Non-destructive determination of the stiffness matrix of a laminated composite structure with lamb wave

Composite Structures 237 (2020) 111956

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Non-destructive determination of the stiffness matrix of a laminated composite structure with lamb wave

T

C. Yilmaza,b,c, S. Topala,b,c, H.Q. Alia,b,c, I.E. Tabrizia,b,c, A. Al-Nadharia,b,c, A. Sulemand, ⁎ M. Yildiza,b,c, a

Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla, 34956 Istanbul, Turkey Integrated Manufacturing Technologies Research and Application Center, Sabanci University, Tuzla, 34956 Istanbul, Turkey c Composite Technologies Center of Excellence, Sabanci University-Kordsa, Istanbul Technology Development Zone, Sanayi Mah. Teknopark Blvd. No: 1/1B, Pendik, 34906 Istanbul, Turkey d Department of Mechanical Engineering, Center for Aerospace Research, University of Victoria, Victoria, BC V8W 3P6, Canada b

ARTICLE INFO

ABSTRACT

Keywords: Lamb wave A0 and S0 modes Thin laminated composites Non-destructive methods

This study presents a combined experimental and numerical study on the determination of the stiffness matrix of transversely isotropic thin laminated composite structures non-destructively with the use of group velocity measurements. The approach relies on the usage of A0 and S0 modes of Lamb waves, triggered the by pencil-lead break, which propagate in the composite material, and enables the evaluation of various elements of the stiffness matrix. It is proven that the experimental Lamb wave method can be reliably utilized for determining the components of the stiffness matrix, Cij . The results are validated with respect to those of mechanical characterization and finite element analysis.

1. Introduction Being able to determine mechanical properties of materials reliably in both cost- and time-effective manner plays a critical role in studying and understanding the overall mechanical performance of these materials. This need is more pronounced for composite structures given that they possess high level of anisotropy and non-homogeneity which affect their mechanical performance substantially. A vital part of the mechanical characterization of composite materials includes determining the elements of stiffness matrix, or elastic-constants, Cij (i,j: 1–6), which are required to establish the relationship between stress and strain in accordance with the generalized Hooke’s law. The elements of stiffness matrix are needed for multi-scale computational analyses and design of composite structures. The mechanical characterization of composite structures is predominantly dependent on destructive testing methods [1,2] despite the considerable amount of research on non-destructive testing [3] and the numerical analysis techniques which can determine the mechanical properties of composite structure from the mechanical properties of constituent materials namely, fiber and matrix. There has been active research on determining the mechanical properties of composite structures through non-destructive evaluation (NDE) methods thereby eliminating the need for time-consuming specimen preparation, and testing. ⁎

Immersion ultrasonic testing method is one of NDE techniques which involves the immersion of test materials in a water-filled tank equipped with a piezoelectric transmitter and a receiver. An ultrasonic wave excited by the transmitter propagates through water and specimen, and is then collected by the receiver. In this method, the velocity values of the ultrasonic wave can be measured to calculate the components of stiffness matrix of the composite material for a given direction. This technique was successfully utilized for the E-glass, Kevlar, Dyneema®[4] and graphite reinforced composites [5,6]. The same approach was also utilized for determining the elastic constants (Young’s modulus, bulk modulus and Poisson’s ratio) of thick glass fiber-reinforced composite laminates [7] and human cortival bone [8]. Kriz et.al. demonstrated that the complete set of the elastic constants of a transversely isotropic thick composite laminates of any fiber volume fraction can be achieved by measuring the ultrasound wave velocities in different directions without the water-filled tank by adhering the transmitter and receiver on the composite laminate [5]. Datta et al. obtained the transverse-isotropic elastic tensor for high strength/lowmodulus and low-strength/high-modulus graphite fibers. They also ultrasonically measured the complete elastic constants of a metal matrix with embedded uniaxial graphite fibers, and then they did an inversemodelling calculation to extract the fibre’s elastic constants [9]. Another group of researchers utilized pulse-echo technique to

Corresponding author at: Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla, 34956 Istanbul, Turkey. E-mail address: [email protected] (M. Yildiz).

https://doi.org/10.1016/j.compstruct.2020.111956 Received 14 October 2019; Received in revised form 6 January 2020; Accepted 17 January 2020 Available online 22 January 2020 0263-8223/ © 2020 Elsevier Ltd. All rights reserved.

