Counter-intuitive results in acousto-elasticity
CCounter-intuitive results in acousto-elasticity a , M. Destrade a,b , R.W. Ogden c a School of Mathematics, Statistics and Applied Mathematics,National University of Ireland Galway,University Road, Galway, Ireland b School of Mechanical & Materials Engineering,University College Dublin,Belfield, Dublin 4, Ireland c School of Mathematics and Statistics,University of Glasgow, Scotland
Abstract
We present examples of body wave and surface wave propagation in deformed solidswhere the slowest and the fastest waves do not travel along the directions of least andgreatest stretch, respectively. These results run counter to commonly accepted theory,practice, and implementation of the principles of acousto-elasticity in initially isotropicsolids. For instance we find that in nickel and steel, the fastest waves are along thedirection of greatest compression, not greatest extension (and vice-versa for the slowestwaves), as soon as those solids are deformed. Further, we find that when some materialsare subject to a small-but-finite deformations, other extrema of wave speeds appear innon-principal directions. Examples include nickel, steel, polystyrene, and a certain hy-drogel. The existence of these “oblique”, non-principal extremal waves complicates theprotocols for the non-destructive determination of the directions of extreme strains.
Keywords:
Acousto-elasticity, surface waves, non-principal waves
1. Introduction
The determination of the direction of greatest tension in a deformed solid is one ofthe main goals of acoustic non-destructive evaluation because, for isotropic solids, thisdirection coincides with the direction of greatest stress. Consider for instance cuttingthrough a membrane under uniaxial tension: cutting parallel to the direction of thetensile force produces a thin cut, while cutting across produces a gaping cut (see Figure1), which can have serious consequences in scaring outcomes after stabbing incidentsor surgery. Finding the direction of greatest stress is also important in geophysics, oilprospecting (Guyer and Johnson, 2009) and structural health monitoring and evaluation(Pao et al., 1984; Kim and Sachse, 2001).In this paper we investigate the propagation of small-amplitude elastic waves in thebody (body acoustic waves – BAWs) and on the surface (surface acoustic waves – SAWs)of a deformed solid, and determine the dependence of their speeds on the angle of prop-agation with respect to the principal directions of pre-strain. It is widely thought that Dedicated to V.I. Alshits
Preprint submitted to Elsevier September 7, 2020 a r X i v : . [ c ond - m a t . s o f t ] S e p igure 1: Cutting through pig skin: a clamped sample of pig skin is put under tension, after 3 cuts havebeen performed, parallel (top), oblique (center) and perpendicular (bottom) to the tensile force. surface waves propagate at their fastest in the direction of greatest stretch and at theirslowest in a perpendicular direction, along the direction of least stretch. This view issupported by intuition and often forms the basis of a non-destructive determination ofthese directions.However, the coupling of acoustics and elasticity is a non-linear phenomenon evenat its lowest order, and it can thus generate counter-intuitive results. The first suchresult is that for some materials, the fastest wave travels along the direction of greatestcompression (and conversely, the slowest wave along the direction of greatest extension).It has been known for some time that a compression in one direction could indeed resultin an increase in the speed of a principal wave instead of the intuitively expected decrease,and Hughes and Kelly (1953) showed experimentally that body wave speeds increase withhydrostatic pressure for polystyrene (see their Figure 3); similar experimental results existfor body waves in railroad steel and surface waves in mild steel; see Kim and Sachse (2001)for a review. Here we extend those results to the consideration of non-principal waves indeformed steel and nickel, and to pre-strains resulting in turn from the application of auniaxial stress and of a pure shear stress.The other counter-intuitive result is that the following statement by Kim and Sachse(2001) is not necessarily true: “The principal stress direction is found where the variationsof the SAW speeds show symmetry about the direction”. Indeed, Tanuma et al. (2013)recently showed that for a small-amplitude SAW traveling in the symmetry plane of atransversely isotropic solid, subject to a small pre-strain, the correction to the wave speeddue to the pre-stress has sinusoidal variations with respect to the angle of propagation,in line with that statement. Explicitly, Tanuma et al. (2013) established the