MREs among smart materials
Diﬀerent stimuli cause some materials called “smart”, “intelligent” or “active” to respond to their environment in a reversible manner, thus producing a useful eﬀect. Common materials that formally have the label of being smart include piezo-electric materials, electro-strictive materials, magneto-strictive materials, electro-rheological materials, magneto-rheological materials, thermo-responsive materials and shape memory alloys. The applied driving forces for field-driven smart materials then can be broadly identified as mechanical fields, electrical fields, magnetic fields and thermal fields. Therefore, an important feature related to smart materials is that they encompass almost all fields of science and engineering.
While electro-active materials usually require high voltages for activation, econom-ically generated magnetic fields (permanent magnets, solenoids) can be used to stimulate magneto-active materials. As a branch of this kind of active materials, electro- and magneto-rheological materials consist of an insulating or non-magnetic matrix (either a fluid, a foam or an elastomer) into which electrically or magnetically polarizable particles are mixed, respectively [Har06].
The first explored [Rab48] and most common magneto-rheological (MR) materials have a fluid matrix. In these MR fluids, the interaction between dipoles, induced by an external magnetic field, causes the particles to form columnar structures par-allel to the uni-axially applied field. As the external magnetic field increases, the mechanical energy needed to yield these chain-like structures increases since it be-comes more and more diﬃcult for the fluid to flow through these formed structures. The obtained field-dependent yield stress, which can be rapidly and reversibly con-trolled, has been exploited in a variety of vibration control or torque transfer devices [Car01, Ber12]. However, the drawback of MR fluids is the settling of their particles and the fact that the devices need to be enclosed. As a solution to this, magneto-rheological foams have been developed, in which the controllable fluid is contained in an absorbent porous matrix. Fluids in these smart devices are thus more resis-tant to gravitational settling. Additionally, no seals or bearings are required and a smaller quantity of fluid is needed [Car00].
A definite answer to the particles settling was the use of an elastomer as the ma-trix. The magnetic particles are mixed into the fluid-like polymer blend but remain locked in place within the cross-linked network of the cured elastomer. Hence, these composite materials are often considered as the solid analogs of magneto-rheological fluids. The first researchers who conducted preliminary tests on these so-called magneto-rheological elastomers1 are Rigbi and Jilken [Rig83]. They studied the be-havior of a ferrite elastomer composite under the combined influence of changing elastic stresses and magnetic fields and described the previously unknown magneto-mechanical eﬀects2.
Classification within MREs
If magnetic fields are applied to the elastomer composite during processing, spe-cial anisotropic properties can be imparted to these materials. They are then called field-structured MREs and it has been observed that they are anisotropic in terms of mechanical, magnetic, electrical, and thermal properties [Car00]. More particularly, an applied uni-axial magnetic field produces chain-like particle structures3: at low particle concentrations the particles initially form chains that slowly coalesce into columns, as shown in Figure 1.1, while at high concentrations the morphology be-comes slightly more complex, with possible branching between columns and a slight loss in anisotropy [Mar98, Boc12]. A bi-axial (e.g. rotating) field produces sheet-like particle structures [Mar00]. Finally, with the generation of triaxial magnetic fields during processing, a variety of isotropic and anisotropic particle structures can be created [Mar04].
MREs can be further classified according to their main constituents, i.e. the non-magnetic matrix and the magnetic filler particles as well as their inherent properties. They usually are composed of a solid, electrically insulating4 matrix, such as silicone rubber, natural rubber, polyurethane or thermoplastic elastomers [Kal05, Hu05, Che07, Wei10, Kal11]. In fact, a great variety of matrix materials covering a wide range of properties in modulus, tensile strength or viscosity can be found on the market and in the literature. Within all these materials however, MREs based on silicone rubber enjoy much popularity due to their excellent processability, to the good compromise between mechanical, thermal and aging properties, as well as to their widespread use in industrial applications. Furthermore, relatively soft matrices with low elastic moduli are achievable in silicon rubbers and tend to ease the magneto-mechanical interaction [Kal05, Gon07, Dig10, Schu14]. Let’s note also that the individual class of “soft magneto-rheological elastomers” has been claimed, which covers MREs whose matrix is elastically soft. This class includes magnetic gels consisting of magnetic particles dispersed in a gel-like polymeric matrix. Gel materials (fluid within a three-dimensional weakly cross-linked network) can indeed be much softer than classical elastomers. However, they usually have a wet and sticky consistency, and when agitated, these materials start to flow (thixotropy), thereby resulting in poor mechanical properties [Zri97, Rai08, Zub12].
