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On the theory of the viscoelastic behavior of soft polymers in moderately large deformations

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Summary

Restricting consideration to deformations in which time shift invariance is preserved, a series of models was developed for the prediction of the viscoelastic behavior of isotropic soft polymers. The models describe the time dependent stress through the Boltzmann superposition integral incorporating into it an appropriately chosen nonlinear measure of strain. The theory was developed in its most general threedimensional form. It requires a single time function, the relaxation modulus of linear viscoelastic theory. In its simplest form the strain measure contains a single material parameter.

Zusammenfassung

Unter der Beschränkung auf Deformationen, für welche Invarianz gegen Verschiebungen in der Zeit vorliegt, wird eine Reihe von Modellen zur Voraussage des viskoelastischen Verhaltens isotroper weicher Polymerer entwickelt. Diese Modelle beschreiben die zeitabhängige Spannung durch ein Boltzmannsches Superpositionsintegral, in das ein geeignet gewähltes nicht-lineares Dehnungsmaß eingeführt wird. Die Theorie wird in der allgemeinen drei-dimensionalen Form dargestellt. Sie verwendet eine einzige Zeitfunktion, nämlich den Relaxationsmodul der linearen Theorie der Viskoelastizität. In der einfachsten Form enthält das Dehnungsmaß nur einen Stoffparameter.

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Abbreviations

\(\underline{\underline B} \) :

Finger's deformation tensor

B :

a body

C 1 ,C 2 :

Mooney-Rivlin constants

\(\underline{\underline C} \) :

Green's deformation tensor

\(\underline{\underline c} \) :

Cauchy's deformation tensor

c :

a proportionality constant

C P :

heat capacity at constant pressure

C V :

heat capacity at constant volume

\(\underline{\underline d} \) :

stretching tensor

da :

elemental deformed area

dA :

elemental undeformed area

ds :

elemental deformed length

dS :

elemental undeformed length

\(\underline{\underline E} \) G :

Green's strain tensor

\(\underline{\underline E} \) c :

Cauchy's infinitesimal strain tensor

\(\underline{\underline E} \) p :

Piola's strain tensor

\(\underline{\underline E} \) tg :

generalized Lagrangean strain tensor

\(\underline{\underline E} \) tn :

Lagrangeann-measure of strain tensor

\(\underline{\underline e} \) A :

Almansi's strain tensors

\(\underline{\underline e} \) s :

Swainger's infinitesimal strain tensor

\(\underline{\underline e} \) F :

Finger's strain tensor

\(\underline{\underline e} \) eg :

generalized Eulerian strain tensor

\(\underline{\underline e} \) en :

Euleriann-measure of strain tensor

\(\underline{\underline F} \) :

deformation gradient tensor

\(\underline{\underline G} \) :

velocity gradient tensor

G :

shear modulus

G e :

equilibrium shear modulus

\(\underline{\underline I} \) :

unit (metric) tensor

III C :

determinant of\(\underline{\underline C} \)

III c :

determinant of\(\underline{\underline c} \)

M i :

memory functions

n :

strain parameter

\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{n} \) :

matrix of the eigenvectors of\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{B} \)

\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{N} \) :

matrix of the eigenvectors\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{C} \)

\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{O} \) :

null matrix

P :

hydrostatic pressure

\(\underline{\underline Q}\) :

orthogonal tensor

\(\underline{\underline R} \) :

rotation tensor of the deformation gradient

r :

a vector

\(\underline{\underline S} \) :

Piola's stress tensor

\(\underline{\underline T} \) :

a tensor

t :

present time

u :

past time

\(\underline{\underline U} \) :

right stretch tensor

V :

volume

\(\underline{\underline V} \) :

left stretch tensor

ν :

velocity

x :

position vector of a particle at timet

\(\underline {\tilde x}\) :

instantaneous position vector of a particle at timeu

X :

position vector of a particle in the reference configuration

α :

index of principal directions

γ :

C P /C v

δ :

Kronecker's delta

η :

Newtonian viscosity

λ x :

stretch ratios in the principal directions

\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\Lambda } \) :

matrix of eigenvalues of\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{C} \) and\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{B} \)

ξ :

a material point or particle

\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\Phi } _i \) :

generalized strain functions

\(\underline{\underline \omega } \) :

vorticity tensor

\(\underline{\underline {\bar \sigma }} \) :

Cauchy's or true stress tensor

∇ :

del operator

subscriptt :

denotes a vector, tensor or matrix relative to the present timet

subscriptu :

denotes a vector, tensor, or matrix relative to the past timeu

superscript* :

denotes rotated coordinates

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Authors and Affiliations

  1. Division of Chemistry and Chemical Engineering, California Institute of Technology, 91125, Pasadena, California, USA

    W. V. Chang & R. Bloch

  2. Department of Chemical Engineering, University of Southern California, 90007, Los Angeles, California, USA

    N. W. Tschoegl

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  2. R. Bloch
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  3. N. W. Tschoegl
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Chang, W.V., Bloch, R. & Tschoegl, N.W. On the theory of the viscoelastic behavior of soft polymers in moderately large deformations. Rheol Acta 15, 367–378 (1976). https://doi.org/10.1007/BF01574493

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