Finite element simulation of the effect of electric boundary conditions on the spherical indentation of transversely isotropic piezoelectric films

For the contact problem of a transversely isotropic piezoelectric film ( − t < z < 0, t is the film thickness) with a rigid, spherical indenter, the mechanical boundary conditions are

$$\begin{eqnarray} &&{u}_{z}(r,0)=h-f(r)\tqs \mathrm{for}~0\leq r\lt a \end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray} &&{\sigma }_{z z}(r,0)=0\tqs \mathrm{for}~r\gt a \end{eqnarray} \tag{ 10 }$$

where h is the indentation depth of the indenter, a is the contact radius and f(r) is the surface profile of the indenter. For an indentation by a rigid spherical indenter of radius R with a ≪ R

$$\begin{eqnarray} &&f(r)\approx \frac{{r}^{2}}{2 R}. \end{eqnarray} \tag{ 11 }$$

For frictionless contact between the indenter and the film, the shear stress is zero, i.e.

$$\begin{eqnarray} &&{\sigma }_{r z}(r,0)=0. \end{eqnarray} \tag{ 12 }$$

The non-slip contact between the film and the rigid substrate gives

$$\begin{eqnarray} &&{u}_{z}(r,-t)={u}_{r}(r,-t)=0. \end{eqnarray} \tag{ 13 }$$

The condition determining the contact radius a is

$$\begin{eqnarray} &&{\sigma }_{z z}(a,0)=0. \end{eqnarray} \tag{ 14 }$$

Two types of indenter are used in the finite element analyses. They are as follows.

Case I. Insulating indenter.

The electric boundary condition on the surface of the piezoelectric material for indentation by an insulating indenter is

$$\begin{eqnarray} &&{D}_{z}(r,0)=0\tqs \mathrm{for}~r\geq 0. \end{eqnarray} \tag{ 15 }$$

Case II. Conducting indenter.

For a conducting indenter without a prescribed electric potential the electric potential within the contact region is a constant. The electric boundary condition on the surface of the piezoelectric material becomes

$$\begin{eqnarray} &&{D}_{z}(r,0)=0\tqs \mathrm{for}~r\gt a \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} &&\frac{\partial \phi (r,0)}{\partial r}=0\tqs \mathrm{for}~r\lt a. \end{eqnarray} \tag{ 17 }$$

For a grounded conducting indenter one has

$$\begin{eqnarray} &&\phi (r,0)=0\tqs \mathrm{for}~r\lt a \end{eqnarray} \tag{ 18 }$$

which has technological advantages for the application of the sharp-instrumentation indentation in characterizing the properties of piezoelectric materials.

The indentation-induced electric charge, Q_p, within the contact area on the top surface of the piezoelectric film can be calculated by integrating the normal component of the electric displacement, D_z(r,0), within the contact area as

$$\begin{eqnarray} &&{Q}_{\mathrm{p}}=2 \pi \int \nolimits _{0}^{a}r{D}_{z}(r,0)\hspace{0.167em} \mathrm{d}r \end{eqnarray} \tag{ 19 }$$

which gives the total charge accumulated in the conducting indenter as

$$\begin{eqnarray} &&{Q}_{\mathrm{i}}=-2 \pi \int \nolimits _{0}^{a}r{D}_{z}(r,0)\hspace{0.167em} \mathrm{d}r. \end{eqnarray} \tag{ 20 }$$

For an insulating substrate

$$\begin{eqnarray} &&{D}_{z}(r,-t)=0 \end{eqnarray} \tag{ 21 }$$

and for a grounded substrate

$$\begin{eqnarray} &&\phi (r,-t)=0. \end{eqnarray} \tag{ 22 }$$

There are six possible combinations of the electric boundary conditions (II, IG, CI, CG, GI and GG). II corresponds to the insulating indenter and the insulating substrate, IG to the insulating indenter and the grounded substrate, CI to the conducting indenter without prescribed electric potential and the insulating substrate, CG to the conducting indenter without prescribed electric potential and the grounded substrate, GI to the grounded indenter and the insulating substrate and GG to the grounded indenter and the grounded substrate.

It needs to be emphasized that the simulation focuses on the contact radius being much larger than the film thickness and there is no transfer of free charges between the indenter and the piezoelectric film.

