सामान्यीकृत मैक्सवेल मॉडल
मैक्सवेल-वीचर्ट मॉडल का योजनाबद्ध
सामान्यीकृत मैक्सवेल मॉडल को मैक्सवेल-विचर्ट मॉडल के रूप में भी जाना जाता है (जेम्स क्लर्क मैक्सवेल और ई विचर्ट के बाद)[1] [2] मैक्सवेल सामग्री समानांतर में इकट्ठी की जाती हैं। यह ध्यान में रखा जाता है कि आराम का समय एक बार में नहीं होता है, बल्कि समय के एक सेट में होता है। अलग-अलग लंबाई के आणविक खंडों की उपस्थिति के कारण, छोटे वाले लंबे समय से कम योगदान देते हैं, एक अलग-अलग समय वितरण होता है। Wiechert मॉडल वितरण को सटीक रूप से दर्शाने के लिए जितने आवश्यक हैं उतने स्प्रिंग-डैशपॉट मैक्सवेल तत्व होने से यह दिखाता है। दाईं ओर का आंकड़ा सामान्यीकृत वीचर्ट मॉडल दिखाता है।[3] [4]
सामान्य मॉडल प्रपत्र ठोस दिया गया N + 1 {\displaystyle N+1} E i {\displaystyle E_{i}} η i {\displaystyle \eta _{i}} τ i = η i E i {\displaystyle \tau _{i}={\frac {\eta _{i}}{E_{i}}}} [citation needed :
General Maxwell Solid Model (
1 )
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}}}}}
= {\displaystyle =}
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}}}}}
This may be more easily understood by showing the model in a slightly more expanded form:
General Maxwell Solid Model (
2 )
σ + {\displaystyle \sigma +} ( ∑ i = 1 N τ i ) ∂ σ ∂ t + {\displaystyle {\left({\sum _{i=1}^{N}{\tau _{i}}}\right)}{\frac {\partial {\sigma }}{\partial {t}}}+} ( ∑ i = 1 N − 1 ( ∑ j = i + 1 N τ i τ j ) ) ∂ 2 σ ∂ t 2 {\displaystyle {\left({\sum _{i=1}^{N-1}{\left({\sum _{j=i+1}^{N}{\tau _{i}\tau _{j}}}\right)}}\right)}{\frac {\partial ^{2}{\sigma }}{\partial {t}^{2}}}} + . . . + {\displaystyle +...+}
( ∑ 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 \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}}}} + . . . + {\displaystyle +...+} ( ∏ i = 1 N τ i ) ∂ N σ ∂ t N {\displaystyle \left({\prod _{i=1}^{N}{\tau _{i}}}\right){\frac {\partial ^{N}{\sigma }}{\partial {t}^{N}}}}
= {\displaystyle =}
E 0 ϵ + {\displaystyle E_{0}\epsilon +} ( ∑ i = 1 N ( E 0 + E i ) τ i ) ∂ ϵ ∂ t + {\displaystyle {\left({\sum _{i=1}^{N}{\left({E_{0}+E_{i}}\right)\tau _{i}}}\right)}{\frac {\partial {\epsilon }}{\partial {t}}}+} ( ∑ i = 1 N − 1 ( ∑ j = i + 1 N ( E 0 + E i + E j ) τ i τ j ) ) ∂ 2 ϵ ∂ t 2 {\displaystyle {\left({\sum _{i=1}^{N-1}{\left({\sum _{j=i+1}^{N}{\left({E_{0}+E_{i}+E_{j}}\right)\tau _{i}\tau _{j}}}\right)}}\right)}{\frac {\partial ^{2}{\epsilon }}{\partial {t}^{2}}}} + . . . + {\displaystyle +...+}
( ∑ 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 \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}}}} + . . . + {\displaystyle +...+} ( E 0 + ∑ j = 1 N E j ) ( ∏ i = 1 N τ i ) ∂ N ϵ ∂ t N {\displaystyle \left({E_{0}+\sum _{j=1}^{N}E_{j}}\right)\left({\prod _{i=1}^{N}{\tau _{i}}}\right){\frac {\partial ^{N}{\epsilon }}{\partial {t}^{N}}}}
उपरोक्त मॉडल के साथ N + 1 = 2 {\displaystyle N+1=2}
Standard Linear Solid Model (
3 )
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}}}}
