H=l ¼ O k3 =l3 : (46) We may then check that the limiting value of the last term in Eq. (37) is finite. In summary, we must have for the fourth-order constants associated with the logarithmic strain E=l ¼ Oð1Þ; F =l ¼ Oðk=lÞ; # 3 3$ H=l ¼ O k =l ; G=l ¼ Oð1Þ; (47) which are necessary and sufficient for incompressibility at this order. For the fourth-order constants associated with the Green strain, we use Eqs. (36) to find ! ¼ Oðk=lÞ; E=l ! ¼ Oðk=lÞ; G=l 3338 4E ð1 % 2mÞE! ! ; 3 E ð1 % 2mÞG! ! : 6 (49) For incompressible solids, I1 ¼ 0 for all stresses, and the fourth-order expansion [Eq. (34)] reduces to W ¼ lI2 þ A I3 þ GI22 ; 3 (50) and only three constants remain.9 Equivalently, in terms of the invariants of the Green strain tensor, the expansion of W reads as Eq. (8), where D! is defined by Eqs. (12) or, equivalently, by A 11l ; D! ¼ G % þ 2 6 (51) making it explicit that it is of the same order as l (and thus, ! as A): (42) Now we find that in the incompressible limit, b0 behaves as E4 b0 ¼ % and, specifically, # $ ! ¼ O k2 =l2 ; F=l # $ ! ¼ O k3 =l3 ; H=l J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010 (48) ! ¼ Oð1Þ: D=l (52) Turning our attention to the stress, we see that all the terms multiplying d in Eq. (37) are absorbed by the arbitrary hydrostatic stress, to give, in the limit, t ¼ %pd þ 2le þ Ae2 þ 4GI2 e: (53) By the Cayley–Hamilton theorem, we have (for an incompressible material where I1 ¼ 0) 1 1 e3 ¼ I2 e þ I3 d; 2 3 (54) and hence we have ! " 2 t ¼ %pd þ 2le þ Ae2 þ 4G 2e3 % I3 d ; 3 (55) and finally, by adjusting the hydrostatic term by introducing p0 ¼ p þ 8GI3 =3, t ¼ %p0 d þ 2le þ Ae2 þ 8Ge3 ; (56) where only the three constants of fourth-order incompressible elasticity appear and no invariant. We recall that the corresponding measure of stress conjugate to the Green strain is given by the connection !t ¼ ðd þ 2! eÞ%1 t, which applies for an isotropic material, yielding !e2 þ 8D! !e3 : !t ¼ %p0 ðd þ 2! eÞ%1 þ 2l! e þ A! (57) Note that the Lagrange multiplier p0 must figure in the expressions for the stress [see also Eq. (32)] but has been omitted in the expression for the stress in several papers.6,11,16,17 The counterpart of the strain–stress relation [Eq. (30)] for the fourth order is very lengthy and is not written here. We note, however, that the contribution to e additional to the first and second-order terms in Eq. (30) for the third order in the stress has the structure M. Destrade and R. W. Ogden: Fourth-order incompressible elasticity Downloaded 10 Jan 2011 to 79.97.235.126. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.jsp a1 t 3 þ a2 T1 t 2 þ ða3 T12 þ a4 T2 Þt þ ða5 T3 þ a6 T1 T2 þ a7 T13 Þd; (58) where a1, a2, … , a7 are constants that are collectively functions of l, k (or E, m), A, B, … , H. With the exception of a1 ¼ A2 =16l5 , these expressions are very lengthy and therefore omitted. However, in the incompressible limit, the strain–stress relation simplifies substantially and the extension of Eq. (31) becomes ! " ! " 3 1 27A 2 1 2 e ¼ 3 ð2E þ 3AT1 Þ t % T1 d % 3 t % T2 d 4E 3 8E 3 ! " 27 1 (59) ðT12 % 3T2 Þð8EG % A2 Þ t % T1 d : þ 