## 1 Introduction to the problem

Metal structures represent one of the greatest achievements of civil engineering as it is a combination of high mechanical properties with a significant reduction in the mass of buildings compared to other construction techniques, such as reinforced concrete, in which the mass is significant.
^{1} However, in steel buildings, there are structural elements, such as connecting plates, which are more stressed than others and subject to compound external loads such as the bi-axial traction [1]. Hence, there is a need to periodically evaluate the state of health of these plates by non-destructive testing and evaluation (NDT/NDE) to quantify the distribution of the deformations [2,3,4] as they could produce excessive internal mechanical tensions [1]. Ferrous materials, such as steels, often highlight inhomogeneities that lead to a phenomenon of electrical run outs that cause unwanted phenomena of uncertainty in the measurements. This could arise from the numerous heat treatments during the manufacture of the product which alter the microstructures causing strong changes in the properties of the material [5]. Furthermore, the metallurgy of the material itself can lead to the modification of the crystal lattice, which leads to the random variation in both magnetic permeability and electrical conductivity [5]. In [6,7], such phenomenon has been studied in the case of finite elasticity, modeling the microstructure in terms of bi-phase materials (derived from the solid-to-solid transformations) and achieving the distribution of deformations as a function of the external bi-axial load applied [8]. Although theoretically interesting and potentially applicable in different fields (necking, shear-band, etc.), this model [6] presents difficulties in obtaining deformations, except for simple geometries and constant boundary conditions along the edges of the plates [1]. Accordingly, the numerical approach results to be the most appropriate to tackle this problem, but the reduced possibility of obtaining analytical existence/uniqueness conditions for this model does not prevent it from being affected by possible ghost solutions [9]. Anyway, the model [6] is theoretically able to produce 2D maps of the amplitude of the mechanical deformation, so that, through the constitutive laws, it is possible to find the 2D maps of internal mechanical stresses [1,10], useful to evaluate the state of health of the plate. Unfortunately, these maps
^{2} require a high computational load for being computed, which makes them useless for real-time applications. In addition, in the framework of small deformations, the evaluation of the plate deformations may not provide useful results, due to the very low values of their amplitudes. Then, it is preferred to know the external bi-axial load producing the detected deformation state to understand if the steel plate has been overstressed. This way (i.e., starting from the distribution of the deformations to achieve the external bi-axial load), we make this inverse problem into a classification one. With these premises, the idea we propose in the present work exploits the fact that, from an experimental point of view, the 2D maps of the mechanical deformations obtainable under the same external bi-axial load conditions [6,7] are equivalent to those obtained by inducing eddy currents (ECs) in the plate and measured by a FLUXSET-type probe [3,11,12] moved by an automatic step-by-step device along *x* and *y* axes with the aim to pick-up the voltage (Figure 1) [13]. On the sidelines, let us propose a further figure (Figure 2) extracted from the center of 2D ECs of Figure 1, which shows a kind of opening in its central zone. Since similar bi-axial loads (which are close to each other in terms of amplitudes) applied to a plate produce similar deformations on it [1,10], they can be grouped into a single class of loads. Furthermore, for each applied load similar to a predetermined load *k*, the related EC map can be evaluated to obtain a “class” of similar images. If the preset bi-axial loads are *k*,…, Class

List of the acronyms

Acronym | Description | Acronym | Description |
---|---|---|---|

NDT/NDE | Non-destructive testing and evaluation | EC | Eddy current |

AC | Alternative current | RMS | Root mean square |

IoF | Index of fuzziness | AFD | Axiomatic fuzzy divergence |

FIS | Fuzzy inference system | FKM | Fuzzy K-means |

SOM | Self-organizing map | I/O | Input/output |

List of the exploited symbols

Symbol | Description | Symbol | Description |
---|---|---|---|

Number of preset bi-axial loads | Isochoric deformation | ||

M | Class in which d is defined | Deformation gradient | |

Stored energy | Eigenvalues of the stretch tensor | ||

Homogeneous, regular and bounded body | Boundary of | ||

Traction vector | Constant traction tensor | ||

Outward normal on | Kirchhoff–Piola tensor | ||

EC map | Gray levels of | ||

Fuzzy membership function | Fuzzified image | ||

Fuzzifiers | Nearest ordinary set | ||

Generic | U | Universe of discourse | |

Two generic images in the class |

## 2 Some theoretical backgrounds

According to Chen’s theory, it is known that energy plays a key role to study problems concerning the stress–deformation relationship [6]. With this premise, let us introduce the following definitions.