followingexpression for the correction to the Rayleigh wave speed v R when the solid is subject toa pre-stress with principal components σ , σ in the plane boundary: v R = v R + A ( σ + σ ) + B ( σ − σ ) cos 2 ψ, where A and B are acousto-elastic coefficients, and ψ is the angle between the direction2f propagation and one of the principal directions of pre-stress. However, their result isonly true when the pre-stress and accompanying pre-strain are infinitesimal. Here weshow that the variations can rapidly lose their sinusoidal regularity beyond that regime,even when a solid is deformed by as little as 1%. Since in non-destructive evaluationand structural health monitoring, the order of magnitude of the pre-stress is not known a priori , we conclude that a complete theoretical and numerical investigation needs to beconducted (as here) prior to the determination of the sought-after principal directions.They will not be found simply by measuring the wave speed in all directions until asymmetry in variation is found.The paper is organized as follows. In the next section we recall the equations gov-erning the propagation of small-amplitude waves in solids subject to a pre-strain of ar-bitrary magnitude. For the constitutive modeling, we focus on isotropic solids with astrain-energy density expressed as a polynomial expansion up to third order in terms ofinvariants of the Green strain tensor. Historically, this is the framework in which theequations of acousto-elasticity have often been written in considering elastic wave prop-agation in a slightly pre-deformed, initially isotropic solid. We refer to, for example, Paoet al. (1984) and Kim and Sachse (2001) for an exposition of the practical and theoret-ical aspects of this technique, which can be dated back to the early efforts of Brillouin(1925) and Hughes and Kelly (1953); see also Destrade and Ogden (2013) for a review ofacousto-elasticity in solids subject to a general homogeneous pre-strain (not necessarilyof infinitesimal amplitude). In Sections 3 and 5, we study body wave and surface wavepropagation, respectively (the latter is more complicated than the former, and we thusdevote Section 4 to a description of our numerical strategy). For both types of waves weuncover examples of solids (steel, Pyrex glass, polystyrene, nickel, hydrogel with a hardcore) where the wave speed does not have its greatest value along the direction of greateststretch, and/or can be extremal along directions which are oblique to the directions of theprincipal stretches. These counter-intuitive results seem to have gone unnoticed before.
2. Governing equations
In this paper we are concerned with the propagation of small-amplitude waves indeformed materials. The equations governing their motion are now well established.Consider a homogeneous elastic solid, held in a state of static homogeneous deformation,which has brought a material point which was at X in the reference configuration toposition x = x ( X , t ) in the current configuration.Let ( X (cid:48) , X (cid:48) , X (cid:48) ) be the coordinates of X with respect to a fixed rectangular Cartesianunit basis vectors ( e (cid:48) , e (cid:48) , e (cid:48) ), and let a pure homogeneous strain be defined by x (cid:48) = λ X (cid:48) , x (cid:48) = λ X (cid:48) , x (cid:48) = λ X (cid:48) , (2.1)with respect to the same basis, where the positive constants λ , λ , λ are the principalstretches of the deformation. Now consider the material to be a half-space occupying theregion x (cid:48) ≥ x (cid:48) = 0 is a principal plane of deformation, which wetake to be free of traction. Now choose a second set of unit basis vectors ( e , e , e ), say,with coordinates ( x , x , x ), so that x = x (cid:48) and the direction of e makes an angle θ e (cid:48) . Then x x x = cos θ θ − sin θ θ x (cid:48) x (cid:48) x (cid:48) . (2.2)A small-amplitude wave traveling in this material is described by the associated me-chanical displacement field u = u ( x , t ), satisfying, in the coordinate system ( x , x , x ),the incremental equations of motion (Ogden, 1997), ρu i,tt = s pi,p = A piqj u j,pq , (2.3)where s pi = A piqj u j,q are the components of the incremental nominal stress tensor, and A piqj are components of the fourth-order tensor of instantaneous moduli A (to bedetailed later), a comma followed by an index i (or t ) denotes partial differentiation withrespect to x i , i = 1 , ,