It is known that the distribution of electric field depends on the geometrical morphology of a material. Puglisi and Zurlo [41] analyzed the effect of surface curvature on electric field localization in thin dielectric films and obtained an approximate expression for the electric field. Yang and Song [42, 43] analyzed the field-induced surface instability of a conducting material with a sinusoidal surface topology by analyzing the effect of the sinusoidal surface on the field and stress distribution. Generally, one needs to include the effect of local curvature on the electric field in the analysis. However, the analysis of the indentation deformation is based on the linear theory of piezoelectricity, which requires that the boundary conditions including the electric boundary conditions be referenced to the undeformed state. In numerical calculations, such as finite element simulation, the electric boundary conditions are actually satisfied at each increment.

The contact radius is a function of the indentation depth, and the closed-form solutions of the contact radius for the indentation of thin films with t ≪ a ≪ R have been obtained only for two cases. Wang et al [36] obtained the dependence of the contact radius on the indentation depth for the indentation of a transversely isotropic piezoelectric film by an insulating indenter with a grounded, rigid substrate as

$$\begin{eqnarray} &&a=\sqrt{2 R h} \end{eqnarray} \tag{ 23 }$$

and for the indentation of a piezoelectric film by a conducting indenter of potential ϕ₀ with a grounded, rigid substrate as

$$\begin{eqnarray} &&a=\sqrt{2 R\left (h+\frac{{e}_{3 3}}{{C}_{3 3}}{\phi }_{0}\right )}. \end{eqnarray} \tag{ 24 }$$

Equation (24) reduces to equation (23) for a grounded, conducting indenter, i.e. ϕ₀ = 0.

Figure 3 shows the variation of the contact radius with the indentation depth for all six different combinations of electrical boundary conditions. The overlap of the curves indicates that the variation of the contact radius with the indentation depth for the spherical indentation of a piezoelectric film is independent of the electric boundary conditions given in section 3. For h/t ≥ 0.01, the relationship between the contact radius and the indentation depth can be described by equation (23), while for h/t < 0.01, equation (23) overestimates the contact radius due to the constraint of a ≫ t. At h/t = 0.01, a = 4.78 × 10⁻³ R = 4.78t. This result suggests that equation (23) can be used to determine the contact radius from the indentation depth when a ≥ 4.78t.

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For the spherical indentation of a piezoelectric half-space the relationship between the contact radius and the indentation depth is [30]

$$\begin{eqnarray} &&a=\sqrt{R h} \end{eqnarray} \tag{ 25 }$$

which is independent of the type of indenter (insulating or grounded, conducting indenter). For comparison, equation (25) is also included in figure 3. Obviously, equation (25) describes the indentation depth–contact radius relationship for h ≪ t and a ≪ t.

The indentation load–depth relationship has been used to determine the reduced contact modulus for the instrumentation indentation technique. According to the results given by Wang et al [36], the relationship between the indentation load and the indentation depth is

$$\begin{eqnarray} &&F=\left ({C}_{3 3}+\frac{{e}_{3 3}^{2}}{{\in }_{3 3}}\right )\frac{\pi R{h}^{2}}{t} \end{eqnarray} \tag{ 26 }$$

for the indentation of a transversely isotropic piezoelectric film by an insulating indenter with a grounded substrate, i.e. the IG case, and

$$\begin{eqnarray} &&F={C}_{3 3}\frac{\pi R{h}^{2}}{t} \end{eqnarray} \tag{ 27 }$$

for the indentation by a grounded, conducting indenter with a grounded substrate, i.e. the GG case. Independently of the electric boundary conditions, the indentation load is proportional to the indenter radius and the square of the indentation depth and inversely proportional to the film thickness, which is similar to the results given by Yang [32, 33] for the spherical indentation of compressible, elastic films with the contact radius much larger than the film thickness. The effect of the electric boundary conditions is represented by the coefficients.

