变分数阶微积分在描述材料力学性质演变方面的应用  

Application of Variable-Order Fractional Calculus to Description of Evolution of Mechanical Properties of Materials

作　　者：李彦青[1] 殷德顺[1] 吴浩[1] 

机构地区：[1]河海大学力学与材料学院,南京210098

出　　处：《三峡大学学报（自然科学版）》2012年第6期63-67,76,共6页Journal of China Three Gorges University：Natural Sciences

基　　金：江苏省自然科学基金资助项目(BK2012810)

摘　　要：材料在受力和变形过程中,力学性质是不断变化的,而现有的整数阶本构模型只能定性描述材料从弹性变为塑性,变为流体,不能定量地描述力学性质的演变.在常分数阶本构模型的基础上,提出一个新的变分数阶本构模型.从理论上讲,该模型可以通过阶数的变化来展示材料力学性质的变化.用该模型,对铜、铝合金以及低碳钢在等应变率拉伸过程中的力学性质进行分析,发现其变化过程均可分为3个阶段:第1阶段为线弹性阶段,第2阶段力学性质发生突变,第3阶段为线性变化阶段,而这与实验中所得现象是符合的,从而得出该模型可以描述力学性质演变.因为阶数的变化范围为0到1,所以该模型可用于其他粘弹性材料在具有时间效应的受力及变形过程中力学性质演变的分析,如岩土及高分子材料的应力松弛与蠕变,等应变率加载等.The mechanical properties of materials are changing when they are under loading, the existing inte- ger models only can depict the properties as from elasticity to plasticity and fluid, and cannot represent the process quantitatively. Based on fractional constitutive model, a new variable-order fractional constitutive model is proposed. Theoretically, the change of the order can show the evolution of mechanical properties of materials. When the tensile tests of copper, low carbon steel and alloy aluminum are simulated using the vari- able-order model, it is found that the changing process of mechanical properties of these ductile metals has three stages in the changing process. The first stage is linear elastic~ the second one is a transient mutation phase~ and the last one is an approximately uniform （linear） changing phase of material feature. These con- clusions are consistent with the phenomenon, which shows that this model can be used to trace the evolution of mechanical properties of materials. Because the value of the order change from 0 to 1, so this model can be used for the analysis of the evolution of mechanical properties of other viscoelastic materials in the deformation and loading process with time effect, such as the rock and high polymer material in the stress relaxation , creep or constant strain rate loading process.

关 键 词：变分数阶微积分 力学性质演变 延性材料 弹塑性 

分 类 号：O34[理学—固体力学] TU502.6[理学—力学]