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Abstract Materials that are self-sensing via the piezoresistive effect have been widely explored for embedded sensing in aerospace structural composites, pressure sensing in touch pads, diagnostics in biomedical implant technology, and many other applications. In this approach, changes in electrical transport of the material are used as an indicator of stress/strain, pressure, or damage. However, engineers and other users of self-sensing materials are typically not directly interested in the electrical state of the material. Rather, they want to know the underlying mechanical state of the material that gives rise to an observed electrical change. Recovering material condition from electrical observations is referred to as the self-sensing inverse problem (SSIP). Prior work by the author has shown that the SSIP can be solved using an electrical impedance tomography (EIT)-generated conductivity map as an input, but this is undesirable because it requires solving a second inverse problem (i.e., the SSIP) on top of the EIT inverse problem. To that end, a direct formulation for the SSIP is herein presented. In this approach, voltage-current data is directly inverted to find the displacement field without using EIT as an intermediate step. Additionally, the direct SSIP is solved within the primal-dual interior point (PDIPM) framework such that an ℓ1-norm can be used on the regularization term, which promotes sparsity in the solution space. This approach is applied to a representative piezoresistive nanocomposite using the Laplace matrix as regularization to promote a spatially smooth solution. From the displacement field, strains and stresses are calculated and compared to a commercial finite element solution with good accuracy. 

Achieving material state awareness (MSA) of a mechanical system, structure, or component empowers engineers to be able to perform accurate failure prognosis and estimate remaining operational lifetime. Piezoresistive materials exhibit a change in electrical conductivity (or resistivity) when subjected to mechanical strain and therefore have the potential to enable functional, self-sensing structures that can provide real-time insight into their mechanical state. While piezoresistive materials have been demonstrated to be able to detect and localize damage by imaging conductivity changes, conductivity changes alone do not directly translate to stresses, strains, or displacements, values that are more meaningful for stress and failure analyses. To this end, the self-sensing inverse problem (SSIP) is a method which reconstructs the strain state that gives rise to an observed conductivity distribution. The SSIP has to date been computationally and experimentally demonstrated to reconstruct the deformed displacement field of simple geometries under load with reasonable accuracy, but mathematical limitations can make obtaining an accurate reconstruction difficult. In this work, we seek to improve the ability of the SSIP to obtain accurate displacement field reconstructions on more complex geometries by incorporating sensor data fusion (SDF) techniques. We herein develop a computational inverse problem for two SDF methods, one fusing electrical resistivity with displacement data and another fusing resistivity with strain data, and demonstrate these methods computationally on shapes representing aircraft components. These results show that incorporating data from relatively few displacement sensors allows the SSIP to reconstruct the displacement field with high accuracy. The results also demonstrate that incorporating strain sensor data improves the accuracy of the displacement reconstruction, though to a lesser degree than displacement data. 

Self-sensing materials with deformation-dependent electrical conductivity have been studied in diverse applications. A limitation of self-sensing materials is that they do not directly report on the underlying condition of the material. That is, it would be much more useful to know the actual stresses, strains or damages that give rise to observed conductivity changes. The process of deducing material condition from electrical data is called theself-sensing inverse problem(SSIP). Previous work has required electrical impedance tomography (EIT) as an intermediate step to estimate the conductivity distribution prior to estimating the spatially varying mechanics. But this is undesirable because EIT is an ill-posed inverse problem and is highly dependent on regularization, which renders the SSIP subject to the same assumptions and limitations of EIT. The contribution of this manuscript is the development of adirectSSIP formulation that omits the EIT step such that mechanics are predicted directly from voltage–current data. The effects of regularization type, regularization norm and formulation (difference versus absolute imaging) are also explored. The direct SSIP formulation is experimentally validated on a soft carbon nanofibre-modified polyurethane pressure sensor and compared to ANSYS-solved mechanics with good agreement.