Introduction

Robots that are flexible, versatile, and light have benefits when used in healthcare, search and rescue, and cooperative human assistance [1, 2]. The requirement of adaptability of rigid robots to such challenging conditions has made soft robots a significant research interest in recent years. The need to construct soft robots with high deformability and compliance has increased the use of unconventional materials and morphologies. The variability and controllability of stiffness are among the major issues in soft robotics [3]. The stiffness of a soft robot must be designed as a compromise between being stiff enough to exert substantial forces on its surroundings and being compliant enough to adapt to the environment [4,5,6,7,8].

Various methods have been proposed to achieve variable stiffness in the literature. Manti et al. [3] summarized on-demand stiffening methods under two main approaches based on soft actuation technologies: the use of active actuators arranged in an antagonistic manner and the use of semi-active actuators. The antagonistic arrangement provides variable stiffness through the use of active elements such as electroactive polymers [9], flexible fluidic actuators [10], and tendon-driven actuators [11, 12]. The active elements are coupled with passive structures [4, 13,14,15,16,17]. Semi-active actuators rely on the change of mechanical properties of the material, including low melting point materials [6, 18,19,20], shape memory materials [21,22,23], viscosity-based materials, and jamming-based materials.

Viscosity-based [e.g., magnetorheological (MR) and electrorheological (ER)] materials can undergo stiffness variations because of the applied magnetic and electrical fields, respectively. ER fluids are less common because of the drawbacks of using ferroelectric particles [24]. The ER elastomers are also not widespread since it does not provide a useful stiffness change [25]. MR fluids (MRFs) [26] are also currently limited by disadvantages like particle settling and controllability issues, which makes another MR material more suitable for tunable stiffness: Magnetorheological Elastomers (MREs) [27, 28]. MREs overcome the problems that accompany the applications of MRFs. They exhibit a unique field-dependent material property when exposed to a magnetic field. Their stiffening and compliance speeds are higher than the other proposed methods, meaning they can provide fast responses (in milliseconds) and are more controllable [28, 29]. These features indicate that MREs have significant potential as a stiffness variation material.

MREs are composed of three main components. These are an elastomer matrix, additives, and magnetic particles [30, 31]. The particles show an MR effect in the presence of a magnetic field, which provides a field-dependent material property to the material, such as controllable modulus and damping. This property is removed from the material in the absence of a magnetic field. The magnetic particles should have low remanent magnetization, high saturation magnetization, and high permeability [32]. MR effect is maximized when the high permeability of the particles is used in the elastomer matrix. It is because the particles easily attract small magnetic leakage fields in the material compound. The most commonly used magnetic particle is Carbonyl Iron Particles (CIPs).

The last method under semi-active actuator is the use of jamming-based materials. Jamming-based technologies have been of interest to soft roboticists since they are simple to scale, easy to fabricate, inexpensive, and provide a high range of stiffness change. Jamming structures have used elements with varying geometries, such as grains [33,34,35], layers [8, 36,37,38], and fibers [39,40,41,42,43]. The mechanical principle of jamming involves harnessing the friction between the elements to make a transition between an unjammed (compliant) state and a jammed (rigid-like) state. When the structure is jammed, the independent elements behave like a one-single piece, thus the stiffness of the structure is increased.

Jamming in soft robotics is induced through different mechanisms such as pneumatic, mechanical, electrostatic, and magnetic [44]. The stiffness of the jamming structures in the literature is mostly controlled using a vacuum. It offers the convenience of achieving consistent isotropic pressure within a jamming structure. However, the use of vacuum has certain drawbacks. First, a pump and an external membrane must be included within the structure, potentially leading design complexities during system integration. Ensuring the effective sealing of the membrane that envelops the stacked elements is crucial without impeding the freedom of movement of the components. Second, particularly the de-stiffening duration required to transition from a rigid state to a soft state is prolonged. Third, maintaining the jammed state requires a continuous power source, which is not an energy-efficient method for achieving jamming transitions. Finally, there is a maximum limit on the vacuum pressure attainable within the enclosing structure, specifically when a perfect vacuum is reached. This limitation leads to an applied pressure equal to the atmospheric pressure, thereby constraining the range of slip thresholds that the structure can achieve. Mechanical actuation methods (e.g., clamps [45], cables [46, 47], and meshes [48]), on the other hand, have no physical upper limit for applied pressure. However, they require mechanical actuators, the suitability of which depends on the specific application when compared to pneumatic pumps. They enable precise control over the direction of the applied pressure. Electrostatic and magnetic jamming methods also offer the benefits of generating directional pressures with high magnitudes. Nevertheless, there are certain limitations associated with these methods. In electrostatic jamming [8, 49, 50], there is a constraint related to volume. The applied pressure diminishes as the distance between the two charged surfaces increases. In addition, high voltages are required, which could restrict their use in applications with safety concerns. In the magnetic jamming studies in the literature [26, 51], jamming is induced on nonmagnetic materials in MRFs. Although high stiffness change is obtained, they face the problems associated with MRFs, such as leakage or particle settling.

