Radial Return Mapping Algorithm Research Articles

Finite element implementation of nonlinear constitutive models for piezoceramic materials

The objective of this work was the finite element implementation of constitutive material models for piezoceramic materials. A phenomenological material model was implemented by means of a customized radial return mapping algorithm into an open source finite element program, and the problem was reduced to the solution of a single nonlinear algebraic equation. This led to a significant reduction in computation time for simulations compared to an implementation by means of an explicit higher order integration scheme for the constitutive differential equations. Furthermore, a microscopically motivated material model was implemented by means of a radial return mapping algorithm based on the backward Euler scheme. Apart from simulations of the principal behavior of the material models and their finite element implementations, simulations of applied examples are discussed which demonstrate the properties of the models and the performance of the implementations. The piezoelectric effect describes a linear relationship between strain and polarization on the one hand, and electric field and stress on the other. Nowadays, this effect is commonly utilized by a class of ferroelectric ceramics, also called piezoceramics for short. The electromechanical coupling properties of these polycrystalline materials are caused by distinct features of their perovskite microstructure. Below the Curie temperature, each grain possesses a substructure of domains, these being regions of uniform orientation of the microdipoles of the crystallographic unit cells. Upon application of electric fields and mechanical stresses of sufficient magnitude, the domains can be reoriented. The macroscopic hysteresis properties resulting from microscopic domain switching processes under a cyclic electric field and mechanical stress loading in such a ferroelectric material are called ferroelectricity and ferroelasticity, respectively. In particular, the so-called poling process caused by strong electric fields leads to the orientation of the domains in the direction of the field resulting in a macroscopic piezoelectric effect from the microscopic piezoelectric contributions of the domains. Piezoceramic materials are used for actuation in various technical fields requiring challenging properties. They facilitate highly accurate positioning, show fast response times, and allow for large actuation forces. Examples of technical applications are diesel injection valves (which in 2005 won the Deutscher Zukunftspreis awarded by the President of the Federal Republic of Germany 1 ), as well as positioners in nano- and microtechnology.

An elasto-plastic contact model applied to nanoindentation

This paper is devoted to the finite element modeling of the nanoindentation problem. The frictional contact between the Berkovitch indenter and the very thin elasto-plastic film is treated by the bi-potential method. The elasto-plastic constitutive equation is integrated by means of the radial return mapping algorithm and the consistent tangent operator is explicitly derived. Numerical results show the validity of the model.

Thermo-Mechanical Modelling of Pebble Beds in Fusion Blankets and its Implementation by a Return-Mapping Algorithm

AbstractIn this investigation, a thermo-mechanical model of pebble beds is adopted and developed based on experiments by Dr. Reimann at Forschungszentrum Karlsruhe (FZK). The framework of the present material model is composed of a non-linear elastic law, the Drucker-Prager-Cap theory, and a modified creep law. Furthermore, the volumetric inelastic strain dependent thermal conductivity of beryllium pebble beds is taken into account and full thermo-mechanical coupling is considered.Investigation showed that the Drucker-Prager-Cap model implemented in ABAQUS can not fulfill the requirements of both the prediction of large creep strains and the hardening behaviour caused by creep, which are of importance with respect to the application of pebble beds in fusion blankets [1]. Therefore, UMAT (user defined material's mechanical behaviour) and UMATHT (user defined material's thermal behaviour) routines are used to re-implement the present thermo-mechanical model in ABAQUS. An elastic predictor radial return mapping algorithm is used to solve the non-associated plasticity iteratively, and a proper tangent stiffness matrix is obtained for cost-efficiency in the calculation. An explicit creep mechanism is adopted for the prediction of time-dependent behaviour in order to represent large creep strains in high temperature. Finally, the thermo-mechanical interactions are implemented in a UMATHT routine for the coupled analysis.The oedometric compression tests and creep tests of pebble beds at different temperatures are simulated with the help of the present UMAT and UMATHT routines, and the comparison between the simulation and the experiments is made.

The least‐squares meshfree method for elasto‐plasticity and its application to metal forming analysis

AbstractA new meshfree method for the analysis of elasto‐plastic deformation is presented. The method is based on the proposed first‐order least‐squares formulation for elasto‐plasticity and the moving least‐squares approximation. The least‐squares formulation for classical elasto‐plasticity and its extension to an incrementally objective formulation for finite deformation are proposed. In the formulation, equilibrium equation and flow rule are enforced in least‐squares sense, i.e. their squared residuals are minimized, and hardening law and loading/unloading condition are enforced pointwise at each integration point. The closest point projection method for the integration of rate‐form constitutive equation is inherently involved in the formulation, and thus the radial‐return mapping algorithm is not performed explicitly. The proposed formulation is a mixed‐type method since the residuals are represented in a form of first‐order differential system using displacement and stress components as nodal unknowns. Also the penalty schemes for the enforcement of boundary and frictional contact conditions are devised and the reshaping of nodal supports is introduced to avoid the difficulties due to the severe local deformation near contact interface. The proposed method does not employ structure of extrinsic cells for any purpose. Through some numerical examples of metal forming processes, the validity and effectiveness of the method are discussed. Copyright © 2005 John Wiley & Sons, Ltd.

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A new meshfree method for the analysis of elasto-plastic deformations is presented. The method is based on the proposed first-order least-squares formulation, to which the moving least-squares approximation is applied. The least-squares formulation for the classical elasto-plasticity and its extension to an incrementally objective formulation for finite deformations are proposed. In the formulation, the equilibrium equation and flow rule are enforced in least-squares sense, while the hardening law and loading/unloading condition are enforced exactly at each integration point. The closest point projection method for the integration of rate-form constitutive equation is inherently involved in the formulation, and thus the radial-return mapping algorithm is not performed explicitly. Also the penalty schemes for the enforcement of the boundary and frictional contact conditions are devised. The mam benefit of the proposed method is that any structure of cells is not used during the whole process of analysis Through some numerical examples of metal forming processes, the validity and effectiveness of the method are presented.

Elasto-plastic contact problem of laying wire rope using FE analysis

The three-dimensional elasto-plastic contact model, formulated with variational equalities, has been described for laying wire rope in this paper. The augmented Lagrangian multiplier method has been applied to calculate both the normal contact force and friction contact force, and the radial return mapping algorithm has been employed to handle stick/slip phases of friction. With a graphical user interface (GUI), a program for calculating boundary conditions of a three-dimensional finite element modelling is developed to create external files, which are in the format of the ANSYS parametric design language (APDL) and the external files can be accessed by ANSYS for nonlinear analysis. FE simulation of a simple wire strand has been given and the effects of friction coefficient and self-rotating ratio on the contact stress have been analysed. Also the effects of them on the forming stress and strain have been discussed.