तरल पदार्थ दिया गया N + 1 {\displaystyle N+1} E i {\displaystyle E_{i}} η i {\displaystyle \eta _{i}} τ i = η i E i {\displaystyle \tau _{i}={\frac {\eta _{i}}{E_{i}}}}
General Maxwell Fluid Model (
4 )
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}}}}}
= {\displaystyle =}
∑ 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}}}}}
This may be more easily understood by showing the model in a slightly more expanded form:
General Maxwell Fluid Model (
5 )
σ + {\displaystyle \sigma +} ( ∑ i = 1 N τ i ) ∂ σ ∂ t + {\displaystyle {\left({\sum _{i=1}^{N}{\tau _{i}}}\right)}{\frac {\partial {\sigma }}{\partial {t}}}+} ( ∑ i = 1 N − 1 ( ∑ j = i + 1 N τ i τ j ) ) ∂ 2 σ ∂ t 2 {\displaystyle {\left({\sum _{i=1}^{N-1}{\left({\sum _{j=i+1}^{N}{\tau _{i}\tau _{j}}}\right)}}\right)}{\frac {\partial ^{2}{\sigma }}{\partial {t}^{2}}}} + . . . + {\displaystyle +...+}
( ∑ 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 \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}}}} + . . . + {\displaystyle +...+} ( ∏ i = 1 N τ i ) ∂ N σ ∂ t N {\displaystyle \left({\prod _{i=1}^{N}{\tau _{i}}}\right){\frac {\partial ^{N}{\sigma }}{\partial {t}^{N}}}}
= {\displaystyle =}
( η 0 + ∑ i = 1 N E i τ i ) ∂ ϵ ∂ t + {\displaystyle {\left({\eta _{0}+\sum _{i=1}^{N}{E_{i}\tau _{i}}}\right)}{\frac {\partial {\epsilon }}{\partial {t}}}+} ( η 0 + ∑ i = 1 N − 1 ( ∑ j = i + 1 N ( E i + E j ) τ i τ j ) ) ∂ 2 ϵ ∂ t 2 {\displaystyle {\left({\eta _{0}+\sum _{i=1}^{N-1}{\left({\sum _{j=i+1}^{N}{\left({E_{i}+E_{j}}\right)\tau _{i}\tau _{j}}}\right)}}\right)}{\frac {\partial ^{2}{\epsilon }}{\partial {t}^{2}}}} + . . . + {\displaystyle +...+}
( 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 \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}}}} + . . . + {\displaystyle +...+} ( η 0 + ( ∑ j = 1 N E j ) ( ∏ i = 1 N τ i ) ) ∂ N ϵ ∂ t N {\displaystyle \left({\eta _{0}+\left({\sum _{j=1}^{N}E_{j}}\right)\left({\prod _{i=1}^{N}{\tau _{i}}}\right)}\right){\frac {\partial ^{N}{\epsilon }}{\partial {t}^{N}}}}
उदाहरण: तीन पैरामीटर द्रव मानक रेखीय ठोस मॉडल के अनुरूप मॉडल तीन पैरामीटर द्रव है, जिसे जेफ़रीज़ मॉडल के रूप में भी जाना जाता है:[5]
Three Parameter Maxwell Fluid Model (
6 )
σ + τ 1 ∂ σ ∂ t = ( η 0 + τ 1 E 1 ) ∂ ϵ ∂ t {\displaystyle \sigma +\tau _{1}{\frac {\partial {\sigma }}{\partial {t}}}=\left({\eta _{0}+\tau _{1}E_{1}}\right){\frac {\partial {\epsilon }}{\partial {t}}}}
संदर्भ
↑ Wiechert, E (1889); "Ueber elastische Nachwirkung", Dissertation, Königsberg University, Germany
↑ 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
↑ Roylance, David (2001); "Engineering Viscoelasticity", 14-15
↑ Tschoegl, Nicholas W. (1989); "The Phenomenological Theory of Linear Viscoelastic Behavior", 119-126
↑ Gutierrez-Lemini, Danton (2013). Engineering Viscoelasticity ISBN 9781461481393 