The state-of-the-art stiffness variation methods in the literature can be compared in terms of transition speed, modes of stiffening, and stiffness variation. MREs are effective in terms of the speed of stiffening/destiffening and are suitable for different modes of stiffening, such as bending, tension, and compression. However, stiffness variation data can only reach high values if a very high magnetic field is applied to the material. Layer jamming technology is appealing due to fabrication, low cost, and stiffness variation. However, the chosen actuation method can negatively affect the jamming structure’s performance. A vacuum is practical because it can maintain constant isotropic pressure inside a jamming structure. However, it calls for a tethered external membrane, which can complicate system integration during design. Moreover, it has slow transition response time in the order of seconds [3].

The use of MREs in soft robotics is reviewed in [52]. It highlights that MREs are used as a magnetostriction-based actuator, a micro-actuator that rotates when exposed to a magnetic field, a pneumatic valve, a soft inchworm robot for terrestrial locomotion, and a soft skin sensor for localizing deformations. In addition, the article shows an example of a variable stiffness surgical manipulator that utilizes MREs and electromagnets in a soft robot design [53]. In our previous work [54], we employed multi-physics finite element analyses to prove that a stiffening module made of MREs could be used to stiffen a soft robot when a magnetic field is applied. In this work, we investigate the unique mechanics of magnetic jamming of MRE layers as a first step to obtain a hybrid stiffening method combining a jamming-based method with a viscosity-based method for stiffness control of soft robots. The primary motivation to combine magnetic jamming with viscoelasticity modification is to achieve an increase in stiffness variation by triggering both methods with a single source of actuation. Recently, we presented a magnetically-controlled hybrid stiffening mechanism that entails layers, fibers, and granules as nonmagnetic scaffolding materials enclosed with a flexible pouch filled with MRF [26]. In our current study, we use MREs as ferromagnetic layer scaffolding materials that avoids the use of MRF and pouch. Thus, we overcome the problems associated with MRF such as leakage or particle settling. However, using MREs in a jamming structure for stiffness variation in soft robots has yet to occur in the literature.

The main focus of this paper is to explore stiffness change both due to jamming and variable viscoelasticity of MRE layers. Although modulus and damping behaviors of MREs in the presence of a magnetic field are well documented in the literature [30,31,32], there are very few studies on how surface friction of MRE materials change under an applied magnetic field [55]. Physical interaction of different MRE samples under a magnetic field has never been studied before. Therefore, we had to investigate the effects of jamming and variable viscosity on stiffness change in a stack of MRE layers before a possible soft robot implementation. Actual implementation of the proposed hybrid stiffening method for soft robotic applications is out of scope of the paper.

Materials and Methods

Conceptual Design

A layer-jamming structure consists of compliant MRE layers and two flexible NdFeB magnets (see Fig. 1). MRE layers were stacked between two flexible magnets with the same magnetic polarization. The stiffness behavior of the structure was explained physically as three stages of deformation [44]. The layers are cohesive during the pre-slip regime—the entire structure functions as one cohesive beam instead of separate, independent layers. The stiffness of the jammed structure is high as long as the shear forces induced in the layers are able to exceed the frictional forces caused by the magnetic forces. Beyond that point, stiffness starts to decline. In this transition regime, slipping starts to occur on the inner slip surfaces of the structure while shear forces are still at a level that cannot exceed the friction force on the outer slip surfaces. All layers are in slip during the full-slip regime, and the structure has the lowest stiffness value.

According to Euler-Bernoulli beam theory, the bending stiffness depends on E (Young’s moduli) and I (area moment of inertia). For the layer jamming structure, the area moment of inertia during the pre-slip region would be Ipreslip = bN3h3/12, whereas it is the sum of the independent layers after the slip (Islip = bNh3/12), resulting N2 stiffness ratio between the unjammed and jammed structure. Where N stands for the number of layers in the jamming structure, b and h are the width and height of each layer, respectively.

