# Numerical Time-Domain Modeling of Lamb Wave Propagation Using Elastodynamic Finite Integration Technique

Authors: Hussein Rappel; Aghil Yousefi-Koma; Jalil Jamali; Ako Bahari
Journal: Shock and Vibration
 (2014)

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## Abstract

This paper presents a numerical model of lamb wave propagation in a homogenous steel plate using elastodynamic finite
integration technique (EFIT) as well as its validation with analytical results. Lamb wave method is a long range inspection technique which is considered to have unique future in the field of structural health monitoring. One of the main problems facing the lamb wave method is how to choose the most appropriate frequency to generate the waves for adequate transmission capable of properly propagating in the material, interfering with defects/damages, and being received in good conditions. Modern simulation tools based on numerical methods such as finite integration technique (FIT), finite element method (FEM), and boundary element method (BEM) may be used for modeling. In this paper, two sets of simulation are performed. In the first set, group velocities of lamb wave in a steel plate are obtained numerically. Results are then compared with analytical results to validate the simulation. In the second set, EFIT is employed to study fundamental symmetric mode interaction with a surface braking defect.


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## Body

## 1. Introduction

Lamb wave testing technique is increasingly used for assessing defects in thin-wall structures like plate and pipes [1–3]. Lamb waves are elastic waves whose wavelength is in the same order as thickness of the structure [4]. One of the main advantages of lamb wave technique is that it allows long-range inspection in contrast to traditional ultrasonic testing, where the coverage is limited to a small area in vicinity of each transducer. Lamb waves were first described theoretically by Horace Lamb in 1917 [5]. These waves arise from coupling between shear and longitudinal waves reflected at the top and bottom edges of a thin wall structure [6]. Lamb wave theory can be found in a number of text books [7]. Defects such as corrosion and fatigue cracks cause changes in effective thickness and local material properties and therefore measurement of variations in lamb wave propagation can be used to assess the integrity of plate [1]. Successful usage of lamb waves in an inspection system needs to understand its schemes of propagation in a waveguide and its scattering at defects. Thus, there is an increasing demand for powerful, flexible, and accurate simulation techniques. First works on numerical simulation of ultrasonic waves were done by Harumi (1986) and Yamawaki and Saito (1992) who calculated and visualized bulk wave propagation [8]. Now, numerical simulation of lamb waves is possible. Common techniques which are used to simulate lamb wave propagation are finite difference time domain (FDTD) [9], finite element method (FEM) [5], boundary element method (BEM) [10], elastodynamic finite integration technique (EFIT) [11, 12], and specialized methods for guided wave calculations such as hybrid methods [13] and semianalytical finite element method (SAFEM) [8].In this work, calculations are based on elastodynamic finite integration technique; historically, finite integration technique was introduced by Weiland in electrodynamics. Fellinger and Langenberg used Weiland’s idea for governing equations of ultrasonic waves in solid, calling it EFIT [14]. EFIT is a grid based numerical time-domain method, using velocity-stress formalism, and easily treats with different boundary conditions which are essential to model ultrasonic wave propagation [12]. Because of its relative simplicity and flexibility, Schubert et al. used EFIT equations to cylindrical coordinates (CEFIT) to simulate axisymmetric wave propagation in pipes with a 2D grid [15]. Schubert also used finite integration technique to simulate elastic wave propagation in porous concrete and showed efficiency of EFIT to model a diverse range of applications [16].Two sets of simulation results are presented in this work using a program developed in MATLAB environment. In the first one, lamb wave propagation in a 2D steel plate is discussed. Results are then compared with analytical results to validate the accuracy of modeling and, in the second example, interaction lamb wave with a surface breaking defect is investigated.


