Instytut Podstawowych Problemów Techniki

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In this paper a selected type of elasto-viscoplastic constitutive equations is considered. The viscoplastic model which was proposed by Marquis is generalized by allowing multiple terms describing the isotropic and the kinematic hardening. Furthermore, the presented model formulation enables one to use an arbitrary equivalent plastic strain function to describe the isotropic hardening behavior. What is more, the backstress evolution equation which was proposed by Marquis was modified so that any equivalent plastic strain function can be used in the recovery term now. The general form of the generalized constitutive model obtained this way was subsequently implemented into the finite element method (FEM). For that purpose, the radial-return mapping algorithm was utilized. The consistent tangent operator was derived for the considered class of models and is presented in the paper. A developed user material subroutine (UMAT) which allows one to use the viscoplastic models under consideration in the FEM program CalculiX is attached in the appendix section. A number of numerical simulations were conducted in order to verify the performance of the developed UMAT code.

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In this study, a new method for determining the material parameters of cyclic plasticity is presented. The method can be applied to evaluate the model parameters from any loading histories measured experimentally. The experimental data require basic processing only to be utilized. The method can be applied to calibrate the parameters of different elastoplastic models such as the Chaboche–Rousselier (Ch–R) constitutive equation or other model formulations which use different rules of isotropic hardening. The developed method was utilized to evaluate the material parameters of copper for a selected group of constitutive models. It is shown that among the considered model formulations a very good description of the mechanical properties of copper is achieved for the Ch–R model with two Voce terms used for simulating the isotropic hardening and two backstress variables utilized for capturing the kinematic hardening behavior. Furthermore, it is demonstrated that a model calibrated using the cyclic tension/compression data is able to properly capture the material response in torsion. Similarly, when the constitutive parameters are determined using the cyclic torsion data the model is able to properly reproduce the material behavior in tension/compression. It is concluded that for the considered type of constitutive equations the material parameters can be identified from a single mechanical test. The proposed methodology was validated using the relations derived analytically.

Słowa kluczowe:

elastoplasticity, cyclic plasticity, Chaboche, material parameters

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In this study, the possibilities of hyperelastic modeling of metallic materials are discussed. The so-called four-parameter model which was postulated by Goel et al. (Int J Solids Struct 48:2977–2986, 2011) in order to simulate the mechanical behavior of metals is analyzed. Experimental data obtained for various materials are utilized to investigate the ability of the aforementioned model to mimic the stress–strain response of metals. For many of the analyzed materials, very good curve-fitting results are achieved. The finite element (FE) implementation of this particular model is not straightforward as the derivatives of the stored energy function are undefined in the natural undeformed state. In this work, a method allowing to overcome this problem is proposed. What is more, it is found that another elastic energy function which was proposed by Knowles (Int J Fract 13:611–639, 1977) is very effective at simulating the highly nonlinear stress–strain characteristics of some metals such as aluminum or brass, for instance. A general-purpose user subroutine UMAT (User’s MATerial) has been developed allowing to implement any first invariant-based hyperelastic models into the FE program CalculiX. The UMAT code is attached in Appendix section. The subroutine is utilized in several FE simulations in order to verify its performance. The available experimental data are used to determine the material parameters of the popular Ramberg–Osgood model. It is found that the discussed hyperelastic models allow to describe the mechanical response of metals more accurately than the Ramberg–Osgood model. Furthermore, it is demonstrated that the hyperelastic models are characterized by better convergence during the FE analysis.

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In this work the finite element (FE) implementation of the small strain cyclic plasticity is discussed. The family of elastoplastic constitutive models is considered which use the mixed, kinematic-isotropic hard ening rule. It is assumed that the kinematic hardening is governed by the Armstrong–Frederick law. The radial return mapping algorithm is utilized to discretize the general form of the constitutive equation. A relation for the consistent elastoplastic tangent operator is derived. To the best of Author’s knowledge, this formula has not been presented in the literature yet. The obtained set of equations can be used to implement the cyclic plasticity models into numerous commercial or non-commercial FE packages. A user subroutine UMAT (User’sMATe rial) has been developed in order to implement the cyclic plasticity model by Yoshida into the open-source FE 15 program CalculiX. The coding is included in Appendix. It can be easily modified to implement any isotropic hardening rule for which the yield stress is a function of the effective plastic strain. The number of the utilized backstress variables can be easily increased as well. Several validation tests which have been conducted in order to verify the code’s performance are discussed.

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In this work a number of selected, isotropic, invariant-based hyperelastic models are analyzed. The considered constitutive relations of hyperelasticity include the model by Gent (G) and its extension, the so-called generalized Gent model (GG), the exponential-power law model (Exp-PL) and the power law model (PL). The material parameters of the models under study have been identified for eight different experimental data sets. As it has been demonstrated, the much celebrated Gent’s model does not always allow to obtain an acceptable quality of the experimental data approximation. Furthermore, it is observed that the best curve fitting quality is usually achieved when the experimentally derived conditions that were proposed by Rivlin and Saunders are fulfilled. However, it is shown that the conditions by Rivlin and Saunders are in a contradiction with the mathematical requirements of stored energy polyconvexity. A polyconvex stored energy function is assumed in order to ensure the existence of solutions to a properly defined boundary value problem and to avoid non-physical material response. It is found that in the case of the analyzed hyperelastic models the application of polyconvexity conditions leads to only a slight decrease in the curve fitting quality. When the energy polyconvexity is assumed, the best experimental data approximation is usually obtained for the PL model. Among the non-polyconvex hyperelastic models, the best curve fitting results are most frequently achieved for the GG model. However, it is shown that both the G and the GG models are problematic due to the presence of the locking effect.

Słowa kluczowe:

hyperelasticity, rubberlike materials, polyconvexity, material parameters

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