The Static and Dynamic Mechanical Properties of Magnetorheological Silly Putty

A novel magnetorheological material defined as magnetorheological Silly Putty (MRSP) is prepared by dispersing soft magnetic particles into Silly Putty matrix with shear stiffening property. Static mechanical properties including creep and stress relaxation and dynamic rheological properties of MRSPs are tested by rheometer.The experimental results indicate that the external magnetic field exerts significant influence on the creep and relaxation behaviors. Moreover, the storage modulus of MRSPs increases sharply in response to the external stimuli of increasing angular frequency automatically and can be enhanced by external magnetic field. Besides, temperature plays a key role in shear stiffening and magnetorheological effect of MRSPs. Furthermore, considering the obstruction to the particle chains formation induced by Silly Putty matrix, a nonperforative particle aggregated chains model is proposed. The model curve is in consistency with experimental data, which means it can describe magnetoinduced behavior of MRSPs well.


Introduction
Silly Putty is an intelligent material that exhibits obvious shear stiffening property.It can be easily kneaded, much like dough, into any shape desired.However, if a shock or impulsive load is applied to the putty, it will stiffen and even shatter.In rheological terms the experimenter is distorting the Silly Putty over a range of Deborah numbers.This nondimensional parameter describes the ratio of the fluid relaxation time scale to that of the experimental time scale.A high Deborah number therefore corresponds to a fast experiment in which the load or impulse is applied over a very short time scale.Due to its unique rheological property, Silly Putty is becoming a promising material for a special damper called shock transmission unit (STU) widely used in multispan bridge [1].
Shear stiffening property of Silly Putty has been investigated by many researchers.Cross [2] proposed a threeelement parameter model to describe the compression of Silly Putty at different compression speeds; Goertz et al. [3] developed a mechanical model for Silly Putty based on three Maxwell elements in parallel to study the influence of the temperature on the rheological property and found that the mechanical properties shifted significantly with temperature.Wang et al. [4,5] prepared a novel shear stiffening gel (STG) with polydimethylsiloxane (PDMS), boric acid, benzoyl peroxide (BPO), and different additives such as calcium carbonate and carbon nanotube.The stiffening mechanism of STG was also explained in respect of chemical bond.
Magnetorheological materials are a class of smart materials that have usually been prepared by dispersing magnetic particles into different polymer matrices.The mechanical properties of magnetorheological materials can be controlled by an externally imposed magnetic field.Due to the controllable rheological property, magnetorheological materials have thus been widely applied in dampers, isolators, and different vibration controllers.However, most solid-state magnetorheological materials cannot present considerable tunability.Interestingly, on the basis of the fabricated STG, Wang et al. [6,7] firstly developed a magnetically responsive shear stiffening gel with excellent shear stiffening performance and magnetorheological effect.This novel magnetically responsive shear stiffening gel could provide credible tunability with external stimuli.
Advances in Materials Science and Engineering Besides, some theoretical models have also been developed to describe the magnetoinduced behavior of magnetorheological materials.Jolly et al. [8] proposed a dipole model based on the magnetic interactions between two adjacent particles.Shen et al. [9] presented a mathematical model to represent the stress-strain relationship of magnetorheological elastomer.The model considered all the dipole interactions in a chain and the nonlinear properties of the host composite.Chen et al. [10] developed a finite-column model to describe the relationships between the microstructures and the viscoelastic properties.However, for magnetorheological Silly Putty (MRSP) of soft magnetic particles dispersing in Silly Putty matrix, when external magnetic field strength increases, the internal structure changes unlike that of magnetorheological elastomers, and the chain structures are not pulling through like magnetorheological fluids.
In this work, several MRSP samples were prepared by dispersing carbonyl iron particles into Silly Putty matrix.Both static and dynamic mechanical properties were tested by rheometer.The static mechanical testing indicated that MRSPs exhibit obvious creep and stress relaxation behaviors and external magnetic field strength plays a key role in creep compliance and relaxation modulus.The dynamic mechanical testing revealed that MRSPs display both good shear stiffening property and high magnetorheological effect; furthermore, influence factors of mechanical properties for MRSPs including temperature and particle concentration were analyzed.Finally, a nonperforative particle aggregated chains model considering the obstruction to the particle chains formation induced by Silly Putty matrix was proposed to describe magnetoinduced behavior of MRSPs.Due to the tunable shear stiffening and magnetorheological effect, MRSPs are candidates for applications in dampers and STUs widely used in vibration control of buildings and bridges.1.The soft magnetic particles carbonyl iron (CI) of average size 3.5 m was purchased from Jiangsu Tianyi Ultra-Fine Metal Powder Co., Ltd., Xuyi, China.