Composite Structures 237 (2020) 111956

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determine the components of stiffness matrix without water-filled tank [10–12]. In this method, test materials are required to be isotropic, and a single sensor is employed to generate and capture the ultrasonic wave which propagates through the specimen and gets reflected from the rear edge of the material. The elastic constants of graphite single crystals, LiNbO3, LiTaO3, silicon and various polycrystalline materials were investigated [10]. This technique was also implemented on the transition metal-carbide ceramics as a function of temperature and hydrostatic pressure [11] and for establishing a correlation between Poisson’s ratio and longitudinal ultrasonic wave velocity in porous materials that are extremely brittle for mechanical testing [12]. In addition to immersion and pulse-echo techniques with ultrasound waves, Lamb wave method is a promising technique that can be implemented for Structural Health Monitoring (SHM), and NDE of laminated composite systems [13]. Discovered by Horace Lamb in 1917 [14], Lamb waves can exist in thin plates with parallel free boundaries [15]. Detailed reviews on the state of the art in the field of guided-wave SHM was provided in the literature [16–18]. Lamb waves have also been utilized for the damage investigation in metallic structures [19–21] where a propagating Lamb wave are used to identify the existence of a crack in the structures. Features of Lamb waves used for NDE and SHM include Time-ofFlight (ToF), mode conversion/generation, change in amplitude/attenuation, velocity, among others. Parametric studies regarding different frequencies and pulse widths in composite plates [22], frequency range and geometries of metallic foam sandwich panels [23] have been performed experimentally and numerically. Finite element based simulations for studying Lamb waves’ interactions with cracks of various shapes and sizes [24,25] and delaminations in composite plates [26–28], T-joints [29], aircraft structures [30] and pipes [31,32] have been studied. Additionally, the local interaction simulation approach (LISA) was developed for the analysis of Lamb waves interactions with geometric irregularities such as notches, holes, or damage sites (delamination) for complex (heterogeneous, anisotropic, attenuative and/or nonlinear) media [33]. These studies in literature have mainly focused on the utilization of Lamb waves generated in the form of sinusoidal (narrow-band) signals to detect the flaws (location, size, edge) experimentally and numerically in the structures rather than mechanical characterization of the undamaged materials. However, Lamb waves can also provide excellent possibilities for determining the stiffness matrix of composite materials non-destructively along fiber and out-of-plane directions due to their unique attribute such that depending on the frequency range, Lamb waves propagate with a number of symmetric and antisymmetric modes. These modes of Lamb waves can be excited with a Hsu-Nielsen (broad-band) source, also known as simple pencil-lead break (PLB). PLB consists of breaking a 0.3 mm-0.5 mm diameter pencil lead with a length of approximately 3 mm ( ± 0.5 mm) from its tip by pressing it against the surface of the piece to be examined. PLB generates an intense acoustic signal, quite similar to a natural acoustic emission source that the sensors detect as a strong burst. In this study, we propose to use lamb waves with PLB as an economical and time-efficient method for obtaining all the components of stiffness matrix of a unidirectional (UD) fiber reinforced thin laminated composite structure (LCS). Two modes of Lamb waves, zero-order symmetric (S0) and zero-order antisymmetric (A0) modes, are excited with a broad-band source in a transversely isotropic thin-LCS. The group velocity values of Lamb waves in different directions are measured and then used in Christoffel’s equation for determining components of stiffness matrix. Numerical analyses are carried out by using a commercial finite element analysis software, Abaqus, to verify the experimental results by simulating the Lamb wave propagation in a thin LCS.

strain is defined with generalized Hooke’s law as ij = cijkl kl where ij is the stress tensor, kl is the strain tensor, and cijkl is the fourth rank stiffness tensor for anisotropic materials. For a transversely isotropic material, the number of components of stiffness tensor reduces to nine with only five of them being independent due to the material symmetry [5]. Using Voight notation, the stiffness tensor for transversely isotropic material can be written in the form of symmetric 6x 6 stiffness matrix Cmn with the non-zero elements being C11, C12, C13, C22, C23, C33, C44, C55 , and C66 where C12 = C13, C22 = C33, C55 = C66 . The components of stiffness matrix, Cmn , can be calculated analytically using Eqs. (1), (2), (5), (8)–(10) based on the experimentally measured elastic constants.

