Copper–graphite composites: thermal expansion, thermal and electrical conductivities, and cross-property connections

Abstract

The paper focuses on thermal and electrical properties of copper–graphite composites. Copper–graphite composites in the range of 0–50 vol% of graphite were prepared from the mixture of copper and graphite powder by the powder metallurgy method. Such composites combine high thermal and electrical conductivities provided by copper matrix and low thermal expansion coefficient and lubricating properties due to the graphite phase. We model thermal and electrical conductivities and thermal expansion coefficient using methods of micromechanics in connection with the material microstructure and measure these quantities experimentally to compare with the results of modeling. Cross-property connections between thermal and electrical properties of the composites are established and experimentally verified.

Appendix

Tensor basis in the space of transversely isotropic fourth-rank tensors. Representation of certain transversely isotropic tensors in terms of the tensor basis

The operations of analytic inversion and multiplication of fourth-rank tensors are conveniently done in terms of special tensorial bases that are formed by the combinations of unit tensor $\delta_{ij}$ and one or two orthogonal unit vectors [44, 45]. In the case of the transversely isotropic elastic symmetry, the following basis is most convenient (it differs slightly from the one used in [45]):

$$ \begin{aligned} T_{ijkl}^{\left( 1 \right)} = \theta_{ij} \theta_{kl} ,\quad T_{ijkl}^{\left( 2 \right)} = {{\left( {\theta_{ik} \theta_{lj} + \theta_{il} \theta_{kj} - \theta_{ij} \theta_{kl} } \right)} \mathord{\left/ {\vphantom {{\left( {\theta_{ik} \theta_{lj} + \theta_{il} \theta_{kj} - \theta_{ij} \theta_{kl} } \right)} 2}} \right. \kern-0pt} 2},\quad T_{ijkl}^{\left( 3 \right)} = \theta_{ij} m_{k} m_{l} , \hfill \\ T_{ijkl}^{\left( 4 \right)} = m_{i} m_{j} \theta_{kl} ,T_{ijkl}^{\left( 5 \right)} = {{\left( {\theta_{ik} m_{l} m_{j} + \theta_{il} m_{k} m_{j} + \theta_{jk} m_{l} m_{i} + \theta_{jl} m_{k} m_{i} } \right)} \mathord{\left/ {\vphantom {{\left( {\theta_{ik} m_{l} m_{j} + \theta_{il} m_{k} m_{j} + \theta_{jk} m_{l} m_{i} + \theta_{jl} m_{k} m_{i} } \right)} 4}} \right. \kern-0pt} 4},T_{ijkl}^{\left( 6 \right)} = m_{i} m_{j} m_{k} m_{l} \hfill \\ \end{aligned} $$

(48)

where $\theta_{ij} = \delta_{ij} - m_{i} m_{j}$ and $\varvec{m} = m_{1} \varvec{e}_{1} + m_{2} \varvec{e}_{2} + m_{3} \varvec{e}_{3}$ is a unit vector along the axis of transverse symmetry.These tensors form the closed algebra with respect to the operation of (non-commutative) multiplication (contraction over two indices):

$$ \left( {{\varvec{T}}^{\left( \alpha \right)} {\varvec{:}}{\varvec{T}}^{\left( \beta \right)} } \right)_{ijkl} \equiv T_{ijpq}^{\left( \alpha \right)} T_{pqkl}^{\left( \beta \right)} $$

(49)

The inverse of any fourth-rank tensor ${\varvec{X}}$, as well as the product ${\varvec{X}}\,{\varvec{:}}\,{\varvec{Y}}$ of two such tensors are readily found in the closed form, as soon as the representation in the basis

$$ {\varvec{X}} = \sum\limits_{k = 1}^{6} {X_{k} {\varvec{T}}^{\left( k \right)} } ,{\varvec{Y}} = \sum\limits_{k = 1}^{6} {Y_{k} {\varvec{T}}^{\left( k \right)} } $$

(50)

are established. Indeed: (a) inverse tensor $\varvec{X}^{ - 1}$ defined by $X_{ijmn}^{ - 1} X_{mnkl}$ $= \left( {X_{ijmn} X_{mnkl}^{ - 1} } \right)\; = J_{ijkl}$ is given by

$$ \varvec{X}^{ - 1} = \frac{{X_{6} }}{{2\Delta }}\varvec{T}^{\left( 1 \right)} + \frac{1}{{X_{2} }}\varvec{T}^{\left( 2 \right)} - \frac{{X_{3} }}{\Delta }\varvec{T}^{\left( 3 \right)} - \frac{{X_{4} }}{\Delta }\varvec{T}^{\left( 4 \right)} + \frac{4}{{X_{5} }}\varvec{T}^{\left( 5 \right)} + \frac{{2X_{1} }}{\Delta }\varvec{T}^{\left( 6 \right)}, $$

(51)

where $\Delta = 2\left( {X_{1} X_{6} - X_{3} X_{4} } \right)$. (b) product of two tensors ${\varvec{X}}\,{\varvec{:}}\,{\varvec{Y}}$ (tensor with $ijkl$ components equal to $X_{ijmn} Y_{mnkl}$) is

$$ \begin{aligned} {\varvec{X}}\,{\varvec{:}}\,{\varvec{Y}} = \left( {2X_{1} Y_{1} + X_{3} Y_{4} } \right){\varvec{T}}^{\left( 1 \right)} + X_{2} Y_{2} {\varvec{T}}^{\left( 2 \right)} + \left( {2X_{1} Y_{3} + X_{3} Y_{6} } \right){\varvec{T}}^{\left( 3 \right)} \\ + \left( {2X_{4} Y_{1} + X_{6} Y_{4} } \right){\varvec{T}}^{\left( 4 \right)} + \frac{1}{2}X_{5} Y_{5} {\varvec{T}}^{\left( 5 \right)} + \left( {X_{6} Y_{6} + 2X_{4} Y_{3} } \right){\varvec{T}}^{\left( 6 \right)} \\ \end{aligned} $$

