Keywords: homogenization method, transient heat transfer, periodic composite structures, finite element method.

Homogenization of the transfer problems in heterogeneous media is still not very well recognized area. While various approximative solutions can be formulated for steady-state problems, then homogenized characteristics for unsteady transfer are still calculated as spatial averages over the entire composite area [1,2]. Such methodology can be efficient, however there is no mathematical proof of their accuracy and, therefore, computational simulation is the only method to demonstrate this efficiency. This idea is studied now on the example of the unsteady heat transfer in various composite materials.

The following algorithm is proposed in the paper to make such a homogenization as general as possible. Steady-state problem for particular composite medium (unidirectional, fiber and particle-reinforced) is homogenized first according to the homogenization technique adequate to the composite type and then, the effective heat capacity is calculated as a spatial average over the composite area (or volume). Such an approach makes possible to introduce multiresolutional homogenization approach where heat conductivity and capacity vary on many geometrical scales of the composite material and/or structure.

It is assumed that (1) there is no mechanical deformation accompanying thermal process, (2) material parameters are temperature independent, (3) interfaces between particular composite components are perfect, and (4) that there are no phase changes and latent heat effects. The governing transient heat transfer equations for the anisotropic composite can be written as

(a)

flow equilibrium conditions:

(b)

temperature (essential) boundary conditions:

(c)

heat flux (natural) boundary conditions:

(d)

initial conditions:

The homogenized heat conductivity coefficient depends on the composite type and for the unidirectional two-scale composite is calculated as [3]

The effective volumetric heat capacity is determined as

what holds true for composites with periodic, quasi-periodic or even non-periodic structure with two or more geometrical scales.

Starting from such equations the variational equations for real and homogenized composites are derived and implemented in the Finite Element Method program ANSYS, release 5.7.1. To demonstrate computational effectiveness of the method, some more popular composites are discretized and subjected to some unsteady thermal heat transfer boundary conditions - temperature spatial distributions, histories, heat fluxes and their computational errors for the real and homogenized structures are compared [4].

The qualitative and quantitative results of performed computational FEM-based modeling confirm the usefulness of the homogenization method proposed to numerical analysis of composite structures with deterministically defined geometry and physical properties. It is shown that even neglecting scale parameter in the mathematical model assumptions, temperatures during transient heat transfer in a real structure are approximated by the homogenized model precisely enough. The phenomenological numerical sensitivity studies with respect to heat conductivities and volumetric capacities contrasts show some influence of the first composite parameter only. It means that for the entire class of composites with the same microgeometry, boundary conditions and varying material parameters, the temperature histories for real and homogenized structures are almost the same.

1

J.L. Auriault, H.I. Ene, "Macroscopic modeling of heat transfer in composites with interfacial thermal barrier". Int. J. Heat & Mass Transfer 37: 2885-2892, 1994. doi:10.1016/0017-9310(94)90342-5

2

A.L. Kalamkarov, A.G. Kolpakov, "Analysis, Design and Optimization of Composite Structures". Wiley, 1997.

3

M. Kaminski, "Homogenized properties of n-components composites". Int. J. Engrg. Sci. 38(4): 405-427, 2000. doi:10.1016/S0020-7225(99)00033-6

4

M. Kaminski, "Probabilistic effective heat conductivity of fiber composites". Int. J. Mech. & Mech. Engrg. 2: 175-207, 1998.

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