Summary
Two yet undiscovered relations between the Eulerian logarithmic strain inV and two fundamental mechanical quantities, the stretching and the Cauchy stress, are disclosed. A new spin tensor and a new objective tensor-rate are accordingly introduced. Further, new rate-form constitutive models based on this objective tensor-rate are established. It is proved that
-
(i).
an objective corotational rate of the logarithmic strain inV can be exactly identical with the stretching and in all strain measures only inV enjoys this property, and
-
(ii).
InV and the Cauchy stress σ form a work-conjugate pair of strain and stress.
These properties of in ν are shown to determine a unique smooth spin tensor called logarithmic spin and by virtue of this spin a new tensor-rate called logarithmic rate is proposed. In all possible rate-form constitutive models relating the same kind of objective corotational rates of an Eulerian stress measure and an Eulerian strain measure, it is proved that the logarithmic rate is the only choice and the strain measure must be the logarithmic strain inV if the stretching, as is commonly assumed, is used to measure the rate of change of deformation. As an illustration, it is shown that all finite deformation responses of the grade-zero hypoelastic model based on the logarithmic rate are completely in agreement with those of a finite deformation elastic model and moreover this simplest rate-form constitutive model based on the logarithmic rate can predict the phenomenon of the known hypoelastic yield at simple shear.
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Xiao, H., Bruhns, O.T. & Meyers, A. Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mechanica 124, 89–105 (1997). https://doi.org/10.1007/BF01213020
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DOI: https://doi.org/10.1007/BF01213020