Dispersion Curves for Infinite Plate Using FEM

1

ASEN 5367

ADVANCED FINITE ELEMENT METHOD

Term Project

Dispersion Curves for Infinite Plate Using FEM

Presented For:

Prof. Carlos A Felippa

Done By:

Hussain AlQahtani (# 800552427)

Contents

1 /

Introduction

/ 3
2 /

Problem Statement

/ 5
3 /

Mathematical Formulations

/ 7
3.1 Displacement / 7
3.2 Strain / 8
3.3 Element Energies / 12
3.4 Hamilton Variational Principle / 13
3.5 Assemblage / 15
3.6 Fourier Transform / 15
3.7 Final Form / 15
4 /

Results and Discussions

/ 16
4.1Dispersion, frequency, and group velocity curves for Ni Plate / 17
4.2Dispersion, frequency, and group velocity curves for Silicon Nitride (Si3N4) Plate / 18
4.3Effects of Anisotropy / 20
5 /

Conclusions and Possible Future Work

/ 21

References

/ 23

Appendix

/ 24

1Introduction

The effect of a sharply applied, localized disturbance in a medium soon transmits or “spreads” to other parts of the medium. This simple fact forms a basis for study of the fascinating subject known as wave propagation. This phenomenon is familiar to everyone in forms such as the transmission of sound in air, the spreading of ripples on a pond of water, the transmission of seismic tremors in the earth, or the transmission of radio waves. These and many other examples could be cited to illustrate the propagation of waves through gaseous, liquid, and solid media and free space.

The physical basis for the propagation of a disturbance ultimately lies in the interaction of the discrete atoms of the solid. Investigations along such lines are more attuned to physics than mechanic, however. In solid and fluid mechanics, the medium is regarded as continuous, so that the properties such as density or elastic constants are considered to be continuous functions representing averages of microscopic quantities.

The practical applications of wave phenomena surely go back to the early history of man. The shaping of stone implements, for example, consists of striking sharp, carefully placed blows along the edges of a flint. Nowadays, the motivations for the current high level of interest in the subject are the many practical applications in science and industry. In the area of structures, for example the interest is mainly in the response to impact or blast loads. Another area in the study of structures involving wave phenomena is that of crack propagation or the interaction of dynamic stress fields with existing cracks, voids, or inclusions in a material. The field of ultrasonics represents another major area of application of wave phenomena. The general aspects of this area involve introducing a very low energy level, high-frequency stress pulse into a material and observing the subsequent propagation and reflection of this energy.

The general subject of waves in the earth covers many interesting propagation phenomena. Earthquakes generate waves that may travel thousands of miles. Study of the propagation of such waves artificially produced has provided the most knowledge on the interior construction of the earth.

As stated above, the phenomenon of waves can exist in almost all applications. In this project, however, a propagation of waves in plate is considered. More precisely, the dispersion curves and frequency spectrum will be computed for infinite plate.

In next section, the problem statement and assumptions will be introduced, followed by the mathematical formulation in section 3. In section 4, numerical results of dispersion and spectrum curves are introduced. The fifth, and last, section of this projects concerns some concluding remarks and possible future work.

2Problem Statement

In this project, an infinite, in x- and y-direction, plate is considered. The plate is assumed to be homogeneous and elastic. Dispersion, frequency, and group velocity curves are to be developed for such plate. Consequently, we consider time-harmonic motion in infinite plate. As shown in Figure 1, the plate consists of N of parallel, homogeneous, and anisotropic layer, which are perfectly bonded together along the length of the plate. A global rectangular coordinate system (x, y, z) is adopted such that x and y axes lie in the mid-plane of the plate, and z-axis parallel to the thickness direction of the plate. The thickness of the plate is discretized using three nodes, each of which has associated of it a displacement vector consisting of displacements in the three main directions, namely, u along x-axis, v along y-axis, and w along the z-axis as shown in Figure 1.

3Mathematical Formulations

3.1Displacement

The displacement at a point within an element is written as,

(1)

where N(z) are standard finite element shape functions:

(2)

(3)

where , and H is the thickness of the element.

The column vector is the nodal displacement:

(4)

and t denotes the time-dependence.

3.2Strain

Using linear elasticity, the strain tensor is derived as follows:

(5)

The derivative matrix can be split into three parts according to derivation indices as follows:

(6)

where the derivative matrices are given as follows:

(7) (8)

and,

(9)

Now, consider the firs term in Equation (6), and substitute for the displacement vector from Equation (1),

(10)

where

and,

In the preceding and following equations, the simplified notations will be used to indicate derivatives, namely, , , and .

Similarly, the second term in Equation (6) is written as:

(11)

where

and

The third term of Equation (6) is written as:

(12)

where,

Rewriting the strain in terms of B matrices, Equation (6) becomes:

(13)

3.3Element Energies

The element strain energy is defined as:

(14)

where h is the thickness of the element.

While the kinetic energy of an element is defined as:

(15)

3.4Hamilton Variational Principle

Hamilton’s principle is an integral principle, which means that it considers the entire motion of a system between t1 and t2. This principle maybe stated as follows:

Among all motions that will carry a conservative system from a given configuration at time t1 to a second given configuration at time t2, that which actually occurs provides a stationary value of the integral.

Mathematically, the principle may be written as,

(16)

Using Equations (14) and (15) in Equation (16), the principle is written as follows:

(17)

The preceding equation is equivalent to

(18)

By performing the integration over the element thickness, the final form of the Hamilton’s principle becomes:

(19)

Where the stiffness and mass matrices are:

The above integrals are calculated numerically using Gaussian integrationrule.

