The hybrid damage index (HDI) is presented as a mean for the damage identification in this paper, which is on the basis of the Kullback-Leibler divergence (KLD) and its approximations. The proposed method is suitable for detecting damage in one-dimensional structure and delamination in laminated composite. The first step of analysis includes obtaining the mode data of the structure before and after the damage, and then the KLD and its approximations are obtained. In addition, the HDI is obtained on the basis of the KLD and its approximations, utilizing the natural frequencies and mode shape at the same time. Furthermore, the modal strain energy (MSE) method is employed to verify the efficiency of the proposed method. Finally, to demonstrate the capability of the proposed method, examples of the beam and laminated composite are applied for checking the present approaches numerically, and the final results validate the effective and accurate performance of the present technique.

As evidenced by the vast literature in the damage detection, the structural health monitoring has become an increasingly crucial issue. To data, significant efforts have been made by researchers in the damage identification. The presence of damage generally produces changes in the structural stiffness matrix. Meanwhile, these changes are accompanied with changes in the structural modal parameters. This phenomenon has been widely noted and used by researchers in distinguishing the damage. However, using different modal parameters correlated with other relevant information in the damage identification may get very various results with varying accuracy. For this reason, seeking a proper selection or combination of dynamic parameters is an imperative purpose.

From the perspective of the damage detection, Park et al. [

Laminated composite have been widely applied to aeronautic structures as well as automobile and wind power equipment; the reason for that is because laminated composite possesses many merits, such as lighter weight, higher stiffness, heat insulation and preservation, and antiradiation. Usually, this kind of the structural material is made of very thin carbon fiber, glass fiber, and so on. One of the advantages over the tradition material is the light weight, typically only 10–15% of that of a steel structure in condition of obtaining the same stiffness and strength.

Under repeated or impact loads in the serving process, these laminated composites are subjected to various forms of the damage as a result of manufacturing defects or in-service factors, mostly the delamination. Such delamination presents a serious threat to the proper performance of the laminated composite and becomes an obstacle to more extensive usage of the laminated composite. Thus the monitoring of the hidden delamination in the laminated composite is critical in the engineering practice. Meanwhile, the use of vibration-based techniques as nondestructive testing methods for the damage monitoring of the laminated composite is a field attracting the interest of many researchers.

Up to now, remarkable efforts have been made by researchers in the identification of delamination; the presence of delamination generally produces changes in the structural physical properties. Likewise, these changes are accompanied with changes in the modal parameters of the structure. This phenomenon has been widely noted and used by researchers for distinguishing the delamination. However, using different modal parameters correlated with other relevant information in delamination may get very various results. For this reason, seeking a proper selection or combination of dynamic parameters is a crucial thing.

With respect to detecting the delamination in the composited composite, Nalawade et al. [

Worden and Burrows [

The KLD is a widely employed tool in the statistics and pattern recognition [

In this study, the KLD and its approximation are applied to identify the damage in the beam and the delamination in the laminated composite. In order to obtain more accurate results of detecting the damage and delamination, especially the multiple damage and delamination, an improving method named the hybrid damage index is adopted. The one-dimensional beam is employed to validate the work of Radzieński et al. [

KLD is a fundamental concept about the expected log-likelihood ratio in statistics, which is a widely used tool in statistics and pattern recognition.

Assuming

Considering a vibration problem of a beam, the

Supposing that the beam structure is divided into

To account for all available modes, the KLD along the beam employed for the damage detection can be expressed as

By the same token, the KLD of the

JD [

in which

The JD of the

To account for all available modes, the JD along the beam employed for the damage detection can be expressed as

Similarly, the JD of the

The

The

To account for all the available modes, the

Similarly, the

The strain energy

Considering a vibration problem of a one-side clamped beam which is subdivided into

Hence the fractional energy of that subregion is

In the same way, the corresponding fractional energy of that subregion for a damaged beam is given as

As a consequence, the damage index for each subregion along the beam is given by

A normalized damage index can be expressed as

For an orthotropic laminated composite with a dimension of

Supposing that the laminated composite structure is subdivided into

Thus the fractional energy of that subregion is

For the laminated composite with delamination, some analogous equations can be defined using the damaged mode shape, which are given by

To account for all the available modes, the delamination index for each subregion along the laminated composite is given by

A normalized damage index is given by

in which

The process of calculating the one-dimensional hybrid damage index is given as follows.