(Set of 2D isochoric deformations) Let us indicate by ^{3} such that *M* the class of isochoric deformations, i.e., the class of deformations such that

(The principal stretches) Once

### 2.1 Stored energy

Indicating by *Y* the strain-energy function of the material, it can be expressed in the following form [6]:

*G*[6]:

^{4}

*G*is defined on

This model describes in detail both equilibrium states and stability. However, it is worth noting that it neglects all the microstructural effects.

In the framework of finite elasticity, a lot of materials support fields characterized by strong gradient of discontinuity where smooth variation in deformations takes place except for particular sections on which the gradient gap can occur.

In the framework of plastic deformations, shear-band formations in

### 2.2 Minimization problem

As known, *Y* depends on

*Y*is not lower semicontinuous with respect to the weak convergence in

*Y*in the deformation admissible set.

Although this approach is theoretically interesting for recovering

Then, we here propose a new fuzzy procedure with low computational complexity that, exploiting the 2D EC maps [3,18], associates an unknown deformation distribution
^{5} to an external bi-axial load.

## 3 Experimental database

### 3.1 Experimental campaign of measurements

It was developed at the “Mediterranea” University of Reggio Calabria (Italy) – Laboratory of Electrotechnics and NDT/NDE. A set of 180 mm ^{6} were subjected to symmetrical and gradually increasing bi-axial loads and, then, investigated. For the training step, the plate was subjected to the load starting from 160 kN to the final load of 200 kN, with steps of 10 kN. Because of the microscopic structure of the material, each external load modifies locally the magnetic properties of the plate. Accordingly, its state of degradation was investigated by analyzing the magnetic changes induced in the structure of the material when the deformations take place. The probe [3] consists of three coils: (1) an exciting coil to induce EC in the specimen (outer diameter = 10 mm, inner diameter = 6 mm and length = 5 mm) and a FLUXSET^{®} sensor (length = 6 mm and diameter = 1.2 mm) which is located between the exciting coil and the plate; (2) a driving coil; and (3) a pick-up coil. Moreover, the approximate lift-offs for both the exciting coil and the sensor are 2 and 0.6 mm, respectively. The pick-up voltage provides a measure, which is proportional to the component of the amplitude of the magnetic field, _{pp} amplitude.

### 3.2 EC imaging

The sensor was assembled to a 0.5 mm step-by-step automatic scanning system moved on a square portion (70 mm side) at the middle of the specimen to map the area where maximum deformation is more likely to occur (e.g., see Figure 2). After each symmetrical bi-axial load application, the specimen was investigated by means of the probe obtaining four 2D signal representatives of the real part, the imaginary part, the module and the phase of the pick-up voltage (mV) at each point of the surface of the specimen. Each 2D signal achieved represents a matrix that, by means of MatLab Toolbox^{®}, has been transformed into a 2D image in which each pixel represents a single measurement sampled by the probe in a specific position of the step-by-step scanning system (e.g., see Figure 1). Therefore, the achieved 2D image
^{7} is significant of the distribution of the deformations on the specimen. For each specimen, four 2D images were obtained related to the real part, imaginary part, module and phase of the pick-up voltage, respectively. Since similar loads provide similar deformations (characterized by similar magnetic properties), a series of gradually increasing bi-axial loads were applied to the plate, obtaining in this way five classes of images (60

## 4 Proposed method

### 4.1 EC maps as fuzzy images: some basic concepts

A 2D EC map with *L* gray levels can be considered as a 2D image, indicated by

Let us consider

### 4.2 Choice of the membership function

The approach presented here to make a suitable membership function exploits the intensification operator to reduce the fuzziness and, at the same time, to enhance the contrast of the image itself. So, if

### 4.3 Fuzziness quantification in a fuzzy image

Once a fuzzy image *ad-hoc*” index of fuzziness (IoF), denoting the degree of ambiguity present in ^{8}:

^{9}

### 4.4 Fuzzy similarities and fuzzy images

Let us indicate by *U* indicates the universe of discourse such that both *U*. To define an FS measure, we need to define a function