3, (or t ) and ρ is the current mass density. We specialize theanalysis to waves that propagate in the x direction, with amplitude variations in the x direction. Hence we seek solutions of the form u = U ( x )e i k ( x − vt ) , (2.4)where U , the amplitude, is a function of x only, k is the wavenumber, and v is the wavespeed. Then the equations of motion reduce to T U (cid:48)(cid:48) ( y ) + i k ( R + R T ) U (cid:48) ( y ) − k ( Q − ρv I ) U ( y ) = , (2.5)where the constant tensors T , R , Q are defined in terms of their components with respectto the basis ( e , e , e ) by T ij = T ji = A i i , R ij = A i j , Q ij = Q ji = A i j , (2.6) I is the identity tensor, and the exponent T denotes the transpose.Without loss of generality, we take λ < λ < λ , so that θ = 0 ◦ corresponds to thedirection of greatest compression and θ = 90 ◦ to the direction of greatest stretch. In thecoordinate system ( x (cid:48) , x (cid:48) , x (cid:48) ) aligned with the principal axes of deformation, there areonly 15 non-zero components of A , given by (Ogden, 1997) A (cid:48) iijj = J − λ i λ j W ij , A (cid:48) ijij = J − ( λ i W i − λ j W j ) λ i / ( λ i − λ j ) , i (cid:54) = j, λ i (cid:54) = λ j , A (cid:48) ijji = J − ( λ j W i − λ i W j ) λ i λ j / ( λ i − λ j ) , i (cid:54) = j, λ i (cid:54) = λ j , A (cid:48) ijij = J − ( λ i W ii − λ i λ j W ij + λ i W i ) / , i (cid:54) = j, λ i = λ j , A (cid:48) ijji = A (cid:48) ijji = J − ( λ i W ii − λ i λ j W ij − λ i W i ) / , i (cid:54) = j, λ i = λ j , (2.7)where J = λ λ λ is the dilatation, W is the strain energy density, W i = ∂W/∂λ i , W ij = ∂ W/∂λ i ∂λ j and there is no sum on repeated indices. In the coordinate system( x , x , x ), the components of A , required to compute the tensors in (2.6), are given by A ijkl = Ω ip Ω jq Ω kr Ω ls A (cid:48) pqrs , (2.8)4here Ω ij is the rotation matrix corresponding to a rotation through the angle θ about x = x (cid:48) .We say that A satisfies the strong-convexity condition (S-C) when A ijkl ξ ij ξ kl > ξ , (2.9)but we remark that this condition does not hold in general, only in the region of defor-mation space corresponding to dead-load stability (see, for example, Ogden, 1997). The strong-ellipticity condition (S-E) reads A ijkl n i n k m j m l > n and m , (2.10)and is implied by strong convexity. For the constitutive modeling of the pre-deformed materials, we focus on generalisotropic compressible elastic solids, with a third-order expansion of the strain-energydensity in powers of the Green strain tensor E , specifically W = λ i + µ i + A i + Bi i + C i , (2.11)where i k = tr (cid:0) E k (cid:1) = 12 k (cid:104)(cid:0) λ − (cid:1) k + (cid:0) λ − (cid:1) k + (cid:0) λ − (cid:1) k (cid:105) , k = 1 , , . (2.12)Here, λ and µ are the Lam´e coefficients of second-order elasticity and A , B , C are theLandau coefficients of third-order elasticity (Landau and Lifshitz, 1986).For our examples, we use material parameters taken from the literature for nickel(Lurie, 2005), steel (Lurie, 2005), polystyrene (Hughes and Kelly, 1953), Pyrex glass(Lurie, 2005), and a certain hydrogel with a hard core (Wu and Kirchner, 2010) allsummarized in Table 1.Material Units λ µ A B C
Nickel 10 bars 7.8 6.12857 − − . . bars 8.1 5.4 − − − bars 0.2889 1.381 − . − . − . bars 2.75 5.583 42 71 − . − − . − . Table 1: Second- and third-order elastic constants for six different materials.
We look at two types of pre-deformations: first, that due to a uniaxial stress andsecond that due to a pure shear stress. A uniaxial pre-stress in the e (cid:48) direction is due toa Cauchy stress for which the only non-zero component is σ = T , say. It leads to anequibiaxial pre-deformation, with corresponding principal stretches λ = λ, λ = λ . (2.13)5ere λ is linked to the compressive stress T through the equation T = J − λ ∂W/∂λ ,whilst λ is found in terms of λ by solving0 = ∂W/∂λ . (2.14)With our choice (2.11) of strain energy density, this turns out to be a quadratic in λ .A pure shear stress is applied parallel to the plane of the boundary so that the onlynon-zero Cauchy stress component is σ = S , say. The corresponding pre-deformation isa combination of simple shear in the x direction and a triaxial stretch (Mihai and Goriely,2011; Destrade et al., 2012). Here it is a simple exercise to check (see, for example, Lurie,2005) that the principal stresses are S , 0, − S , and that the corresponding principaldirections of stress are along (1 , , , , , , − S = J − λ ∂W/∂λ , ∂W/∂λ , − S = J − λ ∂W/∂λ , (2.15)for λ = λ , λ and λ .The range of realistic values for λ is restricted by the existence of a solution of thesystem of equations (2.13) and (2.14) for uniaxial compression, and of the equations in(2.15) for pure shear stress. There is a great variability of this feasible range for λ from onematerial to another. For example, steel can only be sheared for λ from 1 down to 0 . λ ’s by assuming that the materials are subject to uniaxial compressive stressesor pure shear stresses only within the region where S and T are monotone functions of λ . This ensures that our results belong to a physically valid regime.