3.5Assemblage

Next, we assemble the element stiffness and mass matrices into global in the standard manner, and then apply variation to yield the equations of motion:

(20)

where Kij, M, u are the assembled stiffness matrices, the assembled mass matrix, and the column vectors of assembled nodal displacements, respectively.

3.6Fourier Transform

For a wave propagating in the xy-plane, we take the Fourier transform of the displacement as,

(21)

Applying this transformation to Equation (20) leads to the following expression,

(22)

where kx and ky are the wavenumbers in the x and y directions, and  is the circular frequency.

3.7Final Form

For a wave propagating in an arbitrary direction in the xy plane making an angle θ with the x-axis, we can decompose the wavenumbers as,

(23)

Consequently, Equation (22) can be written as,

(24)

Equation (24) is an eigenvalue problem. Solving this problem will determine the dispersion relation for guided waves in infinite plate.

4Results and Discussions

The above derivation is valid for any anisotropic plate. Two materials; isotropic and anisotropic are considered here; namely, Ni and Si3N4, respectively. Spectrum, dispersion and group velocity curves will be computed and plotted for these two materials.

Spectrum curves are the plot of the nondimensional frequencies (F or W=2 F) versus the nondimesional wavenubmers (K), while dispersion curves are the plot of nondimensional wave speed (C) versus either wavenumbers or frequencies.

Group velocity is the velocity of energy propagation, which is defined as . Group velocity curves are the plots of versus either K or W. Again, these curves will be computed and plotted for both materials.

4.1Dispersion, frequency, and group velocity curves for Ni

A Ni plate will be considered first. Table 1 gives the nonzero physical properties of Ni.

Table 1Nonzero Properties for Ni

(kg/m3) / C11
(N/m3) / C12
(N/m3) / C13
(N/m3) / C22
(N/m3) / C23
(N/m3) / C33
(N/m3) / C44
(N/m3) / C55
(N/m3) / C66
(N/m3)
8910 / 299109 / 130109 / 130109 / 299109 / 130109 / 299109 / 85109 / 85109 / 85109

Figure 2 shows spectrum and dispersion curves, respectively, for waveguide modes in the infinite homogenous isotropic Nickel plate. The wave propagation direction is chosen such that  is equal to zero. The plots are W vs. K and C vs. W for spectrum and dispersion curves, respectively. Note that the lowest modes propagate modes propagate at all frequencies, but the higher order modes have cutoff frequencies below which they are evanescent (nonpropagating).

Figure 3 shows group velocity curves for the Ni plate. The two plots are for group velocity versus frequencies and wavenumbers, respectively.

4.2Dispersion, frequency, and group velocity curves for Silicon Nitride (Si3N4)

Similar plots are developed for Si3N4 homogenous plate. Silicon nitride is a hexagonal material widely used MEMS applications. Nonzero physical properties are given in Table 2.

Table 2Nonzero Properties of Si3N4

(kg/m3) / C11
(N/m3) / C12
(N/m3) / C13
(N/m3) / C22
(N/m3) / C23
(N/m3) / C33
(N/m3) / C44
(N/m3) / C55
(N/m3) / C66
(N/m3)
3200 / 574109 / 127109 / 127109 / 433109 / 195109 / 433109 / 108109 / 119109 / 108109

Spectrum and dispersion curves for Silicon Nitride are shown in Figure 4. Similar to Ni plate, the propagation direction is chosen to be along the x-axis, i.e.  is equal to zero.

Group velocity curves for Silicon Nitride plate are shown in Figure 5. Similar to Ni plate, group velocities will be plotted versus K and W, respectively.

4.3Effects of Anisotropy

To see the effect of anisotropy in an infinite homogenous plate, spectrum curves are computed and compared along different directions. For this purpose, these curves are first computed and plotted for the isotropic Ni plate. The first plot in Figure 6 shows spectrum curves when the propagation direction is along the x-axis (=0), while the second plot shows them when the propagation direction makes a 30o angle with the x-axis. Since Ni is an isotropic material the two plots are identical as expected. See Figure 6.

Same curves are computed for the Si3N4 plate as illustrated in Figure 7. In this case, the effect of anisotropy is clear. Spectrum curves are noticeably different along the two propagation directions. This is expected since the Silicon Nitride is a hexagonal material.

5Conclusions and Possible Future Work

Spectrum and dispersion curves are computed for infinite homogenous plates using FEM. Two types of infinite plates were considered, namely Nickel and Silicon Nitride plates. As we have seen, FEM is a powerful tool for constructing dispersion and spectrum curves. Powerfully, FEM can deal with anisotropy very easily, all what we need to do is to modify the elastic stiffness matrix. The method presented here is quite general and can be used to compute curves for other types of plates. For example, curves for multilayered, inhomogeneous, finite width plates can be constructed using this analysis.

As an extension of this work, thermoelasticity will be considered next. This analysis should be modified to account for the thermal effect on dispersion and spectrum curves.

References:

1. Daniel Royer and Eugène Dieulesaint, Elastic Waves in Solids I Free and Guided Propagation, Springer-Verlag: Berlin (2000).

1. James F. Doyle, Wave Propagation in Structures, Springer-Verlag: Berlin (1997).

1. Karl F. Graff, Wave Motion in Elastic Solids, Oxford University Press. (1991).

1. O. M. Mukdadi, S. K. Datta, and M. L. Dunn, “Elastic Guided Waves in a Layered Plate With a Rectangular Cross Section,” Journal of Pressure Vessel Technology, 124, pp. 319-325, 2002.

1. O. M. Mukdadi, Y. M, S. K. Datta, A. H. Shah, and A. J. Niklasson, “Elastic Guided Waves in a Layered Plate With a Rectangular Cross Section,” J. Acoust. Soc. Am., 112(5), pp. 1766-1779, Nov. 2002.

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