Homogenization and multiscale stability analysis in finite magneto-electro-elasticity. Application to soft matter EE, ME and MEE composites

Soft matter electro-elastic (EE), magneto-elastic (ME) and magneto-electro-elastic (MEE) composites exhibit coupled material behavior at large strains. Examples are electro-active polymers and magneto-rheological elastomers, which respond by a deformation to applied electric or magnetic fields, and are used in advanced industrial environments as sensors and actuators. Polymer-based magneto-electric-elastic composites are a new class of tailor-made materials with promising future applications. Here, a magneto-electric coupling effect is achieved as a homogenized macro-response of the composite with electro-active and magneto-active constituents. These soft composite materials show different types of instability phenomena, which even might be exploited for future enhancement of their performance. This covers micro-structural instabilities, such as buckling of micro-fibers or particles, as well as material instabilities in the form of limit-points in the local constitutive response. Here, the homogenization-based scale bridging links long wavelength micro-structural instabilities to material instabilities at the macro-scale. This work outlines a comprehensive framework of an energy-based computational homogenization in electro-magneto-mechanics, which allows a tracking of postcritical solution paths such as those related to pull-in instabilities. It provides variational-based definitions of multiscale structural and material stability phenomena. The starting point is a minimization principle of averaged electro-magneto-elastic energy, that is discretized by using electric and magnetic vector potentials. Next, computationally more effective representations based on scalar potentials are considered by a reformulation of the energy in terms of an averaged enthalpy functional. Structural stability is analyzed based on perturbations of the averaged energy, while local material stability is defined by a generalized quasi-convexity condition. It is shown that the incremental arrays which govern the stability criteria appear within the convenient enthalpy-based representation in a distinct diagonal form, containing mechanical and electro-magnetic partitions. Representative simulations demonstrate a tracking of inhomogeneous EE, ME and MEE composites, showing the development of postcritical zones in the microstructure and a possible unstable homogenized material response.