Maxwell body

Maxwell body

In rheological modeling, the Maxwell body is the series connection of a (linear) Hooke's spring and a damper . From the model properties of these elements,

• the Hooke's law for the spring and ${\displaystyle \sigma =E\cdot \varepsilon }$
• the speed or rate dependency of the damper ${\displaystyle \sigma =\eta \cdot {\dot {\varepsilon }}}$

in combination with the assumption of the addition of strain from the series connection of spring and damper or their rates for the total strain, results directly from insertion ${\displaystyle \varepsilon =\varepsilon _{F}+\varepsilon _{D}}$ ${\displaystyle {\dot {\varepsilon }}={\dot {\varepsilon }}_{F}+{\dot {\varepsilon }}_{D}}$

${\displaystyle {\dot {\varepsilon }}={\frac {{\dot {\sigma }}(t)}{E}}+{\frac {\sigma (t)}{\eta }}}$

as a descriptive differential equation .

The properties of this system can best be shown and discussed if one subjects this strand to a creep or retardation experiment on the one hand and to a relaxation experiment on the other hand and examines the respective reaction.

1. A creep or retardation experiment means the application of a voltage jump to the system , whereby we denote the Heaviside function , i.e. a jump from zero to one . From the descriptive differential equation one obtains the expression through integration with respect to time ${\displaystyle \sigma (t)=\sigma _{0}\,H(t)}$ ${\displaystyle H(t)}$ ${\displaystyle t=t_{0}=0}$ ${\displaystyle t}$

${\displaystyle \varepsilon (t)=\int {\dot {\varepsilon }}{\rm {d}}t={\frac {\sigma _{0}}{E}}+\int \limits _{0}^{t}{\frac {\sigma }{\eta }}{\rm {d}}t={\frac {\sigma _{0}}{E}}+{\frac {\sigma _{0}}{\eta }}\,t}$

for the stretch response of this body under a tension jump of . ${\displaystyle \sigma _{0}}$

This shows us the well-known behavior of a constant response due to the voltage jump at the spring element, but also a (linear) time dependence. This is precisely what describes the unlimited "further stretching" (" creeping ") of this system with the (constant) tension applied here . ${\displaystyle {\frac {\sigma _{0}}{E}}}$ ${\displaystyle \sigma _{0}}$

2. The relaxation experiment shows the response of the system to a strain jump . From the above, descriptive differential equation we immediately see that we only have to solve the homogeneous DGL , because here always applies: ${\displaystyle \varepsilon (t)=\varepsilon _{0}\,H(t)}$ ${\displaystyle {\dot {\varepsilon }}(t)=0}$

${\displaystyle {\frac {{\dot {\sigma }}(t)}{E}}+{\frac {\sigma (t)}{\eta }}=0}$or with as a typical relaxation time . ${\displaystyle \tau ^{\star }{\dot {\sigma }}(t)+\sigma (t)=0}$ ${\displaystyle \tau ^{\star }={\frac {\eta }{E}}}$

The solution of this DGL is an exponential function of the form , whereby the constant of integration results from the initial condition . So the solution is straightforward ${\displaystyle \sigma (t)=c\,e^{-t/\tau ^{\star }}}$ ${\displaystyle \sigma (t=0)=E\varepsilon _{0}}$

${\displaystyle \sigma (t)=E\varepsilon _{0}\,e^{-t/\tau ^{\star }}}$.

The jump in expansion causes a jump in tension . Then the spring contracts and the expansion goes into the damper. This means that the system continues to relax with a given total expansion. This is called "relaxation." ${\displaystyle \varepsilon _{0}}$ ${\displaystyle t=0}$ ${\displaystyle \sigma (t=0)=E\varepsilon _{0}}$

typical stress relaxation

The relaxation curve can be seen here for exemplary characteristic values MPa, MPa.s. The relaxation time is thus s. ${\displaystyle E=4}$ ${\displaystyle \eta =3}$ ${\displaystyle \tau ^{\star }=0.75}$

literature

• H. Altenbach: Continuum Mechanics . Springer, 2012, ISBN 978-3-642-24118-5 .
• P. Haupt: Continuum Mechanics and Theory of Materials . Springer, 2000, ISBN 3-540-66114-X .