DSpace at VNU: Formulas for the Rayleigh wave speed in orthotropic elastic solids

Meccanica (2005) 40: 147–161 DOI 10.1007/s11012-005-1603-6 © Springer 2005 On the Rayleigh Wave Speed in Orthotropic Elastic Solids PHAM CHI VINH and R W OGDEN1,∗ Faculty of Mathematics, Mechanics and Informatics, Hanoi National University, 334, Nguyen Trai Street, Thanh Xuan, Hanoi, Vietnam; Department of Mathematics, University of Glasgow Glasgow G12 8QW, UK (Received: 22 June 2004; accepted in revised form: January 2005) Abstract Recently, a formula for the Rayleigh wave speed in an isotropic elastic half-space has been given by Malischewsky and a detailed derivation given by the present authors This study deals with the generalization of this formula to orthotropic elastic materials and Malischewsky’s formula is recovered as a special case The formula is obtained using the theory of cubic equations and is expressed as a continuous function of three dimensionless material parameters Key words: Rayleigh waves, Wave speed, Surface waves, Orthotropy Introduction Recently, there has been considerable interest in obtaining explicit formulas for the Rayleigh wave speed in an elastic half-space For isotropic materials such formulas have been given by Rahman and Barber [1], Nkemzi [2] and Malischewsky [3] In obtaining his formula Malischewsky used Cardan’s formula for the solution of a cubic equation together with Mathematica A detailed derivation of this formula was given by Pham and Ogden [4] together with an alternative formula See also the recent analysis of Malischewsky [5] For non-isotropic materials, for some special cases of compressible monoclinic materials with symmetry plane x3 = 0, formulas for the Rayleigh wave speed have been found by Ting [6] and Destrade [7] as the roots of quadratic equations, while for incompressible orthotropic materials an explicit formula has been given by Ogden and Pham [8] based on the theory of cubic equations Further, in a recent paper [9] we have obtained explicit formulas for the Rayleigh wave speed in compressible orthotropic elastic solids One of the formulas is analogous to that of Malischewsky in the isotropic case, but we were not able to establish its validity for all relevant ranges of values of the material parameters The main purpose of the present paper therefore is to provide a generalization of Malischewsky’s formula for compressible orthotropic materials that is valid for all appropriate ranges of values of the material parameters We consider a compressible elastic body possessing a stress-free configuration of semi-infinite extent in which ∗ Author for correspondence Tel.: +44-141-330-4550; Fax: +44-141-330-4111; e-mail: rwo@maths gla.ac.uk 148 Pham Chi Vinh and R W Ogden the material exhibits orthotropic symmetry The boundary of the body is taken to be parallel to the (001) mirror plane of the material and we choose rectangular Cartesian axes (x1 , x2 , x3 ) such that the x3 direction is normal to the boundary, the body occupies the region x3 and the Rayleigh wave propagates in the (x1 , x3 ) plane and decouples from any transverse motions (which are not considered here); (see, for example, [10,11]) We recall that for an orthotropic material with the symmetry planes coinciding with the Cartesian coordinate planes the stress-strain relation may be written in the standard compact form σi = cij ej , i, j ∈ {1, , 6}, where σi , ei are the stress and strain components and cij = cj i the elastic constants (cij = for i = j when i = 4, or 6) In terms of the tensor components σij , eij we have σi = σii , i = 1, 2, 3, σ4 = σ23 , ei = eii , i = 1, 2, 3, e4 = 2e23 , σ5 = σ13 , σ6 = σ12 , e5 = 2e13 , e6 = 2e12 (1) (2) For the considered specialization, however, the relevant material constants (those appearing in the equation of motion) are just c11 , c33 , c55 , c13 Necessary and sufficient conditions for the strain energy of the material (under the considered restriction) to