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Dynamic modulus

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Dynamic modulus (sometimes complex modulus<a href="#cite_note-TOU-1">[1]</a>) is the ratio of stress to strain under vibratory conditions (calculated from data obtained from either free or forced vibration tests, in shear, compression, or elongation). It is a property of <a href="Viscoelastic" class="mw-redirect" title="Viscoelastic">viscoelastic</a> materials.

<div class="mw-heading mw-heading2"><h2 id="Viscoelastic_stress–strain_phase-lag">Viscoelastic stress–strain phase-lag</h2></div>

<a href="Viscoelasticity" title="Viscoelasticity">Viscoelasticity</a> is studied using <a href="Dynamic_mechanical_analysis" title="Dynamic mechanical analysis">dynamic mechanical analysis</a> where an oscillatory force (stress) is applied to a material and the resulting displacement (strain) is measured.<a href="#cite_note-2">[2]</a>

<ul><li>In purely <a href="Elasticity_(physics)" title="Elasticity (physics)">elastic</a> materials the stress and strain occur in <a href="Phase_(waves)" title="Phase (waves)">phase</a>, so that the response of one occurs simultaneously with the other.</li>

<li>In purely <a href="Viscosity" title="Viscosity">viscous</a> materials, there is a <a href="Phase_(waves)#Phase_difference" title="Phase (waves)">phase difference</a> between stress and strain, where strain lags stress by a 90 degree (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /2}">

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<annotation encoding="application/x-tex">{\displaystyle \pi /2}</annotation>

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</math><img src="./2b44e3d874a0b229fded7ffce67a0677dd5b8b67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.657ex; height:2.843ex;" alt="{\displaystyle \pi /2}" loading="lazy"> <a href="Radian" title="Radian">radian</a>) phase lag.</li>

<li>Viscoelastic materials exhibit behavior somewhere in between that of purely viscous and purely elastic materials, exhibiting some phase lag in strain.<a href="#cite_note-Meyers-3">[3]</a></li></ul>

Stress and strain in a viscoelastic material can be represented using the following expressions:

<ul><li>Strain: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon =\varepsilon _{0}\sin(\omega t)}">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon =\varepsilon _{0}\sin(\omega t)}</annotation>

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</math><img src="./63b7b4c6a7498f1e2a14cc0bc14f862c0947f229.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.657ex; height:2.843ex;" alt="{\displaystyle \varepsilon =\varepsilon _{0}\sin(\omega t)}" loading="lazy"></li>

<li>Stress: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =\sigma _{0}\sin(\omega t+\delta )\,}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma =\sigma _{0}\sin(\omega t+\delta )\,}</annotation>

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</math><img src="./395604f37d810ec2503a6f1ee72da74119759836.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.423ex; height:2.843ex;" alt="{\displaystyle \sigma =\sigma _{0}\sin(\omega t+\delta )\,}" loading="lazy"> <a href="#cite_note-Meyers-3">[3]</a></li></ul>

where

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<annotation encoding="application/x-tex">{\displaystyle \omega =2\pi f}</annotation>

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</math><img src="./7a1bf35d395c2d52391265e4bbda0aed14f52579.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.317ex; height:2.509ex;" alt="{\displaystyle \omega =2\pi f}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is frequency of strain oscillation,</dd>

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> is time,</dd>

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<annotation encoding="application/x-tex">{\displaystyle \delta }</annotation>

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</math><img src="./c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"> is phase lag between stress and strain.</dd></dl>

The stress relaxation modulus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\left(t\right)}">

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<annotation encoding="application/x-tex">{\displaystyle G\left(t\right)}</annotation>

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</math><img src="./3e19143d64cdc16f1ec7eac6a23bedc0ba1a5e56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.863ex; height:2.843ex;" alt="{\displaystyle G\left(t\right)}" loading="lazy"> is the ratio of the stress remaining at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> after a step strain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

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</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"> was applied at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}">

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<annotation encoding="application/x-tex">{\displaystyle t=0}</annotation>

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</math><img src="./43469ec032d858feae5aa87029e22eaaf0109e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle G\left(t\right)={\frac {\sigma \left(t\right)}{\varepsilon }}}</annotation>

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</math><img src="./c7e00559751db31215c69c069c81aef10182ded2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.163ex; height:5.676ex;" alt="{\displaystyle G\left(t\right)={\frac {\sigma \left(t\right)}{\varepsilon }}}" loading="lazy">,

which is the time-dependent generalization of <a href="Hooke's_law" title="Hooke's law">Hooke's law</a>.