Calculate the change of natural frequencies of the beam before and after the occurrence of the damage as follows:

Certification of the following inequality is done: provided

In order to improve the identification precision and reduce the boundary effect, the spline extrapolation of the “measured” point for the KLD with the number of the size of the filtering window is done.

Reconstructing the KLD of the beam before and after the occurrence of the damage with the weight coefficient

If the number of obtained points is not enough, the cubic extrapolation is adopted to increase the “measured” points with adequate number of points.

Calculate the curvature of the reconstructed

The curvature

Normalize the curvature

Construct the function

The function

Normalize the curvature

To account for all available modes, a hybrid damage index

For the laminated composite, the similar hybrid delamination index

Finally the similar hybrid damage index based on the

In previous preceding sections it has been shown how to utilize the proposed method in order to assess the location of the damage and delamination. Aiming to account for the practical use of results above with the analysis of real cases, following sections are devoted to outlining some application of numerical simulation samples.

To simulate the real experiment environment, we employed the first three modes of the structure before and after the presence of the damage and delamination in the certification of the proposed method by numerical results.

In this section, an isotropic beam is employed to elucidate the performance of the proposed method.

To certify the proposed method, a one-side clamped beam is considered. Material properties are taken from steel where the elastic modulus

The beam is divided into 100 elements, and the single damage is numerically simulated in elements 41 to 45 by reducing the elasticity modulus to the half. Furthermore, in order to detect the multiple damages, two kinds of damage are numerically simulated by reducing the elasticity modulus to the half in elements 31 to 35 and elements 71 to 75, and the reduction of the elasticity modulus in elements 21 to 25, elements 51 to 55, and elements 81 to 85 to half is employed to simulate three kinds of damage, which is named as the first damage scenario. In order to test the ability of the proposed method to differentiate extents of the damage, the elasticity modulus was reduced to forty percent to simulate the second damage scenario. Finally, all the damage scenarios are shown in Figure

Damage scenarios of the beam.

Damage scenario of the single kind of damage

Damage scenario of two kinds of damage

Damage scenario of two kinds of damage

With the aid of the modal analysis in the FE techniques, the first three modes of the intact and the damaged beam are obtained. Owing to the presence of the damage, the change of the mode shape which cannot be visible is induced. The KLD and its approximations are adopted with the intension of detecting the damage in the beam, and then a hybrid damage index based on the KLD and its approximation is applied with the purpose of improving the reliability and accuracy of the above method; all results of the first damage scenario are shown in Figures

Results of the single kind of damage in D1.

MSE of the single kind of damage

Divergence of the single kind of damage

HDI of the single kind of damage

Results of two kinds of damage in D1.

MSE of two kinds of damage

Divergence of two kinds of damage

HDI of two kinds of damage

Results of three kinds of damage in D1.

MSE of two kinds of damage

Divergence of three kinds of damage

HDI of three kinds of damage

Results of the single kind of damage in D2.

MSE of the single beam

Divergence of the single kind of damage

HDI of the single damage

Results of two kinds of damage in D2.

MSE of the single kind of damage

Divergence of two kinds of damage

HDI of two kinds of damage

Results of three kinds of damage in D2.

MSE of the single kind of damage

Divergence of three kinds of damage

HDI of three kinds of damage

The appearance and location of the damage are observed through the abrupt anomaly of the curve. Thereby, the MSE, KLD and its approximations, and HDI based on the KLD and its approximations, respectively, are able to detect the all damage in the first and second damage scenarios in the view of damage identification results from Figures

However, the value of the KLD is almost larger than JD and

It is worth noting out that the error of the multidamage identification is larger than the corresponding error of the single damage identification for KLD and its approximations. Furthermore, the value of peaks of the damage on diverse locations varies; thus different locations of the damage can result in various errors of the damage identification. Generally speaking, the error of the damage identification is smaller when the distance between the location of damage and the clamped end is shorter except for the damage identification of three kinds of damage by the JD and

On the basis of the KLD and its approximation, the hybrid damage index is obtained, and it is found that the error of the damage identification by the HDI is much smaller than the corresponding KLD and its approximations. Hence a noticeable improvement of the damage identification by the hybrid damage index in comparison with the KLD is observed, in the multiple damage detection, peaks in locations of damage alter greater in comparison with the corresponding, which is able to affect the accuracy of the damage detection. However the HDI based on the KLD and its approximations is smaller than the corresponding MSE in the undamaged region. The HDI based on the KLD and its approximations is more sensitive to the MSE.

