1

Modelling of in-plane wave propagation in a plate

using spectral element method and Kane-Mindlin

theory with application to damage detection*

Magdalena Rucka†

Department of Structural Mechanics and Bridge Structures,

Faculty of Civil and Environmental Engineering,

Gdansk University of Technology

Narutowicza 11/12, 80-233 Gdańsk, Poland

ABSTRACT. This paper presents results of experimental and numerical analyses of in-plane

waves propagating in a 5 mm-thick steel plate in the frequency range of 120-300 kHz. For

such a thickness-frequency ratio extensional waves reveal dispersive character. To model in-

plane wave propagation taking into account the thickness-stretch effect, a novel 2D spectral

element, based on the Kane-Mindlin theory, was formulated. An application of in-plane

waves to damage detection is also discussed. Experimental investigations employing a laser

vibrometer demonstrated that the position and length of a defect can precisely be identified by

analyzing reflected and diffracted waves.

Keywords: wave propagation; spectral element method; Kane and Mindlin theory; damage

detection

1. Introduction

Material damage is a potential threat to proper operation of civil, mechanical or

aerospace infrastructure. Various damage detection and structural health monitoring methods

have thus been investigated and developed to improve reliability and safety, and to solve

maintenance problems of engineering structures. Guided wave-based damage detection

methods are widely used in non-destructive testing and they have attracted many researchers’

interest [4, 7, 25, 26].

* Preprint submitted to Archive of Applied Mechanics February 3, 2011

† email: mrucka@pg.gda.pl

2

Modelling of wave propagation in structural elements and structures is a subject of

intensive investigations. One of methods is the frequency-based spectral finite element

method (SFEM) developed by Doyle [6] and then extended by Gopalakrishnan et al. [8] to

anisotropic media. A different approach can be found in Refs. [16] and [17], in which Lee and

Staszewski proposed the local interaction simulation approach (LISA) for wave propagation

in damage detection applications. Modelling of wave propagation can also be performed by

the finite element method (FEM) [1, 20]. The advantage of the finite element technique is the

availability of numerous commercial FEM codes (Ansys 5.3 [20] or ABAQUS EXPLICIT

[1]) and its great ability to analyse structures with complicated geometry. The use of the FEM

to model wave propagation requires very fine mesh; it is recommended to use more than 20

nodes per the shortest wavelength [20]. An expansion of the FEM is the time domain spectral

element method (SEM). The main idea of the SEM is the use of an interpolating polynomial

of high degree. In the SEM, Lagrange polynomials are applied at Gauss-Legendre-Lobbatto

nodes [3]. In comparison with the classical FEM, an important property of the SEM is that the

mass matrix is exactly diagonal, which allows to reduce significantly the algorithm cost [12].

Moreover, in the SEM, the required number of nodes per the shortest wavelength is of order

of 10 or even less.

In this paper, in-plane wave propagation in the form of a wave packet in a steel plate is

studied. In previous work, Żak et al. [27] presented an analysis of in-planes waves in terms of

the spectral element method based on the plane stress theory resulting in non-dispersive

waves. They considered an aluminium plate of thickness 1 mm and an excitation signal of

frequency 100 kHz. Within such frequency-thickness range, an extensional wave is practically

non-dispersive and the plane stress theory provides correct results. For many civil engineering

structures (for example, steel bridge plate girders) plates are thicker and dispersion occurs. To

solve this problem, Peng et al. [21] proposed a three-dimensional (3D) spectral element which

makes the analysis of dispersive waves possible. Such a 3D model is useful in analysing

structures of complicated geometries, but for dispersive wave propagation in plates, an

alternative solution could be a 2D plate model, which results in a substantial reduction of the

computational time in comparison with a 3D model. For flexural waves, the spectral finite

element based on the Mindlin plate theory was developed in Ref. [15], but there is no higher-

order plate spectral element for in-plane waves. One of refined plate theories is the Kane-

Mindlin extensional theory [11] and it is equivalent to the Mindlin-Herrmann rod theory [19],

for which a spectral element formulation was recently developed by Rucka [22, 23]. The

Kane-Mindlin theory includes the out-of-plane stress component and retains the simplicity of

3

the 2D model [14]. In earlier papers, the Kane-Mindlin theory was used in analytical studies

of static and dynamic fracture problems of cracked plates [9, 10, 13, 14]. Wang and Chang

[24] presented a study of plate waves scattered by a cylindrical inhomogeneity and compared

analytical solutions based on the Kane-Mindlin theory with experiments performed on a plate

of thickness 1.02 mm with an added mass. McKeon and Hinders [18] applied the Kane-

Mindlin theory to derive analytical solutions for the scattering of symmetric Lamb waves at a

circular inclusion.

