Interface Waves in Pre-Stressed Incompressible Solids
We study incremental wave propagation for what is seemingly the simplest boundary value problem, namely that constitued by the plane interface of a semi-infinite solid. With a view to model loaded elastomers and soft tissues, we focus on incompressible solids, subjected to large homogeneous static deformations. The resulting strain-induced anisotropy complicates matters for the incremental boundary value problem, but we transpose and take advantage of powerful techniques and results from the linear anisotropic elastodynamics theory. In particular we cover several situations where fully explicit secular equations can be derived, including Rayleigh and Stoneley waves in principal directions, and Rayleigh waves polarized in a principal plane or propagating in any direction in a principal plane. We also discuss the merits of polynomial secular equations with respect to more robust, but less transparent, exact secular equations.
Introduction The term “acousto-elastic effect” describes the interplay between the static deformation of an elastic solid and the motion of an elastic wave. If both the deformation and the motion are of infinitesimal amplitude, then all the governing equations are linearized, see for instance the Chapter by Norris for examples and applications or the experimental results of Pao et al. (1984). If both the deformation and the motion are of finite amplitude, then the resulting governing equations are highly nonlinear, and their resolution is the subject of much research, see the Chapter by Fu for the weakly nonlinear theory and the Chapter by Saccomandi for the fully nonlinear theory.
In between those two situations lies the theory of “small-on-large”, also known as the theory of “incremental” motions, where the wave is an infinitesimal perturbation superimposed onto the large static homogeneous deformation of a generic hyperelastic solid. There, the homogeneous character of the static deformation and the linear character of the incremental equations of motion ensure that the calculations are valid for any strain energy density (to be specified later for applications, if necessary). The next Section of this Chapter briefly recalls the governing equations of incremental motions (see the Chapter by Ogden for their derivation).
It turns out that many similarities can be drawn between the equations of incremental motions and those of linear anisotropic elasticity, with the main difference that in the latter case, the anisotropy is set once and for all for a given crystal whereas in the former case, it is strain-induced and susceptible to great variations from one configuration to another. Using the similarities, we may transpose the so-called Stroh formulation and exploit its many results; on the other hand, when focussing on the differences, we may highlight the influences of the pre-stress and of the choice of a strain-energy density on the propagation of waves. In this Chapter, attention is restricted to waves at the interface of pre-deformed, semi-infinite solids, in contact either with vacuum (Rayleigh waves) or with another solid (Stoneley waves). With a view to model elastomers and biological soft tissues, the solids are considered to be incompressible (mathematically, this internal constraint lightens somewhat the expressions but does not prove essential to the resolution).
Several situations are treated: principal wave propagation in Section 3, principal polarization in Section 4, and principal plane propagation in Section 5. The emphasis is on deriving explicit secular equations in polynomial form, using some simple “fundamental equations” derived at the end of Section 2. Of course, as the setting gets more and more involved, so does the search for a polynomial secular equation; eventually its degree becomes too high for comfort and other techniques are required. The concluding section (Section 6) discusses the pros and cons of such equations, as opposed to exact, non-explicit, secular equations, free of spurious roots.
Consider an isotropic, incompressible, hyperelastic solid at rest, characterized by a mass density and a strain energy function . Then subject it to a large, static, homogeneous deformation (“the pre-strain”) carrying the particle at in the undeformed configuration to the position in the deformed configuration.
Call the corresponding constant deformation gradient and the associated left Cauchy-Green strain tensor. This tensor being symmetric, the directions of its eigenvectors are orthogonal; they are called the principal axes of pre-strain or in short, the principal axes. Also, the eigenvalues of are positive, , , , say, and , , are called the principal stretches. Figure 1 shows how a unit cube with edges aligned with the principal axes, is transformed by the pre-strain.