The magnitude of the applied magnetic force and the layer-to-layer friction coefficient determine the range of these regimes. The magnetized (ON State) and unmagnetized (OFF State) NdFeB magnets were used in three-point bending tests to compare the jammed and unjammed structures’ stiffness.

Sample preparation

MRE Fabrication

The MRE samples were manufactured using PDMS (Sylgard 184, Dow Corning Inc.) and 5 μm CIPs. The matrix material consists of two components: the part (A) and the hardener component (B). These were mixed with the ratio of A:B = 10:1, and 30% by weight of its thinner. Afterward, CIPs were mixed thoroughly into the homogenous mixture until all ingredients were evenly mixed. The mixture was placed into a vacuum chamber for 15 min at -1 bar to ensure proper elimination of air bubbles. Then, the mixture was put into aluminum molds. The curing process was performed at 100 ◦C for 1 h without a magnetic field to have isotropic MRE samples. Five MRE samples containing 0% (PDMS), 10%, 20%, 30%, and 40% volume fractions (f ) were prepared to perform large strain experiments. The size of the MRE samples used in the tests is 60 mm in length, 10 mm in width, and three different thicknesses: 1 mm, 2.5 mm, and 5 mm.

Fig. 1
figure 1

Schematic illustration of the three deformation regions for a multi-layer jamming structure. The magnets (grey) jam the MRE layers (white) with magnetic forces (Fm). (a) In the pre-slip region, there is no slip at the interfaces between the layers, and the bending stiffness is at its maximum. (b) The partial-slip region, where some interfaces begin to slip, and the bending stiffness gradually decreases. (c) All layers are in a slip in the full-slip region, and the structure’s overall stiffness is at its lowest value

Full size image

Flexible Magnet Fabrication

Flexible magnets were manufactured to provide a magnetic field to the MRE layers. The magnets were made of the same elastomer matrix, PDMS (Sylgard 184, Dow Corning Inc.), and hard magnetic neodymium-iron-boron (NdFeB) microparticles (MQP-S-11-9-20001, Magnequench, Co. Ltd.): both were mixed with a mass ratio of 10:1. The fabrication methods in MRE production were also applied in producing the flexible magnets. The magnets’ length, width, and thickness are 60 mm, 10 mm, and 2.5 mm, respectively. The manufactured MRE samples and flexible magnets are shown in Fig. 2. The manufactured flexible magnets were subjected to a large uniform magnetic field of 2.1 T using a dipole electromagnet (5403AC, GMW Associates) (see Fig. 3). Magnetic field measurements were done using a Teslameter (PCE-MFM 2400).

Fig. 2
figure 2

(a) The fabricated MRE specimens with three different thicknesses. (b) An image of a flexible NdFeB magnet

Full size image
Fig. 3
figure 3

Schematic of the magnetization process. A uniform magnetic field is applied through a flexible NdFeB magnet. The magnet and its holder (a non-magnetic 3D-printed part) were placed between the magnetic poles of the electromagnet

Full size image

Experiments

There are two types of experiments conducted in this study. First, tensile tests supported with digital image correlation (DIC) were conducted to characterize the samples. Second, the performance of the jamming structure was evaluated with and without the use of magnetic forces through 3-point bending tests.

Tensile test

Uniaxial tension tests were performed in the absence of a magnetic field to characterize the mechanical behavior of the MRE samples and the NdFeB magnet. The tests were conducted on the 2.5 mm thick samples. The schematic of the experimental setup can be seen in Fig. 4. The setup has a force measuring unit equipped with a 3-kg miniature load cell, a micrometer screw gauge with a resolution of 0.01 mm, and a stepper motor. In this experiment, the MRE samples were clamped using 3D-printed parts. The samples were tested up to 25% engineering strain using the stepper motor running at a constant rotational speed corresponding to 0.2 mm/s tension speed. The tension force was measured using the load cell attached to the moving clamp. The tests were displacement controlled. A DIC system was used to measure strains optically with a high-resolution camera (Pike F505B, Allied Vision Technologies) (see the tests in Online Resources in the form of videos). The Mullins effect, a well-known effect in rubber-like materials, is observed in the MREs since they are sensitive to stress softening [32, 56]. To eliminate this effect, a six-cycle test procedure was performed. The first three cycles were used for stress softening, whereas the last three were used to characterize the material (see Fig. S1 for the influence of Mullins effect in Supplementary Information).

