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Elastic waves interacting with a thin, prestressed, fiber-reinforced surface film
International Journal of Engineering Science 48 (2010) 1604–1609
Contents lists available at ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Elastic waves interacting with a thin, prestressed, fiber-reinforced surface film David J. Steigmann Department of Mechanical Engineering, University of California, Berkeley, CA 94720, United States
a r t i c l e
i n f o
Article history: Available online 24 July 2010 To K.R. Rajagopal, for his courage and intellectual honesty, on the occasion of his 60th birthday Keywords: Thin films Surface waves Anisotropy
a b s t r a c t Elastic surface waves propagating at the interface between an isotropic substrate and a thin, transversely isotropic film are analyzed. The transverse isotropy is conferred by fibers lying parallel to the interface. A rigorous leading-order model of the thin-film/substrate interface is derived from the equations of three-dimensional elasticity for prestressed, transversely isotropic films having non- uniform properties. This is used to study Love waves. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction In this work we derive a rigorous leading-order-in-thickness model for the dynamics of a thin, fiber-reinforced elastic film bonded to an isotropic elastic substrate. The fibers are assumed to lie in the plane of the film and equations of motion are derived for the displacement of the interface. Related work based on a variational argument is given in [1], where references to the pertinent literature may be found. Among these works we direct the reader’s particular attention to [2–4]. Related developments are reported in [5,6]. 2. Basic equations Standard notation is used throughout. Thus, we use bold face for vectors and tensors and indices to denote their components. Latin indices take values in {1, 2, 3}; Greek in {1, 2}. The latter are associated with surface coordinates and associated vector and tensor components. A dot between bold symbols is used to denote the standard inner product. Thus, if A1 and A2 are second-order tensors, then A1 A2 ¼ tr A1 At2 , where tr() is the trace and the superscript t is used to denote the transpose. The notation identifies the standard tensor product of vectors. If C is a fourth-order tensor, then C[A] is the secondorder tensor with orthogonal components CijklAkl. Finally, we use symbols such as Div and D to denote the three-dimensional divergence and gradient operators, while div and r are reserved for their two-dimensional counterparts. Thus, for example, DivA = Aij,jei and divA = Aia,aei, where {ei} is an orthonormal basis and subscripts preceded by commas are used to denote partial derivatives with respect to Cartesian coordinates. The three-dimensional equation of motion without body force is
€; Div P ¼ qu
ð1Þ
P ¼ S þ HS þ C½H
ð2Þ
where
E-mail address: [email protected] 0020-7225/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2010.06.032
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D.J. Steigmann / International Journal of Engineering Science 48 (2010) 1604–1609
is the linear approximation to the Piola stress, S is the (symmetric) residual stress, H = Du is the gradient of the displacement € is the acceleration, and C is the fourth-order tensor of elastic moduli. The moduli possess the usual minor and field uðx; tÞ; u major symmetries, the latter ensuring that
P ¼ U H;
ð3Þ
1 UðH; xÞ ¼ S H þ ðHS H þ H C½HÞ 2
ð4Þ
where
is the quadratic-order approximation to the strain energy per unit volume of the material region R in which explicit dependence on x 2 R is present if the material is non-uniform. Any such dependence occurs through the residual stress and the moduli. Here we take these to be uniform and thus restrict attention to uniform materials. We suppose that traction data
t ¼ Pn
ð5Þ
are assigned on a part of the boundary @R with exterior unit normal n. We impose the strong-ellipticity condition
ðw SwÞv v þ v w C½v w > 0 for all
v w – 0:
ð6Þ