This paper presents results of experimental and numerical analyses of in-plane waves

propagating in a 5 mm-thick steel plate in the frequency range of 120-300 kHz. For such a

thickness-frequency ratio, extensional waves reveal dispersive character. To model in-plane

wave propagation accounting for the thickness-stretch effect, a novel 2D spectral element,

based on the Kane-Mindlin theory, is formulated. Finally, an application of in-plane waves to

damage detection is discussed.

2. Formulation of Kane-Mindlin plate spectral finite element

2.1. Kane-Mindlin plate theory

In-plane wave propagation in plates governed by the equations of plane stress theory is

non-dispersive [7, 27]. An improvement of the plane stress theory can be achieved by

including the thickness-stretch effect. In 1956 Kane and Mindlin [11] developed a higher-

order plate theory taking into account coupling between extensional motion and the first mode

of thickness vibration.

Consider a plate lying in the xy plane bounded by planes z h . The components of

displacements (in-plane displacements u , v and out-of-plane displacement w ) in the Kane-

Mindlin theory are

( , , , ) ( , , ),

( , , , ) ( , , ),

( , , , ) ( , , ) ,

u x y z t u x y t

v x y z t v x y t

w x y z t w x y t z h

(1)

and strains corresponding to the above deformations are

, , , , , .x y z xy xz yz

u v w u v z w z w

x y h y x h x h y

(2)

The kinetic T and strain U energies are

4

2 2 212

(2 2 2 3) ,B

T hρu hρv hρw dxdy (3)

22 2

212

12

2 22 2

12

22 2 2 2 2

4 4 4

2 2 + 2 4 2

3 3

B

B

u vU h G h G G w dxdy

x y h

u v u vh w w dxdy

x y x y

u u v v hG w hG whG hG hG

y y x x x y

,B

dxdy

(4)

where and G are the Lamé constants. Substituting T and U into Hamilton’s principle, the

governing equations of the Kane-Mindlin theory can then be derived

2 2 2 2

2 2

2 2 2 2

2 2

2 2 2

2 2

2 2 2 2 2 2 2 ,

2 2 2 2 2 2 2 ,

2 2 2 22 2 2 ,

3 3 3

x

y

u v w v uh G h hG hG hu p

x x y x x y y

v u w u vh G h hG hG hv p

y x y y x y x

w w u vGh Gh G w hw

x y h x y

(5)

where xp and yp are the external forces in x and y directions, respectively. The constant

was inserted in the expression for the strain energy (4) to compensate the approximation of

displacement fields given in Eqs. (1). There are different rules to set up this parameter, but

due to approximate character of the Kane-Mindlin theory, neither approach can be considered

better than the other. Kane and Mindlin chose the value of 12 by equating the

frequency of pure thickness vibration obtained from the plate equation of motion with the

corresponding frequency obtained from three-dimensional equations [11]. In this study, in

which experimental investigations are reported, the constant was chosen to give the best

compatibility with the experimental data within the frequency range of interest.

2.2. Spectral element formulation

Formulation of the spectral element method is analogous to that of the classical finite

element method [2]. Following classical steps, the weak formulation is obtained in the form

0T T T

B B Bδ dxdy δ dxdy δ dxdy q μq ε Eε q p , (6)

5

where q and ε are the displacements and strains, q and ε are the corresponding virtual

displacements and virtual strains, p is the external force, E is the stress-strain matrix and μ

is the mass density matrix. In the SEM, the domain B can be approximated as a sum of eln

nonoverlaping elements, i.e., 1 ( )eln

e eB B . For the standard 0C elements, interpolation of

the displacement field ( , , )ξ tq in a typical finite element ( )eB is

( ) ( )( , , ) ( , , ) ( , ) ( )e et t t q q N q ,

1

2

( )

( )

( )( )

( )

e

n

t

tt

t

q

qq

q

,

( )

( ) ( )

( )

i

i i

i

u t

t v t

w t

q (7)

where the matrix of interpolation functions ( )( )e ξ,N is

( ) 1 2( ) ( ) ( ) ... ( )e nξ, ξ, ξ, ξ, N N N N , (8)

and the matrix of interpolation functions for a node i is

( , ) 0 0

( , ) 0 ( , ) 0

0 0 ( , )

i

i i

i

N

N

N

N , ( , ) ( ) ( )i p qN N N . (9)

In the above, tilde denotes the approximated quantity, ( , )iN are the Lagrange type

interpolation polynomials, [ 1, 1]ξ, is the parent domain, the index i ( 1,2,...,i n )

denotes nodal values and p qn n n is the number of element nodes, where pn denotes the

number of nodes in direction whereas qn in direction.