For visco-elastic solids, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\left(t\right)}">

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<annotation encoding="application/x-tex">{\displaystyle G\left(t\right)}</annotation>

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</math><img src="./3e19143d64cdc16f1ec7eac6a23bedc0ba1a5e56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.863ex; height:2.843ex;" alt="{\displaystyle G\left(t\right)}" loading="lazy"> converges to the equilibrium shear modulus<a href="#cite_note-4">[4]</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle G=\lim _{t\to \infty }G(t)}</annotation>

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</math><img src="./68e87e79736acb1887ff12098a2b9d850d8980c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.668ex; height:4.009ex;" alt="{\displaystyle G=\lim _{t\to \infty }G(t)}" loading="lazy">.</dd></dl>

The <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of the shear relaxation modulus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(t)}">

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<annotation encoding="application/x-tex">{\displaystyle G(t)}</annotation>

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</math><img src="./e3d6c09ba5569413364689bf4837c7b71ef0892f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.476ex; height:2.843ex;" alt="{\displaystyle G(t)}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {G}}(\omega )={\hat {G}}'(\omega )+i{\hat {G}}''(\omega )}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {G}}(\omega )={\hat {G}}'(\omega )+i{\hat {G}}''(\omega )}</annotation>

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</math><img src="./e1c899f23ed558798b31ac570fc4920e3a443fe1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.809ex; height:3.676ex;" alt="{\displaystyle {\hat {G}}(\omega )={\hat {G}}'(\omega )+i{\hat {G}}''(\omega )}" loading="lazy"> (see below).

<div class="mw-heading mw-heading3"><h3 id="Storage_and_loss_modulus">Storage and loss modulus</h3></div>

The storage and loss modulus in viscoelastic materials measure the stored energy, representing the elastic portion, and the energy dissipated as heat, representing the viscous portion.<a href="#cite_note-Meyers-3">[3]</a> The tensile storage and loss moduli are defined as follows:

<ul><li>Storage: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E'={\frac {\sigma _{0}}{\varepsilon _{0}}}\cos \delta }">

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<annotation encoding="application/x-tex">{\displaystyle E'={\frac {\sigma _{0}}{\varepsilon _{0}}}\cos \delta }</annotation>

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</math><img src="./2a04f494caf829b36584012c15d2af3de025ff9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.729ex; height:5.009ex;" alt="{\displaystyle E'={\frac {\sigma _{0}}{\varepsilon _{0}}}\cos \delta }" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle E''={\frac {\sigma _{0}}{\varepsilon _{0}}}\sin \delta }</annotation>

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</math><img src="./61843c5fe0980cb78c97449cf9b2a895ee67444d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.926ex; height:5.009ex;" alt="{\displaystyle E''={\frac {\sigma _{0}}{\varepsilon _{0}}}\sin \delta }" loading="lazy"> <a href="#cite_note-Meyers-3">[3]</a></li></ul>

Similarly we also define shear storage and shear loss moduli, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'}">

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<annotation encoding="application/x-tex">{\displaystyle G'}</annotation>

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</math><img src="./76634fad5818a777669a77cd8c86d1d816e4c402.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.511ex; height:2.509ex;" alt="{\displaystyle G'}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G''}">

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<annotation encoding="application/x-tex">{\displaystyle G''}</annotation>

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</math><img src="./5a37ec377e9bb29c7dd95a844c1b230fbbebea75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.964ex; height:2.509ex;" alt="{\displaystyle G''}" loading="lazy">.

Complex variables can be used to express the moduli <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle E^{*}}</annotation>

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</math><img src="./a91d2119445b3b65384ba491c4b95f1557571ecc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.848ex; height:2.343ex;" alt="{\displaystyle E^{*}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle G^{*}}</annotation>

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</math><img src="./3885f050db0d7715ebeba11a07383eb252b2da6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.881ex; height:2.343ex;" alt="{\displaystyle G^{*}}" loading="lazy"> as follows:

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<annotation encoding="application/x-tex">{\displaystyle E^{*}=E'+iE''\,}</annotation>

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</math><img src="./fca9c7f0c8abdbfebe58ec6c4ff7a553ca49361e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.386ex; height:2.676ex;" alt="{\displaystyle E^{*}=E'+iE''\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle G^{*}=G'+iG''\,}</annotation>

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</math><img src="./1291d3d1ab39726ea3182b79741054f7c810c4eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.485ex; height:2.676ex;" alt="{\displaystyle G^{*}=G'+iG''\,}" loading="lazy"> <a href="#cite_note-Meyers-3">[3]</a></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> is the <a href="Imaginary_unit" title="Imaginary unit">imaginary unit</a>.

