Identification of material properties of composite

Mechanics of Advanced Materials and Structures

ISSN: 1537-6494 (Print) 1537-6532 (Online) Journal homepage: http://www.tandfonline.com/loi/umcm20

Identification of material properties of composite materials using nondestructive vibrational evaluation approaches: A review Jun Hui Tam, Zhi Chao Ong, Zubaidah Ismail, Bee Chin Ang & Shin Yee Khoo To cite this article: Jun Hui Tam, Zhi Chao Ong, Zubaidah Ismail, Bee Chin Ang & Shin Yee Khoo (2017) Identification of material properties of composite materials using nondestructive vibrational evaluation approaches: A review, Mechanics of Advanced Materials and Structures, 24:12, 971-986, DOI: 10.1080/15376494.2016.1196798 To link to this article: https://doi.org/10.1080/15376494.2016.1196798

Accepted author version posted online: 09 Jun 2016. Published online: 16 Dec 2016. Submit your article to this journal

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Date: 12 December 2017, At: 05:01

MECHANICS OF ADVANCED MATERIALS AND STRUCTURES , VOL. , NO. , – http://dx.doi.org/./..

ORIGINAL ARTICLE

Identification of material properties of composite materials using nondestructive vibrational evaluation approaches: A review Jun Hui Tama , Zhi Chao Onga , Zubaidah Ismailb , Bee Chin Anga , and Shin Yee Khooa Department of Mechanical Engineering, University of Malaya, Kuala Lumpur, Malaysia; b Department of Civil Engineering, University of Malaya, Kuala Lumpur, Malaysia

Downloaded by [University of Malaya] at 05:01 12 December 2017

a

ABSTRACT

ARTICLE HISTORY

Destructive identification approaches are no longer in favor since the advent of nondestructive evaluation approaches, as they are accurate, rapid, and cheap. Researchers are devoted to improving the accuracy, rate of convergence, and cost of such approaches, which depend greatly on the types of vibrational experiments conducted and the types of forward and inverse methods used in numerical section. Therefore, this article presents a review on the development of nondestructive vibrational evaluation approaches in identifying the elastic constants of composite plates, in experimental and numerical manners in order to enlighten researchers with the current trends of nondestructive vibrational approaches.

Received  January  Accepted  May 

1. Introduction Composite materials can be addressed as engineered materials, which are made of two or more materials. Undeniably, in today’s market the demand for composite materials is overwhelming in various industries due to their superior physical and chemical properties, e.g., light weight, high strength, and high corrosion resistance. Consequently, the invention of various types of composite materials is becoming prevalent, and thus, leading to the development of properties identification methods for composite materials. Due to significant impact of composite materials to the market, researchers have been placing emphasis on improving methodologies for elastic characterization of composite materials. Based on Figure 1, generally, the determination of elastic constants of composite materials can be sorted into two main parts, which are destructive technique and nondestructive techniques. Destructive techniques can be classified as classical static approaches that involves static mechanical tests, such as tensile tests, compression tests, bending tests, torsion tests, etc., in order to acquire the stresses and strains of a specimen. Direct identification of elastic constants of composite materials can be done based on the fundamental stress-strain theory. For composite materials, the procedures are more cumbersome and timeconsuming due to the need of several specimens’ analyses. Meanwhile, nondestructive techniques involve two parts, which are the experimental part and the numerical part. In the experimental part, measurements of significant parameters and data extraction will be conducted for subsequent use in the numerical part, while the numerical part involves the use of forward as well as inverse methods for evaluation of elastic properties of composite material. There are two categories in the experimental part, namely, static approaches and dynamic approaches. There are some CONTACT Zhi Chao Ong ©  Taylor & Francis Group, LLC

[email protected]

KEYWORDS

Composites; mechanical properties; nondestructive evaluation; plates; vibration

cases, where static tests are carried out in a nondestructive manner. These tests involve transverse quasi-static loadings on a specimen, in which the induced strains must not exceed 0.5% so that the elasticity of the specimen can be maintained; thus, the deflection can fully recover at the end of the test [1]. The boundary conditions of the plate specimen are usually either simply supported or clamped and the deformational parameters, such as displacements and strains, are acquired without damaging the specimen [1–6]. Unlike the destructive classical static approach, this approach requires the involvement of numerical evaluation in order to determine the elastic properties. The details of the nondestructive static approach procedure and experimental setup were presented in [2] as shown in Figure 2. Similarly, the dynamic approach is nondestructive as well. Basically, there are two types of dynamic approaches, which include wave propagation methods and vibrational methods. Wave propagation methods usually adopt the application of an ultrasonic wave passing through a specimen, where the wave signal velocity and transit time of flight of the wave from the emitting transducer to the receiving transducer will be taken into measurement. The emitting and receiving transducers used in most studies are usually piezoelectric transducers due to their cheap price and light weight. In wave propagation methods, there are bulk-wave-based methods [7–33] and guided-wavebased methods [34–52]. Bulk-wave-based methods are usually used in material properties identification of thick composite materials via through-transmission or back-reflection techniques. Meanwhile, in guided-wave-based methods, Lamb waves, also known as guided waves, are usually generated and remain guided between two parallel free surfaces of a plate or shell. Lamb-wave-based methods are normally the preferred choice for thin-plate analysis. However, in general, due to several disadvantages of wave propagation methods, such as complex dispersive characteristics of waves, the formation of

Department of Mechanical Engineering, University of Malaya, Kuala Lumpur , Malaysia.


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Figure . Overview of composite material properties identification methodologies.

several waveforms in single frequency waves, complex procedures, and the need of active power, vibrational methods appear to be the best alternatives in material identification to eliminate those drawbacks. In vibrational methods, external excitations on the specimens are needed and modal parameters, such as natural frequencies, modal damping, and the mode shapes of the specimen, are extracted from the obtained frequency response functions. Normally, the natural frequencies of the specimen are taken as primary parameters in determining the elastic properties of a composite material. In the numerical part, direct evaluation can be referred to as the direct identification of elastic properties of a material from a derived inverse equation with the experimental resonant frequencies served as the inputs. Meanwhile, nondirect evaluation

Figure . Distributed loading and point loading static test. Reprinted from [], © , with permission from Elsevier.

involves iterations as well as the minimization or maximization of an objective function. As shown in Figure 3, the nondirect evaluation of elastic properties of composite materials involves both forward methods and inverse methods. In forward methods, constitutive parameters, such as natural frequencies, mode shapes, etc., of a specimen are evaluated using inputs of elastic properties of the material depending on respective identification approaches. In inverse methods, minimization of error function based on the difference between experimental and evaluated constitutive parameters are often used to predict the elastic properties and optimization methods are used as catalysts to boost the solution searching process. Generally, two types of nondirect evaluation methods are reviewed in the present article, namely, derivative-based optimization approaches


MECHANICS OF ADVANCED MATERIALS AND STRUCTURES

Figure . Flow chart of numerical evaluation of elastic properties.

