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ABSTRACT
In modern design of composite structures, multiple materials with different properties are bound together. Accurate prediction of the strength of the interface between different materials, especially with the existence of cracks under thermal loading, is demanded in engineering. To this end, detailed knowledge on the distribution of temperature and heat flux is required. This study conducts a systematical investigation on the cracks terminated at material interface under steady-state thermal conduction. A new symplectic analytical singular element is constructed for the numerical modeling. Combining the proposed element with conventional finite elements, the generalized flux intensity factors can be solved accurately.
Nomenclature
| A, B | = | coefficients of the general solution of symplectic eigenvector |
| F | = | vector of coefficients of the general solution of symplectic eigenvector |
| H | = | Hamiltonian operator matrix |
| k | = | thermal conductivity |
| Mi (i = 1, 2, 3) | = | material |
| qr, qθ | = | heat flux densities |
| R | = | chain matrix relates the eigenvectors of two adjacent materials |
| Sr, Sθ | = | symplectic dual variables |
| T | = | temperature |
| Z | = | unknown vector in symplectic solving system |
| α | = | vertex angle of material |
| (r, θ) | = | polar coordinate system |
| ∇2 | = | Laplacian operator |
| μ | = | symplectic eigenvalue |
| = | symplectic eigenvector | |
| ψT, ψr | = | elements in symplectic eigenvector |
| γ | = | coefficients of symplectic eigen expanding terms |
| Θ | = | expression of characteristic equation of symplectic eigenvalue |
Nomenclature
| A, B | = | coefficients of the general solution of symplectic eigenvector |
| F | = | vector of coefficients of the general solution of symplectic eigenvector |
| H | = | Hamiltonian operator matrix |
| k | = | thermal conductivity |
| Mi (i = 1, 2, 3) | = | material |
| qr, qθ | = | heat flux densities |
| R | = | chain matrix relates the eigenvectors of two adjacent materials |
| Sr, Sθ | = | symplectic dual variables |
| T | = | temperature |
| Z | = | unknown vector in symplectic solving system |
| α | = | vertex angle of material |
| (r, θ) | = | polar coordinate system |
| ∇2 | = | Laplacian operator |
| μ | = | symplectic eigenvalue |
| = | symplectic eigenvector | |
| ψT, ψr | = | elements in symplectic eigenvector |
| γ | = | coefficients of symplectic eigen expanding terms |
| Θ | = | expression of characteristic equation of symplectic eigenvalue |
Acknowledgments
The supports of the National Natural Science Foundation of China (No. 11502045, and No. 11372065), the Fundamental Research Funds for the Central Universities (DUT15RC(3)029) are gratefully acknowledged.