This is necessary for the undeformed body to be a minimizer of the total strain energy. It is also necessary for minimizers of the potential energy in standard mixed traction/displacement boundary-value problems [7]. 3. Motion of the film/substrate interface The undeformed film/substrate interface is a plane denoted by X. Let k be the unit vector that orients the interface, directed away from the substrate. The reference placement of the film is described by
x ¼ r þ 1k;
ð7Þ
where r 2 X, k is the fixed orientation of the film, and 1 2 [0, h] in the film, where h is the film thickness. The origin of the position r is assumed to lie on X. In the present work the film-substrate combination is a half space that supports a propagating surface wave whose wavelength l (the reciprocal of the wavenumber k) furnishes the only length scale to which h can be compared. Henceforth we regard h as being small in the sense that h/l 1. In the present section it simplifies matters to adopt l as the unit of length (i.e., l = 1, h 1). 0 Let u(r, 1, t) be the function obtained by substituting (7) into u(x, t), and let r() and () , respectively, stand for the (twodimensional) gradient with respect to r at fixed 1 and the derivative @()/@ 1 at fixed r. Further, let
1 ¼ I k k;
ð8Þ 0
where I is the identity for three-space; this is the projection onto the translation (vector) space X of X. In [1] it is used to derive
H1 ¼ ru;
Hk ¼ u0
ð9Þ
and the consequent orthogonal decomposition
H ¼ ru þ u0 k:
ð10Þ
Using (8) with P = PI, we also obtain
P ¼ P1 þ Pk k
ð11Þ
and write (1) in the form
€; div ðP1Þ þ P0 k ¼ qu
ð12Þ +
where div is the two-dimensional divergence on X. This holds at all points of the thin-film, and in the limit 1 &0 in particular, yielding the interfacial equation of motion
€0; div ðP0 1Þ þ P00 k ¼ q0 u
ð13Þ
where, here and henceforth, the subscript ()0 is used to denote the values of functions on the interface X defined by 1 = 0. If t+ is the traction exerted by the environment on the film, then t+ = P+k where P+ is the stress at 1 = h. A Taylor expansion furnishes
tþ ¼ P0 k þ hP00 k þ oðhÞ:
ð14Þ
Accordingly, (13) and (14) combine to yield 1 1 €0: div ðP0 1Þ þ h ðtþ P0 kÞ þ h oðhÞ ¼ q0 u
ð15Þ
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D.J. Steigmann / International Journal of Engineering Science 48 (2010) 1604–1609
For this to furnish a well-defined problem in the limit of small thickness it is necessary that
tþ P0 k ¼ OðhÞ as h ! 0:
ð16Þ
This is the net traction acting on the lateral surfaces of the film. Let r be the stress in the substrate, assumed to occupy the half-space defined by 1 < 0. Let r0 be the limit of r as 1%0. Then the traction exerted on the substrate by the film at the film-substrate interface X is r0k, whereas that exerted by the substrate on the film is P0k. An elementary pillbox argument yields the exact result
r0 k ¼ P0 k:
ð17Þ
Combining this with (15) and passing to the limit in (15) and (16) yields the leading-order model
€0 div ðhP0 1Þ þ tþ r0 k ¼ hq0 u
and tþ P0 k ¼ 0:
ð18Þ
These equations, together with interfacial continuity of displacements, couple the responses of the film and substrate. In particular, if w(x,t) is the displacement field in the substrate, then
w0 ðr; tÞ ¼ u0 ;
ð19Þ
where w0 ¼ wjX . We assume the substrate to be free of residual stress, so that
r ¼ E½Dw;
ð20Þ
where E is the associated tensor of elastic moduli, possessing the usual minor and major symmetries. Thus, in contrast to the film stress P, r is symmetric. This satisfies the equation of motion
€ Div r ¼ qs w;
ð21Þ
where qs is the mass density of the substrate. The foregoing model agrees precisely with the leading-order system derived elsewhere [1] via a variational argument. We demonstrate that (18)2 can be solved uniquely for the derivative u00 . To this end we fix u0 and define
WðaÞ ¼ Uðru0 þ a kÞ a tþ : Let
ð22Þ
be a parameter and consider the one-parameter family a(). Let g() = W(a()) and let ()
= d()/d. Then (3) yields
€ þ a_ ðW aa Þa; _ g_ ¼ W a a_ and g€ ¼ W a a
ð23Þ
with
W a ¼ P0 k tþ
and ðW aa Þa_ ¼ ðP0 kÞ ;
ð24Þ
wherein P0 is given by (2) and (10), with a() substituted in place of u00 . From (18)2 we find that W is stationary at a ¼ u00 , while (2) furnishes
a_ ðP0 kÞ ¼ ðk S0 kÞa_ a_ þ a_ k C0 ½a_ k:
ð25Þ
