Funciones específicas para usuarios registrados de GForge UCI

σij.j + ρfi = ρüi, 3 equations of motion (i = 1, 2, 3); (6.1) εij =¹₂

(

ui j^. +uj i^.

)

^, 6 independentstrain displacement equations ^; (6.2) σij = λεkkδij + 2μεij, 6 independent constitutive equations (isotropic materials). (6.3) The first two equations are valid for any continuous medium; the specific type of medium concerned is introduced via (6.3). If we eliminate the stress and strain factors from these equations, then we have

μui.jj + (λ + μ)uj.ji + ρfi = ρüi. (6.4) The equations of motion (6.4), which contain only the particle displacements, are the governing partial differential equations for displacement. If the domain in which a solution is sought is infinite, then these equations are sufficient. If the domain is finite, then boundary conditions are needed for a well-posed problem.

The boundary conditions take the form of prescribed tractions and/or displacements on the boundaries of the domain of interest. The general forms of such boundary conditions are as follows:

u(x, t) = u0(x, t) on surface displacements; (6.5) t_i = σjin_j. on surface tractions; (6.6) u(x, t) = u0(x, t) on S1

and ti = σjinj. on S2

as a mixed boundary condition. (6.7)

6.2 The Free Plate Problem

The geometry of the free plate problem is illustrated in Figure 6.4. This problem is governed by the equations of motion (6.4), with boundary conditions of type (6.6). The surfaces at the coordinates X₃ = d/2 = hand X₃ = –d/2 = –h are considered tractionfree.

Ultrasonic excitation occurs at some point in the plate; as ultrasonic energy from the excitation region encounters the upper and lower bounding surfaces of the plate, mode conversions occur (L wave to T wave, and vice versa). After some travel in the plate, superpositions cause the formation of “wave packets,” or what are commonly called guided wave modes in the plate. Based on entry angle and frequency used, we can predict how many different modes can be produced in the plate.

The exact solution of this problem has been obtained through the use of several different approaches. The most popular methods of solution are the displacement

x₃

x_I h

h

Figure 6.4. Geometry of the free plate problem.

potentials and the partial wave techniques (see Achenbach 1984 and Auld 1990, respectively).

6.2.1 Solution by the Method of Potentials

If the displacement vector (field) is decomposed according to Helmholtz decomposition and the result substituted into (6.4), as demonstrated previously, we obtain two uncoupled wave equations. For plane strain, these are

∂

, governing longitudinal waves ; (6.8)

∂

The case of plane strain is not the most general for the problem at hand, but the analysis is greatly simplified in this case. Achenbach (1984) shows that taking the general state of strain as a starting point results in the same set of solutions presented here plus some additional modes (infinitc in number), known as horizontal shear modes, that can exist independently of the other wave modes.

As a result of our assumption of plane strain, the displacements and stresses can be written in terms of the potentials as

u u

where λ and μ are Lamé constants.

We begin the analysis by assuming infinite plane harmonic wave solutions to (6.8) And (6.9) in the form

φ = Φ(x3) exp[i(kx₁ − ωt)]. (6.12) Ψ = Ψ(x3) exp[i(kx₁ − ωt)]. (6.13) Note that these solutions represent traveling waves in the x1 direction and standing waves in the x₃ direction. This is evident from the fact that, although there is a complex exponential term (hence sines and cosines) containing the time variable for the x₁ dependencies, there is only an unknown “static” function of x₃ for the x₃

81 6.2 The Free Plate Problem

dependence. This phenomenon is referred to in many texts as transverse resonance and is exploited in many ways to arrive at a solution. Again, these solutions represent waves that travel along the direction of the plate and that have fixed (as yet, unknown) distributions in the transverse directions.

Substitution of these assumed φ and Ψ solutions into (6.8) and (6.9) yields (6.10) and (6.11). Omitting the term exp[i(kx₁ − ωt)] to simplify all expressions, the results are as follows:

u ik d which are odd (resp. even) functions about x₃ = 0, we split the solution into two sets of modes: symmetric and antisymmetric modes. Specifically, for displacement in the x₁ direction, the motion will be symmetric (with respect to the midplane of the plate) if u1 contains cosines but will be antisymmetric if u1 contains sines. The reverse is true for displacements in the x₃ direction. Thus, we split the modes of wave propagation in the plate into two systems:

Symmetric modes

Antisymmetric modes the plate is symmetric for u and antisymmetric for w. On the other hand, for the antisymmetric modes, the wave structure across the thickness is symmetric for w and hence antisymmetric for u.

It should be noted that this separation of waves into symmetric and antisymmetric modes is an exception rather than a rule. In hollow cylinders, the lack of structure symmetry does not allow this separation. Plate wave modes do exist in an anisotropic plate, but the separation into symmetric and antisymmetric modes is not possible unless the wave propagates along a symmetry axis of the plate. (See Solie and Auld 1973 for an excellent discussion of plate waves in anisotropic plates.)

The constants A1, A2, B1, B2, as well as the dispersion equations, are still unknown.

