ibiomechanic's blog
https://imechanica.org/blog/10115604
enEffective stiffness as a concave function of inhomogeneity properties
https://imechanica.org/node/25546
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p class="MsoNormal">Hi everyone, I am a postdoc at Yale University. I am posting on some of my previous work on fiber networks and heterogeneous materials. We reported that effective stiffness is a concave function of inhomogeneity properties, and effective stiffness becomes smaller with increasing heterogeneity as an example consequence. </p>
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<p class="MsoNormal">Fiber networks are assemblies of connected rods or springs. We used them as models of biological polymers. In my work on networks, we considered the uniaxial testing on a network before and after adding a fiber. Using reciprocity, we derived a relationship between the change in effective stiffness or overall stiffness of the network and the stiffness of a single added fiber</p>
<p class="MsoNormal">Δ<em>E</em>network=<em>a</em>/(<em>c</em>+1/<em>k</em>rod),</p>
<p class="MsoNormal">where <em>a</em> describes the deformation of the network at the points where the new fiber was attached prior to its attachment, and <em>c</em> describes the network response function at the attachment points. Both <em>a</em> and <em>c</em> are positive. According to this relation</p>
<p class="MsoNormal">∂2<em>E</em>network/∂<em>k</em>2rod≤0.</p>
<p class="MsoNormal"><span>This relation indicates that networks with fibers having a distribution of stiffnesses are softer, at the same mean fiber properties. The reason is that the rate of change of </span><em>E</em>network <span>with </span><em>k</em>rod <span>becomes slower at higher </span><em>k</em>rod. <span>The decrease of </span><em>E</em>network <span>from softer fibers is larger than its increase by stiffer fibers. Algebraically an inequality may be derived from the Taylor series of </span><em>E</em>network <span>in the space of </span><em>k</em>rod<span>’s about the average of </span><em>k</em>rod. The Taylor series is regarded as a weak contrast expansion in previous literature on continua. </p>
<p class="MsoNormal">Later I extended this relation to heterogeneous continua. This resulted in a relationship for change of effective stiffness as a function of the mode of deformations of the inhomogeneity. This relationship is also a concave function. The result of continua becoming softer with increasing heterogeneity has been previously reported empirically to the best of my knowledge.</p>
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<p class="MsoNormal">I plan to work on further applications of these relationships in biomaterials.</p>
<p class="MsoNormal">Referense: [<a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4669583/pdf/nihms739638.pdf">preprint_networks</a>, <a href="https://arxiv.org/pdf/1801.05066">preprint_continua</a>, and <a href="https://doi.org/10.1016/j.jmps.2015.11.001">postprint_networks</a>, <a href="https://doi.org/10.1016/j.mechmat.2018.11.007">postprint_continua</a>]</p>
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</div></div></div>Thu, 04 Nov 2021 01:00:45 +0000ibiomechanic25546 at https://imechanica.orghttps://imechanica.org/node/25546#commentshttps://imechanica.org/crss/node/25546