Diﬀerent types of magnetic filler particles have been used in MREs: magneto-strictive or magnetic shape memory particles, as well as hard or soft magnetic par-ticles. A number of researchers employed highly magneto-strictive particles, usually Terfenol-D [Due00]. This alloy of rare earth crystals is the most eﬀective but a cost intensive magneto-strictive material, capable of showing reversible strains in the order of 10−3 in response to an applied magnetic field [Guy94]. Particles with magnetic shape memory have also been used [Sche07] yielding both temperature and magnetic field-driven MRE composites. The dispersion of hard magnetic parti-cles in an elastomeric matrix, magnetized during fabrication, produces anisotropic, magnetically-poled MREs similar to a flexible permanent magnet [Koo12]. However, the most commonly used particles are made of soft ferromagnetic materials such as nickel, cobalt or iron and their alloys [Aus11, Ant11]. In particular, carbonyl iron powder (CIP) with spherical particles has been widely preferred in MREs fabrication [Kal05, Boes07, Dig10, Schu14]. Iron has indeed a high magnetic susceptibility and saturation magnetization, providing high inter-particle interaction forces, as well as a low remanent magnetization required in order to obtain quick and reversible con-trol by the magnetic field in MRE applications. The size of the magnetic particles further distinguishes MREs from ferro-elastomers or ferro-gels5. Standard ferro-gels (as well as ferro-fluids) tend to include nano-sized particles which are magnetic mono-domains. They agglomerate easily and cannot be separated again once they have agglomerated. Following the definition in the review by Carlson and Jolly [Car00], MREs rather embed micron-sized particles possessing a high number of magnetic domains that, overall, are harder to magnetize [Car00, Mar06].
Main MRE research topics and applications
Considering the large amount of possible matrix-filler combinations in the litera-ture, many of the early experiments but also current research have been dedicated to the composition and processing of MREs together with the investigation of the ob-tained microstructures by optical microscopy [Far04a], scanning electron microscopy [Gon05, Chen07] or X-ray microtomography [Bor12]. The processing conditions and (field-)curing mechanisms are crucial parameters for the manufacturing in a labora-tory environment. More specifically, the mixed viscosity – a measure of the thickness of the blended composite constituents –, the temperature and the magnetic field will determine the competition between gravitational settling and the alignment of the particles during curing [Als07, Gue12]. The final structure of the MRE is therefore specific to each of these parameters, which complicates the comparison of experi-mental results in the literature.
Furthermore, extensive studies have been conducted to investigate the dynamic small strain behavior of MREs, especially the influence of an external magnetic field on mechanical properties such as storage, loss and viscoelastic moduli [Shi95, Jol96, Gin00, Lok04, Kal05, Fan11, Kar13]. Since MREs have been shown to alter their dynamic moduli in response to the field, their performance as tunable vibra-tion absorbers and tunable stiﬀness devices has been widely studied and prototype applications have been developed [Eli02, Far04b, Cri09, Mar13, Kim14], such as a prosthetic foot [Tho13] shown in Figure 1.2.
Figure 1.2: a) Picture of a smart prosthetic foot device and b) cross-section of the integrated tunable MRE spring [Tho13]. The motion (red arrows) resistance of the MRE spring (75) can be adjusted by the magnetic flux (green circuit) generated by the coil (74).
In parallel, a large body of research has been devoted to the investigation of the deformation of MRE materials exposed to a magnetic field. The attraction of mag-netically soft and mechanically flexible MREs by magnetic field gradients has been discovered early [Zri96] and the use of this eﬀect in smart high-strain actuators has been proposed [Boes12, Bro12]. Figure 1.3 shows the example of an MRE device en-abling flow control [Ste13]. The deformations induced by a uniform magnetic field have also been studied early [Gin02] but the results are still rather contradictory up to now and cannot yet be explained clearly. We will come back to these aspects in the next section.
