Non-Destructive Damage Assessment of Five Layers Fiber Glass / Polyester Composite Materials Laminated Plate by Using Lamb Waves Simulation

Composite materials are widely used in the engineered assets as aerospace structures, marine and air navigation owing to their high strength/weight ratios. Detection and identification of damage in the composite structures are considered as an important part of monitoring and repairing of structural systems during the service to avoid instantaneous failure. Effective cost and reliability are essential during the process of detecting. The Lamb wave method is an effective and sensitive technique to tiny damage and can be applied for structural health monitoring using low energy sensors; it can provide good information about the condition of the structure during its operation by analyzing the propagation of the wave in the plate. This paper presented the results and conclusions of a theoretical, numerical and experimental study of this method which was used for damage detection in the five layers fiberglass/polyester composite laminated plate, the fiber layer type is plain woven (0/90). The numerical analysis has been carried out by FEM using ABAQUS software for intact and cracked test specimen with the various boundary conditions and excitation frequencies, the results obtained have been confirmed experimentally by using piezoelectric wafer transducers (PZT) as actuator and sensor for both cases of experiments, it has been noted as a good agreement between experimental and numerical results.


INTRODUCTION
Failure of fiber-reinforced composite materials takes several forms; fibers breakage, fibers buckling, delamination and matrix yielding or cracking.Local Failure or micro-failure reduces the stiffness and strength of the composite.Micro-failure does not necessarily lead to sudden collapse because the surrounding matrix supports the fibers; the growth of failure during the operating time is serious in composite structures.Lamb waves at ultrasonic frequencies have the ability to propagate for a long distance in thin elastic plates with little attenuation and have high sensitivity to damage; it can be activated to propagate across the structure to demonstrate their use in pitchcatch behavior to monitor plate integrity.Guided waves (Lamb waves) can be used for nondestructive testing of aluminum aircraft panels, composite laminated plate and another plate-like object.The difficulty in applying Lamb waves includes the continuation of multiple modes and each of them highly dispersive, Graff, 1975 and Rose, 2014, which can make the interpretation of pulse-echo response difficult or impossible.In this paper, it was shown that the selective generation of the S0 wave conquered this difficulty; therefore, selection of pulse characteristics to achieve selective generation should be considered obligatory for this application.The modern use of single piezoelectric wafer transducers, surface-mounted on the plate, has attracted great interest, Kessler, et al., 2003 and Giurgiutiu, and Zagrai, 2000.Simplicity inherent in piezoelectric type wafer transformers is heartening for practical reasons and its low cost.Giurgiutiu, 2002, published a theoretical explanation for the mode selectivity of a PZT wafer transducer, with accompanying experiments, the results were good and important.Nieuwenhuis, et al., 2004, presented a study about using of finite element simulation and experiments to more explore the operation of the wafer transducer and Lamb wave generation for structural health monitoring.The wave velocities and the received voltage signals due to A0 and S0 modes at an output transducer as a function of pulse center frequency have been calculated.The results demonstrate that a piezoelectric wafer transducer can be used for selective excitation of the S0 mode.In this research, Lamb wave technique was used with a pitch-catch configuration to detect plate damage numerically and experimentally, with smart sensor and actuator type piezoelectric lead zirconate titanate (PZT) wafer transducer, with different boundary conditions.

LAMB WAVE THEORY
The theoretical part of Lamb wave propagation started with a summary of the beginning of the 2D equations in a flat plate, which has been followed by the method of Viktorov (1967).Lamb wave dispersion equations have been started from the wave equations in elastic isotropic medium Wilcox, 1998.