3. Results for body waves
For homogeneous body waves, there are no boundary conditions to satisfy and noamplitude variation to consider. Hence we take U ( x ) = U , (3.1)a constant vector, in the governing equation (2.5), resulting in the eigenvalue problem( Q − ρv I ) U = , (3.2)with associated characteristic equation det( Q − ρv I ) = 0, a cubic in ρv .For the body waves traveling along the principal direction corresponding to the leastprincipal stretch λ , i.e. θ = 0 ◦ , we find the three roots ρv = A (cid:48) , A (cid:48) , A (cid:48) , (3.3)and similarly for the body waves along the principal direction corresponding to the largeststretch ratio λ , i.e. θ = 90 ◦ , ρv = A (cid:48) , A (cid:48) , A (cid:48) . (3.4)6) b) Figure 2: The three body-wave speed profiles (plotted as v √ ρ ) for nickel under (a) uniaxial compressivestress with principal compression stretch ratio λ = 0 .
99 (light green curve), 0 .
973 (blue-green curve),0 .
956 (blue curve); (b) pure shear stress with λ = 0 .
99 (light green curve), 0 .
978 (blue-green curve) and0 .
967 (blue curve).
In each set of three roots for ρv , the first root corresponds to a pure longitudinal waveand the next two to pure transverse waves.In general ( θ (cid:54) = 0 , ◦ ), the characteristic equation factorizes into the product of a termlinear in ρv (corresponding to a pure transverse wave polarized along the x direction)and a term quadratic in ρv (with one root corresponding to a pseudo-longitudinal waveand the other to a pseudo-transverse wave); see Norris (1983) for details.Figure 2 depicts the variations of the three body wave speeds (in this and all sub-sequent plots it is v √ ρ that is plotted) in deformed nickel with respect to the angle θ between the direction of greatest compression and the direction of propagation, for differ-ent values of compressive stretch under uniaxial and pure shear stresses. The variationsof the wave traveling with the intermediate speed meet intuitive expectations: this wavetravels at its slowest when θ = 0 ◦ and at its fastest when θ = 90 ◦ . However, this scenariois reversed for the fastest and slowest waves, as soon as the solid is deformed: they travelat their fastest along the direction of greatest compression ( θ = 0 ◦ ) and slowest in theorthogonal direction. Moreover, when a pure shear stress induces a compression of morethan 3%, we notice that the profile for the slowest body wave develops a new minimum;in effect this wave travels at its slowest in a direction which is oblique with respect to theprincipal directions of strain ( θ (cid:39) ◦ ).Figure 3 show the corresponding results for deformed steel . They are similar to thosefor deformed nickel, with the difference that the secondary minimum phenomena occursunder uniaxial compression instead of pure shear stress.In Figures 4 and 5, we study body wave propagation in deformed polystyrene and hydrogel . Here the waves all travel at their fastest along the direction of greatest stretch( θ = 90 ◦ ) and two of the three waves travel at their slowest in the direction of greatestcompression ( θ = 0 ◦ ). There is, however, one wave which travels at its slowest in anoblique direction, for both types of pre-deformations (due to uniaxial stress: figures onthe left; due to pure shear stress: figures on the right). They appear at quite largecompressions (31% for polystyrene, 39% for hydrogel), which are nonetheless compatiblewith the soft nature of these solids and with a physically acceptable material response(i.e. the tension and the shear stress are monotone functions of the stretch).7) b) Figure 3: The three body-wave speed profiles (plotted as v √ ρ ) for steel under (a) uniaxial compressivestress with λ = 0 .
99 (light green curve), 0 .
956 (blue-green curve), 0 .
922 (blue curve); (b) pure shearstress with λ = 0 .
99 (light green curve), 0 .
981 (blue-green curve), 0 .
973 (blue curve). a) b)
Figure 4: The three body-wave speed profiles (plotted as v √ ρ ) for polystyrene under (a) uniaxial com-pressive stress; (b) pure shear stress. The light green curves correspond to λ = 0 .
91, the blue-greencurves to λ = 0 .
8, and the blue curves to λ = 0 . a) b) Figure 5: The three body-wave speed profiles (plotted as v √ ρ ) for hydrogel under (a) uniaxial compressivestress; (b) pure shear stress. The light green curves correspond to λ = 0 .
75 (i.e. 25% maximumcompression) and the dark green curves to λ = 0 .
61 (i.e. 39% maximum compression).
Now we investigate non-principal surface wave propagation in a deformed homoge-neous half-space. There are several methods of resolution available for these problems;8ee, for example, Rogerson and Sandiford (1999); Destrade et al. (2005); Kayestha etal. (2011); Gandhi et al. (2012). Here we adopt a formulation in terms of the surfaceimpedance matrix. In the next section we detail the steps involved in implementing thismethod, based on the analysis of Fu and Mielke (2002).
