Generalized locally-exact homogenization theory for evaluation of electric conductivity and resistance of multiphase materials

2.1 Unit cell overview

A periodic material with unidirectional inhomogeneities (fibers/porosities) along the x₁ axis is considered in this work. Each inhomogeneity is defined within a repeating unit cell (RUC), which is loaded by effective electric-field components Ē_i(i = 1, 2, 3) and satisfied with periodic boundary conditions. In order to generalize the fiber arrangement in the heterogeneous media, several different geometries of the RUCs are considered, including parallelogram, square and hexagonal arrays, Figure 1. For easy mathematical implementation of the boundary conditions, the electric potential and field are decomposed into averaged and fluctuating components, e.g.

Figure 1

Repeating unit cells with (a) hexagonal and (b) parallelogram arrangements

and

where y_j(j = 2, 3) defines the local coordinate system for the microstructural RUCs.

According to the Ohm-Kirchhoff law, the current density components of a piezoresistive media are defined as:

where i, j = 1, 2, 3. σ_ij are the electric conductivity components.

Herein we define the electric current that is the surface integral of the current densities of Eq. (3) along the surface S_i:

The elastic solution of the present boundary value problem is determined by firstly generating the exact analytical solution that satisfies the interior governing equations within fiber and matrix phases and then imposing the periodic boundary conditions on the RUC’s exterior surfaces. The complete solution involves the transformation between the Cartesian and cylindrical coordinates and is hence defined as the inseparable solutions [28]. To solve the exterior problem, a new variational principle is established for the piezoresistive effect of periodic composites [27]:

where ϕ = ϕ⁰ and I = I⁰ are the periodic electric potential and current density constraints imposed on S_ϕ and S_I, respectively. Taking the first variation of Eq. (5) and considering the periodicities across the opposite surfaces of RUCs yield

For a parallelogram microstructure, the explicit periodicities of electric potential and current density take the following form:

where d₂ and d₃ are the length and height of the parallelogram array, Figure 1a. It should be noted that Eq. (7) is degenerated to the square array once φ = π/2. Similarly, in the case of hexagonal array, we have

where d_i(S_j) are the i-th direction projections of the shortest vectors across two opposite faces (j-th surface with its opposite surface) of the unit cell.

2.2 The interior problem

In order to establish the continuity conditions at the perimeter of fiber domain, the boundary value problem is initially solved within the cylindrical coordinate system [29]. Thus,we start with the relationship between electrical potential and fields for each constituent of the composites:

as well as the relationship between the electrical field and current densities (Ohm-Kirchhoff law):

where the superscript i = f , m, and f and m denote the fiber and matrix phases, respectively. σ_z, σ_r and denotes the axial, radial and tangential inplane electrical conductivity components, respectively. For a transversely isotropic media, we have σ_r = σ_θ = σ.

The governing electrical equations (Maxwell equation) of each constituent can be expressed as:

Substitution of Eq. (10) into Eq. (11) yields the Maxwell equation in the cylindrical coordinate in the form:

The solution of Eq. (12) can be obtained by assuming:

Substitution of Eq. (13) into Eq. (12) yields:

where both terms within the bracket are zeros:

Solving the Euler equations, Eq. (15), yields the final solution

leading to the analytical expressions of current densities:

It should be pointed out that the fiber coefficients H n 3 f and H n 4 f are zeros since the electric potential needs to be bounded in the fiber domain. From the continuity conditions at the perimeter of fiber:

the relationship of unknown coefficients between fiber and matrix phases are obtained as

where c 1 = ( σ f + σ m )/ 2 σ m ,  c 2 =1− c 1 .

Besides the present internal series expansion functions, we can also employ the traveling wave variable substitution technique to solve the governing partial differential equation. The basic idea is to replace the coupled real variables by an equivalent independent complex variable with a travelling wave variable, helping obtain the solution of simplified ordinary differential equation directly. Direct comparison between the two techniques will be introduced in the future work.

2.3 The exterior problem

The remaining unknown coefficients H n f can be determined by implementing the proposed variational principle in Eqs. (5, 6), which imposes the surface potential and current densities on the opposite faces of the unit cell with periodic boundary conditions [27, 29]. The two-scale electric-potential representation in Eq. (1) can be reduced to the fluctuating constraints. For instance, we have

for parallelogram (square) array and

for hexagonal array.

Implementation of those conditions into the variational principle, Eq. (6), yields

for parallelogram array and

for hexagonal array.

