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A nonlinear constitutive model by spring, fractional derivative and modified bounding surface model to represent the amplitude, frequency and the magnetic dependency for Magneto-sensitive rubber
Journal of Sound and Vibration 438 (2019) 344–352
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Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
A nonlinear constitutive model by spring, fractional derivative and modified bounding surface model to represent the amplitude, frequency and the magnetic dependency for Magneto-sensitive rubber Bochao Wang ∗ , Leif Kari KTH Royal Institute of Technology, The Marcus Wallenberg Laboratory for Sound and Vibration Research (MWL), Teknikringen 8, 100 44 Stockholm, Sweden
article info
abstract
Article history: Received 2 March 2018 Revised 23 August 2018 Accepted 10 September 2018 Available online 14 September 2018 Handling Editor: Ivana Kovacic
Magneto-sensitive (MS) rubber is a kind of smart material mainly consisting of magnetizable particles and rubber. Inspired by experimental observation that the shear modulus for MS rubber is strongly dependent on amplitude, frequency and magnetic field; while the impact for the magnetic field and strain to the loss factor is relatively small, a new nonlinear constitutive model for MS rubber is presented. It consists of a fractional viscoelastic model, an elastic model and a bounding surface model with parameters sensitive to the magnetic field. To our knowledge, it is the first time that the bounding surface model is incorporated with the magnetic sensitivity and used to predict the mechanical properties for MS rubber. After comparison with the measurement results, it is found that the shear modulus and the loss factor derived from the simulation fit well with the experimental data. This new constitutive model with only eight parameters can be utilized to describe the amplitude, frequency and the magnetic field dependence for MS rubber. It provides a possible new way to understand the mechanical behavior for MS rubber. More importantly, the constitutive model with an accurate prediction property for the dynamic performance of MS rubber is of interest for MS rubber applications in noise and vibration reduction area. © 2018 Elsevier Ltd. All rights reserved.
Keywords: Magneto-sensitive rubber Bounding surface model Frequency dependency Amplitude dependency nonlinear
1. Introduction Vibration is a common phenomenon in the world. For example, vibration exists in mechanical devices under operating and it occurs when structures are subjected to dynamic external loadings. In most cases, vibration is unwanted because it reduces the working life of machine, wastes energy and may cause noise. One commonly used way to reduce vibration is to install some vibration isolators for the target objects. Generally, vibration isolators are made of rubber, which means that the dynamic properties of vibration isolators is fixed once mounted. Therefore, it is impossible to change its characteristics to adapt to various excitation conditions. In that case, when the frequency for the excitation is close to the critical frequency for the system, the vibration attenuation effect of the vibration isolators would be greatly reduced and can even collapse because of frequency mis-tuning problems.
∗ Corresponding author. E-mail address: [email protected] (B. Wang).
https://doi.org/10.1016/j.jsv.2018.09.028 0022-460X/© 2018 Elsevier Ltd. All rights reserved.
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However, magneto sensitive (MS) rubber, a kind of smart material, may find a solution for the mis-tuning problem. It mainly consists of polarized iron particles embedded in a rubber matrix. Under a magnetic field, the mechanical characteristic of MS rubber can be modified rapidly, reversibly and continuously. MS rubber is categorized into either anisotropic or isotropic according to the presence or absence of the magnetic field in the vulcanization process. If MS rubber is used for the fabrication of vibration isolators, an increased vibration reduction effect may be obtained by the stiffness changing capacity of MS rubber. Research on MS rubber started in the late 1990s by Jolly et al. [1] and Ginder et al. [2]. For the last two decades, more researchers have begun the study on MS rubber. The research on MS rubber can be mainly divided into four categories; those are: fabrication, testing, application and modeling. For the fabrication of MS rubber, a wide range of methods aimed at obtaining MS rubber with high MS effect are used. By finite element modeling, Davis [3] predicted that the optimum particle volume fraction for MS rubber to obtain the largest change in modulus is 27%. The result was then verified experimentally [4]. For anisotropic MS rubber, the effects of matrix [5], coupling agent, content of iron particles, plasticizers [6] pre-structure [7] and surfactant [8] on the MR effect were considered. Muniain et al. [9] studied the influence of carbon black and plasticizers on the dynamic properties of isotropic MS natural rubber. Recently, Stanier and his coworkers [10] chose a new method for the fabrication of MS rubber. In their study, nickel-coated carbon fiber was embedded in elastomer and a magnetic field is applied during the vulcanization to obtain a transversely isotropic composite. Subsequently, they investigated the effect of the orientation of the nickel-coated fiber to the magneto mechanical behavior of MS rubber. For the testing of the mechanical properties for MS rubber, the influence of the magnetic field, frequency and temperature on the dynamic stiffness [11,12] and the loss factor [13] of isotropic MS rubber have been studied. In parallel, large efforts have been made to investigate the impact of magnetic field, frequency [14], temperature [15] on the viscoelastic properties of anisotropic MS rubber. Among all the investigations related to testing of MS rubber, Blom and Kari [16] investigated the Fletcher Gent effect [17] of the isotropic MS rubber in a wide frequency range and it revealed that in addition to viscoelastic properties, there is a strong amplitude dependency of MS rubber even for a small strain. The magnetic field sensitivity property of MS rubber makes it attractive for the potential application in semi-active vibration control field. For example, Deng and Gong [18] designed a MS rubber based dynamic vibration absorber. The research result revealed that the adaptively tuned vibration absorber comprised of MS rubber is efficient for tracking the excitation frequency and reducing the vibration of the primary structure. Jung et al. [19] fabricated a base isolation system of MS rubber and installed it on a scaled building structure to investigate the seismic performance of that smart base isolation system. The result showed that compared with conventional base isolation systems, the base isolation system made of MS rubber outperforms in reducing the vibration of the building structure. Similar research was done [20–22] suggesting the feasibility of using MS rubber for improving the vibration attenuation effect and staggering the natural frequency of structure from the loading frequency. Other application of MS rubber in vibration control area includes exploring the potential of using adaptive sandwich beams filled with MS rubber to reduce the vibration of vibrating structural elements like helicopter blades and aircraft wings. For instance, Nayak et al. [23] proposed the governing equation for free vibration of a sandwich beam filled with MS rubber under various boundary conditions. Aguib et al. [24] discussed the effect of different magnetic fields and different percentage of ferromagnetic particles on the vibration response of MS rubber sandwich plates. Furthermore, Dyniewice et al. [25] validated the feasibility of reducing the vibration of a sandwich beam partially filled with MS rubber by changing the magnetic field. There are a number of models developed for the magnetic related mechanical properties of anisotropic MS rubber. Primarily, Jolly et al. [1] proposed a dipole model to represent the magnetic related mechanical properties of the anisotropic MS rubber. Later, Zhu et al. [26] augmented that model by considering the influence of adjacent chains. Dorfmann and Ogden [27–29] derived the governing equations for the deformation of MS rubber by combing Maxwell equations with continuum mechanics and Itskov et al. [30] then extended the model by introducing the mathematically poly-convexity notion. Unfortunately, those models are only valid for the quasi-static case due to the neglect of the viscoelastic effects for MS rubber. To model the viscoelastic behavior of anisotropicMS rubber, a Bouc-Wen and a Kelvin-Voigt