Strains in the Kane-Mindlin plate can be interpolated through the relation:

( ) ( )( , , ) ( , , ) ( , ) ( ),e et t t ε ε B q ( ) ( )( , ) ( , ).e e B DN (10)

In the above, the differential operator matrix D is given by

6

0 0

0 0

10 0

0

10 0

10 0

x

y

h

y x

h x

h y

D 1x=

y

J (11)

where the symbol 1J denotes the inverse of the Jacobian matrix.

Substitution of Eqs. Błąd! Nie można odnaleźć źródła odwołania. and (10) into Eq.

(6) provides that following set of equations which holds on the local element level

( ) ( ) ( ) ( ) ( )e e e e e M q K q p , (12)

where ( )K e and ( )M e are the element matrices, ( )ep is the load vector. To evaluate the

element matrices, numerical integration is employed, and the element matrices are integrated

using the Gauss-Lobatto-Legendre (GLL) quadrature [3]. The formulae for the stiffness

matrix, the mass matrix, and the load vector are

( ) ( ) ( )

1 1

, , det ,

p qn n

T

e p q e p q e p q p q

p q

w wK B EB J , (13)

( ) ( ) ( )

1 1

, , det ,

p qn n

T

e p q e p q e p q p q

p q

w wM N μN J , (14)

( ) ( )

1 1

, , det ,

p qn n

T

e p q e p q p q p q

p q

w wp N p J , (15)

where the stress-strain matrix E and the mass density matrix μ are

2

2 2 2 2 0 0 0

2 2 2 2 0 0 0

2 2 2 2 0 0 0

0 0 0 2 0 0

0 0 0 0 2 3 0

0 0 0 0 0 2 3

h G h

h h G

G h

Gh

Gh

Gh

E (16)

7

2 0 0

0 2 0

0 0 2 3

h

h

h

μ . (17)

In the GLL integration quadrature, the element nodes pξ and q are the same as the

integration points and they are obtained as the roots of the following equations [3]

2

1(1 ) ( ) 0pnξ P ξ dξ , 2

1(1 ) ( ) 0qnP d , (18)

where 1pnP and 1qnP are the Legendre polynomials of degree ( 1)pn and ( 1)qn , and the

associated weights pw and qw are

2

1

2

( 1) ( ( ))

p

p

p p n p

wn n P ξ

, 2

1

2

( 1) ( ( ))q

q

q q n q

wn n P

. (19)

The system of equations of motion is then built using standard aggregation of element

matrices and vectors

1 ( )eln

e eA K K , 1 ( )eln

e eA M M , 1 ( )eln

e eA p p , (20)

yielding the global equation of equilibrium Mq Kq p .

In the SEM approach, the element nodes are irregularly distributed (Fig. 1), in contrast

to the classical FEM with uniformly distributed element nodes. Due to the application of the

GLL rule, interpolation carried out over the GLL nodes leads to the diagonal mass matrix,

and temporal integration of the global equation of motion can thus be efficiently

conducted.

Fig. 1. An 81-node spectral finite element in the parent domain and selected shape function 25 ( , )N

8

3. In-plane wave propagation in a steel plate

3.1. Experimental setup

Wave propagation experiments were performed on a steel plate of dimensions

1000 mm 1000 mm and thickness 2 5 mmh (Fig. 2). The experimentally determined mass

density was found to be 7872 kg/m3. The modulus of elasticity E and the Poisson’s ratio

were also determined experimentally in a force-displacement test using two strain gauges

attached to the specimen of cross-section 20 mm 5 mm in both longitudinal and transverse

directions, and their values were identified as E = 205.35 GPa and = 0.28 GPa. The plate lied

on the flat surface and it was supported by four blocks of Plexiglas. The supporting blocks had

no influence on the registered signals. All four edges of the plate were free but no rigid

movement occurred since excited waves had low amplitudes. Two plates were taken into

investigations: the intact plate and the plate with damage. The rectangular defect of length 250

mm, width 12.5 mm and depth 2.5 mm, obtained by machine cutting, was introduced at the

position shown in Fig. 2a. Such a defect can represent corrosion damage which often occurs in

civil engineering structures subjected to environmental conditions.

