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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Honeycomb sandwich structures are used in a wide variety of applications. Nevertheless, due to manufacturing defects or impact loads, these structures can be subject to imperfect bonding or debonding between the skin and the honeycomb core. The presence of debonding reduces the bending stiffness of the composite panel, which causes detectable changes in its vibration characteristics. This article presents a new supervised learning algorithm to identify debonded regions in aluminum honeycomb panels. The algorithm uses a linear approximation method handled by a statistical inference model based on the maximum-entropy principle. The merits of this new approach are twofold: training is avoided and data is processed in a period of time that is comparable to the one of neural networks. The honeycomb panels are modeled with finite elements using a simplified three-layer shell model. The adhesive layer between the skin and core is modeled using linear springs, the rigidities of which are reduced in debonded sectors. The algorithm is validated using experimental data of an aluminum honeycomb panel under different damage scenarios.

The applications of sandwich structures continue to increase rapidly and range from satellites, spacecrafts, aircrafts, ships, automobiles, rail cars, wind energy systems to bridge construction, among others [

A disadvantage of sandwich structures is that their structural failures, especially in the core, cannot always be detected by traditional non-destructive detection methods. A global technique called vibration-based damage detection has been rapidly expanding over the last few years [

Vibration-based damage assessment methods are classified as model-based or non-model based. Non-model-based methods detect damage by comparing the measurements from the undamaged and damaged structures, whereas model-based methods locate and quantify damage by correlating an analytical model with test data from a damaged structure. Additionally, model-based methods are particularly useful for predicting the system response to new loading conditions and/or new system configurations (damage states), allowing damage prognosis [

Supervised learning algorithms are an alternative to model updating. The objective of supervised learning is to estimate the structure’s health based on current and past samples. Supervised learning can be divided into two classes: parametric and non-parametric. Parametric approaches assumed a statistical model for the data samples. A popular parametric approach is to model each class density as Gaussian [

A new nonparametric method for supervised learning was presented by Gupta

The primary contribution of this research is the development of a real-time damage assessment algorithm for honeycomb panels that uses a linear approximation method in conjunction with the mode shapes and natural frequencies of the structure. The linear approximation is handled by a statistical inference model based on the maximum-entropy principle [

The remainder of this work is structured as follows: Section 2 introduces the proposed damage assessment algorithm and provides previous research on the max-ent linear approximation method. Section 3 describes the construction of the numerical model for the honeycomb sandwich panel. Section 4 presents the experimental structure and the correlation between the experimental and numerical modes. Section 5 describes the setting up of the database. Section 6 presents the case studies and the damage assessment results. Finally, conclusions and forthcoming work are presented in Section 7.

The main problem of vibration-based damage assessment is to ascertain the presence, location and severity of structural damage given a structure’s dynamic characteristics. This principle is illustrated in

Let the observation vector
^{i}_{X,Y}_{X,Y}^{1}, Y^{1}), (X^{2}, Y^{2}), …, (X^{k},^{k}_{Y}_{|}_{X}

Linear approximation takes the

where _{1}_{2}, …,_{N}^{1}(X), X^{2}(X), …, X^{N}

After w is obtained from

where Y^{1}(X), Y^{2}(X), …, Y^{N}

The notion of entropy in information theory was introduced by Shannon as a measure of uncertainty [

Consider a set of _{1}, …, _{N}_{i}_{i}_{i}_{1}, …, _{N}

subject to the constraints:

where
_{r}_{r}_{0}(

The optimization problem (4) assigns probabilities to every _{i}_{i}_{i}_{i}_{i}_{i}/m_{i}

subject to the constraints:

In (5), the variational principle associated with
_{i}_{i}_{i}

Because of its general character and flexibility, we adopt the relative entropy approach for our problem, where the probability _{i}_{i}

subject to the constraints:

where

where
^{i}_{i}^{i}

The solution of the max-ent optimization problem is handled by using the procedure of Lagrange multipliers, which yields [

where

In ^{∗}

which gives rise to the following system of nonlinear equations:

where ∇_{λ} stands for the gradient with respect to λ. Once the converged λ^{∗} is found, the weight functions are computed from

Consequently, the prediction of the dynamic response of the honeycomb panels can be accomplished by equivalent continuum models. In the present study, the honeycomb panels are modeled with finite elements using a simplified three-layer shell model and the adhesive layer between the skin and core is modeled using linear springs. Because the properties of the skin are known, the attention is focused on modeling the effective properties of the adhesive layer and the core material.

A debonded region between the skin and core of a honeycomb panel is similar to a delamination in laminated composites. There are a considerable amount of analytical and numerical methods used to model delaminated composite laminates. Della and Shu [

In this study, the adhesive layer between the skin and core is modeled using linear springs, with reduced rigidity in debonded sectors, as shown in

The numerical model is built in Matlab^{®} using the SDTools Structural Dynamics Toolbox [

The experimental structure consists of a sandwich panel of 0.25 × 0.35 ^{2} made of an aluminum honeycomb core bonded to two aluminum skins, each with a thickness of 0.8

The identified modal parameters of the experimental structure are used to update the numerical model. The parameters that were updated in the numerical model are the following: the density and Young’s Modulus of the skins; the density, bending stiffness and shear correction factor of the core; and the stiffness of the springs representing the adhesive layer. The updated parameters are presented in

Where _{i}

The results show that the correlation between the numerical and experimental modes is almost perfect for the first three modes, with MAC values higher than 0.99. The fifth mode presents the lowest correlation, with a MAC value of 0.83. In this case, the first-order shear approximation may not be sufficient. The maximum difference between the experimental and the numerical natural frequencies is 11%.

The database is built as follows:

Define a set of damage scenarios to be used in the database.