€ > 0 on straight-line paths This is strictly positive by virtue of (6), implying that Waa is positive definite. In particular, g defined by a() = (1 )a1 + a2 with a1, a2 fixed and 2 [0, 1]. These paths belong to the domain of W(), the convex set gen_ Þ > gð0Þ _ _ erated by the linear space of 3-vectors. Integrating with respect to yields gð and gð1Þ gð0Þ > gð0Þ, proving that W(a) is strictly convex; i.e.,
Wða2 Þ Wða1 Þ > W a ða1 Þ ða2 a1 Þ;
ð26Þ
for all unequal pairs a1, a2. Because strictly convex functions have unique stationary points, it follows that
ðru0 ; tþ Þ; u00 ¼ a
ð27Þ
is the value of a at which Wa vanishes. where a In the present work we impose the condition t+ = 0 for all deformations. Eqs. (2) and (18)2, specialized to the case of zero displacement, then require that
S0 k ¼ 0;
ð28Þ
which in turn may be used to simplify (27) for arbitrary displacements. This yields
u00 ¼ A1 0 ðC0 ½ru0 Þk;
ð29Þ
where A0 is the value on X of the acoustic tensor of the film material, defined, for arbitrary v, by
A0 v ¼ ðC0 ½v kÞk:
ð30Þ
D.J. Steigmann / International Journal of Engineering Science 48 (2010) 1604–1609
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That this is positive definite and hence invertible follows from (6), with (28). Accordingly, (18)1 and (29) furnish a system for the interfacial displacement field u0(r, t). Following Spencer [8] we model the film as a transversely isotropic solid. The axes of transverse isotropy are coincident with the direction fields of the (straight) fiber trajectories. We drop the subscript 0 from all notation denoting interfacial values of variables. The components of C relative to the basis {ei} are [1]
C ijkl ¼ kdij dkl þ lT ðdik djl þ dil djk Þ þ aðdij mk ml þ mi mj dkl Þ þ ðlL lT Þðmi mk djl þ mi ml djk þ mj mk dil þ mj ml dik Þ þ bmi mj mk ml ;
ð31Þ
where dij is the Kronecker delta; a, b, k, lT and lL are material constants; and the unit vector m, with components mi, is the fiber axis, assumed here to be uniform and lying in the plane of the film. Spencer [8] shows that lT is the shear modulus for shearing in planes transverse to m, whereas lL is the shear modulus for shearing parallel to m. The remaining material constants in (31) may be interpreted in terms of extensional moduli and Poisson ratios [8]. The general form of the residual stress may be derived by enumerating the strain invariants for transverse isotropy that are linear in the (infinitesimal) strain. These are [8] I H and m m H. Comparison with the linear term in (4) then furnishes
S ¼ ST ðI m mÞ þ SL m m;
ð32Þ
where ST is the constant residual stress in the isotropic plane and SL is the constant residual uniaxial stress along m. In this work we consider the film to be a prismatic body formed by the parallel translation of a midplane X in the direction of its unit normal k(=e3). The fibers are assumed to lie parallel to X, so that m3 = 0 and m = maea. Eq. (28) then yields
S0 ¼ S0 m m;
ð33Þ
where S0 is the residual interfacial stress. The acoustic tensor defined by (30) is
A ¼ kk k þ lT ðI þ k kÞ þ ðlL lT Þm m;
ð34Þ
with eigenvalues lL, lT and k + 2lT. Inequality (6), with (28), requires that these be strictly positive. Eq. (29) then yields
u00 ¼ ðk þ 2lT Þ1 ½kðdivv Þ þ am ðrv Þmk rw;
ð35Þ
where divv = va,a, m (rv)m = va,bmamb and
v ¼ 1u0
and w ¼ k u0
ð36Þ
are the in-plane and transverse interfacial displacements, respectively; i.e.,
u0 ¼ v þ wk:
ð37Þ
4. Example: Love waves We are concerned with the acoustic interaction of the film and substrate, the former coinciding with the r1, r2-plane and the latter with the half-space defined by 1 < 0. Accordingly, we study harmonic surface waves whose amplitudes decay with depth in the substrate. For the sake of simplicity we confine attention to Love waves. For waves propagating along the r1direction, these have the form
ui ¼ di2 Fðr 1 ; 1; tÞ;
Fðr 1 ; 1; tÞ ¼ f ðr 1 ; tÞ expðgk1Þ;
ð38Þ
where g and k are positive constants, and
f ðr 1 ; tÞ ¼ A exp½ikðr1 ctÞ;
ð39Þ
in which c is the wavespeed and A is a constant. The induced displacement of the film/substrate interface is
w ¼ u3 ¼ 0;
v a ¼ ua ¼ da2 f ðr1 ; tÞ:
ð40Þ
For uniform isotropic materials, the film-substrate interaction term is [1]
rk ¼ ls gkf ðr1 ; tÞe2 :
ð41Þ
In the substrate, Eq. (21) is satisfied provided that [2,3]
pffiffiffiffiffiffiffiffiffiffiffiffiffi
g ¼ 1 s2 ; where
s ¼ c=cs < 1 and cs ¼
ð42Þ qffiffiffiffiffiffiffiffiffiffiffiffi ls =qs
is the transverse wavespeed in the substrate.