4. The matrix Riccati method for surface waves
In the following, we replace the tensors T , R , Q , etc., introduced in the previous sectionby their matrix representations with respect to the Cartesian coordinates ( x , x , x ). Ina nutshell, surface wave propagation is governed by the algebraic matrix Riccati equation (Biryukov, 1985; Barnett and Lothe, 1985; Fu and Mielke, 2002; Norris and Shuvalov,2010) = [ Z ( v ) − i R T ] T − [ Z ( v ) + i R ] − Q + ρv I , (4.1)the radiation condition, Im Spec T − [i Z ( v ) − R ] > , (4.2)and the boundary condition of zero incremental traction on x = 0, which is equivalentto det Z ( v ) = 0 . (4.3)Here, the constant 3 × Z ( v ) is the so-called surface impedance matrix . For agiven v , Z ( v ) is a constant Hermitian matrix, of the form Z = Z Z + i Z Z − i Z Z − i Z Z Z + i Z Z + i Z Z − i Z Z , (4.4)say, where the Z k are real constants ( k = 1 , . . . , Z k and v , and uniqueness of thesolution comes from further requiring that Z ( v ) be positive definite , as discussed below.The surface impedance matrix Z ( v ) in a half-space relates the incremental displace-ment u to the incremental traction t on the surface x = constant through the relation-ship, t = − k Z ( v ) u . (4.5)We may rewrite this by noting that the general solution of the homogeneous system ofsecond-order ordinary differential equations with constant coefficients (2.5) for the half-space, is of the form U = e i k E ( v ) x U , where E ( v ) is a constant 3 × U is a constant vector. Then the traction is givenby t i = s i = A iqj u j,q , or t = i k [ R + TE ( v )] u . (4.6)Now write t = − i k V e i( kx − vt ) , where V = − [ R + TE ( v )] U , so that the impedance relation(4.6) reads V = − i Z ( v ) U , with Z ( v ) = − i[ R + TE ( v )] , (4.7)9howing that Z ( v ) is indeed a constant matrix for a half-space. The matrix Z ( v ) corre-sponding to the existence of a surface wave is the one that satisfies the Riccati equation(4.1), the boundary condition (4.3), andIm Spec E ( v ) > , or, equivalenty, (4.2). This condition guarantees the correct decay for U ( x ) = e i k E ( v ) x U as x increases with distance away from the free surface.In the matrix Riccati method, at least two remarkable properties emerge: Z (0) ispositive definite in the region of stability and ∂ Z ( v ) /∂v is negative definite as long asIm Spec T − [i Z ( v ) − R ] >
0. Hence, det Z ( v ) is positive at v = 0 and monotonicallydecreasing as v increases, which means that it is simple to find ˜ v numerically such thatdet Z (˜ v ) = 0. Moreover uniqueness of the surface velocity, calculated by this procedure,is guaranteed. Barnett and Lothe (1985), Fu and Mielke (2002) and Mielke and Fu(2003) have shown these properties, and here we present a somewhat simpler alternativedemonstration (see also Shuvalov et al. (2004); Alshits and Maugin (2005) for furtherimpedance formulations).Recall that the incremental nominal stress has components s pi = A piqj u j,q (withrespect to the non-principal axes) and that the balance of momentum (2.3) reads ρu i,tt = s pi,p . (4.8)Now multiply both sides of this by u ∗ i , the complex conjugate of u i : ρu ∗ i u i,tt = u ∗ i s ji,j = ( u ∗ i s ji ) ,j − u ∗ i,j s ji with summation over i and j. (4.9)Then integrate over the region U = [ x , x + ∆ x ] × [0 , ∞ ] × [ x , x + ∆ x ] in the body,to obtain (cid:90) U ρu ∗ i u i,tt d x d x d x = (cid:90) ∂ U u ∗ i s ji n j d a − (cid:90) U u ∗ i,j s ji d x d x d x , (4.10)where n is the outward unit normal vector to the boundary ∂ U and d a the associatedarea element. Now substitute u ( x , x , x ) = U ( x )e i k ( x − vt ) to arrive at − k v (cid:90) ∞ ρ U ∗ ( y ) · U ( y ) d y = u ∗ i t i (cid:12)(cid:12)(cid:12) y = ∞ y =0 − (cid:90) ∞ A jilk u ∗ i,j u l,k d y, (4.11)where we have introduced the components t i , defined in (4.6), of the traction t on planesnormal to the x -axis. Observe that the above equation is independent of x and x .Finally, assume that the wave amplitude decays away from the free surface, so that U ( ∞ ) = . Then substitute for t i from (4.5) and rearrange to obtain k U ∗ (0) · Z ( v ) U (0) = (cid:90) ∞ A jilk u ∗ i,j u l,k d x − k v (cid:90) ∞ ρ U ∗ ( x ) · U ( x ) d x . (4.12) Here and in the following we write the scalar product of two vectors as a · b rather than in the matrixform a T b . Z depends on v because (i) U can be chosen independently of v since forany choice of displacement field U , a traction field V can be determined by equation (4.7)such that momentum is balanced, and (ii) v cancels out in the products u ∗ i,j u l,k . Therefore,by differentiating with respect to v , we obtain U ∗ (0) · d Z ( v )d v U (0) = − kv (cid:90) ∞ ρ U ∗ ( x ) · U ( x ) d x < , (4.13)while writing equation (4.12) at v = 0 gives k U ∗ (0) · Z (0) U (0) = (cid:90) ∞ A jilk u ∗ i,j u l,k d x , (4.14)for any choice of U (0). Clearly d Z / d v is negative definite by (4.13) and, from thestrong-convexity condition (2.9) and (4.14), Z (0) is positive definite if at least one ofthe components of u i,j is non-zero. Below we show that Z (0) is positive definite when thedeformation is within the region of (dead-load) stability.For a material in the reference configuration , strong-convexity is considered to be anecessary physical requirement, and it implies that Z (0) is positive definite and that thedecay condition (4.2) holds at v = 0. For a pre-stressed material , strong-convexity is notexpected in general. However, Z (0) is positive definite for a deformation in the region ofdead-load stability. Let the magnitude of the finite deformation be parameterized by α ,with α = 0 corresponding to no deformation (for instance, α can be the amount of shearin a simple shear pre-deformation, or the elongation λ − Z depends on α as well as on v and the boundary condition of noincremental surface-traction (the secular equation) takes the formdet Z ( v, α ) = 0 . (4.15)Assume that for α = 0 the strong-convexity condition (2.9) is satisfied, so that Z (0 ,
0) ispositive definite. As α is increased and the deformation moves into the region of dead-loadstability consider the change in the eigenvalues of Z (0 , α ); these eigenvalues are positiveuntil α reaches a critical value α ∗ , say, when at least one eigenvalue becomes zero anddet Z (0 , α ∗ ) = 0. At this point the half-space supports a standing-wave solution givenby (2.4) with v = 0 (at the boundary of the dead-load stability region), and the materialhas buckled (that is, it is unstable, at least in the linearized sense). For waves alongthe principal direction, this buckling criterion can be shown to be the same as found inDowaikh and Ogden (1991). For α > α ∗ we say that the half-space is unstable withrespect to surface-wave perturbations (Fu and Mielke, 2002).We are only interested in surface waves in the stable region 0 < α < α ∗ where Z (0 , α )is positive definite, and we define an implicit curve v → Z ( v, α ) by using the Riccatiequation (4.1). As long as Im Spec T − (i Z ( v, α ) − R ) > v untildet Z ( v, α ) = 0. If along this curve Im Spec T − (i Z ( v, α ) − R ) ≤ Z ( v, α )reaches zero, then there is no surface-wave.
5. Results for surface waves
We transform the above analysis into a numerical method by choosing A for whichthere is a positive definite Z (0) satisfying equation (4.1). Then, as v is increased, we11alculate the implicit curve for Z ( v ) from Z (0) up to Z (ˆ v ) where det Z (ˆ v ) = 0, all thewhile verifying that Im Spec T − (i Z − R ) >
0. From that point on, we calculate anotherimplicit curve that satisfies equations (4.1) and (4.3) by varying A (for instance, byvarying the angle of propagation with respect to the principal axes or by varying theamplitude of the pre-deformation). If at some point T − (i Z − R ) ≤
0, then to confirmthat there is no surface-wave calculate the implicit curve for v (cid:55)→ Z ( v ) that departsfrom Z (0) and if, for some v , T − (i Z ( v ) − R ) ≤
0, then no surface-wave exists; if not,then varying A has caused a discontinuous jump in the velocity, which may indeed bepossible.Using this method we now present Surface Acoustic Wave (SAW) velocity profiles inseveral materials subject to either a uniaxial compressive stress or a pure shear stress,applied in the plane parallel to the free surface x = 0.Figure 6 depicts the variations of the surface wave speed with the angle of propagationwith respect to the principal directions of strain in nickel subject to a uniaxial compressivestress. In the early stages of compression, from 1% to 3% compressive stretch say, “thevariations of the SAW speeds show symmetry about the [principal] direction[s]” as statedby Kim and Sachse (2001), with the proviso that the SAW travels at its fastest along thedirection of greatest compression θ = 0 ◦ and at its slowest along the direction of greateststretch θ = 90 ◦ (in line with the behavior of the body waves in nickel, as shown in theprevious section). However, as the material is further compressed (compression beyond10%), secondary extrema develop: for λ ≥ . θ (cid:39) ◦ direction and the slowest SAW travels in the θ (cid:39) ◦ direction. A similar phenomenonoccurs when nickel is subject to a pure shear stress, as shown in Figure 7: then the slowestwave travels at the oblique angle θ (cid:39) ◦ when the material is compressed by as little as3.6%; see figure on the right. Λ (cid:61)
Λ (cid:61) Θ Ρ v Λ (cid:61)
Λ (cid:61) Θ Ρ v Figure 6: Speed profiles for surface waves (plotted as v √ ρ ) in nickel subject to uniaxial compressivestress, with pre-stretch λ decreasing from 0.998 to 0.964 (on the left) and from 0.907 to 0.873 (on theright). As the color of the curves changes from green to blue, λ is decreased by regular increments of0.0057 from one curve to the next. For deformed steel , we observe similar characteristics for the SAW velocity profileunder uniaxial compression and pure shear stress as for deformed nickel, as shown inFigure 8.
