Bending of hard-magnetic soft beams: A finite elasticity approach with anticlastic bending

Highlights

• An analytical solution for the bending of hard-magnetic soft beam has been derived.

• Mooney–Rivlin model has been used to capture the non-linearity of the material.

• Coupled anticlastic bending has also been incorporated into the framework.

• Prony series approximation was used to encapsulate the time-dependent response.

Abstract

Soft-materials that respond to external stimuli are very useful for application in soft robotics, stretchable electronics, biomedical applications etc. Recent times have seen a surge in research of magnetically activated polymers due to their wide range of material properties, applications and important features such as non-contact, fast and non-invasive actuation. Furthermore, these applications ask for Hard-magnetic particles, which can retain their magnetization even after the applied magnetic field has been removed. In this work, we focus on developing a thermodynamically-consistent analytical solution to the large bending deformation of hard-magnetic soft hyperelastic beams under the influence of an applied uniform magnetic field. The principal stress in the cross-sections, which arise due to anticlastic bending were also calculated. Lastly, a Prony series approximation was used to encapsulate the time dependent response of the material properties of the soft beam. The model was verified by comparing the results to previously developed experimental, numerical and analytical results.

Introduction

Recent times have seen a surge in the study of soft materials. These involve a wide range of materials such as photoactive materials (Yu et al., 2019, Bai and Bhattacharya, 2020, Mehta et al., 2020, Dunn and Maute, 2009), magneto-active soft materials (Zhao et al., 2019, Rao et al., 2010, Saxena et al., 2015), pH-responsive polymers (Kocak et al., 2017, Dai et al., 2008), electro-active polymers (Vogel et al., 2014), thermally activated polymers (Kim et al., 2009, Zhao et al., 2020, Zare et al., 2019, Baniasadi et al., 2021) etc. Specifically, magnetically activated materials are of great interest due to their non-contact, fast and non-invasive actuation (Mehnert et al., 2017, Zhao et al., 2019, Bastola and Hossain, 2020). Magnetically activated polymers find much usage in bioengineering applications due to their biocompatibility, hydrophilic nature, non-toxicity, biodegradability, similar mechanical response as biological tissue and magnetic actuation (Paradossi et al., 2003). The past decade has seen a wide range of research being conducted on magneto soft materials (elastomers and polymers) such as materials modelling (Zabihyan et al., 2020, Rao et al., 2010, Mehnert et al., 2017, Pivovarov and Steinmann, 2016, Hossain et al., 2015, Saxena et al., 2014, Kumar and Sarangi, 2019), numerical simulations (Pivovarov and Steinmann, 2016), synthesis (Walter et al., 2017, Walter et al., 2014) and characterization (Kim et al., 2018, Walter et al., 2017, Walter et al., 2014).

Usually, these magnetically activated polymers are composed of a soft-matrix material embedded with ferromagnetic materials with low coercivity (called soft-magnetic particles) (Borcea and Bruno, 2001, Zrínyi et al., 1996). Soft-magnetic particles are materials that are easily magnetized and demagnetized. They usually possess low coercivity (less than 1000 $A∕m$). These materials are often used in applications where the material is magnetized in order to operate and then demagnetized after the operation, such as in the case of an electromagnet. A crucial limitation while dealing with soft-magnetic materials is the loss of their magnetism upon removing the external magnetic field due to their low coercivity (Garcia-Gonzalez, 2019, Wang et al., 2020). The application and usage of these soft-magnetic materials are quite limited as they have restricted movement and cannot undergo complex deformations. Therefore, to address these limitations, researchers have started using hard-magnetic materials. Hard-magnetic materials retain their magnetism even after the external magnetic field is removed. They have a coercive force greater than $10kA∕m$. In order to completely demagnetize hard-magnetic particles, a very high opposite magnetic field is required. Embedding hard-magnetic particles into soft materials enables programmable and complex geometries under the influence of a magnetic field (Lum et al., 2016, Kim et al., 2018). These hard-magnetic soft-polymers have been actively used as soft robots to navigate through complex and constrained environments, which can be used in various medical applications (Kim et al., 2018, Runciman et al., 2019, Cianchetti et al., 2018, Yang et al., 2018).