Figure 4 shows the indentation load–depth curves for all six different combinations of electrical boundary conditions. For comparison, the results calculated from equations (26) and (27) are also included. The finite element results are in good agreement with the analytical results for the corresponding boundary conditions. The electric boundary conditions have some effect on the dependence of the indentation load on the indentation depth, since the curves slightly differ from one another. The indentation load in general increases with the square of the indentation depth for all six cases under the simulation conditions. To produce the same indentation depth, the smallest indentation load is needed for the indentation by a grounded, conducting indenter with a grounded substrate, while the largest indentation load is needed for the indentation by an insulating indenter with an insulating substrate. Comparison of the dependence of the indentation load on the indentation depth for all six cases reveals that the force responses can be divided into two categories; one is associated with the grounded, conducting indenter and the other is related to the indentation by an insulating indenter or a conducting indenter without a prescribed potential.

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It is worth mentioning that the finite element analysis of the spherical indentation of a transversely isotropic piezoelectric half-space by a conducting or an insulating indenter was performed by using the ABAQUS finite element code [40]. The simulation results are in good accord with the results given by the analytical relationships between the indentation load and the indentation depth as summarized by Wang et al [36] and given by Ding et al [24], which have validated the finite element code used in this work.

From equations (26) and (27) and the simulation results shown in figure 4, one can express the relationship between the indentation load and the indentation depth for all six cases as

$$\begin{eqnarray} &&F=\chi \frac{\pi R{h}^{2}}{t} \end{eqnarray} \tag{ 28 }$$

where χ is a function of the material properties of the piezoelectric material and the electric boundary conditions. Equation (28) gives the contact stiffness, S, as

$$\begin{eqnarray} &&S=\frac{\mathrm{d}F}{\mathrm{d}h}=2 \chi \frac{\pi R h}{t}=\frac{\chi A}{t} \end{eqnarray} \tag{ 29 }$$

for the spherical indentation of transversely isotropic piezoelectric films with the contact radius much larger than the film thickness. Here, $\chi ={C}_{3 3}+{e}_{3 3}^{2}/{\in }_{3 3}$ for the indentation by an insulating indenter with a grounded substrate and χ = C₃₃ for the indentation by a grounded, conducting indenter with a grounded substrate. The contact stiffness is proportional to the contact area A and inversely proportional to the film thickness in contrast to the result for the spherical indentation of semi-infinite piezoelectric materials. Equation (29) provides an approach to measure the electromechanical properties of piezoelectric films by controlling the electric boundary conditions.

For the indentation of a transversely isotropic piezoelectric film with the contact size much larger than the film thickness, the distribution of normal stress over the contact area (0 ≤ r ≤ a) is [36]

$$\begin{eqnarray} \fl {\sigma }_{z z}(r,0)&=&-\left ({c}_{3 3}+\frac{{e}_{3 3}^{2}}{{\in }_{3 3}}\right )\frac{{a}^{2}-{r}^{2}}{2 R t}\nonumber\\ &&=~-\left ({c}_{3 3}+\frac{{e}_{3 3}^{2}}{{\in }_{3 3}}\right )\frac{2 R h-{r}^{2}}{2 R t} \end{eqnarray} \tag{ 30 }$$

for the spherical indentation by an insulating indenter with a grounded substrate (i.e. the case of IG) and

$$\begin{eqnarray} &&{\sigma }_{z z}(r,0)=-\frac{{c}_{3 3}({a}^{2}-{r}^{2})}{2 R t}=-\frac{{c}_{3 3}(2 R h-{r}^{2})}{2 R t} \end{eqnarray} \tag{ 31 }$$

for the spherical indentation by a grounded indenter with a grounded substrate (i.e. the case of GG). Both the stress distributions in the contact zone are quadratic functions of the radial variable r. Figure 5 shows the finite element results for the distribution of the normal stress in the contact zone for all six cases with a maximum indentation depth of 50 nm. For comparison, the analytical results of equations (30) and (31) are also included in figure 5. For the spherical indentation by an insulating indenter or a grounded indenter with a grounded substrate, the magnitudes of the normal stress obtained from the FEM simulation are in good accord with the corresponding analytical results for r/t ≥ 7 and slightly larger than the corresponding analytical results for r/t < 7. Such behavior is due to the condition used in deriving equations (30) and (31), which requires a/t ≫ 1, even though it has no significant effect on the load–displacement relationship. As shown in figure 5, the piezoelectric material experiences compressive stress, as expected, with the maximum compressive stress being at the contact center. For the same indentation depth and a spherical indenter of the same size the indentation by an insulating indenter with an insulating, rigid substrate produces the largest compressive stress in the contact zone, while the indentation by a grounded indenter with a grounded, rigid substrate produces the smallest compressive stress. Due to the singularity of the electric field, the indentation by the conducting indenter creates a stress singularity at the contact edge, which is likely to lead to structural damage.