Friction tests

The experimental setup used in the tensile tests was adapted to measure friction coefficients between the MRE samples and the NdFeB magnet. The schematic of the experimental setup is shown in Fig. 5. In this experiment, 3D-printed parts were glued to the MRE samples and the NdFeB magnet. One of the 3D-printed parts was fixed to the ground, while the other was attached to the load cell. A known mass M was placed onto the upper 3D-printed part. The material at the top was displaced 10 mm horizontally at a constant speed of 0.1 mm/s while the resulting force was measured. The tests were performed for ten pairs consisting of the MRE samples and the NdFeB magnet, and MRE-to-MRE interactions. The number of repetitions for each test is three.

Fig. 4
figure 4

Schematic of the experimental setup for uniaxial tensile test. The sample was sprayed with white paint to create a random speckle pattern. The distal end of the sample was fixed, while the proximal end was attached to the load cell. The tests were performed for each MRE sample and the NdFeB magnet

Full size image
Fig. 5
figure 5

Schematic of the friction test setup. Kinetic coefficients of friction µ between MRE-to-MRE and MRE-to-NdFeB are calculated by taking the average of the measured force in the dynamic range

Full size image

3-point bending test

The three-point bending test was used to evaluate the layer jamming structure (see Fig. 6). The structure consists of two flexible magnets at the top and bottom and the MRE layers at the core. The magnets attract each other and jam the MRE layers since they are placed so that their pole directions are the same. In the OFF State, non-magnetized flexible magnets were used, while magnetized flexible magnets were used in the ON State. The tests were performed for 30 cases, a combination of five different MRE volume fractions (f ), two states (ON State and OFF State), and three different test configurations. The configurations are a single-layer MRE with 5 mm thickness, two MRE layers with 2.5 mm thickness, and five MRE layers with 1 mm thickness. The force measuring unit was placed vertically down to measure forces. The videos of the three-point bending test when f = 0.3 can be seen in Online Resources. The number of repetitions for each test is five. The samples were preconditioned by bending a few times with bare hands before each test to mitigate the Mullins effect.

Fig. 6
figure 6

Schematic of 3-point bending test. N stands for the number of layers in the jamming structure. The MRE layers (white) and the flexible magnets (grey) are deflected while the resultant force (F) is measured. The thickness of the overall jamming structure is 10 mm, and the deflection (δ) is 5 mm

Full size image

Finite Element Analysis

Finite element analyses were performed for three purposes: material characterization, magnetic simulations, and multilayer jamming simulations. First, a finite element model was developed to validate our tensile test data with DIC measurements. The simulations were performed using the Static Structural module of ANSYS to characterize the MRE samples with 0 (PDMS), 10, 20, 30, and 40% volume fractions (f ) and the flexible NdFeB magnet. The materials were modeled in 3D and stretched up to 25% engineering strain, as in the experiments. Curve-fitting was performed on the experimental data using a hyperelastic model.

Second, magnetic simulations were performed on a three-layer jamming structure model using the Magnetostatic module of ANSYS in 3D. The model consists of two flexible magnets, an MRE material at the core, and an air domain. The geometry of an overall model is shown in Fig. 7. The thickness of the MRE is 5 mm, and the size of the magnets is 60 mm in length, 10 mm in width, and 2.5 mm in thickness. The magnetic properties of the NdFeB magnet were obtained directly from the manufacturer. They were calculated based on an estimation using the volumetric loading magnetic powder. The retentivity and coercivity values were set to 447 mT and 296 kA/m (Hc), respectively. Magnetic parameters such as MRE permeability with 10, 20, 30, 40% (f ) are retrieved from [32]. The values are 1.6, 2.2, 3.7, and 6, respectively. The permeability of PDMS is assumed to be 1. The magnetic simulations were performed for 20 cases, a combination of five different MRE volume fractions (f ), and four different deflections. The deflections are 0 mm, 2.5 mm, 4 mm, and 5 mm. In situations where the structure is deflected, we have established local cylindrical coordinate frames for both the upper and lower NdFeB magnets. This enables us to determine the polarization direction of these magnets. The force in the radial direction represents the average magnetic jamming force. These values were interpolated for the full deflection range.

