Chapter 4
Visco-Elasticity
I’d say the differences are more interesting than the similarities
at this point.
— Philip Glass, b. 1973 —
Visco-elasticity is the paradigm for rate-dependent, either (asymptotically) reversible
(in the case of solids) or non-reversible (in the case of fluids) material behavior.
Thereby experimental evidence for various classes of materials, in particular for
polymers (typically above the glass transition temperature) and fluids, suggests to
introduce the total stress or parts of it (as the response to an external load) as being
rate-dependent. Rate-dependence is rooted in sub-scale time-dependent relaxation
processes in the material, take as an example chain disentanglements in polymeric
materials with a chain network sub-scale structure. From a convex analysis point
of view, rate-dependence is engraved in the smoothness of the convex dissipation
potential and its dual. The elementary rheological model to capture viscous, i.e. ratedependent material behavior in terms of a one-to-one relation between the stress
and the rate of strain is the viscous dashpot. The rheological model for a viscous
fluid consisting of a viscous dashpot only is denoted the Newton model. Parallel and serial arrangements of a viscous dashpot with an elastic spring render the
Kelvin model for visco-elastic solids and the Maxwell model for visco-elastic fluids,
respectively. Further, serial or parallel arrangements of either several Kelvin models
or several Maxwell models are established as the Generalized-Kelvin model and the
Generalized-Maxwell model, respectively. Particular three parameter sub-cases of
the corresponding specific Generalized-Kelvin or Generalized-Maxwell model are
denoted as the Standard-Linear-Solid (SLS) Kelvin or Maxwell model, respectively,
and the Standard-Linear-Fluid (SLF) Kelvin or Maxwell model (which shall not be
considered here), respectively.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
P. Steinmann and K. Runesson, The Catalogue of Computational Material Models,
https://doi.org/10.1007/978-3-030-63684-5_4
77
Visco-Elasticity
I’d say the differences are more interesting than the similarities
at this point.
— Philip Glass, b. 1973 —
Visco-elasticity is the paradigm for rate-dependent, either (asymptotically) reversible
(in the case of solids) or non-reversible (in the case of fluids) material behavior.
Thereby experimental evidence for various classes of materials, in particular for
polymers (typically above the glass transition temperature) and fluids, suggests to
introduce the total stress or parts of it (as the response to an external load) as being
rate-dependent. Rate-dependence is rooted in sub-scale time-dependent relaxation
processes in the material, take as an example chain disentanglements in polymeric
materials with a chain network sub-scale structure. From a convex analysis point
of view, rate-dependence is engraved in the smoothness of the convex dissipation
potential and its dual. The elementary rheological model to capture viscous, i.e. ratedependent material behavior in terms of a one-to-one relation between the stress
and the rate of strain is the viscous dashpot. The rheological model for a viscous
fluid consisting of a viscous dashpot only is denoted the Newton model. Parallel and serial arrangements of a viscous dashpot with an elastic spring render the
Kelvin model for visco-elastic solids and the Maxwell model for visco-elastic fluids,
respectively. Further, serial or parallel arrangements of either several Kelvin models
or several Maxwell models are established as the Generalized-Kelvin model and the
Generalized-Maxwell model, respectively. Particular three parameter sub-cases of
the corresponding specific Generalized-Kelvin or Generalized-Maxwell model are
denoted as the Standard-Linear-Solid (SLS) Kelvin or Maxwell model, respectively,
and the Standard-Linear-Fluid (SLF) Kelvin or Maxwell model (which shall not be
considered here), respectively.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
P. Steinmann and K. Runesson, The Catalogue of Computational Material Models,
https://doi.org/10.1007/978-3-030-63684-5_4
77