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<article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" article-type="research-article">
<front>
<journal-meta>
<journal-id journal-id-type="publisher-id">Front. Mater.</journal-id>
<journal-title>Frontiers in Materials</journal-title>
<abbrev-journal-title abbrev-type="pubmed">Front. Mater.</abbrev-journal-title>
<issn pub-type="epub">2296-8016</issn>
<publisher>
<publisher-name>Frontiers Media S.A.</publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="doi">10.3389/fmats.2018.00076</article-id>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Materials</subject>
<subj-group>
<subject>Original Research</subject>
</subj-group>
</subj-group>
</article-categories>
<title-group>
<article-title>Magnetic Field Induced Surface Micro-Deformation of Magnetorheological Elastomers for Roughness Control</article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name><surname>Chen</surname> <given-names>Shiwei</given-names></name>
<xref ref-type="aff" rid="aff1"><sup>1</sup></xref>
<uri xlink:href="http://loop.frontiersin.org/people/564887/overview"/>
</contrib>
<contrib contrib-type="author">
<name><surname>Li</surname> <given-names>Rui</given-names></name>
<xref ref-type="aff" rid="aff2"><sup>2</sup></xref>
</contrib>
<contrib contrib-type="author">
<name><surname>Li</surname> <given-names>Xi</given-names></name>
<xref ref-type="aff" rid="aff2"><sup>2</sup></xref>
</contrib>
<contrib contrib-type="author" corresp="yes">
<name><surname>Wang</surname> <given-names>Xiaojie</given-names></name>
<xref ref-type="aff" rid="aff3"><sup>3</sup></xref>
<xref ref-type="corresp" rid="c001"><sup>&#x0002A;</sup></xref>
</contrib>
</contrib-group>
<aff id="aff1"><sup>1</sup><institution>Chongqing University of Science and Technology, Academy Mathematics and Physics</institution>, <addr-line>Chongqing</addr-line>, <country>China</country></aff>
<aff id="aff2"><sup>2</sup><institution>Chongqing University of Posts and Telecommunications, Chinese Academy of Sciences</institution>, <addr-line>Changzhou</addr-line>, <country>China</country></aff>
<aff id="aff3"><sup>3</sup><institution>Institute of Advanced Manufacturing Technology, Hefei Institute of Physical Science, Chinese Academy of Sciences</institution>, <addr-line>Changzhou</addr-line>, <country>China</country></aff>
<author-notes>
<fn fn-type="edited-by"><p>Edited by: Norman M. Wereley, University of Maryland, College Park, United States</p></fn>
<fn fn-type="edited-by"><p>Reviewed by: Xufeng Dong, Dalian University of Technology (DUT), China; Xiaomin Dong, Chongqing University, China</p></fn>
<corresp id="c001">&#x0002A;Correspondence: Xiaojie Wang <email>xjwang&#x00040;iamt.ac.cn</email></corresp>
<fn fn-type="other" id="fn001"><p>This article was submitted to Smart Materials, a section of the journal Frontiers in Materials</p></fn></author-notes>
<pub-date pub-type="epub">
<day>19</day>
<month>12</month>
<year>2018</year>
</pub-date>
<pub-date pub-type="collection">
<year>2018</year>
</pub-date>
<volume>5</volume>
<elocation-id>76</elocation-id>
<history>
<date date-type="received">
<day>21</day>
<month>05</month>
<year>2018</year>
</date>
<date date-type="accepted">
<day>05</day>
<month>12</month>
<year>2018</year>
</date>
</history>
<permissions>
<copyright-statement>Copyright &#x000A9; 2018 Chen, Li, Li and Wang.</copyright-statement>
<copyright-year>2018</copyright-year>
<copyright-holder>Chen, Li, Li and Wang</copyright-holder>
<license xlink:href="http://creativecommons.org/licenses/by/4.0/"><p>This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.</p></license>
</permissions>
<abstract><p>In this work, we propose mesoscopic model to investigate the surface micro structures of magnetorheological elastomers (MREs) under a magnetic field. By comparing the surface roughness changes of MREs, we found that the surface micro-deformation of MREs, not the field-induced hardness, mainly accounts for the controllable friction characteristics of MREs. The results also demonstrate that the field-induced friction of MREs depends on the particle contents as well as the initial surface roughness. The model predicts that the maximum relative roughness change of MREs occurs when the MRE has particle volume fraction of around 9%, which is validated by experimental results.</p></abstract>
