For the solution of peridynamic equations of motion, a meshless approach is typically used instead of utilizing semi-analytical or mesh-based approaches. In contrast, the literature has limited analytical solutions. This study develops a novel analytical solution for one-dimensional peridynamic models, considering the effect of damping. After demonstrating the details of the analytical solution, various demonstration problems are presented. First, the free vibration of a damped system is considered for under-damped and critically damped conditions. Peridynamic solutions and results from the classical theory are compared against each other, and excellent agreement is observed between the two approaches. Next, forced vibration analyses of undamped and damped conditions are performed. In addition, the effect of horizon size is investigated. It is shown that for smaller horizon sizes, peridynamic results agree well with classical results, whereas results from these two approaches deviate from each other as the horizon size increases.

Peridynamics is a new approach to continuum mechanics. There has been rapid progress in peridynamics research, especially in recent years. In this review study, recent advances in peridynamics research were summarised. A large number of studies were considered and classified into different categories ranging from additive manufacturing, artificial intelligence and machine learning, composite materials, fatigue, functionally graded materials, impact, reduced order modelling, structural health monitoring, topology optimisation, and many more. Finally, some future directions were highlighted.

In this study, a two-dimensional "double-horizon peridynamics" formulation was presented for membranes. According to double-horizon peridynamics, each material point has two horizons: inner and outer horizons. This new formulation can reduce the computational time by using larger horizons and smaller inner horizons. To demonstrate the capability of the proposed formulation, various different analytical and numerical solutions were presented for a rectangular plate under different boundary conditions for static and dynamic problems. A comparison of peridynamic and classical solutions was given for different inner and outer horizon size values.