# A Multi-Level Optimization Method for Elastic Constants Identification of Composite Laminates
## 1 Introduction
They have also shown that the coefficients of variation, of the measured strains/displacements falling in the range of 5%, can lead to acceptable elastic constants (\(E_{1}\) = Young's modulus in fiber direction, \(E_{2}\) = Young's modulus in the direction perpendicular to the fiber direction, \(\
u_{12}\) = Poisson' ratio, and \(G_{12}\) = in-plane shear modulus) with coefficients of variation less than 15%. Therefore, it is obvious that the use of strains for material constants identification of composite plates is viable and may produce good estimations of the elastic constants. Nevertheless, the material constants identification via the plate bending testing approach is also time consuming and requires the design of a special clamping fixture. Furthermore, it may be difficult to model the actual boundary conditions of the plate, which have been tested in the analysis of the plate deformation. Thus, the search for simple testing procedures and for the accurate identification of elastic constants, for engineering applications, still remains an important topic of research. To tackle this problem, [PERSON] and his associates [17; 18; 19] attempted to test composite axial members or beams to identify the elastic constants. In particular, [PERSON] and [PERSON] [18] have used one angle-ply laminated member in uniaxial tensile testing to determine the elastic constants of the laminate. The testing of this type of laminate required the use of a pair of special mechanical end grips to restrain the two ends of the member so that the St. Venant's end effects can be reduced and the deformations at the laminate center can be similar to the shearing and extension of an unrestrained laminate. If the end grips cannot allow the in-plane shearing of the laminate to occur freely, it may not be able to attain good estimations of the elastic constants. However, if the conventional/hydraulic grips are used to perform the tensile testing, in which the two ends of a laminated member are totally clamped, two symmetric angle-ply composite laminates with different fiber angles can be used to identify the elastic constants [19]. While the use of two symmetric angle-plylaminates can produce good results, in terms of time, it is still expensive to fabricate the laminates and perform the tests. Therefore, as far as engineering applications are concerned, it is worthwhile to investigate whether it is possible to use only one composite laminate in the simple uniaxial tensile testing to identify the elastic constants of the laminate. It is noted that if a non-angle-ply composite laminate is used in the identification process, the search for accurate elastic constants may become very difficult and the previously proposed optimization techniques for elastic constants identification using one or two angle-ply laminates may be unable to produce good results in an efficient and effective way. Therefore, a new and effective optimization technique should be developed to extract elastic constants of a non-angle-ply composite laminate. In this paper, a simple yet effective multi-level optimization method for elastic constant identification of composite materials is presented. The solutions of a number of well-known minimization problems using the proposed multi-level optimization method, as well as several other existing global minimization algorithms, are first given to demonstrate the capability of the proposed method to search for the global minima. The elastic constants identification of a [0\({}^{\circ}\)/\(\pm\)45\({}^{\circ}\)]\({}_{\text{s}}\) composite laminate is then performed to illustrate the applications of the proposed method. The accuracy and feasibility of the proposed method to identify elastic constants of different composite materials are studied by way of several numerical and experimental examples.