(e.g., feasible directions method, Bayesian estimation method, Newton’s method, and nonlinear least squares method) and meta-heuristic optimization approaches (e.g., genetic algorithm and response surface methodology). In a derivative-based optimization approach, the procedures are as follows: Initially, the number of iterations, improvement error, and the minimum value of error function are specified. The benchmark or experimental modal parameters (natural frequencies, mode shapes, or/and damping properties) are specified as well in the error function. To execute the algorithm, a set of elastic parameters need to be assigned. The algorithm begins with the forward evaluation of modal parameters from the given set of elastic properties. The error between benchmark and evaluated modal parameters is evaluated through the use of the established error function, followed by the computation of derivative of the evaluated modal parameters with respect to each elastic parameter. Thereafter, the step vector is evaluated from the derived inverse algorithm, involving the use of evaluated derivative matrix and error function values. This computed step vector signifies the difference between the assigned elastic properties and the actual elastic properties that indirectly decides the error between evaluated and benchmark modal parameters. The next iteration begins with the addition of the step vector to the previous assigned elastic parameters. The iterations are carried on until either of the termination criteria is satisfied. The termination criteria include the minimum value of error function, the maximum number of iteration, and the minimum value of evaluated elastic properties improvement. The elastic properties are eventually identified once the running algorithm is terminated. A meta-heuristic optimization approach does not involve any derivative evaluation. There are slight differences in the procedures if compared to the former approach. Instead of assigning only a set of elastic parameters at once, specific number of populations and boundaries of search region need to initially be defined. For the first iteration, sets of elastic parameters are randomly picked within the specified search region. In this approach, minimization or maximization of the error function is involved. The general concept of this optimization approach can be explained in the sense that an individual set of elastic

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parameters with better error function value will be prioritized and the search will emphasize more on its surrounding area in the next iteration. The iterations persist until either of the stopping criteria as mentioned before is achieved. In the end, the optimized elastic properties are evaluated once the iterations are terminated. A few publications have been done on the review with regard to elastic parameters identification methods, for instance, a review of mechanical properties identification methods based on full-field measurements presented by Avril and colleagues [53] and a review of recent progress of elastic characterization methods developed at the University of Calabria presented by Pagnotta [54]. In the present article, focus will be placed on reviewing the development of nondestructive vibrational experimental and numerical methods as highlighted in Figure 1. The novelty of the present article comprises the review of recent application of vibrational experimental methods integrated with the use of different numerical forward and inverse methods. The selection of different types of vibrational experimental methods as well as forward methods and inverse methods in the numerical evaluation section would incur significant influences on the accuracy, rate of convergence, and cost of an identification approach. Hence, presentation of experimental and numerical approaches in the manner of forward and inverse methods is necessary.

2. Experimental measurements 2.1. Vibrational methods Numerous researches have been performed in the field of elastic constants identification for composite materials by applying vibrational methods. A vibrational method is usually regarded as experimental modal analysis, as explained in [55]. It is an investigation on vibration behaviors of elastic structures. It involves experimental methods in investigating the oscillation behavior of component structures by defining a system with its modal parameters: its natural frequencies, natural mode shapes, and natural damping. There are generally two types of excitations, which consist of impulse excitation [56–71] and continuous variable excitation [72–84]. In impulse excitation, usually an impulser (hammer, etc.) is used to strike the specimen mechanically and elastically, as demonstrated in the ASTM standard for testing isotropic materials [85]. Continuous variable excitation commonly involves the use of loudspeakers or shakers fed by a variable frequency oscillator. In accordance with the development of fast Fourier transform (FFT) analyzers and virtual instruments, an impulse excitation technique has been widely used since the late 1970s. Impulse technique is often the primary choice in the study of material elastic properties identification due to its simplicity and inexpensive procedures, as shown in [56–70]. Currently, there is only an ASTM standard procedure provided for testing isotropic materials but none for anisotropic materials. Despite this, multiple studies have been done using such a procedure applied on orthotropic and anisotropic laminated materials, as shown in most of the mentioned publications. With regard to the layerwise identification for a multi-layered laminated plate, instead of acquiring only single test vibration data, several plates’ vibration tests are needed depending on the number of materials used


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Figure . Schematic of the impulse technique experimental set-up. Reprinted from [], © , with permission from Elsevier.

in the multi-layered laminate plate [66, 86]. The more types of materials used in a multi-layered plate, the higher the number of vibration tests are required. A vibrational experiment that involves impulse technique is commonly regarded as impact testing as depicted in Figure 4 [69], in which an impact hammer is usually used as the exciter and contactless eddy current proximity transducers (displacement) [57], accelerometers [56, 58, 59, 62, 65, 67–70], microphones [60, 61], or laser Doppler vibrometer (LDV) [63, 64, 66] are used as the response detectors. However, the drawbacks of this technique are its low reproducibility of input characteristics (manual excitation) and nonapplicability to light and brittle objects. In a continuous variable excitation technique, shaker excitation is commonly used and often taken as the comparison target for impact excitation. Both of these excitations are developed since the creation of FFT analyzers. As shown in Figure 5 [73], shaker testing is in fact used to overcome the drawbacks of impact testing, such as the use of impact hammer, which might damage delicate surfaces, and its limited frequency range of excitation, etc. Common types of shakers, such as

electro-dynamic shakers and hydraulic shakers, are used together with a stinger, which is a long slender rod, in order to specify the direction of the excitation force applied. There are several types of broadband signals for shaker measurements with FFT analyzers, which include transient signal, true random signal, pseudo random signal, burst random signal, fast sine sweep (chirp) signal, and burst chirp signal. The shaker excitation technique is widely used in various studies, including in the field of material properties identification [72–76]. Accelerometers [75], LDV [73, 74], and noninterferometric [72] transducers are the common response-measuring transducers used in a shaker-based experiment. Although shakers exhibit better applicability in broadband frequency-wise, complications in setting up the experiment as well as the mass loading effects of attached accelerometers on the specimen seem to impede the widespread usage. Besides shaker excitation, an alternative technique, which is known as acoustic excitation has been developed as well as demonstrated in Figure 6 [78]. In the acoustic-based experiment, a loudspeaker fed by a signal generator with a power



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Figure . Schematic of the shaker-based experimental set-up. Reprinted from [], © , with permission from Elsevier.