C11 = E1 (1

(1)

23 32 )

C12 = E1 (

21

+

31 23)

= E2 (

12

+

32 13)

(2)

C13 = E1 (

31

+

21 32 )

= E3 (

13

+

12 23)

(3)

C22 = E2 (1

13 31)

(4)

C33 = E3 (1

12 21)

(5)

C44 = (C22

(6)

C23)/2 = G23

C55 = G13

(7)

C66 = G12

(8)

=

1 1

12 21

E2 = E3,

12

=

23 32

13,

31 13

2

21 32 13

G12 = G13

(9) (10)

where Ei is the Young’s modulus in the direction of i, ij is the Poisson’s ratio which corresponds to the contraction along the j-axis when an expansion is applied in i-axis. Also, Gij is the shear modulus in j-axis on the plane with its normal in the direction of i. Lamb waves are surface-guided waves propagating within a thin elastic plate in two fundamental modes, namely, symmetric (S) and anti-symmetric (A) with respect to mid-plane as schematically depicted in Fig. 1. They might have several orders (ie. zero order (S0, A0), first order (S1, A1), second order (S2, A2) etc…) for symmetric and antisymmetric modes depending on the frequency interval as shown in Fig. 2(a). Herein, due to the frequency interval of excitation load (Fig. 3(b)), only two zero-order modes (S0 and A0) exist in the lowband frequency region (0–400 kHz) as seen in Fig. 2(a). A representative Lamb wave excitation load used in this study is given in Fig. 3(a). The phase velocities of these modes show no sign of dispersion in this band unlike the other modes of Lamb wave. The phase velocity, Vp is related to the group velocity, Vg through the equation,

Vg =

Vp2

Vp

(fh)

dVp d (fh)

. Given that the phase velocity is not dispersive in the

range of 0 400 kHz, the term including the first order derivative of Vp with respect to frequency can be eliminated, leading to the condition of Vg = Vp as shown in Fig. 2(b). The relationship between the stiffness matrix, Cmn and ultrasonic group velocity for an anisotropic material is given by the Christoffel’s equation, Cmn = Vg2, d [34] where Vg , d is the ultrasonic group velocity measured in the direction of corresponding component of stiffness matrix, is the material density, and subscript d denotes the modes of

2. The relation between stiffness matrix and lamb wave Fig. 1. Modes of Lamb waves, symmetric and antisymmetric.

For linear-elastic materials, the relationship between stress and 2

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Fig. 2. Lamb wave dispersion curves, (a) phase velocity of the first four modes, and (b) group velocity of the zero order symmetric and antisymmetric modes, calculated by GUIGUW software [35].

ultrasonic wave. In this study, only the zero-order symmetric and antisymmetric Lamb wave modes are considered for the calculation of stiffness matrix. As the focus of this study is on a thin composite laminate with unidirectional fibers as shown in Fig. 4, it can be treated as a transversely isotropic material. Here, the axis x1 is defined along the direction of fibers in the composite, and all three axes are mutually perpendicular to each other. The relation between the zero-order symmetric Lamb wave mode and the components of stiffness matrix, C11 and C22 are given as follows [36]:

C11 = Vg2, s

(11)

C22 = Vg2, s

(12)

Fig. 4. Schematic of the Cartesian coordinate system defined with respect to fiber direction in the LCS.

3. Experimental study In this work, E-glass stitched UD fabric, purchased from Metyx (Turkey) with a trade code of L300 E10B-0, is used as a reinforcement for the composite plate. Araldite LY564 epoxy/XB3403 hardener system, purchased from Hunstman (USA), is used as the matrix material. A thin-composite plate of dimensions 600 × 300 × 1.8 mm is produced by using vacuum infusion method, with a cure cycle of 15 h at 80 °C. While the half of the plate is used for Lamb wave experiments, the other half is cut into specimens to obtain elastic constants (E1, E2, E3, 12, 13, 23, G12, G13 , and G23) and density as given in Table 1. Elastic constants given are determined performing tensile and Iosipescu shear tests following the procedure defined in ASTMD3039 and ASTMD5379, respectively where experimentally constructed stressstrain graphs are curve fitted between strain points of 1000 µ and 3000 µ . Further information can be found on the sample preparation and test procedures in the earlier studies of the authors [38,39]. Table 2 presents the components of stiffness matrix, Cij , analytically determined using Eqs. (1), (2), (5), (8)–(10) based on the experimentally measured elastic constants. Lamb waves are generated by the mechanical break of pencil lead with the size and hardness of 0.5 mm and 2H, respectively (also widely

where Vg , s is the group velocity of zero-order symmetric (S0 ) Lamb wave measured in the x1 and x2 directions, respectively. Next, C44 and C66 can also be calculated solely by zero-order antisymmetric Lamb wave mode and represented by the following equations:

C66 = Vg2, as

(13)

C44 = Vg2, as

(14)

where Vg , as is the group velocity of zero-order antisymmetric ( A0 ) Lamb wave mode measured in the x1 and x2 directions, respectively. For the last independent component, C12 , the following relation can be considered [37]:

Vg2, s =

{

1 C22 + C11 ± [(C22 2

1

}

C11) 2 + 4(C12 + C66)2]2 + C66

(15)

where Vg , s is the group velocity of zero-order symmetric (S0 ) wave propagating along 45o with respect to fiber direction in x1 x2 plane. Once components C11, C22, C44 and C66 are evaluated, C12 can be calculated.