<kwd-group>
<kwd>MRE</kwd>
<kwd>magnetostrictive</kwd>
<kwd>roughness controling</kwd>
<kwd>tunable interface</kwd>
<kwd>magneto induced deformation</kwd>
</kwd-group>
<contract-num rid="cn001">11502038</contract-num>
<contract-num rid="cn001">11572320</contract-num>
<contract-num rid="cn001">11372366</contract-num>
<contract-num rid="cn002">2017M610386</contract-num>
<contract-sponsor id="cn001">National Natural Science Foundation of China<named-content content-type="fundref-id">10.13039/501100001809</named-content></contract-sponsor>
<contract-sponsor id="cn002">China Postdoctoral Science Foundation<named-content content-type="fundref-id">10.13039/501100002858</named-content></contract-sponsor> <counts>
<fig-count count="6"/>
<table-count count="0"/>
<equation-count count="5"/>
<ref-count count="27"/>
<page-count count="8"/>
<word-count count="3760"/>
</counts>
</article-meta>
</front>
<body>
<sec sec-type="intro" id="s1">
<title>Introduction</title>
<p>Polymers are hyperplastic materials that can be subjected to a large recoverable deformation under a relatively small loading. In taking advantage of this natural property, researchers have developed adaptive materials called magnetorheological elastomers (MREs), by embedding magnetic particles (micron size) in the polymer matrix (Jolly et al., <xref ref-type="bibr" rid="B13">1996</xref>; Ginder et al., <xref ref-type="bibr" rid="B10">1999</xref>). The magnetizing particles in the matrix would interact with each other under an external magnetic field, and consequently induce deformation and modulus changes of MREs. With these unique controllable properties, MREs could be possible candidates for a variety of applications, such as in soft actuators (B&#x000F6;se et al., <xref ref-type="bibr" rid="B3">2012</xref>), vibration absorber (Menzel, <xref ref-type="bibr" rid="B22">2015</xref>; Chen et al., <xref ref-type="bibr" rid="B4">2016</xref>), and magnetic sensor (Sutrisno et al., <xref ref-type="bibr" rid="B24">2015</xref>) etc.</p>
<p>Recently, several experimental studies (Lee et al., <xref ref-type="bibr" rid="B15">2013</xref>; Lian et al., <xref ref-type="bibr" rid="B17">2015</xref>, <xref ref-type="bibr" rid="B18">2016</xref>; Li et al., <xref ref-type="bibr" rid="B16">2018</xref>) have shown that the tribological properties of MREs also change with external magnetic fields, which would possibly extend the MRE&#x00027;s application in interfacial friction control. However, the mechanism of field-induced friction behavior of MREs is still unclear. Lee et al. (<xref ref-type="bibr" rid="B15">2013</xref>) employed a homemade linear sliding tester to evaluate the friction characteristics of MREs, and found that the friction coefficient of MRE decreased with the external magnetic field. Later, the same group (Lian et al., <xref ref-type="bibr" rid="B17">2015</xref>, <xref ref-type="bibr" rid="B18">2016</xref>) applied a reciprocating friction tester to characterize four MRE samples, which were prepared by embedded particles in different matrix materials, and found the same phenomenon for all test conditions. The authors argued that the friction reduction of MREs under a magnetic field was due to the field-stiffening effect. According to the classic contact theory, the large stiffness of materials will result in less interfacial friction (Popov, <xref ref-type="bibr" rid="B23">2010</xref>). However, in a previous study (Li et al., <xref ref-type="bibr" rid="B16">2018</xref>), we observed that even though the friction coefficient of MREs increases with external magnetic fields in some cases, the MREs are getting stiffer. Accordingly, this phenomenon challenges existing theoretical interpretation.</p>
<p>In fact, the surface micro-scale morphology may mainly contribute to the tribological characteristics of the MREs in a magnetic field. However, few studies (Gong et al., <xref ref-type="bibr" rid="B11">2012</xref>) have been carried out to investigate the surface micro-deformation of the MREs under magnetic fields.</p>
