Noether Theorem for Fractional Singular Systems

Noether theorems for two fractional singular systems are discussed.One system involves mixed integer and Caputo fractional derivatives,and the other involves only Caputo fractional derivatives.Firstly,the fractional primary constraints and the fractional constrained Hamilton equations are given.Then,the fractional Noether theorems of the two fractional singular systems are established,including the fractional Noether identities,the fractional Noether quasi-identities and the fractional conserved quantities.Finally,the results obtained are illustrated by two examples.

Toward an understanding of dwell fatigue damage mechanism of bimodal Ti-6Al-4V alloys

Dwell fatigue effect is a long-standing problem threatening the long-term service reliability for fan blades and fan disks of an aircraft engine.To understand the basic mechanism of dwell fatigue damage,pure fatigue and 60 s dwell fatigue properties of bimodal Ti-6Al-4V alloys with different volume fractions of the primaryα(α_(p))phase were examined comparatively.The results showed that both pure fatigue and dwell fatigue life decreased with increasing the volume fraction of theα_(p)phase and the dwell fatigue life was lower than the pure fatigue one.The quasi-in-situ test results and the quantitative characterization of damage behaviors of the local microstructure units defined by theα_(p)-secondaryα(α_(s))combination reveal that theα_(s)phase close to theα_(p)phase with extensively slip activities was gradually damaged under dwell fatigue loading,while that under pure fatigue loading was undamaged,demonstrating that the dwell loading induced the damage of theα_(s)phase,and further reduced the fatigue life.A stress relaxation-based model is proposed to describe the physical mechanism on dwell fatigue damage of the bimodal Ti-6Al-4V alloy,i.e.the elastic deformation of theα_(s)phase caused by the strain incompatibility would be gradually transformed into plastic deformation during the dwell stage,and thus promotes fatigue damage.The model provides new insights into the microscopic process of stress/strain transfer between the soft and hard microstructure units under dwell fatigue loading.