建構於標準線性固體模型的黏彈波動方程式之推導並運用於頻散之分析

Abstract

本研究針對黏彈性(viscoelasticity)材料中之波動現象進行理論推導與數值分析，發展一套以馬克士威形式之標準線性固體模型(Maxwell form of the standard linear solid model)為基礎之黏彈性波動方程式，並進一步引入分數階微積分以模擬材料之記憶效應與能量耗散特性。相較於傳統的整數階模型，分數階導數可更準確地描述材料在應力或應變歷史影響下的非局部反應行為，有效捕捉波在黏彈性介質中，因傳遞所產生之衰減(attenuation)與頻散(dispersion)等物理現象。 首先，回顧了黏彈性材料與波動理論之基礎，並建構含有兩組彈性模數與一黏滯係數的馬克士威標準線性固體架構。透過動量守恆定律與傅立葉轉換，本研究推導出黏彈性波動方程在頻率域下之形式，進而導出複數波數並分離其實部與虛部，透過聯立方程式得到相速度與衰減因子的解析解。此外，本研究也透過無因次化標準化處理，使得不同材料之模擬結果具可比性。 在數值模擬部分當中，本文分析了天然橡膠(NR)、生醫矽膠(GC-5)與活體豬腦等材料，觀察其在不同分數階指數β下波動行為的改變。結果顯示當β趨近0時，材料表現近似彈性；當β趨近1時，黏性效應增強，波速顯著頻散且能量快速衰減，完全符合物理理論之趨勢。由此可知，分數階指數β為一有效之控制參數，能靈活調控材料的耗散行為與頻率響應特性。 本研究所建立之模型，不僅克服了Kelvin-Voigt模型在高頻極限下相速度無窮之不合理問題，亦於物理表徵能力與擬合彈性上展現高度優勢。該模型具潛力應用於結構健康監測、生醫超聲影像、地震模擬與材料識別等多元領域，提供一套具備理論深度與實用價值的波動描述框架。

This study presents a theoretical and numerical investigation of wave propagation in viscoelastic materials, focusing on the development of a viscoelastic wave equation based on the Maxwell form of the standard linear solid (SLS) model. To account for memory effects and energy dissipation, fractional calculus is incorporated into the formulation. Compared with traditional integer-order models, the use of fractional derivatives allows for a more accurate representation of nonlocal responses influenced by stress or strain history, effectively capturing key physical phenomena such as attenuation and dispersion during wave transmission in viscoelastic media. The study begins with a review of the theoretical foundation of viscoelasticity and wave mechanics, followed by the construction of a Maxwell form of the SLS framework comprising two elastic moduli and one viscosity coefficient. By applying momentum conservation and Fourier transform techniques, the wave equation is reformulated in the frequency domain, and the complex wavenumber is derived. Real and imaginary parts of the wavenumber are then separated to obtain analytical expressions for phase velocity and attenuation via coupled equations. Additionally, a dimensionless normalization process is implemented to ensure comparability across different materials. In the numerical simulations, three representative materials, natural rubber (NR), biomedical silicone gel (GC-5), and in vivo porcine brain tissue, which were analyzed to evaluate wave behavior under varying fractional-order indices β. The results demonstrate that as β approaches 0, the material behaves almost elastically; as β approaches 1, viscous effects become prominent, leading to increased dispersion and rapid energy attenuation, consistent with expected physical trends. The fractional-order index β is thus shown to be an effective control parameter for tuning frequency-dependent dissipative behavior. The proposed model not only addresses the unrealistic high-frequency velocity divergence observed in Kelvin-Voigt model, but also exhibits superior capability in physical representation and curve-fitting accuracy. It provides a robust and flexible framework for modeling wave phenomena in viscoelastic media, with potential applications in structural health monitoring, biomedical ultrasound imaging, seismic simulation, and material characterization.