Schematic of Maxwell–Wiechert model The Generalized Maxwell model also known as the Maxwell–Wiechert model (after James Clerk Maxwell and E Wiechert[1] [2] 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
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))))
The general form for the model for solids is given by[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))))
Following the above model with
N
+
1
=
2
{\displaystyle N+1=2}
standard linear solid model :
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))))
Fluids
Given
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))))
The general form for the model for fluids is given by:
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))))
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]
Three Parameter Maxwell Fluid Model (
6 )
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))))
