Abstract
Coupled magneto-mechanical wrinkling has appeared in many scenarios of engineering and biology. Hence, soft magneto-active (SMA) plates buckle when subject to critical uniform magnetic field normal to their wide surface. Here, we provide a systematic analysis of the wrinkling of SMA plates subject to an in-plane mechanical load and a transverse magnetic field. We consider two loading modes: plane-strain loading and uni-axial loading, and two models of magneto-sensitive plates: the neo-Hookean ideal magneto-elastic model and the neo-Hookean magnetization saturation Langevin model. Our analysis relies on the theory of nonlinear magneto-elasticity and the associated linearized theory for superimposed perturbations. We derive the Stroh formulation of the governing equations of wrinkling, and combine it with the surface impedance method to obtain explicitly the bifurcation equations identifying the onset of symmetric and antisymmetric wrinkles. We also obtain analytical expressions of instability in the thin- and thick-plate limits. For thin plates, we make the link with classical Euler buckling solutions. We also perform an exhaustive numerical analysis to elucidate the effects of loading mode, load amplitude, and saturation magnetization on the nonlinear static response and bifurcation diagrams. We find that antisymmetric wrinkling modes always occur before symmetric modes. Increasing the pre-compression or heightening the magnetic field has a destabilizing effect for SMA plates, while the saturation magnetization enhances their stability. We show that the Euler buckling solutions are a good approximation to the exact bifurcation curves for thin plates.
Highlights
Soft magneto-active (SMA) materials, such as magneto-active elastomers, are a promising kind of smart materials that can respond to magnetic field excitation
Where T = F−1τ is the total nominal stress tensor, the nominal magnetic field vector BL = F−1B and the nominal magnetic induction vector HL = FTH are the Lagrangian counterparts of B and H, respectively, and p is a Lagrange multiplier related to the incompressibility constraint, which can be determined from the equilibrium equations and boundary conditions
The material properties used in the numerical computations are taken as G = 10 kPa, χ = 0.4 and μ0ms = 0.5 T, which are obtained from experiments with a class of magnetorheological elastomers (Psarra et al, 2017, 2019) consisting of a soft silicone mixed with iron particles at a volume fraction of 20%
Summary
Soft magneto-active (SMA) materials, such as magneto-active elastomers, are a promising kind of smart materials that can respond to magnetic field excitation. Pao and Yeh (1973) used a general theory of magneto-elasticity to re-examine this problem and to yield an identical antisymmetric buckling equation for thin plates Following those works, many investigations looked at the same problem, trying to improve mathematical models to explain the discrepancy between experimental results and theoretical predictions (Wallerstein and Peach, 1972; Popelar, 1972; Dalrymple et al, 1974; Miya et al, 1978; Gerbal et al, 2015; Singh and Onck, 2018). Some mathematical expressions or derivations are provided in Appendices A-C
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