In this paper,a nonlinear wave equation with variable coefficients is studied,interestingly,this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material densities.With the aid of Lou’s direct method1,the nonlinear wave equation with variable coefficients is reduced and two sets of symmetry transformations and exact solutions of the nonlinear wave equation are obtained.The corresponding numerical examples of exact solutions are presented by using different coefficients.Particularly,while the variable coefficients are taken as some special constants,the nonlinear wave equation with variable coefficients reduces to the one with constant coefficients,which can be used to describe the propagation of the travelling waves in general cylindrical rods composed of generally hyperelastic materials.Using the same method to solve the nonlinear wave equation,the validity and rationality of this method are verified.
Coupled thermo-mechanical model was used to investigate the effects of the pin diameter, the shoulder diameter and the in conical angle on the heat generations, the material deformations and the energy histories in friction stir welding（FSW） of AA2024-T3 alloy. Results indicate that the shoulder-plate contact area takes more important contribution to the heat generation than the pin-plate contact area. The increase of the shoulder diameter or the decrease of the pin diameter can lead to the increase of the welding temperature in FSW, but the change of shoulder size is more important. Compared to the cases in FSW of AA6061-T6, the input power is obviously increased in FSW of AA2024-T3 and the ratio of the plastic dissipation to the friction dissipation becomes decreased.
Understanding the mechanisms of hard–soft material interaction under impact loading is important not only in the defense industry but also in daily life.However,traditional mesh-based spatial discretization methods that are time consuming owing to the need for frequent re-meshing,such as the finite element method and finite difference method,can hardly handle large deformation involving failure evolution in a multi-phase interaction environment.The objective of this research is to develop a quasi-meshless particle method based on the material point method for the model-based simulation of the hard–soft material interaction response.To demonstrate the proposed procedure,scenarios of a hard–soft material impact test are considered,where a force is applied to layers of materials and a hard bar with an initial velocity impacts a target with layers of different materials.The stress wave propagation and resulting failure evolution are simulated and compared with available data.Future research tasks are then discussed on the basis of the preliminary results.
LIU Han TaoJIANG ShanCHEN ZhenGAN YongCHANG Jian ZhongWANG Yan HuaTONG Zhi Hui