Fig. 7
figure 7

(a) 3D finite element model of the three-layer jamming structure for magnetic simulation. An MRE layer with 5 mm thickness was placed between two NdFeB magnets, and the structure was surrounded by air. (b) Front view of the model. The simulations were performed for four different deflections. For each case, a cylindrical coordinate frame was assigned to determine the polarization direction of the top and bottom magnets

Full size image

Third, finite element simulations were conducted in 2D to predict the mechanical behavior of the N-layer jamming structure. The model consists of two NdFeB layers, N-2 MRE layers, and three rollers (see Fig. 8). The layers and the supports were modeled with 2D plane-stress elements with material properties, dimensions, and boundary conditions the same as in the experiments. The interaction between two adjacent MRE layers and the interaction between NdFeB-MRE layers were modeled using the contact elements with a constant coefficient of friction values stated in Table 1. The magnetic force values obtained from magnetic simulations were applied to the structures’ top and bottom surfaces. The top layer was deflected 5 mm incrementally while the resulting force was measured. Large deflection and quasi-static solutions were enabled in the simulations.

Results and Discussion

The stress vs. strain curves of the MREs and the NdFeB magnet tested under tensile tests are shown in Fig. 9. The results indicate that the mechanical behavior of the materials is nonlinear. The Neo-Hookean model was selected to obtain the mechanical properties of the MRE samples and the NdFeB magnet since it is the simplest form of all hyperelastic models and suitable for 25% strain. The strain energy function of the Neo-Hookean model is expressed as

$$W=\frac\mu2\left({\overline I}_1-3\right)+\frac1{D_1}\left(J-1\right)^2$$
(1)

where µ is the shear modulus, D1 is the incompressibility factor, J is the Jacobian of the deformation gradient, and \(\overline I_1\) is the first invariant of the strain tensor.

Fig. 8
figure 8

Finite element model for N-layer jamming structure. The top roller was displaced 5 mm in -y direction. The bottom rollers were fixed. The magnetic force density values obtained from magnetic simulations were applied to surfaces A and B. Frictional contact type was used in all interfaces. The surfaces C and D were defined as contact surfaces for the MRE-to-NdFeB and MRE-to-MRE interfaces, respectively

Full size image
Fig. 9
figure 9

Stress vs. strain for the MRE samples with different volume fractions (f ) and the NdFeB magnet. The experimental and FEA results are represented as solid and dashed lines, respectively

Full size image

The material constants µ and D1 are obtained from the curve fitting on the experimental tensile test data and DIC measurements, respectively. Additionally, see Fig. S2 for friction coefficient constants in Supplementary Information. The material properties are shown in Table 1.

$$D_1=\frac{6\left(1-2v\right)}E$$
(2)

where ν is Poisson’s ratio (see Fig. S3 for DIC results in Supplementary Information), E is Young’s modulus of the materials, and is assumed to be constant.

Table 1 Material properties for the MRE samples and the NdFeB magnet
Full size table

The results of the magnetic surface force of the flexible NdFeB magnets are shown in Fig. 10. The results show that magnetic force increases when a higher volume fraction of MRE is used at the core. For instance, the flexible magnet offers an average magnetic force of 0.49 N, 0.52 N, 0.54 N, 0.55 N, 0.56 N for f = 0 (PDMS), f = 0.1, f = 0.2, f = 0.3, f = 0.4, respectively when there is no deflection. In addition, the magnetic force decreases as the model deflection (δ) increases. For instance, the average magnetic force of the top magnet is decreased from 0.52 N to 0.28 N, whereas the average magnetic force of the bottom magnet is decreased from 0.52 N to 0.17 N when f = 0.1. Furthermore, it can be inferred from Fig. 10 that magnetic force profile losses its uniformity as the structure is deflected.

Figure 11 shows the results of the 3-point bending test of the multilayer jamming structure. Slopes of the resulting force vs. deflection curve give bending stiffness values of the structure. The results of the unjammed structures indicate that the resulting force shows a linear behavior (R2 ≥ 0.993) with deflection for each case. The results also indicate that the pre-slip stiffness of the jammed structure is higher than that of the unjammed structure for each case. It is an expected result since magnetic forces jam the MRE layers and the overall stiffness of the structure increases. Table 2 summarizes the stiffness values for each case. The results indicate that the stiffness of the jamming structure in the OFF State decreases when the number of layers is increased. For instance, when f = 0.2, the stiffness of the structure with 3-layer, 4-layer, and 7-layer are calculated as 0.52 N/mm, 0.38 N/mm, and 0.30 N/mm, respectively. The FEA results have very similar results for this case. The values are 0.53 N/mm, 0.35 N/mm, and 0.29 N/mm, respectively.