ð43Þ
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D.J. Steigmann / International Journal of Engineering Science 48 (2010) 1604–1609
According to (40)2, we have div v = 0 identically at the interface. Using (35) and Eqs. (76) and (82) of [9] we find, after much manipulation, omitted here for the sake of brevity, that (18)1 becomes
hSðrrv Þm m þ hlT Dv þ hcr½m ðrv Þm þ hdfm ½ðrrv Þm mg þ hðlL lT Þ½ðm Dv Þm þ ðrrv Þm m rk ¼ hqv tt ;
ð44Þ
where
c¼
ka þ a þ lL lT ; k þ 2lT
d¼b
a2
ð45Þ
k þ 2l T
and where (rrv)m m = vc,abmambec and Dv = vc,aaec. Substituting (40)2 reduces this to
hf
00
n o Sm21 e2 þ cm1 m2 e1 þ dm21 m2 m þ ðlL lT Þ m2 m þ m21 e2 þ lT e2 ls gkf e2 ¼ hq€f e2 ;
ð46Þ
wherein the primes and dots refer to derivatives of f with respect to r1 and t, respectively. This is equivalent to its projections onto e1 and e2, given respectively by
00 hf m1 m2 c þ dm21 þ lL lT ¼ 0
ð47Þ
and
hf
00
Sm21 þ dm21 m22 þ lL ls gkf ¼ hq€f :
ð48Þ
With f given by (39), Eq. (47) requires that either the expression in parentheses vanishes or that m1m2 vanishes. For typical data on carbon-fiber/epoxy-resin composites [8], we find that the first alternative has no solution for real-valued m1. The remaining alternative yields the two possibilities m = ±e1 and m = ±e2, corresponding to waves propagating parallel and transverse to the fibers, respectively. In the first case; i.e., for waves traveling along the fibers, (48) reduces to 2
hkðS þ lL Þ þ ls g ¼ hqkc ;
ð49Þ
yielding
g ¼ hk rs2
S þ lL
ls
;
ð50Þ
where r = q/qs, and
s ¼ c=cs < 1;
where cs ¼
qffiffiffiffiffiffiffiffiffiffiffiffi ls =qs
ð51Þ
is the transverse shear-wave speed in the substrate. Eliminating g between (49) and (50) yields the dispersion relation
pffiffiffiffiffiffiffiffiffiffiffiffiffi S þ lL ; 1 s2 ¼ rs2
ð52Þ
ls
where = hk. The theory used here to obtain this relation purports to be valid only if ingly, we assume that s2 = 1 2C2 + o(2), solve for C and obtain
s1
1, so that 1 s2 = O(2). Accord-
2 1 2 S þ lL r ; 2 ls
ð53Þ
to leading order. The case of waves traveling in the direction transverse to the fibers leads to the same results, but with S omitted. We observe that the relevant film stiffness in both cases is the longitudinal modulus lL. This is to be expected because the transverse modulus lT pertains to shearing in the isotropic plane orthogonal to m, whereas deformations of the type considered induce shearing in a plane containing m. It is easy to show, from (31), that such shear deformations generate a shear stress equal to 2lL times the shear strain, provided that m is oriented along either of the axes of strain; i.e., along ±e1 or ± e2, as in the foregoing. Acknowledgment The author would like to thank the referees for helpful comments leading to substantial improvement of the article. References [1] D.J. Steigmann, R.W. Ogden, Surface waves supported by thin-film/substrate interactions, IMA J. Appl. Math. 72 (2007) 730–747. [2] J.D. Achenbach, Wave Propagation in Elastic Solids, North-Holland, Amsterdam, 1973. [3] I. Murdoch, The propagation of surface waves in bodies with material boundaries, J. Mech. Phys. Solids 24 (1976) 137–146.
D.J. Steigmann / International Journal of Engineering Science 48 (2010) 1604–1609
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[4] A.V. Pichugin, G.A. Rogerson, An asymptotic membrane-like theory for long-wave motion in a pre-stressed elastic plate, Proc. Roy. Soc. Lond. A458 (2002) 1447–1468. [5] A.L. Shuvalov, A.G. Every, On the long-wave onset of dispersion of the surface-wave velocity in coated solids, Wave Motion 45 (2008) 857–863. [6] T.C.T. Ting, Steady waves in an anisotropic elastic layer attached to a half-space or between two half-spaces: a generalization of Love waves and Stoneley waves, Math. Mech. Solids 14 (2009) 52–71. [7] R.J. Knops, L.E. Payne, Uniqueness Theorems in Linear Elasticity, Springer Tracts in Natural Philosophy 19 (1971). [8] A.J.M. Spencer, Constitutive theory for strongly anisotropic solids, in: A.J.M. Spencer (Ed.), Continuum Theory of the Mechanics of Fibre-Reinforced Composites, CISM Courses and Lectures, vol. 282, Springer, Wien, 1984, pp. 1–32. [9] D.J. Steigmann, Linear theory for the bending and extension of a thin, residually stressed, fiber-reinforced lamina, Int. J. Eng. Sci. 47 (2009) 1367–1378.
