Computation of effective thermo-piezoelectric properties of porous ceramics via asymptotic homogenization and finite element methods for energy-harvesting applications

Abstract

We consider the linear thermo-piezoelectric properties of a ceramic matrix with cylindrical empty pores distributed periodically. The asymptotic homogenization method is applied to an elliptical tensor-weighted boundary value problem in the Stress-Charge-Entropy formulation of the constitutive relations with rapidly oscillating coefficients and free boundary conditions on the surfaces of the pores. For different shapes of the pore cross section, we solve the local problems via finite element method to compute the effective coefficients as functions of the physical properties of the matrix, the shape of the pore cross section and their volume fraction. The numerical results show excellent agreement with analytical formulae. When the effective coefficients are transformed to the Strain-Charge-Entropy formulation of the constitutive relations, they become independent of the shape of the cross section, which further validates the importance of the analytical formulae. We compute the piezoelectric and pyroelectric figures of merit for energy-harvesting applications, which depend on the effective coefficients and are compared with recent experimental results. This contribution could be useful for fine-tuning the properties of this class of materials for energy-harvesting applications.

Appendix

We illustrate the adimensionalization process in the problem $L_2^q$ (expression (19)) as its associated effective coefficients were the most deviated when the material properties were entered in the program in their SI units without prefixes.

The tilde $\widetilde{\cdot }$ indicates an adimensionalized coefficient. The subscript 0 indicates the adimensionalization/rescaling constants. The following parameters are introduced.

$$\begin{aligned} \widetilde{e}_{ijk}= & {} \frac{e_{ijk}}{e_0},\quad \widetilde{\kappa }^\varepsilon _{ij} = \frac{\kappa ^\varepsilon _{ij}}{\kappa _{0}},\quad \widetilde{c}_{ijkl} = \frac{c_{ijkl}}{c_0} \end{aligned}$$

(32)

$$\begin{aligned} \widetilde{g}_{k}^q= & {} \frac{g_{k}^q}{g_0},\quad \widetilde{y}_{j} = \frac{y_{j}}{L_0},\quad \widetilde{\pi }^q = \frac{\pi ^q}{L_0} \end{aligned}$$

(33)

By substituting these definitions in the first equation of (19) and dividing by $e_0$, we obtain:

$$\begin{aligned} \frac{\partial }{\partial \widetilde{y}_j}\left( \widetilde{e}_{qij} + \frac{c_0g_0}{L_0 e_0}\widetilde{c}_{ijkl}\frac{\partial \widetilde{g}_k^{q}}{\partial \widetilde{y}_l}+ \widetilde{e}_{lij}\frac{\partial \widetilde{\pi }^{q}}{\partial \widetilde{y}_l}\right) = 0 \end{aligned}$$

(34)

Substituting in the second equation and dividing by $\kappa _0$:

$$\begin{aligned} \frac{\partial }{\partial \widetilde{y}_j}\left( \widetilde{\kappa }^\varepsilon _{jq} - \frac{e_0g_0}{\kappa _0 L_0}\widetilde{e}_{jkl}\frac{\partial g_k^{q}}{\partial \widetilde{y}_l} + \widetilde{\kappa }^\varepsilon _{jl}\frac{\partial \pi ^{q}}{\partial \widetilde{y}_l}\right) = 0 \end{aligned}$$

(35)

In order to obtain the first equation with only adimensionalized coefficients, we set:

$$\begin{aligned} \frac{c_0g_0}{L_0 e_0} = 1 \implies g_0 = \frac{L_0 e_0}{c_0}. \end{aligned}$$

(36)

By introducing the expression of $g_0$ found above and forcing the second equation to have only adimensionalized coefficients, we obtain the following relationship between the rescaling constants:

$$\begin{aligned} \frac{e_0g_0}{\kappa _0 L_0} = \frac{e_0^2 L_0}{\kappa _0 c_0 L_0} = 1 \implies c_0 = \frac{e_0^2}{\kappa _0}, \end{aligned}$$

(37)

and a local problem that is symbolically equivalent to the original formulation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial }{\partial \widetilde{y}_j}\left( \widetilde{e}_{qij} + \widetilde{c}_{ijkl}\frac{\partial \widetilde{g}_k^{q}}{\partial \widetilde{y}_l}+\widetilde{e}_{lij}\frac{\partial \widetilde{\pi }^{q}}{\partial \widetilde{y}_l}\right) = 0,\quad i = 1,2,3\\ \frac{\partial }{\partial \widetilde{y}_j}\left( \widetilde{\kappa }^\varepsilon _{jq} - \widetilde{e}_{jkl}\frac{\partial \widetilde{g}_k^{q}}{\partial \widetilde{y}_l} + \widetilde{\kappa }^\varepsilon _{jl}\frac{\partial \widetilde{\pi }^{q}}{\partial \widetilde{y}_l}\right) = 0\\ \left( \widetilde{e}_{qij} + \widetilde{c}_{ijkl}\frac{\partial \widetilde{g}_k^{q}}{\partial \widetilde{y}_l}+\widetilde{e}_{lij}\frac{\partial \widetilde{\pi }^{q}}{\partial \widetilde{y}_l}\right) n_j = 0,\quad i = 1,2,3\\ \left( \widetilde{\kappa }^\varepsilon _{jq} - \widetilde{e}_{jkl}\frac{\partial \widetilde{g}_k^{q}}{\partial \widetilde{y}_l} + \widetilde{\kappa }^\varepsilon _{jl}\frac{\partial \widetilde{\pi }^{q}}{\partial \widetilde{y}_l}\right) n_j = 0. \end{array}\right. } \end{aligned}$$

(38)

The relationship (37) binds any of the three rescaling constants to the values of the other two. This gives us the freedom to select their values in a way that the resulting adimensionalized problem (38) does not have such a large difference in the orders of magnitude of the constants. The effective coefficients in terms of the adimensionalized constants have the following forms:

$$\begin{aligned} {\left\{ \begin{array}{ll} \widehat{e}_{qij} = e_0\left\langle \widetilde{e}_{qij} + \widetilde{c}_{ijkl}\frac{\partial \widetilde{g}_k^{q}}{\partial \widetilde{y}_l}+ \widetilde{e}_{lij}\frac{\partial \widetilde{\pi }^{q}}{\partial \widetilde{y}_l}\right\rangle ,\quad i = 1,2,3\\ \widehat{\kappa }_{jq} = \kappa _0\left\langle \widetilde{\kappa }^\varepsilon _{jq} - \widetilde{e}_{jkl}\frac{\partial \widetilde{g}_k^{q}}{\partial \widetilde{y}_l} + \widetilde{\kappa }^\varepsilon _{jl}\frac{\partial \widetilde{\pi }^{q}}{\partial \widetilde{y}_l}\right\rangle . \end{array}\right. } \end{aligned}$$

(39)