Damage detection in CFRP composite beams based on vibration analysis using proper orthogonal decomposition method with radial basis functions and cuckoo search algorithm

Abstract

This paper presents a new fast approach for crack identification using vibration analysis based on model reduction using proper orthogonal decomposition method with radial basis function (POD-RBF). The method is formulated as an inverse problem for detecting the position, length and depth of crack in Carbon Fibre Reinforced Polymer (CFRP) composite structures, where Genetic and Cuckoo search algorithms are used to minimize the cost function based on numerical and experimental natural frequencies. The results show that the POD-RBF combined with Cuckoo search algorithm is an efficient and a feasible methodology of predicting the width, depth and position of double rectangular notches in CFRP beams. The stability of this technique is tested by introducing a white Gaussian noise in the frequencies data input. The results show that the proposed approach is stable when the noise level is lower than 6%.

Introduction

Composite materials are nowadays increasingly used as alternative to conventional materials, especially in the aerospace industry, because of their high strength, specific rigidity, and their mechanical properties being adjustable within wide limits. The composite materials are often subjected to various undetectable damages; i.e. cracks in fibres, matrix, and interfaces between fibres and matrix, which are very common as fatigue failure mode in composites [1]. The presence of cracks in beams of non-homogeneous material has been investigated both theoretically and experimentally in a number of works [2], [3].

Damage identification techniques, using model reduction based on the proper orthogonal decomposition (POD) method, have the main objective to estimate the crack length and its position in a structure using boundary displacements as input data. An optimization algorithm, such as Genetic Algorithm (GA) or Particle Swarm Optimization (PSO), is then applied for the minimization of the error function expressed as the difference between the boundary displacements of the actual crack and those of the estimated crack [4], [5]. The inverse damage detection and localization based on model reduction using a finite element model of bi-dimensional monolithic composite beam reinforced by a graphite-epoxy was used to define a numerical model of a tested structure, in which different scenarios of damage were considered by stiffness reduction [6]. The accuracy of the method was verified through different damage configurations. The inverse problem based on optimization techniques for damage detection in different fields of structures is used by [7], [8].

The proposed approach was the radial point interpolation method (RPIM) presented in Ref. [9] for the analysis of concrete structures using an elastic continuum damage constitutive model. A theoretical model of describing the behaviour of CFRP cantilever beams was developed on the basis of previous research, which investigated free vibration of beams with single and multiples cracks [10] or notches [11], [12]. The dynamic response of structural elements was modified due to real damage resulting from defects, loss of integrity and cracking of FRP material by overloading during service life [13], [14], [15].

The application of three-dimensional spectral element method (SEM) into propagation problems in plate structures to predict damage location was reported in literature. The main feature of SEM is that the mass matrix is diagonal. This is because of the choice of Lagrange interpolation function supported on the Gauss–Lobatto–Legendre (GLL) points in conjunction with the GLL integration [16]. The proposed model can detect damages in those structures.

A large number of works based on cracks in composite materials and/or notched CFRP elements can be found in Ref. [17]. The damage detection and localization of damage in structures by applying concepts derived from the theory of proper orthogonal decomposition (POD) was investigated by simulating tests on two beams and provided promising results [18]. POD provided the most efficient way of capturing the dominant components of an infinite-dimensional process with only (often surprisingly) few modes. Various applications of POD to structural dynamics problems were carried out in the literature [19], [20], [21].

The inverse problem algorithm for detecting and locating damage in piezoelectric structure using extended finite element method was presented in Ref. [22]. The inverse problem of detecting multiple voids in piezoelectric structure was used in Ref. [23] along with Extended Finite Element method (XFEM). The numerical method based on combination of classical shape derivative and of the level-set method for crack front propagation used in structural optimization was utilized to minimize objective function. Although Finite Element Method (FEM) is commonly used in conjunction with damage assessment in structures, other advanced numerical techniques such as isogeometric analysis [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37] may offer useful advantages in saving computational efforts. A design methodology based on a combination of isogeometric analysis (IGA) was presented by Ghasemi et al. [38] for topology optimization of piezoelectric/flexoelectric materials. The accuracy of the IGA model was validated through numerical examples by including a beam under a point load and a truncated pyramid under compression with different electrical boundary conditions. A new approach was presented by Rabczuk et al. [39] for modelling discrete cracks in meshfree methods, in which the crack could be arbitrarily oriented, but its growth was represented discretely by activation of crack surfaces at individual particles. The crack was modelled by a local enrichment of the test and trial functions with a sign function and the model was applied to several 2D problems and compared to experimental data. A new robust and efficient approach for modelling discrete cracks in meshfree methods was described by Rabczuk et al. [40]. They applied this method to several two- and three-dimensional problems in statics and dynamics. A dual-horizon peridynamics formulation, which allowed for simulations with dual-horizon with minimal spurious wave reflection and also analyze the crack pattern of random point distribution and the multiple materials issue in peridynamics, was presented by Ren et al. [41]. A dual-horizon peridynamics (DH-PD) formulation that naturally included varying horizon sizes and completely solved the ‘ghost force’ issue and two-dimensional and three-dimensional examples including the Kalthoff–Winkler experiment and plate with branching cracks was tested to demonstrate the capability of the method presented in Ref. [42].

A crack propagation algorithm, which was independent of particular constitutive laws and specific element technology, was presented by Areias et al. [43]. The crack paths were implicitly defined from the localized region. A realistic 3D finite strain analysis and crack propagation with tetrahedral meshes required mesh refinement/division as presented by in Ref. [44]. Areias et al. [45] proposed an alternative, simpler algorithm for FEM based on computational fracture in brittle, quasi-brittle and ductile materials based on edge rotations. Areias et al. [46] proposed an algorithm according the previous research by including the injection of continuum softening elements directly in the process region and presented an extension of the classical smeared approach to fracture.

Our proposed approach, based on POD-RBF combined with CS algorithm and GA, has never been applied to composite structures and in this paper we present the first attempt to do so. It is applied to damage detection of double notches in CFRP composite beams. The POD-RBF approach is used to reduce the dimensions of snapshot matrix calculated by FEM with different notch positions and dimensions.