model were aligned in parallel to represent the viscoelastic behavior of MS rubber [19]. The main drawback of that model is the high computational cost for identifying the material parameters. To solve that problem, a modified Kelvin-Voigt model for which the parameters in stiffness and damping element are all related to the magnetic field, frequency and strain were proposed [31]. However, the agreement between the simulation and experiment result for the loss factor needs to be improved. More importantly, for all the models mentioned above, the measurements are performed based on the anisotropic MS rubber where the iron particles are aligned in chains. For isotropic MS rubber, the distribution of iron particles and the mechanical behavior are completely different. Based on that physical fact, Rudykh et al. [32,33] investigated the effect for distribution of magnetizable particles on the magneto-mechanical coupling effect and stability for MS rubber. The Fletcher-Gent effect, which represents the dependence of the modulus on the amplitude of the applied strain for MS rubber was generally omitted for the model mentioned above. However, according to the research by Blom and Kari [16], the amplitude dependent stiffness behavior is an important feature for MS rubber and should not be ignored. To describe the dynamic behavior of MS rubber completely, a model which incorporated the Fletcher-Gent effect for MS rubber is needed. In 2011, a nonlinear constitutive model that includes the effect of strain, frequency and magnetic field for the mechanical behavior for isotropic MS rubber was proposed by Blom and Kari [34]. In their model, a fractional derivative was used to describe the viscoelastic properties and a smooth frictional stress model was adopted to describe the Fletcher-Gent effect for MS rubber. In fact, there are other methods that are capable of capturing the Fletcher-Gent effect for filled rubber. One possible way is to use the bounding surface model, which was proposed by Dafalias and Popov [35] in 1977. The innovation of the bounding surface model is that the plastic modulus could vary continuously and the specialized case for bounding surface model could be
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used for rate sensitive plastic materials. Since then, the bounding surface model has been widely used to simulate the behavior of cohesive soils [36]. In 2014 a model [37] that tried to capture the Fletcher-Gent effect [17] of filled rubber by using the bounding surface model was presented. The accurate simulation result revealed that the bounding surface model is ideal for modeling the amplitude dependency of filled rubber. However, for the work done by Österlöf et al. [37], only frequency dependency and amplitude dependency for filled rubber were modeled. For MS rubber with iron particles, in addition to frequency and amplitude dependency, there are also the magnetic dependency. Inspired by the work of Refs. [34] and [37], a new constitutive model for MS rubber is developed in this paper. The novelty of this model is that it incorporates the bounding surface model with magnetic dependency to model the magnetic field-dependent amplitude dependence of MS rubber. Compared with the smooth frictional stress model presented by Blom and Kari [34], the bounding surface model used in this paper has the advantage of taking irregular cyclic loading in the three dimensional case into consideration. Thus, for the model developed in this paper, further extension to three dimensions is available. Comparison with the experimental result reveals that this newly built model could well replicate the magnetic, strain amplitude and frequency dependent properties for MS rubber. For the possible application of MS rubber in vibration controlling area, if the shear modulus and the loss factor of MS rubber are replicated well, the response of the anti-vibration components made from MS rubber could be predicted more accurately and the uncertainty for the model could be reduced considerably. The constitutive model presented in this paper for MS rubber plays a significant role in the precise implementation of the control strategy and in obtaining a satisfactory vibration reduction effect for the probable use of MS rubber in adaptive vibration control areas. 2. Experiment result The ingredients for the MS rubber used in this paper are based on Blom and Kari [16]. During vulcanization, no external magnetic field is applied resulting in a random distribution of iron particles in the rubber, generating isotropic MS rubber. The experimental studies about the mechanical behavior of MS rubber were done by Lejon and Kari [38]. For the details about the measurement setup and method, the reader is referred to Ref. [38]. In this paper only room temperature is considered, thus measurement results with only 20 ◦ C were chosen for the subsequent parameter identification. MS rubber works in shear deformation mode. Four levels of the applied magnetic field with values of 0, 0.3, 0.55 and 0.8 T are considered. The dynamic shear strains are 0.0005, 0.0015 and 0.005, respectively. The excitation signal is a stepped sine excitation with a frequency interval of 10 Hz ranging from 200 to 900 Hz. The test results for the magnitude and loss factor of the shear modulus versus frequency with different amplitude of strains are shown in the solid lines from Figs. 3–5. It should be noted that the “bumps” for the measurement magnitude and loss factor of the shear modulus in Figs. 3–5 are caused by the resonance of the test machine. 3. Constitutive modeling 3.1. General models The rheological model developed in this paper is shown in Fig. 1. The total stress is assumed to consist of three parts dependent on time t
𝜏 = 𝜏e + 𝜏ve + 𝜏f ,
(1)
where 𝜏 , 𝜏 e , 𝜏 ve and 𝜏 f are total, elastic, viscoelastic and frictional shear stress, respectively. The elastic shear stress is obtained by
𝜏e (t) = Ge · 𝛾 (t) ,
(2)
where 𝛾 (t ) is the shear strain and Ge is the elastic shear modulus. The viscoelastic shear stress is described by a relaxation convolution integral
𝜏ve (t) =
t
b d 𝛾 (s) ds, Γ(1 − a) dt ∫0 (t − s)a
(3)
where a and b are the parameters in the relaxation convolution integral, Γ is the gamma function and 𝛾 (s) = 0 for s < 0. The frictional shear stress is described by the bounding surface model. Normally, for bounding surface model, a bounding surface F and a yield surface f are included [35]. Those are defined as f (𝜎ij − 𝛼ij ) = 0
(4)
F (𝜎 ij − 𝛽ij ) = 0,
(5)
and
with a bar over 𝜎 ij indicates stresses on the bounding surface. The symbols 𝛼 ij and 𝛽 ij are back-stresses which serve as the center of the two surfaces, respectively.
B. Wang and L. Kari / Journal of Sound and Vibration 438 (2019) 344–352
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Fig. 1. Rheological model for MS rubber.
During loading, these two surfaces may move, change size and even contact with each other with the limitation that the bounding surface always encircles the yield surface. The key idea for the bounding surface model is that the plastic modulus is determined by the distance between the two surfaces. For the one dimensional case, with a yield surface of vanishing size, the evolution law simplifies to
𝜏̇ f = H𝛾̇ = 𝛼̇
(6)
̇ 𝛽̇ = Hp 𝛾,
(7)
and
with H being the plastic modulus for the yield surface f and Hp being the plastic modulus for the bounding surface F. The symbols 𝛼 and 𝛽 represent the back-stresses for the yield surface and bounding surface in one dimensional case, respectively. A dot over 𝛾 represents the change of strain for new time step. The expression for the plastic modulus is determined according to Österlöf et al. [37], that is
(
H(𝛿, 𝛿in ) = Hp
1+
𝛿
𝛿in − 𝛿
)
,
(8)
where 𝛿 is the current distance to the bounding surface and 𝛿 in is the distance to the bounding surface at the previous turning point. 3.2. Numerical calculation method In the following, the value of variables at time step t = nΔt is denoted by (·)n Δ t where Δt is the constant time step and n ∈ ℕ. The subscript (·)(n +1) Δ t represents the next time step. For the calculation of the elastic shear stress, only the current state of strain is needed. To calculate the viscoelastic shear stress, after using the truncation definition for fractional differentiation [39], the viscoelastic shear stress becomes
𝜏ve[(n+1)Δt] =
n b ∑ cj (a)n+1−j 𝛾(n+1−j)Δt (Δt )a
(9)
j=0
and c0 (a) = 1, cj (a) =
j−1−a cj−1 (a). j
(10)
To calculate the frictional shear stress, the Euler forward method is used. Based on the work by Österlöf et al. [37], firstly, a trial function 𝜏f∗ which corresponds to the change of shear stress in each time step is calculated,
𝜏f∗[(n+1)Δt] = HnΔt 𝛾̇ [(n+1)Δt] , the parameter Hn Δ t represents the plastic modulus at time t = nΔt.
