## 2 Basic Theory of the Non-Ordinary State-Based Peridynamics Model
### Correspondence Material Model for Thermomechanical Problems  
Based on the correspondence material model in the PD theory, the force vector state \(\underline{\mathbf{t}}\) and heat flow scalar state \(\underline{\tau}\) are related to the non-local deformation gradient \(\overline{\mathbf{F}}\) and the non-local temperature gradient \(\overline{\mathbf{G}}\)[43], that are  
\[\overline{\mathbf{F}}=\int_{\Omega_{\mathbf{x}_{i}}}\underline{\omega}\big{ }\langle\boldsymbol{\xi}_{ij}\rangle\underline{\mathbf{Y}}\big{\langle} \boldsymbol{\xi}_{ij}\big{\rangle}\bigotimes\underline{\mathbf{X}}\big{\langle} \boldsymbol{\xi}_{ij}\big{\rangle}\mathrm{d}V_{\mathbf{x}_{j}}\cdot\mathbf{K}, \tag{3}\]  
\[\overline{\mathbf{G}}=\int_{\Omega_{\mathbf{x}_{i}}}\underline{\omega}\big{ }\langle\boldsymbol{\xi}_{ij}\rangle\underline{Q}\big{\langle}\boldsymbol{ \xi}_{ij}\big{\rangle}\bigotimes\underline{\mathbf{X}}\big{\langle} \boldsymbol{\xi}_{ij}\big{\rangle}\mathrm{d}V_{\mathbf{x}_{j}}\cdot\mathbf{K}, \tag{4}\]  
where \(\underline{\omega}\) is the influence function and \(\mathbf{K}=\int_{\Omega_{\mathbf{x}_{i}}}\underline{\omega}\big{\langle} \boldsymbol{\xi}_{ij}\big{\rangle}\underline{\mathbf{X}}\big{\langle} \boldsymbol{\xi}_{ij}\big{\rangle}\bigotimes\underline{\mathbf{X}}\big{\langle} \boldsymbol{\xi}_{ij}\big{\rangle}\mathrm{d}V_{\mathbf{x}_{j}}\) is the shape tensor defined by [PERSON] [18]. In this work, the influence function is chosen as \(\underline{\omega}\big{\langle}\boldsymbol{\xi}_{ij}\big{\rangle}=\left(1- \frac{|\boldsymbol{\xi}_{ij}|}{\delta}\right)\).  
Considering the assumption of small deformation in this work, the force vector state \(\underline{\mathbf{t}}\) and heat flow scalar state \(\underline{\tau}\) can be given by [48],  
\[\underline{\mathbf{t}}[\mathbf{x}_{i},t]\big{\langle}\boldsymbol{\xi}_{ij} \big{\rangle}=\underline{\omega}\big{\langle}\boldsymbol{\xi}_{ij}\big{\rangle} \boldsymbol{\sigma}_{i}\cdot\mathbf{K}_{i}^{-1}\cdot\boldsymbol{\xi}_{ij}, \tag{5}\]  
\[\underline{\tau}[\mathbf{x}_{i},t]\big{\langle}\boldsymbol{\xi}_{ij} \big{\rangle}=-\underline{\omega}\big{\langle}\boldsymbol{\xi}_{ij}\big{\rangle} \mathbf{q}_{i}\cdot\mathbf{K}_{i}^{-1}\cdot\boldsymbol{\xi}_{ij}, \tag{6}\]  
where \(\boldsymbol{\sigma}\) is the Cauchy stress tensor and \(\mathbf{q}\) is the heat flux density vector.  
For mechanical problems, the stress-strain relationship that involves plastic behaviors and temperature effects is given by [48],  
\[\boldsymbol{\sigma}=\mathbb{C}:\Big{(}\boldsymbol{\varepsilon}-\boldsymbol{ \varepsilon}^{\mathrm{P}}-\boldsymbol{\varepsilon}^{T}\Big{)}, \tag{7}\]  
where \(\mathbb{C}\) is the elastic tensor, \(\boldsymbol{\varepsilon}=\frac{1}{2}\big{(}\overline{\mathbf{F}}-\mathbf{I} \big{)}^{\mathrm{T}}+\frac{1}{2}\big{(}\overline{\mathbf{F}}-\mathbf{I}\big{)}\) is the total strain tensor, \(\boldsymbol{\varepsilon}^{\mathrm{P}}\) is the plastic strain tensor, \(\boldsymbol{\varepsilon}^{T}=\alpha\varGamma\mathbf{I}\) is the thermal strain tensor, and \(\alpha\) is the coefficient of thermal expansion.  
For thermal problems, the heat flux is given by [48],  
\[\mathbf{q}=-\mathbf{k}\cdot\overline{\mathbf{G}}, \tag{8}\]  
where \(\mathbf{k}\) is the thermal conductivity tensor.