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The dependence of the indentation-induced electric potential at the contact center on the indentation depth is depicted in figure 6. The magnitude of the electric potential at the contact center increases with increasing indentation depth, which is in accordance with the mechanism of piezoelectric behavior. For the same indentation depth the indentation by the insulating indenter with a grounded, rigid substrate induces the largest electric potential at the contact center, while the indentation of the piezoelectric film by the conducting indenter with an insulating, rigid substrate induces the smallest electric potential. These results suggest that the electric boundary conditions determine the electric response of the piezoelectric film during indentation. For h/t ≥ 0.02 the indentation-induced electric potential at the contact center is a linear function of the ratio h/t for any contact radius larger than the film thickness, i.e.

$$\begin{eqnarray} &&{\phi }_{\mathrm{c}}=\alpha +\kappa \frac{h}{t} \end{eqnarray} \tag{ 32 }$$

where ϕ_c is the electric potential at the contact center and α and κ are constants depending on the material properties and the electric boundary conditions. Through the best curve-fitting, one can calculate the α and κ constants from the FEM results. For the indentation by a conducting indenter without a prescribed potential the indentation-induced potential on the indenter then can be calculated from equation (32) if the constants α and κ have been determined from the numerical calculation or from the indentation test of piezoelectric films by characterizing the variation of the contact potential with the indentation depth.

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According to the definition of piezoelectric charge coefficients, d_ij [44],

$$\begin{eqnarray} &&{d}_{i j}=\frac{\partial {\varepsilon }_{j}}{\partial {E}_{\mathrm{i}}} \end{eqnarray} \tag{ 33 }$$

in which ε_j is the matrix representation of ε_ij with ε₁ = ε₁₁, ε₂ = ε₂₂, ε₃ = ε₃₃, ε₄ = 2ε₂₃, ε₅ = 2ε₁₃ and ε₆ = 2ε₁₂. For the indentation of piezoelectric films by the conducting indenter with the grounded substrate and the contact radius much larger than the film thickness the compressive strain underneath the indentation can be approximated as

$$\begin{eqnarray} &&{\varepsilon }_{3}\approx -\frac{h}{t-h} \end{eqnarray} \tag{ 34 }$$

and the magnitude of the electric field intensity as

$$\begin{eqnarray} &&{E}_{3}\approx \frac{{\phi }_{\mathrm{c}}}{t-h}=\frac{\alpha }{t-h}+\frac{\kappa }{t}\frac{h}{t-h}=\frac{\alpha }{t-h}-\frac{\kappa }{t}{\varepsilon }_{3}. \end{eqnarray} \tag{ 35 }$$

Substituting equations (34) and (35) into (33) and using the condition of h ≪ t, the piezoelectric charge coefficient, d₃₃, can be determined by

$$\begin{eqnarray} &&{d}_{3 3}\approx -\frac{t}{\kappa }\approx -\left (\frac{\mathrm{d}{\phi }_{\mathrm{c}}}{\mathrm{d}h}\right )^{-1}. \end{eqnarray} \tag{ 36 }$$

The nominal piezoelectric charge coefficients as determined from the indentation of a piezoelectric film by a conducting indenter with a grounded, rigid substrate and the contact radius much larger than the film thickness are inversely proportional to the derivative of the electric potential with respect to the indentation depth.

Due to the electromechanical coupling in piezoelectric materials, the indentation deformation by the conducting indenter will polarize the piezoelectric material and lead to accumulation of electric charges on the conducting indenter. The total charge accumulated on the surface of the conducting indenter can be calculated from equation (20). Figure 7 shows the variation of the total charge with the indentation depth for the indentation by the conducting indenter. The total electric charge increases with increasing indentation depth due to the piezoelectric coupling. For the same indentation depth the largest total charge accumulated on the conducting indenter is induced when the indenter is grounded, while the smallest total charge is induced for the indentation with an insulating substrate. The electric boundary conditions play an important role in controlling the accumulation of electric charges on the conducting indenter.

