## 1 Introduction
These works lay a solid theoretical foundation for modeling thermal shock problems involving brittle fractures. Currently, the study of thermal-shock-induced fracture mainly use the BB-PD [37; 38] and the OSB-PD models [39; 40]. Although these studies have demonstrated the good performance of the PD model in dealing with thermal shock problems, half of them have focused on fracture in brittle materials, which leads to difficulties in applying these methods for modeling ductile materials. Therefore, it is very promising to use the advantage of the NOSB-PD model to study more complex materials under thermal shocks. The fracture criterion is also a non-negligible part in the simulation of the fracture problem. In fracture mechanics, the critical energy release rate is one of the most commonly used material fracture properties that determines when a material will fracture [41]. According to the linear elastic fracture mechanics (LEFM), the value of the critical energy release rate can be measured by the geometry of the specimen and external loading parameters at a specific crack configuration [42]. In the PD framework, the situation is somewhat different. As a non-local meshless approach, the PD model can calculate the energy release rate by calculating the fractured bonds cut at the surface [16; 43]. In other words, the relationship between the energy release rate and the elongation of the bond can be established. Thus, the most commonly used fracture criterion in the PD model is the bond stretch criterion. However, this criterion is seldom applied in the NOSB-PD framework, where the stress [44; 45] and strain criteria [46; 47] are more applicable. For brittle materials, either the stress or strain criterion can describe the fracture behavior well. For ductile materials, the fracture behavior is not only related to stress and strain, but also to damage behavior. This leadsto the fact that different materials often require different fracture criteria. Therefore, the development of a fracture criterion for both ductile and brittle materials is very promising for applications. Based on the advantage of the NOSB-PD in combining with the CCM theory, this work aims to develop a coupled thermomechanical non-ordinary state-based peridynamics (NOSB-PD) model for accurately predicting crack propagation arising from thermal shocks in ductile and brittle solids. In this model, a unified multiaxial constitutive model with damage is proposed to describe both ductile and brittle behaviors, in which the distinction is achieved by using different physical variables representing different fracture mechanisms. To suppress the numerical oscillations due to the zero-energy modes, a damage-related fracture criterion is considered instead of the traditional criterion (e.g., the bond criterion). Furthermore, full explicit time integration is used for the accurate response of coupled thermodynamic effects. This paper is organized as follows. The basic theory of the NOSB-PD model for the thermodynamic problem is described in Section 2, including the stabilization method for thermal problems. The multiaxial plastic constitutive model involving damage is given in Section 3. The time integration and stress integration strategies are shown in Section 4. The ability of the proposed model to simulate thermodynamic problems and predict the crack propagation in ductile or brittle solids under thermal shocks is verified by several represented examples in Section 5. Final conclusions are drawn in Section 6.