Fig. 3. A representative load excitation signal (a) in time domain, (b) in frequency domain. 3

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propagates during each time step ( t ) should be smaller than the distance between two neighboring elements [15].

Table 1 Material properties of thin-LCS. E1 (GPa)

E2 (GPa)

E3 (GPa)

12

29

9

9

0.23

23

13

0.23

0.3

G12 (GPa)

G13 (GPa)

G23 (GPa)

(kg/m3)

3.4

3.4

2.5

1682

t<

C12 (GPa)

C13 (GPa)

C22 (GPa)

C23 (GPa)

C33 (GPa)

C44 (GPa)

C55 (GPa)

C66 (GPa)

30.4

3.1

3.1

10.2

5.2

10.2

2.5

3.4

3.4

(16)

fmax

where fmax is the max frequency of the load defined in simulation to excite the Lamb wave. In this study, fmax is selected as 150 kHz, which leads to a time step size of 3 × 10 7 s. A total time of 5 × 10 4 s is chosen for all simulations, being sufficient for observing the Lamb wave propagation in the composite structure. The element size should be small enough to capture the wavelength. Hence, Blakes criterion (Eq. (17)) is used for the element size [15] such that

Table 2 Analytically determined components of the stiffness matrix. C11 (GPa)

1 20

Lmax <

referred to as Hsu-Nielsen source), and detected by wide band (WD) piezo electric sensors (PICO 200–750 kHz Lightweight Miniature AE Sensor, Mistras) connected to Mistras PCI-2 acoustic emission (AE) setup. A Mistras 0/2/4 preamplifier with a gain of 20-dB is used between AE sensor and AE set-up to amplify the signal output by tenfold. As seen from Fig. 5(a)–(c), two AE sensors are attached to the top surface of the composite plate with hot-melt silicon adhesive. The distance between consecutive AE sensors is set to 50 mm. Lamb wave velocities in three different directions, namely, x1, x2 , and x are measured by placing the AE sensor pair parallel, perpendicular and at angle of 45o with respect to reinforcing glass fibers as shown in Fig. 5(a)–(c).

min

nmin

=

C nmin *fmax

(17)

where Lmax is the maximum element size, min is the minimum of the all the wavelengths, nmin is the number of elements per wavelength, which is selected as 10 for this work, and C is the dilatational wave speed (m/ s). The element size in x1 and x2 directions is 2 mm while being 1 mm in x3 direction. The general-purpose quadratic brick element C3D20R with reduced integration is chosen as the element type. All numerical simulations are performed in an eight-core processors workstation. For each simulation, parallelization of processors with GPGPU acceleration is utilized. Lamb waves are reported to undergo significant attenuation (reduction in its amplitude) even in a very short distance in viscoelastic media (i.e. cross-ply glass/epoxy laminate). Hence, during the finite element analysis of Lamb wave, the medium’s damping effect needs to be taken into account. To this end, we have used the Rayleigh damping model [40] in which the damping ratio of the material is related material constants, and , needed to be determined experimentally

4. Numerical simulation of the lamb wave Implicit solver of Abaqus finite element software is employed for the numerical simulation of Lamb wave propagation in a transversely isotropic LCS. For ensuring an accurate numerical simulation, appropriate element size and time step should be utilized. The numerical time is determined through referring to Courant-Friedrichs-Lewy (CFL) condition given in Eq. (16), which requires that the distance that Lamb wave

=

1 2

where

+ and

(18) are specifically mass and stiffness proportional

Fig. 5. Experimental procedure for measuring lamb wave velocity in three different directions, (a) parallel, (b) perpendicular, and (c) at an angle of 45° with respect to the fiber direction in LCS. 4