be positive definite are cii > 0, i = 1, 3, 5, c11 c33 − c13 > (3) It is convenient to pursue the analysis in terms of three dimensionless material parameters, defined by α = c33 /c11 , γ = c55 /c11 , δ = − c13 /c11 c33 , (4) so that, in accordance with (3), α > 0, γ > 0, < δ < (5) These parameters may also be expressed in terms of other elastic constants For example, if ν13 , ν31 are the Poisson’s ratios in the (x1 , x3 ) plane then α = ν31 /ν13 , δ = − ν13 ν31 , (6) while γ = δG13 /E1 , where G13 is the shear modulus associated with the (x1 , x3 ) plane and E1 the Young’s modulus for the x1 direction A generalization of Malischewsky’s formula for the Rayleigh wave speed is obtained for values of these parameters such that either α 1, < δ < 1, < γ < α or < α < 1, < δ < 1, < γ < α, γ (α − 1) + 2αδ > In this formula, which is based on the theory of cubic equations, each cubic root takes its principal value We also obtain an alternative formula in which the cubic roots take their secondary values Formulas for the Rayleigh wave speed for the remaining ranges of the parameter values are also investigated For the case of isotropy (for which Malischewsky’s formula applies) the values of the parameters are such that α = 1, < γ < 3/4, δ = 4γ (1 − γ ), as we will note in Section Rayleigh Wave Speed 149 The Secular Equation The equations of motion have been examined in detail previously (see, for example, [9] and references therein) and are not therefore repeated here We begin with the form of the secular equation given by Chadwick [10], namely √ c55 − ρc2 c13 − c33 c11 − ρc2 + c33 c55 ρc2 (c11 − ρc2 )(c55 − ρc2 ) = 0, (7) where c is the Rayleigh wave speed and ρ the mass density of the material As discussed previously (see, for example, [9]), the wave speed must satisfy the inequalities inequality < ρc2 < min{c11 , c55 } (8) Note that for c11 , c33 , c55 , c13 satisfying (3), Chadwick [10] showed that equation (7) has a unique (real) solution satisfying (8) and corresponding to a surface wave We now introduce the notation x = ρc2 /c55 (9) Then, from (8), we deduce that 0< x < σ if < c55 c11 , 0< x < σ < if < c11 < c55 , (10) where, for convenience, we have also introduced the notation σ ≡ 1/γ (11) It follows that x ∈ (0, σˆ ), σˆ ≡ min{1, σ } Equation (7) can now be written in the form √ √ α − x(σ δ − x) = x − γ x (12) (13) with x ∈ (0, σˆ ) and the parameters α, γ , δ satisfying the inequalities (5) A Formula for the Rayleigh Wave Speed From equation (13), after squaring and rearranging, we obtain the cubic equation F (x) ≡ (γ − α)x + (α + 2ασ δ − 1)x − ασ δ(σ δ + 2)x + ασ δ = (14) for x PROPOSITION In the interval (0, σ∗ ), where σ∗ = min{1, σ δ}, equation (14) has a unique real solution, x0 say, that corresponds to the Rayleigh wave 150 Pham Chi Vinh and R W Ogden Proof According to Chadwick [10], in the interval (0, σˆ ), equation (13) has a unique real solution x0 corresponding to the Rayleigh wave From (13) it follows that x0 ∈ (0, σ∗ ) By (5), < δ < 1, and hence (0, σ∗ ) ⊂ (0, σˆ ) Thus, in the interval (0, σ∗ ), equation (13) has a unique real solution x0 , and since in this interval equations (13) and (14) are equivalent, the proposition is established For the values of α and γ such that α = γ , equation (14) may be written F1 (x) ≡ x + a2 x + a1 x + a0 = 0, (15) where a0 = ασ δ , γ −α a1 = ασ δ(σ δ + 2) , α−γ a2 = α − + 2ασ δ γ −α (16) From (15) and (16), we then have F1 (0) = ασ δ , γ −α F1 (1) = γ −1 , γ −α F1 (σ δ) = σ δ (δ − 1) γ −α (17) Let E denote the three-dimensional Euclidian space of points (α, γ , δ) and let M = M(α, γ , δ) denote a point of E We define the following subsets of E: = {M ∈ E : α > 0, γ > 0, < δ < 1}, : α < γ }, : α > γ }, = {M ∈ = {M ∈ : α = γ }, 1}, = {M ∈ : γ = {M ∈ = {M ∈ : < γ < 1}, 1, d > 0}, 11 = {M ∈ : d < 0}, 12 = {M ∈ : α 13 = {M ∈ : α < 1, d > 0, (α − 1)γ + 2αδ > 0}, 14 = {M ∈ : α < 1, d > 0, (α − 1)γ + 2αδ < 0}, = {M ∈ : d = 0}, : (α − 1)γ + 2αδ > 0}, = {M ∈ : (α − 1)γ + 2αδ < 0} = {M ∈ (18) In (18) we have used the notation d = a22 − 3a1 , (19) where a1 and a2 are given by (16) Note that 4d is the discriminant of the quadratic F1 (x) Thus, when γ = α and d 0, F1 (x) is a monotone function and equation (15) has a unique real root Note also that a1 > in and that a2 = when (α − 1)γ + 2αδ = 0; therefore, for a point M ∈ such that (α − 1)γ + 2αδ = 0, d < 0, i.e M ∈ 11 and hence M ∈ We now state a theorem concerning a formula for the Rayleigh wave speed THEOREM In the region ∗ = 11 ∪ 12 ∪ wave speed c, with x0 = ρc2 /c55 , is given by ρc2 /c55 = 13 ∪ 1, x0 , and hence the Rayleigh √ √ α − + 2ασ δ 3 + sign(−d) sign(−d)[R + D] − −R + D, 3(α − γ ) (20) Rayleigh Wave Speed 151 where the (complex) roots take their principal values, the principal argument of a complex w, Arg w, is taken in the interval (−π, π ], and R and D are given by R = 9a1 a2 − 27a0 − 2a23 /54, D = 4a0 a23 − a12 a22 − 18a0 a1 a2 + 27a02 + 4a13 /108, (21) in terms of a0 , a1 , a2 , as defined in (16) For the case of an isotropic material we have c11 = c33 = λ + 2µ, c55 = µ, c13 = λ and hence α = 1, δ = 4γ (1 − γ ), γ = µ/(λ + 2µ), < γ < 3/4, where λ and µ are the Lam´e moduli From these and equations (16), (19) and (21) we obtain d = 48(γ − 1/6), R = 8(45γ − 17)/27, D = 64(11 − 62γ + 107γ − 64γ )/27 (22) Using (22) it is easy to show that formula (20) reduces to Malischewsky’s formula given in [3], namely − h3 (η) + sign[h4 (η)] sign[h4 (η)]h2 (η) , where the functions hi (η), i = 1, 2, 3, 4, are given by ρc2 /µ = (23) h1 (η) = 33 − 186η + 321η2 − 192η3 , h2 (η) = 45η − 17 + h1 (η), h3 (η) = 17 − 45η + h1 (η), h4 (η) = −η + 1/6 (24) with η = γ in our notation We also note that (23) can be rewritten in the form (20) in the region ω∗ = {M : α = 1, δ = 4γ (1 − γ ), < γ < 3/4} ⊂ ∗ , in which c55 = µ, γ = η and d, R and D are specialized according to (22) Thus, the form of (23) does not change when passing from ω∗ to ∗ In order to prove Theorem we need a number of lemmas LEMMA (a) ∩ = ∅; (b) ∩ = ∅; (c) ∩ = ∅ Proof (a) This follows immediately from the definitions of and (b) Since a1 < in it follows that d > ∀M ∈ Hence the result (c) From (17), since σ δ < 1, equation (15) has at least two distinct real roots when M ∈ But, for d = 0, F1 (x) is monotonic so that the result is established From Lemma 1, therefore, the surface is located in We also note that the set is in , with the values of α necessarily restricted according to < α < We now introduce the sets defined by (1) 1, < γ < 1, < δ < 1, d > 0}, 12 = {M : α (2) 1, < δ < 1, d > 0}, 12 = {M : α > 1, α > γ (1) ∗∗ = 12 ∪ 13 It is clear that F1 (0) < 0, 12 = (1) 12 ∪ F1 (1) > 0, (2) 12 , (25) and from (17) we have F1 (σ∗ ) > in ∗∗ (26) 152 Pham Chi Vinh and R W Ogden LEMMA The set ∗∗ is a connected set Proof Let G(α0 ) be the intersection of ∗∗ with the plane P (α0 ) defined by α = α0 > 0, where α0 is a constant Then, ∗∗ = α0 >0 G(α0 ) We shall show below that G(α0 ) defines a region in the (δ, γ ) plane, as illustrated as in Figure From Figure we see that G(α0 ) is a connected set, and G(α0 ) contains the set defined by T (α0 ) ={M ∈ G(α0 ) : 2/3 < δ < 1, < γ < if α0 1, < γ < α0 if < α0 < 1} (27) It is clear that the strip α0 >0 T (α0 ) is a connected set Thus, two arbitrary points M1 (α1 , γ1 , δ1 ) and M2 (α2 , γ2 , δ2 ) in ∗∗ can be connected by a simple curve M1 M3 M4 M2 , with M3 ∈ T (α1 ), M4 ∈ T (α2 ), M1 M3 ∈ G(α1 ), M2 M4 ∈ G(α2 ) and M3 M4 in the strip Hence, the set ∗∗ is connected We now establish the property of G(α0 ) stated above Let (α0 ) denote the intersection ∩ P (α0 ) From (16) and (19) it can be seen that, in (α0 ), the equation d = may be written as a quadratic equation for γ , namely g(γ ) ≡ [(α0 − 1)2 + 6α0 δ]γ − α0 δ(4 − 3δ + 2α0 )γ + α02 δ = 0, (28) where δ ∈ (0, 1) is considered as a parameter and d > if and only if g(γ ) > We denote the curve (28) in the (δ, γ ) plane by (α0 ) We note the following facts, that may