(11)
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Secondly, the variable l which determines the loading direction for the new step is obtained by
𝜏 ∗ f[(n+1)Δt]
l[(n+1)Δt] = | |𝜏 ∗
|
|.
(12)
f[(n+1)Δt] ||
If l[(n+1)Δt] · l(nΔt) > 0, the loading direction and the distance to the bounding surface remains the same. If l[(n+1)Δt] · l(nΔt) < 0, it means that the direction of the load has changed. If the direction of the loading has changed, the internal state variables (𝛿 and 𝛿 in ) update according to
𝛿in[(n+1)Δt] = Syield l[(n+1)Δt] − (𝜏f(nΔt) − 𝛽(nΔt) )
(13)
𝛿[(n+1)Δt] = Syield l[(n+1)Δt] − (𝜏f(nΔt) − 𝛽(nΔt) + 𝜏f∗[(n+1)Δt] ),
(14)
and
with 𝜏f (nΔt) means the frictional stress calculated at time t = nΔt. The symbol Syield is the initial distance from the vanishing sized yield surface to the bounding surface. The plastic modulus updates as
( H[(n+1)Δt] = Hp
1+
𝛿[(n+1)Δt] 𝛿in[(n+1)Δt] − 𝛿[(n+1)Δt]
)
.
(15)
Lastly, the stresses for the two surfaces updates as
𝜏̇ f[(n+1)Δt] = H[(n+1)Δt] 𝛾̇ [(n+1)Δt] ,
(16)
𝛽̇ [(n+1)Δt] = Hp 𝛾̇ [(n+1)Δt] ,
(17)
𝜏f[(n+1)Δt] = 𝜏f(nΔt) + 𝜏̇ f[(n+1)Δt]
(18)
𝛽[(n+1)Δt] = 𝛽(nΔt) + 𝛽̇ [(n+1)Δt] ,
(19)
and
where 𝜏 f(n Δ t) and 𝛽 (n Δ t) represent the value at time t = nΔt. 3.3. Magnetic field sensitivity for the model Based on the experimental observations by Lejon and Kari [38] and utilizing the control variates method, the magnetic sensitivity term for the three sub-models were determined as follows. Firstly, when the frequency is constant, the dynamic properties of MS rubber reflects a magnetic sensitivity and the magnetic sensitivity is strongly correlated with the amplitude dependency. It suggests that the amplitude dependent parameters in the bounding surface model should include the magnetic sensitivity. Secondly, if strain amplitude is fixed, the variation for the magnitude for shear modulus between zero magnetic field and maximum magnetic field at different frequency seems to be approximately constant. It means that there is no magnetic dependency for the parameters a and b in the viscoelastic model. Lastly, the loss factor, which is the quotient between the loss modulus and the storage modulus, does not fluctuate considerably for different magnetic field. It means that the changing of the parameters for the bounding surface model with related to magnetic field should be balanced by the change of the elastic modulus Ge with related to magnetic field. In summary, only three parameters in the model which are Ge , Syield and Hp should contain the magnetic sensitivity term. According to the nonlinear theory of magnetoelastic interaction developed by Dorfmann and Odgen [27–29], in one dimensional case, the magneto-sensitivity for shear modulus is quadratically dependent on the magnetic field applied. Therefore, the increase of Ge should be proportional to the square of the magnetic field. Furthermore, experimental observation has shown that the dissipated energy of MS rubber increases with the magnetic field, suggesting that for the bounding surface model, the values of Syield and Hp should positively related to the magnetic field. Similarly, a square of M is chosen to reflect the magnetic dependency of Syield and Hp . Thus, the three parameters reflecting the magnetic sensitivity are explicated as
[ Ge =
Syield =
(
1+
M Ms
[
(
1+
]
)2
M Ms
𝛿1 Ge0 , )2
(20)
]
𝛿2 Syield0
(21)
B. Wang and L. Kari / Journal of Sound and Vibration 438 (2019) 344–352
and
[ Hp =
(
1+
M Ms
)2
349
]
𝛿3 Hp0 ,
(22)
where Ge0 , Syield0 and Hp0 are the state values of MS rubber when magnetic field is zero. The parameter Ms represents the saturation magnetic field which is the upper limit for the applied magnetic field. Parameters 𝛿 1 , 𝛿 2 and 𝛿 3 are material constant that need to be decided. 4. Result and discussion A nonlinear least squares fit method is implemented to obtain the parameters for zero magnetic field. The special function “lsqnonlin” was used for the parameter identification in MatlabⓇ (MATLAB Release 2015b, The MathWorks, Inc., Natick, Massachusetts, United States). Considering the computational cost, the measurement values for the shear modulus at four levels of shear strain amplitudes with frequency ranging from 200 to 900 Hz with a frequency interval of 100 Hz used. The objective function (squared absolute error) for the nonlinear square fit method is obj =
n {[ ∑
]
[
G∗meas (i) − G∗sim (i) · conj G∗meas (i) − G∗sim (i)
]}
,
(23)
i=1
where G∗meas (i) and G∗simu (i) are the ith measurement and simulation dynamic shear modulus for MS rubber. The symbol conj [·] is the complex conjugate. The objective function is minimized to obtain the material parameters for MS rubber. The relative error is evaluated as
√ {[ ] [ ]} √ √ n √∑ G∗meas (i) − G∗sim (i) · conj G∗meas (i) − G∗sim (i) {[ ∗ ] [ ]} . err = √ Gmeas (i) · conj G∗meas (i) i=1
(24)
In order to ensure the accuracy for the modeling, during the simulation phase, ten rounds of loading with the time increment Δt = (27 fexc )−1 are applied, where fexc is the frequency for the harmonic loading. In order to identify the magnitude and loss factor for the dynamic shear modulus under steady motion state, only the last six rounds of simulation data were used. The numerical calculation is based on the Euler forward method, therefore, the maximum truncation global error for the simulation is proportional to the time size (3.9063 × 10− 5 s), corresponding to the case where the frequency of the external load is 200 Hz. After halving the time step, it is found that the identified material parameters for the constitutive model of MS rubber differ only in the fourth digit, which means that the set time step meets the accuracy requirement for the simulation. After parameter identification and optimization, the parameters under zero magnetic field case are obtained as Ge0 = 3.040 × 106 Nm− 2 , a = 0.2999, b = 0.2679 × 106 Nsa m− 2 , Syield0 = 0.2170 × 105 Nm− 2 and Hp0 = 0.2002 × 106 Nm− 2 . The absolute error under zero magnetic field case with frequency ranging from 200 to 900 Hz for 24 sets of experimental data is 1.1632 × 106 Nm− 2 . The relative error for the novel model fits to the experimental result in zero magnetic field case is 3.294%. The results are shown in Fig. 2, where the magnitude and the loss factor of shear modulus versus frequency are plotted. As described above, the peaks for the magnitude and the loss factor of the shear modulus of MS rubber are due to the influence of the resonance for the measurement setup. Solid lines and symbols represent the measurement results and simulation result, respectively in Fig. 2. Apparently, the coincidence between experiments and modeling results is very good. After obtaining the material parameters for MS rubber at zero magnetic field state, the second round of least square fit method is applied to obtain the parameters representing the magnetic sensitivity of MS rubber. In this case, four levels of magnetic field (0, 0.3, 0.55 and 0.8 T) accompanying different frequency and shear strains are used. The four magnetic related parameters are obtained as 𝛿 1 = 0.2248, 𝛿 2 = 0, 𝛿 3 = 0.7670 and Ms = 0.5615 T. The absolute error for four different kind of magnetic field case with frequency ranging from 200 to 900 Hz is 1.1636 × 106 Nm− 2 . The relative error for the novel model fits
Fig. 2. The magnitude and loss factor of the dynamic shear modulus without the effect of magnetic field. Solid lines are experimental data, symbols are simulations. Three kinds of strain (0.0005, 0.0015, 0.005) were considered.