Preparing and Testing the MRSPs.
The dilatant compound as matrix and different volume fractions of CI particles as fillers were homogeneously mixed by a two-roll mill (Nantong Hailite Rubber Machinery Inc., China, model XK-160) at room temperature.For mechanical mixing method, up to six different volume fractions had been considered in this study: 0, 6.98, 10.11, 15.84, 20.80, and 27.29% and the prepared samples were marked as MRSP-0, MRSP-1, MRSP-2, MRSP-3, MRSP-4, and MRSP-5 in sequence.For particle filled materials like MRSPs, dispersion quality is crucial in the rheological properties [11].Therefore, in this work, Hitachi S4800 scanning electron microscope (SEM) was used to observe the internal microstructure of MRSPs.MRSP sample and the SEM image are displayed in Figure 1.It can be obtained that CI particles are uniformly dispersed in Silly Putty matrix Static mechanical properties including creep and stress relaxation and dynamic rheological properties were carried out using a commercial rheometer (Physica MCR 302, Anton Paar Co., Austria).During the testing procedure, a parallel plate PP20 with a diameter of about 20 mm was used and the gap of 1 mm was kept all the time.At the same time, a controllable magnetic field was generated by an external coil.Besides, all the testing samples were kept at approximately the same volume value.Most of all, as a kind of rate-dependent material, MRSPs exhibit obvious transition from polymeric fluids on the condition of static state or low shear rate to solidlike polymers under high shear rate.Furthermore, the applied shear strain was very small, especially for dynamic testing, the shear strain was only 0.1%.So, no matter for polymeric fluid state or solid-like state of MRSPs, the fracture or wall-slip effects were not considered in this work.

Creep and Relaxation Behavior of MRSPs. Figures 2(a)
and 2(b) show the creep characteristic of MRSP-4 on the condition of different initial stress and zero magnetic field at room temperature.Creep phase (0-300 s) and recovery phase (300-900 s) are explicitly demonstrated in Figure 2(a).MRSP presents more obvious deformation (maximum strain 20.6% of 300 Pa) but lower creep compliance with higher initial stress as shown in Figure 2(b).In the creep phase, creep compliance () of MRSP increases with time gradually.
Figures 3(a) and 3(b) show the effect of different external magnetic field strength on creep characteristic of MRSP-4 with the initial stress 300 Pa.It proves that external magnetic field strength can substantially confine creep deformation and creep compliance of MRSP.With an external magnetic field of 23.9 kA/m, the maximum strain sharply decreases to only 0.375% comparing with the value 20.6% of zero magnetic field.That is because CI particles arrange in chain structure with external magnetic field, thus inducing the generation of magnetoinduced modulus which contributes to the reciprocal of creep compliance ().
Relaxation modulus of MRSP-4 as a function of time under different initial strain and external magnetic field strength is displayed in Figures 4(a) and 4(b), respectively.Without external magnetic field, MRSP exhibits obvious relaxation behavior especially under larger initial strain.Taking initial strain of 3.5% as example, relaxation modulus falls from 507 kPa to 95.8 kPa in less than one second.Besides, as shown in Figure 4(b), external magnetic field can improve relaxation modulus but delay the starting time of relaxation behavior.Interestingly, with the increasing external magnetic field strength, relaxation modulus increases with time firstly and then decreases to generate relaxation.When the external magnetic field strength reaches 156 kA/m, the relaxation would not generate until time of 1 s.

