Thermal cracking simulation of functionally graded materials using the combined finite–discrete element method

Abstract

The functionally graded materials (FGMs), characterized by spatially varying material properties, have been used in a wide range of engineering applications (i.e., aerospace, nuclear reactor and microelectronics) in high temperature and high-temperature gradient environments. The prediction of crack behavior under severe temperature conditions is essential for the safety and long-term service life of such critical components. In this paper, a novel thermal–mechanical coupling model for FGMs is proposed, which consists of a thermal part for the temperature field computation and the combined finite–discrete element method part for the crack evolution modeling. The spatially dependent material characteristics of the FGMs are captured in this model, together with typical property variation functions (quadratic, exponential and trigonometric). The accuracy and robustness of the proposed coupled TM model are validated by numerical tests. Then, this model is applied to investigate the thermal cracking process in FGMs under different kinds of thermal loads. The influence of the crack interaction on crack growth pattern is also discussed. The results show that the proposed method is useful to the fracture mechanics analysis and design of the FGMs structures.

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Acknowledgements

This work was supported by the National Key Research and Development Program of China (Grant No. 2017YFC1501300), NSERC/Energi Simulation Industrial Research Chair program and Ph.D. Short-time Mobility Program of Wuhan University. The authors would like to thank Prof. Munjiza for the use of the Y code. The proposed Y-FGM™ is based on the Y code. The authors would also like to thank the anonymous reviewers for their constructive comments and suggestions.

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Appendix A

Appendix A

Under the steady-state condition, the heat flux q is the same in the domain, and the temperature remains unchanged. Thus, the thermal gradient at coordinate x can be calculated as:

$$ J(x) = \frac{{q_{{}} }}{k(x)} $$
(25)

where k(x) = k0f(x), here f(x) is the thermal conductivity variation function.

Considering the temperature boundary condition on the left and right sides of the plate, we can have:

$$ T_{0} { + }\int_{0}^{l} {J(x)} dx = T_{l} $$
(26)

where Tl and T0 are temperature at x = l and x = 0, respectively.

Substituting the variation functions [Eqs. (46) into Eq. (24)], the exact solution for the constant heat flux and steady temperature field for different variation functions can be obtained, as illustrated in Table 3. Then, the effective heat conductivity (EHC) [49], defined as the amount of heat flux flow into unit area under unit temperature difference, can be calculated as:

$$ k_{eq} = \frac{q}{{T_{l} - T_{0} }} $$
(27)
Table 3 The heat flux and temperature distribution in steady stage

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Sun, L., Grasselli, G., Liu, Q. et al. Thermal cracking simulation of functionally graded materials using the combined finite–discrete element method. Comp. Part. Mech. 7, 903–917 (2020). https://doi.org/10.1007/s40571-019-00290-9

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Keywords

  • Functionally graded materials (FGMs)
  • Thermal–mechanical coupling problem
  • Thermal cracking
  • Crack interaction
  • Combined finite–discrete element method (FDEM)