Ordinary state-based peridynamic modelling for fully coupled thermoelastic problems

  • Yan Gao
  • Selda OterkusEmail author
Open Access
Original Article


An ordinary state-based peridynamic model is developed for transient fully coupled thermoelastic problems. By adopting an integral form instead of spatial derivatives in the equation of motion, the developed model is still valid at discontinuities. In addition, the ordinary state-based peridynamic model eliminates the limitation on Poisson’s ratio which exists in bond-based peridynamics. Interactions between thermal and structural responses are also considered by including the coupling terms in the formulations. These formulations are also cast into their non-dimensional forms. Validation of the new model is conducted by solving some benchmark problems and comparing them with other numerical solutions. Thin plate and block under shock-loading conditions are investigated. Good agreements are obtained by comparing the thermal and mechanical responses with those obtained from boundary element method and finite element solutions. Subsequently, a three-point bending test simulation is conducted by allowing crack propagation. Then a crack propagation for a plate with a pre-existing crack is investigated under pressure shock-loading condition. Finally, a numerical simulation based on the Kalthoff experiment is conducted in a fully coupled manner. The crack propagation processes and the temperature evolutions are presented. In conclusion, the present model is suitable for modelling thermoelastic problems in which discontinuities exist and coupling effects cannot be neglected.


State-based peridynamics Fully coupled Thermoelasticity Crack propagation 

List of symbols


Cross-sectional area for one-dimensional problems \(\left( \hbox {m}^{2} \right) \)


Specific heat capacity under constant volume \([{\hbox {J}/\left( {\hbox {kg\,K}} \right) }]\)


Young’s modulus \(\left( {\hbox {Pa}} \right) \)


Critical energy release rate \(\left( {\hbox {J/m}^{{2}}} \right) \)


Thickness for two-dimensional problems \(\left( \hbox {m} \right) \)

\(K_\theta \)

Bulk modulus \(\left( {\hbox {GPa}} \right) \)


Thermal conductivity \([{\hbox {W}/\left( {\hbox {mK}} \right) }]\)

\(\mathbf{u}\left( \mathbf{x},t \right) \)

Displacement of point \(\mathbf{x}\) at time \(t \left( \hbox {m} \right) \)

\(\dot{\mathbf{u}}\left( \mathbf{x},t \right) \)

Velocity of point \(\mathbf{x}\) at time \(t \left( {\hbox {m/s}} \right) \)

\(\ddot{\mathbf{u}}\left( \mathbf{x},t \right) \)

Acceleration of point \(\mathbf{x}\) at time \(t \left( \hbox {m/s}^{2} \right) \)

\(u_x \left( \mathbf{x},t \right) \)

Scalar value of horizontal displacement of point \(\mathbf{x}\) at time \(t \left( \hbox {m} \right) \)

\(\alpha \)

Linear thermal expansion coefficient \(\left( {\hbox {K}^{-1}} \right) \)

\(\beta _\mathrm{cl}\)

Thermal modulus in classical mechanics \(\left( {\hbox {Pa/K}} \right) \)

\(\Theta _0\)

Reference temperature \(\left( \hbox {K} \right) \)

\(\lambda ,\;\mu \)

Lamé constants \(\left( {\hbox {GPa}} \right) \)

\(\nu \)

Poisson’s ratio

\(\rho \)

Density \(\left( {\hbox {kg/m}^{3}} \right) \)


Volume of point \(\mathbf{x}' \left( {\hbox {m}^{3}} \right) \)

\(\Theta \left( \mathbf{x},t \right) \)

Absolute temperature of point \(\mathbf{x}\) at time \(t \left( \hbox {K} \right) \)

\(T\left( \mathbf{x},t \right) \)

Temperature change of point \(\mathbf{x}\) at time \(t \,\,\left( \hbox {K} \right) \)

\(\dot{T}\left( {\mathbf{x},t} \right) \)

Time rate of temperature change of point \(\mathbf{x}\) at time \(t \left( {\hbox {K/s}} \right) \)

\(u_y \left( \mathbf{x},t \right) \)

Scalar value of vertical displacement of point \(\mathbf{x}\) at time \(t \,\,\left( \hbox {m} \right) \)



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Authors and Affiliations

  1. 1.Department of Naval Architecture, Ocean and Marine EngineeringUniversity of StrathclydeGlasgowUK

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