## 2. The Elastodynamic Finite Integration Technique for Linear Elastics

### 2.1. Governing Equations

The governing equations of elastic waves in a general media are the Cauchy equation of motion and equation of deformation rate. These equations are given in integral form for a finite volumeVand surface S as follows:
(1)∂∂t∫VρvidV=∫STijnidS+∫VfidV,(2)∂∂t∫VsijklTkldV=12∫S(vinj+vjni)dS,
where v is the particle velocity vector, T is stress tensor, ρ is density, n is the outward normal vector on surface S, f is the body force vector, and s is the compliance tensor. The inverse of s is the stiffness tensor c. Thus, using stiffness tensor, deformation rate equation can be expressed in another form. Consider
(3)∂∂t∫VTkldV=12∫Scklij(vinj+vjni)dS=∫ScklijvinjdS.In the case of isotropic materialc can be written as [17]
(4)cijkl=λδijδkl+μ(δikδjl+δilδjk),
where λ and μ are lame constants.


### 2.2. Spatial Discretized Form of Two Dimensional EFIT

Consider the Cartesian coordinate{x,y,z} and ultrasonic wave which propagates in two dimensional xz-plane. To apply FIT to (1) and (2), squares shown in Figure 1 are used as integral volume V, assuming constant v and T for each volume.Figure 1
Definition of integration cells for stress and velocity components. The geometry consists of four true material cells and four pseudomaterial cells [12].The final results for discretized form are(5)∂vx(x0,z0)∂t=1ρ[Txx(x0+(Δx/2),z0)-Txx(x0-(Δx/2),z0)Δx+Txz(x0,z0+(Δz/2))-Txz(x0,z0-(Δz/2))Δz+fxΔxΔzTxx(x0+(Δx/2),z0)-Txx(x0-(Δx/2),z0)Δx].A same manner of integration equation (1) about a vz integration cell centered at (x1,z1) results in
(6)∂vz(x1,z1)∂t=1ρ[Txz(x1+(Δx/2),z1)-Txz(x1-(Δx/2),z1)Δx+Tzz(x1,z1+(Δz/2))-Tzz(x1,z1-(Δz/2))Δz+fzΔxΔzTxz(x1+(Δx/2),z1)-Txz(x1-(Δx/2),z1)Δx].Now, using the normal stress equations, integration of (3) about Txx and Tzz centered at (x2,z2) yields
(7)∂Txx(x2,z2)∂t=(λ+2μ)[vx(x2+(Δx/2),z2)-vx(x2-(Δx/2).z2)Δx]+λ[vz(x2,z2+(Δz/2))-vz(x2,z2-(Δz/2))Δz],∂Tzz(x2,z2)∂t=(λ+2μ)[vz(x2,z2+(Δz/2))-vz(x2,z2-(Δz/2))Δz]+λ[vx(x2+(Δx/2),z2)-vz(x2-(Δx/2),z2)Δx].Finally, integration of (3) over Txz integration cell centered at (x3,z3) the intersection for material cells results in
(8)∂Txz(x3,z3)∂t=μ[vx(x3,z3+(Δz/2))-vx(x3,z3-(Δz/2))Δz+vz(x3+(Δx/2),z3)-vz(x3-(Δx/2),z3)Δx].As shown in Figure1, to simplify indexing into stress and velocity arrays of staggered grids when programming the numerics and to keep the same array sizes for all quantities, pseudomaterial cells are used. These cells have the same material properties as the true material they are added to but are not part of physical simulations.


### 2.3. Time Discretization

Central differences are used to discretize the equations in time domain which results in the velocity and stress components being staggered in time byΔt/2 [15]. Consider
(9)vn=vn-1+Δtv˙n-(1/2),Tn+(1/2)=Tn-(1/2)+ΔtT˙n,
where Δt is time interval, superscript n is integer number of time step, and dot {·} denotes the time differentiation.Equations (5)–(8) are solved at all points in simulation space and, by use of (9), the simulation proceeds in time in a “leap frogging” manner. A specific stability condition and adequate spatial resolution must be satisfied to guarantee EFIT convergence and accurate answers [15].