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constants, respectively, and is the circular frequency which is equal to 2 f , and f is taken to be the maximum frequency of the wave packet. As described in the experimental study section, two AE sensors with 50 mm apart, aligned along the fiber direction are placed on the top surface of the laminate and a stress wave is created with Hsu-Nielsen source. The stress wave propagates along the plate and hits sensors. Then, amplitudes of signal are measured to calculate constants ( and ), which are found to be 29369 (rad/s) and 1.1903 × 10 8 (s/rad) for the composite plate under investigation. For numerical simulation of lamb wave propagation in the composite structure, solid model of the composite plate with the dimensions of 600 × 300 × 1.8 mm is generated. During the simulation, the load excitation signal given in Fig. 3(a) is applied on a given node within a solid model. The second and third nodes away (15 and 65 mm, respectively) from the load application point are chosen as sensor points that are shown in Fig. 6(a). The solid model is composed of six individual plies having transversely isotropic symmetry. Each ply has a thickness of 0.3 mm and is reinforced by unidirectional fibers as shown in Fig. 6(b). After the solid model is created, elastic constants and density ( ) are assigned to each ply. Unlike the physical experiment, in numerical simulations, it is possible to generate the S0 and A0 modes of the Lamb wave separately in a composite plate. A0 mode is generated by loading the composite plate from its top surface while S0 mode is triggered by applying the point load from the top and bottom surfaces simultaneously. The numerical separation of both A0 and S0 modes from each other becomes very beneficial in understanding the behavior of these modes of the Lamb wave. To be able to calculate the components of stiffness matrix based on the numerical simulation of Lamb wave propagation in composite plates, in total, two sets of simulations are performed, namely, one set is for A0 mode whereas the other set is for S0 mode. Each set includes three different orientations of the sensor pair on the composite plate with respect to the unidirectional fiber, namely, parallel, perpendicular and at an angle of 45o configurations thereby enabling one to determine Lamb wave velocities along these directions for both A0 and S0 modes.

Fig. 6. (a) The model of the composite plate with load application and sensor points, (b) the stack plot of composite plate (red lines indicating the direction of fibers, and t the thickness of plies in m).

5

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5. Result and discussion

primary difference between these two Lamb wave modes is that the S0 mode is denser in the number of peaks and troughs than that of A0 mode for the given time interval. This can also be observed from Fig. 7 given that S0 exhibits obviously more packed waveform than the A0 mode. In the course of experimental process, when Hsu-Nielsen source is utilized on one side of the plate, both S0 and A0 modes of Lamb wave are triggered simultaneously. However, these modes propagate with different velocities as shown in Fig. 10, enabling one to differentiate them from one another for the calculation of components of the stiffness matrix.

5.1. Excitation of the lamb wave modes The propagation of Lamb wave in S0 and A0 modes on x1 x2 plane obtained from the numerical model can be viewed in Fig. 7 for various instances of time. Both modes propagate with an elliptical trajectory since the velocities of the Lamb wave vary in different directions due to the material anisotropy such that composite plate has significantly high elastic modulus along the fiber direction. It can be seen from Fig. 7 that the wavefront of S0 propagates much faster than that of A0. The through-thickness deformation of the composite plate due to the S0 and A0 modes of Lamb wave propagating in x1-direction can be viewed from Fig. 8 for different instants of time where the magnitude of deformation in x1 x3 plane is provided. It is clearly seen from figure that the deformation of composite plate by S0 Lamb wave mode along x3 -direction is symmetric whereas for the A0 mode, it is antisymmetric, indicating the reliability of numerical simulation. Fig. 9(a) and (b) give the representative Lamb wave forms for S0 and A0 modes, captured by a sensor pair aligned along x1-direction. The

5.2. Stiffness matrix of laminate The velocities of A0 and S0 modes of Lamb wave along x1, x2 and 45° directions with respect to reinforcing fibers are calculated using the simple relation, V = X / t where t and X is the time of flight of the wave and the distance (50 mm.) between two sensors, respectively and are tabulated in Table 3. Here, t is obtained from the difference in the arrival times of the same wave to the two consecutive sensors as

Fig. 9. Capturing of propagation Lamb waves by the sensor pair aligned along the x1 direction for (a) S0 mode and (b) A0 mode. 6

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observations are reported in literature [6]. Since the components of stiffness matrix are proportional to the square of velocities of Lamb waves in the corresponding directions, a small error in the velocity measurement can result in a significant deviation in the computed components of stiffness matrix. Inspecting the magnitude of the error regarding experimental data, the accumulation of error is again taken place due to the usage of more than one Cij components simultaneously, i.e., for the evaluation of C12 and C23 in Eqs. (15) and (5), respectively. However, the Lamb wave experiment with pencil-lead breaking still proves to be practical for the quick evaluation of the elements of the stiffness matrix. 6. Conclusion Fig. 10. S0 and A0 mode of Lamb wave propagating together and captured by two consecutive sensors with a separation distance of 50 mm.