Contents lists available at ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Elastic waves interacting with a thin, prestressed, fiber-reinforced surface film David J. Steigmann Department of Mechanical Engineering, University of California, Berkeley, CA 94720, United States
a r t i c l e
i n f o
Article history: Available online 24 July 2010 To K.R. Rajagopal, for his courage and intellectual honesty, on the occasion of his 60th birthday Keywords: Thin films Surface waves Anisotropy
a b s t r a c t Elastic surface waves propagating at the interface between an isotropic substrate and a thin, transversely isotropic film are analyzed. The transverse isotropy is conferred by fibers lying parallel to the interface. A rigorous leading-order model of the thin-film/substrate interface is derived from the equations of three-dimensional elasticity for prestressed, transversely isotropic films having non- uniform properties. This is used to study Love waves. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction In this work we derive a rigorous leading-order-in-thickness model for the dynamics of a thin, fiber-reinforced elastic film bonded to an isotropic elastic substrate. The fibers are assumed to lie in the plane of the film and equations of motion are derived for the displacement of the interface. Related work based on a variational argument is given in [1], where references to the pertinent literature may be found. Among these works we direct the reader’s particular attention to [2–4]. Related developments are reported in [5,6]. 2. Basic equations Standard notation is used throughout. Thus, we use bold face for vectors and tensors and indices to denote their components. Latin indices take values in {1, 2, 3}; Greek in {1, 2}. The latter are associated with surface coordinates and associated vector and tensor components. A dot between bold symbols is used to denote the standard inner product. Thus, if A1 and A2 are second-order tensors, then A1 A2 ¼ tr A1 At2 , where tr() is the trace and the superscript t is used to denote the transpose. The notation identifies the standard tensor product of vectors. If C is a fourth-order tensor, then C[A] is the secondorder tensor with orthogonal components CijklAkl. Finally, we use symbols such as Div and D to denote the three-dimensional divergence and gradient operators, while div and r are reserved for their two-dimensional counterparts. Thus, for example, DivA = Aij,jei and divA = Aia,aei, where {ei} is an orthonormal basis and subscripts preceded by commas are used to denote partial derivatives with respect to Cartesian coordinates. The three-dimensional equation of motion without body force is
€; Div P ¼ qu
ð1Þ
P ¼ S þ HS þ C½H
ð2Þ
where
E-mail address: [email protected] 0020-7225/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2010.06.032
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D.J. Steigmann / International Journal of Engineering Science 48 (2010) 1604–1609
is the linear approximation to the Piola stress, S is the (symmetric) residual stress, H = Du is the gradient of the displacement € is the acceleration, and C is the fourth-order tensor of elastic moduli. The moduli possess the usual minor and field uðx; tÞ; u major symmetries, the latter ensuring that
P ¼ U H;
ð3Þ
1 UðH; xÞ ¼ S H þ ðHS H þ H C½HÞ 2
ð4Þ
where
is the quadratic-order approximation to the strain energy per unit volume of the material region R in which explicit dependence on x 2 R is present if the material is non-uniform. Any such dependence occurs through the residual stress and the moduli. Here we take these to be uniform and thus restrict attention to uniform materials. We suppose that traction data
t ¼ Pn
ð5Þ
are assigned on a part of the boundary @R with exterior unit normal n. We impose the strong-ellipticity condition
ðw SwÞv v þ v w C½v w > 0 for all
v w – 0:
ð6Þ
This is necessary for the undeformed body to be a minimizer of the total strain energy. It is also necessary for minimizers of the potential energy in standard mixed traction/displacement boundary-value problems [7]. 3. Motion of the film/substrate interface The undeformed film/substrate interface is a plane denoted by X. Let k be the unit vector that orients the interface, directed away from the substrate. The reference placement of the film is described by