5 32E 3 As in the third-order case, it is seen immediately that tr(e) ¼ 0. In deriving Eq. (59), we have used the Cayley–Hamilton theorem for t to eliminate t3. B. Acoustoelasticity of surface acoustic waves (SAWs) Hayes and Rivlin22 computed the acoustoelastic coefficient for SAW propagation. For a wave propagating in the direction of uniaxial pre-stress of (small) magnitude r, it is defined by Eq. (60) where v is now the SAW speed. Using the results of Tanuma and Man,23,24 we present it in the form K ¼ 1 % ac22 % bc23 % cc33 % dc44 (61) for a compressible isotropic elastic solid, where the c’s are defined in terms of the Lamé constants by IV. EXAMPLES A. Acoustoelasticity of bulk acoustic waves Hughes and Kelly18 used the acoustoelastic effect to evaluate experimentally the third-order elasticity constants, by measuring the speed of infinitesimal bulk homogeneous plane waves propagating in a solid subject to a small prestress. We summarize their results in Table I, where we use the layout of Norris19 and the definition K¼ in an incompressible solid. The formulas for shear waves are in accord with those established using different means by Gennisson et al.20 and by Destrade et al.21 in the case of uniaxial pre-stress. For hydrostatic pre-stress, we note that the speeds of the shear waves are unaffected by the hydrostatic stress, which is also to be expected for an incompressible material. c22 ¼ ðk þ 2lÞ½%8ðk þ lÞ þ 2ð5k þ 6lÞX & %ð2k þ 3lÞX2 =D; c23 ¼ 4kð1 % XÞ½4ðk þ lÞ % ðk þ 2lÞX'=D; c33 ¼ ½1 % lX=ðk þ 2lÞ'=D; c44 ¼ %8ðk þ 2l % lXÞ½2ðk þ lÞ % ðk þ 2lÞX'=D; % dðqv2 Þ %% dr %r¼0 (60) of the acoustoelastic coefficient K, where q is the mass density in the unstrained state and r is the pre-stress (which is either hydrostatic or uniaxial). The incompressible counterparts to these formulas are readily established by use of Eqs. (22) and (29). They appear in the last column of the table. In particular, it is seen that the speed of longitudinal waves is infinite, as expected ' & D ¼ ðk þ lÞ 8ð3k þ 4lÞ % 16ðk þ 2lÞX þ 3ðk þ 2lÞX2 ; (62) and the nondimensional quantity X is the unique real positive root to Rayleigh’s cubic,25 X3 % 8X2 þ 8 3k þ 4l kþl X % 16 ¼ 0; k þ 2l k þ 2l (63) TABLE I. Acoustoelastic coefficients for bulk acoustic waves in compressible and incompressible solids (here 3j ¼ 3k þ 2l). Stress Mode Propagation Polarization Hydrostatic Longitudinal Arbitrary kn Hydrostatic Transverse Arbitrary \n Uniaxial Longitudinal k Stress kn Uniaxial Longitudinal \ Stress kn Uniaxial Transverse k Stress \n Uniaxial Transverse \ Stress k Stress Uniaxial Transverse \ Stress \ Stress J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010 K (compressible) 1 % ½7k þ 10l þ 2A! þ 10B! þ 6C!' 3j 1 % ½3k þ 6l þ A! þ 3B!' 3j ( ) 1 kþl ! % k þ 2B! þ 2C! þ 2 ð2k þ 5l þ A! þ 2BÞ 3j l ( ") ! A! 2 ! ! k k þ 2l þ þ B! % BþC% 2 3j l ( ) 1 k þ 2l ! ! 4ðk þ lÞ þ AþB % 3j 4l ( ) 1 k þ 2l ! ! % k þ 2l þ AþB 3j 4l ( ) 1 k ! ! 2k þ A % B 3j 2l K (incompressible) 1 0 1 1 ( % 1þ ) A! 12l A! 12l A! 1þ 6l % M. Destrade and R. W. Ogden: Fourth-order incompressible elasticity 3339 Downloaded 10 Jan 2011 to 79.97.235.126. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.jsp and a, b, c, and d depend on the second- and third-order constants according to a¼ ð4k þ 3lÞðk þ 2lÞ 2ðk þ lÞA! þ ð3k þ 2lÞl ð3k þ 2lÞl ! þ 2ð1 % 2mÞ b ¼ 3E þ ð1 % 2mÞ3 ðB! þ CÞ ! ! þ 6m2 A: ( ð1 þ mÞðA! þ 2BÞ 2ð2k þ 3lÞB! 2C! þ ; þ ð3k þ 2lÞl 3k þ 2l b¼ ðk þ lÞk ðk þ 2lÞB! 2C! þ þ ; ð3k þ 2lÞl ð3k þ 2lÞl 3k þ 2l ð2k þ lÞðk þ 2lÞ kA! % ð3k þ 2lÞl ð3k þ 2lÞl ! ! 2ðk % lÞB 2C þ ; % ð3k þ 2lÞl 3k þ 2l B! kþl ðk þ 2lÞA! : þ þ ð3k þ 2lÞl 4ð3k þ 2lÞl 3k þ 2l 3 ! b ¼ ð4l þ AÞ; 2 (64) Substantial simplifications occur when the incompressible limits (22)–(29) apply. In particular, X is now a definite number,25 namely X ) 0.9126, the root of X3 % 8X2 þ 24X % 16 ¼ 0, and c22 ¼ %c23 =2 ¼ c33 ¼ 2ðX % 1Þð4 % XÞ=ð26 % 16X þ 3X2 Þ. Because of these latter relationships, the limit of K remains finite since, even though each of a, b, and c goes to infinity as Oðk=lÞ, the combination a % 2b þ c tends ! to a finite limit, specifically a % 2b þ c ! A=ð3lÞ, while ! d ! A=ð12lÞ. The final result is that the acoustoelastic coefficient for SAWs in incompressible media is (65) as established differently by Destrade et al.21 We emphasize that this result applies for the situation in which the stress is uniaxial. It is interesting to note in passing that in the corresponding plane strain problem [in the (1, 2) plane with stress r in the x1 direction], it can be shown that K ¼ 1 % X=2, ! which is independent of the third-order constant A. C. Solitary waves in rods Porubov26 showed that the propagation of nonlinear strain waves in an elastic rod with free lateral surface is governed by the so-called double-dispersive equation (DDE). For solids with the third-order strain-energy density Eq. (13), the DDE is vtt % a1 vxx % a2 ðv2 Þxx % a3 vxxtt þ a4 vxxxx ¼ 0; W ¼ C10 ½trðCÞ % 3' þ C01 ½trðC%1 Þ % 3'; 3340 where C ¼ 2! e þ d, is the right Cauchy-Green strain tensor and C10 and C01 are constants, is equivalent to the strainenergy density Eq. (28), with the connections A! ¼ %8ðC10 þ 2C01 Þ: (71) Clearly, it follows that here we have b ¼ % 32C01. However, it is also well known that the governing equations of motion for Mooney–Rivlin solids are strongly elliptic when27 C01 > 0. Provided that this condition is satisfied, we deduce that b < 0; (72) for all incompressible third-order solids and, therefore, that a compressive solitary wave emerges from an initial localized compressive input.26 In fact, Porubov and Maugin28 show that fourth-order elasticity is required to account for the possibility of simultaneous compressive and tensile solitary waves. They use the Murnaghan29 counterpart to Eq. (5), where the expansion is carried out in terms of the principal invariants of e!, which we write as i1 ¼ tr e!; 1 i2 ¼ ½ðtr e!