amplifier is used as the source of excitation. This excitation is more advantageous over shaker excitation in the sense that the source of excitation and the specimen are contactless, thus avoiding mass loading implication. Moreover, it is useful when dealing with relatively low internal damping structure. Similar to shaker excitation, several types of broadband signals can be assigned. The application of this technique is considerably popular in identifying the elastic properties of materials [77–84]. Normally, acoustic loudspeakers are used as the exciter and LDV [77–80] is used as the detector. Further, interferometric transducers are also used to measure the dynamic response [82, 83]. In terms of signal to noise ratio, the value is found to be comparatively low at low amplitudes, indicating that the noise level is relatively high. Despite this, the accuracy of the natural frequencies educed from a narrow frequency band is barely affected. Impulse technique, shaker excitation technique, and acoustic excitation technique incorporated with diverse measuring

Figure . Schematic of the acoustic-based experimental set-up. Reprinted from [], © , with permission from Elsevier.

techniques, such as conventional contact accelerometers, contactless noninterferrometric transducers, LDV, and microphones, can be seen to be equally high in demand and widely used due to their respective advantages and disadvantages that complement each other, in which the selection of either technique is based on one’s prior requirements in the research. In fact, there is no discernible difference in procedures for each technique when handling a particular type of plate, e.g., isotropic, orthotropic, anisotropic, and laminated plates. Certainly, when dealing with anisotropic and laminated plates, more than one single experimental test is required. Among those aforementioned techniques, impulse technique appears to be the most popular approach currently in the research of material properties identification because of the ease of implementation, inexpensive procedures, and involvement of simple structures, e.g., plates. In terms of accuracy, those mentioned techniques provide reliable results only if proper and correct procedures are conducted.

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ρhω2 = H ab

3. Numerical evaluation 3.1. Forward methods


In a vibrational approach, forward methods are needed to determine the modal parameters, e.g., natural frequencies, mode shapes, or/and damping, of a structure for following use in the identification process. The details of Rayleigh’s method, Rayleigh–Ritz method, finite element method (FEM), and Fourier method, including the determining parameters, types of plates used with respective geometrical shapes, and boundary conditions as well as principles of methods used will be discussed and presented. 3.1.1. Rayleigh’s method In Rayleigh’s method, the maximum kinetic and potential energies of a plate are needed to compose the Rayleigh’s quotient. Referring to the study done by Dickinson [87], for an orthotropic plate, the potential energy, Vmax , and kinetic energy, Tmax , can be expressed as: 2 2 ∂ W ∂ 2W ∂ 2W 1 a b Vmax = + 2v D Dx xy y 2 0 0 ∂x2 ∂x2 ∂y2

2 2 2 ∂ 2W ∂ W ∂W 2 + 4D + N xy x ∂y2 ∂x∂y ∂x ∂W 2 + Ny dydx, (1) ∂y +Dy

Tmax =

1 ρhω2 2

0

a

b

W 2 dydx,

(2)

0

where Dx = Ex h3 /12(1 − v xy v yx ), Dy = Dx Ey /Ex , and Dxy = Gxy h3 /12. Ex and Ey are the elastic moduli, Gxy is the shear modulus, v xy and v yx are Poisson’s ratios, W (x, y) is the transverse displacement of the plate, h denotes the thickness of the plate, ρ is the density of plate material, Nx and Ny are the constant in-plane forces per unit width, and the double integral is taken over the area of the plate. The Rayleigh’s quotient is defined as the ratio of total potential to kinetic energy for a mode of vibration, which can be expressed as: Vmax Tmax ⎡ 2 2 2 ⎤ ∂ 2W ∂ 2W ∂ 2W ∂ W D + 2v D + D xy y ∂x2 ∂y2 y ∂y2 a b ⎢ x ∂x2 ⎥ 2 ⎦ dydx ⎣ 2 0 0

∂W 2 ∂ 2W ∂W +4Dxy ∂x∂y + Nx ∂x + Ny ∂y . = ab ρh 0 0 W 2 dydx (3)

R (ω) = ω2 =

Let H = v xy Dy + 2Dxy and the assumed shape function W (x, y) = θ (x)φ(y) × constant, where θ (x) and φ(y) are characteristic beam functions satisfying boundary conditions on x = 0, a and y = 0, b, respectively, and can be rearranged into:

0

0

+4

Dx (θ φ)2 H 2

Dxy {(θ H

+ 2θ θ φφ +

Dy H

(θ φ )2

φ ) − θ θ φφ } + NHx (θ φ)2 + ab 2 0 0 (θ φ) dydx

Ny H

(θ φ )2

dydx .

(4) By adopting the tables of beam characteristic functions provided by Felgar [88], the frequently-used orthotropic plate frequency parameter can be presented as: λ2ortho =

Dy 4 a2 Dx 4 b2 ρhω2 a2 b2 = G G + x π 4H H a2 H y b2 Dxy Ny a2 Nx b2 +2 Hx Hy + 2 Jx Jy − Hx Hy + 2 Jx + 2 Jy , H π H π H (5)

where the coefficients Gx , Gy , Hx , Hy , Jx , and Jy are referred to those given by Warburton [89]. Several studies have been done utilizing Rayleigh’s method for forward evaluation of natural frequencies of isotropic [87, 89–94] and orthotropic composite plates [59, 87, 89, 90, 94–99] in the field of elastic properties identification, in which both thin plates [87, 89–92, 94, 96–99] as well as thick plates [59] are investigated. Most of the studies are done involving rectangular plates with free-free boundary conditions. Warburton was the first to propose the use of characteristic beam vibration functions in Rayleigh’s method to study the vibration of thin, isotropic plates [90]. His work was further investigated by Hearmon [95], applied to specially orthotropic composite plates, in which the load is exerted either in parallel or perpendicular to the plates’ fibers. Even though Warburton’s expression allowed straightforward calculation of plates’ natural frequencies, the accuracy can be tremendously reduced if one or more free edges are involved in the investigations. Since the presence of free edge(s) affects the reliability of Warburton’s equation, Alfano and Pagnotta [91] proposed the use of Warburton’s equation with appropriate correction factors, determined from the results done by a finite element code to identify the elastic properties of thin isotropic rectangular plate with free edges. The two elastic constants, namely, elastic modulus and Poisson’s ratio of the isotropic plate were finally identified using the equation derived from Warburton’s equation involving at least two of the first four experimental natural frequencies. Later, Alfano and Pagnotta [92] adopted the use of polynomial interpolating functions to form a relationship between more accurate correction factors and the variation of Poisson ratio in order to identify the Poisson’s ratio and elastic modulus of the similar plate. Those less accurate correction factors were excluded from the interpolating functions. In [98], the Rayleigh’s method based on the classical lamination theory was used for computing the natural frequencies of a thin rectangular orthotropic plate with free edges. Validation of the proposed method was done using thin isotropic, orthotropic, and laminated plates. McIntyre and Woodhouse [96] used a similar method to identify the elastic and damping constants of thin, orthotropic composite plates. In fact, this approach was conducted in the basis of frequencies