Note that because the solid is incompressible, its volume is preserved through any deformation so that here,
The first two principal invariants of strain are defined as
In the Cartesian coordinate system aligned with the principal axes, is diagonal. Calling , , , the unit vectors in the , , directions, respectively, we have
and the computation of , there gives
For an isotropic solid, may be given as a function of the invariants: or equivalently, as a symmetric function of the principal stretches: , according to what is most convenient for the analysis or according to how has been determined experimentally. With the first choice, the constant Cauchy stress (“the pre-stress”) necessary to maintain the solid in its state of finite homogeneous deformation is
where is a Lagrange multiplier due to the constraint of incompressibility (a yet arbitrary constant scalar to be determined from initial and boundary conditions.) With the second choice, the non-zero components of relative to the principal axes are written as
Consider a half-space filled with an incompressible hyperelastic solid subject to a large homogeneous deformation. We take the Cartesian coordinate system (, , ) to be oriented so that the boundary is at , and we study the propagation in the direction of an infinitesimal interface wave in the solid.
This wave is inhomogeneous as it progresses in an harmonic manner in a direction lying in the interface, while its amplitude decays with distance from the boundary.
We call the mechanical displacement associated with the wave, and the increment in the Lagrange multiplier due to incompressibility. The incremental nominal stress tensor has components
where the comma denotes partial differentiation with respect to the coordinates , , . Here, is the fourth-order tensor of instantaneous elastic moduli, with components
Note that due to the symmetry above, has in general 45 independent components. In the principal axes coordinate system () however, there are only 15 independent non-zero components; they are (Ogden, 2001):
(no sums on repeated indexes here), where and .
Finally, the governing equations are the incremental equations of motion and the incremental constraint of incompressibility; they read
Now everything is in place to solve an interface wave problem. We take and in the form
where is the wave number, and are functions of the variable only, is the unit vector in the direction of propagation, and is the speed. Clearly by (7), has a similar form, say
where is a function of only.
the prime denotes differentiation with respect to the variable , and are the tractions acting on planes parallel to the boundary, with components . Here the matrix has the following block structure
where , , and are square matrices. This is the so-called Stroh formulation. In effect, many of the results established thanks to the Stroh (1962) formalism in linear anisotropic elasticity can formally be carried over to the context of incremental dynamics in nonlinear elasticity, as shown by Chadwick and Jarvis (1979a), Chadwick (1997), and Fu (2005a, b).
The solution to the first-order differential system (13) is an exponential function in ,
where is a constant vector and is a scalar. Then the following eigenvalue problem emerges: . Its resolution is in two steps.
First, find the eigenvalues by solving the propagation condition,
for , and keep those ’s which satisfy the decay condition. For instance, when the solid fills up the half-space, the decay condition is
ensuring that the solution (15) is localized near the interface and vanishes away from it. The penetration depth of the interface wave is clearly related to the magnitude of : the smaller this quantity is, the deeper the wave penetrates into the solid.
The propagation condition (16) is a polynomial in with real coefficients and it has only complex roots (Fu, 2005b), which come therefore in pairs of complex conjugate quantities. Hence, half of all the roots to the propagation condition qualify as satisfying the decay condition. Let be the eigenvector corresponding to the qualifying root .
Now proceed to the second step, which is to construct the general localized solution to the equations of motion, as
for some arbitrary constants . Then compute this vector at the interface and apply the boundary conditions. The vector is often decomposed as follows,
where and are square matrices and is the vector with components . For instance, the archetype of interface waves is the Rayleigh (1885) surface wave, which propagates at the interface between a solid half-space and the vacuum, leaving the boundary free of tractions. Mathematically, the corresponding boundary condition is that , leading to
This (complex) form of the secular equation is however not the optimal form, and it might lead to unsatisfactory answers to the questions of existence and uniqueness of the wave (see Barnett (2000) for an historical account of this point). From the Stroh formalism, and its application to the present context, we learn that it is much more efficient to work with the surface impedance matrix (Ingebrigsten and Tonning, 1969) than with the matrices and ; this matrix is defined by
is a real equation, in contrast to (20). Moreover, if there is a root to this equation in the subsonic regime (where is less than the speed of any bulk wave), then it is unique; also, the existence of a root is equivalent to the existence of a surface wave.