easily be verified (i) For any given positive value of α0 , equation (28) has negative discriminant for δ ∈ (2/3, 1) and therefore has no real solution for such values of δ On the other hand, it has two distinct positive real roots, denoted γ1 , γ2 (> γ1 ), for δ ∈ (0, 2/3), and has a unique (double) positive real root, denoted γ0 , when δ = 2/3 (ii) The curve (α0 ) is located in (α0 ) = ∩ P (α0 ) (according to Lemma 1), and it is a continuous curve (a) (b) (c) Figure Plot of the curve d = 0, i.e (α0 ), in (δ, γ ) space with γ (vertical axis) against δ (horizontal axis) for (a) α0 > 1, (b) α0 = 1, (c) < α0 < In (a) and (c) the curve encloses the region d < 0; in (b) the curve and the γ axis enclose the region d < In (c) the line defined by (α0 − 1)γ + 2α0 δ = is also shown; it cuts the curve d = at its maximum point δ = δ0 ≡ (1 − α0 )/2, γ = α0 Within the square (0, 1) × (0, 1) in (a) and (b) the connected region outside the curve d = is the set G(α0 ) In (c), G(α0 ) is the region in the rectangle (0, 1) × (0, α0 ) outside the curve d = and below the line (α0 − 1)γ + 2α0 δ = Note that d is not defined at the point (δ0 , α0 ), which is shown as a filled circle Rayleigh Wave Speed 153 (iii) For α0 1, < γ1 < γ2 < for all δ ∈ (0, 2/3] Note that g(1) = (α0 − − α0 δ)2 + 3α0 δ > for all α0 > 0, δ > For < α0 < we have < γ1 < γ2 < α0 for all δ ∈ (0, 2/3], δ = (1 − α0 )/2 When δ = (1 − α0 )/2, γ2 = α0 The point M(α0 , α0 , (1 − α0 )/2) belongs to the line (α0 − 1)γ + 2α0 δ = but not to (α0 ) (iv) For any α0 = 1, γ1 and γ2 tend to zero as δ → For α0 = 1, γ1 tends to zero while γ2 tends to as δ → (v) Clearly, g(γ ) < for all γ ∈ (γ1 , γ2 ) This means that, in respect of Figure 1(a) and (c), d < inside (α0 ), while for Figure 1(b) d < in the domain bounded by (α0 ) and the γ axis The facts (i)–(v) show that the set G(α0 ) has the structure shown in Figure 1, and the proof of Lemma is completed LEMMA (a) Let r = −2R; then r = F1 (xN ), where xN is the inflection point on the cubic curve y = F1 (x) (b) When d > 0, F1 (x) has maximum and minimum stationary points, which we denote by xmax and xmin , respectively (c) In ∗∗ , we have < xmax < xmin (29) Proof The assertion (b) is clear Using (15) and (21)1 it is easy to see by direct calculation that (a) is true From (15), we have F1 (x) = 3x + 2a2 x + a1 (30) When d > 0, F1 (x) has two distinct real zeros, namely xmin , xmax From (16) and the definition of ∗∗ , it is clear that the following inequalities hold in ∗∗ : xmin xmax = a1 /3 > 0, xmin + xmax = −2a2 /3 > (31) Hence, (c) LEMMA In ∗∗ , R < if D Proof We note that for all M ∈ ∗∗ , d(M) > Suppose that there exists a point M1 ∈ ∗∗ such that D(M1 ) but R(M1 ) If R(M1 ) = then r(M1 ) = Since d(M1 ) > 0, then by Lemma 3(a) and (b), equation (15) has three distinct real roots at M1 Thus, by Remark 1(iii) below as will be shown shortly, it follows that D < This contradicts the assumption D(M1 ) Next, consider R(M1 ) > (and hence r(M1 ) < 0) If D(M1 ) = then from d(M1 ) > 0, (26), (29), Lemma 3(a) and r(M1 ) < we deduce that equation (15) has two distinct real roots in the interval (0, σ∗ ) This contradicts Proposition Thus D(M1 ) > It is not difficult to verify that the point M2 (1, 3/4, 3/4) ∈ ∗∗ and D(M2 ) < Since M1 , M2 ∈ ∗∗ , then by Lemma we can connect M1 and M2 by a simple continuous curve, which we denote by L12 ∈ ∗∗ Since D is a continuous function on L12 and D(M1 ) > 0, D(M2 ) < 0, there must exist a point M0 ∈ L12 , M0 = M1 , M2 such that D(M0 ) = and D(M) > for all M ∈ L10 (except M0 ), where L10 is the part of L12 connecting M1 and M0 Analogously to above arguments, one can see that 154 Pham Chi Vinh and R W Ogden R does not vanish at any point M ∈ L10 Since R is a continuous function on L10 and R(M1 ) > 0, then R(M) > for all M ∈ L10 Hence R(M0 ) > 0, i.e r(M0 ) < This together d(M0 ) > 0, D(M0 ) = 0, (26), (29) and Lemma 3(a), (b) shows that equation (15) has two distinct real roots in the interval (0, σ∗ ) But this contradicts Proposition 1, and the proof of Lemma is therefore completed We are now in a position to prove Theorem Proof of Theorem In terms of the variable z defined by z = x + a2 /3, (32) equation (15) becomes z3 − 3q z + r = 0, (33) where q = (a22 − 3a1 )/9 = d/9, r = −2R (34) By the theory of cubic equation the three roots of equation (33) are given by the Cardan’s formula (see, for example, [12]) z1 = S + T , 1√ z2 = − (S + T ) + i 3(S − T ), 2 1√ z3 = − (S + T ) − i 3(S − T ), (35) 2 where i2 = −1, √ S = R + D, D = R + Q3 , √ T = R − D, R = − r, Q = −q , (36) and D is given by (16) and (21) It is noted that R and D in (36) are given by the values defined in (21) Remark (i) For the cube root of a real, √ negative number, we take the real, negative result (ii) When D < and hence R + D is complex, then T = S ∗ , where S ∗ is the complex conjugate of S (iii) The nature of three roots of equation (33) depends on the sign of the discriminant D In particular, if D > 0, equation (33) has one real root and two complex conjugate roots; if D = 0, it has three real roots, at least two of which are equal; if D < 0, it has three distinct real roots Let z0 denote the real root of equation (33) corresponding to x0 (defined in Proposition 1) and the Rayleigh wave speed given by (20) In order to prove Theorem we examine the distinct cases associated with different subsets of ∗ First, we consider 11 On 11 , we have d < From (36)3,5 , and the fact that D > we have √ √ R + D > 0, −R + D > (37) Since D > equation (33) has a unique real solution, namely z0 = z1 , given by (35)1 and (36), in which the radicals are understood as real roots Since the value of the Rayleigh Wave Speed 155 real root of a positive real number coincides with the principal value of its corresponding complex root, it is clear that inequalities (37) together with (16), (32) ensure that (20) is valid Second, on we make use of the following lemma LEMMA On 1, R This result is established below If R < 0, then, by (36)3,5 , D > 0, so that equation (33) has a unique real solution and (20) is valid If R = then D = 0, and it is clear from (35), (36) that equation (33) has a unique (triple) real root z0 = and in this case (20) is also valid We now show that R on Suppose that M0 (α0 , γ0 , δ0 ) is an arbitrary point of , so that d(M0 ) = If R(M0 ) > 0, then D(M0 ) > by (36)3 Since D is a continuous function in the open set ⊃ (according to Lemma 1), and M0 ∈ , there exists a sufficiently small neighborhood U0 (M0 ) = {M : (α − α0 )2 + (γ − γ0 )2 + (δ − δ0 )2 < κ } of the point M0 , with κ a sufficiently small positive number, such that U0 (M0 ) ⊂ and D(M) > for all M ∈ U0 (M0 ) Defining U = ∗∗ ∩ U0 (M0 ), we have d(M) > 0, D(M) > for all M ∈ U , and hence, by Lemma 4, R(M) < for all M ∈ U Since R is continuous on ⊃ U , and M0 is a boundary point of U , we conclude that R(M0 ) But this contradicts the assumption that R(M0 ) > (2) Next, noting that 12 ∪ 13 = ∗∗ ∪ (2) ∗∗ and 12 , we examine formula (20) on 12 separately On ∗∗ , by definition, d > If D > 0, equation (33) has a unique real solution, namely z0 = z1 , given by (35)1 and (36) Since, by Lemma 4, R < and d > 0, it follows from (36)3,5 that √ √ − R + D > 0, −R + D > (38) Taking into account (16), (32), (38) and the fact that the value of the real root of a positive real number coincides with the principal value of its corresponding complex root, it is easy to see that (20) is valid If D = 0, then, by Lemma 4, R < Taking into account (36)3−5 , we have r = −2R = 2q (q > 0), and equation (33) becomes (z − q)2 (z + 2q) = 0, (39) whose solutions are q (double root) and −2q Bearing in mind that equation (15) has a unique real solution in the interval (0, σ∗ ), it follows from (26)1 and (29) that the Rayleigh wave speed is determined from the smallest real root of (15) in this case, and thus z0 is the smallest real root of equation (33), i.e z0 = −2q and (20) is applicable In the case D < equation (33) has three distinct real roots, and hence so does equation (15) From Proposition 1, (26)1 and (29), it is clear that x0 is the smallest real root of equation (15), and thus z0 is the smallest real root of (33) When D < the three real roots of (33) are given by (35) and (36), in which complex cubic (square) roots can take one of three (two) possible values such that T = S ∗ In our case we take the principal value and we shall indicate that z2 , as expressed by (35)2 156 Pham Chi Vinh and R W Ogden is the smallest real root of (33) Throughout the remainder of this section, for simplicity, we take the complex roots as their principal values From (36) we have S = R + i −R − Q3 , T = S ∗ (40) √ √ Let 3θ denote the principal argument of R + i −D Since −D > 0, 3θ ∈ (0, π ), and the phase angle corresponding to the principal value of S is θ ∈ (0, π/3) From (40) this implies that |S| = q, and hence S and T can be expressed as S = qeiθ , T = qe−iθ , (41) where θ ∈ (0, π/3) satisfies the equation cos 3θ = −r/2q , (42) which is obtained by substituting z = S + T = 2q cos θ into equation (33) Note that D < implies that |−r/2q | < 1, which ensures equation (42) has a unique solution in the interval (0, π/3) From (35) and (41) it may be verified that z1 = 2q cos θ, z2 = 2q cos(θ + 2π/3), z3 = 2q cos(θ + 4π/3) (43) With reference to (43), taking into account that θ ∈ (0, π/3), it is clear that z1 > z3 > z2 , i.e z2 is the smallest real root of (33) Therefore, z0 = 2q cos(θ + 2π/3) (44) It is clear that to prove (20) for the case d > 0, D < 0, we need the equality √ √ 3 − −R + D − −(R + D) = 2q cos (θ + 2π/3), (45) where the √ roots are complex √ roots taking their principal values Indeed, we have Arg(R + D) = 3θ , Arg(R − D) = −3θ √ √ , 3θ ∈ (0, π ) being the solution of (42) Thus, Arg[−(R + D)] = 3θ − π, Arg[−(R − D)] = −3θ + π Note that √ by Arg w we denote the principal argument of the complex number w Since |R ± D| = q it follows that √ −(R + D) = qei(θ −π/3) , √ −(R − D) = qei(−θ +π/3) (46) From (46) it follows that: √ √ 3 − −R + D − −(R + D) = −2q cos (θ − π/3) = 2q cos (θ + 2π/3) (47) and (45) is established On (2) 1, and hence < σ∗ < 1, and from (17) we have 12 we have γ F1 (0) < 0, F1 (σ∗ ) > 0, F1 (1) in (2) 12 (48) It follows from (48) that on (2) 12 (15) has at least two distinct real roots, and therefore D and, on account of Proposition 1, x0 is the smallest real root It is noted that when equation (15) has two distinct real roots, i.e D = 0, then R < By arguments Rayleigh Wave Speed 157 analogous to those used for ∗∗ , for which d > 0, D 0, one can see that formula (20) is valid in (2) 12 , and the proof of Theorem is completed Finally in this section, we illustrate the dependence of the wave speed on the parameters α, γ , δ in Figure 2, in which ρc2 /c55 is plotted against γ > for several values of α > and one value of δ (results for other values of δ are very similar) These are the continuous curves Also plotted, for comparison, is the corresponding result for an isotropic material (dashed curve), for which α = 1, < γ < 3/4 and δ depends on γ This crosses the α = curve at values of γ corresponding to δ = 0.8, which are marked by the bullet points For the isotropic case the left-hand limit (γ = 0) corresponds to incompressibility Alternative Formulas 4.1 The Region 14 ∪ THEOREM In the region given by x0 = ρc2 /c55 = 14 ∪ 2, x0 , and hence the Rayleigh wave speed, is √ √ α − + 2ασ δ 3 + R + D + R − D, 3(α − γ ) (49) where the (complex) roots take their principal values, the principal argument of a complex number is taken in the interval (−π, π], and R and D are given by (16) and (21) To prove Theorem we need the following lemmas Figure Plot of the curves ρc2/c55 against γ for δ = 0.8 and α = 0.2, 1, 10, 100 For other values of δ ∈ (0, 1) the picture is very similar The dashed curve is the corresponding result for an isotropic material (for which < γ < 0.75) Note that this cuts the α = curve at values of γ for which δ = 0.8 158 Pham Chi Vinh and R W Ogden LEMMA In 14 , R > if D Proof This lemma is similar to Lemma 4, but its proof is much simpler Indeed, since d > 0, a1 > 0, a2 > on 14 , the maximum and minimum stationary points of the function F1 (x) are such that xmax < xmin < (50) These inequalities, together Lemma 3(a) and F1 (0) < 0, ensure the validity of Lemma LEMMA On 2, R This is similar to the Lemma Its proof is analogous to that of Lemma and it uses Lemma To establish (49) on 14 , we follow the procedure used to establish Theorem on 0, x0 is the ∗∗ , but instead of the Lemma we use Lemma 6, noting that when D largest real root of equation (15) The proof of Theorem on is similar to that of Theorem on , but Lemma replaces Lemma 4.2 The Regions and On use of (17), taking account of Proposition 1, we see that, on , equation (15) has at least two distinct real roots (hence D 0), and x0 is the intermediate one Since α = γ , equation (14) reduces to the quadratic (γ + 2δ − 1)x − δ(σ δ + 2)x + σ δ = (51) On use of Proposition 1, one can show that equation (51), with γ + 2δ − = 0, has two distinct real roots and that x0 is the smaller (larger) root when γ + 2δ − > (< 0) For the case γ + 2δ − = 0, i.e δ = (1 − γ )/2 (so that δ > ⇒ σ > 1), the Rayleigh wave speed is given by ρc2 /c55 = (σ − 1)/(σ + 3) (52) From the facts mentioned above, it is not difficult to check the validity of the following theorem THEOREM (a) On 2, the Rayleigh wave speed is given by ρc2 /c55 = √ √ α − + 2ασ δ 3 + e4πi/3 R + D + e−4πi/3 R − D, 3(α − γ ) (53) where each radical is understood as a complex root taking its principal value, R and D ( 0) are determined by (16) and (21), and the principal argument of a complex number is taken in the interval (−π, π ] Rayleigh Wave Speed 159 (b) on the Rayleigh wave speed is given by ρc2 /c55 = δ(σ δ + 2) − δ σ (σ δ + − 4δ) 2(γ + 2δ − 1) for γ + 2δ − = (54) and ρc2 /c55 = (σ − 1)/(σ + 3) for γ + 2δ − = (55) 4.3 A Formula Using the Second Values of Cube Roots Let w be a nonzero complex number with its principal argument taken in the interval [0, 2π), i.e Arg w < 2π Let n and m be given integers such that n 2,√1 m n We define the m th value of the complex root of order n of w, denoted m,n w, by √ Arg w (m − 1)2π + w = n |w| exp i n n m,n (56) Corresponding to the value m = we have the principal (first) value of the order-n root In this subsection, by using the second value of the complex cube roots, we obtain an alternative formula for the Rayleigh wave speed in the region ∗ Remark From the definition of the m th value of the complex root of order n of a complex number, it is clear that the second value of a complex cube root of a negative real number coincides with its real cube root THEOREM In the region by x0 = ρc2 /c55 = ∗, x0 , and hence the Rayleigh wave speed, is given √ √ α − + 2ασ δ 3 + sign(d) sign(d)[R + D] + R − D, 3(α − γ ) (57) where the (complex) cube roots take their second values while the square (complex) roots take their principal values, the principal argument of a complex number is taken in the interval [0, 2π), and R and D are given by (16) and (21) Proof Analogously to the proof of the Theorem 1, we consider Theorem on each subset of the set ∗ First, we consider 11 As is known from Section 3, on 11 d < 0, D > and √ √ − R + D < 0, R − D < (58) Since D > equation (33) has a unique real solution, namely z0 , given by (35)1 and (36), the radials being understood as real roots Noting Remark 2, it is clear that inequalities (58) together with (16) and (32) ensure that (57) is valid Next, consider By Lemma 5, on R If R < 0, then, by (36)3,5 , D > 0, and hence equation (33) has a unique real solution and (57) is again valid If R = then D = 0, and it is clear from (35) and (36) that equation (33) has a (triple) unique real root z0 = and in this case (57) is also valid 160 Pham Chi Vinh and R W Ogden Third, we consider ∗∗ For D 0, the proof of (57) is