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Advances in Numerical Analysis
Volume 2012 (2012), Article ID 731591, 24 pages
Signorini Cylindrical Waves and Shannon Wavelets
Department of Mathematics, University of Salerno, 84084 Fisciano (SA), Italy
Received 25 February 2012; Accepted 16 April 2012
Academic Editor: Doron Levy
Copyright © 2012 Carlo Cattani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Hyperelastic materials based on Signorini’s strain energy density are studied by using Shannon wavelets. Cylindrical waves propagating in a nonlinear elastic material from the circular cylindrical cavity along the radius are analyzed in the following by focusing both on the main nonlinear effects and on the method of solution for the corresponding nonlinear differential equation. Cylindrical waves’ solution of the resulting equations can be easily represented in terms of this family of wavelets. It will be shown that Hankel functions can be linked with Shannon wavelets, so that wavelets can have some physical meaning being a good approximation of cylindrical waves. The nonlinearity is introduced by Signorini elastic energy density and corresponds to the quadratic nonlinearity relative to displacements. The configuration state of elastic medium is defined through cylindrical coordinates but the deformation is considered as functionally depending only on the radial coordinate. The physical and geometrical nonlinearities arising from the wave propagation are discussed from the point of view of wavelet analysis.
In this paper, cylindrical waves arising from the nonlinear equation of hyperelastic Signorini materials [1–6] are studied. In particular, it will be shown that cylindrical waves can be easily given in terms of Shannon wavelets.
Hyperelastic materials based on Signorini’s strain energy density [7, 8] were recently investigated [1–6, 9, 10], because of the simple form of the Signorini potential, which has the main advantage to be dependent only on three constants, including the two classical Lamé constants . Hyperelastic materials and composites are interesting for the many recent advances both in theoretical approaches and in practical discoveries of new composites, having extreme behaviors under deformation .
However, Signorini hyperelastic materials, as a drawback, lead to some nonlinear equations, to be studied in cylindrical coordinates [11–13]. The starting point, for searching the solution of these equations, is the Weber equation, which is classically solved by the special functions of Bessel type. Thus the main advantage of three parameters’ potential is counterbalanced by the Bessel function approximation. It has been recently shown  that Bessel functions locally coincide with Shannon wavelets, thus enabling us to represent cylindrical waves by the multiscale approach [9, 15, 16] of Shannon wavelets [14, 17–20]. In this way, Shannon wavelets might have some physical meaning through the cylindrical waves propagation.
In recent years wavelets have been successfully applied to the wavelet representation of integrodifferential operators [16–24], thus giving rise to the so-called wavelet solutions of PDE (see, e.g., [16, 21, 22]) and integral equations (see, e.g., [20, 23, 24]).
In fact, wavelets enjoy many interesting features such as the localization, the multiscale representation, and the fast decay to zero (either in space or in frequency domain), which are a useful tool in many different applications, (see, e.g., [17–20] and references therein).
We will see that Shannon wavelets can approximate very well the Bessel functions, thus being the most suitable tool for investigating cylindrical waves. Shannon wavelets are analytically defined functions, infinitely differentiable, and sharply bounded in the frequency domain. Their derivatives can be defined to any order by a simple analytical function [17–20], thus enabling us to approximate a function and its derivatives and easily performing the projection of differential operators.
This paper is organized as follows. Section 2 deals with some preliminary remarks on the elastic materials in generalized coordinates. In Section 3, Signorini density energy is defined and the basic equations in cylindrical coordinates for wave propagation in materials are given. The main properties of Shannon wavelets, reconstruction of a function, and connection coefficients are shortly described in Section 4. In Section 5 the similarities and distinctions between Bessel functions and Shannon wavelets are given. Section 6 deals with some remarks on perturbation method. In the same section the Shannon wavelet solution of the nonlinear wave propagation is given and the corresponding nonlinear effects are commented.
2. Preliminary Remarks
Let , be the (Lagrangian) cylindrical coordinate system , and with being the corresponding vector length and metric.
The Cauchy-Green strain tensor is defined as with being the displacement vector (in each point of the continuum).
The covariant derivatives of a vector are
and can be easily computed by means of the Christoffel’s symbols
Concerning the deformation, it can be classified according to the nonvanishing components of the displacement vector. We have cylindrical waves [4, 9, 10, 26–29] when When the components of the Cauchy-Green tensor are known, we can easily evaluate the three invariants:
which, in dealing with hyperelastic materials, enable us to compute the potential.
so that, by neglecting displacements of order higher than three, we have Signorini potential, which belongs to the polynomial hyperelastic model (also called generalized Rivlin model) [26–29], is defined as [1–9, 14] with
Therefore from the previous equation, by taking into account (2.10), we have the approximation
that isand, according to (2.8),
3. Cylindrical Waves Equation
Taking into account that
and, according to (2.11), it is
and, by (2.14), the only unvanishing derivatives are
In the strain components, we will neglect those terms with order higher than 3, so that the only unvanishing components of are By using (2.8),(2.10), and (3.6) we finally get the Kirchoff tensor in terms of displacements:
From (3.1) the only nontrivial equation is the first one:
where the coefficients depend on Signorini parameters :
In the following, we will search solutions in the following form:
where time-harmonic waves are separated by the longitudinal waves , so that
and is the solution of the following equation: with .
Equation (3.12) gives the more general model of cylindrical wave propagation for Signorini hyperelastic materials. At the r.h.s. there appear nonlinear terms up to the third order in while the coefficients depend on both inverse up to the 4th power and the physical parameters . In the following we will search the Shannon wavelet solution of (3.12), by neglecting terms in the r.h.s., by showing that Shannon wavelets are linked with Bessel functions.