Shear Stiffening Property of MRSPs.
The MRSP exhibits apparent temperature dependent property.The response of storage modulus   to a change in shear angular frequency  for MRSP-4 at different temperature is shown in Figure 5.The angular frequency is applied from 1 to 100 rad/s.When the angular frequency is increased, the shear rate on the MRSP also increases.As the temperature decreases, the storage modulus increases more rapidly within a certain angular frequency (0-20 rad/s) that demonstrates excellent shear stiffening property.To quantify the shear stiffening property, absolute shear stiffening effect (ASTe) and relative shear stiffening effect (RSTe) referring to characterization of magnetorheological effect [6] are introduced in (1).  max is the maximum storage modulus of MRSP-4 induced by the shear angular frequency and   min is the initial storage modulus.The details about MRSP-4 at different temperature are listed in Table 2: MRSP-4 displays maximum ASTe 0.805 MPa at 10 ∘ C and maximum RSTe 6600% at 60 ∘ C as shown in Table 2. Due to the larger initial storage modulus at lower temperature, RSTe of MRSP increases with increasing temperature, but the ASTe is on the contrary.stress.On the condition of no external magnetic field, CI particles only play a role in reinforcing the Silly Putty matrix and enhance the storage modulus of MRSPs.The storage modulus of the MRSP-4 as a function of angular frequency at different magnetic field strength is shown in Figure 6(b).The shear stiffening property can be controlled through the external magnetic field due to the formation of magnetic chains of CI particles in MRSP.All the relevant details to quantify the shear stiffening effect of MRSP from Figures 6(a) and 6(b) are listed in Table 3.
Pure Silly Putty MRSP-0 displays maximum RSTe and MRSP-5 with CI volume fraction of 27.29% exhibits maximum ASTe as shown in Table 3. Due to the reinforcing effect of CI particles, more soft magnetic fillers can induce larger  initial storage modulus at 0.1 rad/s which contribute to the descending RSTe.Meanwhile, when the external magnetic field strength is 23.9 kA/m, the ASTe of MRSP-4 with CI volume fraction of 20.8% reaches to 1.01 MPa.But as the magnetic field strength is increased, both ASTe and RSTe decrease.Therefore, it can be concluded that the state of larger magnetic field strength is not conducive to the shear stiffening property of MRSP, but lower magnetic field strength can improve ASTe of MRSP.Overall, MRSP is a dilatant material where the viscosity increases faster than the strain rate.There are two mechanisms (and hence two characteristic time scales) at work in this material [1,12].The high molecular weight PDMS has a characteristic polymeric relaxation time  relax .However due to the boric acid there are also transient boron mediated "crosslinks" arising from associating boron linkages.These act to give the MRSP a behavior more like an elastic solid than a liquid.However since these "crosslinks" are dynamic (with a characteristic time  assoc , i.e., much shorter than  relax ) the material is not permanently locked in place and can consequently flow under the correct conditions.Therefore at time scales longer than  assoc the MRSP behaves like a high molecular weight polymeric fluid (with a characteristic relaxation of  relax ).Over time scales much shorter than  assoc MRSP behaves like a crosslinked elastic solid.
The mechanisms of shear stiffening behind MRSPs are very complex, but our goal here is to discover a relatively simple function to accurately fit the storage modulus versus angular frequency at different temperature, CI contents, and external magnetic field strength.Galindo-Rosales et al. [13,14] used an apparent viscosity function with 11 parameters to describe the three characteristic regions of shear thickening fluid (STF), including the slight shear thinning at low shear rates, followed by a sharp increase in viscosity over the critical shear rate, and a subsequent, but pronounced, shear thinning region at high shear rates.Furthermore, Tian et al. [15] proposed a modified model shown in (2) to overcome the drawback of 11 parameters function that if any  1 ,  2 , or  3 was a decimal, the value in the bracket must be positive; otherwise the equation failed: A new function is proposed in (3) through imitating part of (2) to describe the shear stiffening property of MRSPs.In (3),   min represents the storage modulus at initial experimental angular frequency  min ;   max represents the maximum storage modulus at final experimental angular frequency  max ; parameter  refers to the slope of the shear stiffening behavior in log-log plot; parameter  is responsible for the ratio of changed storage modulus induced by frequency to the initial storage modulus (RSTe); finally, parameter  is introduced to overcome the shortcoming that the final experimental point ( max ,   max ) in curves cannot be fitted:  Figure 7 shows the fitting curves of all the angular frequency scanning testing data on the condition of different temperature, CI contents, and external magnetic field strength.It is obvious that (3) fits to the experimental results perfectly.Levenberg-Marquardt algorithm (LMA) is applied in all the fitting procedures.This algorithm combines the Gauss-Newton and steepest descent methods to obtain the values of the parameters by an iterative chi-square minimisation technique.Fitting is considered to have converged when the difference between the  2 values obtained in two successive iterations is smaller than a given tolerance, chosen here as 10 −10 .
Table 4 lists all the fitting values of parameters in (3).It is observed from related coefficient  that a high fitting precision is obtained in each fitting result.From each section, parameter  is inversely proportional to the value of RSTe; interestingly, when there is an external magnetic field applied (Section III), due to higher initial storage modulus and lower changed storage modulus, parameter  increases with the value magnetic field strength obviously.Parameter  reflects the slope of shear stiffening behavior.In log-log plot, the slope difference in each section is not very distinct which leads to tiny change or even the same values of parameter .Besides, parameter  is relatively independent of experimental data that means the value changes a little in each section.

