# Department of Mechanics: Seminar: Abstract Gil 2019

### A new framework for large strain electromechanics based on Convex Multi-Variable strain energies

#### Antonio J. Gil, Professor, Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea University, SA1 8EN, UK

**Room B-366, Faculty of Civil Engineering, CTU in Prague**

**Wednesday, 5 June 2019, 10:30-11:30**

Dielectric Elastomers (DE) are a class of Electro-Active Polymers with outstanding actuation properties. Voltage induced area expansions of 1980% on a DE membrane have been recently reported. In this case, the electromechanical instability is harnessed as a means for obtaining these electrically induced massive deformations with potential applications in soft robots. Computational simulation in this context becomes extremely challenging and must be addressed *ab initio* by the definition of well-posed constitutive models.

In this presentation, we postulate a new Convex Multi-Variable (CMV) variational framework for the analysis of these materials exhibiting massive deformations [2, 3, 4]. This extends the concept of polyconvexity [1] to strain energies which depend on non-strain based variables introducing other physical measures such as the electric displacement. A new definition of the electro-mechanical internal energy is introduced, being expressed as a Convex Multi-Variable (CMV) function of a new extended set of electromechanical arguments. Crucially, this new definition of the internal energy enables the most accepted constitutive inequality, namely ellipticity, to be extended to the entire range of deformations and electric fields and, in addition, to incorporate the electromechanical energy of the vacuum, and hence that for ideal dielectric elastomers, as a degenerate case. Spurious numerical instabilities can then effectively be removed from the model whilst maintaining real physical instabilities. Hyperbolicity, variational principles, and Finite Element functional spaces will be shown prior to demonstrating the potential of the new paradigm through extremely challenging numerical examples involving wrinkling and the onset of instabilities [5, 6].

**References**

[1] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Archive for Rational Mechanics and Analysis 63 (1976), 337–403.

[2] A.J. Gil, R. Ortigosa, A new framework for large strain electromechanics based on convex multi-variable strain energies: Variational formulation and material characterisation, CMAME 302 (2016), 293–328.

[3] R. Ortigosa, A.J. Gil, A new framework for large strain electromechanics based on convex multi-variable strain energies: Finite Element discretisation and computational implementation, CMAME 302 (2016), 329–360.

[4] R. Ortigosa, A.J. Gil, A new framework for large strain electromechanics based on convex multi-variable strain energies: Conservation laws, hyperbolicity and extension to electro-magneto-mechanics, CMAME 309 (2016), 202–242.

[5] R. Poya, A.J. Gil, R. Ortigosa, A high performance data parallel tensor contraction framework: Application to coupled electro-mechanics, CPC 216 (2017), 35–52.

[6] R. Poya, A.J. Gil, R. Ortigosa, R. Sevilla, J. Bonet, W. Wall, A curvilinear high order finite element framework for electromechanics: from linearised electro-elasticity to massively deformable dielectric elastomers, CMAME 329 (2018), 75–117.