Thermal shock analysis of 2D cracked solids using the numerical manifold method and precise time integration
Introduction
Engineering equipment such as the aero-engines, gas turbines and pressure vessels is frequently exposed to thermal shock. Under certain conditions, thermally induced stresses may cause the fracture and failure of cracked facilities. Consequently, the study of the behavior of cracked solids under thermal shock is of great importance. In light of the significance, a lot of work has been carried out during the past several decades. Many researchers focused on the analytical solutions. Oliveira and Wu [1] determined the thermal stress intensity factors (TSIFs) of axial cracks in hollow cylinders under thermal shock using the closed form weight function formula. Noda and Ashida [2] adopted the successive approximation, Fourier integral and Bessel series to solve transient thermal annular crack problem in an infinite transversely isotropic cylinder. Lee and Kim [3] calculated the TSIFs of elliptical surface cracks in thin-walled and thick-walled cylinders with the modified Vainshtok's weight function method. Noda and Wang [4] applied the approach of singular integration equation to tackle the thermal shock responses of the functionally graded materials (FGMs) with collinear cracks. Shahani and Nabavi [5] used the finite Hankel transform and the weight function method to compute the TSIFs of internal semi-elliptical crack in a thick-walled cylinder subjected to transient thermal stresses. Wang and Li [6] analyzed the transient thermoelastic fracture of piezoelectric material with the Laplace transformation and integral equation method.
Although widely applied, analytical solutions are generally limited to simple configurations (e.g., single or periodic cracks in infinite or semi-infinite mediums under regular initial or boundary conditions). For problems with finite dimensions, arbitrary cracks or under complex loadings, numerical tools such as the finite element method (FEM), the boundary element method (BEM) and the extended finite element method (XFEM) are much more popular. Emmel and Stamm [7] computed the transient TSIFs for cracked rectangular plate and hollow cylinder with the FEM. Magalhaes and Emery [8] inspected the effects of transient thermal loads on crack propagation in brittle film-substrate structure by the FEM. Through the three-dimensional elastic-plastic FEM, Kim et al. [9] evaluated the integrity of vessel with subclad crack under pressurized thermal shock. Prasad et al. [10] applied the dual BEM and path-independent J-integral to investigate the two-dimensional (2D) transient thermoelastic crack problems. Considering crack closure conditions, Giannopoulos and Anifantis [11] obtained both steady and transient TSIFs of 2D bimaterial interfacial cracks by the BEM. With the uncoupled thermoelastic theory, Zamani and Eslami [12] implemented the XFEM to model the effect of both mechanical and thermal shocks on 2D cracked solids. Rokhi and Shariati [13] studied the response of cracked FGMs under thermal shock with the XFEM in the framework of coupled thermoelasticity.
In the past two decades, considerable efforts have been put on to the development of the numerical manifold method (NMM) proposed by Shi [14]. The soul of the NMM lies in the use of finite cover concept, which has been adopted by Bathe and his coauthors [15], [16], [17] to improve the FEM solutions most recently. Benefiting from the use of dual cover systems (i.e., the mathematical cover and the physical cover) [14], [18], the NMM is very powerful in discontinuous analysis (e.g., in solving crack or inclusion problems). The major highlights of the NMM for crack modeling can be summarized in four aspects: (1) the mathematical cover can be independent of all domain boundaries including cracks; (2) the discontinuity of physical field across crack faces can be manifested in essence; (3) the local property at crack tip zone can be well captured through the use of associated local functions in the approximation, and (4) higher-order approximations can be achieved through the use of higher-order local functions on a fixed mathematical cover. To date, the NMM has been successfully improved to solve various stationary cracks and crack growth problems in homogeneous [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30] and heterogeneous materials [31], [32], [33], [34], [35].
In the present paper, the NMM is further extended to tackle 2D stationary crack problems under thermal shock. The solution procedure is generally divided into two parts: Firstly, the transient heat diffusion problem is analyzed and the corresponding NMM discrete equations are solved with the precise time integration method (PTIM) [36], which has absolute stability, immunity to oscillations and time-step-independent precision. Subsequently, the calculated temperature fields at selected instants are imported into the thermoelastic part to extract the TSIFs.
The remaining of the paper is addressed as follows. In Section 2, the governing equations and associated boundary and/or initial conditions are provided. The NMM formulations for both transient heat conduction and thermoelastic analysis are derived in Section 3. Section 4 presents the details of the PTIM for transient heat conduction analysis and the spatial integral scheme as well. To verify the proposed method, several numerical examples are tested in Section 5. Finally, the concluding remarks are addressed in Section 6.
Section snippets
Statement of problems
As shown in Fig. 1, consider a cracked isotropic homogenous physical domain enclosed by the boundary in the unidirectional coupling transient linear thermoelasticity (i.e., the thermal loading affects the displacement, strain and stress fields, but not vice versa). Ignoring both the heat source and the body force, the governing equations for this problem are [10]where Ο is the density and c is the specific heat at constant pressure. denotes partial derivative.
NMM approximations
In the NMM, to solve a crack problem, the mathematical cover (MC), composed of a series of mathematical patches (MPs), is firstly built. Broadly speaking, the MP, formed by mathematical elements, can be of any shape and overlapped. The MC may be independent of all domain boundary (including cracks) but must be large enough to cover the whole cracked domain. On each MP, a partition of unity (PU) [37] weight function is defined. Next, the physical patches (PPs), the collection of which is termed
Computer implementations
To solve Eq. (23), difference methods such as the forward difference scheme (i.e., Euler scheme), the central difference scheme (i.e., Crank-Nicolson scheme) and the backward difference scheme are mostly adopted [40]. However, the forward scheme is conditionally stable and may also be oscillating. As for the central and backward scheme, they both are unconditionally stable, whereas the former is also non-immune to oscillation although its accuracy is higher at a given time step and mesh
Numerical examples
In this part, to verify the proposed method, three numerical examples are conducted. Firstly, a pure mode II edge crack in a rectangular plate is tested; then, two embedded mixed mode I-II cracks in a square plate is considered; finally, a symmetrical branched crack in a square plate is studied. Throughout the computations, constant polynomial basis (see Eqs. (12), (13)) is considered. The same material with,, ,, and is investigated. The temperature boundary
Concluding remarks
In this work, the numerical manifold method, combined with the precise time integration scheme, has been developed to tackle 2D crack problems subjected to thermal shock. The temperature and displacement discontinuity across crack surfaces were naturally represented due to the introduction of dual cover systems in the NMM. The singularity of field gradients at crack tip was described through the use of asymptotic basis in the local functions on singular PPs. The time integration in transient
Acknowledgements
The present work was supported by the National Natural Science Foundation of China (Grant Nos. 11462014, 11572282), the Provincial Natural Science Foundation of Jiangxi, China (Grant No. 20151BAB202003), the Science and Technology Program of Educational Committee of Jiangxi Province of China (Grant No. GJJ14526).
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