複合材料積層板及樑結構之彈性常數識別

Abstract

本文提出結合複合材料力學及數值最佳化之方法來識別已成形加工之複合材料積層板及樑結構的彈性常數值。考慮複合材料積層板結構，在承受主結構方向即x-y方向之軸向拉力時，或複合材料積層樑結構做三點彎曲試驗，應變與複合材料積層結構之各勁度間的關係，以及利用所產生之應變來作為建立複合材料積層板及樑結構之材料彈性常數識別的數值最佳化目標函數，進而識別出複合材料積層板結構之材料彈性常數E1、E2、剪力模數G12及蒲松比□12。其方法是先量測出複合材料積層板結構受軸向拉力後之軸向應變、橫向應變及剪應變值，或複合材料積層樑結構受力矩後之軸向應變、橫向應變及剪應變值，利用這些應變值建立成為數值最佳化之目標函數，以複合材料積層板或樑結構的各項彈性常數值為變數，再求取有限制條件下，目標函數的總域極小值，藉搜尋不同路徑以識別出複合材料積層板及樑結構的各項彈性常數值，數值最佳化的方法中，其設計變數是以隨機多起始點同時搜尋的方法將其運用在數值最佳化中，隨機多起始點同時搜尋的方法是利用可變區間內各獨立變數以隨機取樣的方法找出各獨立變數的起始點，以此方式同時做為搜尋軌跡的開始，以找到目標函數之區域極小值。上述是利用擴增拉格蘭吉乘子法，來得到一個無限制條件的新目標函數，再結合多起始點軌跡搜尋法及貝氏逼近法和貝氏論點以及總域極小值之數值最佳化演算法，找到誤差函數之極小值，便可精確而且迅速識別出複合材料積層結構的各項彈性常數。 第二部份是研究兩階段識別法來識別當複合材料積層板或樑結構，只有軸向應變及橫向應變值時，則必需使用兩階段式識別法來複合材料積層板或樑結構之彈性常數識別法，依前述之方法先以應變規及其量測儀器來量測□45°對稱堆疊的複合材料積層板或樑結構的軸向應變及橫向應變值，做第一階段之複合材料積層板或樑結構之彈性常數識別，再固定其所識別出之剪力模數G12及蒲松比□12，然後再量測□□° (非□45°) 對稱堆疊的複合材料積層板或樑結構的軸向應變及橫向應變值，做第二階段之複合材料積層板或樑結構之彈性常數識別，而識別出複合材料積層板或樑結構之彈性常數E1及E2。 本文將以Graphite/epoxy (Gr/ep)及Glass/epoxy (Gl/ep)為材料，分別以不同之堆疊方式為例子，來證實本方法之可行性及精確性，再以Gr/ep之複合材料做實驗來驗證之。

A simple yet effective nondestructive evaluation technique for determining four elastic constants of symmetric angle-ply plates/beams is presented. For elastic constants identification of composite materials, using three measured strains of a single angle-ply laminate subjected to tensile testing, or three strains measured in, respectively, axial, lateral, and 45 degree directions from a symmetric angle-ply composite beam subjected to three-point bending testing are used to identify the elastic constants of the material. In the proposed method, the trial material constants of the angle-ply laminate/beam are used in the laminate/beam analysis to predict the strains in the laminate/beam. An error function is established to measure the difference between the experimental and theoretical predictions of the strains. The identification of material constants is then formulated as a constrained minimization problem in which the material constants are determined to make the error function a global minimum. The accuracy and capability of the proposed method are demonstrated by means of a number of examples on the material constants identification of angle-ply laminates with different layups. The accuracy of the proposed technique is studied by means of a number of examples on the elastic constants identification of graphite/epoxy (Gr/ep) or glass/epoxy (Gl/ep) symmetric angle-ply beams. The excellent results obtained in this study have demonstrated the feasibility and applications of the proposed technique. In this thesis, a two-level optimization method for elastic constants identification of symmetric angle-ply laminates/beams is also presented. Measured axial and lateral strains of two symmetric angle-ply laminates/beams with different fiber angles are used in the proposed method to identify four elastic constants of the composite material. In the first-level optimization process, the theoretically and experimentally predicted axial and lateral strains of a [(45°/-45°)2]s laminate are used to construct the error function which is a measure of the differences between the experimental and theoretical predictions of the axial and lateral strains. In the second-level optimization process, the shear modulus and Poisson’s ratio determined in the previous level of optimization are kept constant while the Young’s moduli of the second angle-ply laminate with fiber angles other than 45° are identified using the same minimization technique that has been used in the previous level. The accuracy of the proposed method are studied by means of a number of examples on the material constants identification of symmetric angle-ply laminates/beams made of different composite materials.