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Fig. 3. The magnitude and loss factor of the dynamic shear modulus with shear strain equals to 0.0005. Solid lines are experimental data, symbols are simulations.
Fig. 4. The magnitude and loss factor of the dynamic shear modulus with shear strain equals to 0.0015. Solid lines are experimental data, symbols are simulations.
to the experimental result in zero magnetic field case is 3.296%. Interestingly, it is found that there is no effect of magnetic field on the parameter Syield for the bounding surface model. The application of the magnetic field only makes the plastic modulus in the bounding surface model stiffer. The comparison between the measurement data and the simulation results are shown from Figs. 3–5 with shear strain amplitude 0.0005, 0.0015 and 0.005, respectively. Figs. 3–5 reveal that the simulation results of the shear modulus fit well with the measured shear modulus. The tendency is that the shear modulus increases with the applied magnetic field and the shear modulus become stiffer for smaller shear strain. This tendency is similar to the measurement observation, which shows that the modified bounding surface model depicts the magnetic sensitive plastic properties of MS rubber very well. For the modeling of the loss factors, the modeled results resemble the experiment accurately. In Figs. 6 and 7, the modeled hysteresis curves are compared with the corresponding measurement data with two extreme magnetic fields (0 and 0.8 T). The shear strain and frequency were chosen to be 0.005 and 200 Hz respectively in Fig. 6, which represents the minimum value of the shear modulus for MS rubber. The chosen shear strain and frequency were 0.0005 and 900 Hz in Fig. 7 respectively, resulting in the maximum value of the shear modulus for MS rubber. By comparing the hysteresis curve between experiment and simulation from Figs. 6 and 7, it is found that these two curves fit well with each other. At increased applied magnetic field, the slope of the hysteresis curve becomes steeper. Combined with the result in Figs. 3–5, it demonstrates that the derivation about the magnetic sensitivity in the developed model is reasonable.
Fig. 5. The magnitude and loss factor of the dynamic shear modulus with shear strain equals to 0.005. Solid lines are experimental data, symbols are simulations.
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Fig. 6. Comparison between measurement shear strain-stress curve (solid lines) and modeled shear strain-stress curve (symbols) with shear strain 0.005 and frequency 200 Hz at applied magnetic fields with values of 0 and 0.8 T.
Fig. 7. Comparison between measurement shear strain-stress curve (solid lines) and modeled shear strain-stress curve (symbols) with shear strain 0.0005 and frequency 900 Hz at applied magnetic fields with values of 0 and 0.8 T.
5. Conclusion A nonlinear model which depicts the dynamic strain amplitude, frequency and magnetic field dependency for MS rubber is developed. It is assumed that the viscoelastic properties only depend on frequency, irrespective of the magnetic field. Moreover, for the modeling of frictional properties of MS rubber, a modified bounding surface model with parameters sensitive to magnetic field is utilized. Due to the merit of bounding surface model that can take three-dimensional irregular circular loading into account, further extension of this nonlinear constitutive model to three-dimension and a more accurate and comprehensive prediction of the mechanical behavior for MS rubber would be accessible. After introducing the numerical calculation method, two rounds of non-linear least squares fits are applied to obtain the modeling material parameters. The analyzed simulation results are compared with measurement date and a good coincidence is reached. This model with only eight parameters (Ge0 , a, b, Syield0 , Hp0 , 𝛿 1 , 𝛿 3 and Ms ) can be used to predict the elasticity, viscoelasticity, plasticity and the magnetic sensitivity for MS rubber. For the possible application of isotropic MS rubber in vibration isolators with shear deformation, this model could be used to accelerate the design of component before real experimental testing. Furthermore, a constitutive model which precisely predicts the dynamic properties of MS rubber could be used to facilitate the potential application of MS rubber in vibration isolators and other devices where capacities for adaptive change of stiffness are required. Acknowledgments The authors gratefully acknowledge Dr. Jonas Lejon for providing the measurement data for MS rubber. In addition, the China Scholarship Council is gratefully acknowledged for the financial support through the KTH-CSC programme. References [1] [2] [3] [4] [5]
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Kari, Amplitude and frequency dependence of magneto-sensitive rubber in a wide frequency range, Polym. Test. 24 (2005) 656–662. [17] W.P. Fletcher, A.N. Gent, Nonlinearity in the dynamic properties of vulcanized rubber compounds, Rubber Chem. Technol. 27 (1954) 209–222. [18] H.X. Deng, X.L. Gong, Adaptive tuned vibration absorber based on magnetorheological elastomer, J. Intell. Mater. Syst. Struct. 18 (2007) 1205–1210. [19] H.J. Jung, S.H. Eem, D.D. Jang, J.H. Koo, Seismic performance analysis of a smart base-isolation system considering dynamics of mr elastomers, J. Intell. Mater. Syst. Struct. 22 (2011) 1439–1450. [20] Y.C. Li, J.C. Li, A highly adjustable base isolator utilizing magnetorheological elastomer: experimental testing and modelling, J. Vib. Acoust. 137 (2015) 011009. [21] J. Yang, S.S. Sun, T.F. Tian, W.H. Li, H.P. Du, G. Alici, M. Nakano, Development of a novel multi-layer mre isolator for suppression of building vibrations under seismic events, Mech. Syst. Signal Process. 70–71 (2016) 811–820. [22] R. Brancati, G. Di Massa, S. Pagano, A Vibration Isolator Based on Magneto-rheological Elastomer, Springer International Publishing, Cham, pp. 483–490. [23] B. Nayak, S. Dwivedy, K. Murthy, Dynamic analysis of magnetorheological elastomer-based sandwich beam with conductive skins under various boundary conditions, J. Sound Vib. 330 (2011) 1837–1859. [24] S. Aguib, A. Nour, H. Zahloul, G. Bossis, Y. Chevalier, P. Lançon, Dynamic behavior analysis of a magnetorheological elastomer sandwich plate, Int. J. Mech. Sci. 87 (2014) 118–136. [25] B. Dyniewicz, J.M. Bajkowski, C.I. Bajer, Semi-active control of a sandwich beam partially filled with magnetorheological elastomer, Mech. Syst. Signal Process. 60–61 (2015) 695–705. [26] Y.S. Zhu, X.L. Gong, H. Dang, X.Z. Zhang, P.Q. Zhang, Numerical analysis on magnetic-induced shear modulus of magnetorheological elastomers based on multi-chain model, Chin. J. Chem. Phys. 19 (2006) 126–130. [27] A. Dorfmann, R. Ogden, Magnetoelastic modelling of elastomers, Eur. J. Mech. Solid. 22 (2003) 497–507. [28] A. Dorfmann, R. Ogden, Some problems in nonlinear magnetoelasticity, Z. Angew. Math. Phys. ZAMP 56 (2005) 718–745. [29] A. Dorfmann, I. Brigadnov, Constitutive modelling of magneto-sensitive cauchy-elastic solids, Comput. Mater. Sci. 29 (2004) 270–282. [30] M. Itskov, V.N. Khiêm, A polyconvex anisotropic free energy function for electro- and magneto-rheological elastomers, Math. Mech. Solid 21 (2016) 1126–1137. [31] M. Norouzi, S.M.S. Alehashem, H. Vatandoost, Y.Q. Ni, M.M. Shahmardan, A new approach for modeling of magnetorheological elastomers, J. Intell. Mater. Syst. Struct. 27 (2016) 1121–1135. [32] E. Galipeau, S. Rudykh, G. deBotton, P.P. Castañeda, Magnetoactive elastomers with periodic and random microstructures, Int. J. Solid Struct. 51 (2014) 3012–3024. [33] S. Rudykh, K. Bertoldi, Stability of anisotropic magnetorheological elastomers in finite deformations: a micromechanical approach, J. Mech. Phys. Solid. 61 (2013) 949–967. [34] P. Blom, L. Kari, A nonlinear constitutive audio frequency magneto-sensitive rubber model including amplitude, frequency and magnetic field dependence, J. Sound Vib. 330 (2011) 947–954. [35] Y.F. Dafalias, E.P. Popov, Cyclic loading for materials with a vanishing elastic region, Nucl. Eng. Des. 41 (1977) 293–302. [36] A. Nieto-Leal, V.N. Kaliakin, Improved shape hardening function for bounding surface model for cohesive soils, J. Rock Mech. Geotech. Eng. 6 (2014) 328–337. [37] R. Österlöf, H. Wentzel, L. Kari, N. Diercks, D. Wollscheid, Constitutive modelling of the amplitude and frequency dependency of filled elastomers utilizing a modified boundary surface model, Int. J. Solid Struct. 51 (2014) 3431–3438. [38] J. Lejon, L. 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Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