Similar results also exist for other types of interface waves as seen in the course of this Chapter. For instance, the boundary conditions for Stoneley (1924) interface waves are that displacements and tractions are continuous across the boundary between two rigidly bonded semi-infinite solids. Then where the asterisk refers to quantities for the solid in . Equivalently, , , from which comes , leading to
the optimal form of the secular equation for Stoneley interface waves. Here is the surface impedance matrix for the solid in the half-space, defined as
Explicit secular equations
The derivation of a secular equation, preferably in the optimal form involving the surface impedance matrix, is no sinecure in general. The problematic step lies in the resolution of the propagation condition (16).
For principal wave propagation (, , are aligned with the principal axes), the propagation condition factorizes into the product of a term linear in and a term quadratic in . Here we can compute the roots explicitly, keep the qualifying ones (see (17)), and solve the boundary value problem in its entirety. Many problems falling in this category have been solved over the years, and some are presented in Section Interface Waves in Pre-Stressed Incompressible Solids.
For a non-principal wave with propagation direction and attenuation direction both in a principal plane (the saggital plane () is a principal plane but is not a principal axis), the propagation condition factorizes into the product of a term linear in and a term quartic in . We treat this case in Section Interface Waves in Pre-Stressed Incompressible Solids. Although it is possible to write down formally the qualifying roots of the quartic (Fu, 2005a; Destrade and Fu, 2006; Fu and Brookes, 2006), the formulas involved are cumbersome to interpret.
For a wave propagating in a principal plane but not in a principal direction ( is aligned with a principal axis but neither nor are aligned with principal axes), the propagation condition is a cubic in . We treat this case in Section Interface Waves in Pre-Stressed Incompressible Solids. Now it is a daunting task to find analytical expressions for the roots satisfying the decay condition (17).
Finally, for wave propagation in any other case, the propagation condition is a sextic in , unsolvable analytically according to Galois theory.
These observations suggest that, except in the case of principal waves, numerical procedures are required in order to make progress. It is indeed the case that sophisticated tools and efficient numerical recipes have been developed by Barnett and Lothe (1985), Fu and Mielke (2002), and several others, with most satisfying results. However it is also the case that some interface wave problems can be solved analytically, up to the derivation of the secular equation in explicit polynomial form. The first steps in that direction were taken by Currie (1979), and his advances were later refined by Taylor and Currie (1981) and Taziev (1989), revisited by Mozhaev (1995) and by Ting (2004), and extended by Destrade (2003).
The equations that turn out to be fundamental in the derivation of explicit polynomial secular equations are
where and is an integer. Their derivation is most simple. First, it can be shown by induction (Ting, 2004) that has a block structure similar to that of , that is
with , . It then follows that
is symmetric for all . Now take the scalar product of both sides of the governing equation (13) by to get
finally add its complex conjugate to this equality to end up with
and, by integration between the interface (at ) and infinity (where and , and thus , vanish), arrive at (25).
For instance, the boundary condition for Rayleigh surface waves is that there are no incremental tractions at the interface; thus , and the fundamental equations (25) reduce to
Here we take () to coincide with the principal axes (). The pre-deformation is thus
Figure 2 summarizes the situation with respect to the waves’ characteristics near the interface. Bear in mind that the wave analysis is linear and gives no indication about the amplitude; moreover, a half-space has no characteristic length so that the secular equation is non-dispersive and the wavelength remains undetermined.
For principal waves, the fields (11) and (12) are independent of , because and . Also, recall from (Interface Waves in Pre-Stressed Incompressible Solids) that the non-zero components of in the () coordinate system of principal axes are
and also , , , , , , , , and , whose expressions are not needed in this Section.