Propagation of Wave in Isotropic Elastic Medium
A Cartesian axes system in the x, y, and z-directions was defined in the materials being studied to discover the wave equation.When ρ is the density of material and with considering the equilibrium cube element of material and by using Hooke's law to link the relationship between stresses and strains, the differential equation for the displacement field has been deduced, Rose, 2014 as: Where u, v and w are the displacements in the x, y, and z-direction, λ and μ were the Lamé constants of the material, ƒ was the centrifugal force and t represented the time.This was the motion differential equations (Navier's-Lamés displacement equations) which referred to as the wave equations.In this research, the centrifugal force was considered zero ƒ = 0, Lammering, et al., 2017. (2) The solution of the wave equations has been considered here with waves propagating in a continuous homogeneous plane at x-direction.In those waves, every wave-front was an infinitely plane parallel with the y z plane, and the displacement field was independent of the y and zdirection.For this reason, a harmonic spatial reliance was described by exp(ikx), where k indicated to the wave-number and i = √− 1 .The solution can be in two forms, depending on the movement of the particles, parallel or vertical on the occurrence of the wave propagation, longitudinal and transverse waves, respectively, as shown in Fig. 1.The displacement field in the longitudinal wave is then given by: Where Ax is constant, ω is angular frequency.Substitution the Eq.(3) into Eq.( 1), the result is: Where   is the longitudinal velocity (phase velocity) it is wave crests propagate, in this case, the particle motion has been constrained in the direction of wave propagation.
The displacement field in the transverse wave is then given by: Where,   and   are constants, unlike the displacement field for longitudinal waves.Substitution the Eq. ( 5) into Eq.(1) result gives: Where ct the phase velocity of the propagating transverse wave, it referred to shear wave velocity because of the particle motion is at right angles to the direction of propagation.
Eq. ( 1) can be rewritten in potential function form to decompose the displacement field  ⃑ into two fields, associated with a longitudinal and transverse wave which is non-rotational and nondivergent, respectively.If the displacement field was the grad of scalar field∅, the vector showed the displacements field is non-rotational.And, if the displacement field was the curl vector field  ⃑ , the displacement field shown to be non-divergent.Then, the displacement field for any particle can be written, Wilcox, 1998, as:

Wave Propagation in Flat Plate
The plate with the Cartesian axis was defined as the (x, y) axes in the plane of the plate and the zaxis perpendicular to the plane of it.The plate was extending to infinitely in the x and y directions, the z-axis was located in midway through the plate thickness (h) as shown in Fig. 2. Hence, the boundary conditions which wave in the plate must satisfy has been given by: Lamb wave propagates in the x-direction only, this assumption has been considered.This means the wave-fronts are assumed to be infinitely in the y-direction as illustrated in schematic Fig. 2 that means the behaviors of the waves are not dependent on the y-direction, hence   = 0.The second condition, does not exist displacement in the y direction, hence the solution set include only actual Lamb wave and does not contain transverse wave (SH) that rotation about z axis, so v=0.Then there was rotation about axes in the y direction only, therefore the  ⃑ include components in the y direction (   ) only.The wave equations, Eq. ( 8) and ( 9), will be as Wilcox, 1998: Defining longitudinal and transverse wavenumbers, kl and kt respectively, as follows: Assuming a temporal harmonic dependence of exp(iωt) enables the wave equations to be further reduced to:

Solution with Symmetric and Anti -Symmetric Mode
The calculation is concerned with waves propagating along the plate, hence a new wavenumber (k) has been introduced in this direction.When the waves crests are straight, a spatial harmonic solution of the potentials functions governed by exp(ikx) was assumed (this was the difference between the solution for straight crests Lamb waves and that for circular crests Lamb waves), Wilcox, 1998.The potential functions satisfy to reduce wave equations, Eq. ( 15) and ( 16) as: Where A1, A2, B1, and B2 are four constants determined by the boundary conditions, There with the in-plane (u) and out-of-plane displacements (w) become: In Eq. ( 19) and Eq. ( 20) the field quantities contained sine and cosine terms with amount z, which are odd and even functions in z, respectively.So, the displacements u and w are divided into the symmetric and anti-symmetric part.If u comprises cosine terms, then the displacements motion during x-direction is symmetric concerning the mid-plane of the plate.So, the displacements motion of in-plane was anti-symmetric for sin terms of u.The same applies to the z-direction.
Based on the field of displacement u of wave propagation, can differentiate between a symmetrical and an anti-symmetrical mode, as shown in Fig. 3.
For the symmetric mode, one obtains: And the anti-symmetric waves are given as: To determine the constants Ai and Bi (for i = 1, 2) as well as the angular wavenumber k in Eq. ( 23) and Eq. ( 24) the stress boundary conditions are included at the surfaces of the infinitely extended plate.And as explained previously from the displacement field, the strain field may be calculated and finally, Hooke's law, And the anti-symmetric ones: On free surfaces, the stresses are: Resulting in the homogeneous equation system of the symmetric wave of Eq. ( 25): And the anti-symmetric wave of Eq. ( 26): From the previous equations, Eq. ( 27) and ( 28), and by finding the determinants of them, the formulae of determinate the symmetric and anti-symmetric angular wavenumber ksym and kanti with given frequency has been created, which are denoted to Lamb wave equations, Lammering, et al., 2017, as: The equation can be separated based on values of ksym or kanti as: for symmetric modes ( 30) This is the Lamb frequency equation for the propagation of symmetric anti-symmetric modes in a plate.