A nonlinear constitutive model by spring, fractional derivative and modified bounding surface model to represent the amplitude, frequency and the magnetic dependency for Magneto-sensitive rubber Bochao Wang ∗ , Leif Kari KTH Royal Institute of Technology, The Marcus Wallenberg Laboratory for Sound and Vibration Research (MWL), Teknikringen 8, 100 44 Stockholm, Sweden
article info
abstract
Article history: Received 2 March 2018 Revised 23 August 2018 Accepted 10 September 2018 Available online 14 September 2018 Handling Editor: Ivana Kovacic
Magneto-sensitive (MS) rubber is a kind of smart material mainly consisting of magnetizable particles and rubber. Inspired by experimental observation that the shear modulus for MS rubber is strongly dependent on amplitude, frequency and magnetic field; while the impact for the magnetic field and strain to the loss factor is relatively small, a new nonlinear constitutive model for MS rubber is presented. It consists of a fractional viscoelastic model, an elastic model and a bounding surface model with parameters sensitive to the magnetic field. To our knowledge, it is the first time that the bounding surface model is incorporated with the magnetic sensitivity and used to predict the mechanical properties for MS rubber. After comparison with the measurement results, it is found that the shear modulus and the loss factor derived from the simulation fit well with the experimental data. This new constitutive model with only eight parameters can be utilized to describe the amplitude, frequency and the magnetic field dependence for MS rubber. It provides a possible new way to understand the mechanical behavior for MS rubber. More importantly, the constitutive model with an accurate prediction property for the dynamic performance of MS rubber is of interest for MS rubber applications in noise and vibration reduction area. © 2018 Elsevier Ltd. All rights reserved.
Keywords: Magneto-sensitive rubber Bounding surface model Frequency dependency Amplitude dependency nonlinear
1. Introduction Vibration is a common phenomenon in the world. For example, vibration exists in mechanical devices under operating and it occurs when structures are subjected to dynamic external loadings. In most cases, vibration is unwanted because it reduces the working life of machine, wastes energy and may cause noise. One commonly used way to reduce vibration is to install some vibration isolators for the target objects. Generally, vibration isolators are made of rubber, which means that the dynamic properties of vibration isolators is fixed once mounted. Therefore, it is impossible to change its characteristics to adapt to various excitation conditions. In that case, when the frequency for the excitation is close to the critical frequency for the system, the vibration attenuation effect of the vibration isolators would be greatly reduced and can even collapse because of frequency mis-tuning problems.
∗ Corresponding author. E-mail address: [email protected] (B. Wang).
https://doi.org/10.1016/j.jsv.2018.09.028 0022-460X/© 2018 Elsevier Ltd. All rights reserved.
B. Wang and L. Kari / Journal of Sound and Vibration 438 (2019) 344–352
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However, magneto sensitive (MS) rubber, a kind of smart material, may find a solution for the mis-tuning problem. It mainly consists of polarized iron particles embedded in a rubber matrix. Under a magnetic field, the mechanical characteristic of MS rubber can be modified rapidly, reversibly and continuously. MS rubber is categorized into either anisotropic or isotropic according to the presence or absence of the magnetic field in the vulcanization process. If MS rubber is used for the fabrication of vibration isolators, an increased vibration reduction effect may be obtained by the stiffness changing capacity of MS rubber. Research on MS rubber started in the late 1990s by Jolly et al. [1] and Ginder et al. [2]. For the last two decades, more researchers have begun the study on MS rubber. The research on MS rubber can be mainly divided into four categories; those are: fabrication, testing, application and modeling. For the fabrication of MS rubber, a wide range of methods aimed at obtaining MS rubber with high MS effect are used. By finite element modeling, Davis [3] predicted that the optimum particle volume fraction for MS rubber to obtain the largest change in modulus is 27%. The result was then verified experimentally [4]. For anisotropic MS rubber, the effects of matrix [5], coupling agent, content of iron particles, plasticizers [6] pre-structure [7] and surfactant [8] on the MR effect were considered. Muniain et al. [9] studied the influence of carbon black and plasticizers on the dynamic properties of isotropic MS natural rubber. Recently, Stanier and his coworkers [10] chose a new method for the fabrication of MS rubber. In their study, nickel-coated carbon fiber was embedded in elastomer and a magnetic field is applied during the vulcanization to obtain a transversely isotropic composite. Subsequently, they investigated the effect of the orientation of the nickel-coated fiber to the magneto mechanical behavior of MS rubber. For the testing of the mechanical properties for MS rubber, the influence of the magnetic field, frequency and temperature on the dynamic stiffness [11,12] and the loss factor [13] of isotropic MS rubber have been studied. In parallel, large efforts have been made to investigate the impact of magnetic field, frequency [14], temperature [15] on the viscoelastic properties of anisotropic MS rubber. Among all the investigations related to testing of MS rubber, Blom and Kari [16] investigated the Fletcher Gent effect [17] of the isotropic MS rubber in a wide frequency range and it revealed that in addition to viscoelastic properties, there is a strong amplitude dependency of MS rubber even for a small strain. The magnetic field sensitivity property of MS rubber makes it attractive for the potential application in semi-active vibration control field. For example, Deng and Gong [18] designed a MS rubber based dynamic vibration absorber. The research result revealed that the adaptively tuned vibration absorber comprised of MS rubber is efficient for tracking the excitation frequency and reducing the vibration of the primary structure. Jung et al. [19] fabricated a base isolation system of MS rubber and installed it on a scaled building structure to investigate the seismic performance of that smart base isolation system. The result showed that compared with conventional base isolation systems, the base isolation system made of MS rubber outperforms in reducing the vibration of the building structure. Similar research was done [20–22] suggesting the feasibility of using MS rubber for improving the vibration attenuation effect and staggering the natural frequency of structure from the loading frequency. Other application of MS rubber in vibration control area includes exploring the potential of using adaptive sandwich beams filled with MS rubber to reduce the vibration of vibrating structural elements like helicopter blades and aircraft wings. For instance, Nayak et al. [23] proposed the governing