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Generalized Maxwell model
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<h2 id="general-model-form">General model form</h2> <h3>Solids</h3> <p>Given N + 1 {\displaystyle N+1} elements with moduli E i {\displaystyle E_{i}} , viscosities η i {\displaystyle \eta _{i}} , and relaxation times τ i = η i E i {\displaystyle \tau _{i}={\frac {\eta _{i}}{E_{i}}}} </p><p>The general form for the model for solids is given by : </p> General Maxwell Solid Model (1) <p> σ + {\displaystyle \sigma +} ∑ n = 1 N ( ∑ i 1 = 1 N − n + 1 . . . ( ∑ i a = i a − 1 + 1 N − ( n − a ) + 1 . . . ( ∑ i n = i n − 1 + 1 N ( ∏ j ∈ { i 1 , . . . , i n } τ j ) ) . . . ) . . . ) ∂ n σ ∂ t n {\displaystyle \sum _{n=1}^{N}{\left({\sum _{i_{1}=1}^{N-n+1}{...\left({\sum _{i_{a}=i_{a-1}+1}^{N-\left({n-a}\right)+1}{...\left({\sum _{i_{n}=i_{n-1}+1}^{N}{\left({\prod _{j\in \left\{{i_{1},...,i_{n}}\right\}}{\tau _{j}}}\right)}}\right)...}}\right)...}}\right){\frac {\partial ^{n}{\sigma }}{\partial {t}^{n}}}}} </p><p> = {\displaystyle =} </p><p> E 0 ϵ + {\displaystyle E_{0}\epsilon +} ∑ n = 1 N ( ∑ i 1 = 1 N − n + 1 . . . ( ∑ i a = i a − 1 + 1 N − ( n − a ) + 1 . . . ( ∑ i n = i n − 1 + 1 N ( ( E 0 + ∑ j ∈ { i 1 , . . . , i n } E j ) ( ∏ k ∈ { i 1 , . . . , i n } τ k ) ) ) . . . ) . . . ) ∂ n ϵ ∂ t n {\displaystyle \sum _{n=1}^{N}{\left({\sum _{i_{1}=1}^{N-n+1}{...\left({\sum _{i_{a}=i_{a-1}+1}^{N-\left({n-a}\right)+1}{...\left({\sum _{i_{n}=i_{n-1}+1}^{N}{\left({\left({E_{0}+\sum _{j\in \left\{{i_{1},...,i_{n}}\right\}}{E_{j}}}\right)\left({\prod _{k\in \left\{{i_{1},...,i_{n}}\right\}}{\tau _{k}}}\right)}\right)}}\right)...}}\right)...}}\right){\frac {\partial ^{n}{\epsilon }}{\partial {t}^{n}}}}} </p> <h4>Example: <a href="/facts/Standard_linear_solid_model/zPVQNDYE">standard linear solid model</a></h4> <p>Following the above model with N + 1 = 2 {\displaystyle N+1=2} elements yields the <a href="/facts/Standard_linear_solid_model/zPVQNDYE">standard linear solid model</a>: </p> <a href="/facts/Standard_linear_solid_model/zPVQNDYE">Standard Linear Solid Model</a> (3) <p> σ + τ 1 ∂ σ ∂ t = E 0 ϵ + τ 1 ( E 0 + E 1 ) ∂ ϵ ∂ t {\displaystyle \sigma +\tau _{1}{\frac {\partial {\sigma }}{\partial {t}}}=E_{0}\epsilon +\tau _{1}\left({E_{0}+E_{1}}\right){\frac {\partial {\epsilon }}{\partial {t}}}} </p> <h3>Fluids</h3> <p>Given N + 1 {\displaystyle N+1} elements with moduli E i {\displaystyle E_{i}} , viscosities η i {\displaystyle \eta _{i}} , and relaxation times τ i = η i E i {\displaystyle \tau _{i}={\frac {\eta _{i}}{E_{i}}}} </p><p>The general form for the model for fluids is given by: </p> General Maxwell Fluid Model (4) <p> σ + {\displaystyle \sigma +} ∑ n = 1 N ( ∑ i 1 = 1 N − n + 1 . . . ( ∑ i a = i a − 1 + 1 N − ( n − a ) + 1 . . . ( ∑ i n = i n − 1 + 1 N ( ∏ j ∈ { i 1 , . . . , i n } τ j ) ) . . . ) . . . ) ∂ n σ ∂ t n {\displaystyle \sum _{n=1}^{N}{\left({\sum _{i_{1}=1}^{N-n+1}{...