analysis and damping factors of low vibration modes, enabling the application on many other materials. Meanwhile, in [59] the applicability of a similar method on thick orthotropic plates with free edges was investigated as well. Instead of solely utilizing the previous classical lamination theory, the through-the-thickness shear and rotatory inertia were taken into account in Rayleigh’s method for forward evaluation of the plate’s resonant frequencies. In summary, the Rayleigh’s method can be used to study the vibration of isotropic and orthotropic composite plates. When dealing with isotropic problems, only two material properties were involved, while a minimum of four material properties were needed when involving orthotropic structures. Rayleigh’s method is adopted in the modeling of the dynamic behavior of rectangular plates, mainly because of its ease of implementation. However, its restriction in providing only information about the lowest or the first resonant frequency, as well as its mediocre accuracy, are the main reasons that lead to the development of the Rayleigh–Ritz method. 3.1.2. Rayleigh–Ritz method Rayleigh–Ritz method is an extension of Rayleigh’s method. Instead of using a single assumed function (static deflection shape), several assumed functions are superimposed in order to obtain a closer approximation. This method allows the evaluation of higher modes’ natural frequencies depending on the arbitrary number of assumed functions used. Referring to [57], a similar Rayleigh’s quotient is expressed as: R(ω) = ω2 =

W (x, y) =

Vmax , ab ρh 0 0 W 2 dydx

p q

Amn φm (x)ψn (y).

(6)

(7)

m=1 n=1

In Eqs. (6) and (7), Amn denotes the coefficients and W (x, y) represents the transverse displacement or the mode shape of the plate, which is composed of several admissible assumed functions, φm and ψn , as well as ω indicates the natural frequency, which can eventually be obtained by solving the corresponding eigenvalue problem. In practice, the Rayleigh–Ritz method is said to be applicable to all plate problems, linear or nonlinear, since the principle of virtual displacements applies to all plate problems [100]. The application of Rayleigh–Ritz technique for forward computation of natural frequencies in the study of elastic properties determination can be seen in [57, 63, 65, 67, 68, 101–105]. Most of the investigations are done on thin rectangular orthotropic plates [57, 63, 101–103, 105, 106], followed by thin rectangular laminated plates [65, 67] and thick rectangular sandwich plates [68, 104] with combinations of free-clamped edges [57, 63], free edges [101–106], and elastically-restrained edges [65, 67, 68]. For isotropic or especially orthotropic plates problems, a trigonometric series provides simple solutions since the stiffness and mass matrices are diagonal. For problems with boundary conditions other than simple support or for problems involving generally orthotropic laminates, the application of the trigonometric series is complicated. Thus, the polynomial series came into existence and has been extensively used in computing the natural frequencies of plates due to its straightforward algebraic manipulation [65, 67, 68, 104, 107]. The use of this

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series was limited in the mid-1980s due to complications in setting up a series compromising the geometric boundary conditions. The solution to this difficulty is to generate higher terms of the series, based on starting polynomials specific to the boundary conditions proposed by Bhat [107]. The Gram–Schmit process was used to produce orthogonal sets of polynomials. The outcomes showed that the orthogonal polynomials led to better convergence. Lee and Kam adopted a set of Legendre’s orthogonal polynomials to predict the natural frequencies of partially restrained thin laminated plates [65], elastically restrained thin laminated plates [67], and free-edged thick laminated sandwich plates [68]. The difference between partially and elastically restrained was the inclusion of additional strain energy stored in the center spring in elastically restrained laminated plates. Compared to partially restrained, elastically restrained laminated plates showed more accurate results (elastic constants) with lower errors. In the absence of center support, only five measured natural frequencies were needed in the identification process for better accuracy. However, for elastically restrained laminated plates with the number of the unknown spring constants and elastic constants of the elastic supports larger than two, more than seven measured natural frequencies were required to determine the mechanical properties. Meanwhile, Rebillat and Boutillon [104] used basic orthogonal polynomials similar to reference [107] to estimate the resonant frequencies of a thick sandwich plate with free-free boundary condition. In [107], the characteristic orthogonal polynomials were proven providing good results for lower modes in free-edged plates. On the other hand, Al-Obeid and Cooper [108] stated that the Rayleigh– Ritz approaches are designated for the symmetric balanced and unbalanced cases. A more general shape function was introduced to curb some previously unsolved orthotropic composite plate boundary conditions, e.g., CSCF, CCCF, and CFCF, where F, S, and C represent the free, simply-supported, and clamped conditions, respectively. Instead of using the Ashton approach [109], a summation of polynomial products was adopted as it is comparatively simpler. When compared with previous Rayleigh–Ritz approaches, improved convergence properties were obtained. Apart from using a polynomial series, the use of vibrating beam characteristic functions has also been widely utilized to obtain the natural frequencies of plate structures in the study of elastic properties identification [57, 102, 105, 106]. The selection of the characteristic equations of a vibrating beam is normally made based on the boundary conditions of the plate to represent its assumed mode shapes. Further, a complication was found, wherein the computed natural frequencies using such a method were not closely matched with the reference experimental natural frequencies when involving orthotropic rectangular plates with combinations of free-clamped edges [57]. However, the percentage of discrepancy between experimental and predicted resonant frequencies was found to be much smaller (at most around 5%) for lower modes when applying this similar method on free-edged orthotropic rectangular plates [102]. A similar method was utilized to obtain the natural frequencies of thin specially orthotropic plates with free edges. The error was reduced by introducing influence coefficients in the frequency expression [105]. A major challenge of the Rayleigh–Ritz method lies in the selection of appropriate shape functions for specific problems as this determines the reliability of the outcomes. Moreover, due to the limited flexibility of Rayleigh–Ritz

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method, which is subjected to boundary constraints, FEM has thus come into existence to circumvent the shortcoming.