equation for free vibration of a sandwich beam filled with MS rubber under various boundary conditions. Aguib et al. [24] discussed the effect of different magnetic fields and different percentage of ferromagnetic particles on the vibration response of MS rubber sandwich plates. Furthermore, Dyniewice et al. [25] validated the feasibility of reducing the vibration of a sandwich beam partially filled with MS rubber by changing the magnetic field. There are a number of models developed for the magnetic related mechanical properties of anisotropic MS rubber. Primarily, Jolly et al. [1] proposed a dipole model to represent the magnetic related mechanical properties of the anisotropic MS rubber. Later, Zhu et al. [26] augmented that model by considering the influence of adjacent chains. Dorfmann and Ogden [27–29] derived the governing equations for the deformation of MS rubber by combing Maxwell equations with continuum mechanics and Itskov et al. [30] then extended the model by introducing the mathematically poly-convexity notion. Unfortunately, those models are only valid for the quasi-static case due to the neglect of the viscoelastic effects for MS rubber. To model the viscoelastic behavior of anisotropicMS rubber, a Bouc-Wen and a Kelvin-Voigt model were aligned in parallel to represent the viscoelastic behavior of MS rubber [19]. The main drawback of that model is the high computational cost for identifying the material parameters. To solve that problem, a modified Kelvin-Voigt model for which the parameters in stiffness and damping element are all related to the magnetic field, frequency and strain were proposed [31]. However, the agreement between the simulation and experiment result for the loss factor needs to be improved. More importantly, for all the models mentioned above, the measurements are performed based on the anisotropic MS rubber where the iron particles are aligned in chains. For isotropic MS rubber, the distribution of iron particles and the mechanical behavior are completely different. Based on that physical fact, Rudykh et al. [32,33] investigated the effect for distribution of magnetizable particles on the magneto-mechanical coupling effect and stability for MS rubber. The Fletcher-Gent effect, which represents the dependence of the modulus on the amplitude of the applied strain for MS rubber was generally omitted for the model mentioned above. However, according to the research by Blom and Kari [16], the amplitude dependent stiffness behavior is an important feature for MS rubber and should not be ignored. To describe the dynamic behavior of MS rubber completely, a model which incorporated the Fletcher-Gent effect for MS rubber is needed. In 2011, a nonlinear constitutive model that includes the effect of strain, frequency and magnetic field for the mechanical behavior for isotropic MS rubber was proposed by Blom and Kari [34]. In their model, a fractional derivative was used to describe the viscoelastic properties and a smooth frictional stress model was adopted to describe the Fletcher-Gent effect for MS rubber. In fact, there are other methods that are capable of capturing the Fletcher-Gent effect for filled rubber. One possible way is to use the bounding surface model, which was proposed by Dafalias and Popov [35] in 1977. The innovation of the bounding surface model is that the plastic modulus could vary continuously and the specialized case for bounding surface model could be
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used for rate sensitive plastic materials. Since then, the bounding surface model has been widely used to simulate the behavior of cohesive soils [36]. In 2014 a model [37] that tried to capture the Fletcher-Gent effect [17] of filled rubber by using the bounding surface model was presented. The accurate simulation result revealed that the bounding surface model is ideal for modeling the amplitude dependency of filled rubber. However, for the work done by Österlöf et al. [37], only frequency dependency and amplitude dependency for filled rubber were modeled. For MS rubber with iron particles, in addition to frequency and amplitude dependency, there are also the magnetic dependency. Inspired by the work of Refs. [34] and [37], a new constitutive model for MS rubber is developed in this paper. The novelty of this model is that it incorporates the bounding surface model with magnetic dependency to model the magnetic field-dependent amplitude dependence of MS rubber. Compared with the smooth frictional stress model presented by Blom and Kari [34], the bounding surface model used in this paper has the advantage of taking irregular cyclic loading in the three dimensional case into consideration. Thus, for the model developed in this paper, further extension to three dimensions is available. Comparison with the experimental result reveals that this newly built model could well replicate the magnetic, strain amplitude and frequency dependent properties for MS rubber. For the possible application of MS rubber in vibration controlling area, if the shear modulus and the loss factor of MS rubber are replicated well, the response of the anti-vibration components made from MS rubber could be predicted more accurately and the uncertainty for the model could be reduced considerably. The constitutive model presented in this paper for MS rubber plays a significant role in the precise implementation of the control strategy and in obtaining a satisfactory vibration reduction effect for the probable use of MS rubber in adaptive vibration control areas. 2. Experiment result The ingredients for the MS rubber used in this paper are based on Blom and Kari [16]. During vulcanization, no external magnetic field is applied resulting in a random distribution of iron particles in the rubber, generating isotropic MS rubber. The experimental studies about the mechanical behavior of MS rubber were done by Lejon and Kari [38]. For the details about the measurement setup and method, the reader is referred to Ref. [38]. In this paper only room temperature is considered, thus measurement results with only 20 ◦ C were chosen for the subsequent parameter identification. MS rubber works in shear deformation mode. Four levels of the applied magnetic field with values of 0, 0.3, 0.55 and 0.8 T are considered. The dynamic shear strains are 0.0005, 0.0015 and 0.005, respectively. The excitation signal is a stepped sine excitation with a frequency interval of 10 Hz ranging from 200 to 900 Hz. The test results for the magnitude and loss factor of the shear modulus versus frequency with different amplitude of strains are shown in the solid lines from Figs. 3–5. It should be noted that the “bumps” for the measurement magnitude and loss factor of the shear modulus in Figs. 3–5 are caused by the resonance of the test machine. 3. Constitutive modeling 3.1. General models The rheological model developed in this paper is shown in Fig. 1. The total stress is assumed to consist of three parts dependent on time t
𝜏 = 𝜏e + 𝜏ve + 𝜏f ,
(1)
where 𝜏 , 𝜏 e , 𝜏 ve and 𝜏 f are total, elastic, viscoelastic and frictional shear stress, respectively. The elastic shear stress is obtained by
𝜏e (t) = Ge · 𝛾 (t) ,
(2)
where 𝛾 (t ) is the shear strain and Ge is the elastic shear modulus. The viscoelastic shear stress is described by a relaxation convolution integral
𝜏ve (t) =
t
b d 𝛾 (s) ds, Γ(1 − a) dt ∫0 (t − s)a
(3)
where a and b are the parameters in the relaxation convolution integral, Γ is the gamma function and 𝛾 (s) = 0 for s < 0. The frictional shear stress is described by the bounding surface model. Normally, for bounding surface model, a bounding surface F and a yield surface f are included [35]. Those are defined as f (𝜎ij − 𝛼ij ) = 0
(4)
F (𝜎 ij − 𝛽ij ) = 0,
(5)
and
with a bar over 𝜎 ij indicates stresses on the bounding surface. The symbols 𝛼 ij and 𝛽 ij are back-stresses which serve as the center of the two surfaces, respectively.