analogous to that of (20) for D 0, and we use Remark For D < 0, as is known from Section 3, in this case equation (33) has three distinct real roots and among them z0 is the smallest real root When D < three distinct real roots of (33) are given by (35) and (36) with complex cube (square) roots taking one of three (two) possible values such that T = S ∗ Throughout the remainder of this subsection, the cube (square) roots take their second (principal) values It is noted that if Arg S = θ ∈ [0, 2π) then Arg S ∗ = 2π − θ Analogously to Section 3, we also have z0 = 2q cos(θ + 2π/3), (59) where θ ∈ (0, π/3) is the solution of equation (42) It is clear that to ensure (57) is valid we have to show that √ √ R + D + R − D = 2q cos (θ + 2π/3) (60) √ √ Indeed, we have Arg (R + D) = 3θ , Arg (R − D) = 2π − 3θ Thus, by (56) 3 √ R + D = qei(θ +2π/3) , √ R − D = qei(4π/3−θ) = qe−i(2π/3+θ) (61) and (60) follows The proof of Theorem is completed For isotropic materials, taking into account (22), equation (57) reduces to − sign[h4 (η)] sign[h4 (η)](17 − 45η − h1 (η)) + 45η − 17 − h1 (η) , (62) where η = µ/(λ + 2µ), < η < 3/4), h1 (η) and h4 (η) are given by (24) and the cube roots take their second values Together, the formulas obtained by Pham and Ogden [4] and Malischewsky [3] provide alternative formulas for the Rayleigh wave speed for compressible isotropic elastic materials In conclusion, we emphasize that the results obtained in this paper can be used for other types of anisotropy Indeed, Royer and Dieulesaint [11] have shown that with respect to surface (Rayleigh) waves, the results established for the orthotropic case may be applied to 16 different symmetry classes, including cubic, tetragonal and hexagonal anisotropy ρc2 /µ = References Rahman, M and Barber, J.R., ‘Exact expression for the roots of the secular equation for Rayleigh waves’, ASME J Appl Mech 62 (1995) 250–252 Nkemzi, D., ‘A new formula for the velocity of Rayleigh waves’, Wave Motion 26 (1997) 199– 205 Malischewsky, P.G., ‘Comment to “A new formula for velocity of Rayleigh waves” by D Nkemzi [Wave Motion 26 (1997) 199–205]’, Wave Motion (2000) 93–96 Pham, C.V and Ogden R.W., ‘On formulas for the Rayleigh wave speed’, Wave Motion 39 (2004) 191–197 Malischewsky Auning, P.G., ‘A note on Rayleigh-wave velocities as a function of the material parameters’, Geofisica Internacional 43 (2004) 507–509 Ting, T.C.T., ‘A unified formalism for elastostatics or steady state motion of compressible or incompressible anisotropic elastic materials’, Int J Solids Struct 39 (2002) 5427–5445 Rayleigh Wave Speed 161 10 11 12 Destrade, M., ‘Rayleigh waves in symmetry planes of crystals: explicit secular equations and some explicit wave speeds’, Mech Materials 35 (2003) 931–939 Ogden, R.W and Pham, C.V., ‘On Rayleigh waves in incompressible orthotropic elastic solids’, J Acoust Soc Am 115 (2004) 530–533 Pham, C.V and Ogden R.W., ‘Formulas for the Rayleigh wave speed in orthotropic elastic solids’, Arch Mech 56 (2004) 247–265 Chadwick, P., ‘The existence of pure surface modes in elastic materials with orthorhombic symmetry’, J Sound Vib 47 (1976) 39–52 Royer, D and Dieulesaint, E., ‘Rayleigh wave velocity and displacement in orthorhombic, tetragonal, hexagonal, and cubic crystals’, J Acoust Soc Am 76 (1984) 1438–1444 Cowles, W.H and Thompson, J.E., Algebra, Van Nostrand, New York, 1947  ... obtain an alternative formula in which the cubic roots take their secondary values Formulas for the Rayleigh wave speed for the remaining ranges of the parameter values are also investigated For. .. (0, σˆ ) and the parameters α, γ , δ satisfying the inequalities (5) A Formula for the Rayleigh Wave Speed From equation (13), after squaring and rearranging, we obtain the cubic equation F (x)... (see, for example, [10,11]) We recall that for an orthotropic material with the symmetry planes coinciding with the Cartesian coordinate planes the stress-strain relation may be written in the