Eigenfunction expansion method of upper triangular operator matrixand application to two-dimensional elasticity problems based onstress formulation

This paper studies the eigenfunction expansion method to solve the two dimensional （2D） elasticity problems based on the stress formulation. The fundamental system of partial differential equations of the 2D problems is rewritten as an upper tri angular differential system based on the known results, and then the associated upper triangular operator matrix matrix is obtained. By further research, the two simpler com plete orthogonal systems of eigenfunctions in some space are obtained, which belong to the two block operators arising in the operator matrix. Then, a more simple and conve nient general solution to the 2D problem is given by the eigenfunction expansion method. Furthermore, the boundary conditions for the 2D problem, which can be solved by this method, are indicated. Finally, the validity of the obtained results is verified by a specific example.

Vibration of quadrilateral embedded multilayered graphene sheets based on nonlocal continuum models using the Galerkin method

Free vibration analysis of quadrilateral multilayered graphene sheets（MLGS） embedded in polymer matrix is carried out employing nonlocal continuum mechanics.The principle of virtual work is employed to derive the equations of motion.The Galerkin method in conjunction with the natural coordinates of the nanoplate is used as a basis for the analysis.The dependence of small scale effect on thickness,elastic modulus,polymer matrix stiffness and interaction coefficient between two adjacent sheets is illustrated.The non-dimensional natural frequencies of skew,rhombic,trapezoidal and rectangular MLGS are obtained with various geometrical parameters and mode numbers taken into account,and for each case the effects of the small length scale are investigated.

Elastic deformation behavior of CuZrAlNb metallic glass matrix composites with different crystallization degrees

The room temperature brittleness has been a long standing problem in bulk metallic glasses realm.This has seriously limited the application potential of metallic glasses and their composites.The elastic deformation behaviors of metallic glass matrix composites are closely related to their plastic deformation states.The elastic deformation behaviors of Cu48-xZr48Al4Nbx（x=0,3at.%）metallic glass matrix composites（MGMCs）with different crystallization degrees were investigated using an in-situ digital image correlation（DIC）technique during tensile process.With decreasing crystallization degree,MGMC exhibits obvious elastic deformation ability and an increased tensile fracture strength.The notable tensile elasticity is attributed to the larger shear strain heterogeneity emerging on the surface of the sample.This finding has implications for the development of MGMCs with excellent tensile properties.

EXPLAINING THE CHANGES IN ALPS BY IDENTIFYING ALPS-IMPORTANT COEFFICIENTS:AN EMPIRICAL APPLICATION OF CHINA

This paper establishes the relation between APLs and direct input coefficients through Sherman-Morrison formulation.On such a basis,elasticity matrix can be calculated for each element of the APLs matrix,which measures the percentage change in the APLs-matrix element brought by one percentage change in every direct input coefficient.Hence,the percentage change in each APLs-matrix element caused by the real percentage change in each coefficient with other coefficients fixed can be drawn,from which it is easily to find out the APLs-important coefficients and is useful to explain the reason for change in the matrix.The empirical application studies the Chinese economy.What's more,the method is applied under different level of aggregation.The comparison between the APLs matrix of 1997 and 2002 allows the authors to visualize the elements that change dramatically.Then the methodology above is applied to explain the change from the perspective of direct input coefficients and find out the important coefficients to the Chinese APLs matrix.