On stress and strain descriptions of continuum mechanics

本研究為了描述有記憶性連體的力學行為，提出參數描述法，並利用微分形式理論定義質量純量值3次微分形式(質量形式)與面力向量值2次微分形式(面力形式)，爾後重新推導質量守恆式、力平衡式與力矩平衡式。經由虛功原理，我們可以得到與面力形式功對偶的共軛變數為伸長向量值1次微分形式(伸長形式)，之後建立彈性組成方程式以及處理客觀性的問題。文中也指出其他描述法在處理有記憶性連體的不足之處。最後藉由簡單剪切變形、實心圓柱扭轉與實心圓柱伸長扭轉膨脹這三個範例，說明參數描述法配合微分形式在應用上的可行性與在計算上的簡潔性。We propose a parametric description and use the theory of differential forms to study the mechanical behavior of continuum with memory. We define mass scalar-valued differential 3-forms (mass form) and traction vector-valued differential 2-forms (traction form), and re-derive the conservation laws of mass, linear momentum, and angular momentum. According to the principle of virtual work, the stretch vector-valued differential 1-form (stretch form) are obtained as the work-conjugate variable of the traction form. The elastic constitutive law in term of the traction and stretch forms and the objectivities of the forms and the law are established. The shortcomings of other descriptions are also pointed out. In order to verify the accessibility and the simplicity of the parametric description three examples, simple shear deformation, pure torsion of solid cylinder, and simultaneous extension, inflation, and torsion of solid cylinder, are demonstrated