An alternative approach to finite deformation

Abstract

In elastoplasticity formulation constitutive relations are usually given in rate form, i.e. they represent relations between stress rate and strain rate. The adopted constitutive laws have to stay independent in relation to the change of frame of reference, i.e. to stay objective. While the objectivity requirement in a material description is automatically satisfied, in an Eulerian description, especially in the case of large deformations, the objectivity requirement can be violated even for objective quantities. Thus, instead of a material time derivative in the Eulerian description objective time derivatives have to be implemented. In this work the importance of the objective rate implementation in the constitutive relations of finite elastoplasticity is clarified. Likewise, it shows the overview of the most frequently used objective rates nowadays, their advantages and shortcomings, as well as the distinctive features of the recently introduced logarithmic rate.

On instability of constitutive models for isotropic elastic-perfectly plastic material behaviour at finite deformations

“…The corotational rates that are examined here are the Jaumann rate, the Green-Naghdi rate and the logarithmic rate (the Log-rate). Their corresponding spin tensors 16 are given as…”

“…represent a unique work conjugate pair (see Trajkovic´-Milenkovic´and Bruhns 16 for details and further references). Using these quantities, the relation ( 16) becomes F ¼ ðq 0 � aÞ : ðq 0 � aÞ � gð�Þ ð 19Þ…”

“…2,3 Definition of a suitable constitutive model for finite elastoplasticity with the objective rates implementation was the topic of numerous theoretical studies at turn of the century [4][5][6][7][8][9][10][11][12][13] as well as in the recent detailed numerical studies. [14][15][16][17][18][19] It is proved on theoretical models that the application of the objective rates incorporated in the material models of contemporary commercial packages for structural analysis gives physically unrealistic results in the case of large, monotonic as well as cyclic deformations, in which the shear deformation is predominant. It is also shown that only constitutive relations based on the recently discovered logarithmic rate [8][9][10][11][12][14][15][16][17][18][19] predict the theoretically expected behaviour of materials.…”

“…[14][15][16][17][18][19] It is proved on theoretical models that the application of the objective rates incorporated in the material models of contemporary commercial packages for structural analysis gives physically unrealistic results in the case of large, monotonic as well as cyclic deformations, in which the shear deformation is predominant. It is also shown that only constitutive relations based on the recently discovered logarithmic rate [8][9][10][11][12][14][15][16][17][18][19] predict the theoretically expected behaviour of materials.…”