3.1.3. FEM FEM has gained its usage prevalence in the modeling of plate structure for the purpose of material properties identification several years ago and multiple studies have been done applying this approach in plate material characterization. Basically, finite element models are created in order to obtain the numerical modal parameters using given elastic properties as inputs. Natural frequencies or/and mode shapes are computed from the model for subsequent application in material determination process, as it can be observed in [66, 69, 70, 77, 78, 83, 86, 110–130]. Studies are often done on rectangular [66, 69, 70, 77, 83, 86, 110–117, 119–124, 130] and square [77, 78, 110, 115, 116, 118, 125, 126, 128–130] plates. Further, there are also few studies done on plates of various geometrical shapes, as shown in [115, 116]. In the aspect of boundary condition, plates with free-free edges [66, 69, 70, 77, 78, 83, 86, 110–128, 130], simplysupported edges [110], and clamped edges [83, 129] are often taken into investigation. Generally, laminated plates are utilized in previous studies [86, 110, 112–114, 117–119]. Thickness of plates also plays a cardinal role in deciding the use of appropriate plate theory in finite element modeling. Hence, collection of previous studies can be classified into thin isotropic [115, 116], thin anisotropic [121, 123], thin orthotropic [66, 120, 129], thin laminated plates [69, 70, 86, 110, 112, 118, 127], as well as thick anisotropic [122] and thick laminated plates [77, 78, 112–114, 117, 119, 124–126, 128, 130–132]. For thin laminated plates, the created finite element model is usually based on Love–Kirchhoff theory or classical lamination plate theory. For thick laminated plates, Mindlin theory, first-order shear deformation theory, or higher-order theory are used in the finite element modeling. Pagnotta and Stigliano [115, 116] developed 2D and 3D models of isotropic plates of various shapes, based on a theory that assumed small deformations and linear elastic characteristics of the material. The plate model was assumed to be undergoing vibration in the absence of damping as well as made of homogeneous and isotropic material and with geometry and nominal sizes, which resemble the actual plate with a low degree of discrepancy. The accuracy of the model can be judged based on the accuracy of the natural frequencies, in which the main affecting factors were the inhomogeneity of the material as well as the low degree of resemblance between the dimensions of the model and those of the actual plate. In [110], FEM is used to determine the natural frequencies of isotropic, orthotropic, and anti-symmetric angle ply plates. Since only two elastic properties were needed in the isotropic problem, the equation of fundamental frequency was much simpler compared to orthotropic and anti-symmetric angle ply plates. For anti-symmetric angle ply plate, the properties of each layer were included in the governing equation. At least four resonant frequencies were required to accurately determine the two unknown elastic properties of isotropic plates as stated by Pagnotta and Stigliano [115]. Sol and colleagues constructed a finite element model of a medium thick anisotropic plate based on Reissner–Mindlin theory for the identification of anisotropic plate rigidities. The presence of local material anisotropy was found to have influences on the numerical natural frequencies; hence, a single test

was not advisable when dealing with inhomogeneous materials. Besides, in the research done by Lauwagie and co-workers [66], Love–Kirchhoff theory was adopted in developing the finite element model of orthotropic plates, whereby eight order polynomial Lagrange functions were used as the shape functions. The main drawback of this theory consisted of the limited thickness of the plate. Thus, an adapted Resonalyser procedure was used based on a more complex 3D finite element model in identifying the material properties of moderately thick isotropic and orthotropic materials. However, this procedure was not encouraged when the homogeneity of the material was the subject of investigation. Later, they developed a novel method based on an extension of the ‘Resonalyser’ procedure to identify the elastic properties of a plate with arbitrary number of layers, each with respective elastic properties [86]. The classical lamination theory was used to surmount the nonuniqueness problem. Single test vibration data was not sufficient for subsequent use in the identification process. Instead, several test plates’ vibration data were needed, depending on the number of layered materials to be identified. The larger the number of layered materials used, the larger the number of plate’s configurations required, and thus, the more the plate’s vibration data needed. Maletta and Pagnotta [112] proposed the use of an overdetermined number of natural frequencies of an anisotropic plate to identify its elastic properties. Neither mode shapes nor modal indices of particular modes were needed since frequencies were correlated simply by their number in a sequential order of magnitude. This method was said to be applicable on orthotropic and laminated plates. Later, Matter and co-workers [77, 78, 130] developed finite element models, resembling multilayered composite plates, based on a variable p-order shear deformation theory (PSDT). This generalized higher-order theory was selected due to the use of inherent structure, thick or moderately thick multilayered plates. The shape functions representing the throughthickness displacement were said to be flexible, and thus, adaptable to accuracy and computation time requirements. As proven in [133], the simulations for a thick multilayered composite plate model with an order of third and above applying PSDT are well correlated with those from a third-order layer-wise plate theory as well as the computational effort was much reduced. Araujo and co-workers [113, 114] adopted FEM to construct an active plate model with surface-bonded piezoelectric patches, based on a displacement field using third-order expansions in the thickness coordinate of the in-plane displacements, and a constant transverse displacement. This model allowed the analysis of arbitrary thin and thick plate and shell structures with more accurate results. Hwang and colleagues [69, 70] adopted a FEM to model laminated composite plates. The use of more resonant frequencies was recommended to improve accuracy. As of today, FEM is very popular, and thus, leading the market in the field of elastic characterization due to its flexibility and robustness; but when comparing the accuracy to an analytical method, such as Fourier method, FEM’s accuracy certainly comes second. 3.1.4. Fourier method In the studies done by Ismail et al. [134, 135], an accurate analytical forward method based on the Fourier series was utilized to generate the natural frequencies of orthotropic plates with general elastic boundary conditions for incoming use in


material characterization: D11

∂ 4W ∂ 4W ∂ 4W + 4D16 3 + 2(D12 + 2D66 ) 2 2 4 ∂x ∂x ∂y ∂x ∂y

∂ 4W ∂ 4W + D22 4 − ρhω2W (x, y) = q(ω, x, y), 2 ∂x∂y ∂y ∞ ∞ W (x, y) = Amn cos(λam x) cos(λbn y) +4D26

+

4 l=1

m=0 n=0

ξbl (y)

∞

l cm

m=0

cos(λam x) +

ξal (x)

∞

(8)

dml

× cos(λbn y) .

m=0


(9) Equation (8) represents the motion for an orthotropic plate, assuming that a harmonic excitation is imparted. A relationship between the elastic constants, Dij , and the plate’s deflection displacement, W (x, y), is established, in which the displacement deflection function is expressed as a more robust form of Fourier series expansion, as shown in Eq. (9). It is noted that for a thin orthotropic plate, D11 = E1 h3 /12(1 − v 12 v 21 ), D22 = E2 h3 /12(1 − v 12 v 21 ), D12 = v 12 D22 , D66 = G12 h3 /12, and D16 = 0, D26 = 0, in which E1 and E2 are the elastic moduli, h is the plate thickness, v 12 and v 21 are the Poisson’s ratios, and G12 is the shear modulus. Basically, the displacement function was composed of a 2D Fourier cosine series supplemented with number of terms in the form of a 1D series. Direct evaluation of the series expansions for all the related derivatives was done via term-by-term differentiations of the displacement series and a classical solution was derived by allowing the series to exactly fulfill the governing differential equation and all the boundary conditions at every field and boundary point. Eventually, the modal parameters, including the natural frequencies, were computed by resolving a standard matrix eigenvalue problem. The salient advantage of this approach consists in its solution exactness and accuracy, in which the governing differential equation and the boundary conditions are satisfied entirely and exactly on a point-wise basis. This method is also applicable to plates with arbitrary boundary conditions, loading features, and/or load conditions. Extensive studies have been done applying Fourier method on plate vibration analysis; however, as of today, the use of this method in material characterization is not common. Though, with its outstanding accuracy and versatility, investigations into the field of material identification utilizing this Fourier method can be promising. 3.2. Inverse method Inverse method plays an important role in determining elastic properties of a material as the accuracy, speed, and cost of the identification method reflect its reliability. Genetic algorithm (GA), feasible directions method, response surface methodology (RSM), Bayesian estimation method, Newton’s method, and nonlinear least squares method (Levenberg–Marquardt method) are considered the most popular algorithms used as inverse methods in the field of material identification at present. In this context, the aforementioned inverse methods with the use of different error functions will be discussed thoroughly.