<p>Although theoretical modeling of the deformation behavior of MREs has been studied by many others, these models originated from either the continuum mechanics (Dorfmann and Ogden, <xref ref-type="bibr" rid="B9">2004</xref>; Kankanala and Triantafyllidis, <xref ref-type="bibr" rid="B14">2004</xref>), or the multiscale theories (Davis, <xref ref-type="bibr" rid="B8">1999</xref>; Cremer et al., <xref ref-type="bibr" rid="B6">2015</xref>) mostly focus on prediction of the magnetic induced properties of MRE in macro conditions. The models from the point of view of continuum mechanics (Allahyarov et al., <xref ref-type="bibr" rid="B1">2014</xref>; Menzel, <xref ref-type="bibr" rid="B21">2014</xref>) hardly take into account the surface morphology of MREs. And almost all of the representative volume unit models (Yin et al., <xref ref-type="bibr" rid="B26">2002</xref>; Ivaneyko et al., <xref ref-type="bibr" rid="B12">2014</xref>) based on multiscale theories ignore the surface micro profile of MREs. Therefore, all the reported theoretical work on modeling of MREs can only predict the average bulk deformation of MREs. In order to fully understand the field-induced friction behavior of MREs, it is essential to develop a new model to analyze the surface micro structures of MRE under external magnetic fields.</p>
<p>Meanwhile, experimental studies have been performed to measure and analyze the deformation of MREs in magnetic fields. A testing platform with a CCD camera was established by Zr&#x000ED;nyi et al. (<xref ref-type="bibr" rid="B27">1996</xref>) to measure deformation of ferrogels under non-uniform magnetic fields. The effects of compress force and particle volume fraction on magnetic field induced deformation of MRE were investigated by Martin et al. (<xref ref-type="bibr" rid="B20">2006</xref>) and Danas et al. (<xref ref-type="bibr" rid="B7">2012</xref>) by using similar testing platforms. However, most experimental studies were only interested in measuring the average deformation of MRE samples. Gong et al. (<xref ref-type="bibr" rid="B11">2012</xref>) constructed a digital holographic interferometry to achieve so called full-field deformation of MREs. However, they did not investigate the transition of surface micro structures of MREs under magnetic fields.</p>
<p>To fully understand how the external magnetic field affects the surface roughness of MRE, and how the initial roughness and particle volume fractions impact on the variation of the roughness, a 2 dimension mesoscopic model is proposed in this paper. By employing the Monte-Carlo method (Tsang et al., <xref ref-type="bibr" rid="B25">2004</xref>), our model could incorporate the initial surface irregularities of MRE. Based on this, the changes of MREs surface roughness in magnetic field are analyzed by utilizing magneto-mechanical coupling (FEM) algorithms. Besides, several MRE samples with different particle fractions and initial surface roughness are prepared and their roughness are measured by a white light interferometer to reveal the mechanism of magneto induced roughness changes.</p>
</sec>
<sec id="s2">
<title>Modeling</title>
<p>In the proposed model, we assume that the radius of magnetic particle sizes follow normal distribution, and the particles are tightly bonded to the matrix. In addition, the effect of polymer networks on particle interaction is negligible in the theoretical simulation (Davis, <xref ref-type="bibr" rid="B8">1999</xref>; Yin et al., <xref ref-type="bibr" rid="B26">2002</xref>; Ivaneyko et al., <xref ref-type="bibr" rid="B12">2014</xref>). Since the magnetic induced deformation of MRE is usually tiny (Zr&#x000ED;nyi et al., <xref ref-type="bibr" rid="B27">1996</xref>; Martin et al., <xref ref-type="bibr" rid="B20">2006</xref>; Danas et al., <xref ref-type="bibr" rid="B7">2012</xref>), both magnetic particles and polymer matrixes are considered as standard incompressible liner elastic materials. The Young&#x00027;s modulus of particles is 200Mpa, and the Young&#x00027;s modulus of matrixes is 1Mpa.</p>