Þ2 % trð! e2 Þ'; 2 i3 ¼ det e!: (73) The Murnaghan strain-energy function is then given by mðm % 1Þ 2 a3 ¼ R ; 2 J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010 (70) (66) for the strain function v ¼ v(x,t), where x is the space variable in the direction of propagation, t is time, and the subscripts denote partial differentiation, and E b a2 ¼ ; a1 ¼ ; q 2q mER2 : a4 ¼ % 2q (69) and the a’s simplify accordingly. The analysis of Porubov can then be carried through, particularly to study solitary waves and solitons. It should be pointed out that the sign of b can be determined for all incompressible third-order solids. Indeed, it is well known that, at the same level of approximation, the model described by the Mooney–Rivlin strainenergy density l ¼ 2ðC10 þ C01 Þ; A! ; K ¼ 1 þ 0:9126 12l (68) Now, in the incompressible limit described by Eqs. (22)–(29), this nonlinear parameter reduces to the very simple expression c¼% d¼ Here E is Young’s modulus, m is Poisson’s ratio, R is the rod radius, q is the mass density, and b is the nonlinear parameter. [Note that this b is different from the b defined in Eq. (21).] Explicitly, W¼ (67) k þ 2l 2 l þ 2m 3 i1 % 2li2 þ i1 % 2mi1 i2 þ ni3 þ m1 i41 2 3 þ m2 i21 i2 þ m3 i1 i3 þ m4 i22 ; (74) M. Destrade and R. W. Ogden: Fourth-order incompressible elasticity Downloaded 10 Jan 2011 to 79.97.235.126. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.jsp where m, l, … , m4 are constants. The correspondence between the Landau and the Murnaghan constants is easy to establish. We find the connections or equivalently, in terms of the incompressible Landau constant A! and the constant D! defined in Eq. (12), ! c ¼ 14l þ 6A! þ 18D: A! ! m ¼ þ B; 2 ! n ¼ A; (81) ! l ¼ B! þ C; D. Nonlinear plane waves ! m1 ¼ E! þ F! þ G! þ H; ! m2 ¼ %3E! % 2F! % 4G; ! m4 ¼ 4G; ! m3 ¼ 3E; (75) and thus, the incompressible limits m=l ¼ Oðk=lÞ; # n=l ¼ Oð1Þ; # $ m1 =l ¼ O k3 =l3 ; 2 2 Wochner et al.16,30 also use the notation b to identify the coefficient of cubic nonlinearity for shear waves. Again, this b is different from that used previously in this paper and is given by " # ! þ B!Þ2 3 ð k þ 2l þ A=2 k þ 2l þ A! þ 2B! þ 2G! % : b¼ kþl 4l $ l=l ¼ O k =l ; # $ m2 =l ¼ O k2 =l2 ; m3 =l ¼ Oðk=lÞ; m4 =l ¼ Oðk=lÞ; (82) (76) with the specific limits ð1 % 2mÞm3 ! 4E; ð1 % 2mÞm ! %E=3; ð1 % 2mÞm4 ! 2E=3: According to the first, second, and fifth limits in Eqs. (12), ! this quantity should collapse to b ¼ 3ð2l þ AÞ=ð4lÞ for 16 incompressible solids. In fact, the true limit is b¼ (77) Now, when fourth-order terms are taken into account, the corresponding DDE has an extra term28 and becomes vtt % a1 vxx % a2 ðv2 Þxx % a3 vxxtt þ a4 vxxxx % a5 ðv3 Þxx ¼ 0; ! " 3 A! þ 2D! 1þ ; 2 2l (83) ! as defined in Eq. (12), should remain finite, because D, although until now, this behavior had not been proved rigorously. An alternative means of finding the correct limit is to rewrite the expression for b in terms of the constants associated with the logarithmic strain, as (78) where a1, a2, a3, and a4 are still given by Eqs. (67) and a5 ¼ c=(3q), with c (different from the c in Sec. II), is given by 2 2 5 2 2 Ec ¼ E % 8l ð1 % 2mÞ ð1 þ mÞ % 32m m ð1 % 2mÞð1 þ mÞ 3 % 8n2 m2 ð1 % 2mÞð1 þ mÞ þ 4lð1 % 2mÞ3 ( fE % 4mð1 þ mÞ½2mð1 þ mÞ % n'g þ 8mðE þ 4nm2 Þ ( ð1 % 2mÞð1 þ mÞ2 þ 12nEm2 þ 8m1 Eð1 % 2mÞ4 % 8m2 Eð1 % 2mÞ2 ð2 % mÞm þ 8m3 Eð1 % 2mÞm2 2 2 þ 8m4 Eð2 % mÞ m : (79) Clearly, the limits [Eqs. (76) and (77)] do not give a definite incompressible