In addition to the presented stiffness values for the multilayer MRE jamming structure, change in stiffness values is calculated for each volume fraction and the number of layers. Let K be the ratio of the stiffness of the jammed structure to the stiffness of the unjammed structure. The results are shown in Fig. 12. The results show that simulation results are larger than experimental results. The reason is that the magnetic properties of the NdFeB magnet are less than the simulation values due to the attenuation effect caused by the losses in the production stage. The simulation results also indicate that K increases with the increasing number of layers stacked within the jamming structure. For instance, when f = 0.4, the stiffness ratio of the jamming structure with 3-layer, 4-layer, and 7-layer is calculated as 2.21, 3.95, and 5.63, respectively. However, this situation is not valid for the experiment results for f = 0.1 and f = 0.2. The value K is slightly decreasing from N = 3 to N = 4. The main reason for this situation is that the number of friction contact types differs. There are two different friction interactions when N = 4 (MRE-to-MRE and MRE-to-NdFeB), whereas there is only one (MRE-to-NdFeB) when N = 3. The results show that the best performance was obtained when N = 7 and f = 0.4.

Fig. 10
figure 10

Top view of magnetic force contours of the flexible magnets. Each image shows the average force values in the upper left corner. T and B represent the top and the bottom magnets, respectively

Full size image
Fig. 11
figure 11

Finite element analysis (dashed lines) and experimental (solid lines) results for N-layer jamming structures. Blue and red color stand for ON States and OFF States, respectively. The structures were loaded in three-point bending for the magnetized NdFeB magnet and the non-magnetized NdFeB magnet for each MRE with different volume fractions

Full size image

The main reason to use MREs in the middle layers is to benefit from both viscoelasticity change in MREs under a magnetic field and increase in magnetic jamming force. We can observe changes in stiffness caused by two factors: the jamming and MR effects. We have also tested the jamming structures in which PDMS layers are used as nonmagnetic materials. This approach enabled us to eliminate the MR effect, and examine the jamming effect alone that arises exclusively from the magnetic forces generated by the NdFeB magnets. For instance, when N = 7, the experimental K value of the PDMS is 2.35, which is solely due to jamming. On the other hand, the corresponding K value is 3.66 for f = 0.4. This indicates that when MREs are used instead of PDMS, stiffening change is higher. This increase in stiffness change can be attributed to both the MR effect and the increase in magnetic jamming force due to iron particles in the MRE layers. Although we could not estimate the individual contribution of each factor, these results show the benefit of using MREs in a layer jamming structure.

The stiffness change is theoretically proportional to the square of the number of layers involved in layer jamming. According to this theory, the resulting stiffness ratios would be K = 9, 16, 49 for N = 3, 4, 7 respectively. However, experimentally obtained values for f = 0.4 are 2.55, 3.17, and 3.66 for N = 3, 4, and 7, respectively. It is obvious that the N2 relationship does not work. The main reason for this may be due to the way jamming force is applied. Unlike pneumatic jamming, no uniform pressure is acting on the entire outer surface of our jamming structure. The reason we are unable to obtain uniform magnetic jamming forces upon loading is due to the non-uniform magnetic profile of the jamming structure. The magnetic profile is almost uniform when there is no deflection; however, it is decreased when the structure is deflected, as stated in the top left of the subfigures of Fig. 10. This loss of uniformity occurs because the top and bottom magnets have different poles due to their distinct radii of curvature.

The results also show that this increased stiffness is only sustained under a limited range of loads. For 12 cases, the mean slip point for the experiments and finite element simulations was calculated as 0.35 mm and 0.16 mm, respectively. Full slip occurs when deflection is 0.88 mm for the experiments and the simulations. Although the experiments and the simulations experience the full-slip region at almost the same points, the simulations reach the slip point earlier. It is because we assumed constant coefficient of friction, and used coefficient of kinetic friction in our simulations instead of coefficient of static friction.