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3.2.1. Genetic algorithm (GA) Basically, GA is a useful instrument for finding optimal solutions in scientific, technical, and production problems. It is a class of stochastic search algorithms that possess the global search ability and requires no initial guesses during the optimization process conducted. The use of GA is very popular in the field of material characterization. An optimization objective function is required in the GA as the fitness function. Normally, the objective function that is usually used in a vibrational identification approach is defined as the discrepancy between the measured and calculated natural frequencies as it can be seen in [69, 70, 110, 112, 115, 116, 136–139]. Elastic properties of various types of plates have been studied using GA, including orthotropic plates [112, 136] and laminated composite plates [70, 110, 112, 136–139]. Rectangular plates with free [69, 70, 112, 136–138] and simply-supported [110] boundary conditions are often the subjects of investigation. Referring to [110, 112, 115, 116, 136, 137], binary genetic algorithms can be said to be the preferred choice for researchers due to its simplicity. As discussed in [112], it was stated that the involvement of the mode shape and the modal indices in the objective function was not necessary as simple correlation of frequencies by their number in a sequential order of magnitude was said to be sufficient. The distinct advantage of such function can be seen in reducing the effect of large experimental errors that may happen at higher frequencies. Determination of four and five elastic constants was proven sufficient for thin laminated plates and thick laminated plates, respectively. Instead of using a minimization objective, a maximization objective function was selected as required by the GA [110]. The difference between unity and the relative error discrepancy between the experimental and numerical natural frequencies was included in the maximization objective function. Orthotropic and laminated plates’ elastic properties were then determined with the aid of GA. The study indicated that the analytical approach yielded more accurate results (elastic properties) than that of using the finite element approach; however, the analytical approach was applicable to certain boundary conditions only, unlike the finite element method, which was more flexible. Further, the inverse method involving simple GA was proven feasible for orthotropic and laminated rectangular plates of arbitrary thickness as well as free edges. In the investigation of laminated plates [136, 137], results showed that the deviation between the identified and reference Poisson’s ratio was relatively large if compared to other properties, due to low sensitivity of resonant frequencies towards the Poisson’s ratio. The sensitivity of the resonant frequencies to the transverse shear moduli was also found to be low in variation of natural frequencies. Thus, in order to identify the Poisson’s ratios and transverse shear moduli that were sensitive to the change in natural frequencies, the optimum plate geometric parameters, such as the plate side aspect ratio, orthotropy angles, and thickness, which were related to the natural frequencies need to be identified. In this study, two-step identification processes were conducted. The first step was the identification of the plate geometric parameters as mentioned above, while the second step involved the ordinary material properties identification procedures adopting the optimized geometric parameters determined in the first step. Genetic algorithm was utilized in both of the steps. From the investigation, the use of a plate


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geometric parameter optimization procedure engaging genetic algorithm was proven necessary for achieving better identification results. On the other hand, four types of objective functions as well as optimization methods have been investigated in elastic properties determination [115, 116]. The efficiency of each objective function coupled with each type of optimization method was investigated and compared in terms of the number of FEM code runs. All of the objective functions were applicable to GA and the results denote that the accuracy of the GA solution was the most promising due to its global search ability; however, its convergence speed was the lowest among them. The application of hybrid GA with developed efficiency and accuracy in the discipline of material identification can be observed in [69, 70, 138, 139]. Referring to [70], a hybrid algorithm was used in the basis of a real-parameter GA (RGA) [140] to identify the material properties of rectangular laminated composite plates with free edges. RGA was utilized in the crossover and mutation processes; in the meantime, simulated annealing was used as well in another mutation processes. Adaptive mechanisms are used to modify the probabilities of the crossover and mutation operators in order to enhance the hillclimbing ability in the search of optimum solution. Only the first six measured resonant frequencies were used in an algorithm to identify the four elastic constants of a thin laminated plate due to the difficulty in acquiring a large number of experimental natural frequencies. Besides, higher modes of natural frequencies were not necessary since only four elastic properties are needed. Similarly, the estimation of the Poisson’s ratios was inconsistent, unlike the other elastic constants, which exhibit high repeatability. However, the discrepancy on the Poisson’s ratios could be reduced if more resonant frequencies could be considered in the objective function. In [69], a similar hybrid genetic algorithm was used as well to identify the effective elastic properties of a rectangular woven composite plate and two rectangular printed circuit boards with free edges. Distinctively, a two-step procedure was used in the determination process. The first step was conducted by taking into account the first four measured frequencies, in which the free three were fixed and the fourth was free to match in order to identify the missing frequencies. After the identification of missing frequencies was done in the first step, the identification of the effective elastic properties of the plates was continued using more measured frequencies according to the mode sequence. Comparisons of the effective elastic constants obtained in the first step and second step were done, indicating that the second step produced better results in terms of improved objective function values as well as improved repeatability in the Poisson’s ratios, due to the inclusion of more frequencies in the objective function. Recently, the use of GA has become prevalent in determining elastic properties due to its global search ability and high accuracy, thus, leading to competitive investigations. 3.2.2. Feasible direction interior-point method The use of feasible directions method has rooted its reign of recognition in the field of material identification since the 1990s for its simplicity in coding and efficiency, in which penalty functions, active set strategies, or quadratic programming subproblems are not involved in the solutions [60, 61, 113, 114, 117, 141]. It is very useful for problems with objective function or constraint functions that are not defined at infeasible points, e.g.,