From these observations follows that the third equation of motion (10) reduces to
where by (7),
Hence the movement along the principal axis is governed by an equation which depends only on . For this equation, governing what is termed the anti-plane motion, we take the trivial solution: , and we focus on the in-plane motion. According to (10), it is governed by
where by (7),
We eliminate in favour of the pre-stress: by (6) at , we have and so by (Interface Waves in Pre-Stressed Incompressible Solids),
It then follows from the second equation above that
Proceeding similarly for , , we find eventually that the governing equations are indeed in the form (13), where and , , and are given by
respectively, where we used the following short-hand notations (no sums),
The propagation condition (16) reduces to a quadratic in ,
Notice how , though present in , does not appear explicitly in this equation.
Calling , , the roots of the quadratic, we have
The roots , of the biquadratic satisfying the decay condition (17) are in one of the two following forms; either: , , where , , or: , , where . Whatever the case, , , and is a purely imaginary quantity. From the first inequality we deduce that (Dowaikh and Ogden, 1990)
is a real quantity. From the second, and using the definitions of and , we find
We compute the eigenvectors and of corresponding to and as any column of the matrix adjoint to and to , respectively. Choosing the third column, we find
and is a non-dimensional measure of the pre-stress.
We can now construct the and matrices, and the surface impedance matrix . It turns out to be
which is indeed Hermitian because is a purely imaginary quantity and is real.
Dowaikh and Ogden (1990) established this form of the secular equation for principal surface waves in pre-stressed incompressible solids, following other works by Hayes and Rivlin (1961), Flavin (1963), Willson (1973a, b), Chadwick and Jarvis (1979a), Guz (2002, review with an extensive bibliography), and many others. It is of course consistent with Lord Rayleigh’s own analysis of surface waves in linear isotropic incompressible solids. To check this, let the solid be un-stressed () and un-deformed (); then reduces to , where is the infinitesimal shear modulus; also, and is the real root of , that is giving , as found by Rayleigh (1885).
For Stoneley interface waves, the secular equation is (23) where
This equation was studied in great detail by Dowaikh and Ogden (1991) and by Chadwick (1995). It is consistent with the analysis of Stoneley (1924) of interface waves in linear isotropic incompressible solids. A remarkable feature of this secular equation for principal Stoneley interface waves in deformed incompressible solids – first noted by Chadwick and Jarvis (1979b) – is that the pre-stress does not appear explicitly in it, in contrast to the equation for surface waves (50). This quantity, which is continuous across the interface (), disappears in the addition of the two surface impedance matrices. Of course it still plays an implicit role, in determining the pre-strain.
Dowaikh and Ogden (1990, 1991), Chadwick (1995), and Guz (2002, review) have covered almost every aspect of principal interface wave propagation and more information can be found in their respective articles. In the next Subsection we rapidly work out two examples of surface waves.
First we present an example taken from the literature on elastomers, where the Mooney-Rivlin strain energy function is often encountered. It is given by
where and are positive constants with the dimensions of a stiffness. The Mooney-Rivlin material enjoys special properties with respect to wave propagation (the neo-Hookean material, which corresponds to the special case , enjoys even more special properties as is seen in Section Interface Waves in Pre-Stressed Incompressible Solids). For instance, once subjected to a large homogeneous pre-strain, it permits the propagation of bulk waves in every direction; these waves can be infinitesimal, but also of arbitrary finite amplitude (Boulanger and Hayes, 1992); they can be homogeneous plane waves but also inhomogeneous plane waves (Destrade, 2000, 2002). The quantities (Interface Waves in Pre-Stressed Incompressible Solids) are also quite special; they are
where , and thus they satisfy
These relationships mean that the biquadratic (43) factorizes to
and that the secular equation (50) reduces to
Hence one qualifying root is , whatever the values of the material constants and . The other root is . When there is no pre-stress normal to the boundary (), then is the real root of , that is giving
a result first established by Flavin (1963). Here and so that the penetration depth of the surface wave is fixed and is completely independent of the pre-strain and of the material parameters and . We say that the penetration depth is universal relative to the class of Mooney-Rivlin materials.
