The eigenvalue problems of the buckling loads and natural frequencies of a braced beam on an elastic foundation are investigated. sented. The eigenvalues vary with the different The exact solutions for the eigenvalues are preparameters and are especially sensitive to the brace location. As the beam of a continuous system has infinite eigenvalues and these eigenvalues are influenced differently by a brace, the eigenvalues show rich variation patterns. Because these eigenvalues physically correspond to the structure buckling loads and natural frequencies, the study on the eigenvalues variation patterns can offer a design guidance of using a lateral brace of translation spring to strengthen the structure.
Cu samples were subjected to high-pressure torsion (HPT) with up to 6 turns at room temperature (RT) and liquid nitrogen temperature (LNT),respectively.The effects of temperature on grain refinement and microhardness variation were investigated.For the samples after HPT processing at RT,the grain size reduced from 43 μm to 265 nm,and the Vickers microhardness increased from HV52 to HV140.However,for the samples after HPT processing at LNT,the value of microhardness reached its maximum of HV150 near the center of the sample and it decreased to HV80 at the periphery region.Microstructure observations revealed that HPT straining at LNT induced lamellar structures with thickness less than 100 nm appearing near the central region of the sample,but further deformation induced an inhomogeneous distribution of grain sizes,with submicrometer-sized grains embedded inside micrometer-sized grains.The submicrometer-sized grains with high dislocation density indicated their nonequilibrium nature.On the contrary,the micrometer-sized grains were nearly free of dislocation,without obvious deformation trace remaining in them.These images demonstrated that the appearance of micrometer-sized grains is the result of abnormal grain growth of the deformed fine grains.
XIE ZiLing1,2,XIE JiJia2,HONG YouShi2 & WU XiaoLei2 1 College of Architecture and Civil Engineering,University of Wenzhou,Wenzhou 325035,China
Recently, people are confused with two opposite variations of elastic modulus with decreasing size of nano scale sample： elastic modulus either decreases or increases with decreasing sample size. In this paper, based on intermolecular potentials and a one dimensional model, we provide a unified understanding of the two opposite size effects. Firstly, we analyzed the microstructural variation near the surface of an fcc nanofilm based on the Lennard-Jones potential. It is found that the atomic lattice near the surface becomes looser in comparison with the bulk, indicating that atoms in the bulk are located at the balance of repulsive forces, and the elastic moduli decrease with the decreasing thickness of the film accordingly. In addition, the decrease in moduli should be attributed to both the looser surface layer and smaller coordination number of surface atoms. Furthermore, it is found that both looser and tighter lattice near the surface can appear for a general pair potential and the governing mechanism should be attributed to the surplus of the nearest force to all other long range interactions in the pair potential. Surprisingly, the surplus can be simply expressed by a sum of the long range interactions and the sum being positive or negative determines the looser or tighter lattice near surface respectively. To justify this concept, we examined ZnO in terms of Buckingham potential with long range Coulomb interactions. It is found that compared to its bulk lattice, the ZnO lattice near the surface becomes tighter, indicating the atoms in the bulk are located at the balance of attractive forces, owing to the long range Coulomb interaction. Correspondingly, the elastic modulus of one-dimensional ZnO chain increases with decreasing size. Finally, a kind of many-body potential for Cu was examined. In this case, the surface layer becomes tighter than the bulk and the modulus increases with deceasing size, owing to the long range repulsive pair interaction, as well as the cohesive many-body interaction caused by