Fig. 2. Experimental setup for measurements of in-plane waves in a steel plate: (a) geometry of tested plate and

measurement points; (b) photograph of hardware and the plate with damage

The experimental setup is shown in Fig. 2b. The piezoelectric (PZT) plate actuator

Noliac CMAP11 of dimensions 5 mm 5 mm 2 mm was bonded at the edge of the plate, at

x = 0 and y = 500 mm, to excite in-plane waves. The Tektronix function generator with the

amplifier created an excitation signal in the form of a five-peak sine modulated with a

Hanning window. The Hanning window provided smoothed tone burst in order to reduce

excitation of side frequencies [7]. Velocity signals were detected and registered in 17 points

evenly distributed along the left edge of the plate (Fig. 2a) by the scanning head PSV-I-400 of

the Polytec Scanning Laser Vibrometer PSV-3D-400-M.

9

3.2. Dispersion curves

Group velocity dispersion curves were experimentally determined for the intact plate.

A velocity signal was measured on the plate edge (at the same position as that of the source x

= 0 and y = 500 mm) for frequencies varying from 120 to 300 kHz with the increment of 10

kHz. Figure 3 presents examples of registered signals for frequencies 120, 200 and 250 kHz.

The excitation force applied normal to the plate edge resulted in propagation of both in-plane

waves: an extensional wave (P wave) and a shear horizontal wave (SH wave). Since the

measurements were made on the plate edge, a strong Rayleigh wave (R wave) was also

observed. In the measured signals, the first reflection was the non-dispersive Rayleigh wave

while the second reflection was the first symmetric S0 mode. Figure 3 reveals dispersive

character of the measured S0 mode. The SH wave was not directly registered on the plate

edge. Based on the time-of-flight, the group velocities of the S0 mode and R wave were

determined. The group velocity of the SH-wave SHc was then calculated through the relation

(1 ) / (0.87 1.12 )SH Rc c , where Rc is the group velocity of the Rayleigh wave.

Fig. 3. Time history of the experimentally measured in-plane waves in the intact plate for the determination of

dispersion curves: (a) 120 kHz; (b) 200 kHz; (c) 250 kHz

10

Fig. 4 shows experimental and analytical dispersion curves (for both the plane

stress theory and the Kane-Mindlin theory) for the considered 5 mm-thick steel plate. The

plane stress theory captures two modes: fundamental extensional mode and fundamental

shear horizontal mode. Shear horizontal mode is the SH0 mode (non-dispersive for

isotropic body), but the extensional mode approximates the S0 mode only at low

frequencies because it reveals non-dispersive character and in general the plane stress

theory cannot model the dispersion behaviour of the S0 mode properly. For the Kane-

Mindlin theory three modes exist: the first and the second extensional modes and the

fundamental SH0 mode. The two extensional modes of the Kane-Mindlin theory correctly

approximate the dispersion behaviour of S0 and S1 Lamb modes. The parameter in the

Kane-Mindlin theory was chosen to give the best fit to the experimentally measured wave group

velocity for the frequency range 120-300 kHz. It was determined by applying the method of least

squares and its values was set to 0.73 . Note in Fig. 4 that the Kane-Mindlin analytical

dispersion curve for the S0 mode fits the experimental data, moreover it agrees with the exact

Lamb mode in the selected frequency range 120-300 kHz.

Fig 4. Dispersion curves for the considered 5 mm steel plate: experimental results and analytical solutions for the

plane stress and the Kane-Mindlin theories

3.3. Numerical model of wave propagation for defect detection

Numerical modelling of in-plane wave propagation in plates was performed by

applying the time domain spectral element method. The plate was meshed to 80 80 spectral

finite elements, each element with 9 9 81 GLL nodes (Fig. 1). The defect was modelled

11

using 20 elements with height reduced by 2.5 mm. The highest frequency used in numerical

simulations was 250 kHz and for this frequency the applied mesh guaranteed 9.1 nodes per

the shortest wavelength. Damping was not considered in this model. Temporal integration

was performed using the Newmark scheme with the time step set at 85 10 st . This

algorithm uses accelerations as the primary variables and takes the advantage of the diagonal

structure of mass matrix [5].

In Fig. 5, experimental signals for in-plane wave propagation are compared with

numerical results for the plane stress and Kane-Mindlin theories. Amplitudes of

experimental and numerical signals were normalized to 1 and only signal envelopes were

plotted for clarity. The velocity signal 9 ( )v t was measured on the left edge of the plate at

the same position as the actuator, i.e., x = 0 and y = 500 mm (Fig. 2a). When the plane

stress theory is used, experimental data are not compatible with the numerical ones. It is

visible in Fig. 5 that reflections (from defect or from plate edge) of extensional waves in the

numerical signal are delayed with respect to reflection of extensional waves in the

experimental signal. Considering the Kane-Mindlin plate theory, we note that numerical

simulations are in good agreement with experimental data so that this theory guarantees

better approximation for the S0 mode than the plane stress theory.