## 2.1. Governing Equations

The governing equations of elastic waves in a general media are the Cauchy equation of motion and equation of deformation rate. These equations are given in integral form for a finite volumeVand surface S as follows:
(1)∂∂t∫VρvidV=∫STijnidS+∫VfidV,(2)∂∂t∫VsijklTkldV=12∫S(vinj+vjni)dS,
where v is the particle velocity vector, T is stress tensor, ρ is density, n is the outward normal vector on surface S, f is the body force vector, and s is the compliance tensor. The inverse of s is the stiffness tensor c. Thus, using stiffness tensor, deformation rate equation can be expressed in another form. Consider
(3)∂∂t∫VTkldV=12∫Scklij(vinj+vjni)dS=∫ScklijvinjdS.In the case of isotropic materialc can be written as [17]
(4)cijkl=λδijδkl+μ(δikδjl+δilδjk),
where λ and μ are lame constants.


## 2.2. Spatial Discretized Form of Two Dimensional EFIT

Consider the Cartesian coordinate{x,y,z} and ultrasonic wave which propagates in two dimensional xz-plane. To apply FIT to (1) and (2), squares shown in Figure 1 are used as integral volume V, assuming constant v and T for each volume.Figure 1
Definition of integration cells for stress and velocity components. The geometry consists of four true material cells and four pseudomaterial cells [12].The final results for discretized form are(5)∂vx(x0,z0)∂t=1ρ[Txx(x0+(Δx/2),z0)-Txx(x0-(Δx/2),z0)Δx+Txz(x0,z0+(Δz/2))-Txz(x0,z0-(Δz/2))Δz+fxΔxΔzTxx(x0+(Δx/2),z0)-Txx(x0-(Δx/2),z0)Δx].A same manner of integration equation (1) about a vz integration cell centered at (x1,z1) results in
(6)∂vz(x1,z1)∂t=1ρ[Txz(x1+(Δx/2),z1)-Txz(x1-(Δx/2),z1)Δx+Tzz(x1,z1+(Δz/2))-Tzz(x1,z1-(Δz/2))Δz+fzΔxΔzTxz(x1+(Δx/2),z1)-Txz(x1-(Δx/2),z1)Δx].Now, using the normal stress equations, integration of (3) about Txx and Tzz centered at (x2,z2) yields
(7)∂Txx(x2,z2)∂t=(λ+2μ)[vx(x2+(Δx/2),z2)-vx(x2-(Δx/2).z2)Δx]+λ[vz(x2,z2+(Δz/2))-vz(x2,z2-(Δz/2))Δz],∂Tzz(x2,z2)∂t=(λ+2μ)[vz(x2,z2+(Δz/2))-vz(x2,z2-(Δz/2))Δz]+λ[vx(x2+(Δx/2),z2)-vz(x2-(Δx/2),z2)Δx].Finally, integration of (3) over Txz integration cell centered at (x3,z3) the intersection for material cells results in
(8)∂Txz(x3,z3)∂t=μ[vx(x3,z3+(Δz/2))-vx(x3,z3-(Δz/2))Δz+vz(x3+(Δx/2),z3)-vz(x3-(Δx/2),z3)Δx].As shown in Figure1, to simplify indexing into stress and velocity arrays of staggered grids when programming the numerics and to keep the same array sizes for all quantities, pseudomaterial cells are used. These cells have the same material properties as the true material they are added to but are not part of physical simulations.


## 2.3. Time Discretization

Central differences are used to discretize the equations in time domain which results in the velocity and stress components being staggered in time byΔt/2 [15]. Consider
(9)vn=vn-1+Δtv˙n-(1/2),Tn+(1/2)=Tn-(1/2)+ΔtT˙n,
where Δt is time interval, superscript n is integer number of time step, and dot {·} denotes the time differentiation.Equations (5)–(8) are solved at all points in simulation space and, by use of (9), the simulation proceeds in time in a “leap frogging” manner. A specific stability condition and adequate spatial resolution must be satisfied to guarantee EFIT convergence and accurate answers [15].