x ¼ r þ 1k;
ð7Þ
where r 2 X, k is the fixed orientation of the film, and 1 2 [0, h] in the film, where h is the film thickness. The origin of the position r is assumed to lie on X. In the present work the film-substrate combination is a half space that supports a propagating surface wave whose wavelength l (the reciprocal of the wavenumber k) furnishes the only length scale to which h can be compared. Henceforth we regard h as being small in the sense that h/l 1. In the present section it simplifies matters to adopt l as the unit of length (i.e., l = 1, h 1). 0 Let u(r, 1, t) be the function obtained by substituting (7) into u(x, t), and let r() and () , respectively, stand for the (twodimensional) gradient with respect to r at fixed 1 and the derivative @()/@ 1 at fixed r. Further, let
1 ¼ I k k;
ð8Þ 0
where I is the identity for three-space; this is the projection onto the translation (vector) space X of X. In [1] it is used to derive
H1 ¼ ru;
Hk ¼ u0
ð9Þ
and the consequent orthogonal decomposition
H ¼ ru þ u0 k:
ð10Þ
Using (8) with P = PI, we also obtain
P ¼ P1 þ Pk k
ð11Þ
and write (1) in the form
€; div ðP1Þ þ P0 k ¼ qu
ð12Þ +
where div is the two-dimensional divergence on X. This holds at all points of the thin-film, and in the limit 1 &0 in particular, yielding the interfacial equation of motion
€0; div ðP0 1Þ þ P00 k ¼ q0 u
ð13Þ
where, here and henceforth, the subscript ()0 is used to denote the values of functions on the interface X defined by 1 = 0. If t+ is the traction exerted by the environment on the film, then t+ = P+k where P+ is the stress at 1 = h. A Taylor expansion furnishes
tþ ¼ P0 k þ hP00 k þ oðhÞ:
ð14Þ
Accordingly, (13) and (14) combine to yield 1 1 €0: div ðP0 1Þ þ h ðtþ P0 kÞ þ h oðhÞ ¼ q0 u
ð15Þ
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D.J. Steigmann / International Journal of Engineering Science 48 (2010) 1604–1609
For this to furnish a well-defined problem in the limit of small thickness it is necessary that
tþ P0 k ¼ OðhÞ as h ! 0:
ð16Þ
This is the net traction acting on the lateral surfaces of the film. Let r be the stress in the substrate, assumed to occupy the half-space defined by 1 < 0. Let r0 be the limit of r as 1%0. Then the traction exerted on the substrate by the film at the film-substrate interface X is r0k, whereas that exerted by the substrate on the film is P0k. An elementary pillbox argument yields the exact result
r0 k ¼ P0 k:
ð17Þ
Combining this with (15) and passing to the limit in (15) and (16) yields the leading-order model
€0 div ðhP0 1Þ þ tþ r0 k ¼ hq0 u
and tþ P0 k ¼ 0:
ð18Þ
These equations, together with interfacial continuity of displacements, couple the responses of the film and substrate. In particular, if w(x,t) is the displacement field in the substrate, then
w0 ðr; tÞ ¼ u0 ;
ð19Þ
where w0 ¼ wjX . We assume the substrate to be free of residual stress, so that
r ¼ E½Dw;
ð20Þ
where E is the associated tensor of elastic moduli, possessing the usual minor and major symmetries. Thus, in contrast to the film stress P, r is symmetric. This satisfies the equation of motion
€ Div r ¼ qs w;
ð21Þ
where qs is the mass density of the substrate. The foregoing model agrees precisely with the leading-order system derived elsewhere [1] via a variational argument. We demonstrate that (18)2 can be solved uniquely for the derivative u00 . To this end we fix u0 and define
WðaÞ ¼ Uðru0 þ a kÞ a tþ : Let
ð22Þ
be a parameter and consider the one-parameter family a(). Let g() = W(a()) and let ()
= d()/d. Then (3) yields
€ þ a_ ðW aa Þa; _ g_ ¼ W a a_ and g€ ¼ W a a
ð23Þ
with
W a ¼ P0 k tþ
and ðW aa Þa_ ¼ ðP0 kÞ ;
ð24Þ
wherein P0 is given by (2) and (10), with a() substituted in place of u00 . From (18)2 we find that W is stationary at a ¼ u00 , while (2) furnishes
a_ ðP0 kÞ ¼ ðk S0 kÞa_ a_ þ a_ k C0 ½a_ k:
ð25Þ
€ > 0 on straight-line paths This is strictly positive by virtue of (6), implying that Waa is positive definite. In particular, g defined by a() = (1 )a1 + a2 with a1, a2 fixed and 2 [0, 1]. These paths belong to the domain of W(), the convex set gen_ Þ > gð0Þ _ _ erated by the linear space of 3-vectors. Integrating with respect to yields gð and gð1Þ gð0Þ > gð0Þ, proving that W(a) is strictly convex; i.e.,