Both of the variational principles yield the final solution for the unknown fiber coefficients:

where E ¯ = E ¯ 2 E ¯ 3 T and H^f is a vector comprised of the unknown fiber coefficients. In addition, A and B are the matrices in which the elements are functions of the geometrical and conductivity constants of the microstructures.

2.4 Homogenization equations

Distinct from the finite-element-based techniques that usually adopt the volume integration of the fiber and matrix domains in the homogenization, herein we derive the explicit analytical expression for the homogenization constitutive equations. Firstly, we define a generalized Hill electric-field concentration matrix A^f [30] to relate the averaged fiber electric field and macroscopic electric field:

The averaged fiber electric field can be obtained in a closed form by integrating its local expressions in Eq. (9) over the fiber domain, leading to the explicit expression of Eq. (25) by imposing one unit macroscopic electric field component at a time, with the other components kept zeros, and we have:

Finally, the homogenized constitutive equation for the two-phase unidirectional piezoresistive composites is

where σ^* is the homogenized piezoresistive matrix where the effective piezoresistive conductivity constants are given as:

3.1 Convergence study

The success of the present technique depends on the fast convergence of the series expansion of the internal solutions as well as the periodic boundary implementation. Thus,we first demonstrate the robustness of the developed approach in the calculation of the homogenized and localized electric behavior of composite materials with different microstructural arrangements.

The homogenized electric conductivities of composite materials with 20% volume fraction are calculated as a function of the maximum number of harmonic terms in Eq. (16) of the electric potential representation, Figure 2, which are normalized by the corresponding converged results when 15 harmonic terms are employed. Two cases with relatively large ⧸material property contrasts are considered, namely, σ^fσ^m = 100 and 0.001, representing conductive and nonconductive fibers embedded in conductive matrices, respectively. Unit cells with hexagonal, square and parallelogram arrangements are used in the calculation. The angle between the two connected edges of the parallelogram unit cell is taken as 75^∘. The ability to generate the homogenized properties and local electric field distributions of/within an arbitrary parallelogram unit cell is a major advantage of the present elasticity-based approach. It should be noted that the discretization of arbitrary parallelogram microstructures cannot be easily achieved based on the finite-element and finite-volume-based numerical techniques that require complicated mesh discretization techniques to accomplish.As observed, the normalized effective properties converge to “1” with around 8 harmonic terms for both cases. No difference is shown further increasing harmonic terms in the electric potential representation, Eq. (16). This indicates that the convergence rate of the locally-exact homogenization theory with the Fourier series expansion is very rapid and the effective electric conductivities can be calculated accurately with only a few harmonics.

Figure 2

Convergence of the normalized effective conductivity as a function of harmonic number for different fiber/matrix property ratios and fiber arrangements: (a) σ^f /σ^m = 100; (b) σ^f /σ^m = 0.001

We proceed to illustrate the convergence of localized current density distributions within hexagonal and parallelogram composite microstructures during a unit electric loading Ē₂ = 1V/m as it is the most demanding. The material system investigated herein is a conductive fiber 1000 S/m embedded in a less conducive matrix 200 S/m. The volume fractions for both unit cells are prescribed as 20%. The angle between two connected edges of the parallelogram unit cell is taken as 75^∘. The small fiber diameter relative to the overall unit cell dimensions and relatively large property contrast result in high electric field gradients at the fiber/matrix interface, providing a demanding test of the methods’ accuracy. Figure 3 illustrates the transverse local current density distributions J₂ for three different harmonics, N = 2, 5, 10, respectively. For 2 harmonics and within both microstructures, the basic features of the local current density distributions are reasonably well-captured. The periodicity feature of the parallelogram unit cell in y₃ direction is, however, poorly predicted. 5 harmonics yield improved results for the current density distribution that differs from that obtained based on 2 harmonics. Further increasing harmonics to 10 produces converged local current density field but no fundamental differences are observed between 5 and 10 harmonics, as also observed in Figure 2. Thus, the convergence of the LEHT is very fast and doesn’t require a large number of harmonic terms.

Figure 3

Convergence of the local current-density field J₂ (y₂, y₃) (A/m2) of unit cells under electric loading Ē₃ = 1 V/m and with different maximum harmonic numbers

3.2 Generalized Eshelby problem

Next, the locally-exact homogenization theory with electric capability is validated against the generalized Eshelby solution that describes an inclusion embedded into an infinite matrix,which is subjected to far-field current density loading J 2 ∞ ≠ 0.