B. Wang and L. Kari / Journal of Sound and Vibration 438 (2019) 344–352
347
Fig. 1. Rheological model for MS rubber.
During loading, these two surfaces may move, change size and even contact with each other with the limitation that the bounding surface always encircles the yield surface. The key idea for the bounding surface model is that the plastic modulus is determined by the distance between the two surfaces. For the one dimensional case, with a yield surface of vanishing size, the evolution law simplifies to
𝜏̇ f = H𝛾̇ = 𝛼̇
(6)
̇ 𝛽̇ = Hp 𝛾,
(7)
and
with H being the plastic modulus for the yield surface f and Hp being the plastic modulus for the bounding surface F. The symbols 𝛼 and 𝛽 represent the back-stresses for the yield surface and bounding surface in one dimensional case, respectively. A dot over 𝛾 represents the change of strain for new time step. The expression for the plastic modulus is determined according to Österlöf et al. [37], that is
(
H(𝛿, 𝛿in ) = Hp
1+
𝛿
𝛿in − 𝛿
)
,
(8)
where 𝛿 is the current distance to the bounding surface and 𝛿 in is the distance to the bounding surface at the previous turning point. 3.2. Numerical calculation method In the following, the value of variables at time step t = nΔt is denoted by (·)n Δ t where Δt is the constant time step and n ∈ ℕ. The subscript (·)(n +1) Δ t represents the next time step. For the calculation of the elastic shear stress, only the current state of strain is needed. To calculate the viscoelastic shear stress, after using the truncation definition for fractional differentiation [39], the viscoelastic shear stress becomes
𝜏ve[(n+1)Δt] =
n b ∑ cj (a)n+1−j 𝛾(n+1−j)Δt (Δt )a
(9)
j=0
and c0 (a) = 1, cj (a) =
j−1−a cj−1 (a). j
(10)
To calculate the frictional shear stress, the Euler forward method is used. Based on the work by Österlöf et al. [37], firstly, a trial function 𝜏f∗ which corresponds to the change of shear stress in each time step is calculated,
𝜏f∗[(n+1)Δt] = HnΔt 𝛾̇ [(n+1)Δt] , the parameter Hn Δ t represents the plastic modulus at time t = nΔt.
(11)
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Secondly, the variable l which determines the loading direction for the new step is obtained by
𝜏 ∗ f[(n+1)Δt]
l[(n+1)Δt] = | |𝜏 ∗
|
|.
(12)
f[(n+1)Δt] ||
If l[(n+1)Δt] · l(nΔt) > 0, the loading direction and the distance to the bounding surface remains the same. If l[(n+1)Δt] · l(nΔt) < 0, it means that the direction of the load has changed. If the direction of the loading has changed, the internal state variables (𝛿 and 𝛿 in ) update according to
𝛿in[(n+1)Δt] = Syield l[(n+1)Δt] − (𝜏f(nΔt) − 𝛽(nΔt) )
(13)
𝛿[(n+1)Δt] = Syield l[(n+1)Δt] − (𝜏f(nΔt) − 𝛽(nΔt) + 𝜏f∗[(n+1)Δt] ),
(14)
and
with 𝜏f (nΔt) means the frictional stress calculated at time t = nΔt. The symbol Syield is the initial distance from the vanishing sized yield surface to the bounding surface. The plastic modulus updates as
( H[(n+1)Δt] = Hp
1+
𝛿[(n+1)Δt] 𝛿in[(n+1)Δt] − 𝛿[(n+1)Δt]
)
.
(15)
Lastly, the stresses for the two surfaces updates as
𝜏̇ f[(n+1)Δt] = H[(n+1)Δt] 𝛾̇ [(n+1)Δt] ,
(16)
𝛽̇ [(n+1)Δt] = Hp 𝛾̇ [(n+1)Δt] ,
(17)
𝜏f[(n+1)Δt] = 𝜏f(nΔt) + 𝜏̇ f[(n+1)Δt]
(18)
𝛽[(n+1)Δt] = 𝛽(nΔt) + 𝛽̇ [(n+1)Δt] ,
(19)
and
where 𝜏 f(n Δ t) and 𝛽 (n Δ t) represent the value at time t = nΔt. 3.3. Magnetic field sensitivity for the model Based on the experimental observations by Lejon and Kari [38] and utilizing the control variates method, the magnetic sensitivity term for the three sub-models were determined as follows. Firstly, when the frequency is constant, the dynamic properties of MS rubber reflects a magnetic sensitivity and the magnetic sensitivity is strongly correlated with the amplitude dependency. It suggests that the amplitude dependent parameters in the bounding surface model should include the magnetic sensitivity. Secondly, if strain amplitude is fixed, the variation for the magnitude for shear modulus between zero magnetic field and maximum magnetic field at different frequency seems to be approximately constant. It means that there is no magnetic dependency for the parameters a and b in the viscoelastic model. Lastly, the loss factor, which is the quotient between the loss modulus and the storage modulus, does not fluctuate considerably for different magnetic field. It means that the changing of the parameters for the bounding surface model with related to magnetic field should be balanced by the change of the elastic modulus Ge with related to magnetic field. In summary, only three parameters in the model which are Ge , Syield and Hp should contain the magnetic sensitivity term. According to the nonlinear theory of magnetoelastic interaction developed by Dorfmann and Odgen [27–29], in one dimensional case, the magneto-sensitivity for shear modulus is quadratically dependent on the magnetic field applied. Therefore, the increase of Ge should be proportional to the square of the magnetic field. Furthermore, experimental observation has shown that the dissipated energy of MS rubber increases with the magnetic field, suggesting that for the bounding surface model, the values of Syield and Hp should positively related to the magnetic field. Similarly, a square of M is chosen to reflect the magnetic dependency of Syield and Hp . Thus, the three parameters reflecting the magnetic sensitivity are explicated as
[ Ge =
Syield =
(
1+
M Ms
[
(
1+
]
)2
M Ms
𝛿1 Ge0 , )2
(20)
]
𝛿2 Syield0
(21)
B. Wang and L. Kari / Journal of Sound and Vibration 438 (2019) 344–352
and
[ Hp =
(
1+
M Ms
)2
349
]
𝛿3 Hp0 ,
(22)
where Ge0 , Syield0 and Hp0 are the state values of MS rubber when magnetic field is zero. The parameter Ms represents the saturation magnetic field which is the upper limit for the applied magnetic field. Parameters 𝛿 1 , 𝛿 2 and 𝛿 3 are material constant that need to be decided. 4. Result and discussion A nonlinear least squares fit method is implemented to obtain the parameters for zero magnetic field. The special function “lsqnonlin” was used for the parameter identification in MatlabⓇ (MATLAB Release 2015b, The MathWorks, Inc., Natick, Massachusetts, United States). Considering the computational cost, the measurement values for the shear modulus at four levels of shear strain amplitudes with frequency ranging from 200 to 900 Hz with a frequency interval of 100 Hz used. The objective function (squared absolute error) for the nonlinear square fit method is obj =
n {[ ∑
]
[
G∗meas (i) − G∗sim (i) · conj G∗meas (i) − G∗sim (i)
]}
,
(23)
i=1
where G∗meas (i) and G∗simu (i) are the ith measurement and simulation dynamic shear modulus for MS rubber. The symbol conj [·] is the complex conjugate. The objective function is minimized to obtain the material parameters for MS rubber. The relative error is evaluated as
√ {[ ] [ ]} √ √ n √∑ G∗meas (i) − G∗sim (i) · conj G∗meas (i) − G∗sim (i) {[ ∗ ] [ ]} . err = √ Gmeas (i) · conj G∗meas (i) i=1
(24)
