基于分布阶导数的本构方程模型的理论分析

Theoretical Analysis on the Model of Constitutive Equation Based on Distributed Order Derivatives

研究基于分布阶导数的固体型黏弹材料的本构方程,方程中涉及到关于应变的分数阶导数的阶的积分.用分数阶导数算子_0D_t~α,Laplace变换及其数值逆方法,讨论了本构方程模型的松弛模量和蠕变柔量,谐变应力下应变的瞬态响应和滞后圈的形成.用分数阶导数算子_-∞D_t~α和待定系数方法,研究了模型在谐变应力下的稳态响应.模型能够合理地表示材料的黏弹特性,参数能够特征黏性或弹性的强弱.

The constitutive equation of solid-like viscoelastic materials based on distributed order derivatives was considered, and in the equation the integral on the order of fractional derivatives of strain was involved. By using a fractional derivative operator oD7 , Laplace transform and its numerical inverse method, relaxation modulus, creep compliance, instantenous response of strain under harmonic stress and the formation of hysteresis loop are discussed. By using a fractional derivative operator-∞Dαt and the method of undetermined coefficients, the steady response of the model to harmonic stress is investigated. The model can reasonably show the viscoelastic properties of materials, and the parameteres can characterize the strength of viscosity or elasticity.