According to the transformation equations between the Cartesian and cylindrical coordinates, the boundary conditions can be re-expressed as:

Similar to the analytical solution of Eq. (17), the expression for a transversely isotropic piezoresistive material is given as:

Compared against Eq. (17), herein we only adopt the first-order (n=1) series representation in Eq. (30) since the higher-order terms are ignored when comparing against the boundary conditions in Eq. (29). At infinity (r → ∞), comparison of Eq. (30) with the boundary expression in Eq. (29) yields

while the rest unknown coefficients are obtained by implementing continuity conditions between fiber and matrix phases:

which generates systems of equations:

from which all of the remaining unknown coefficients are solved.

To simulate the far-field loading condition in the modified generalized Eshelby problem, a square repeating unit cell with a dilute volume fraction, namely, 5% in the present case, is employed in the generalized locally-exact simulation. A macroscopic current density is then J̄₂ = 1 A/m² applied to the unit cell. Figure 4 presents a comparison of the inplane local current density distributions, J₂ (y₂, y₃) and J₃ (y₂, y₃), in the Cartesian coordinate system near the fiber for a composite with a conductive fiber, based on the exact solution of an infinite plate under farfield loading J 2 ∞ = 1 A / m 2 and the present LEHT technique. Figure 5 presents similar results in the case of a nonconductive fiber. As observed, the results generated by the locally-exact homogenization theory correlates well with the modified Eshelby solutions almost everywhere in the domain of interest. It should be pointed out that a current-density concentration factor (CDCF) of two is observed in at the top and bottom vicinities of the nonconductive fiber, meaning the current density could be magnified two times under far-field loading condition for a conductive matrix with a nonconductive fiber or porosity. In particular, the transverse local current density within the fiber is uniform while the local current density J₃ (y₂, y₃) is zero in the fiber domain. The small departures between the locally-exact and Eshelby solutions that are seen occur in the vicinity of the boundary of the unit cell, which is due to the differences in the applied boundary conditions. While far-field conditions are imposed in the infinity of the Eshelby problem, the periodical boundary conditions are applied to the mirror faces of the unit cell in the LEHT theory. Nonetheless, these differences are absent when a smaller fiber volume fraction is employed in the LEHT solution.

Figure 4

Comparison of local current densities for a composite unit cell reinforced with conductive fibers based on the exact solution of an infinite plate under far-field loading J 2 ∞ = 1 A / m 2 and the present LEHT technique

Figure 5

Comparison of local current densities for a composite unit cell reinforced with nonconductive fibers based on the exact solution of an infinite plate under far-field loading J 2 ∞ = 1 A / m 2 and the present LEHT technique

3.3 Numerical validations and new results

To assess the accuracy of the locally-exacthomogenization theory in calculating the homogenized and localized behavior at moderate and higher volume fractions, the homogenized version of the FVDAM theory is extended to fulfill this demand. The FVDAM is among a few micromechanics theories that yield accurate results for both the effective and localized behavior, the accuracy of which is proved to be comparable to the Q8 finite-element method but with greater efficiency and stability. Herein,we use the FVDAM as a gold standard for comparison with our newly extended locally-exact theory.

Figure 6 firstly compares the present simulations agains the experimental measurements for the electric conductivities in the longitudinal and transverse directions of a porous nickel. The conductivity parameter of a homogenenous nickel is σ_M = 1.41 × 10⁷⁻¹m⁻¹ [21]. It is seen that most of the measurements are conducted for nickel with porosity volume fractions between 0.2 and 0.4. Both hexagonal and square microstructures are employed for the predictions where the geometric effect doesn’t play an important role when the porosity volume fraction is lower than 0.5. However, good agreement is still obtained against the experimental data, indicating that the present simulation tool is reliable in predicting effective conductivities with high volume fractions that may not be easily conducted in the laboratories.

Figure 6

Validation of the present simulations against the experimental measurement [21] for the effective conductivities of a porous nickel

Figure 7 illustrates comparison of the complete set of effective electric conductivities of composites with three different matrix properties over a wide range of volume fractions. The conductive fibers’ arrangement is modeled as either square or hexagonal arrays. The specific values for the investigated material systems are listed in Table 1, where the fiber electric conductivities are fixed as 1000 S/m with the matrix properties varied from 10 S/m to 200 S/m. The correlation is seen to be excellent for all the material systems and microstructural geometries using 10 harmonics, whose choice was motivated by the results presented in Figure 2. It should be noted that only 5 seconds and 7 seconds are required to generate a full set of effective properties for composites with fourteen different volume fractions under an uncompiled MATLAB environment on PC Intel(R) Core(TM) i5-3320M CPU @ 2.60GHz.