3.1. Linear Equation
In fact, Bessel function of order is defined as the solution of the Weber equation:
In particular, when , the more general solution of
The Taylor series for Bessel function is with being gamma function.
So, for integer values of n, being , there result
It can be easily seen that are even functions and are odd functions, while both are localized functions with some decay to zero (Figure 1).
A good approximation of in the interval can be already obtained by the third-order polynomial, while with the 7th power polynomial we can have a good approximation in : Analogously a good approximation of is obtained in the interval by a second order polynomial, whereas with the 6th power polynomial we have a good approximation in
3.2. Second-Order Equation
Equation (3.12) gives rise to many interesting nonlinear equations for cylindrical waves. In fact, up to the second-order nonlinearities, it becomes
So, by keeping only the first term of the right-hand side, which is equivalent to neglect terms , we have
3.3. Third-Order Equation
Up to the third-order nonlinearities, and neglecting all terms , (3.12) gives We will give the solutions of (), (), and () by using Shannon wavelets. In order to do so, we need first to show that, in a sufficient large neighborhood of zero, Shannon wavelets are equivalent to the Bessel function. We can also see that at the same approximation the Taylor polynomial for Shannon wavelets is one order lower than the Taylor polynomial for the corresponding Bessel function, so that Shannon wavelets are more efficient from computational point of view.
4. Shannon Wavelet
Shannon scaling function and wavelet function are localized functions with some decay to zero (like Bessel functions), defined as The corresponding families of translated and dilated instances wavelet [17–20], on which is based the multiscale analysis, are with . In the following, we will denote
Both families of Shannon scaling and wavelet are -functions, with a slow decay to zero, so that
For each and , the inner product is defined as
where the bar stands for the complex conjugate.
with being the Kroenecker symbols.
Let be a function such that the integrals
exist and have finite values; it can be shown that the series
on the right side converges to . For a fixed upper bound we simply have the approximation (for the error estimate see )
4.1. Differentiable Properties of Shannon Wavelets
when , and , and with
According to (4.10) the Taylor series of the scaling and Shannon wavelet, nearby the origin, are
5. Similarities between Bessel Functions and Shannon Wavelets
Since Bessel functions are , they can be easily represented in terms of Shannon wavelets as follows: In particular, around they nearly coincide with the Shannon scaling and wavelet, so that the even can be well approximated by the scaling Shannon functions (Figure 2), while the odd Bessel functions can be approximated by the Shannon wavelets (Figure 3).
Although this approximation for both is restricted to an interval, we can assume that in the interval , where the perturbation method is applied, Bessel functions substantially coincide with the Shannon wavelet families; in other words, Shannon scaling functions and Shannon wavelets are solution of the Weber equation in the interval .
According to (4.13) and (4.15) the Taylor expansion (in ) for the scaling wavelet is so that at the sixth order Analogously, for the Shannon wavelet , in , it is up to the sixth order; By comparing the Taylor expansion for Bessel functions, as given by (), and () and the Taylor expansion of Shannon wavelets (5.3) and (5.4), we can see that a good approximation of the Bessel can be obtained by a lower-order polynomial approximation of the Shannon wavelet (Figures 4 and 5).
so that for it is
The series at the r.h.s is an alternating series which converges to zero, since, according to Leibniz rule, it is
Analogously, we can show the same result for the wavelet approximation (4.15)1 of the Bessel function .
By using the approximation (5.1) we can assume as solution of the Weber equation () the Shannon wavelet The derivatives of this function, according to (4.10), are and up to the second order, where, the explicit values of the connection coefficients are (4.13): The derivatives (5.10) have two components along two orthogonal functions, so that the projection with respect to gives
while, with respect to , we get
It can be easily shown by a direct computation that it is also
6. Perturbation Method
In order to compute the cylindrical waves solution of the nonlinear equations () and () we will consider the perturbation method . This method is based on the assumption that the solution of the nonlinear problem
with and being the linear and nonlinear parts of the differential operator, can be expressed as a converging series, which depends on a small parameter :
such that is the solution of the linear problem:
The other terms of the series are computed by solving the recursive set of (of linear) equations:
6.1. Second-Order Nonlinearity
where is the solution of the linear equation: When is known, is computed as the solution of Moreover as initial condition is taken, and the perturbation is on time so that the small parameter can be identified with time and the solution of () can be written as
The general solution of (6.8) implies some cumbersome hypergeometric series and Laguerre polynomials (see, e.g., ); however, it should be noticed that since the r.h.s. of (6.8) is obtained from (3.12), by neglecting all terms we can approximate also the l.h.s with the same hypotheses so that can be searched as solution of
By using the values of the connection coefficients (), for and the orthogonality property of wavelets, we have The solution of (6.12) is searched in the form
Equation (6.12) becomes
that is,The solution is
If we assume that at the initial time , the nonlinear effect is neglectable, in a such a way that so that
which simplifies the previous form of into
There follows that the general solution of (6.5) is
and the explicit solution of () becomes (see Figure 6)
As expected the evolution of the initial profile (Figure 6) shows the main nonlinear effect of large (increasing) amplitude. The initial profile is deformed by showing the increasing amplitude.
6.2. Third-Order Nonlinearity
Let us search the solution of the third-order nonlinear equation () (where ): The solution of (6.12) can be written as
where is given by (5.8). Inserting this wavelet function in the right-hand side of (6.23), with the same approximation as in the previous case, and taking into account (5.9) and (), the function will be obtained by solving
By taking into account the values of the connection coefficients (), we have The solution of (6.23) is searched in the form
Equation (6.27) becomes
So, by assuming the same hypotheses of the previous quadratic case, and with the same computations, we have
The general solution of (6.23) is