摘要
针对经典的本构模型不能准确地描述粘弹性材料的力学性质,且考虑热量对粘弹性材料性质的影响,本文采用了分数微分型本构模型来描述粘弹性材料的本构关系,研究了分数微分型热粘弹性体的动力耦合问题。论文重点研究了用分数微分本构模型描述的热粘弹性杆的热动力学行为。主要分别研究了经典本构模型和分数微分本构模型描述的热粘弹性杆动力耦合作用下的热动力学行为,并将它们的热力学行为进行比较。此外,研究了分数微分型热粘弹性体热动力学耦合问题控制微分方程的差分格式。最后分析了三种不同初值条件下轴对称圆柱形粘弹性阻尼器热动力行为。主要成果及结论包括:
热动力耦合情况下,耦合效应随着耦合系数、粘性系数和弹性模量的增大而减弱,耦合效应表现的热粘弹阻尼作用减弱,各物理量振动的衰减速度变缓慢。
分数阶数越大,各物理量振动衰减得越慢,耦合效应越弱;与整数阶描述的热动力耦合问题相比,分数阶所描述的各物理量所受的耦合作用较大,产生的振幅峰值也较大。
系统的阻尼作用对速度的影响最大;温差对位移、应变梯度和速度等力场的影响很大;与材料阻尼相比,由耦合效应产生的热粘弹阻尼在热动力耦合过程中起主导作用。
用分数算子去描述热粘弹性体动力耦合问题的研究至少在国内还很少见,这是本文的创新之处,所以本文的工作为以后研究分数微分型热粘弹性体动力耦合问题奠定了坚实的基础。
Aiming at the mechanics properties of viscoelastic materials can't be described accurately by classical constitutive model,and considering the properties of viscoelastic materials are affected by heat,fractional derivative viscoelastic constitutive model is adopted to describe the constitutive relations of viscoelastic materials in this paper.The coupling problems of fractional derivative thermoviscoelastic objects are studied here.The thermodynamic behaviors of thermoviscoelastic pole,which is described by fractional constitutive model in the dynamic coupling effect,is mainly studied.The thermodynamic behaviors of thermo-viscoelastic poles, which are described by calssical constitutive model and fractional constitutive model separately, are studied mainly,and their thermodynamic behaviors are compared.Moreover,the difference format of governing differential equations of fractional derivative viscoelastic object is studied in the thermodynamic coupling effect.Finally,thermodynamic behaviors of axis-symmetry columniform viscoelastic damper are analyzed under three kinds of intial value conditions.The main results and conclusions are following:
(1) Under the conditions of thermodynamic coupling,with the increase of coupling coefficient,viscosity coefficient and elasticity modul,the coupling effect become weaker.So the thermoviscoelastic damp effect become weaker,the speed of vibration decaying become slower.
(2) The fractional rank is greater,the speed of vibration decaying become slower and the coupling effect become weaker.Comparing to the classical thermodynamic coupling problems, the physics quantities,decribed by fractional constitutive model,suffer geater coupling effect, the peak value of vibration produced are greater.
(3) The velocity is effected the most by the damp effect of system.The difference in temperatuer is effect great to displacement,the gradient of stress and velocity.Comparing to matrials' damp,the thermoviscoelastic damp produced by coupling effect plays more important role in the course of thermodynamic coupling.
The study about thermodynamic coupling problems of viscoelastic body described by fractional differential calculus in this dissertation is still very rare at least in the domestic.So the research has provided a solid foundation for further study to the fractional thermo-dynamic coupling problems.
引文
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