सामान्यीकृत मैक्सवेल मॉडल
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मैक्सवेल-वीचर्ट प्रारूप का योजनाबद्ध
सामान्यीकृत मैक्सवेल प्रारूप को मैक्सवेल-विचर्ट प्रारूप के रूप में भी जाना जाता है (जेम्स क्लर्क मैक्सवेल और ई विचर्ट के बाद)[1] [2] मैक्सवेल तत्व समानांतर में एकत्रित होते हैं। यह ध्यान में रखा जाता है कि छूट एक बार में नहीं, बल्कि समय के समुच्चय में होता है। अलग-अलग लंबाई के आणविक खंडों की उपस्थिति के कारण, छोटे वाले लंबे समय से कम योगदान देते हैं, अलग-अलग समय वितरण होता है। वीचर्ट प्रारूप वितरण को सही रूप से दर्शाने के लिए जितने आवश्यक हैं उतने स्प्रिंग-डैशपॉट मैक्सवेल तत्व होने से यह दिखाता है। दाईं ओर का आंकड़ा सामान्यीकृत वीचर्ट प्रारूप दिखाता है।[3] [4]
सामान्य प्रारूप 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}}}}
ठोस के लिए प्रारूप का सामान्य रूप किसके द्वारा दिया गया है :
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 प्रारूप 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}}}}
तरल पदार्थ के प्रारूप के लिए सामान्य रूप निम्न द्वारा दिया गया है:
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 प्रारूप के अनुरूप प्रारूप तीन पैरामीटर द्रव है, जिसे जेफ़रीज़ प्रारूप 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 