Set

Parameterized the ^{j}

Build the numerical model associated with the

Construct a feature vector X^{j}

Add the pair of vectors (X^{j}^{j}

The damage scenarios used to construct the database consist of panels with circular-shaped debonded regions at one of the 117 points that are depicted in

Damage is modeled by circular-shaped debonded regions centered at some of the 117 points that are shown in ^{j}

The first six global mode shapes and natural frequencies that are shown in ^{j}

where _{j}

Since each mode shape is a vector of dimension 117 × 1 and each vector of natural frequencies has dimension 6×1, the feature vectors have dimension 123×1. A disadvantage of high-dimensional feature spaces, as the present case, is that points that are scattered in those spaces are usually far from each other. Thus, neighborhood methods that are based purely on distance become less useful. Nevertheless, Gupta [

The algorithm is tested for the three experimental damage scenarios shown in

The results of the max-ent linear approximation algorithm are compared with those obtained by solving the nonnegative least-squares problem posed in

Perform an experimental modal analysis of the damaged panel and identify the first six global mode shapes and natural frequencies.

Construct the feature test point X using

Read the feature vectors in the database.

Compute weights.

Select parameter _{i}

Solve the system of nonlinear equations presented in

Compute the weight functions from

Select the

Build the matrices

Compute the weights using the Matlab function

Read the observation vectors in the database and estimate the experimental damage from (

Both methods use the ten closest neighbors to the test point. The time needed to solve the linear approximation problem (stages 4 and 5 above) is 0.7 and 0.03 seconds for the max-ent and nonnegative least-squares approaches, respectively. _{i}

This article presented a new methodology to identify debonded regions in aluminum honeycomb panels using a linear approximation method handled by a statistical inference model based on the maximum-entropy principle. The algorithm was validated using experimental data from an aluminum honeycomb panel subjected to different damage scenarios.

The honeycomb panels were modeled with finite elements using a simplified three-layer shell model. The adhesive layer between the skin and core was modeled using linear springs with the rigidity being reduced at debonded locations. This numerical model predicted the first six modes of the undamaged and damaged panels with reasonable accuracy. Nevertheless, the numerical model can be improved by using higher order shear approximations.

In the three experimental cases, the linear approximation using the max-ent technique was successful in assessing the experimental damage. The detected damage closely corresponds with the experimental damage in all cases. In addition, the damage state of the panels is assessed in less than one second thereby providing the possibility of real-time damage assessment.

The results show that the proposed algorithm can assess debonded regions with sizes between 0.038

The proposed damage assessment algorithm provides only two options for a spring in the adhesive layer: either healthy or debonded. This can be improved by setting the output for each spring as a number associated with a debonding probability that ranges from 0 to 1.

Lastly, further research is needed to adapt this algorithm to cases with multiple debonded regions and to test its performance in more realistic configurations than a free plate.

Valentina del Fierro was supported by CONICYT grant CONICYT-PCHA/Magï£¡ster Nacional/2013-221320691. The authors acknowledge the partial financial support of the Chilean National Fund for Scientific and Technological Development (Fondecyt) under Grants No. 11110389 and 11110046.

Viviana Meruane and Valentina del Fierro built the updated numerical model, designed the experiment, processed and analyzed the experimental data, and tested the damage assessment algorithm. Alejandro Ortiz-Bernardin programmed the max-ent linear approximation algorithm and adapted it to the application case. The article was written by Viviana Meruane and Alejandro Ortiz-Bernardin.

The authors declare no conflict of interest

Principle of a vibration-based damage assessment algorithm.

Scheme of a honeycomb sandwich panel.

Lateral view of the numerical model: (a) undamaged panel, (b) panel with a debonded region.

Finite element model of the sandwich panel.

Fabrication of the experimental panel.

Experimental set-up.

Numerical and experimental undamaged mode shapes.

Numerical and experimental mode shapes with a debonded region at the center of the panel.

Example of an observation vector Y^{j}

Experimental damage scenarios introduced to the panel; debonded regions are enclosed by circles.

Damage assessment results for the first damage scenario.

Damage assessment results for the second damage scenario.

Damage assessment results for the third damage scenario.

Properties of the honeycomb core.

Property | Value |
---|---|

Cell size | 19.1 |

Foil thickness | 5 × 10^{−5} |

Thickness | 10 |

Density | 20.8 ^{3} |

Compressive strength | 0.448 |

Shear strength in longitudinal direction (_{xy} |
0.345 |

Shear modulus in longitudinal direction (_{xy} |
89.63 |

Shear strength in width direction (_{yz} |
0.241 |

Shear modulus in width direction (_{yz} |
41.37 |

Updated properties of the honeycomb sandwich panel.

Component | Property | Value |
---|---|---|

Skin | Density | 2.53 × 10^{3} ^{3} |

Young’s modulus | 6.14 × 10^{10} | |

| ||

Core | Density | 31.1 ^{3} |

Young’s modulus | 2.41 × 10^{10} | |

Shear correction factor | 0.67 | |

| ||

Adhesive layer | Stiffness healthy layer | 3.42 × 10^{8} ^{−1}/^{2} |

Stiffness debonded layer | 4.96 × 10^{7} ^{−1}/^{2} |

Running time for each stage of the linear approximation method.

Stage | Time (s) | ||
---|---|---|---|

Max-ent | Nonnegative least-squares | ||

4 | (a) | 5.91×10^{−1} |
4.30×10^{−3} |

(b) | 9.89×10^{−2} |
1.24×10^{−4} | |

(c) | 1.00×10^{−2} |
2.90×10^{−3} | |

5 | 8.7×10^{−5} |
8.7×10^{−5} |