考虑非局部效应和记忆依赖微分的广义热弹问题

现有的广义热弹理论主要适用于求解时间尺度极短但空间尺度仍属宏观尺度的广义热弹问题的动态响应,而当所研究的弹性体的特征几何尺寸也属微尺度时,弹性体的力学响应将呈现出强烈的尺寸相关性,现有的广义热弹理论不再适用. 本文基于通过非局部效应和记记依赖微分修正的广义热弹性理论,研究了两端固定、受移动热源作用的有限长热弹杆的动态响应. 建立了问题的控制方程,给出了问题的初始条件及边界条件,运用拉普拉斯变换及其数值反变换,对方程进行了求解. 数值计算中,首先考察了时间延迟因子对模型所预测各物理量分布的影响;然后对比了模型中的时间延迟因子在两种不同类别核函数下(通过归一化条件修正和未修正形式)对各物理量分布的影响效应;最后考察了考虑新的可以描述尺寸效应的非局部因子对无量纲温度、位移及应力的影响,并用图形进行了示例. 结果表明, 时间延迟因子增大,各物理量的峰值变大,传播距离变小,且时间延迟因子在归一化条件修正过的核函数下影响更加显著;非局部参数几乎不影响无量纲温度的分布,轻微影响无量纲位移的分布,但对无量纲应力的峰值的影响显著. 

A GENERALIZED THERMOELASTIC PROBLEM WITH NONLOCAL EFFECT AND MEMORY- DEPENDENT DERIVATIVE

The existing generalized thermoelastic theory is mainly applicable to obtain the dynamic responses of the problems in which the time scale is extremely short while the spatial scale is still macro-scale. Nevertheless, when the characteristic length scale of elastic body is also of micro-scale, the dynamic responses of the elastic body will take on intense size-dependent effect, and the existing generalized thermoelastic theory will be no longer suitable for such problems. In present work, based upon the generalized thermoelasticity with nonlocal effect and memory-dependent derivative, the dynamic response of a finite thermoelastic rod fixed at both ends and subjected to a moving heat source is investigated. The corresponding governing equations of the problem are formulated and the initial conditions as well as the boundary conditions are specified. Then, the governing equations are solved by means of Laplace transform and its numerical inversion. In calculation, first, the influence of the time-delay factor on the distributions of the considered physical quantity was examined. Then, the influence of the time-delay factor on the distributions of the considered variables under two kinds of kernel functions (i.e. normalized form and unmodified form) was compared. Last, the influence of the nonlocal factor on the dimensionless temperature, displacement and stress is considered and illustrated graphically. The results show that: with the increase of the time-delay factor, the heat wave propagation velocity becomes smaller, the peak values of the physical quantities become larger, and the influence of the time delay factor on the considered variables is more significant in the case with the kernel function modified by normalized condition than that with unmodified kernel function; The non-local parameter barely affects the distribution of the dimensionless temperature, slightly affects the distribution of the dimensionless displacement, while markedly affects the peak values of the dimensionless stress.