Finally, an important part of the literature on MREs is concerned with the modeling of these materials. The approaches therein can basically be partitioned in micro-mechanical or structural models on the one hand and continuum or phe-nomenological models on the other hand. In the continuum approach that is the theoretical basis for what has been done in this thesis, the material constitutive behavior needs to be characterized with the help of magneto-mechanical experi-ments. In Section 1.6, a review on the available experimental data will show that the magneto-elastic coupled behavior up to high strain and high magnetic fields has not been well explored yet. Hence the design of smart devices capable of high defor-mations has been limited so far and very few applications allow the MRE material to deform up to 20 percent [Cri09, Tho13, Ste13].
Physical phenomena in MRE behavior
In this section, selected physical phenomena involved in MREs behavior are dis-cussed in more detail, mainly from an experimental point of view. To remain within the scope of this work, the focus is especially set on the quasi-static or low-frequency behavior of these materials while accounting for large strains and high magnetic fields.
Mechanical behavior of MREs as particle-filled composites
Some purely mechanical phenomena characteristic of particle-filled composites have to be clarified when investigating MREs, even before considering magneto-mechanical aspects. It is well known that the highly nonlinear macroscopic behavior of filled polymers in response to a mechanical load is aﬀected by matrix-filler and filler-filler interactions. Diﬀerent phenomena have been experimentally observed6 and can be linked to changes in the microstructure.
Figure 1.4: Illustration of the Mullins eﬀect for a carbon black filled rubber (par-ticle volume fraction Φvol ∼ 0.2) from [Mie00, Dor04]. Two virgin samples are either uni-axially stretched directly up to high strain (dashed line) or cycled by increasing the strain amplitude after each set of cycles (solid lines). The stable upload curve for the set of cycles of lowest strain amplitude is highlighted in red.
Under large deformations, a softening (characterized by a lower resulting stress for the same applied strain) and a permanent deformation between first and subsequent load cycles can be observed in filled rubbers and MREs. This phenomenon depends on the maximum strain applied and is called “Mullins eﬀect” [Mul57, Mie00, Dor04, Coq06a, Dia09]. It is illustrated in Figure 1.4. The highest softening occurs after the first load cycle and leads to a stable response after few cycles (c.f. Figure 1.4, stable upload curve in red), aside from a smaller fatigue eﬀect [Schu14]. The propensity of the Mullins eﬀect, as well as the composite stiﬀness are increased with a higher filler content (see Figure 1.5). Let’s note here that, in the MRE literature, the filler content is (with a small number of exceptions, e.g. [Als07]) expressed in terms of the particle volume fraction Φvol.
Figure 1.5: Qualitative summary of MREs elastic modulus E in the small strain region on the left (from [Boeh01, Bel02]) and stable stress-strain behavior up to high strain on the right (from [Schu14]) for diﬀerent particle volume fractions Φvol.
Along with the particle content, the particle size and shape have a significant influ-ence on the composite mechanical behavior. Below a critical particle size, usually in the nano-scale, the stiﬀness can be greatly enhanced whereas for a large particle size, above 10 microns, the filler influence can even be degrading [Leb02, Ram04]. Irregular and elongated particles can as well change the mechanical stress transfer throughout the material and thus the material stiﬀness, depending on their orien-tation within the host matrix [Fu08].
Another important aspect mainly influencing the mechanical strength is the inter-facial adhesion between the filler particles and the matrix [Dek83, Coq04, Fu08]. Indeed, at a critical stress level, debonding acts as a distinct failure phenomenon in a polymer containing rigid inclusions due to stress concentrations at the weak particle-matrix interface [Gen84, Cre01], which could be of great influence on the magneto-mechanical coupling in soft MREs.