limit for c. In particular, the terms proportional to (1 % 2m)m2 and to m4 tend to infinity, and the limits of the terms proportional to (1 % 2m)3lm and to (1 % 2m)2m2 are not known. When c is written in terms of the Landau constants [using the inverse of Eq. (75)], similar ambiguities arise. However, if the expression for c is further transformed in terms of A, B, … , H, the nonlinear constants associated with the logarithmic strain, then it is a simple matter to find its unequivocal incompressible limit. In the end, we find that c tends to c¼ 11E % 3A þ 18G; 3 J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010 (80) ( 3ð1 þ mÞ E 2ð1 þ mÞð1 % 2mÞ 2G % % b¼ 2E 6ð1 þ mÞ E ! "2 # A E þB% : ( 2 2ð1 þ mÞ (84) Then the limit is clearly b ¼ 9ð2G % E=9Þ=ð4EÞ, which is the same as Eq. (83). Note that Wochner et al.16 also provide bl, the Gol’dberg31 coefficient of nonlinearity for longitudinal waves, as 3 A! þ 3B! þ C! : bl ¼ % k þ 2l 2 (85) If, as is incorrectly reported in Refs. 8, 14, and 15, C! were to ! ¼ Oðk=lÞ, then bl would remain finite in the behave as C=l incompressible limit, suggesting that longitudinal homogeneous plane waves are possible. As we have established in ! behaves as Oðk2 =l2 Þ, and bl therefore blows Eq. (29), C=l up in the incompressible limit, as should be expected, thereby precluding the existence of such waves. V. CONCLUDING REMARKS Using the logarithmic strain measure, we are able to determine the exact behavior of the elastic constants of second, third, and fourth orders in the incompressible limit, as collected below. For the second-order Lamé constants M. Destrade and R. W. Ogden: Fourth-order incompressible elasticity 3341 Downloaded 10 Jan 2011 to 79.97.235.126. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.jsp k ! 1; l ! E=3; (86) as is well known; for third-order Landau constants, ! ¼ Oð1Þ; ! ¼ Oðk=lÞ; A=l B=l # $ ! ¼ O k2 =l2 ; C=l (87) and for the fourth-order Landau constants, ! ¼ Oðk=lÞ; E=l ! ¼ Oðk=lÞ; G=l # $ ! ¼ O k2 =l2 ; F=l # $ ! ¼ O k3 =l3 : H=l (88) For the constants which vary linearly with k as it goes to infinity, the specific limits are ð1 % 2mÞB! ! %E=3; ð1 % 2mÞE! ! 4E=3; ð1 % 2mÞG! ! E=6; (89) as the Poisson ratio m ! 1=2. We have used these limits to show that it is easy to take known results of elastic wave propagation in compressible materials to the corresponding incompressible limits. In fact, other types of internal constraints could be accounted for just as easily.32,33 We conclude the paper with two remarks. The first is technical: It should not be forgotten that the hydrostatic term in the stress-logarithmic strain relation of an incompressible solid is an arbitrary Lagrange multiplier, to be determined from initial/boundary conditions. This Lagrange multiplier also appears in the stress-Green strain relation, see Eqs. (32) and (57). Omission of this term would therefore lead to incorrect solutions of the equations. The second is semantic: One gets the impression from reading the acoustics literature that “soft” and “incompressible” are two interchangeable adjectives. 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