Fig. 5. Propagation of in-plane wave of frequency 250 kHz – comparison between experimental and numerical

velocity signals: (a) intact plate; (b) plate with damage

12

3.4. Wave propagation in damaged plate

In this experiment, the wave packet was imposed along the x axis at the node 9,

whereas the velocity responses were measured at nodes 1 to 17 evenly distributed along the

left edge of the plate (Fig. 2a). The frequency of the incident wave was chosen as 250 kHz.

This frequency was found to be the most effective for the considered specimens and applied

instrumentation. As the reference state, the intact plate was first examined. Experimental and

numerical signals for this case are illustrated in Fig. 6 in the time and spatial domains. The

fronts of the S0 mode and Rayleigh waves were registered on the plate edge.

To visualize wave patterns occurring on the plate edge, a C-scan was performed. The

C-scan, based on the numerical velocity signals (Kane-Mindlin theory), provided a two-

dimensional xy plane view at the selected time instant t = 0.12 ms. The force applied

perpendicular to the plate edge resulted in propagation of the S0 and SH0 modes with

cylindrical fronts, as is indicated in Fig. 7. Moreover a Rayleigh surface wave (R wave) was

visible on the plate edge.

The second example concerned the plate with damage (Fig. 8). In both the numerical

and experimental results, the fronts of the S0 modes reflected form the damage were visible.

Two fronts of the S0 modes were caused by the 1st and 2nd reflection from the damage; they

are marked by solid lines in Fig. 8. The first reflection in the numerical signal occurred at the

time instant equal to 0.1149 ms. Knowing the plate geometry and the group velocity of the S0

mode (4973.17 m/s) localization of defect can be identified as 286 mm. In the case of

experimental signal, reflection occurred at the time instant equal to 0.1141 ms and the

velocity of the S0 mode was 5009.78 m/s, thus the identified position of damage was 286 mm.

Two additional wave fronts, marked with dashed line in Fig. 8, were caused by the S0 mode

diffraction at the defect ends. This S0 mode arose from a mode conversion upon interaction of

the SH0 mode with the defect, which is visible in Fig. 7b. The fronts of diffracted waves can

be used to estimate the length of the defect. Moreover, in the experimental signals, an

additional reflection appeared. It was the R wave reflected at the defect (dotted line in

Fig. 8b). This reflection was identified as coming from imperfect work of the equipment. The

amplifier created the additional wave packet (of amplitude about 0.01 of the incident wave) at

the moment of arriving the S0 mode reflected from defect. This wave packet created

propagation of an additional R wave, which provide the supplementary indicator of damage

existence.

13

Fig. 6. Propagation of in-plane wave of frequency 250 kHz in the time and spatial domains in the intact plate: (a)

numerical simulations based on Kane-Mindlin theory; (b) experimental results

14

Fig. 7. C-scan of numerical in-plane waves based on the Kane-Mindlin theory (at the time instant t = 0.12 ms):

(a) intact plate; (b) plate with damage

15

Fig. 8. Propagation of in-plane wave of frequency 250 kHz in the time and spatial domains in the plate with

damage: (a) numerical simulations based on Kane-Mindlin theory; (b) experimental results

16

4. Conclusions

In this paper, the spectral Kane-Mindlin finite element was successfully developed and

applied to numerical simulations of in-plane wave propagation in a steel plate. Application of

the Kane-Mindlin spectral finite element guarantees that the mass matrix is diagonal so that

the temporal integration can be efficiently performed. Moreover, higher order Kane-Mindlin

theory provides an accurate description of dispersive behaviour of the S0 mode which was

proved by the comparison with experimentally measured signals, and it also allows an

analysis of the S1 mode.

The detection of damage was considered by analyzing wave speeds and reflection

times in the recorded velocity signals. Modelling of in-plane wave propagation by the Kane-

Mindlin spectral finite element predicted proper times of reflections from damage so that the

numerical model used in structural health monitoring systems should employ the SEM

formulation based on the Kane-Mindlin theory. Measurements of time velocity signals in

several points (17 points in this study) provided information of wave propagation in time-

spatial plane. As a result, the interaction of waves with boundaries or potential discontinuities

could be observed more precisely. These experimental investigations demonstrated that

position and length of the defect could clearly be identified by the reflected and diffracted

waves.

Acknowledgments

This work was partially supported by the project POIG 01.01.02-10-106/09-00.

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