Wða2 Þ Wða1 Þ > W a ða1 Þ ða2 a1 Þ;
ð26Þ
for all unequal pairs a1, a2. Because strictly convex functions have unique stationary points, it follows that
ðru0 ; tþ Þ; u00 ¼ a
ð27Þ
is the value of a at which Wa vanishes. where a In the present work we impose the condition t+ = 0 for all deformations. Eqs. (2) and (18)2, specialized to the case of zero displacement, then require that
S0 k ¼ 0;
ð28Þ
which in turn may be used to simplify (27) for arbitrary displacements. This yields
u00 ¼ A1 0 ðC0 ½ru0 Þk;
ð29Þ
where A0 is the value on X of the acoustic tensor of the film material, defined, for arbitrary v, by
A0 v ¼ ðC0 ½v kÞk:
ð30Þ
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That this is positive definite and hence invertible follows from (6), with (28). Accordingly, (18)1 and (29) furnish a system for the interfacial displacement field u0(r, t). Following Spencer [8] we model the film as a transversely isotropic solid. The axes of transverse isotropy are coincident with the direction fields of the (straight) fiber trajectories. We drop the subscript 0 from all notation denoting interfacial values of variables. The components of C relative to the basis {ei} are [1]
C ijkl ¼ kdij dkl þ lT ðdik djl þ dil djk Þ þ aðdij mk ml þ mi mj dkl Þ þ ðlL lT Þðmi mk djl þ mi ml djk þ mj mk dil þ mj ml dik Þ þ bmi mj mk ml ;
ð31Þ
where dij is the Kronecker delta; a, b, k, lT and lL are material constants; and the unit vector m, with components mi, is the fiber axis, assumed here to be uniform and lying in the plane of the film. Spencer [8] shows that lT is the shear modulus for shearing in planes transverse to m, whereas lL is the shear modulus for shearing parallel to m. The remaining material constants in (31) may be interpreted in terms of extensional moduli and Poisson ratios [8]. The general form of the residual stress may be derived by enumerating the strain invariants for transverse isotropy that are linear in the (infinitesimal) strain. These are [8] I H and m m H. Comparison with the linear term in (4) then furnishes
S ¼ ST ðI m mÞ þ SL m m;
ð32Þ
where ST is the constant residual stress in the isotropic plane and SL is the constant residual uniaxial stress along m. In this work we consider the film to be a prismatic body formed by the parallel translation of a midplane X in the direction of its unit normal k(=e3). The fibers are assumed to lie parallel to X, so that m3 = 0 and m = maea. Eq. (28) then yields
S0 ¼ S0 m m;
ð33Þ
where S0 is the residual interfacial stress. The acoustic tensor defined by (30) is
A ¼ kk k þ lT ðI þ k kÞ þ ðlL lT Þm m;
ð34Þ
with eigenvalues lL, lT and k + 2lT. Inequality (6), with (28), requires that these be strictly positive. Eq. (29) then yields
u00 ¼ ðk þ 2lT Þ1 ½kðdivv Þ þ am ðrv Þmk rw;
ð35Þ
where divv = va,a, m (rv)m = va,bmamb and
v ¼ 1u0
and w ¼ k u0
ð36Þ
are the in-plane and transverse interfacial displacements, respectively; i.e.,
u0 ¼ v þ wk:
ð37Þ
4. Example: Love waves We are concerned with the acoustic interaction of the film and substrate, the former coinciding with the r1, r2-plane and the latter with the half-space defined by 1 < 0. Accordingly, we study harmonic surface waves whose amplitudes decay with depth in the substrate. For the sake of simplicity we confine attention to Love waves. For waves propagating along the r1direction, these have the form
ui ¼ di2 Fðr 1 ; 1; tÞ;
Fðr 1 ; 1; tÞ ¼ f ðr 1 ; tÞ expðgk1Þ;
ð38Þ
where g and k are positive constants, and
f ðr 1 ; tÞ ¼ A exp½ikðr1 ctÞ;
ð39Þ
in which c is the wavespeed and A is a constant. The induced displacement of the film/substrate interface is
w ¼ u3 ¼ 0;
v a ¼ ua ¼ da2 f ðr1 ; tÞ:
ð40Þ
For uniform isotropic materials, the film-substrate interaction term is [1]
rk ¼ ls gkf ðr1 ; tÞe2 :
ð41Þ
In the substrate, Eq. (21) is satisfied provided that [2,3]
pffiffiffiffiffiffiffiffiffiffiffiffiffi
g ¼ 1 s2 ; where
s ¼ c=cs < 1 and cs ¼
ð42Þ qffiffiffiffiffiffiffiffiffiffiffiffi ls =qs
is the transverse wavespeed in the substrate.