Dispersion Curves
The phase velocity cp for a given angular excitation frequency ω and the associated angular wavenumber k have been calculated by the following relation Wilcox, 1998: The phase velocity is the velocity at which the wave peaks of a continuous wave at a single frequency propagate, this leads to computing the dispersion curves.The phase velocity describes the velocity of wave peaks in a continuous single frequency wave, but a wave packet of finite duration and therefore of finite bandwidth will propagate at a different velocity.This is called the group velocity, cg , and is defined, Wilcox, 1998 and Lammering, et al., 2017, as : (33) Rearranging Eq. ( 33) yields an expression for the group velocity in terms of the phase velocity and the angular frequency.

Lamb Waves in Plate of Multiple Layers
The Lamb waves dispersion equation has been given above were exact to the state of the flat isotropic plate.Elastic wave in multi-layer as polyester composite was of a large interest, and this subject was intensively studied in latest years along with the exponential increase of applications of advanced composite materials in the various manufacturing sector.The orthotropic natures of multi-layer structure introduce many singular phenomena as directionally depending on wave velocity and the difference in group and phase velocities.
Considering a plate comprised of homogeneous layers as shown in Fig. 4, the propagation of Lamb waves inside the plate includes not only scattering on the upper and lower surfaces but reflection and refraction between layers.Expanding Eq. ( 1) to an N-layered laminate, the displacement field, (u, v and w) within each layer must satisfy the Navier's displacement equations, and for the n th layer, Rose, 2014: The equation has been described by the superscript for each layer.Once more, the Helmholtz decomposition was the majority efficient method to analyze displacement fields and to obtain the stress, strain, and displacement in each odd layer.The propagation of Lamb waves in multi-layered structures cannot be described analytically and requires the numerical approach.The software which was used for most dispersion calculations uses the alternative global matrix method, as it is numerically more stable at high-frequency thickness values, Demcenko, and Mazeika, 2010.
Figure 4. Geometry and the coordinate system of an N-layered plate, Rose, 2014.

LAMB WAVE SIMULATION USING ABAQUS SOFTWARE
In wave propagation analysis, the mesh and time increment size is highly dependent on the type of wave to be captured, the time increment size must be sufficient to capture the smallest period of interest and the element size should be small enough to capture wavelength but not smaller than one increment.Below are some important paragraphs which need clarification for the purpose of calculating the time increment size and element size and simulation of Lamb wave propagation using Abaqus software.

Elastic Wave Characteristics
Lamb wave propagation is dependent on the density and elastic modulus of a medium (Elastic wave propagation) with the longitudinal and transverse wave; the particle displacement is parallel and perpendicular to the direction of wave propagation respectively as shown in (2) into Eq.( 37), the result is:

Symmetric and Anti-Symmetric Mode
Typically, symmetric and anti-symmetric wave generated simultaneously for the general elastic medium is shown in Fig. 5. Since the existence of many modes in the response of high frequencies makes analysis difficult, isolating S0 and A0 from other modes (S1, A1, S2……) is recommended to make analysis possible.From material properties dispersion curves of Lamb wave are calculated and from phase velocity dispersion curve the operating frequency selected,  In this research, (f•d) product is less than 0.5 MHz.mm.S0 mode is the fastest wave as shown in dispersion curve in Fig. 6, is typically used as a signal for health monitoring.

Determining Element Edge Length
In order to capture propagation of Lamb wave, 10~20 elements per wavelength is a good mesh in order to show the form of wave properly.To choose proper element size first step is to calculate transverse wave speed.In this paper, the mechanical properties of the composite plate are shown in the Table 1.Therefore, typical (min ~ max) element size (  ) would be (3*10 -4 ~ 7*10 -4 ).

Frequenc y range For only So and Ao
The element size also can be calculated with longitudinal wave speed (  = 3153.2/).The impulse of excitement signal has a wavelength of (~2×Cl×dt) where dt indicates the impulse time increment size, it is associated with operating frequency.10 ~20 elements per wavelength is a good element size of mesh.