- Property of reflexivity, which establishes that each image is similar to itself with the maximum degree, must be verified:
$\begin{array}{c}\forall F\left({I}_{x}\right)\in U\Rightarrow \text{FS}\left(F\right({I}_{x}),F({I}_{x}\left)\right)=\underset{\forall F\left({I}_{x}\right),F\left({I}_{y}\right)\in U}{\mathrm{max}}\\ \phantom{\rule{0ex}{0ex}}\text{FS}(F{\left(I\right)}_{x},F{\left(I\right)}_{y})=1.\end{array}$ - FS must be symmetric. In other words, it should not depend on the order in which the fuzzy images are taken into account. That is,
$\text{FS}\left(F\right({I}_{x}),F({I}_{y}\left)\right)=\text{FS}\left(F\right({I}_{y}),F({I}_{x}\left)\right).$ - In addition,where
$\forall F\left({I}_{x}\right)\in U\Rightarrow \text{FS}\left(F\right({I}_{x}),\overline{F\left({I}_{x}\right)})=0,$ represents the complementary fuzzy image of$\overline{F\left({I}_{x}\right)}$ such that its membership values are computed as$F\left({I}_{x}\right)$ $1-{m}_{{I}_{x}}\left({a}_{ij}\right).$ - Finally,
, if$\forall F\left({I}_{x}\right),F\left({I}_{y}\right),F\left({I}_{z}\right)\in U$ ^{10}

### 4.5 Suitable formulations of fuzzy similarities

For our purposes, with respect to the four required properties specified in Section 4.4, we need to exploit FS formulations with low computational load particularly useful for real-time applications. With this goal in mind, excellent candidates are the Chaira FSs [14,15] that, properly adapted to the case under study, take the following form:

### 4.6 Construction of the $\tilde{N}$ classes starting from the $\tilde{N}$ external bi-axial load conditions

Let *k*th class of loads considered similar to each other. In addition,

## 5 Representative 2D EC image of each class of loads: a fuzzy image fusion approach

In order to get,

- We initially subdivide
and$F\left({I}_{k}^{s}\right)$ ,$F\left({I}_{k}^{z}\right)$ , into$\forall s,\phantom{\rule{0ex}{0ex}}z\in P$ *T*not overlapped subimages ( pixels). Let them be$10\times 10$ and$F{\left({I}_{k}^{s}\right)}_{{t}_{1}}$ ,$F{\left({I}_{k}^{z}\right)}_{{t}_{2}}$ .${t}_{1},{t}_{2}\in H=\{1,\dots ,T\}$ - Then, we computeand
$\mathrm{F}{S}_{\chi}(F{\left({I}_{k}^{s}\right)}_{1},F{\left({I}_{k}^{z}\right)}_{1}),\phantom{\rule{0ex}{0ex}}\chi =1,2,3,4$ .$s,z\in P$ - Letbe the pair of subimages providing
$(F{\left({{I}_{k}}^{\overline{s}}\right)}_{1},F{\left({{I}_{k}}^{\overline{z}}\right)}_{1}),\phantom{\rule{0ex}{0ex}}\overline{s},\overline{z}\in P,$ $\mathrm{max}\phantom{\rule{0ex}{0ex}}\left\{{\text{FS}}_{\chi}\right(F{\left({{I}_{k}}^{\overline{s}}\right)}_{1},F{\left({{I}_{k}}^{\overline{z}}\right)}_{1}\mathrm{)\}}.$ - Indicating by
the portion of$F{\left({I}_{k}\right)}_{1}$ corresponding (in terms of pixel positions,$F\left({I}_{k}\right)$ ) to the subimages$(i,j)$ and$F{\left({I}_{k}^{s}\right)}_{1}$ , we pose:$F{\left({I}_{k}^{z}\right)}_{1}$ in order that each pixel belonging to${\left[F{\left({I}_{k}\right)}_{1}\right]}_{(i,j)}={\left(1+{e}^{-\frac{{\left[F{\left({I}_{k}^{s}\right)}_{1}\right]}_{(i,j)}+{\left[F{\left({I}_{k}^{z}\right)}_{1}\right]}_{(i,j)}}{2}}\right)}^{-1}$ is achieved by the sigmoidal smoothing approach on the arithmetic average of the corresponding pixels of both$F{\left({I}_{k}\right)}_{1}$ and$F{\left({I}_{k}^{s}\right)}_{1}$ .$F{\left({I}_{k}^{z}\right)}_{1}$ - Repeat steps (1)–(4),
obtaining the fuzzy image$\forall {t}_{1},{t}_{2}\in H$ related to the$F\left({I}_{k}\right)$ th class of external bi-axial loads.$k$ - To achieve all of the fuzzy images related to all classes, it is sufficient to repeat steps (1)–(5),
.$\forall k=\mathrm{1,}\dots ,\tilde{N}+1$

## 6 A brief overview of the techniques used to compare the results

To compare the quality of the results obtained by the proposed procedure, the classification has also been carried out through fuzzy inference systems (FISs; Mamdani and Sugeno types), Fuzzy *K*-means clustering technique and self-organizing map (SOM).