(52)

If $x_{3}$ is the axis of transverse symmetry, general transversely isotropic fourth-rank tensor, being represented in this basis

$$ \Psi_{ijkl} = \sum {\psi_{m} T_{ijkl}^{m} } $$

has the following components:

$$ \begin{aligned} \psi_{1} = {{\left( {\Psi_{1111} + \Psi_{1122} } \right)} \mathord{\left/ {\vphantom {{\left( {\Psi_{1111} + \Psi_{1122} } \right)} 2}} \right. \kern-0pt} 2};\psi_{2} = 2\Psi_{1212} ;\psi_{3} = \Psi_{1133} ;\psi_{4} = \Psi_{3311} ; \hfill \\ \psi_{5} = 4\Psi_{1313} ;\quad \psi_{6} = \Psi_{3333} \hfill \\ \end{aligned} $$

(53)

In particular:

• Tensor of elastic compliances of the isotropic material $S_{ijkl} = \sum {s_{m} T_{ijkl}^{m} }$ has the following components

  $$ s_{1} = \frac{1 - \nu }{{4G\left( {1 + \nu } \right)}};s_{2} = \frac{1}{2G};s_{3} = s_{4} = \frac{ - \nu }{{2G\left( {1 + \nu } \right)}};s_{5} = \frac{1}{G};s_{6} = \frac{1}{{2G\left( {1 + \nu } \right)}} $$

  (54)

• Tensor of elastic stiffness of the isotropic material by $C_{ijkl} = \sum {c_{m} T_{ijkl}^{m} }$ has components

  $$ c_{1} = \lambda + G;c_{2} = 2G;c_{3} = c_{4} = \lambda ;c_{5} = 4G;c_{6} = \lambda + 2G, $$

  (55)

  where $\lambda = {{2G\nu } \mathord{\left/ {\vphantom {{2G\nu } {\left( {1 - 2\nu } \right)}}} \right. \kern-0pt} {\left( {1 - 2\nu } \right)}}.$

• Unit fourth-rank tensors are represented in the form

  $$ J_{ijkl}^{\left( 1 \right)} = {{\left( {\delta_{ik} \delta_{lj} + \delta_{il} \delta_{kj} } \right)} \mathord{\left/ {\vphantom {{\left( {\delta_{ik} \delta_{lj} + \delta_{il} \delta_{kj} } \right)} 2}} \right. \kern-0pt} 2} = \frac{1}{2}T_{ijkl}^{1} + T_{ijkl}^{2} + 2T_{ijkl}^{5} + T_{ijkl}^{6} $$

  (56)
  $$ J_{ijkl}^{\left( 2 \right)} = \delta_{ij} \delta_{kl} = T_{ijkl}^{1} + T_{ijkl}^{3} + T_{ijkl}^{4} + T_{ijkl}^{6} $$

  (57)

• Compliance contribution tensor H of a spheroidal inhomogeneity, with bulk and shear moduli $K_{1}$ and $G_{1}$, is represented in this basis by coefficients (see [20], for example):

  $$ \begin{aligned} h_{1} = \frac{1}{{2\Delta }}\left[ {\delta K + \frac{4}{3} \delta G + q_{6} } \right];h_{2} = \frac{1}{{2\delta G + q_{2} }};h_{3} = h_{4} = - \frac{1}{\Delta }\left[ {\delta K - \frac{2}{3}\delta G + q_{3} } \right]; \hfill \\ h_{5} = \frac{4}{{4\delta G + q_{5} }};h_{6} = \frac{2}{\Delta }\left[ {K + \frac{1}{3}\delta G+ q_{1} } \right] \hfill \\ \end{aligned} $$

  (58)

  where the following notations are used

  $$ \delta K = {{K_{1} K_{0} } \mathord{\left/ {\vphantom {{K_{1} K_{0} } {\left( {K_{0} - K_{1} } \right)}}} \right. \kern-0pt} {\left( {K_{0} - K_{1} } \right)}};\begin{array}{*{20}c} {} & {\delta G = } \\ \end{array} {{G_{1} G_{0} } \mathord{\left/ {\vphantom {{G_{1} G_{0} } {\left( {G_{0} - G_{1} } \right)}}} \right. \kern-0pt} {\left( {G_{0} - G_{1} } \right)}} $$
  $$ \Delta = 2\left[ {3\delta G\delta K + \delta K\left( {q_{1} + q_{6} - 2q_{3} } \right) + \frac{\delta G}{3}\left( {4q_{1} + q_{6} + 4q_{3} } \right) + \left( {q_{1} q_{6} - q_{3}^{2} } \right)} \right]\; $$

  (59)

  and

  $$ \begin{aligned} q_{1} = G\left[ {4\kappa - 1 - 2\left( {3\kappa - 1} \right)f_{0} - 2\kappa f_{1} } \right],q_{2} = 2G\left[ {1 - \left( {2 - \kappa } \right)f_{0} - \kappa f_{1} } \right], \hfill \\ q_{3} = q_{4} = 2G\left[ {\left( {2\kappa - 1} \right)f_{0} + 2\kappa f_{1} } \right],q_{5} = 4G\left( {f_{0} + 4\kappa f_{1} } \right) \hfill \\ q_{6} = 8G\left( {\kappa f_{0} - \kappa f_{1} } \right), \hfill \\ \end{aligned} $$

  (60)

  where $\kappa$, $f_{0} \left( \gamma \right)$, and $f_{1} \left( \gamma \right)$ are given by (18) and (9).