Figure 7

Comparison of the homogenized conductivity parameters between the present technique and the FVDAM for different arrays

Figure 8 illustrates comparison of the complete set of homogenized electric conductivities in the case of nonconductive fibers or pores in three different matrices in square and hexagonal arrays over a wide range of volume fractions, simulating porous materials. In the locally-exact micromechanics simulation, the electric conductivities of the matrices are the same as those in the proceeding case

Figure 8

Comparison of the homogenized conductivity parameters of a porous material between the present technique with the FVDAM with different arrays

while the electric conductivities of the fiber phase are prescribed as one-tenth of the corresponding values of the matrix phases. In the FVDAM simulation, the fiber phase is excluded from the composite such that a real porosity is modeled. Once again, the locally-exact homogenization exhibits remarkable correlation with the finite-volume predictions in the low and intermmediate volume fractions. A cursory examination at higher volume volumes indicates that the locally-exact homogenization generates slightly larger effective conductivities. The small differences are due to the way in which the porosity is treated. The contribution from the fiber phase is negligible at low and intermmediate volume fractions but becomes noticeable when the fiber-volume fraction is sufficiently large to contribute some load-bearing capability despite the small conductivities. The locally-exact results are therefore consistent with this observation, illustrating that the method is sufficiently sensitive to correctly capture these small effects.

The localized current density distributions, J₂ (y₂, y₃) and J₃ (y₂, y₃), generated by the locally-exact homogenization and FVDAM simulation of a hexagonal unit cell with 20% volume fraction during transverse electric loading Ē₂ = 1 V/m are compared in Figure 9. The electric conductivities of the fiber and matrix phases are 1000,200 S/m, respectively. As expected, no visible difference is observed between the two approach’s predictions. Similarly, Figure 10 illustrates comparison of local electric field distributions, E₂ (y₂, y₃) and E₃ (y₂, y₃), within a square unit cell inserted with a nonconductive fiber in the case of LEHT and a porosity in the case of FVDAM. Given the negligible contributions from the nonconductive fiber to electric field in the vicinity of fiber/matrix interface, the LEHT generates generally good correlation with the FVDAM in the matrix domain.

Figure 9

Comparison of the local current-density fields of a composite unit cell with 20% volume fraction generated by the LEHT and FVDAM under electric loading Ē₂ = 1 V/m

Figure 10

Comparison of the local electric fields of a porous composite unit cell with 20% volume fraction generated by the LEHT and FVDAM under loading Ē₂ = 1 V/m

To quantitatively highlight the accuracy of the locally-exact theory relative to the FVDAM results, Table 2 compares effective electric conductivity for rhombic unit cells with two different characteristic angles, φ = 50^∘ and 70^∘, and three different fiber volume fractions, 0.1, 0.3 and 0.5, thus covering a wide range of reinforcement content. Similar research is conducted in Table 3 for the porous materials when the fiber phase is excluded for the same microstructure. Table 4 presents comparison of the effective

electric conductivity for rhombic unit cells with two large property contrasts, σ_f / σ_m = 10,100, and three different volume fractions, 0.3, 0.5 and 0.7 at a fixed angle φ = arccos(1/4). The electric conductivities predicted by the locally-exact theory with N = 10 converge to the FVDAM results to within approximately less than 0.1% and remain stable thereafter.

Despite a few publications appeared in the literature that address the homogenized behavior of a rhombic unit cell, substantially less work was done concerning the localized field distributions within a rhombic microstructure [31, 32]. Figure 11 presents local electric current density distributions, J₂ (y₂, y₃) and J₃ (y₂, y₃), during a transverse electric loading E₂ = 1 V/m for a rhombic unit cell with a connecting angle of φ = 50^∘ and 30% volume fraction, generated by the locally-exact homogenization theory. Figure 12 illustrates electric field distributions, E₂ (y₂, y₃) and E₃ (y₂, y₃), when the connecting angle increases to φ = 70^∘ and volume fraction increases to 60%. Not surprisingly, the two approaches predict nearly identical results almost everywhere in the interest of domain. We note that the above two cases are very demanding for the finite-volume and finite-element approaches because of extensive mesh refinements are required. However, no meshes are involved in the present simulations.

Figure 11

Comparison of the local current-density fields of a composite unit cell with 30% volume fraction generated by the LEHT and FVDAM under loading E₂ = 1 V/m

Figure 12

Comparison of the local electric fields of a composite unit cell with 60% volume fraction generated by the LEHT and FVDAM under electric loading E₂ = 1 V/m