彈性波於多層域均質與非均質材料之暫態波傳理論解析與數值計算

Abstract

當一均佈動力載荷施加於均質或非均質層狀介質的表面時，本文以拉普拉斯轉換技巧，分析其域內的一維暫態波傳問題。對於均質多層域的暫態響應，採用三種不同的分析及數值計算方式：廣義射線法，Durbin 數值拉普拉斯逆轉換以及有限元素法。由矩陣形式Bromwich展開所組成的廣義射線解為一精確解，展開後級數的每一項代表經過界面相同次數的穿透或反射波。若不執行級數展開，將轉換域下的矩陣解直接採用Durbin 數值逆轉換，所得之結果為一混合解析與數值解，適合計算層狀介質的長時間暫態響應。有限元素解則為一純數值解，可以多點計算並快速分析複雜結構物的暫態響應，但對於高頻或是急遽變化的響應則會出現震盪形式的數值誤差，三種數值計算的結果皆有良好的驗證。 而關於非均質材料的暫態問題，則是使用拉普拉斯轉換技巧配合數值逆轉換的混合解析與數值解，首先探討多項式函數形式的功能性梯度單層域，在雙邊自由與單邊固定的邊界條件，於表面受動力載荷下的應力波傳分析，而與多層均質材料模擬單層功能性梯度板的暫態響應，亦有良好的一致性。本文進一步分析層狀功能性梯度材料，討論其域內的暫態彈性波傳，在數值計算上，則以三層功能性梯度材料為例，將其退化為廣泛應用的雙層相異質材料夾功能性梯度材料之暫態問題，並研究其單邊與雙邊的不連續情況對於暫態響應之影響。 隨機、週期與連續分佈型三種類型的多層均質材料在本文亦有深入研究。文中並以複合材料力學的等效材料方式進行化簡，分析多層域暫態響應並探討等效材料在暫態波傳分析的適用性。

In this study, one-dimensional transient wave-propagation in homogeneously and inhomogeneously multilayered media are analyzed by Laplace transform technique. The numerical calculations for homogeneously multilayered media are performed by three methods: generalized ray method, numerical Laplace inversion method (Durbin’s formula), and finite element method (FEM). The analytical result of generalized ray solution for multilayered structures is composed of matrix-form Bromwich expansion in the transform domain. Every term represents a group of waves which is transmitted or reflected through the interface. The numerical inversion of Laplace transform by Durbin’s formula is also used to calculate the transient responses. This numerical Laplace inversion technique has the advantage of calculating the long-time transient responses for complicated multilayered structures. FEM result also agrees well with the calculations by generalized ray method and numerical Laplace inversion. For the transient-wave problem of inhomogeneously multilayered media, we use Laplace transform technique and the numerical Laplace inversion (Durbin’s formula) to calculate the dynamic behavior of the polynomial FGM (functionally graded material) slab. In addition, the FGM slab is approximated as a multilayered medium with homogeneous material in each layer. The transient responses of FGM formulation and multilayered solution are discussed in detail. Furthermore, transient-wave in inhomogeneously multilayered media is analyzed. In the numerical calculation, three-layered functionally graded media is used for analysis and the degenerative problem of an FGM bounded to two elastic homogeneous materials is discussed. Finally, the numerical calculations of the transient responses for randomly distributed, periodically distributed, and continuously distributed multilayered media are performed to investigate if the effective material concept is suitable for dynamic analysis.