# Design Study on Customised Piezoelectric Elements for Energy Harvesting in Total Hip Replacements

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Finite Element Model

_{app}[25] was assumed with a maximum Young’s modulus of 20 GPa, please see [15] for more detail.

^{®}Xeon

^{®}Gold 6248 CPU processors (2.50 GHz) and 192 GB RAM per node. A sparse direct solver was used.

#### 2.2. Postprocessing

_{Imp}), where a local stress concentration is induced by integration of the EHS. This parameter was considered uncritical when not exceeding 290.9 MPa which was the global stress maximum for the unmodified geometry in the same loading situation, shown in [15]. For the piezoelectric element, the von Mises stress (σ

_{Piez}) was evaluated in the mid plane, since the contact situation leads to local stress singularities at the top and bottom end faces. The limit was defined as 30.0 MPa. This value was specified by the manufacturer as maximum preload for using the element with a constant force [32].

_{33}(t) by solving the differential equation for different load resistance R

_{layer}and the piezoelectric constants ${\mathsf{\epsilon}}_{33}^{\mathrm{T}}$ and d

_{33}. In the design study, the base area A of the element was variable, as well as the height h and the resulting number of layers n

_{layer}. The layer thickness was kept constant at h

_{layer}= 0.05 mm as for the off-the-shelf piezoelectric elements specified by the manufacturer. The used input parameters for the piezoelectric element are listed in Table 2. Since the off-the-shelf piezoelectric elements had passive top and bottom layers of around 0.175 mm thickness at a height of 2.5 mm for each single element (i.e., 2.15 mm of active height and 43 layers for a single element and 4.35 mm of active height and 86 layers for the stacked element, respectively) we assumed an according passive proportion for the customised geometries. If necessary, we reduced the active height to obtain an integer number of layers, thereby choosing for a conservative power approximation.

_{33}from the FEA [18]. The resulting load profile F

_{33}(t) approximated the force profile on the piezoelectric element, requiring only the simulation of a single load step (Figure 3a exemplary shows the scaled force profile for the final design). The generated power for different load resistance R was calculated by

#### 2.3. Geometry Variants

**Recalculation of previous data:**For the customised geometries the individual representation of the multilayer elements would exceed the computational capacity, as mentioned above. For comparison with our previous data and as reference basis for new simulations, we recalculated the off-the-shelf configurations (single and stacked multilayer piezoelectric element, see Figure 2) with the approach described above, neglecting the layers in the simulation, and considering them only for the power calculation.

**New geometries:**In addition to the off-the-shelf stacked piezoelectric element with a ring base area, we considered a full cylinder stacked multilayer element.

**Iterative design study**: On basis of the initial customised geometry, we conducted a design study to optimise the power output. Starting with the initial cavity design, we successively changed the piezoelectric element’s height to find the geometry providing the maximum power output, but ensuring compliance with the defined critical stress values (Table 3). The larger the piezoelectric element’s height, the shorter its extension towards the cavity needed to be defined to fit in the UHMW-PE housing (see Figure 2b, A to G). The best design point was then used as input for a new cavity geometry where we changed the cavity height or depth (see Figure 2a for the definition). Again, the piezoelectric element’s height was changed to comply with our specified stress values. After five new cavity geometries and over 20 simulated design points, we stopped the study. All design point geometries are presented and listed in Figure A1 and Table A1, Appendix A.

## 3. Results

_{33}was notably larger, and the stress in the piezoelectric element rose compared to the single element. The full cylinder also absorbed more load, however, the stress in the piezoelectric element was smaller and thereby less voltage and power were generated. All the three design variants fulfilled the stress criteria for the metallic implant component and for the piezoelectric element (see Table 5).

## 4. Discussion

_{33.}This has already been shown in our previous work and can be explained by the linear relationship between force and generated voltage which in contrast is contained as squared value in the calculation of the power (see Equation (2)) [15]. The difference for the stress in the piezoelectric element especially for the larger stacked configuration may result from the different height of the finite elements in the very area. The element height was notably smaller when the individual layers were modelled. With respect to the minor absolute deviation, the difference is acceptable and the approach still considered straightforward since the reduction of the model size is necessary to allow the simulation of larger piezoelectric elements of increased height or increased base area. To minimise the mesh density influence for a good comparability of the different design points in the iterative design study, the finite element height for the piezoelectric elements was kept constant for all design points.

_{33}can be withstood. This can be realised by increasing the height or base area to maximise the force transmission on the piezoelectric element. Therefore, the overall design aspect of the EHS with the mediating function of the UHWM-PE housing allows perfectly matching the piezoelectric elements geometry while maintaining the cavity geometry. This is demonstrated by the initial customised geometry; increasing the base area using the maximum available space for the same height largely raised the generated power but showed a tolerable stress level. The potential of a higher piezoelectric element, pushing the stress to the specified allowed limit, is not even exploited.

_{33}and the new configuration of the piezoelectric element, the generated voltage maximum was higher and the best matching resistance was slightly shifted to a lower value.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Variants of the geometry study for the customised geometries showing the different cavity designs from left to right (the changed cavity parameters are underlined), all design points are ordered according to the height of the piezoelectric elements (top down). The arrows indicate the proceeded progress. The evaluated stresses (MPa) and power (mW) are shown in bars based on the maximum reached values. Green bars indicate a value below the critical value, and red above. For quantitative date, see Table A1.

