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On isotropic tensor-functions and the measure of deformation

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Zusammenfassung

Es wird gezeigt, dass eine isotrope Beziehung zwischen zwei symmetrischen Tensoren zweiten Ranges auf die Form

gebracht werden kann, wobei

sowie dieA n,B n,C n ähnliche Potenzreihen in den Hauptinvarianten vonx rs sind. Die Konstantenk müssen durch geeignete Versuche bestimmt werden, die über den Zusammenhang zwischen den physikalischen Grössenx rs undy rs Auskunft geben. Die

können aber auch Konstanten sein. Dies trifft beispielsweise zu, wennx rs undy rs verschiedene Masse der Deformation sind. In diesem Falle kann man zum Beispiel das logarithmische Hencky-Mass durch das Almansi-Mass ausdrücken. Tut man dies, dann ergeben sich aus einer quasilinearen Spannungs-Fluss-Beziehung gewisse Effekte zweiter Ordnung in zähen Flüssigkeiten, auch dann, wenn das Medium unelastisch und die Zähigkeit gering ist. Damit liesse sich ein Zentripetalpump-Effekt in Luft, der neulich beobachtet worden ist, theoretisch erklären.

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Abbreviations

A :

indicatingAlmansi

A n,B n,C n :

coefficients

C :

indicatingCauchy

e :

strain

G :

indicatingGreen

g rs :

metric tensor

H :

indicatingHencky

i, j, k :

principal directions

K n :

constant coefficient

l :

length

l 0 :

initial length

l 1 :

final length

p rs :

stress tensor

S :

amount of shear, also indicatingSwainger

V :

volume

v :

velocity

x rs ,y rs :

symmetrical tensors of rank two

α:

coefficient

Δl :

elongation

δ:

Kronecker's delta

η:

coefficient of viscosity

ηc :

coefficient of cross viscosity

λ:

Lamé's constant

μ:

shear modulus

I, II, III:

principal invariants

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Authors and Affiliations

  1. Israel Institute of Technology, Haifa, Israel

    Meir Hanin & Markus Reiner

Authors
  1. Meir Hanin
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  2. Markus Reiner
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Additional information

The research reported in this document has been sponsored in part by the Air Research and Development Command, United States Air Force, under Contract AF 61 (514)-871, through the European Office, ARDC.

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Hanin, M., Reiner, M. On isotropic tensor-functions and the measure of deformation. Journal of Applied Mathematics and Physics (ZAMP) 7, 377–393 (1956). https://doi.org/10.1007/BF01606325

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