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Research Papers

# On the Accuracy of a Four-Node Delaminated Composite Plate Element and Its Application to Damage Detection

[+] Author and Article Information
Cesar F. Casanova

Department of Structural & Continuum
Mechanics,
Universitat Politècnica de València,
Spain;
Department of Applied Physics,
Spain
e-mail: ceferca@cam.upv.es

A. Gallego

Department of Applied Physics,
Spain
e-mail: antolino@ugr.es

M. Lázaro

Department of Structural & Continuum
Mechanics,
Universitat Politècnica de València,
Spain
e-mail: malana@mes.upv.es

The DKQ element is a C0 element.

$|∫ΩfgdΩ|≤(∫Ωf2dΩ)1/2(∫Ωg2dΩ)1/2$

For C1 bending elements, the trial solution w = {w0, θx0, θy0} $∈ H2$ Sobolev space; analogously it can be proven that the composite element also passes the “Patch Test.”

Contributed by the Design Engineering Division of ASME for publication in the Journal of Vibration and Acoustics. Manuscript received August 1, 2012; final manuscript received March 5, 2013; published online June 19, 2013. Assoc. Editor: Marco Amabili.

J. Vib. Acoust 135(6), 061003 (Jun 19, 2013) (10 pages) Paper No: VIB-12-1215; doi: 10.1115/1.4023994 History: Received August 01, 2012; Revised February 26, 2013

## Abstract

This paper presents a new four-node composite element, which incorporates nd delaminations through its thickness. Based on the extended finite element method (X-FEM) technology, the element is particularized on a CLT (classical laminate theory). Delamination is considered in the kinematic equations with additional degrees of freedom. The result is a four-node quadrilateral element requiring only two single FEM (finite element method) formulations, a bending one and a membrane one. An important result is that this formulation has the same accuracy as when separate elements are considered (“four region approach”). It is furthermore proven that the delaminated element passes the “patch test” if the selected FEM formulations to build the element pass the test in the pure single problems, making this methodology very attractive to develop other fractured elements. To illustrate this result, two benchmark problems were studied: first a complete delaminated cantilever plate, and second a complete delaminated circular plate. The element was tested in the context of SHM (structural health monitoring). Frequency shifts, damage indexes, and changes in mode shapes and frequency response functions (FRF) were obtained to quantify the severity of damage due to delamination.

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## Figures

Fig. 1

Four regions approach

Fig. 2

(a) Delaminated cantilever beam. (b) FEM model.

Fig. 3

(a) Relative error in L2 norm for separate elements (DKQ) and delaminated element. (b) Relative error in eigenfrequencies (1–4). Lines (DKQ), dots (delaminated element).

Fig. 4

(a) Delaminated circular clamped plate (quarter represented). (b) FEM model.

Fig. 5

(a) Relative error in L2 norm for separate elements (DKQ) and delaminated element. (b) Relative error in eigenfrequencies (1–3). Lines (DKQ), dots (delaminated element).

Fig. 6

Delaminated FEM model of a cantilever beam. Green elements represent the delamination.

Fig. 7

Eigenvector 6 of displacements for extensions I, II, and III. (a) Composite [0 90]3S. (b) Composite [0 90 45 −45 0 90]S. The undamaged one is represented by 0. The vertical dashed line represents the center of the delamination.

Fig. 8

Eigenvector 6 of curvatures for extensions I, II, and III. (a) Composite [0 90]3S. (b) Composite [0 90 45 −45 0 90]S. The undamaged one is represented by 0. The vertical dashed line represents the center of the delamination.

Fig. 9

Frequency shifts (%) for composite [0 90]3S for modes 5, 6, and 7. (a) Influence of the delamination extension. Delamination center at x=0.45 L. (b) Influence of delamination location, with extension 10%.

Fig. 10

Eigenvector 6 of displacements for different delamination locations. Extension is 10%.

Fig. 11

Composite clamped plate. Sections where eigenvectors are obtained (F1 and F2). Actuator (A1). Sensor (S1). Delamination (D).

Fig. 12

Eigenvectors of second mode at section x = 0.13 m (F1). Extensions I, II, and III. (a) Displacements w. (b) Curvatures κx.

Fig. 13

Eigenvectors of second mode at section y = 0.58 m (F2). Extensions I, II, and III. (a) Displacements w. (b) Curvatures κy.

Fig. 14

FRF at location x=0.75 m and y=0.30 m. (a) Extensions 0, I, II, and III. (b) Extensions 0, IV, V, and VI.

Fig. 15

Selected bandwidths of FRF (to compute damage indexes). (a) Extensions 0, I, II, and III. (b) Extensions 0, IV, V, and VI. Marked eigenfrequency 1139 Hz.

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