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**Figure 1.**(

**a**) Femoral bone segment with modified hip stem implanted; in the detail view (cross section) the components of the EHS are visible including a cylindrical piezoelectric element. (

**b**) Loading of an arbitrary cylindrical piezoelectric element with the force F

_{33}transmitted through the implant and the UHMW-PE housing. The piezoelectric element in this picture is formed by stacking two single multilayer elements with passive top and bottom layers.

**Figure 2.**(

**a**) Modified hip stem with EHS (UHWM-PE housing in transparent view and exemplary customised piezoelectric element) with detail view. Arrows indicate the cavity depth (blue) and height (black). (

**b**) Different piezoelectric element geometries; (

**A**) off-the-shelf single element, (

**B**) off-the-shelf stacked element, (

**C**) full cylinder stacked element, (

**D**) initial customised geometry, (

**E**) to (

**G**) further customised geometries of different heights and depth for first cavity geometry.

**Figure 3.**(

**a**) Force profile of F

_{33}(t) acting on the piezoelectric element for the final design, scaled on the basis of Bergmann et al. [18] and our approach described in [15] (red) and the according calculated generated voltage for the best matching resistance R = 0.08 MΩ (blue). (

**b**) Calculated power output according to Equation (2) for different load resistance R for the final design with a maximum of 729.9 µW at R = 0.08 MΩ.

Component | Young’s Modulus (GPa) | Poisson’s Ratio ^{1} (-) |
---|---|---|

Metallic implant | 195 [26] | 0.3 |

Bone cement mantle | 2.3 [27] | 0.3 |

UHMW-PE housing | 0.83 [28] | 0.46 [28] |

PZT piezoelectric element | 52.4 [29] | 0.35 [29] |

Femoral bone | 6850 × ρ_{app}^{1.49} [25],max. value 20 GPa [30,31] | 0.3 |

^{1}a Poisson’s ratio of 0.3 was assumed, if no other data was available.

Parameter | Value |
---|---|

A | Variable |

h | Variable, minus passive layers |

h_{layer} | 0.05 mm |

n_{layer} | Variable, defined by h and h_{layer} |

${\mathsf{\epsilon}}_{33}^{\mathrm{T}}/{\epsilon}_{0}$ | 1751 [29] |

d_{33} | 3.996E − 10 m/V [29] |

Symbol | Output Parameter | Limit |
---|---|---|

σ_{Imp} | Stress maximum at cavity base (metallic implant component) | 290.9 MPa |

σ_{Piez} | Stress maximum in the piezoelectric element’s mid plane | 30.0 MPa |

F_{33} | Contact force acting on the piezoelectric element’s end face | n/a |

P | Approximated power output | To be maximised |

**Table 4.**Comparison of output data when considering or neglecting the individual layers of the piezoelectric element during FEA.

Variant | σ_{Imp} (MPa) | σ_{Piez} (MPa) | F_{33} (N) | P (µW) | |
---|---|---|---|---|---|

Single | Layers | 269.4 | 14.0 | 141.6 | 8.1 |

No layers | 269.6 | 14.1 | 140.6 | 8.0 | |

Difference (%) | 0.08% | 0.64% | −0.67% | −1.33% | |

Stack | Layers | 267.2 | 16.9 | 196.3 | 31.1 |

No layers | 267.6 | 15.1 | 192.0 | 29.8 | |

Difference (%) | 0.14% | −10.71% | −2.20% | −4.36% |

**Table 5.**Output results of the different geometry variants and the initial and final geometry variants for the iterative design study.

Variant | |||||
---|---|---|---|---|---|

Single | Stacked | Stacked | Initial Final | ||

Off-The-Shelf | Full Cylinder | Customised Geometry | |||

σ_{Imp} (MPa) | 269.6 | 267.6 | 266.6 | 256.6 | 273.4 |

σ_{Piez} (MPa) | 14.1 | 15.1 | 11.9 | 16.9 | 29.7 |

F_{33} (N) | 140.6 | 192.0 | 207.7 | 536.0 | 1218.7 |

P (µW) | 8.0 | 29.8 | 26.1 | 77.8 | 729.9 |

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**MDPI and ACS Style**

Lange, H.-E.; Bader, R.; Kluess, D. Design Study on Customised Piezoelectric Elements for Energy Harvesting in Total Hip Replacements. *Energies* **2021**, *14*, 3480.
https://doi.org/10.3390/en14123480

**AMA Style**

Lange H-E, Bader R, Kluess D. Design Study on Customised Piezoelectric Elements for Energy Harvesting in Total Hip Replacements. *Energies*. 2021; 14(12):3480.
https://doi.org/10.3390/en14123480

**Chicago/Turabian Style**

Lange, Hans-E., Rainer Bader, and Daniel Kluess. 2021. "Design Study on Customised Piezoelectric Elements for Energy Harvesting in Total Hip Replacements" *Energies* 14, no. 12: 3480.
https://doi.org/10.3390/en14123480