in the field of material properties determination for laminated composite plates with surface-bonded piezoelectric patches [113, 114, 117]. As stated in [117], it must be noted that simultaneous identification of elastic constants, piezoelectric and dielectric coefficients was impractical, thus separate evaluation of each category was required using the proposed method. This study also ascertained the feasibility of the proposed method for determining elastic properties of multi-material laminated composite plates. Moreover, in [113, 114], two approaches were adopted and compared. The first method was the gradient based optimization technique, specifically, the use of Gauss–Newton algorithm (gradient-based) integrated with the feasible arc interior point algorithm (FAIPA) for unconstrained and constrained optimizations, respectively. The second method was the use of a metamodeling method based on artificial neural network (ANN) technique. Both approaches produced convincing results as the evaluated material properties were found to be reasonable. However, in a close comparison, the error residuals found in using the FAIPA technique was slightly lower than the ANN technique. In the meantime, the convergence rate for FAIPA was discovered to be 1.2 to 2.4 times higher than ANN. Therefore, overall, FAIPA technique can be claimed as the better technique. 3.2.3. Response surface methodology (RSM) RSM is a collection of statistical and mathematical techniques, which are beneficial for developing and optimizing processes. Extensive studies have been done in identifying the material properties of laminated composite plates utilizing the integration of the design of experiments and RSM [62, 84, 125, 126, 142]. The use of RSM significantly improves the rate of convergence (approximately 50–100 times), thus leading to the dominance of such a technique over the conventional identification technique. In [125], the elastic properties of laminated composite plates with two different fiber-surface treatments were studied, comprised of the use of epoxy dispersion with aminosilane to enhance fiber/matrix adhesion (EP) and the use of polyethylene to eliminate fiber/matrix adhesion (PE). Besides a tremendous increase in computational time, only one vibration test of the plate sample was needed to sufficiently determine the entire elastic constants by utilizing the proposed RSM; the outcomes indicated that EP composite possessed higher transverse stiffness due to excellent fiber/matrix adhesion. Moreover, in [126, 142], the transverse shear modulus was found experiencing relatively large deviation for thin plates due to unnoticeable transverse deformation. Meanwhile, the transverse shear modulus of moderately thick plates was also found to be erroneous perhaps due to the use of the RESINT program, which only considered the main terms of the regression equation during the approximation process. In comparison to the terms containing the inplane elastic constants, the terms containing the transverse shear modulus were of second order, thus causing error in a transverse shear modulus. Although the use of RSM provides a substantial increase in convergence rate, its accuracy can still be further improved. As discussed in [127, 143], hybrid RSM and particle swarm optimization (PSO) method was developed and utilized in material identification of composite plates for better accuracy. As demonstrated in Table 1 [143], hybrid RSM-PSO imparted the most accurate results with the smallest errors among the different methods, while the convergence rate was reasonably


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Table . System identification of orthotropic plate using hybrid RSM–PSO method in time domain with % noise []. Predicted parameters System parameters E (N/m ) E (N/m ) ν  G (N/m ) Avg. error (%) Max. error (%) CPU time (s)

Actual parameters

RSM

Improved PSO

Hybrid RSM–PSO

GA (no noise)

. ×  . ×  . . × 

. ×  . ×  . . ×  . . .

. ×  . ×  . . ×  . . .

. ×  . ×  . . ×  . . .

. ×  . ×  . . ×  . . —

Reprinted from [], © , with permission from Elsevier.


good but not as good as that for solely RSM with the largest error. Lately, researchers can be seen placing their attention into the investigation of using this method in material identification mainly because of its prominent speed of convergence and accuracy. 3.2.4. Bayesian estimation method In a Bayesian parameter estimation expression, weighting coefficients on the parameters and the responses are taken into consideration. The deviation between the initial model estimates and the experimental data is reduced by minimizing a weighted error function. As discussed in [106], the Bayesian estimation method was chosen for the characterization of unidirectional laminated composite plates with free-free boundary conditions. As mentioned, an error function was used, signifying the discrepancy between the theoretical and experimental responses (natural frequencies) as well as the difference between the initial and updated parameters (estimated elastic constants) with the presence of two weighting matrices. A sensitivity matrix, denoting the derivatives of the natural frequencies with respect to the four stiffnesses, was evaluated as well for subsequent use in the identification process. The iteration process started with an initial guess of elastic constants until the updated elastic constants fell within 0.1% of the previous values. The complete procedures were explained in [106]. Generally, the number of iteration in Bayesian estimation would not exceed 10 times. In fact, the use of confidences on the initial estimates improved convergence and ensured a unique solution for the updated parameters. The comparison between the confidence matrices for initial estimates and the updated parameters signifies the accuracy of the identified elastic properties. Overall, the results of the elastic properties determined using Bayesian estimation were approaching the results obtained in static testing. Hence, Bayesian estimation using modal data can be said to be one of the alternatives for static tests in material identification.

Moreover, Daghia et al. [119] proposed a very different approach in determining the elastic constants of free-edged thick laminated plates within a Bayesian framework, applying two estimators, which were the Bayesian estimator (B) and the minimum variance estimator (MVE). Although the B estimator showed better performances in terms of efficiency and robustness, it was biased by the a-priori information. As the deviation was reducing, the parameters were biased to remain close to the initial guesses. In contrast, the MVE estimator exhibited higher accuracy without any bias, but it was only sensitive to local minima and lower in convergence rate. The MVE estimator was then modified to improve its convergence rate by introducing a twostep procedure. From Table 2 [119], the modified MVE estimator can be seen improving in convergence rate with outstanding accuracy. Bayesian estimation method in parameter identification is quite common; however, researches done on plate material identification using this method are quite limited, mainly due to its mediocre accuracy and rate of convergence. 3.2.5. Newton method Ismail et al. [134, 135] proposed the use of Newton–Raphson multivariate iterative method incorporated with Fourier series method to form an inverse Fourier series method, and thus, determining the elastic properties of orthotropic plates with general boundary conditions. Validations were done on plates with three boundary conditions, comprised of combinations of free and clamped edges. Newton–Raphson multivariate method formulations started with a weighted error function describing the square difference between experimental and predicted natural frequencies. The derivative of the error function with respect to the derivative of each parameter to be identified was expected to be zero. In the formulation, the sensitivity matrix denoting the derivative of the eigenvalues with respect to the derivative of the parameters to be identified was utilized as well. The use of four, five, six, and

Table . Final estimates and number of iterations performed for cross-ply laminate [].

Initial Deviation B MVE MVE-modified Stage  Stage  Target

E (GPa)

E (GPa)

ν

G (GPa)

G (GPa)

G (GPa)

No. iter.

 . . 

  . .

. . . .

 . . 

 . . .

 . . 

 

.  

. . .

. . .

.  

. . 

.  

Reprinted from [], © , with permission from Elsevier.

 

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J. H. TAM ET AL.