In the KLD and its approximations, the KLD is larger than the JD and

The damage extent of the first damage scenario is deeper than the second damage scenario; meanwhile, the peak in the damage location in the first damage scenario is larger than the corresponding peak in the second damage scenario, which affects the error of the damage detection.

In this section, an orthotropic laminated composite is employed to demonstrate the effectiveness and robustness of the proposed method.

In this study, a 16-layer square laminated composite with a side length of 240 mm and a total thickness of 3.2 mm is considered. The ply orientation distribution along the plate thickness is

After the delamination occurs in the laminated composite, the delamination region is between two separate sublaminates, which are called the upper and lower sublaminates, respectively. In this section, the delamination is numerically simulated by the method of merging nodes within the delamination region, which has the same coordinate but belongs to the upper and lower sublaminates.

When the laminated composite is in motion, elements of the upper and lower sublaminates may overlap or even penetrate into each other within the delamination region, whereas it is physically impossible. In order to avoid this phenomenon, the virtual spring element is added between penetrated nodes in the upper and lower sublaminates within the delamination region. And the stiffness coefficient of the virtual spring element is set to 0.1 as same as in [

In the delamination scenario 3, the area of the single delamination is a square with a side length of 100 mm. In addition, the area of two kinds of delamination is both a square and with a side length of 60 mm, and the configuration of three kinds of delamination is all a square with a side length of 40 mm. Furthermore, in the delamination scenario 4, the area of the single delamination is

Delamination scenarios of the laminated composite.

Delamination scenario of the single delamination

Delamination scenario of two kinds of delamination

Delamination scenario of three kinds of delamination

On the basis of the technique of the modal analysis, the first three mode shapes of the laminated composite before and after the presence of the delamination in the damage scenarios 3 and 4 are obtained, and then the KLD and its approximation are calculated. In order to reduce and remove the error, the hybrid damage index based on the KLD and its approximation is applied with the intension of identifying the delamination in laminated composite more accurately, and the method of modal strain energy is employed so as to verify the efficiency of the proposed method; finally, all results of the delamination identification are shown in Figures

Results of the single kind of delamination identification in D3.

KLD of the single kind of delamination

JD of the single kind of delamination

SD of the single kind of delamination

KLDHDI of the single kind of delamination

JDHDI of the single kind of delamination

SDHDI of the single kind of delamination

MSE of the single kind of delamination

Results of the single kind of delamination identification in D3.

KLD of two kinds of delamination

JD of two kinds of delamination

SD of two kinds of delamination

KLDHDI of two kinds of delamination

JDHDI of two kinds of delamination

SDHDI of two kinds of delamination

MSE of two kinds of delamination

Results of the delamination identification in D3.

KLD of three kinds of delamination

JD of three kinds of delamination

SD of the single kind of delamination

KLDHDI of the single kind of delamination

JDHDI of the single kind of delamination

SDHDI of the single kind of delamination

MSE of the single kind of delamination

Results of the single kind of delamination identification in D4.

KLD of the single kind of delamination

JD of the single kind of delamination

SD of the single kind of delamination

KLDHDI of the single kind of delamination

JDHDI of the single kind of delamination

SDHDI of the single kind of delamination

MSE of the single kind of delamination

Results of two kinds of delamination identification in D4.

KLD of two kinds of delamination

JD of two kinds of delamination

SD of two kinds of delamination

KLDHDI of two kinds of delamination

JDHDI of two kinds of delamination

SDHDI of two kinds of delamination

MSE of two kinds of delamination

Results of two kinds of delamination identification in D4.