Pyrex glass also exhibits a local minimum under pure shear stress, when λ (cid:39) . λ = 0 .
97, i.e. under a compressionof 3% (figures not shown to save space). 12 (cid:61)
Λ (cid:61) Θ Ρ v Λ (cid:61)
Λ (cid:61) Θ Ρ v Figure 7: Speed profiles for surface waves (plotted as v √ ρ ) in nickel subject to pure shear stress, withpre-stretch λ decreasing from 0.998 to 0.970 (on the left) and from 0.964 to 0.959 (on the right). As thecolor of the curves changes from green to blue, λ is decreased by regular increments of 0.0057 from onecurve to the next. a) Λ (cid:61)
Λ (cid:61) Θ Ρ v b) Λ (cid:61)
Λ (cid:61) Θ Ρ v Figure 8: Speed profiles for surface waves (plotted as v √ ρ ) in steel subject to (a) uniaxial compressivestress, with pre-stretch λ decreasing from 0.990 to 0.905 (on the left) and (b) pure shear stress, withpre-stretch λ decreasing from 0.99 to 0.95 (on the right). As the color of the curves changes from greento blue, λ is decreased by regular increments of 0.0056 from one curve to the next. SAWs in deformed polystyrene behave in a more orderly way, as they travel at theirfastest along the direction of greatest stretch θ = 90 ◦ and at their slowest along θ = 0 ◦ .Although the first derivative of the velocity profile is not a monotone function of theangle, no secondary extremum develops, in contrast to the behavior of the body wavesin the same material (see previous section).Finally, SAW propagation in deformed hydrogel is also almost regular under uniaxialcompression even at a relatively large strain (up to 40%); see Figure 10(a). However, twosecondary extrema develop under pure shear stress, with the secondary minimum in anoblique direction, eventually becoming an absolute minimum; see Figure 10(b).
6. Conclusion
Clearly, the existence of oblique slowest waves greatly complicates the determination ofthe principal directions of strain in a deformed body. Finding the direction where a wavetravels at its slowest or fastest is not a guarantee of having determined the direction ofgreatest compression or tension, or that it is indeed a principal direction. In our examples,13)
Λ (cid:61)
Λ (cid:61) Θ Ρ v b) Λ (cid:61)
Λ (cid:61) Θ Ρ v Figure 9: Speed profiles for surface waves (plotted as v √ ρ ) in polystyrene subject for (a) uniaxialcompressive stress, with pre-stretch λ decreasing from 0.908 to 0.602 (on the left) and (b) pure shearstress, with pre-stretch λ decreasing from 0.908 to 0.602 (on the right). As the color of the curves changesfrom green to blue, λ is decreased by regular increments of 0.028 from one curve to the next. a) Λ (cid:61) .639
Λ (cid:61) Θ Ρ v b) Λ (cid:61) .583
Λ (cid:61) Θ Ρ v Figure 10: Speed profiles for surface waves (plotted as v √ ρ ) in hydrogel subject to (a) uniaxial com-pressive stress, with pre-stretch λ decreasing from 0.750 to 0.639 (on the left) and (b) pure shear stress,with pre-stretch λ decreasing from 0.750 to 0.583 (on the right). As the color of the curves changes fromgreen to blue, λ is decreased by regular increments of 0.028 from one curve to the next. we have found that the slowest body wave can sometimes be along an oblique directionand similarly for surface waves. However, we found that the fastest body waves do indeedtravel along a principal direction, a criterion which can thus be used to determine principaldirections, at least in deformed nickel, steel, polystyrene and hydrogel. Unfortunately,this characteristic does not carry over to the case of surface waves, as the example ofnickel subject to pure shear stress shows, where the fastest surface wave is oblique. Theoverall conclusion is that, for a given solid, a full analysis of wave speed variation withangle of propagation, such as that conducted in this paper, is required. Acknowledgements