Magnetorheological Property of MRSPs.
MRSP is also a novel magnetorheological material whose mechanical properties are highly dependent on the externally applied magnetic field.Figure 8 shows the response of storage modulus to a change in magnetic flux density for MRSP-4 at different temperature.The angular frequency applied on MRSP-4 is 10 rad/s and magnetic flux density increases from 0 T to 0.894 T. Similar to its shear stiffening property, magnetorheological effect of MRSP is also influenced by temperature.Meanwhile, absolute magnetorheological effect (AMRe) and relative magnetorheological effect (RMRe) are introduced in (4) to quantify the magnetorheological property.  max is the maximum storage modulus of MRSP-4 induced by the external magnetic field and   min is the initial storage   modulus.The details obtained from Figure 8 are listed in Table 5: The overall trends displayed in Table 5 are similar to those presented in Table 2.When the temperature is increased, RMRe of MRSP-4 enhances significantly, but the AMRe is on the contrary.Higher temperature can soften the Silly Putty matrix, thus inducing the lower initial storage modulus and less resistance to the regular arrangement of CI particles.Figures 9(a) and 9(b) show the magnetic field-dependent storage modulus of MRSPs with different CI contents at angular frequency 20 rad/s and 100 rad/s, respectively.These experimental curves demonstrate that there is a tendency for the storage modulus of MRSPs to increase with the increase of the magnetic flux density firstly and then saturate with further increases of the magnetic flux density.CI content is observed from Table 6 to have direct influence on AMRe of MRSPs,  which means that higher CI content induces greater AMRe.The magnetoinduced storage modulus is mainly attributed to the particle chains formed under external magnetic field.However, greater AMRe is not certain to keep up with higher RMRe.For example, when the applied angular frequency is 20 rad/s, AMRe of MRSP-4 is 1.614 MPa less than that of MRSP-5, but RMRe of MRSP-4 reaches to maximum value of 216.35%.Moreover, it can also be concluded that magnetorheological effect of MRSPs on lower shear rate state is greater than that of higher shear rate state.