<p>Figure <xref ref-type="fig" rid="F1">1A</xref> shows the statistic sizes of 175 particles which are uniformly distributed in matrix. Meanwhile, the 2D Gaussian random contour surfaces are generated by employing the Monte-Carlo method (Tsang et al., <xref ref-type="bibr" rid="B25">2004</xref>) representing the initial surface micro profile of the MRE model (Figure <xref ref-type="fig" rid="F1">1B</xref>). In the model, the surface profiles of MRE can be linear superposed by different sinusoidal curves. The coordinate of the scatter point in the profile curve can be obtained as:</p>
<disp-formula id="E1"><label>(1)</label><mml:math id="M1"><mml:mtable class="eqnarray" columnalign="right center left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:mfrac><mml:msubsup><mml:mrow><mml:mo>&#x02211;</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow><mml:mrow><mml:mfrac><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msubsup><mml:mi>F</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>&#x003C9;</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:msub><mml:mrow><mml:mi>&#x003C9;</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msup></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<fig id="F1" position="float">
<label>Figure 1</label>
<caption><p>The two dimension mesoscopic model of MRE. <bold>(A)</bold> Left: statistic distribution of particle sizes, the total number of particles <italic>N</italic> &#x0003D; 175 and the average of the particle radius is 2.5 &#x003BC;m. Right: the particles are uniform distributed in a (250 &#x000D7; 130 &#x003BC;m) matrix area, and the particles volume fraction of MRE &#x003D5;&#x02248;10.5%. <bold>(B)</bold> The generated initial surface micro profile of the MRE model. The surface roughness of the MRE (<italic>Ra</italic>) is 1.67 &#x003BC;m.</p></caption>
<graphic xlink:href="fmats-05-00076-g0001.tif"/>
</fig>
<p>Where <italic>n</italic> is the scatter point number, (<italic>x</italic><sub><italic>n</italic></sub>,<italic>y</italic><sub><italic>n</italic></sub>) represents the coordinate of the <italic>n</italic>-th scatter point in the profile curve, <italic>F</italic>(&#x003C9;<sub><italic>j</italic></sub>) is the Fourier transform pair of <italic>f</italic> (<italic>x</italic><sub><italic>n</italic></sub>), and can be expressed as:</p>
<disp-formula id="E2"><label>(2)</label><mml:math id="M200"><mml:mrow><mml:mi>F</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>&#x003C9;</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>&#x003C0;</mml:mi></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi>&#x00394;</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msqrt></mml:mrow></mml:mfrac><mml:msqrt><mml:mrow><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>&#x003C9;</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msqrt><mml:mo>&#x022C5;</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>N</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mi>N</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mtext>&#x00A0;&#x00A0;;</mml:mtext><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mo>&#x02212;</mml:mo><mml:mfrac><mml:mi>n</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo>,</mml:mo><mml:mo>&#x022EF;</mml:mo><mml:mo>,</mml:mo><mml:mtext>&#x0200B;</mml:mtext><mml:mo>&#x02212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mi>N</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mtext>&#x000A0;&#x000A0;</mml:mtext><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mfrac><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow></mml:mrow></mml:math></disp-formula>
<p>Where <italic>N</italic>(0,1) represents a random number of normal distribution with a mean value of 0 and a variance of 1. &#x00394;<italic>x</italic> is the distance between the nearest two scatter points in <italic>x</italic>-th direction. <italic>S</italic>(&#x003C9;<sub><italic>j</italic></sub>) is the power spectral density of random surface profiles, and obeys gauss distribution function. And it can be expressed as follows:</p>
