# Note on modelling directionality in piezoresistivity

Balakrishnan, Sreenath and Deshpande, Kaustubh and Ananthasuresh, GK (2012) Note on modelling directionality in piezoresistivity. In: National Conference on Smart Materials Structures and Systems, September 21-22, 2012, Coimbatore, India.

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## Abstract

When computing the change in electrical resistivity of a piezoresistive cubic material embedded in a deforming structure, the piezoresistive and the stress tensors should be in the same coordinate system. While the stress tensor is usually calculated in a coordinate system aligned with the principal axes of a regular structure, the specified piezoresistive coefficients may not be in that coordinate system. For instance, piezoresistive coefficients are usually given in an orthogonal cartesian coordinate system aligned with the <100> crystallographic directions and designers sometimes deliberately orient a crystallographic direction other than <100> along the principal directions of the structure to increase the gauge factor. In such structures, it is advantageous to calculate the piezoresistivity tensor in the coordinate system along which the stress tensors are known rather than the other way around. This is because the transformation of stress will have to be done at every point in the structure but piezoresistivity tensor needs to be transformed only once. Here, using tensor transformation relations, we show how to calculate the piezoresistive tensor along any arbitrary Cartesian coordinate system from the piezoresistive coefficients for the <100> coordinate system. Some of the software packages that simulate the piezoresistive effect do not have interfaces for calculation of the entire piezoresistive tensor for arbitrary directions. This warrants additional work for the user because not considering the complete piezoresisitive tensor can lead to large errors. This is illustrated with an example where the error is as high as 33%. Additionally, for elastic analysis, we used hybrid finite element formulation that estimates stresses more accurately than displacement-based formulation. Therefore, as shown in an example where the change in resistance can be calculated analytically, the percentage error of our piezoresistive program is an order of magnitude lower relative to displacement-based finite element method.

Item Type: Conference Paper Copyright of this article belongs to Institute for Smart Structures and Systems. Piezoresistivity; Coordinate Transformation; Fourth Order Tensor; Hybrid Finite Elements Division of Mechanical Sciences > Mechanical Engineering 19 Mar 2014 04:55 19 Mar 2014 04:55 http://eprints.iisc.ernet.in/id/eprint/48661