Keywords: mixed numerical-experimental methods, parameter identification, composite laminates, HSDT shell finite elements, material properties, modal analysis.

Fibre-reinforced composites are being increasingly used as alternatives for conventional materials primarily because of their high strength, specific stiffness, light weight and adjustable properties. However, before using this type of material with confidence in industrial applications, a thorough characterization of the constituent material properties is needed. Unfortunately, with laminates resulting from a stacking of layers, the constitutive properties can not be accurately estimated by performing experimental tests on one lamina and by extrapolating the results to a multilayered composite according to the fibre orientation and the stacking sequence of the plies. An elegant way to circumvent this lack consists in using mixed numerical-experimental methods which constitute powerful tools for estimating unknown constitutive coefficients in a numerical model of a composite structure from static and/or dynamic experimental data collected on the real structure. Starting from the measurement of quantities such as the displacements, stresses, or natural frequencies and mode shapes, these methods allow, by comparing numerical and experimental observations, the progressive refinement of the estimated material properties in the corresponding numerical model. In this domain, dynamic mixed techniques have gained in importance owing to their simplicity and efficiency.

In this paper, a new mixed numerical-experimental identification method based on the modal response of thick laminated shells is presented. This technique is founded on the minimization of the discrepancies between both the natural frequencies and mode shapes computed with a highly accurate composite shell finite ele- ment model with adjustable elastic properties and the corresponding experimental quantities derived with a precise contact-free measurement setup. In the case of thick shells, the constitutive parameters that can be identified are the two in-plane Young's moduli and , the in-plane Poisson's ratio and the in-plane and transverse shear moduli , and . To determine these six parameters, a typical set of 10 to 15 measured eigendata is selected, and the over-determined optimization problem is solved with a nonlinear least squares algorithm.

In order to maximize the quality of the identification, free-free boundary conditions and a non-contacting modal measurement method are chosen for the experimental determination of the eigenparameters. To obtain optimal experimental conditions, the specimens are suspended by thin nylon yarns and excited by a calibrated acoustic source (loudspeakers) while the dynamic response is measured with a scanning laser vibrometer. The measured frequency response functions are then treated in a modal curve fitting software to obtain a high quality set of modal data.

As the accuracy of this inverse method is directly depending on the precision of the finite element model, a family of very efficient thick laminated shell finite elements based on a variable p-order approximation of the through-the-thickness displacement with a full 3D orthotropic constitutive law has been developed. In these elements with degrees of freedom per node, varying the degree of approximation of the model allows to adjust the needs in accuracy and/or computation time. It has been shown that for thick and highly orthotropic plates, the formulation exhibits a good convergence on the eigenfrequencies with and a nearly exact solution for . In comparison to other 3D solid or thick shell elements, such as layerwise models, the presented elements show an equivalent precision of the computed eigenfrequencies and are computationally less expensive for laminates with more than 8 plies.

A classical Levenberg-Marquardt nonlinear least squares minimization algorithm is used to solve the inverse problem of finding the elastic constitutive parameters which are best matching the experimental modal data. Original multiple objective functions are used for comparing the computed and measured values. In comparison to other estimation methods, the current technique generates improved results, since the functionals are not only based upon the relative differences between the natural frequencies but also on the diagonal and off-diagonal terms of the so-called modal assurance criterion norm on the mode shapes, and upon geometrical properties of the mode shapes such as the nodal lines. As for other iterative minimization methods, the derivatives of the objective functions must be computed precisely and with a minimum of computational cost in order to accelerate the convergence of the procedure. In the present work, these derivatives with respect to the identification parameters require the computation of the eigenfrequencies and eigenmodes derivatives. A comparison of various methods for computing these modal derivatives have shown that a direct finite difference scheme constitutes in general a reasonable compromise. Investigations on the convergence properties of the minimization algorithm have shown that the procedure usually requires between 3 and 5 iterations to reach a residual error of less than 0.1%.

Finally, two real identification examples are presented, one for thick orthotropic E-glass/polypropylene (G/PP) specimen and another for a relatively thick unidirectional graphite/polyether-ketone-ketone (AS4/PEKK) plate. The robustness and the convergence of the present identification method are studied and the identification results are compared to those obtained with classical static tests. An estimation of the errors and uncertainties on the final estimated parameters is also evaluated. It can be concluded that overall the present identification method is able to accurately determine the in-plane Young's and shear moduli and to a lesser extent the transverse shear moduli and the in-plane Poisson's ratio.

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