Finally, filler networks such as the particle chains in field-structured MREs, as well as possible aggregates and agglomerates due to the fabrication process, further af-fect the composite mechanical behavior. Some amount of rubber can indeed get trapped inside the particle networks, leading to an increase of the eﬀective filler volume fraction and hence of the stiﬀness, as long as the filler network is not bro-ken down [Wan99, Yat01, Leb02]. Field-structured MREs also exhibit the highest mechanical stiﬀness when the load is applied in the direction of the particle align-ment [Bel02, Var05, Schu14], similar to what is classically observed in uni-directional fiber-reinforced composites [Cam10].
Magnetic response of MREs
Before investigating the macroscopic magnetic properties of MREs, important vari-ables, commonly used in the magnetics-related literature, will be reviewed for clarity. In vacuum, the magnetic field b0 = µ0h0 is proportionally related to the magnetic permeability of free space µ0 = 4π10−7 [N/A2] multiplied by the externally applied field h0. If a finite ferromagnetic7 body is now exposed to this excitation field h0, the body becomes magnetized and itself provokes a perturbation field h1, also known as demagnetizing field or stray field [Guy94, Kan04]. The macroscopic magnetic constitutive relation then can be defined following the SI (Système International d’Unités) system of measurement as: = µ0 (h0 + h1 + m) = µ0 (h + m) , (1.1)
where b = b0 + b1 is the total magnetic field, also called magnetic induction or magnetic flux density and expressed in Tesla [T = N/Am] and b1 = µ0 (h1 + m) is the magnetic perturbation field. h = h0 +h1 is the total h-field, also called magnetic field intensity or magnetic field strength, expressed in [A/m]. The state of magnetic polarization within the body is described by the magnetization field m [A/m]. The nonlinear relation between h and m for a ferromagnetic material can be determined experimentally and usually takes the form of a hysteresis loop (see Figure 1.6a).
Table of contents :
1.1 MREs among smart materials
1.2 Classification within MREs
1.3 Main MRE research topics and applications
1.4 Physical phenomena in MRE behavior
1.4.1 Mechanical behavior of MREs as particle-filled composites
1.4.2 Magnetic response of MREs
1.4.3 Magnetic field-dependent modulus
1.4.4 Deformation under magnetic field
1.5 Approaches to the modeling of MREs
1.5.1 Micro-mechanically based modeling of MREs
1.5.2 Phenomenological continuum description
1.6 Previous experimental characterizations of magneto-elastic properties at finite strain
1.6.1 Overview of experimental studies on MREs
1.6.2 Magneto-mechanical experimental characterization using continuum models
1.7 Scope and organization of the present work
2 Materials and samples
2.2 Sample shape for coupled magneto-mechanical testing
2.3 Materials and fabrication procedure
2.3.1 Materials selection
2.3.2 Fabrication procedure
2.3.3 Molds and stands
2.4 Study of interfacial adhesion
2.4.1 Samples preparation
2.4.2 Scanning electron microscopy
2.4.3 Macroscopic mechanical tests
2.4.4 Discussion of results
3 Magneto-mechanical characterization
3.1 General theoretical framework
3.1.1 Governing equations
3.1.2 Free energy function for transversely isotropic MREs
3.2 Constitutive parameter identification
3.2.1 Reduced form of the free energy function
3.2.2 Free energy density and response functions
3.2.3 Different test cases
3.3 Magneto-mechanical characterization setup
3.3.2 Tension system
3.3.3 Mechanical diagnostics
3.3.4 Magnetic measurements
3.4 Experiments and parameter identification
3.4.1 Testing protocol
3.4.2 Experimental results
3.4.3 Parameter identification results
4 Numerical implementation
4.2 Total Lagrangian variational formulation of the fully-coupled magnetomechanical problem
4.2.1 General framework for a non-uniform applied magnetic load
4.2.2 Framework for the prototype device simulation
4.3 Axisymmetric FEM formulation
4.3.1 Discretized variational principle
4.3.2 Axisymmetric space
4.3.3 FEM implementation
4.4 Problem geometry/mesh, initial/boundary conditions and material parameters
4.5 Simulation results
4.6 Experimental validation
4.6.1 Haptic surface prototype
5 Conclusion and future work
5.2 Future work
A.1 Derivation of the traction response
A.2 Coefficients of the force vector and the stiffness matrix