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A 2D discrete heat transfer model considering the thermal resistance effect of fractures for simulating the thermal cracking of brittle materials

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Abstract

A concise, two-dimensional discrete heat transfer model is presented, which considers the thermal resistance effects of cracks. The discrete heat transfer model discretizes the continuum into individual triangular elements, and the adjacent triangular elements transfer heat through corresponding joint elements. Then, the principle of selecting the heat exchange coefficient of the joint element is determined through a parameter sensitivity analysis. Moreover, the 2D discrete heat transfer model is combined with the finite discrete element method to build a thermomechanical coupled model for simulating brittle material thermal cracking. The thermal stress and thermal shock problems are studied by using the thermomechanical coupled model, and the numerical results are compared with either the analytical solution or experimental results to validate the 2D discrete heat transfer model and thermomechanical coupled model. The discrete heat transfer model and thermomechanical coupled model provide a powerful tool for solving heat transfer problems in discontinuous media as well as the thermal cracking problem in brittle materials.

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Abbreviations

\(q_{i}\) :

Heat flow rate along the \(i\) direction

\(k_{ij}\) :

Thermal conductivity tensor

\(T\) :

Temperature

\(t\) :

Time

\(Q_{\text{total}}\) :

Total heat flow into the mass \(M\) per unit time

\(C_{p}\) :

Specific heat

\(M\) :

Mass

\(A\) :

Area of the triangular element

\(n_{i}\) :

Outward normal unit vector

\(\bar{T}^{m}\) :

Average temperature of edge \(m\)

\(\Delta x_{j}^{m}\) :

Coordinate difference between the two nodes of edge \(m\)

\(\varepsilon_{ij}\) :

Two-dimensional permutation tensor

q x :

Heat flow rate along the x direction

q y :

Heat flow rate along the y direction

\(n_{i}^{(n)}\) :

Outward normal unit vector

\(L^{(n)}\) :

Length of the edge opposite to node \(n\)

\(Q_{\Delta 123 \to 1}\) :

Heat flow flowing into node 1 from triangular element Δ123

\(T_{i}\) :

Temperature at node i

\(h_{j}\) :

Heat exchange coefficients of the joint element

\(Q_{\Delta 456 \to 1}\) :

Heat flow flowing into node 1 from triangular element Δ456

\(Q_{\Delta 789 \to 1}\) :

Heat flow flowing into node 1 from triangular element Δ789

\(T_{1}^{t + \Delta t}\) :

Temperature of node 1 at \(t + \Delta t\)

\(T_{1}^{t}\) :

Temperature of node 1 at \(t\)

\(k\) :

Thermal conductivity coefficient of triangular element

\(L_{\text{e}}\) :

Element size

\(r_{T}\) :

Reduction coefficient

\(\Delta \sigma_{ij}\) :

Stress increment

\(K\) :

Bulk modulus

\(G\) :

Shear modulus

\(\alpha\) :

Thermal expansion coefficient

\(K^{*}\) :

Plane strain problem \(K^{*} = K\), plane stress problem \(K^{*} = 6KG/(3K + 4G)\)

\(n_{j}\) :

Outward normal unit vector of the triangular element edges

\(T_{1}\) :

Temperature at the left boundary

\(T_{2}\) :

Temperature at the right boundary

\(\kappa\) :

Thermal diffusion coefficient (\(\kappa { = }k/\rho C_{p}\) when \(k_{x} = k_{y}\))

\(W\) :

Width

\(L\) :

Length

\(x\) :

The distance from the left boundary

\(\rho\) :

The density

\(T_{0}\) :

The initial temperature

\(E\) :

Elastic modulus

\(v\) :

Poisson’s ratio

\(f_t\) :

Tensile strength

\(h_f\) :

Heat exchange coefficient between the water and solid

α :

Thermal expansion coefficient

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Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11872340, 11602006; the Beijing Natural Science Foundation under Grant No. 1174012; the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (CUG170657); the Chaoyang District Postdoctoral Science Foundation project funded under Grant No. 2016ZZ-01-08; and the National Natural Science Foundation of China under Grant No. 41731284.

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Authors and Affiliations

  1. Faculty of Engineering, China University of Geosciences, Wuhan, 430074, China

    Chengzeng Yan & Yu-Yong Jiao

  2. International Joint Research Center for Deep Earth Drilling and Resource Development, China University of Geosciences, Wuhan, 430074, China

    Chengzeng Yan & Yu-Yong Jiao

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Correspondence to Yu-Yong Jiao.

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Yan, C., Jiao, YY. A 2D discrete heat transfer model considering the thermal resistance effect of fractures for simulating the thermal cracking of brittle materials. Acta Geotech. 15, 1303–1319 (2020). https://doi.org/10.1007/s11440-019-00821-x

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