This study presents a fast and inexpensive test method to determine components of the stiffness matrix of a transversely isotropic LCS nondestructively. In the numerical simulation and experimental part, S0 and A0 modes of Lamb waves are used due to their non-dispersive nature in a certain frequency interval, and the velocities of S0 and A0 modes of Lamb waves are measured in different directions for the evaluation of stiffness matrix components. We herein showed that these components (some with high accuracy) could be calculated by using a regular mechanical pencil-lead breakage as an external source and an AE set-up. This method eliminated the excessive destructive testing procedures to determine the whole set of stiffness matrix components for the thin composite laminates. In addition, the pencil-lead break proved to be an effective source to excite Lamb waves compared to a typical waveform generator and an amplifier. This alternative Lamb wave source further simplified the procedure with a reduction in the cost of the experimental set-up. Further, numerical evaluation of all components of the stiffness matrix was performed through FEA, and the results were compared with both the reference values from mechanical characterization steps and the experimental Lamb wave analysis. In this work, the distinguishable properties of S0 and A0 Lamb wave modes were successfully used in the FEA, allowing a clearer visualization of the wave propagation in the model. Two stiffness elements were obtained with relatively higher amount of error with respect to the reference values, due to an error accumulation arising from the use of other components in the corresponding formulations. Future work will include different contact and sensor types to eliminate the potential reasons for the deviation of both the experimental and the FEA results from the reference values.

Table 3 Velocity values (m/s) of S0 and A0 modes in three different directions.

FEA Experiment

Modes

x1

x2

at 45°

S0 A0 S0 A0

4208 1398 4051 1419

2311 1365 2584 1208

2486 – 2659 –

illustrated in Fig. 9(a) and (b). Table 4 provides the components of the stiffness matrix obtained using three different approaches. The second column includes the stiffness matrix components determined analytically using experimentally obtained elastic constants in Eqs. 1,2,5,8,9. The third and fourth columns lists the components calculated based on numerically and experimentally obtained A0 and S0 mode velocity values given in Table 3 using Eqs. (11)–(15). In Table 4, the percent difference between the components of the stiffness matrix determined using numerical and experimental Lamb wave propogation and mechanical characterization (‘reference’) is denoted with ‘ENum ’ and ‘EExp ’, respectively. A good agreement is achieved between the reference values and numerically obtained results except C23 and C44 showing a considerable amount of deviation (46% and 24% error). C44 corresponds to out-of-plane shear modulus G23, calculated by using the A0 mode of Lamb wave, which is relatively more dispersive than the S0 mode (Fig. 2(b)). The dispersive nature of the corresponding group velocity influences the computational result of C44 when implemented in Eq. (14). Therefore, the rise of the phase velocity of A0 mode in the low-frequency regime might be the underlying reason for this error. The mentioned numerical deviation in C44 affects the computed result of C23 , which is obtained by using Eq. (5). Similar

CRediT authorship contribution statement C. Yilmaz: Conceptualization, Methodology, Validation, Formal analysis, Writing - original draft. S. Topal: Formal analysis, Investigation, Writing - review & editing. H.Q. Ali: Investigation. I.E. Tabrizi: Investigation, Writing - review & editing. A. Al-Nadhari: Investigation, Writing - review & editing. A. Suleman: Supervision, Writing - review & editing. M. Yildiz: Supervision, Writing - review & editing.

Table 4 Components of the stiffness matrix determined using mechanical characterization, numerical and experimental lamb wave propagation for the composite plate used in this study. Method/Cij

C11 C12 C13 C22 C23 C33 C44 C55 C66

Mechanical Characterization

Lamb wave

Error (% )

Reference

Numerical

Experimental

Enum

Eexp

30.4 3.1 3.1 10.2 5.2 10.2 2.5 3.4 3.4

29.8 2.9 2.9 9 2.8 9 3.1 3.3 3.3

27.6 3.8 3.8 11.3 6.3 11.2 2.5 3.4 3.4

2 6.5 6.5 11.8 46.2 11.8 24 3 3

9.2 22.6 22.6 9.8 21.2 9.8 0 0 0

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors gratefully acknowledge the funding provided by The Scientific and Technological Research Council of Turkey (TUBITAK) to the first author under the 2214-A International Research Fellowship Program for Ph.D. students. 7

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