<disp-formula id="E3"><label>(3)</label><mml:math id="M3"><mml:mtable class="eqnarray" columnalign="right center left"><mml:mtr><mml:mtd><mml:mi>S</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>&#x003C9;</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>h</mml:mi><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msqrt><mml:mrow><mml:mi>&#x003C0;</mml:mi></mml:mrow></mml:msqrt></mml:mrow></mml:mfrac><mml:mo class="qopname">exp</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mi>&#x003C9;</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:mfrac></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<p>Where <italic>h</italic> is the expectation of profiles&#x00027; amplitudes, and <italic>l</italic><sub>cd</sub> is the expectation of distance between two peaks of profiles.</p>
<p>In this work, we use parameter <italic>R</italic><sub><italic>a</italic></sub> to describe the roughness of the MREs&#x00027; surface micro profiles. It can be expressed as follows:</p>
<disp-formula id="E4"><label>(4)</label><mml:math id="M4"><mml:mtable class="eqnarray" columnalign="right center left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:mfrac><mml:mstyle displaystyle="true"><mml:munderover accentunder="false" accent="false"><mml:mrow><mml:mo>&#x0222B;</mml:mo></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:munderover></mml:mstyle><mml:mo>|</mml:mo><mml:mi>y</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:mfrac><mml:mstyle displaystyle="true"><mml:munderover accentunder="false" accent="false"><mml:mrow><mml:mo>&#x0222B;</mml:mo></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:munderover></mml:mstyle><mml:mi>y</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mi>x</mml:mi><mml:mo>|</mml:mo><mml:mi>d</mml:mi><mml:mi>x</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
</sec>
<sec id="s3">
<title>Simulation Results and Discussions</title>
<p>A sequential FEM<sup>25</sup> is adopted in this paper to solve the magneto-mechanical coupling problem and obtain the micro deformation of MRE model in Figure <xref ref-type="fig" rid="F1">1</xref> by using commercial software COMSOL Multiphysics. Besides, in the mesoscopic scale, the applied magnetic field can be assumed to follow a uniform distribution. Therefore, in the magneto module (MFNC module in COMSOL), the magnetic scalar potential is equal zero on the boundary <italic>ab</italic>. The magnetic field applied on the MREs is in y-direction, and the particles will be magnetized in the same direction, and their magnetization is set to 1,000 kA/m. in the simulation. The other boundaries are considered as magnetic insulation. Meanwhile, in the structural mechanical module (SOLID module in COMSOL), we use <bold>U</bold> &#x0003D; (<italic>u</italic>,<italic>v</italic>) to describe the displacement field of MRE. The boundary condition is chosen in such a way where u &#x0003D; 0 for boundary ad, and v &#x0003D; 0 for boundary ab, and other boundaries are free to move. Also, the magnetic force <bold>F</bold><sub><italic>i</italic></sub> on each particle surface &#x02202;&#x003A9;<sub><italic>i</italic></sub> can be calculated from the local field <bold>H</bold> with the aid of the Maxwell stress tensor <bold>&#x003C3;</bold><sup>M</sup> as follow (Ly et al., <xref ref-type="bibr" rid="B19">1999</xref>):</p>
<disp-formula id="E5"><label>(5)</label><mml:math id="M5"><mml:mtable class="eqnarray" columnalign="right center left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>F</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>&#x003C0;</mml:mi><mml:msub><mml:mrow><mml:mi>&#x003BC;</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:msub><mml:mrow><mml:mo>&#x0222E;</mml:mo></mml:mrow><mml:mrow><mml:mi>&#x02202;</mml:mi><mml:msub><mml:mrow><mml:mi>&#x003A9;</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mo>&#x003C3;</mml:mo></mml:mstyle></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msup><mml:mo>&#x000B7;</mml:mo><mml:mstyle mathvariant="bold"><mml:mtext>n</mml:mtext></mml:mstyle><mml:mtext>ds</mml:mtext><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>&#x003C0;</mml:mi></mml:mrow></mml:mfrac><mml:msub><mml:mrow><mml:mo>&#x0222E;</mml:mo></mml:mrow><mml:mrow><mml:mi>&#x02202;</mml:mi><mml:msub><mml:mrow><mml:mi>&#x003A9;</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>&#x003BC;</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>H</mml:mtext></mml:mstyle><mml:mstyle