Homogenization, macroscopic instabilities and domain formation in magnetoactive composites: Theory and applications

Abstract

This work deals with the stability and macroscopic response of soft magnetoelastic composites under combined magnetic and mechanical loading. It is known that, as in the purely mechanical setting, soft composites with suitably designed microstructures can undergo macroscopic (long wavelength) instabilities under certain special loading conditions. The mechanism for such instabilities, and their effects on the macroscopic response of the composites, are less well understood. We investigate here both the onset the instabilities, as well as the macroscopic response after the instabilities, by means of a ‘twinning’ or domain formation mechanism. For this purpose, we propose a generalized Maxwell construction, which builds on the work of Ball and James (1987) for shape-memory alloys and properly accounts for mechanical and magnetic compatibility requirements between ‘twins’ or domains. Thus, generalizing the work of Avazmohammadi and Ponte Castañeda (2016) for the purely mechanical response of reinforced elastomers, this is accomplished by means of the relaxation or quasiconvexification of the ‘principal’ homogenized solution, i.e., the solution prior to the onset of an instability. The methodology is then applied to two-phase magnetoelastic laminates under plane-strain loading conditions with a magnetic field applied in the plane of deformation. We find that the principal solution generally loses (strict) global rank-one convexity before it loses strong ellipticity, as the laminate finds stability by forming ‘twinned’ lamellar domains. In addition, we use the estimates to characterize the ‘phase diagrams’ for the laminates and to describe in detail their macroscopic response when the magnetic field is applied either parallel or perpendicular to the layers. We find that the formation of the lamellar domains leads to a softer mechanical response – including perfectly soft modes of deformation – which can be either facilitated or impeded, depending on the direction of the applied magnetic field. This mechanism opens up novel strategies for the design of magnetoactive elastomer sensors and actuators.