<div class="mw-heading mw-heading3"><h3 id="Ratio_between_loss_and_storage_modulus">Ratio between loss and storage modulus</h3></div>

The ratio of the loss modulus to storage modulus in a viscoelastic material is defined as the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \delta }">

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<annotation encoding="application/x-tex">{\displaystyle \tan \delta }</annotation>

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</math><img src="./60037a84b6a15450308cb179eecc9e4690f4959c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.795ex; height:2.343ex;" alt="{\displaystyle \tan \delta }" loading="lazy">, (cf. <a href="Loss_tangent" class="mw-redirect" title="Loss tangent">loss tangent</a>), which provides a measure of damping in the material. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \delta }">

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<annotation encoding="application/x-tex">{\displaystyle \tan \delta }</annotation>

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</math><img src="./60037a84b6a15450308cb179eecc9e4690f4959c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.795ex; height:2.343ex;" alt="{\displaystyle \tan \delta }" loading="lazy"> can also be visualized as the tangent of the phase angle (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">

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<annotation encoding="application/x-tex">{\displaystyle \delta }</annotation>

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</math><img src="./c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy">) between the storage and loss modulus.

Tensile: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \delta ={\frac {E''}{E'}}}">

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<annotation encoding="application/x-tex">{\displaystyle \tan \delta ={\frac {E''}{E'}}}</annotation>

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</math><img src="./78e8b112bc03707110098c56a82dbb4dfe315cca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.661ex; height:5.509ex;" alt="{\displaystyle \tan \delta ={\frac {E''}{E'}}}" loading="lazy">

Shear: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \delta ={\frac {G''}{G'}}}">

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<annotation encoding="application/x-tex">{\displaystyle \tan \delta ={\frac {G''}{G'}}}</annotation>

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</math><img src="./188ea0f800b4ba557279b2bad63e9ca71e814b5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.694ex; height:5.676ex;" alt="{\displaystyle \tan \delta ={\frac {G''}{G'}}}" loading="lazy">

For a material with a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \delta }">

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<annotation encoding="application/x-tex">{\displaystyle \tan \delta }</annotation>

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</math><img src="./60037a84b6a15450308cb179eecc9e4690f4959c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.795ex; height:2.343ex;" alt="{\displaystyle \tan \delta }" loading="lazy"> greater than 1, the energy-dissipating, viscous component of the complex modulus prevails.

<ul><li><a href="Dynamic_mechanical_analysis" title="Dynamic mechanical analysis">Dynamic mechanical analysis</a></li>

<li><a href="Elastic_modulus" title="Elastic modulus">Elastic modulus</a></li>

<li><a href="Palierne_equation" title="Palierne equation">Palierne equation</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-TOU-1"><a href="#cite_ref-TOU_1-0">^</a> The Open University (UK), 2000. T838 Design and Manufacture with Polymers: Solid properties and design, page 30. Milton Keynes: The Open University.

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080916080329/http://las.perkinelmer.com/content/ApplicationNotes/APP_FilmsandCoatings.pdf">"PerkinElmer "Mechanical Properties of Films and Coatings""</a> (PDF). Archived from <a rel="nofollow" class="external text" href="http://las.perkinelmer.com/content/ApplicationNotes/APP_FilmsandCoatings.pdf">the original</a> (PDF) on 2008-09-16. Retrieved 2009-05-09.</cite>

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<li id="cite_note-Meyers-3">^ <a href="#cite_ref-Meyers_3-0">a</a> <a href="#cite_ref-Meyers_3-1">b</a> <a href="#cite_ref-Meyers_3-2">c</a> <a href="#cite_ref-Meyers_3-3">d</a> <a href="#cite_ref-Meyers_3-4">e</a> Meyers and Chawla (1999): "Mechanical Behavior of Materials," 98-103.

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFRubinstein,_Michael,_1956_December_20-2003" class="citation book cs1">Rubinstein, Michael, 1956 December 20- (2003). Polymer physics. Colby, Ralph H. Oxford: Oxford University Press. p. 284. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>019852059X</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/50339757">50339757</a>.</cite><code class="cs1-code">{{cite book}}</code>: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)

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