mathvariant="bold"><mml:mtext>H</mml:mtext></mml:mstyle><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo>|</mml:mo><mml:mo>|</mml:mo><mml:mstyle mathvariant="bold"><mml:mtext>H</mml:mtext></mml:mstyle><mml:mo>|</mml:mo><mml:msup><mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mrow><mml:mstyle mathvariant="bold"><mml:mn>2</mml:mn></mml:mstyle></mml:mrow></mml:msup><mml:mstyle mathvariant="bold"><mml:mtext>I</mml:mtext></mml:mstyle></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>&#x000B7;</mml:mo><mml:mstyle mathvariant="bold"><mml:mtext>n</mml:mtext></mml:mstyle><mml:mtext>ds</mml:mtext></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<p>Where <bold>n</bold> is the unit normal vector on &#x02202;&#x003A9;<sub><italic>i</italic></sub>, &#x003BC;<sub>i</sub>, &#x003BC;<sub>0</sub> is the permeability for the particles and matrix, respectively, <bold>I</bold> is the 2th rank unit tensor.</p>
<p>The simulation results of the MRE model under an applied magnetic field of 1,000 kA/m are shown in Figure <xref ref-type="fig" rid="F2">2</xref>. Figure <xref ref-type="fig" rid="F2">2A</xref> gives the results of magnetic flux density distribution in the simulation domain. Figure <xref ref-type="fig" rid="F2">2B</xref> shows the stress field of MRE induced by the magnetic field. These results illustrate that the magnetized particles in the matrix would interact with each other though magnetic force and create the magnetic induced stress in the MRE. As a result, the MRE will deform irregularly.</p>
<fig id="F2" position="float">
<label>Figure 2</label>
<caption><p>Simulation results with proposed model <bold>(A)</bold> Magnetic flux density distribution. <bold>(B)</bold> Stress field. <bold>(C)</bold> Magnetic induced displacements of the surface profile in Y-direction. <bold>(D)</bold> The new surface profile after the magnetization of 1,000 KA/m is applied on. And the roughness of the surface is 1.53 &#x003BC;m.</p></caption>
<graphic xlink:href="fmats-05-00076-g0002.tif"/>
</fig>
<p>Figure <xref ref-type="fig" rid="F2">2C</xref> shows the positive field-induced displacements of MRE which are consistent with previous studies is shown. However, the irregularity of the surface deformation by the magnetic field is ignored by most theoretical predictions (Davis, <xref ref-type="bibr" rid="B8">1999</xref>; Yin et al., <xref ref-type="bibr" rid="B26">2002</xref>; Dorfmann and Ogden, <xref ref-type="bibr" rid="B9">2004</xref>; Kankanala and Triantafyllidis, <xref ref-type="bibr" rid="B14">2004</xref>; Chen et al., <xref ref-type="bibr" rid="B5">2013</xref>; Allahyarov et al., <xref ref-type="bibr" rid="B1">2014</xref>; Andriushchenko et al., <xref ref-type="bibr" rid="B2">2014</xref>; Menzel, <xref ref-type="bibr" rid="B21">2014</xref>; Cremer et al., <xref ref-type="bibr" rid="B6">2015</xref>). The cause of the irregular deformation of MREs can be explained by the magnetic induced stress field shown in Figure <xref ref-type="fig" rid="F2">2B</xref>. It can be seen that the magnetic stress is not uniformly distributed in the matrix. The stress is larger where particles are assembled, while it is smaller where the particles are scattered. Figure <xref ref-type="fig" rid="F2">2D</xref> shows the new surface profile of MRE after the magnetic field is applied. It is obtained by superposing the initial profile of Figure <xref ref-type="fig" rid="F1">1B</xref> and the magnetic induced displacement of Figure <xref ref-type="fig" rid="F2">2C</xref> together. Figure <xref ref-type="fig" rid="F2">2D</xref> shows that the roughness of the MRE is 1.53 &#x003BC;m. Comparing with the initial roughness <italic>Ra</italic> &#x0003D; 1.67 &#x003BC;m, the roughness of the MRE is decreased. The simulation results are consistent with previous experimental observations (Lian et al., <xref ref-type="bibr" rid="B17">2015</xref>, <xref ref-type="bibr" rid="B18">2016</xref>; Li et al., <xref ref-type="bibr" rid="B16">2018</xref>) where the magnetic fields reduce the friction coefficient of MREs. However, when we evaluate the relative roughness changes as a function of