The second example is taken from the biomechanics literature. From a series of uniaxial tests on human aortic aneurysms, Raghavan and Vorp (2000) deduced that the following strain energy density gave a satisfying fit with the data plots,
where, typically, MPa, MPa (Karduna et al. (1997) use the same expression to model the response of passive myocardium.) A uniaxial pre-stress is , , leading through (6) to the following equi-biaxial pre-strain,
where is calculated from
Finally, using the following expressions for the relevant moduli,
it is a simple matter to solve the secular equation (50) numerically and plot the variations of the squared wave speed, scaled with respect to the squared bulk wave speed , with the pre-stretch . Figure 3 displays these variations; for comparison purposes, Figure 3 shows the variations of the scaled squared wave speed in the case of a Mooney-Rivlin material in uniaxial stress; in that later case the graph is independent of the material parameters and because by (57), . The dashed lines indicate the speed of Lord Rayleigh’s squared speed in the isotropic (no pre-strain) case where , . Notice how different the responses of two solids are in that neighbourhood. Note also that for high compressive stretches, the squared speeds eventually falls off to zero; this happens at for all Mooney-Rivlin materials, as shown by Biot (1963), and at for the soft biological tissue model above. Beyond that critical compression stretch, , leading to a purely imaginary , an amplitude which then grows exponentially with time according to (11), and a breakdown of the linearized analysis. The search for critical compression stretches is an extremely active area of research, clearly linked to the geometric stability analysis of solids.
Waves polarized in a principal plane
In this Section we study the case where is aligned with the principal axis but neither (propagation direction) nor (attenuation direction) are aligned with principal axes. The components in the () coordinate system of the instantaneous elastic moduli tensor are related to the components , given by (Interface Waves in Pre-Stressed Incompressible Solids), in the () coordinate system of principal axes through the tensor transformations
and is the angle between and .
In particular we find that the non-zero components in the forms and are , , , , , and . The mechanical fields (11) and (12) are independent of because here. As a consequence, the third equation of motion (10) reduces to
where by (7),
Hence the movement along the principal axis is governed by an equation which depends only on , and for this anti-plane motion we take the trivial solution: .
The equations governing the in-plane motion have been derived in the case of a general plane pre-strain by Fu (2005a) and solved for surface waves in the case of a pre-strain consisting in a triaxial stretch followed by a simple shear by Destrade and Ogden (2005). Instead of treating these cases again, we revisit the case relative to one of the most important pre-strain fitting into the present context, that of finite simple shear, presented originally by Connor and Ogden (1995) for surface waves.
Figure 4 sketches what happens to a unit cube when a solid is subject to the simple shear of amount ,
Here the principal axes are and , which make an angle with and with , respectively. That angle, and the corresponding principal stretches are (e.g. Chadwick (1976)),
These relations highlight a major difference between this homogeneous pre-strain and the triaxial pre-stretch (31): here the orientation of the principal axes with respect to the plane interface changes as the magnitude of the pre-strain changes.
Governing equations for simple shear pre-strain
The deformation (65) is an example of plane strain. As we focus on two-partial incremental waves in this Section, we may take advantage of formulas established by Merodio and Ogden (2002) in a similar context. The components of the deformation gradient tensor and of the left Cauchy-Green strain tensor for 2D pre-strain and 2D incremental motions, in the () coordinate system (aligned with the () system), are
Then the 2D version of the constitutive equation (5) is
where is the Lagrange multiplier due to the incompressibility constraint. Also, the components of are (Merodio and Ogden, 2002),
where , . With the help of (67), we find that in the () coordinate system, the non-zero components relevant to in-plane motion are