The objective of this article is the evaluation of the validity range of two objective rates, the nowadays most frequently used Jaumann rate and the recently introduced logarithmic rate, in material behaviour prediction due to large deformations. For that purpose the model subjected to large monotonic uniaxial elastoplastic deformation has been examined as well as the initially prestressed models which are afterwards exposed to the cyclic process of combined elastic lengthening and shearing. It is shown that whenever the participation of shear deformation to the total deformation stays in the small deformation range both rates are almost equal in constitutive relations implementation since the outputs for both rates are nearly congruent. Otherwise, if the influence of rotations is significant the Jaumann rate gives theoretically and experimentally unexpected results whereas the logarithmic rate results are stable and consistent with the notion of elasticity, i.e. without any residual stresses at the end of the elastic deformation cycles.
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.
A solution for the elastoplastic deflection of cantilever beams with linearly variable circular cross-section loaded by shear force at the free end, which is suitable for practical use, has not yet been developed. A semi-analytical solution for such a problem is proposed in this paper. The solution involves beams made of homogenous and isotropic materials with bilinear elastoplastic strain hardening behavior. The Bernoulli–Euler formula is used for determining the elastic deflection. However, for the plastic domain of material behavior, the differential equation of beam bending does not have a solution in closed form. Therefore, an incremental procedure for determining the curvature of the plastified region of the beam is suggested. Deflection of the cantilever beam is calculated via integration of the approximated function of the beam curvature. The proposed semi-analytical solution is validated using experimental results of the seismic energy dissipation device components which have been selected as a sample of a real engineering system. Also, validation is done via finite element analysis of six different cantilever beam models with varying geometric and material characteristics. A satisfying agreement between the proposed semi-analytical results and the subsequent experimental and numerical results is herein achieved, confirming its reliability.
Constitutive relations which describe engineering materials behaviour during the finite elastoplastic deformations are usually presented in the form of rates of stresses and strains. One of the possible approaches in the constitutive relations formulation is the additive decomposition of the total deformation rate into its elastic part and its plastic part. The elastic deformation rate contributes to any elastoplastic deformation at any stage. Hence, its exact and well-considered formulation is of particular importance and it has to be properly predicted by the corresponding material law. This is of great importance in particular when deformation cyclic processes are considered, in which case small errors may accumulate, even if the total deformation is small. The implementation of the most frequently used corotational rates, i.e. the Jaumann rate and the Green-Naghdi rate, in the hypo-elastic constitutive relations regarding small and moderate rotations gives accurate results for low number of repeated deformation cycles. With increased number of cycles, however, the implementation of these rates results in different and physically non-admissible material responses. This instability in results is particularly observable during the cyclic deformations with large rotations, which is the main subject of this work. In contrast to the aforementioned objective rates, the results of the logarithmic rate implementation into the hypo-elastic constitutive relations for the case of pure elastic deformation describe a physically stable process.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.