KLD of three kinds of delamination

JD of three kinds of delamination

SD of three kinds of delamination

KLDHDI of three kinds of delamination

JDHDI of two kinds of delamination

SDHDI of two kinds of delamination

MSE of three kinds of delamination

The presence and location of the delamination can be obtained through the sudden change of the corresponding curve; thus it can come to the conclusion that the KLD and its approximations are able to identify all the delamination in the delamination scenarios 3 and 4. The MSE is able to identify the single delamination and two kinds of delamination, whereas it is not able to identify three kinds of delamination in the delamination scenarios 3 and 4. Furthermore, the HDI based on the KLD and its approximation is able to identify all the delamination in the delamination scenarios 3 and 4. As a consequence, the HDI based on the KLD and its approximations is more sensitive to the delamination than the MSE.

It is worth noting out that the error of the delamination identification by the KLD and its approximation is relatively large. The JD and

The MSE and HDI based on the KLD and its approximations are both able to identify the single delamination in the delamination scenario 3, whereas the HDI based on the KLD (KLDHDI) has a relative large error. The reason for this is that the values at

The MSE have a much larger error of the delamination identification in comparison with the HDI when they are employed to identify two kinds of delamination in the delamination scenario 4. From Figure

The value of the HDI based on the KLD and its approximations in the region between two kinds of delamination is nearly zero when it is employed to identify two kinds of delamination, whereas the corresponding value of the MSE is much larger. It is found that the HDI have a better ability than the MSE in identifying two kinds of delamination.

However, the value of the HDI changes much in different delamination locations, when it is employed to identify multiple kinds of delamination, and it is a direction that needs improvement of the HDI based on the KLD and its approximations.

Generally speaking, it is worth pointing out that the HDI have a better ability to identify delamination than the MSE.

For the real modal testing, it is expected that there would be some deviations due to measurement noise. In order to study the efficiency of the proposed method in the noisy environment, the noise is added with the method in [

JDHDI with noise.

JDHDI of three kinds of delamination with 5% noise

JDHDI of three kinds of delamination with 10% noise

JDHDI of three kinds of delamination with 15% noise

JDHDI of three kinds of delamination with 20% noise

From the result it is found that the noise mainly affects the value of JDHDI in the intact region of the laminated composite. When the amount of the noise is 5 percent, JDHDI varies a little greater than the one with noise absent in the intact region, whereas the JDHDI is almost the same in the delamination region; the analogous conclusion also can be obtained with another level of noise. Furthermore, the change of the JDHDI becomes greater in the undelaminationed region, as the amount of noise increases; meanwhile the relative large percent

It indicates that the noise mainly affects the error of the delamination in a permissible tolerance, and the results of the delamination identification give a reasonable agreement between the damage identified and the damage assumed. Finally, it is found that the proposed method is an effective and robust method for the delamination identification.

In this paper the hybrid damage index for structural damage detection based on the KLD and its approximations has been proposed, and the following conclusions can be drawn from the above study.

The KLD and its approximations are able to detect the damage in all damage scenarios in this paper, whereas the error of the damage identification is relatively large. As a consequence, the KLD and its approximations are suitable for the beam and the laminated composite.

The localized ability on the damage location of the hybrid damage index is stronger than the KLD and its approximations, and the corresponding error of the damage detection by the hybrid damage index is still less than the KLD and its approximations. Hence the hybrid damage index outperforms the corresponding KLD and its approximation.

The MSE method is not able to detect three kinds of delamination for the laminated composite; meanwhile the corresponding error of the damage detection is relatively large, whereas the hybrid damage index based on the KLD and its approximations not only is able to detect the damage in all damage scenarios, but also induces a small error of the damage identification. Thus the hybrid damage index based on the KLD and its approximations is a better damage index in comparison with the modal strain energy method.

The applicability of the KLD and hybrid damage index to different types of structures and different sizes of damage is under experimental validation and it will be illustrated in the future work.

Results of the present work refer to the beam and laminated composite; moreover they can be easily extended to more complex structures and more complicated boundary conditions.

In conclusion, the present consequences provide a foundation of using the hybrid damage index based on the KLD and its approximations as an efficient tool in the damage and delamination identification in the beam and laminated composite, respectively. Moreover, the enhancement for reliability and precision of the proposed method is undergoing in our next research.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The research described in this paper has been supported by the National Natural Science Foundation of China (Grant nos. 51035007 and 51175401) and the Research Fund for the Doctoral Program of Higher Education of China (no. 20110201130001).