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The significance of second and higher degrees terms in the continuum mechanics of infinitesimally small strains

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The systems of coordinates

OnlyCartesian systems will be used

Simplification

In order to avoid difficulties and complications which have no relation to the problems here under consideration strains have been restricted to two dimensions.

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Abbreviations

A, B, R :

orthogonal tensors of rotation

Q, S, s :

symmetrical tensors of elongation

a andp :

attached symbols indicating “ante” and “post” rotation

α, β, γ :

angles of rotation

D :

subscript indicating diagonal form

ε :

infinitesimal quantity of first degree

\(\left| {\begin{array}{*{20}c} {\begin{array}{*{20}c} .. \\ .. \\ \end{array} } \\ \end{array} } \right|\) :

abbreviated form for a tensor of two dimensions\(\left| {\begin{array}{*{20}c} {\begin{array}{*{20}c} . & . & 0 \\ . & . & 0 \\ 0 & 0 & . \\ \end{array} } \\ \end{array} } \right|\)

[ ]:

these brackets round the index of an equation means that the equation refers only to orders of magnitude

r :

rank of approach to degeneracy

~:

superscript indicating transposition of indices

orthogonality:

\(\tilde M\) =M −1

symmetry:

\(\tilde N\) =N

antisymmetry:

\(\tilde L\) = −L

f (T) :

\(f(T) = \left| {\begin{array}{*{20}c} {f(T_1 )} \\ 0 \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ {f(T_2 )} \\ 0 \\ \end{array} } \right.\left. {\begin{array}{*{20}c} 0 \\ 0 \\ {f(T_3 )} \\ \end{array} } \right|\)

References

  • Weissenberg, K. Arch. Sci. Phys. Nat.11, 22 (1935).

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  • Lagally, M. andW. Franz, Vorlesungen über Vektorrechnung, 6. Aufl., Mathematik und ihre Anwendungen in Phys. und Technik, pp. 246ff. (Leipzig 1959).

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Authors and Affiliations

  1. Ridderlaan 111, Den Haag, Holland

    K. Weissenberg

Authors
  1. K. Weissenberg
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Weissenberg, K. The significance of second and higher degrees terms in the continuum mechanics of infinitesimally small strains. Rheol Acta 14, 477–483 (1975). https://doi.org/10.1007/BF01525301

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  • DOI: https://doi.org/10.1007/BF01525301

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