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Decreasing of the initial shear modulus with increasing axial strain explained by means of a plastic-hypoelastic model

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Summary

The purpose of this work is to examine in detail the possibility to explain the decreasing of the initial shear modulus with increasing axial strain, observed first by Feigen, by means of the plastic-hypoelastic stress-strain relation suggested by Lehmann and by the author of the present paper.

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Abbreviations

ɛ ij :

components of the infinitesimal strain tensor dilatation

\(\varepsilon '_{ij} = \varepsilon ij - \frac{1}{3}\theta \delta _{ij} \) :

strain deviator

σ ij :

components of the stress tensor

θ:

spherical part of the stress tensor

\(\sigma '_{ij} = \sigma _{ij} - \frac{1}{3}\Theta \delta _{\iota j} \) :

stress deviator

τ 2=σ ′ ij σ ′ ij :

second invariant of the stress deviator

ɛ=ɛ 33 :

axial strain

e=ɛ 13 :

shear component of the strain tensor

γ=2ɛ 13 :

shear strain

σ=σ 33 :

axial stress

s=σ 13 :

shear stress

T ij :

components of Cauchy's stress tensor

F ij :

components of the deformation gradient

L ij :

components of the velocity gradient (Eulerian coordinates)

\(D_{ij} = \frac{1}{2}(L_{ij} + L_{ji} )\) :

components of the rate of deformation tensor

\(W_{ij} = \frac{1}{2}(L_{ij} - L_{ji} )\) :

components of the spin tensor

\(d_{ij} = D_{ij} - \frac{1}{3}D_{kk} \delta _{ij} \) :

components of the rate of deformations deviator

\(t_{ij} = T_{ij} - \frac{1}{3}T_{kk} \delta _{ij} \) :

components of Cauchy's stress deviator

T=T 33 :

axial Cauchy's stress

References

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  1. Institut für Mechanik, Technische Hochschule Darmstadt, Hochschulstrasse 1, D-6100, Darmstadt, Germany

    P. Mazilu

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  1. P. Mazilu
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Mazilu, P. Decreasing of the initial shear modulus with increasing axial strain explained by means of a plastic-hypoelastic model. Acta Mechanica 56, 93–115 (1985). https://doi.org/10.1007/BF01306026

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  • DOI: https://doi.org/10.1007/BF01306026

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