ð43Þ
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D.J. Steigmann / International Journal of Engineering Science 48 (2010) 1604–1609
According to (40)2, we have div v = 0 identically at the interface. Using (35) and Eqs. (76) and (82) of [9] we find, after much manipulation, omitted here for the sake of brevity, that (18)1 becomes
hSðrrv Þm m þ hlT Dv þ hcr½m ðrv Þm þ hdfm ½ðrrv Þm mg þ hðlL lT Þ½ðm Dv Þm þ ðrrv Þm m rk ¼ hqv tt ;
ð44Þ
where
c¼
ka þ a þ lL lT ; k þ 2lT
d¼b
a2
ð45Þ
k þ 2l T
and where (rrv)m m = vc,abmambec and Dv = vc,aaec. Substituting (40)2 reduces this to
hf
00
n o Sm21 e2 þ cm1 m2 e1 þ dm21 m2 m þ ðlL lT Þ m2 m þ m21 e2 þ lT e2 ls gkf e2 ¼ hq€f e2 ;
ð46Þ
wherein the primes and dots refer to derivatives of f with respect to r1 and t, respectively. This is equivalent to its projections onto e1 and e2, given respectively by
00 hf m1 m2 c þ dm21 þ lL lT ¼ 0
ð47Þ
and
hf
00
Sm21 þ dm21 m22 þ lL ls gkf ¼ hq€f :
ð48Þ
With f given by (39), Eq. (47) requires that either the expression in parentheses vanishes or that m1m2 vanishes. For typical data on carbon-fiber/epoxy-resin composites [8], we find that the first alternative has no solution for real-valued m1. The remaining alternative yields the two possibilities m = ±e1 and m = ±e2, corresponding to waves propagating parallel and transverse to the fibers, respectively. In the first case; i.e., for waves traveling along the fibers, (48) reduces to 2
hkðS þ lL Þ þ ls g ¼ hqkc ;
ð49Þ
yielding
g ¼ hk rs2
S þ lL
ls
;
ð50Þ
where r = q/qs, and
s ¼ c=cs < 1;
where cs ¼
qffiffiffiffiffiffiffiffiffiffiffiffi ls =qs
ð51Þ
is the transverse shear-wave speed in the substrate. Eliminating g between (49) and (50) yields the dispersion relation
pffiffiffiffiffiffiffiffiffiffiffiffiffi S þ lL ; 1 s2 ¼ rs2
ð52Þ
ls
where = hk. The theory used here to obtain this relation purports to be valid only if ingly, we assume that s2 = 1 2C2 + o(2), solve for C and obtain
s1
1, so that 1 s2 = O(2). Accord-
2 1 2 S þ lL r ; 2 ls
ð53Þ
to leading order. The case of waves traveling in the direction transverse to the fibers leads to the same results, but with S omitted. We observe that the relevant film stiffness in both cases is the longitudinal modulus lL. This is to be expected because the transverse modulus lT pertains to shearing in the isotropic plane orthogonal to m, whereas deformations of the type considered induce shearing in a plane containing m. It is easy to show, from (31), that such shear deformations generate a shear stress equal to 2lL times the shear strain, provided that m is oriented along either of the axes of strain; i.e., along ±e1 or ± e2, as in the foregoing. Acknowledgment The author would like to thank the referees for helpful comments leading to substantial improvement of the article. References [1] D.J. Steigmann, R.W. Ogden, Surface waves supported by thin-film/substrate interactions, IMA J. Appl. Math. 72 (2007) 730–747. [2] J.D. Achenbach, Wave Propagation in Elastic Solids, North-Holland, Amsterdam, 1973. [3] I. Murdoch, The propagation of surface waves in bodies with material boundaries, J. Mech. Phys. Solids 24 (1976) 137–146.
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[4] A.V. Pichugin, G.A. Rogerson, An asymptotic membrane-like theory for long-wave motion in a pre-stressed elastic plate, Proc. Roy. Soc. Lond. A458 (2002) 1447–1468. [5] A.L. Shuvalov, A.G. Every, On the long-wave onset of dispersion of the surface-wave velocity in coated solids, Wave Motion 45 (2008) 857–863. [6] T.C.T. Ting, Steady waves in an anisotropic elastic layer attached to a half-space or between two half-spaces: a generalization of Love waves and Stoneley waves, Math. Mech. Solids 14 (2009) 52–71. [7] R.J. Knops, L.E. Payne, Uniqueness Theorems in Linear Elasticity, Springer Tracts in Natural Philosophy 19 (1971). [8] A.J.M. Spencer, Constitutive theory for strongly anisotropic solids, in: A.J.M. Spencer (Ed.), Continuum Theory of the Mechanics of Fibre-Reinforced Composites, CISM Courses and Lectures, vol. 282, Springer, Wien, 1984, pp. 1–32. [9] D.J. Steigmann, Linear theory for the bending and extension of a thin, residually stressed, fiber-reinforced lamina, Int. J. Eng. Sci. 47 (2009) 1367–1378.