Identification par recalage modal et fréquentiel des propriétés constitutives de coques en matériaux composites

Abstract

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 such as marine, automotive or aerospace structural components, a thorough characterization of the constituent material properties is needed. Because of the number and the inherent variability of the constitutive properties of composite materials, the experimental characterization is quite cumbersome and requires a large number of specimens to be tested. 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 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 work, a new mixed numerical-experimental identification method based on the modal response of thick laminated shells is presented. This technique is founded on the minimisation of the discrepancies between the eigenvalues and eigenmodes computed with a highly accurate composite shell finite element model with adjustable elastic properties and the corresponding experimental quantities. In the case of thick shells, the constitutive parameters that can be identified are the two in-plane Young's moduli E1 and E2, the in-plane Poisson's ratio ν12 and the in-plane and transverse shear moduli G12, G13 and G23. To determine these six parameters, a typical set of 10 to 15 measured eigenfrequencies and eigenmodes is selected, and the over-constrained optimisation 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 (mode shapes and frequencies). As the accuracy of this inverse method directly depends 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, varying the degree of approximation of the model allows to adjust the needs in accuracy and/or computation time. It is shown that for thick and highly orthotropic plates, the formulation exhibits a good convergence on the eigenfrequencies with p = 3 and a nearly exact solution for p = 7. 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 minimisation 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. They are based upon the relative differences between the eigenfrequencies, upon 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. In this work, the convergence properties of the minimisation algorithm are also investigated. It can be observed that usually the minimisation requires between 3 and 6 iterations to reach a residual error of less than 0.2 %. Finally, real identification examples are presented, for various thin to thick unidirectional carbon fiber plates and for a relatively thick cross-ply glass – polypropylene specimen. 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. It can be concluded that overall, when the test specimens are moderately thick, the present identification method can accurately determine the in-plane Young's and shear moduli as well as the transverse shear moduli and the in-plane Poisson's ratio. It is also seen that the stability of the method is excellent as long as the number of measured modes is reasonably larger than the number of parameters to be identified.