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Meshless method for nonlinear heat conduction analysis of nano-composites

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Abstract

Carbon nanotubes (CNTs) may become ideal reinforcing materials for high-performance nano-composites due their exceptional properties. Still, much work is needed to be done before the potentials of CNT based composites can be fully realized. The evaluation of effective material properties of nano-composites is one of many difficult tasks. Simulations using continuum mechanics approach can play a significant role in the analysis of these composites. In the present work, nonlinear heat conduction analysis of CNT based composites has been carried out using continuum mechanics approach. Element free Galerkin method has been applied as a numerical tool. Thermal conductivities of nanotube and polymer matrix are assumed to vary quadratically with temperature. Picard and quasi-linearization schemes have been utilized to obtain the solution of a system of nonlinear equations. Cylindrical representative volume element has been used to evaluate the thermal properties of nano-composites. Present simulations show that the temperature dependent matrix thermal conductivity has a significant effect on the equivalent thermal conductivity of the composite, whereas temperature dependent nanotube thermal conductivity has a small effect on the equivalent thermal conductivity of the composite. The results obtained by Picard method have been found almost similar with those obtained by quasi-linearization approach.

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Abbreviations

k m :

thermal conductivity of polymer (epoxy resin) matrix (W/m-K)

k o :

reference thermal conductivity of polymer (epoxy resin) matrix (W/m-K)

k c :

thermal conductivity of carbon nanotube (W/m-K)

k′o :

reference thermal conductivity of carbon nanotube (W/m-K)

k e :

equivalent thermal conductivity of composite (W/m-K)

L :

length of cylindrical RVE (nm)

L c :

nanotube length (nm)

m :

number of terms in the basis

n :

number of nodes in the domain of influence

N :

number of iterations

N′:

number of time steps

q :

heat flux (W/m2)

r o :

outer radius of CNT (nm)

R o :

radius of cylindrical RVE (nm)

t :

time (s)

Δt :

time step size (s)

t c :

thickness of CNT (nm)

\(T^{h} ({\mathbf{r}})\) :

MLS approximation function for temperature

w :

weight function used in MLS approximation

\(\bar{w}\) :

weighting function used in weak form

α:

penalty parameter

β 2 :

matrix thermal conductivity parameter

β′2 :

nanotube thermal conductivity parameter

Γ:

boundary of the domain

Ω m :

computational domain for matrix

Ω c :

computational domain for nanotube

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Acknowledgments

This work was supported by CLUSTER of Ministry of Education, Culture, Sports, Science and Technology, Japan.

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Authors and Affiliations

  1. Department of Mechanical Systems Engineering, Shinshu University, Nagano, 380-8553, Japan

    Indra Vir Singh, Masataka Tanaka & Morinobu Endo

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  1. Indra Vir Singh
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  2. Masataka Tanaka
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  3. Morinobu Endo
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Corresponding author

Correspondence to Indra Vir Singh.

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Singh, I.V., Tanaka, M. & Endo, M. Meshless method for nonlinear heat conduction analysis of nano-composites. Heat Mass Transfer 43, 1097–1106 (2007). https://doi.org/10.1007/s00231-006-0194-7

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