Thermoelasticity: modeling the stress change due to thermal diffusion

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Thermal diffusion through a solid body (or fluid/gas/vapor diffusion through a porous medium) causes temperature (or fluid/gas/vapor pressure) change with time that alters strain and stress. Such temperature, strain and stress variations can cause cracking, failure or phase change, etc inside the considered body (Johnson et al., 1978; Behar et al., 1997; Zalba et al., 2003; De Schutter, 2002).

The temperature (or fluid pressure) change during the diffusion process can be modeled by solving a time and space differential equation (Bergman and Incropera, 2011). Such problem can be directly solved in time domain by using the Finite Difference Method (FDM) in which the time and space derivative terms are approximated by the finite different ones (Ozisik, 1994; Smith, 1985). It can be also solved by transforming the solutions obtained in Laplace's domain to time domain (see the module Modeling thermal diffusion using Laplace's transform).

At a given time step, temperature field obtained by the diffusion model can be used to model the stress field by considering the classical thermoelastic coupled constitutive model (Biot, 1956; Green and Lindsay, 1972; Zimmerman, 2000).

Let's consider an example of a cylinder hole (circular hole on the transversal plane) located in an infinite medium that has a homogeneous farfield temperature (or fluid pressure for the case of fluid diffusion). A local temperature is applied on the inner surface of the hole. Temperature and stress in the area arround the hole will change with time due to the diffusion process and the thermoelastic coupled behavior. The diffusivity of the medium's material around the hole is required to model the temperature field and thermoelastic properties are necessary to model the stress field. Below, a tool is provided to simulate that problem. Note that the positive sign convention is conidered for tenion stresses.

Zalba, B., Marı́n, J. M., Cabeza, L. F., & Mehling, H. (2003). Review on thermal energy storage with phase change: materials, heat transfer analysis and applications. Applied thermal engineering, 23(3), 251-283.
Behar, F., Vandenbroucke, M., Tang, Y., Marquis, F., & Espitalie, J. (1997). Thermal cracking of kerogen in open and closed systems: determination of kinetic parameters and stoichiometric coefficients for oil and gas generation. Organic Geochemistry, 26(5), 321-339.
De Schutter, G. (2002). Finite element simulation of thermal cracking in massive hardening concrete elements using degree of hydration based material laws. Computers & Structures, 80(27), 2035-2042.
Johnson, B., Gangi, A. F., & Handin, J. (1978, January). Thermal cracking of rock subjected to slow, uniform temperature changes. In 19th US Symposium on Rock Mechanics (USRMS). American Rock Mechanics Association.
Bergman, T. L., & Incropera, F. P. (2011). Fundamentals of heat and mass transfer. John Wiley & Sons.
Ozisik, N. (1994). Finite difference methods in heat transfer. CRC press.
Smith, G. D. (1985). Numerical solution of partial differential equations: finite difference methods. Oxford university press.
Biot, M. A. (1956). Thermoelasticity and irreversible thermodynamics. Journal of Applied Physics, 27(3), 240-253.
Zimmerman, R. W. (2000). Coupling in poroelasticity and thermoelasticity. International Journal of Rock Mechanics and Mining Sciences, 37(1), 79-87.
Green, A. E., & Lindsay, K. A. (1972). Thermoelasticity. Journal of Elasticity, 2(1), 1-7.

Input the parameters then click Compute button to run the simulation.

Hole radius (m)

Borehole temperature (°C)

Farfield temperature (°C)

Diffusivity (mm2/s)

Young modulus (Gpa)

Poisson ratio

Thermal expansion coefficient (1e-6/°C)

Time (day)

Output data