Axial waves in rods

anti-symmetric modes. The examples of group velocity dispersion curves for Lamb modes as well as shear horizontal modes in a 5-mm steel plate (investigated next in Chapter 6) are given in Fig. 2.5.

Fig. 2.5. Group velocity dispersion curves for symmetric and antisymmetric Lamb modes and shear horizontal modes for a plate of thickness h = 5 mm, E = 205.35 GPa, ρ = 7872 kg/m³, ν ^{= 0.28}

Since the exact treatment of Lamb wave motion in structural elements is rather compli-cated, several models of rods, beams and plates are derived in the following sections. These models provide approximated description of wave motion. However, application of higher order theories ensures that obtained results cover with the exact guided Lamb modes.

2.2. Axial waves in rods 2.2.1. Elementary rod theory

The elementary wave theory for a thin rod assumes the presence of 1-D (one-dimensional) uniform axial stress only and neglects the lateral contraction (Doyle 1997).

Consider a rod of length L, axial stiffness EA and mass per unit length ρ where A A, denotes the cross-sectional area. The time-varying axial displacement of the rod is denoted as u x t_x( , ), where x is the spatial variable and t denotes the temporal variable. The axial strain corresponding to the deformation u x t_x( , ) is given by:

x xx

u ε =∂x

∂ . (2.11)

The kinetic energy T and the strain energy U are formulated as follows:

1 2

The governing equation can be derived using the Hamilton’s principle (cf. Achenbach 1975, Doyle 1977, Nowacki 1970):

( )

By substituting the energies (2.12) and (2.13), as well as the work of external forces (2.15) into the Hamilton’s principle (2.14), the governing equation can be written as:

2

To obtain the spectrum relation, the displacement is assumed to have the solution in the form:

( )

( , ) ˆ ^{i kx t}

x x

u x t =

∑

u e^{− −ω , (2.17)}

where the summation is over the angular frequency ω, k denotes the wavenumber, 1

i= − , and the amplitude spectrum ˆux is frequency dependent. Substitution of Eq. (2.17) into the homogeneous differential equation of motion (2.16) yields the characteristic equa-tion for determining k:

2 2

k = Eρ ω . (2.18)

Therefore, the spectrum relation for the elementary rod is given by the expression:

o

k c

= ω , (2.19)

where the velocity c_o = E ρ is called the thin-rod velocity. The dispersion relation for the elementary rod theory is:

p o

2.2. Axial waves in rods 19

Fig. 2.6. Spectrum relation for the elementary, Love and Mindlin-Herrmann rod theories (for a steel rod of cross-section A = 6×6 mm, E = 200.11 GPa, ρ = 7556 kg/m³, ν ^{= 0.33)}

Fig. 2.7. Dispersion relation for the elementary, Love and Mindlin-Herrmann rod theories (for a steel rod of cross-section A = 6×6 mm, E = 200.11 GPa, ρ = 7556 kg/m³, ν ^{= 0.33):}

a) in terms of phase velocity; b) in terms of group velocity

where the phase velocity c_p and the group velocity c_g are constant with respect to the frequency and equal to the thin-rod velocity c_o. Therefore, the result of the wave equation (2.16) is a non-dispersive signal, i.e. the signal that does not change shape as it propagates.

The plot of the wavenumber variation with the frequency, called the spectrum relation, is shown in Fig. 2.6. The dispersion relation, i.e. the plot of the wave velocity against the frequency, is presented in Fig. 2.7. The spectrum and dispersion relations are calculated for a rod investigated in Chapter 4. The results of the elementary rod theory are compared with the exact Lamb modes calculated from Eq. (2.4). In the frequency range 0–800 kHz, pre-sented in Fig. 2.6 and Fig. 2.7, there exist three Lamb symmetric modes (S₀,S₁,S₂), while the elementary rod reveals only one propagating mode S0. Moreover, the elementary theory coincides with the Lamb S₀ mode only at low frequencies.

2.2.2. Love rod theory

An improvement of the elementary one mode rod theory can be achieved by taking into consideration the effects of the lateral inertia. The rod not only deforms in longitu-dinal direction, but it also contracts due to the Poisson’s ratio effect. The transverse strain ε_t and the axial strain ε_xx are connected through the relation ε_t ^{= −}νε_xx^{, where} ν denotes the Poisson’s ratio. Such modified theory is called the Love theory after its investigator (Love 1920). In the Love theory, the strain energy is the same as for the elementary theory, given by Eq. (2.13), whereas the kinetic energy takes into account the component related with the lateral deformation (see Love 1920, Nowacki 1972, Doyle 1997): able parameter, introduced after Doyle (1997). Reasoning that the lateral deformation is represented not sufficiently accurately in the Love theory, the kinetic energy term associ-ated with transverse motion is modified by the parameter K . Thus the governing differ-_L ential equation becomes:

The spectrum relation for the Love theory is given by:

2 2 2

and it is nonlinear to the frequency ω , therefore, the Love theory is characterized by dispersive waves, i.e. waves for which the wave speed changes with frequency, what is illustrated in Fig. 2.6 and Fig. 2.7. The adjustable parameter K_L was set as 1.05 in this example. It was determined by the method of the least squares to give the best fit with the exact S0 Lamb mode in the frequency range 50–300 kHz. The one mode Love model can give a reasonable approximation for the S0 Lamb mode; however, it should be noted, that the Love theory is unable to coincide exactly with the first symmetric Lamb mode in such wide frequency range.

2.2. Axial waves in rods 21

2.2.3. Mindlin-Herrmann rod theory

The one mode Love rod theory takes into account contraction of a rod, but it retains the Poisson’s ratio relation between the axial and transverse strains. A more general ap-proach introduces the Mindlin-Herrmann theory (Mindlin and Herrmann 1952), in which the lateral contraction (ψ x,t is assumed to be independent of the axial deformation )

( )

The strains corresponding to the above deformations are:

x

The kinetic and strain energies are, respectively:

(

^{2 2}

)

and K₂^{M H−} are adjustable parameters set to compensate the approximate form of the dis-placement field (2.24) and they are associated with the lateral contraction energies. Differ-ent rules to establish the correction factors K₁^{M H−} and K₂^{M H−} were considered by Doyle (1997) and Martin et al. (1994), nevertheless due to approximate character of the consid-ered theory, neither approach can be judged more right than the other. Marin et al. (1994) proposed to select K₁^{M H−} and K₂^{M H−} based upon comparison with the 2-D finite element results. In studies, in which experimental investigations are performed, the parameters

1

KM H⁻ and K₂^{M H−} can be chosen to give the best correspondence with the experimental results in the considered frequency range (Rucka 2010a, 2010b).

The governing equations for the Mindlin-Herrmann rod theory follow from the Hamil-ton’s principle as:

Substitution of Eqs. (2.29) into the homogeneous differential equations of motion (2.28) results in the characteristic equation for determining the wavenumber k:

( )

The equation (2.30) is quadratic in respect of k , and therefore there are two propagating ² modes in the Mindlin-Herrmann rod theory. The first mode is characterized by decreasing speed with the frequency. The second mode appears above the cut-off frequency ω : _c

W dokumencie Guided wave propagation in structures : modelling, experimental studies and application to damage detection (Stron 18-23)