the initial surface roughness of MREs, the magnetic fields do not always induce negative changes in surface roughness. Figure <xref ref-type="fig" rid="F3">3A</xref> presents the simulation results of the relative roughness changes <italic>Ra</italic><sub><italic>r</italic></sub> with initial surface roughness of MREs. The relative roughness change is defined as: <italic>Ra</italic><sub><italic>r</italic></sub> &#x0003D; (<italic>Ra</italic><sub><italic>m</italic></sub>-<italic>Ra</italic>)/<italic>Ra</italic>, where <italic>Ra</italic><sub><italic>m</italic></sub>, is the roughness under a magnetic field, <italic>Ra</italic> is the initial surface roughness. We can see that when the initial profile roughness <italic>Ra</italic> &#x0003C; 1 &#x003BC;m, the relative roughness changes under a magnetic field are positive. This suggests that the smooth surface profiles of MREs will become rough under magnetic field; in other words, the friction coefficient of MREs will increase by an applied magnetic field. We suggested that this phenomenon is a result of that the roughness of the magnetic induced deformation itself will be the dominant role of <italic>Ra</italic><sub><italic>r</italic></sub>, When initial profile is smooth. And we can also explain it by an extreme assumption; we can imagine a MRE which has an absolute smooth surface that should become rough when superimposed on inhomogeneous deformation, which is the result of external magnetic loading.</p>
<fig id="F3" position="float">
<label>Figure 3</label>
<caption><p>Magnetic induced relative roughness <italic>Ra</italic><sub><italic>r</italic>.</sub> <bold>(A)</bold> Relationship with initial roughness. <bold>(B)</bold> Relationship with particle volume fraction.</p></caption>
<graphic xlink:href="fmats-05-00076-g0003.tif"/>
</fig>
<p>Further, we investigate how the particle contents in MREs affect the relative roughness changes under the same applied magnetic field. We set the initial surface roughness of MREs to be 1.67 &#x003BC;m, and analyze the field-induced surface profiles for the MREs with different particle volume fractions. Figure <xref ref-type="fig" rid="F3">3B</xref> shows the simulation results of the relative roughness changes as a function of particle volume fractions of MREs.</p>
<p>It can be seen that there is an optimal particle volume fraction for the relative roughness changes, in which the <italic>Ra</italic><sub><italic>r</italic></sub> has the maximum absolute value. It is interesting to note that optimal particle volume fraction for the <italic>Ra</italic><sub><italic>r</italic></sub> is around 9%, but is not 27%, in which the magnetorheological effect is predicted the strongest (Davis, <xref ref-type="bibr" rid="B8">1999</xref>). It is suggested that it is a result of the particle distribution which varies with the particle volume fraction. When the particle volume fraction of MREs is low (&#x0003C;9%), the interactions between particles would be increased with its volume fraction, then the irregularity of the magnetostriction is also increased with it. However, when the particle volume fraction is higher than 9%, the distance between particles would be decreased and particle distribution becomes more uniform. Then the irregularity of the magnetostriction is reduced although the particle interactions are increasing. This finding could give an explanation as to why the MRE with 10% particle fraction has the largest tunable friction coefficients in reference Li et al. (<xref ref-type="bibr" rid="B16">2018</xref>)</p>
</sec>
<sec id="s4">
<title>Experimental Validation</title>
<p>In this section, two kinds of isotropic MREs are manufactured by uniformly embedded magnetic particles into silicon rubber (HT-18; Shanghai Tongshuai Co., Ltd, China). As shown in Figure <xref ref-type="fig" rid="F4">4</xref>, One kind of MREs is the samples are prepared by using the glass mold, and the other is made by the iron mold. So, the latter has rougher surface than the former. A white light interferometer (ContourGT-K, Germany BRUKER company), of which the vertical resolution can reach 0.01 nm is used to obtain the three-dimensional morphology of all MRE samples.</p>