𝐶 𝐿
Table 2 shows parameters along excitation frequency.The lower frequency which can travel longer needs high magnitude to avoid elastic wave attenuation.Therefore, frequency and magnitude of excitation signal need to be carefully considered.Excitation can be strain/displacement and stress/load.If PZTs were used for excitatory, the stress will be used as an excitation signal.

EXPERIMENTAL LAMB WAVE ANALYSIS
The damage detection using Lamb wave can be classified into two types, passive (pulse-echo) and active (pitch-catch) system as shown in Fig. 7.In the active system, Lamb wave is excited into structures using actuator and then sensed back by the sensor with damage information at the other side of the wave's paths.In a passive configuration, both source and sensor are located on the same side of the aim (object), and the sensor receives the echoed wave's signals from the aim.In this experimental work, Lamb wave at ultrasonic frequencies passes through the thin elastic composite laminated plate to monitor plate integrity using pitch-catch configuration at different boundary conditions with and without a crack.Exciter and sensor include PZT wafer-type transducers.(8)-Rig.

Lamb Wave Excitation
In the experimental study and numerical simulation, the PZT actuator was excited by lamb wave signal, it was driven by a windowed sinusoid of the form: Where,  0 = 5 volt and  represents the angular frequency.The frequency () in the experimental test was varied from 10 to 50 kHz.The excitation was generated by computer control using LabVIEW.Fig. 10 illustrates the waveform of this excitation.Creation an analog signal with Lamb waveform is illustrated in Fig. 11 and Fig. 12 shows the response signal of the composite plate before and after filtration (Bessel filter).

LAMB WAVE PROPAGATION RESULTS
This section presents the results of numerical simulation and experimental procedures for Lamb wave propagation in a plate with and without a crack.

Numerical Simulation Results
The first set of results obtained from the FEM model created in Abaqus software is shown in Fig. 13.This figure illustrates the simulation of the Lamb wave's behavior and analyzes its travel across the composite laminate plate with and without crack, at (0~350) microsecond intervals and (100 kHz) with (CCCC) boundary condition.The results of these analyses were taken from the movie files which moved the vertical displacement of the nodes over time as the waves propagated from the loaded nodes.The FEM results did not show much of a change between the intact plate and defect plate for the time of travel, however, there was a notably reflected wave traveling from the crack region after the wave passed over it.The results prefer Black-White spectrum for wave propagation because easy to see its pattern.
The second set of results is Time-trace of the voltage signal from the PZT sensor; it is a convenient way to show differences between recorded signals, it is amplitude representation with (x, y) direction.From the results, it can be noticed that for every wave path that passes the crack region, the corresponding wave package will be delayed in phase because the S0 wave is very dispersive at low-frequency range, and slight loss of stiffness in the crack region leads to a marked change in phase velocity.Finally, the oscillations in the curve progression are a result of the interference of wave groups.

Experimental Results
The magnitude of the diagnostic Lamb wave signal and response signal of the composite laminated plate increases proportionately with an increase in the excitation voltage.Usually, activation of a PZT actuator at 1~10 V can create a response signal of 2~25 mV, with accompanying noise at the level of 0.2~5 mV, Zhongqing, and Lin, 2009.Fig. 15 illustrates the result of experimental verification for the response signal of the specimen when the excitation voltage was (1V).In this experimental work, the Lamb wave signals were excited with (5V) amplitude and various value of frequency.In this experimental work, the form of damaged plates has been investigated as a simple plate.The damage was a crack created in the composite laminated plate by hand using electric cutting tool with dimensions (20 mm length and 0.4 mm width) through whole plate thickness in the middle of the plate.The cutting tool was passed in the crack on the other side of the plate to adjust the edge and depth of the crack.The experimental results for time-trace of the voltage signal from the PZT sensor shown in Fig. 16 and Fig. 17, it illustrates the comparison between the intact and defect plate signals with (CCCC) and (CFFF) boundary condition, respectively.The comparison shows a clear difference between the two signals due to the presence of the crack, the reason was explained in the paragraph above.