Table . Summary of identification method with respect to types of plate. Types of plate

Direct/non-direction identification method

Isotropic plates

Direct identification method r Single vibration test data is sufficient. r The Warburton’s equation in Rayleigh method can be used to determine the elastic properties directly from the derived inverse equation with the experimental resonant frequencies used as the rinputs. References: [, –, ] Non-direct identification method r Single vibration test data is sufficient. r Forward method with high flexibility is required. r This method is needed when dealing with shapes of plates. rarbitrary Inverse algorithms are required. r Iterations are needed, usually will involve or maximization of an error function. rminimization References: [, ] Non-direct identification method r Single vibration test data is sufficient. r Forward methods are required. r Inverse algorithms are required. r Iterations are needed, usually will involve or maximization of an error function. rminimization References: [, , , –, , , , , , ] Non-direct identification method r Several vibration test data is required due to rinhomogeneity. Forward methods with high flexibility are required. r Inverse algorithms with global search ability are rrecommended. Iterations are needed, usually will involve minimization or maximization of an error function. r References: [, , , , , , –, –, , –, –]


Orthotropic plates

Anisotropic and laminated plates; e.g., unidirectional, transversely-isotropic, symmetrical, non-symmetrical, and multi-layered plate, composed of different layer materials.

seven natural frequencies with three weighting conditions, consisting of absolute error, relative error, and error in between the two, was investigated. From the findings, the root mean squared error of the sets of the identified elastic parameters decreased in values, as the number of frequencies used increased. Moreover, the use of the absolute error and the relative error in the formulation was found to have improved the identification results as well. Currently, the integrated use of Newton method with the Fourier method to determine material properties is considerably new and not popular, since Newton method is a root-finding, which is only useful for simple identification problems, but not very practical for complex material identification problems. 3.2.6. Nonlinear least squares method (Levenberg–Marquardt method) Cugnoni et al. [130] utilized the nonlinear least squares algorithm (Levenberg–Marquardt algorithm) to identify the elastic properties of multilayered laminated composite plates, based on a global error function composed of the natural frequencies error function, the error function related to mode shapes, as well as the nodal lines error function. The error function related to mode shapes can be expressed in terms of modal assurance criteria (MAC) error function and in terms of mode shapes error function. In fact, MAC error function can be further decomposed into two classical error norms, which are the diagonal and off-diagonal MAC error norms. Meanwhile, mode

shapes error function defined the sum difference between the numerical and experimental of mode shapes in absolute values. Considering the difficulty in obtaining an accurate Poisson’s ratio estimation using conventional error function (natural frequencies only), information on the nodal lines of the mode shapes, which were sensitive to the Poisson’s ratio, should be taken into account in the determination process. On top of that, the nodal lines error function was developed via interpolation of the mode shapes into grey-shaded 2D images. Eventually, these five error functions were combined and weighted to form a general error function, which would be minimized using the Levenberg–Marquardt algorithm in the identification process. The results indicated that the advantages of the proposed method consist of its prominent convergence rate as well as its robustness. Only little iteration were needed in the determination in-plane Young’s and shear moduli with an uncertainty of 2 to 5%. Meanwhile, the transverse moduli of thin plates showed an uncertainty of about 10% and the deviation diminished as the thickness-to-span ratio of the plate increases. Later, Matter et al. [77, 78] further improved the method proposed in [130] by proposing a two-stage identification method. There were two ways of conducting the two-stage identification method, which included the identification by parameter subset and the identification by progressive refinement. The identification by parameter subset was conducted by separating the evaluation of Poisson’s ratio from the identification of the Young’s and shear moduli, whereas identification by progressive refinement was meant by determining the entire set of elastic properties via two consecutive steps from rough prediction to refined estimation. From the findings, this two-stage identification method showed transcendent identification results, especially the Poisson’s ratio, and exhibits an excellent convergence rate if compared to the conventional one-step optimization method.

4. Conclusions and recommendations Material is one of the most important components in the development of industries from various fields. Materials are manufactured in different qualities and properties and, at present, composite materials are considered one of the most demanding materials due to outstanding characteristics. Without a reliable technique for measuring or acquiring the properties, the quality of composite materials could neither be judged nor discovered. Therefore, the need of a reliable material identification technique is essential. Material identification methodology can be seen improving in the aspects of accuracy, time, and cost from tedious destructive tests in the past to the present rapid nondestructive vibrational evaluation approaches. Both experimental and numerical procedures are involved in nondestructive vibrational evaluation approaches. In the experimental section, impulse technique appears to be the most popular approach currently in the research of material properties identification on account of the ease of implementation, inexpensive procedures, and involvement of simple structures, e.g., plates. In the section of numerical evaluation, a direct identification method is usually designated for simple structures, such as beams, rods, as well as isotropic rectangular/square plates. In cases with regards to shape variations for the same type of plate,



a nondirect identification method is needed. For orthotropic, anisotropic, and laminated plate, due to involvement of several elastic constants, nondirect method draws the solutions and the summary is presented in Table 3. In view of forward methods, due to high flexibility and applicability to any boundary conditions and geometrical shapes, FEM has always been the primary choice for forward computation of modal parameters in material identification. Nevertheless, in terms of accuracy, Fourier method is comparatively better than FEM. Although the use of Fourier method as the forward approach in the discipline of material identification is rare, the development of this method in the near future can be assuring for its high stability, accuracy, and robustness. As for inverse methods, genetic algorithm is developing rapidly and its contributions in the field of material identification are tremendous as it yields accurate results; hence, drawing more attention in studies. However, the main concern over genetic algorithm is the rate of convergence. If compared to the combined use of RSM with other algorithms, the convergence rate of the latter is much more prominent; therefore, this contributes to the recent widespread use of such a method. The concept is convincing, especially the integrated use of RSM with a PSO method because of its stochastic nature and excellent speed of convergence. Meanwhile, only a few studies have been done utilizing the feasible direction interior-point method as well as Bayesian estimation method due to mediocre accuracy and average rate of convergence. Similarly, Newton’s method could hardly gain appreciation from researchers in the study of material identification as it is not very useful or helpful in complex identification problems, since it is merely a root-finding algorithm. However, integration of Newton’s method with Fourier method to form an inverse method is quite new in the market. Apart from that, recently, a nonlinear least squares method has been used to deal with the combined use of error functions (e.g., natural frequencies, mode shapes, nodal lines, damping). As of today, the combined use of error functions is uncommon. In spite of that, it is believed to have the potential for further development in coming years due to its remarkable outcomes. Overall, this study focuses on nondestructive vibrational evaluation approaches in determining the material properties of composite plates. The commonly used nondestructive vibrational evaluation approaches in the field of material identification have been highlighted in this review. This review presents previous and current trends to identify material properties of composite plates in a nondestructive manner and it may incite new trends of nondestructive approaches in the future. A possible future research can be the development of new nondestructive approaches based on hybrid use of the Fourier method and PSO method via combined use of natural frequencies, mode shapes, nodal lines, and damping as a global error function, applied on composite plates with arbitrary shapes and mixed boundary conditions for improving accuracy, rate of convergence, cost, as well as versatility and flexibility.

Funding The authors wish to acknowledge the financial support and advice given by University of Malaya Research Grant (RP022D-13AET), Fundamental Research Grant Scheme (FP010-2014A), Advanced Shock and Vibration Research (ASVR) Group of University of Malaya, Postgraduate Research Fund (PG009-2015A) and other project collaborators.

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