## 3. Propagation of Lamb Wave in a Steel Plate

In this part, the propagation of lamb wave in a steel plate is simulated using 2D-EFIT. The steel plate has the lengthl=300 mm and the thickness d=2 mm. Table 1 shows material properties used in this paper.Table 1
Material properties used for simulation.
Property
Value
Density
7700 (kg/m3)
Elastic modulus
195 Gpa
Lame constantλ
96.95 Gpa
Lame constantμ
76.17 GpaAs excitation source, point sources at top and bottom borders of plate are used. Figure2 shows location of applied loads.Steel sheet with lengthl=300 mm and thickness d=2 mm. (a) Excitation pattern for symmetric mode. (b) Excitation pattern for axisymmetric mode.
(a)
(b)Using excitation patterns shown in Figure2 and dispersion diagram for steel plate (Figure 3), single mode lamb wave is generated which makes signal interpretation easier.Dispersion diagram for a plate steel (a) phase velocity curve and (b) group velocity curve.
(a)
(b)Using 2D-EFIT code developed in MATLAB, propagation of lamb wave in the steel plate is simulated. To guarantee stability and accuracy of results,Δx and Δz are chosen 0.2 mm and Δt is 20 ns. The simulation results using EFIT-tool for symmetric and axisymmetric modes are presented in Figure 4, where the ultrasonic wave field in the plate at time T=80 μs is shown (excitation pulse is a raised cosine with five cycles with center frequency of 500 kHz).Lamb wave propagation in a steel plate at timeT=80 μs. The snapshot represents normal component of particle velocity (vz): (a) symmetric mode and (b) axisymmetric mode.
(a)
(b)As shown in Figure4, for the fundamental symmetric mode (S0), the lamb wave field is symmetric about half plane line and, for the fundamental axisymmetric mode (A0), normal component of particle velocity vz has the same value for every particle with same longitudinal position. From dispersion curve, we find that S0 travels faster than A0 which is validated by simulation results (see Figure 4).In order to check EFIT accuracy, group velocities obtained from simulation are compared with analytical results at both symmetric and axisymmetric modes (Figures5 and 6).Figure 5
Analytical group velocities comparison with 2D-EFIT results for fundamental axisymmetric mode.Figure 6
Analytical group velocities comparison with 2D-EFIT results for fundamental symmetric mode.Figures5–7 show good agreement for simulation results with analytical ones; also Figure 7 shows error dependence on frequency for axisymmetric mode is less than symmetric mode.Figure 7
Error comparison for symmetric and axisymmetric modes.


## 4. Reflection of the Fundamental Symmetric Mode (S0) from a Defect

In this section, interaction of theS0 mode with a defected steel plate is analyzed. The results presented here were used for a sizing study of rectangular surface braking defect with different depths and opening length 2 mm on a steel plate (Figure 8).Figure 8
Schematic of model used for studying interaction lamb wave with a defect in a steel plate.The same method used in the proceeding section is used to generate single mode with center frequency of500 kHz. However, as the lamb wave interacts with a defect, the axisymmetric mode will be generated. To study lamb wave interaction with a defect, the ratio of the maximum amplitudes for two modes RS0/RA0 is then calculated and compared at different depths (Figure 9).Figure 9
Ratio of reflection coefficients as function of crack depth.Figure10 shows the ultrasonic wave field in the defected plate at time T=60 μs; the defect depth is 0.4 mm. As shown in Figure 10, because symmetric modes travel faster than axisymmetric ones, mode separation happens after lamb wave interaction with defect.Figure 10
Lamb wave propagation in a steel plate with defect at timeT=60 μs. The snapshot represents normal component of particle velocity (vz(m/s)).


## 5. Conclusion

EFIT was used for studying lamb wave propagation in a steel plate using a program developed in MATLAB environment. Two sets of simulation results were presented in this paper. In the first example, group velocities of lamb wave for different frequencies were obtained using numerical signals and then the results were compared with analytical results; the comparison shows, for both fundamental symmetric and axisymmetric modes, the group velocity values are in good agreement with theoretical ones. In the second example, reflection ofS0 mode from a defect is studied and ratio of reflection coefficients was obtained as a function of crack depth which shows that as the crack depth increases the ratio RS0/RA0 increases. Each calculation presented in this paper was done on ordinary PC (Core i5, 2.4 GHz, 4 GB RAM).



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