<fig id="F4" position="float">
<label>Figure 4</label>
<caption><p>MRE samples <bold>(A)</bold> the glass mold <bold>(B)</bold> the MRE generated in glass mold <bold>(C)</bold> iron mold <bold>(D)</bold> the MRE generated in iron mold.</p></caption>
<graphic xlink:href="fmats-05-00076-g0004.tif"/>
</fig>
<p>Figure <xref ref-type="fig" rid="F5">5</xref> gives the white light interferometer test results of a 10% particle volume fraction MRE that is generated by iron mold. Figure <xref ref-type="fig" rid="F5">5A</xref> gives the initial surface contour of the MRE, Figure <xref ref-type="fig" rid="F5">5B</xref> gives the surface contour of MRE under a uniform vertical magnetic field of 450 mT, and Figure <xref ref-type="fig" rid="F5">5C</xref> gives the 2 dimension picture of the MRE surface profiles with/without a uniform vertical magnetic field. It can be seen that the roughness of MRE sample decreases from 2.76 to 2.36 &#x003BC;m when a magnetic field is applied. However, for the MRE samples made by glass mold, the magnetic field increases the surface roughness (see Figure <xref ref-type="fig" rid="F6">6</xref>).</p>
<fig id="F5" position="float">
<label>Figure 5</label>
<caption><p>White light interferometer test results of a typical MRE (10% particle volume fraction, generated by iron mold) <bold>(A)</bold> the initial surface contour, the roughness is 2.76 &#x003BC;m <bold>(B)</bold> the surface contour under a magnetic field of 450 mT, the roughness is 2.36 &#x003BC;m <bold>(C)</bold> the 2-demension profiles of MRE.</p></caption>
<graphic xlink:href="fmats-05-00076-g0005.tif"/>
</fig>
<fig id="F6" position="float">
<label>Figure 6</label>
<caption><p>Comparisons of roughness of MREs for different particle volume fractions with/without magnetic field.</p></caption>
<graphic xlink:href="fmats-05-00076-g0006.tif"/>
</fig>
<p>Figure <xref ref-type="fig" rid="F6">6</xref> shows the results of surface roughness of MREs with different particle contents made by iron mold with/without magnetic field. The results of roughness of MRE samples with 10% particle volume fraction made by glass mold are also presented in the Figure <xref ref-type="fig" rid="F6">6</xref>. As can be seen, the initial roughness of MREs mainly depends on the roughness of corresponding molds. And MREs fabricated by the steel mold has the similar roughness. Besides, Figure <xref ref-type="fig" rid="F6">6</xref> shows that the MRE with 10% particle volume fraction has the largest relative roughness change, which conforms to the theoretical prediction. This is different to MREs made by iron mold, as the surface roughness of MREs generated by glass mold increases with magnetic fields.</p>
</sec>
<sec sec-type="conclusions" id="s5">
<title>Conclusion</title>
<p>In summary, a mesoscopic model that considers deformation of surface micro-structures of MREs has been established to predict the surface roughness of MREs under magnetic fields. The model can explain how the field-induced friction of MRE changes as a function of particle contents, and how the initial surface roughness affects the changes. In addition, the proposed model has been verified experimentally. These findings may contribute to the area of interfacial friction control, in which controllable friction surfaces or techniques are expected to apply for the design of high efficient smart devices and mechanical systems.</p>
</sec>
<sec id="s6">
<title>Author Contributions</title>
<p>RL and XL contribute the experimental parts of this works. XW gives some advises of this manuiscript and farbricates the MRE. SC writes the paper and establishes the surface model of MRE.</p>
<sec>
<title>Conflict of Interest Statement</title>
<p>The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.</p>
</sec>
</sec>
</body>
<back>
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<fn fn-type="financial-disclosure"><p><bold>Funding.</bold> Financial supports from the National Natural Science Foundation of China (Grant NO.11502038, NO.11572320, NO.11372366). And it also supports from China Postdoctoral Science Foundation (NO. 2017M610386).</p>
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