Английская Википедия:Generalized Maxwell model
The generalized Maxwell model also known as the Maxwell–Wiechert model (after James Clerk Maxwell and E Wiechert[1][2]) is the most general form of the linear model for viscoelasticity. In this model several Maxwell elements are assembled in parallel. It takes into account that the relaxation does not occur at a single time, but in a set of times. Due to the presence of molecular segments of different lengths, with shorter ones contributing less than longer ones, there is a varying time distribution. The Wiechert model shows this by having as many spring–dashpot Maxwell elements as are necessary to accurately represent the distribution. The figure on the right shows the generalised Wiechert model.[3][4]
General model form
Solids
Given <math>N+1</math> elements with moduli <math>E_i</math>, viscosities <math>\eta_i</math>, and relaxation times <math>\tau_i=\frac{\eta_i}{E_i}</math>
The general form for the model for solids is given by Шаблон:Citation needed: Шаблон:Equation box 1{\partial{t}^{n}} } </math>
<math> = </math>
<math>E_0\epsilon+</math> <math> \sum^{N}_{n=1}{ \left({ \sum^{N-n+1}_{i_1=1}{ ... \left({ \sum^{N-\left({n-a}\right)+1}_{i_a=i_{a-1}+1}{ ... \left({ \sum^{N}_{i_n=i_{n-1}+1}{ \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}} } </math> |cellpadding = 6 |border = 1 |border colour = black |background colour = white}}
Шаблон:Divhide Шаблон:Equation box 1\right)} \frac{\partial{\sigma}}{\partial{t}} + </math> <math> {\left({\sum^{N-1}_{i=1}{ \left({\sum^{N}_{j=i+1}{ \tau_i\tau_j }}\right) }}\right)} \frac{\partial^{2}{\sigma}}{\partial{t}^{2}} </math> <math>+...+</math>
<math> \left({ \sum^{N-n+1}_{i_1=1}{ ... \left({ \sum^{N-\left({n-a}\right)+1}_{i_a=i_{a-1}+1}{ ... \left({ \sum^{N}_{i_n=i_{n-1}+1}{ \left({ \prod_{j\in\left\{{i_1,...,i_n}\right\}}{ \tau_j } }\right) } }\right) ... } }\right) ... } }\right) \frac{\partial^{n}{\sigma}}{\partial{t}^{n}} </math> <math>+...+</math> <math> \left({ \prod^{N}_{i=1}{ \tau_i } }\right) \frac{\partial^{N}{\sigma}}{\partial{t}^{N}} </math>
<math> = </math>
<math>E_0\epsilon+</math> <math> {\left({\sum^{N}_{i=1}{\left({E_0+E_i}\right)\tau_i}}\right)} \frac{\partial{\epsilon}}{\partial{t}} + </math> <math> {\left({\sum^{N-1}_{i=1}{ \left({\sum^{N}_{j=i+1}{ \left({E_0+E_i+E_j}\right) \tau_i\tau_j }}\right) }}\right)} \frac{\partial^{2}{\epsilon}}{\partial{t}^{2}} </math> <math>+...+</math>
<math> \left({ \sum^{N-n+1}_{i_1=1}{ ... \left({ \sum^{N-\left({n-a}\right)+1}_{i_a=i_{a-1}+1}{ ... \left({ \sum^{N}_{i_n=i_{n-1}+1}{ \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}} </math> <math>+...+</math> <math> \left({ E_0+\sum_{j=1}^{N} E_j }\right) \left({ \prod^{N}_{i=1}{ \tau_i } }\right) \frac{\partial^{N}{\epsilon}}{\partial{t}^{N}} </math> |cellpadding = 6 |border = 1 |border colour = black |background colour = white}} Шаблон:Divhide
Example: standard linear solid model
Following the above model with <math>N+1=2</math> elements yields the standard linear solid model: Шаблон:Equation box 1{\partial{t}}=E_0\epsilon+\tau_1\left({E_0+E_1}\right)\frac{\partial{\epsilon}}{\partial{t}} </math> |cellpadding = 6 |border = 1 |border colour = black |background colour = white}}
Fluids
Given <math>N+1</math> elements with moduli <math>E_i</math>, viscosities <math>\eta_i</math>, and relaxation times <math>\tau_i=\frac{\eta_i}{E_i}</math>
The general form for the model for fluids is given by: Шаблон:Equation box 1{\partial{t}^{n}} } </math>
<math> = </math>
<math> \sum^{N}_{n=1}{ \left({ \eta_0+\sum^{N-n+1}_{i_1=1}{ ... \left({ \sum^{N-\left({n-a}\right)+1}_{i_a=i_{a-1}+1}{ ... \left({ \sum^{N}_{i_n=i_{n-1}+1}{ \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}} } </math> |cellpadding = 6 |border = 1 |border colour = black |background colour = white}}
Шаблон:Divhide Шаблон:Equation box 1\right)} \frac{\partial{\sigma}}{\partial{t}} + </math> <math> {\left({\sum^{N-1}_{i=1}{ \left({\sum^{N}_{j=i+1}{ \tau_i\tau_j }}\right) }}\right)} \frac{\partial^{2}{\sigma}}{\partial{t}^{2}} </math> <math>+...+</math>
<math> \left({ \sum^{N-n+1}_{i_1=1}{ ... \left({ \sum^{N-\left({n-a}\right)+1}_{i_a=i_{a-1}+1}{ ... \left({ \sum^{N}_{i_n=i_{n-1}+1}{ \left({ \prod_{j\in\left\{{i_1,...,i_n}\right\}}{ \tau_j } }\right) } }\right) ... } }\right) ... } }\right) \frac{\partial^{n}{\sigma}}{\partial{t}^{n}} </math> <math>+...+</math> <math> \left({ \prod^{N}_{i=1}{ \tau_i } }\right) \frac{\partial^{N}{\sigma}}{\partial{t}^{N}} </math>
<math> = </math>
<math> {\left({\eta_0+\sum^{N}_{i=1}{E_i\tau_i}}\right)} \frac{\partial{\epsilon}}{\partial{t}} + </math> <math> {\left({\eta_0+\sum^{N-1}_{i=1}{ \left({\sum^{N}_{j=i+1}{ \left({E_i+E_j}\right) \tau_i\tau_j }}\right) }}\right)} \frac{\partial^{2}{\epsilon}}{\partial{t}^{2}} </math> <math>+...+</math>
<math> \left({ \eta_0+ \sum^{N-n+1}_{i_1=1}{ ... \left({ \sum^{N-\left({n-a}\right)+1}_{i_a=i_{a-1}+1}{ ... \left({ \sum^{N}_{i_n=i_{n-1}+1}{ \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}} </math> <math>+...+</math> <math> \left({ \eta_0+ \left({ \sum_{j=1}^{N} E_j }\right) \left({ \prod^{N}_{i=1}{ \tau_i } }\right) }\right) \frac{\partial^{N}{\epsilon}}{\partial{t}^{N}} </math> |cellpadding = 6 |border = 1 |border colour = black |background colour = white}} Шаблон:Divhide
Example: three parameter fluid
The analogous model to the standard linear solid model is the three parameter fluid, also known as the Jeffreys model:[5] Шаблон:Equation box 1{\partial{t}}=\left({\eta_0+\tau_1 E_1\frac{\partial}{\partial t}}\right)\frac{\partial{\epsilon}}{\partial{t}} </math> |cellpadding = 6 |border = 1 |border colour = black |background colour = white}}
References
- ↑ 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
- ↑ Шаблон:Cite book