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For the indentation of the piezoelectric film with a grounded substrate the simulation results shown in figure 7 suggest that, for h/t ≥ 10⁻³, the total charge accumulated on the conducting indenter is a power function of the ratio of h/t. Using the best curve-fitting one obtains

$$\begin{eqnarray} &&{Q}_{\mathrm{i}}=\beta \left (\frac{h}{t}\right )^{n} \end{eqnarray} \tag{ 37 }$$

with n = 2 for the indentation by a grounded indenter with a grounded substrate and n = 1.5 for the indentation by a conducting indenter without a prescribed potential and with a grounded substrate. Here β is a constant depending on the material properties and the electric boundary conditions.

Using the result given by Wang et al [36] for the grounded indenter and the grounded substrate with the poling direction being anti-parallel to the loading direction, the total charge is found to be

$$\begin{eqnarray} &&Q=\frac{{e}_{3 3}\pi R}{t}{h}^{2}. \end{eqnarray} \tag{ 38 }$$

For comparison, the calculated results from equation (38) are also included in figure 6. Obviously, the simulation results are in agreement with those calculated from equation (38) for h/t ≥ 10⁻². For the spherical indentation of piezoelectric films by a grounded, conducting indenter with a grounded substrate and the contact radius much larger than the film thickness equation (38) can be used to determine the piezoelectric constant e₃₃.

For the indentation by a grounded indenter with a grounded, rigid substrate and a ≫ t one can use the result given by Wang et al [36] to obtain the distribution of the normal component of the electric displacement in the contact zone as

$$\begin{eqnarray} &&{D}_{z}(r,0)=\mp \frac{{a}^{2}-{r}^{2}{e}_{3 3}}{2 R t}\tqs \mathrm{for}~0\leq r\leq a \end{eqnarray} \tag{ 39 }$$

in which the sign of  ±  depends on the relative direction between the loading direction and the poling direction (axisymmetric axis). For the current configuration with the loading direction anti-parallel to the poling direction the ' − ' is used in equation (39). The normal component of the electric displacement is a quadratic function of the radial variable r. Figure 8 depicts the distribution of the normal component of the electric displacement for all the six different boundary conditions with the maximum indentation depth of 50 nm. For comparison, the results obtained from equation (39) are also included in figure 8; these are slightly lower than the FEM results near the contact center, while there is good agreement in the region away from the contact center.

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For the indentation by an insulating, rigid indenter the normal component of the electric displacement is zero, as expected. There exists a transition of the normal component of the electric displacement from a negative value to a positive value for the other four types of electric boundary condition, i.e. CI, CG, GI and GG. Field singularity is observable for the indentation by a conducting indenter due to non-zero potential on the indenter, which leads to the stress singularity at the contact edge as shown in figure 5. It is worth mentioning that the field singularity has not been reported for the indentation of piezoelectric films, even though the same behavior has been observed from the finite element simulation of semi-infinite piezoelectric materials by Liu and Yang [40].

Figure 9 shows the distribution of electric potential on the contact surface between the indenter and the piezoelectric film for all the six electric boundary conditions with a maximum indentation depth of 50 nm. The electric potential for the grounded indenter is zero in the contact zone, as expected. The distribution of electric potential depends on the electrical boundary conditions. The indentation by the conducting indenter produces uniform, non-zero electric potential on the contact surface. The magnitude of the indentation-induced potential is a function of the electric condition of the rigid substrate with the grounded substrate generating a smaller magnitude of electric potential than that of the insulating substrate at the contact center. Non-uniform electric potential is created for the indentation by the insulating indenter, which is relatively independent of the electric condition of the substrate.

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The above analyses have established the rationale of using the sharp-instrumented indentation to characterize the complicated electromechanical interaction of piezoelectric films. The load–displacement relationship and the load–potential relationship can be used to quantify the piezoelectric behavior of piezoelectric films similarly to the work by Kollosche and Kofod [45] by using the contact technique to examine the dependence of the breakdown strength of soft elastomers on Young's modulus. Careful comparison between the FEM results and experimental results will be explored in the future.