Gradient grain structure materials with superior mechanical properties of high strength and high toughness have attracted widespread attention. Gradient materials can effectively improve toughness by constructing a microstructure from fine to coarse crystals inside the material, which has gradually become a hotspot of attention in the academic and engineering communities. In this paper, based on the crystal plasticity intrinsic theory, dislocation density is introduced as a characterization quantity, and cohesive units are added at grain boundaries to simulate damage fractures. The results of this study reveal the fracture damage mechanism of gradient crystal structure materials, providing new ideas and methods for the design of gradient materials.

A low stiffness makes long-span suspension bridges sensitive to loads, and this sensitivity is particularly significant for wind-induced nonlinear vibrations. In the present paper, nonlinear vibrations of suspension bridges under the combined effects of static and vortex-induced loads are explored using the nonlinear partial differential–integral equation that models the plane bending motion of suspension bridges. First, we discretized the differential–integral equation through the Galerkin method to obtain the nonlinear ordinary differential equation that describes the vortex-induced vibrations of the bridges at the first-order symmetric bending mode. Then, the approximate analytical solution of the ordinary differential equation was obtained using the multiple scales method. Finally, the analytical solution was applied to reveal the relationships between the vibration amplitude and other parameters, such as the static wind load, the frequency of dynamic load, structural stiffness, and damping. The results show that the static wind load slightly impacts the bridge’s vibrations if its influence on the natural frequency of bridges is ignored. However, the bridge’s vibrations are sensitive to the load frequency, structural stiffness, and damping. The vibration amplitude, as a result, may dramatically increase if the three parameters decrease.

With the development of modern structures in the direction of higher and more complexity, the existence of multiple factors in the design and requirements of structures can lead to structures prone to nonlinear properties. The tuned mass damper (TMD), a widely implemented passive control mechanism, plays a crucial role in the engineering field by effectively reducing vibrations within primary structures. Nevertheless, its deployment frequently induces nonlinear dynamics due to the substantial displacements resulting from TMD operation or the integration of limiting devices. This research delineates a computational framework for a single-degree-of-freedom nonlinear primary system regulated by a nonlinear tuned mass damper (NTMD), designed to emulate near-fault seismic phenomena via a sinusoidal load. The study concentrates on the nonlinear attributes of both the NTMD and the primary system. Utilizing the complex variable averaging method in conjunction with the multiscale technique, complex variable equations and slow invariant manifolds are formulated for the system under 1:1 resonance conditions, with their accuracy and validity substantiated through numerical simulations. Expanding upon the derived complex variable equations and slow invariant manifolds, this study examines the impact of nonlinear coefficients within the NTMD and the primary system on both the damping performance of the NTMD and the stability of the primary system. Furthermore, this research delves into the effects of mass ratio fluctuations on the damping effectiveness of the TMD and the control efficiency of the primary system, as well as the emergence of jump phenomena in the presence of significant nonlinear coefficients. The analytical outcomes underscore the critical need to account for the inherent nonlinearities in both the TMD and the primary system, which can have detrimental effects. By considering the mass ratio as a key design parameter, optimizing it can enhance the TMD’s vibration suppression capabilities and the primary system’s control behavior, while also reducing the likelihood of jump phenomena and improving overall structural stability.