In order to ensure the accuracy for the modeling, during the simulation phase, ten rounds of loading with the time increment Δt = (27 fexc )−1 are applied, where fexc is the frequency for the harmonic loading. In order to identify the magnitude and loss factor for the dynamic shear modulus under steady motion state, only the last six rounds of simulation data were used. The numerical calculation is based on the Euler forward method, therefore, the maximum truncation global error for the simulation is proportional to the time size (3.9063 × 10− 5 s), corresponding to the case where the frequency of the external load is 200 Hz. After halving the time step, it is found that the identified material parameters for the constitutive model of MS rubber differ only in the fourth digit, which means that the set time step meets the accuracy requirement for the simulation. After parameter identification and optimization, the parameters under zero magnetic field case are obtained as Ge0 = 3.040 × 106 Nm− 2 , a = 0.2999, b = 0.2679 × 106 Nsa m− 2 , Syield0 = 0.2170 × 105 Nm− 2 and Hp0 = 0.2002 × 106 Nm− 2 . The absolute error under zero magnetic field case with frequency ranging from 200 to 900 Hz for 24 sets of experimental data is 1.1632 × 106 Nm− 2 . The relative error for the novel model fits to the experimental result in zero magnetic field case is 3.294%. The results are shown in Fig. 2, where the magnitude and the loss factor of shear modulus versus frequency are plotted. As described above, the peaks for the magnitude and the loss factor of the shear modulus of MS rubber are due to the influence of the resonance for the measurement setup. Solid lines and symbols represent the measurement results and simulation result, respectively in Fig. 2. Apparently, the coincidence between experiments and modeling results is very good. After obtaining the material parameters for MS rubber at zero magnetic field state, the second round of least square fit method is applied to obtain the parameters representing the magnetic sensitivity of MS rubber. In this case, four levels of magnetic field (0, 0.3, 0.55 and 0.8 T) accompanying different frequency and shear strains are used. The four magnetic related parameters are obtained as 𝛿 1 = 0.2248, 𝛿 2 = 0, 𝛿 3 = 0.7670 and Ms = 0.5615 T. The absolute error for four different kind of magnetic field case with frequency ranging from 200 to 900 Hz is 1.1636 × 106 Nm− 2 . The relative error for the novel model fits
Fig. 2. The magnitude and loss factor of the dynamic shear modulus without the effect of magnetic field. Solid lines are experimental data, symbols are simulations. Three kinds of strain (0.0005, 0.0015, 0.005) were considered.
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Fig. 3. The magnitude and loss factor of the dynamic shear modulus with shear strain equals to 0.0005. Solid lines are experimental data, symbols are simulations.
Fig. 4. The magnitude and loss factor of the dynamic shear modulus with shear strain equals to 0.0015. Solid lines are experimental data, symbols are simulations.
to the experimental result in zero magnetic field case is 3.296%. Interestingly, it is found that there is no effect of magnetic field on the parameter Syield for the bounding surface model. The application of the magnetic field only makes the plastic modulus in the bounding surface model stiffer. The comparison between the measurement data and the simulation results are shown from Figs. 3–5 with shear strain amplitude 0.0005, 0.0015 and 0.005, respectively. Figs. 3–5 reveal that the simulation results of the shear modulus fit well with the measured shear modulus. The tendency is that the shear modulus increases with the applied magnetic field and the shear modulus become stiffer for smaller shear strain. This tendency is similar to the measurement observation, which shows that the modified bounding surface model depicts the magnetic sensitive plastic properties of MS rubber very well. For the modeling of the loss factors, the modeled results resemble the experiment accurately. In Figs. 6 and 7, the modeled hysteresis curves are compared with the corresponding measurement data with two extreme magnetic fields (0 and 0.8 T). The shear strain and frequency were chosen to be 0.005 and 200 Hz respectively in Fig. 6, which represents the minimum value of the shear modulus for MS rubber. The chosen shear strain and frequency were 0.0005 and 900 Hz in Fig. 7 respectively, resulting in the maximum value of the shear modulus for MS rubber. By comparing the hysteresis curve between experiment and simulation from Figs. 6 and 7, it is found that these two curves fit well with each other. At increased applied magnetic field, the slope of the hysteresis curve becomes steeper. Combined with the result in Figs. 3–5, it demonstrates that the derivation about the magnetic sensitivity in the developed model is reasonable.
Fig. 5. The magnitude and loss factor of the dynamic shear modulus with shear strain equals to 0.005. Solid lines are experimental data, symbols are simulations.
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Fig. 6. Comparison between measurement shear strain-stress curve (solid lines) and modeled shear strain-stress curve (symbols) with shear strain 0.005 and frequency 200 Hz at applied magnetic fields with values of 0 and 0.8 T.
Fig. 7. Comparison between measurement shear strain-stress curve (solid lines) and modeled shear strain-stress curve (symbols) with shear strain 0.0005 and frequency 900 Hz at applied magnetic fields with values of 0 and 0.8 T.
5. Conclusion A nonlinear model which depicts the dynamic strain amplitude, frequency and magnetic field dependency for MS rubber is developed. It is assumed that the viscoelastic properties only depend on frequency, irrespective of the magnetic field. Moreover, for the modeling of frictional properties of MS rubber, a modified bounding surface model with parameters sensitive to magnetic field is utilized. Due to the merit of bounding surface model that can take three-dimensional irregular circular loading into account, further extension of this nonlinear constitutive model to three-dimension and a more accurate and comprehensive prediction of the mechanical behavior for MS rubber would be accessible. After introducing the numerical calculation method, two rounds of non-linear least squares fits are applied to obtain the modeling material parameters. The analyzed simulation results are compared with measurement date and a good coincidence is reached. This model with only eight parameters (Ge0 , a, b, Syield0 , Hp0 , 𝛿 1 , 𝛿 3 and Ms ) can be used to predict the elasticity, viscoelasticity, plasticity and the magnetic sensitivity for MS rubber. For the possible application of isotropic MS rubber in vibration isolators with shear deformation, this model could be used to accelerate the design of component before real experimental testing. Furthermore, a constitutive model which precisely predicts the dynamic properties of MS rubber could be used to facilitate the potential application of MS rubber in vibration isolators and other devices where capacities for adaptive change of stiffness are required. Acknowledgments The authors gratefully acknowledge Dr. Jonas Lejon for providing the measurement data for MS rubber. In addition, the China Scholarship Council is gratefully acknowledged for the financial support through the KTH-CSC programme. References [1] [2] [3] [4] [5]
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