They can be determined by applying the traction-free boundary condition, which reduces to

σ31 = σ33 ≡ 0 at x3 = ±d/2 = ±h (for convenience) (6.23) in the case of plane strain. The resulting displacement, stress, and strain fields depend upon the type of mode (i.e., symmetric or antisymmetric). However, applying the boundary conditions will give a homogeneous system of two equations for the appropriate two constants A₂, B₁ (for the symmetric case) and A₁, B₂ (antisymmetric case). For homogeneous equations we require that the determinant of the coefficient matrix vanish in order to ensure solutions other than the trivial one. From (6.23) we thus have

( )sin

After some manipulation, this may be rewritten as tan( )

The denominator on the RHS of (6.25) can be further simplified by using wave velocities and the definitions of p and q from (6.16). From the definition of cL, we obtain

λ=cL2ρ−2 . µ (6.26)

83 6.2 The Free Plate Problem

Using (6.16) and c_T² =µ ρ yields/

Now, substituting (6.30) into an initial form of the dispersion equation (6.25), we obtain a transcendental equation

tan( )

2 2 2 for symmetric modes (6.31)

Proceeding along analogous lines, we can show that tan( )

4 2 for antisymmetric modes (6.32)

Recall that p and q are as defined in (6.16).

For a given ω and derived k. the displacements can be calculated using the expressions for u and w in (6.21) and (6.22). More explicit expressions are given in Auld (1990).

These equations are known as the Rayleigh–Lamb frequency relations, and they were first derived at the end of the nineteenth century. These equations can be used to determine the velocity (or velocities) at which a wave of a particular frequency (fh or fd product) will propagate within the plate. Equations of this nature are known as dispersion relations. Although the equations look simple, they can be solved only by numerical methods.

6.2.2 The Partial Wave Technique

Although the method just presented is quite simple and elegant, its usefulness is restricted to isotropic plates: only then will the governing equations (6.4) be in such a simple form. For the problem of plate waves in anisotropic plates, the only suitable technique is the partial wave (or transverse resonance) technique. As pointed out in Solie and Auld (1973). the partial wave technique has two major advantages over the method of displacement potentials: (1) it leads more directly to wave solutions, and (2) it provides more insight into the physical nature of the waves.

Keep in mind that the formulation of the free plate problem has in no way changed: we are merely trying another solution method. In the partial wave technique, we try to construct solutions to the problem defined by (6.4) and (6.6) from simple exponential-type waves that reflect back and forth between the boundaries of the plate (see Figure 6.5).

We begin by assuming that each of the waves depicted in Figure 8.5 can be expressed as

u_j = aj exp [ik(x + lzz)], (6.33)

where j = x, y, z and lz = kz/kx (we are now using x, y, and z, instead of the xi, to denote position). Also, we are solving the more general problem, that is, with no assumption of plane strain. Note that the x component of each assumed partial wave is the same, which is exactly the statement of Snell’s law. This is studied in detail in Chapter 5.

Substituting these solutions in (6.33) into a form of Christoffel equations,

(kilcIjkJj − ρω²δij)uj = 0, (6.34)

(which is equivalent to using equation (6.4) after a plane wave solution is assumed) yields a linear homogeneous system of three equations in the three polarization components for each partial wave. The coefficients are functions of the material properties of the plate and also of the (unknown) phase velocity of the plate wave mode. Requiring the determinant to vanish for nontrivial solutions yields a sixth-order equation for lz that defines the propagation direction of the six partial waves (see Figure 6.5).

Now we know the direction of propagation for each of the partial waves. Hence we can take a linear combination of them in the form

uj Cn jn ik x l z j x y z thereby satisfy the traction-free boundary conditions (equation (6.6) evaluated at z

= ±h). Note that the traction-free condition must be satisfied along the entire upper and lower surfaces, so that the partial waves must reflect in such a manner that they reconstruct themselves after returning to the top of the plate. That is, they must be

(1)

(a) SH partial waves

(b) SV partial waves L waves

Figure 6.5. Types of partial waves used in the isotropic problem.

85 6.3 Numerical Solution of the Rayleigh–Lamb Frequency Equations

standing waves in the transverse direction (hence the term “transverse resonance”).

This explains why, when using the method of displacement potentials, we assumed that the potentials (6.12) and (6.13) had not only a static dependence on x₂ but were also allowed to propagate in the x1 direction.

The last step in our problem is to substitute this assumed linear combination of partial waves into the boundary condition equations. This gives a system of six (remember that we no longer assume plane strain, so to (6.23) we must add σxz = 0) homogeneous linear equations in which the coefficients Cn are now functions of the density, the elastic constants of the plate, and the product hk. Requiring the determinant of this “boundary condition matrix” to vanish (and thus yield nontrivial solutions for the wave amplitudes) gives us the dispersion relations that we seek.

Just as in the solution found by the method of potentials, an infinite number of modes are defined by our dispersion relations. In this case, however, we pick up the additional modes that were lost in that previous method by our assumption of plane strain. (We did not actually need to make that assumption in the method of potentials, but did so for simplicity.) The dispersion relations for these extra modes, known as shear horizontal (SH) modes, are

(Mπ)² = (ωh/cT)² − (kh)². (6.36) These modes are plotted in Figure 6.6. As can be seen, they are simple hyperbolas in a (kh, ωh)-plane. Complete details of a horizontal shear wave solution in a plate are presented in Chapter 15.

The dispersion relations governing the symmetric and antisymmetric “in-plane”

modes (known as Lamb waves) are of course the same as the ones resulting from the method of potentials, equations (6.31) and (6.32). Figure 6.7 shows a sample plot of the dispersion curves for these modes. As mentioned previously, the dispersion equations for the Lamb wave modes are simple in appearance but can be solved only by numerical methods.

6.3 Numerical Solution of the Rayleigh–Lamb Frequency Equations

In document Desarrollo Colaborativo de Software con la herramienta GForge. (página 34-38)