Fig. 12
figure 12

Stiffness ration of the MRE jamming structure

Full size image
Table 2 Stiffness [N/mm] values of the multilayer jamming structure
Full size table

Conclusion

In this paper, we investigated the behavior of magnetically-induced jamming of magnetorheological elastomer layers as a first step towards a hybrid variable stiffness method. The MRE and NdFeB layers were characterized using uniaxial tensile tests supported with a DIC system. The bending stiffness of the jamming structure was tested using 3-point bending experiments, and finite element simulations validated the experiments. We proved that a higher stiffness variation was achieved in the pre-slip regime when MREs were used instead of PDMS. This shows the benefit of the hybrid approach combining magnetic layer jamming with viscoelasticity change. The results show that this increased stiffness was only sustained under a limited range of loads. It is because shear forces at the interfaces between the MRE layers caused by magnetic forces are quite low. Furthermore, we showed that the stiffness ratio increases when the number of MRE layers is increased within the multi-layer jamming structure.

The stiffness variation can be further improved in some ways with future research. First, the thickness of the manufactured MRE samples used in the jamming structure can be decreased using more advanced fabrication techniques such as micro molding and extrusion-based 3D printing. Therefore, more MRE layers can be stacked within the multi-layer structure. Second, the CIP arrangement plays an important role in determining the mechanical properties of the MREs. The MR effect is boosted when anisotropic MRE samples are used within the jamming structure such that the magnetization direction of the NdFeB magnets and the direction of the CIPs are the same. In addition, the MR effect is increased when an elastomer matrix with a lower modulus is chosen to fabricate the MREs.

In this study, we have shown that the use of MRE layers results in an increased stiffness change due to both the MR effect and the enhancement in magnetic jamming force caused by iron particles within the MRE layers. However, we were unable to quantify how much each method contributes to the total stiffness change. To quantify the contribution of the viscoelasticity change (i.e., the MR effect) alone, it is necessary to perform a 3-point bending test on an individual MRE layer exposed to the same magnetic field conditions as in our experiments. However, setting up an external magnetic field that precisely matches the magnetic field provided by the flexible NdFeB magnets is practically not possible. The sharp corners of the MRE sample which are exposed to more field intensity than the inner regions, can cause bending of the MRE layer under the applied magnetic field.

Although our final goal is to use the hybrid stiffening method (combining a jamming-based and a viscosity-based method) for stiffening of soft robots, the proposed mechanism in its present form cannot be used in soft robots due to certain disadvantages. The most significant drawback of this study is the jamming transition that is induced mechanically. The non-magnetized NdFeB magnets must be changed with the magnetized NdFeB magnets to make a transition from OFF State to ON State. Since the aim of this work was to focus on investigating the effect of magnetic jamming of MRE layers, we used permanent NdFeB magnets to generate magnetic fields. However, magnetic fields can be generated electronically using electro-magnets or electro-permanent magnets (EPMs) for a possible implementation of electronically controlled jamming and stiffening [26]. As a future work, we propose an approach for potential implementation of the stiffness control of the jamming structure, involving electronically activated EPMs. An EPM is a solid-state device whose magnetic flux can be switched between the ON and OFF states due to a discrete electrical pulse. It is composed of two permanent magnets with identical retentivity but different coercivity, two iron caps, and a coil system. McDonald et al. [57] manipulated the material properties of MRFs by utilizing EPMs, enabling the regulation of the pressure within soft actuators. EPMs present a promising solution for providing magnetic jamming of MREs. The jamming elements can be placed between the EPM caps. When the EPM is in its ON state, it can generate a substantial magnetic field in that region, facilitating the MR effect. For the jamming effect, two identical EPMs can be employed, with their facing caps having opposing magnetic poles. The proposed method introduced here will reduce the time required for stiffening and de-stiffening to mere milliseconds. Additionally, the proposed actuation mechanism for the jamming transition via EPMs is characterized by its low operational energy consumption. Thanks to the EPMs, energy is consumed only for couple of milliseconds during the switching between the states. Therefore, the proposed magnetic jamming structure with multi-layer MREs is a promising method for stiffness variation of soft robots.

Change in stiffness in layer jamming structures is valid only in one direction. On the other hand, fiber jamming structures are effective in soft robotic applications because variable bending stiffness in more than one direction is achieved [44]. We are currently working on a follow-up project in which MRE fibers will be used in an actual soft robot, and the jamming transition will be triggered using electro-permanent magnets. In this future work, MRE fibers will be inserted axially in the central channel of a soft robot. They will be enclosed by EPMs such that magnetic polarization is in the radial direction. Thus, the bending stiffness of the robot will be controlled in more than one direction.