To get a more fundamental understanding of these hard-magnetic soft materials, we require an understanding of their mechanical behaviour. The earliest known theoretical work on magneto coupled mechanical deformation was performed in the late 20th century (Pao, 1978, Eringen and Maugin, 1990, Maugin, 1989). Following these models, micro-mechanical modelling of the same materials was performed over 15 years ago (Brigadnov and Dorfmann, 2003, Dorfmann and Ogden, 2004). The continuum mechanics based models for magnetically activated soft materials is relatively more recent. The previous works by Saxena et al. proposed a continuum mechanics based model for magneto-viscoelasticity where a crucial assumption was made to split the applied magnetic field itself into elastic and viscous components (Saxena et al., 2013). The following work by Garcia-Gonzalez relaxed this assumption (Garcia-Gonzalez, 2019). The article formulates a constitutive model for the finite deformation of soft material matrix with embedded hard-magnetic particles within a thermodynamically consistent framework. Viscous dissipation was coupled with hyperelasticity, and magnetic contribution to the mechanical behaviour through Maxwell’s equations. The constitutive equations related the magnetic fields with the magnetic moments and stress components. The complete constitutive model was implemented into a finite element framework. Although the framework is valid for any hyperelastic energy function, the article is focused on a compressible Neo-Hookean model. The framework was applied to four numerical examples where the influence of the magnetic properties on the mechanical behaviour of smart structures was evaluated, namely (1) Uniaxial compression in an uniform constant magnetic field, (2) Uniaxial tensile test under varying magnetic fields, (3) Tests to observe the time dependent behaviour by varying magnetic and mechanical boundary conditions and (4) Inertial effects on uniaxial tensile tests under the influence of time dependent magnetic fields. Garcia-Gonzalez and Landis (2020) further modified the model which allowed diffusion of solvent across the polymer matrix. In the absence of any free current, the equilibrium equations and Maxwell’s equations were coupled with mass conservation to account for variation in solvent concentration. The energy function was modified to include the viscous effects produced due to the solvent–polymer interactions (Flory, 1953). The entire model was then implemented into a general FE framework.

Apart from theoretical modelling, significant experimental research for these hard-magnetic soft-materials was performed by Kim et al. where Hard-magnetic soft materials were synthesized by embedding neodymium–iron–boron (NdFeB) into polymeric gels (Kim et al., 2019). Unlike soft magnetic materials, like pure iron, NdFeB has high coercivity and therefore, can preserve its magnetization once they are magnetically saturated. The primary matrix, in which these NdFeB particles are embedded, was synthesized by a soft elastomeric composite. Moreover, the research by Zhao et al. (2019) presented a general magneto-elastic continuum-level framework to describe the deformation of such materials under external magnetic fields. Their model used additive decomposition of Cauchy stress tensor into magnetic and elastic components to couple the magnetic effects with deformation. This model was validated using experimental results of a beam embedded with hard-magnetic particle bending under a constant magnetic field (Kim et al., 2018, Zhao et al., 2019).

The primary limitation with all the above mentioned models is that such numerical models with finite element simulation requires high computational cost. Furthermore, these frameworks are not favourable for solving inverse problem in mechanics where calculating the magnitude of external magnetic field is of greater importance than predicting the future configuration with provided initial conditions. To overcome this limitation, Wang et al. proposed an analytical solution to hard-magnetic soft-beams under the influence a constant magnetic fields (Wang et al., 2020). The analytical solution was also compared with numerical and experimental results (Kim et al., 2018, Zhao et al., 2019). Although the proposed model can be used to predict and calculate the bending of the beam and the necessary magnetic field, the elastic contribution of the beam was modelled using a linearly elastic constitutive model. Furthermore, the model was primarily used to predict the behaviour of catheters coated with magnetic hydrogels. However, the model does not calculate the stresses across the beam’s cross section, which is essential to observe the stresses in the underlying catheters. The variation of the principal stresses across the cross-sectional area of the catheter are important to compute localized stresses. The latest known research was presented by Chen et al. where they further expanded the model for functionally graded hard-magnetic linearly elastic beams (Chen et al., 2020). The elastic modulus ($E$) of the entire material was approximated as

$$E=E_{0}exp\left(\frac{2.5\psi }{1-1.35\psi }\right)$$

where $E_0$ is the modulus of an unadulterated elastomer without embedding the Hard-magnetic particle. $\psi$ is the volume fraction of the hard-magnetic particle. But this research is also limited to pure linearly elastic materials. Furthermore, as mentioned previously, the model fails to calculate the cross-sectional stresses, which are crucial for calculating the stresses in the underlying catheters.

To address this issue, a theoretical model has been developed to obtain the analytical solution for the bending of a hard-magnetic soft-beam under a constantly applied magnetic field. The hyperelasticity of the soft material has been captured using the Mooney–Rivlin constitutive model. The principal stresses have been calculated using an anticlastic bending approximation of the beam. Furthermore, Prony series has been used to incorporate the material property change due to relaxation. The model has been verified by comparing the results to previous experimental, numerical and analytical solutions. To reiterate, the novelty of this work is the derivation of an analytical solution for hard-magnetic soft hyperelastic beam bending under the influence of a constantly applied magnetic field (Zhao et al., 2019). The theoretical framework for deriving the necessary equations for hard-magnetic particles in a soft matrix was obtained by splitting the total Helmholtz energy into elastic and magnetic components. The constitutive model was verified to be thermodynamically consistent. The cross-sectional principal stresses which arise due to anticlastic bending was plotted for different cross-sections. These cross-sectional principal stresses are crucial for calculating localized stresses. The time dependent variation of the material parameters was also modelled using Prony series.

This manuscript has been organized as follows: Section 2 presents the theoretical model developed and the final analytical solution. Section 3 contains the results and comparison of the proposed model. Finally, Section 4 concludes the study.