\left({\sum _{i_{a}=i_{a-1}+1}^{N-\left({n-a}\right)+1}{...\left({\sum _{i_{n}=i_{n-1}+1}^{N}{\left({\prod _{j\in \left\{{i_{1},...,i_{n}}\right\}}{\tau _{j}}}\right)}}\right)...}}\right)...}}\right){\frac {\partial ^{n}{\sigma }}{\partial {t}^{n}}}}} </p><p> = {\displaystyle =} </p><p> ∑ n = 1 N ( η 0 + ∑ i 1 = 1 N − n + 1 . . . ( ∑ i a = i a − 1 + 1 N − ( n − a ) + 1 . . . ( ∑ i n = i n − 1 + 1 N ( ( ∑ j ∈ { i 1 , . . . , i n } E j ) ( ∏ k ∈ { i 1 , . . . , i n } τ k ) ) ) . . . ) . . . ) ∂ n ϵ ∂ t n {\displaystyle \sum _{n=1}^{N}{\left({\eta _{0}+\sum _{i_{1}=1}^{N-n+1}{...\left({\sum _{i_{a}=i_{a-1}+1}^{N-\left({n-a}\right)+1}{...\left({\sum _{i_{n}=i_{n-1}+1}^{N}{\left({\left({\sum _{j\in \left\{{i_{1},...,i_{n}}\right\}}{E_{j}}}\right)\left({\prod _{k\in \left\{{i_{1},...,i_{n}}\right\}}{\tau _{k}}}\right)}\right)}}\right)...}}\right)...}}\right){\frac {\partial ^{n}{\epsilon }}{\partial {t}^{n}}}}} </p> <h4>Example: three parameter fluid</h4> <p>The analogous model to the <a href="/facts/Standard_linear_solid_model/zPVQNDYE">standard linear solid model</a> is the three parameter fluid, also known as the Jeffreys model:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> Three Parameter Maxwell Fluid Model (6) <p> σ + τ 1 ∂ σ ∂ t = ( η 0 + τ 1 E 1 ∂ ∂ t ) ∂ ϵ ∂ t {\displaystyle \sigma +\tau _{1}{\frac {\partial {\sigma }}{\partial {t}}}=\left({\eta _{0}+\tau _{1}E_{1}{\frac {\partial }{\partial t}}}\right){\frac {\partial {\epsilon }}{\partial {t}}}} </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Wiechert, E (1889); "Ueber elastische Nachwirkung", Dissertation, Königsberg University, Germany <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Wiechert, E (1893); "Gesetze der elastischen Nachwirkung für constante Temperatur", Annalen der Physik, Vol. 286, issue 10, p. 335–348 and issue 11, p. 546–570 <a href="https://doi.org/10.1002/andp.18932861011" target="_blank">https://doi.org/10.1002/andp.18932861011</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Roylance, David (2001); "Engineering Viscoelasticity", 14-15 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Tschoegl, Nicholas W. (1989); "The Phenomenological Theory of Linear Viscoelastic Behavior", 119-126 <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Gutierrez-Lemini, Danton (2013). Engineering Viscoelasticity. Springer. p. 88. ISBN 9781461481393. <a href="9781461481393" target="_blank">9781461481393</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>