Partial funding from a Royal Society International Joint Project grant and from theHardiman Scholarship programme at the National University of Ireland Galway are grate-fully acknowledged. Helpful discussions with Alexander Shuvalov are also acknowledged.14 eferences
V.I. Alshits, G.A. Maugin. Dynamics of multilayers: elastic waves in an anisotropic gradedor stratified plate. Wave Motion (2005) 357–394.D.M. Barnett, J. Lothe. Free surface (Rayleigh) waves in anisotropic elastic half-spaces:the surface impedance method, Proc. Roy. Soc. Lond. A (1985), 135–152.L. Brillouin. Sur les tensions de radiation, Ann. Phys. ser. 10 (1925), 528–586.S.V. Biryukov. Impedance method in the theory of elastic surface waves, Sov. Phys.Acoust. (1985), 350–354.M. Destrade, J.G. Murphy, G. Saccomandi. Simple shear is not so simple, Int. J. Non-Linear Mech. (2012), 210–214.M. Destrade, R.W. Ogden. On stress-dependent elastic moduli and wave speeds, IMA J.Appl. Math., in press. DOI:10.1093/imamat/hxs003M. Destrade, M. Ottenio, A.V. Pichugin, G.A. Rogerson. Non-principal surface waves indeformed incompressible materials, Int. J. Eng. Sci. (2005), 1092–1106.M.A. Dowaikh, R.W. Ogden. On surface waves and deformations in a compressible elastichalf-space, Stability Appl. Analysis Cont. Media (1991), 27–45.Y.B. Fu, A. Mielke. A new identity for the surface impedance matrix and its applicationto the determination of surface-wave speeds, Proc. Roy. Soc. Lond. A (2002),2523–2543.N. Gandhi, J.E. Michaels, S.J. Lee. Acoustoelastic Lamb wave propagation in biaxiallystressed plates, J. Acoust. Soc. Am. (2012), 1284–1293.R.A. Guyer, P.A. Johnson. Nonlinear Mesoscopic Elasticity . Wiley-VCH, Weinheim(2009).D.S. Hughes, J.L. Kelly. Second-order elastic deformation of solids, Phys. Rev. (1953),1145–1149.P. Kayestha, A.C. Wijeyewickrema, K. Kishimoto. Wave propagation along a non-principal direction in a compressible pre-stressed elastic layer, Int. J. Solids Struct. (2011), 2141–2153.K.Y. Kim, W. Sachse. Acoustoelasticity of elastic solids, in Handbook of Elastic Propertiesof Solids, Liquids, and Gases , Levy, Bass, Stern (Editors), , 441–468. Academic Press,New York (2001).L.D. Landau, E.M. Lifshitz. Theory of Elasticity , 3rd ed. Pergamon, New York (1986).A.I. Lurie.
Theory of Elasticity . Springer, Berlin (2005).A. Mielke, Y.B. Fu. A proof of uniqueness of surface waves that is independent of theStroh Formalism, Math. Mech. Solids (2003), 5–15.15.A. Mihai, A. Goriely. Positive or negative Poynting effect? The role of adscititiousinequalities in hyperelastic materials. Proc. R. Soc. Lond. A (2011), 3633–3646.A.N. Norris. Propagation of plane waves in a pre-stressed elastic medium, J. Acoust. Soc.Am. (1983), 1642–1643.A.N. Norris, A.L. Shuvalov. Wave impedance matrices for cylindrically anisotropic radi-ally inhomogeneous elastic solids. Q. J. Mech. Appl. Math. (2010), 401–435.R.W. Ogden, Nonlinear Elastic Deformations . Dover, New York (1997).Y.-H. Pao, W. Sachse, H. Fukuoka. Acoustoelasticity and ultrasonic measurements ofresidual stresses, in W.P. Mason and R.N. Thurston, eds.,
Physical Acoustics , Vol. 17,pages 61–143. Academic Press (1984).G.A. Rogerson, K.J. Sandiford. Harmonic wave propagation along a non-principal direc-tion in a pre-stressed elastic plate, Int. J. Eng. Sci. (1999), 1663–1691.A.L. Shuvalov, O. Poncelet, M. Deschamps. General formalism for plane guided waves intransversely inhomogeneous anisotropic plates, Wave Motion (2004) 413–426.K. Tanuma, C.-S. Man, W. Du. Perturbation of phase velocity of Rayleigh waves in pre-stressed anisotropic media with orthorhombic principal part, Math. Mech. Solids, inpress. DOI:10.1177/1081286512438882M.S. Wu, H.O.K. Kirchner. Nonlinear elasticity modeling of biogels, J. Mech. Phys. Solids58