Modelling on Magnetoinduced
Behavior.Before applying external magnetic field, CI particles are homogeneously dispersed in Silly Putty matrix, but when there is an external magnetic field applied, particles are arranged in chain structure instantaneously as shown in Figure 10.In the chain structure, CI particles are simplified to spherical particles with the radius  and the additional acting force between two particles caused by magnetic field leads to the increment of the storage modulus.However, the viscosity of Silly Putty matrix is much larger than general liquids which can cause tremendous obstruction to the formation of pulling-through particle chains.So a nonperforative particle aggregated chains model is displayed in Figure 10 to explain the magnetoinduced mechanism.In the model, MRSP is divided into several square columns with the length of side 2 and the height ℎ.At zero magnetic field strength, particles are randomly dispersed in each square column and MRSP exhibits isotropy.When a low magnetic field is applied, particles aggregate to several apart chains along the direction of magnetic field in each square column with different length   and the total length of the chains included in a square column can be marked as ∑   .The chains in parallel are sparse due to the low magnetic field strength.However, when the magnetic field strength increases, the aggregated chains in each square column turn unstable and a small disturbance (shearing) may drift some particles in a square column away to aggregate to another apart chain in a new square column.So the chains in parallel become close and even to form particle columns extremely, but the total chains length in a square column becomes short.
For the square columns, the effective permeability  sceff can be calculated by using Maxwell Garnett mixing rule as [16,17] where  CI and  SP are the relative permeability of CI particles and Silly Putty matrix, respectively. is the particles volume fraction in each square column.In the granular basic unit, the volume fraction  can be calculated as  = (4/3) 3 /(2) 3 = /6.So  can be expressed as below: where ∑   /ℎ is relevant to external magnetic field strength.Besides, if  sc is the volume fraction of the square columns in MRSP and can be expressed as where   is the particles volume fraction in MRSP.So, the effective permeability  eff along the direction of square columns can be calculated as follows according to the parallel connection rule [18,19] When shear strain is applied on MRSP under the external magnetic field, the relative permeability of MRSP changes which can induce the generation of additional force, that is, magnetoinduced shear stress.The magnetoinduced shear stress can be expressed as where  0 is the vacuum permeability,  is the shear strain, and  0 is the external magnetic field.The shear strain can be marked as The magnetoinduced shear stress can be obtained by substituting ( 8) and ( 10) into (9) as Therefore, the storage modulus   can be calculated as below: Furthermore, the relative permeability of CI particles  CI is the function of external magnetic field strength, which can be obtained from Frohlich-Kennelly equation [20]:  CI ( 0 ) = 1 + ( 0 − 1)   / 0 ( 0 − 1) +   / 0 , where  0 is the relative permeability of CI particles at zerofield strength and   is the magnitude of the saturation magnetization.
Besides, the variable ∑   /ℎ can be seen as the function of external magnetic field strength, and the possible function can be expressed as below, and , , and  are the parameters:

Figure 1 :Figure 2 :
Figure 1: Prepared MRSP sample and SEM image of internal microstructure.

Figure 6 (Figure 3 :Figure 4 :
Figure 3: The creep characteristic of MRSP-4 on the condition of different external magnetic field strength and the same initial stress 300 Pa: (a) creep and recovery phase and (b) creep compliance in creep phase.

Figure 5 :
Figure 5: The effect of temperature on shear stiffening property for MRSP-4.

Figure 6 :
Figure 6: The storage modulus of MRSPs as a function of angular frequency at 25 ∘ C (a) different CI contents and (b) different external magnetic field strength for MRSP-4.

Figure 7 :
Figure 7: Fitting curves of all the experimental data.

Figure 8 :
Figure 8: The effect of temperature on magnetorheological property for MRSP-4.

Figure 9 :
Figure 9: The storage modulus of MRSPs as a function of magnetic flux density at 25 ∘ C (a) angular frequency 20 rad/s and (b) angular frequency 100 rad/s.

Figure 10 :
Figure 10: Schematic diagram of chain-formation mechanism for MRSPs under magnetic field and the nonperforative particle aggregated chains model.
Dow Corning 3179 Dilatant Compound purchased from Dow Corning Co., Ltd. was used as the Silly Putty matrix.The composition of the silicone material was as outlined in Table 2.1.Materials.

Table 1 :
The composition of Silly Putty matrix.

Table 2 :
max ,   min , ASTe, and RSTe% of MRSP-4 in the angular frequency scanning test at different temperatures.

Table 4 :
All the fitting values of parameters.

Table 6 :
max ,   min , AMRe, and RMRe% of MRSPs with externally applied magnetic field at different angular frequencies.