### 6.1 FISs: Mamdani and Sugeno approaches

An FIS realizes an input/output (I/O) mapping by means of inference based on ^{®} Fuzzy Toolbox it has been made a Mamdani FIS, whose bank of fuzzy rules is constituted by 25 fuzzy rules with four multiple antecedents,

### 6.2 Fuzzy *k*-means approach and SOM maps

Here, a variation in the well-known unsupervised learning algorithm [24,27] for clustering problems has been implemented by the MatLab^{®} Fuzzy Clustering Toolbox. Fixing *a priori* six clusters (the six classes considered as output), they will be centered far away from each other. Then, each point of a given data set (^{®} SOM Toolbox.

## 7 Some relevant results

The procedure described in Sections 4 and 5 has been implemented on Intel Core 2 CPU 1.47 GHz and MatLab^{®} R2017a. Preliminarily, for each class of external bi-axial loads, the ranges of IoFs have been computed. Their high values highlight high fuzziness in the images belonging in each class as well as the values of IoF related to each

Ranges of IoF related to the images belonging to each class and to each

Class | Range of IoF | IoF of | Class | Range of IoF | IoF of |
---|---|---|---|---|---|

0.92–0.97 | 0.95 | 0.79–0.83 | 0.80 | ||

0.88–0.91 | 0.90 | 0.80–0.86 | 0.84 | ||

0.79–0.85 | 0.83 | Without loads | 0.83–0.91 | 0.89 |

Performance of classification compared to other soft computing techniques

Technique | CPU time (s) | Pick-up voltage (real part) (%) | Pick-up voltage (imaginary part) (%) | Pick-up voltage (module) (%) | Pick-up voltage (phase) (%) |
---|---|---|---|---|---|

0.45 | 99.6 | 99.5 | 99.6 | 99.6 | |

0.43 | 99.4 | 99.4 | 99.4 | 99.2 | |

0.34 | 99.8 | 99.9 | 99.7 | 99.8 | |

0.25 | 99.6 | 99.5 | 99.6 | 99.6 | |

FIS (Mamdani) | 0.45 | 99.1 | 99.4 | 99.4 | 99.3 |

FIS (Sugeno) | 1.05 | 99.9 | 99.9 | 99.8 | 99.9 |

FKM | 1.23 | 99.7 | 99.5 | 99.7 | 99.6 |

SOM | 0.99 | 99.8 | 99.6 | 99.8 | 99.7 |

## 8 Conclusions and perspectives

In this article, a novel approach based on FS computations for the assessment of the integrity of steel plates under an external bi-axial load is proposed. Through EC testing, 2D electrical maps related to the 2D distribution of deformations on a plate have been experimentally achieved and grouped into different classes satisfying the principle according to which similar external bi-axial loads produce similar deformation distributions. Fuzzifying properly all the obtained 2D maps and evaluating their fuzziness using a suitable index, 2D maps representative of each class are obtained exploiting a new approach based on the fuzzy image fusion. Finally, four different FS formulations have been presented and used to determine the association of an unknown load (and, therefore, the distribution of deformations generated by it) with a specific class of external loads. By the proposed approach, on one hand, the determination of the deformations in a steel plate is translated in terms of a suitable classification problem and, on the other hand, the theoretical computation of mechanical deformations is translated into an experimental campaign of EC measurements with portable instrumentation (useful for *in situ* real-time applications). The obtained results can be considered encouraging because they have put in evidence performances of classification comparable with other well-established soft computing procedures characterized by a higher computational complexity. However, a better formalization of the proposed procedure, assisted by a more careful selection of FS formulations, will surely improve the performance of the proposed approach. In addition, the use of step function voltage to excite the probe is desirable because a single-step function contains a set of frequencies and, since the penetration depth depends on the frequency, they acquired simultaneously information from a range of depth. Last but not least, it seems useful to take into account vagueness and/or uncertainty phenomena due to inhomogeneity of the morphological structure of the material, which inevitably affects the distributions of magnetic permeability and electrical conductivity of the specimen.

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## Footnotes

^{1}

Seismic actions on a building are proportional to the mass of the building itself, whereby high masses involve higher seismic stresses.

^{2}

There is no experimental instrumentation able to derive such deformations so that it is necessary to get them analytically.

^{3}

An isochoric deformation is a first-order tensor, i.e., a vector.

^{4}

^{5}

The EC distributions in a plate are closely linked to the plate’s deformation *d*.

^{6}

Hot rolled steel for open profiles: specific weight

^{7}

The 2D image is suitably nuanced through the MatLab shading interp function.

^{8}

^{9}

It is possible, because

^{10}

That is,