Comparison between Numerical and Experimental Signal Results
During numerical and experimental testing, it was noticed that choosing the location of the sensor has a great effect on the possibility of a detection of crack.Therefore, to increase the probability of defect detection many testing measurements should be made at different sensor positions.In this numerical and experimental results one sensor was used and the crack wa created in the middle of the distance between the sensor and actuator.Fig. 18 illustrates the comparison between numerical and experimental results for the amplitude and travel times of response signal in the plate with and without crack with (CCCC) boundary condition.Good agreement was found between the signals at (1.3 ms) of data, the deviations between the results resulting from some possible measurement errors such as; noise effects, non-uniformity in the properties of the specimen, nonuniform surface finishing, and variations in thickness voids.From Fig. 18 a, the average amplitude is (0.0124 V) and (0.0154 V) for numerical and experimental signal respectively, and the error ratio is (19.48 %) and From Fig. 18 b, the average amplitude is (0.0151 V) and (0.0136 V) for numerical and experimental signal respectively, and the error ratio is (11.03%).This proves that the transmitter-receiver damage detection using Lamb wave is viable and implementable.

CONCLUSIONS
1.The study in this paper was focused only on the crack detection in the multi-layer composite plate [0 0 /90 0 ]5.However, other types of damage can be detected with the same analysis and experiments in composite or metal materials.This is possible in future 2. The operation of a PZT has been analyzed using a combination of finite element simulation and experiments to excite and detect the Lamb waves which were used to detect damage.3.For every wave path that passes through the crack region, the corresponding wave package will be delayed in phase because the S0 wave is very dispersive at low-frequency range, and slight loss of stiffness in the crack region leads to a marked change in phase velocity.4. The results show that Lamb wave propagation is a good technique for the identification of damage in the composite laminated plate with high accuracy; also it is sensitive to small damage.5. Through experimental tests and numerical simulations in this study, it was found that the PZT wafer transducer can be used for the eclectic irritation of the S0 mode that use to damage detection.6.As a future works in the field of composite laminated structures, can be taken into consideration studying the identification the size and location of damage in composite structures using Lamb waves with multiple actuators and sensors, also, more work latent in the future for different types of built-up structures, other than those tested during the current search, as joined sections and sandwich structures, etc.

Figure 2 .
Figure 2. Diagram of the flat isotropic plate, showing the orientation of axes, wave propagation direction and wavefronts, Wilcox, 1998.

Fig. 1 ,
Longitudinal and transverse wave speed depends on Young's and shear modulus, respectively.
Demcenko, and Mazeika, 2010 and Maghsoodi, et al., 2014, as shown in Fig. 6.The velocity of the Lamb wave mode is a function of (f•d) where f indicates the frequency and d indicates the plate thickness, expressed in MHz.mm.

Figure 7 .
Figure 7.The damage detection in active and passive systems.The testing model (composite laminated plate) with dimensions (285 mm long, 285 mm width and 3.05 mm thickness) was made up polyester reinforced by five-layer woven fiberglass [0 0 /90 0 ]5 by using hand lay-up process.Samples are made from manufactured composite laminated plate and tested to evaluate mechanical properties.4.1 Experimental SetupThe experimental set-up was established to obtain all necessary damage detection testing results.This set-up makes use of National Instruments LabVIEW software.A program was created to control the waveform type, frequency, and amplitude.The diagram view of the experimental setup as illustrated in Fig. 8.A function analog input (NI 9215), function analog output (NI 9263) and data acquisition (NI cDAQ-9178) selected from National Instruments (NI) is used for data.

Figure 12 .
Figure 12.Response signal for (CFFF) intact plate before and after filtration.
Fig. 14 illustrates the comparison of numerical results between the signals for the intact plate and defect plate with crack dimensions (20 mm long, 0.4 mm width and 3.05 mm thickness), the excitation signals with (5V) amplitude and (1 kHz) frequency with various boundary conditions.The response signals in two states show a clear change in amplitude and phase at S0 mode.Although there is a significant change caused by damage in signals amplitude, this study focuses only on delaying the time of the signal because the time of Lamb wave propagation can be measured more minutely in the experimental test, while the amplitude of the signals may vary due to the different bonding conditions of the PZT sensors.

Figure 18 .
Comparison between numerical and experimental results (amplitude and travel times) of the response signal at (1.3 ms) of data :(a) intact plate, (b) defect plate with crack with (CCCC).

Table 1 .
The elastic constants of the woven composite specimen.
3.4 Determining Step SizeAdequate integration time step is important for accuracy, typically 20 points/cycle of the highest frequency results is reasonable, deciding time step needs two criteria; Courant, Fredrichs and Lewy criterion andMoser criterion, Moser, et al., 1999.

Table 2 .
The parameters along excitation frequency.