Rate-independent elastoplasticity at finite strains and its numerical approximation

2016 ◽  
Vol 26 (12) ◽  
pp. 2203-2236 ◽  
Author(s):  
Alexander Mielke ◽  
Tomáš Roubíček
Keyword(s):  
Kinematic Hardening ◽  
Time Discretization ◽  
Energy Functional ◽  
Implicit Time ◽  
Large Strains ◽  
Gradient Plasticity ◽  
Lavrentiev Phenomenon ◽  
Penalization Method ◽  
Energetic Solutions ◽  
Nonsimple Materials

Gradient plasticity at large strains with kinematic hardening is analyzed as quasistatic rate-independent evolution. The energy functional with a frame-indifferent polyconvex energy density and the dissipation is approximated numerically by finite elements and implicit time discretization, such that a computationally implementable scheme is obtained. The nonself-penetration as well as a possible frictionless unilateral contact is considered and approximated numerically by a suitable penalization method which keeps polyconvexity and simultaneously bypasses the Lavrentiev phenomenon. The main result concerns the convergence of the numerical scheme toward energetic solutions. In the case of incompressible plasticity and of nonsimple materials, where the energy depends on the second derivative of the deformation, we derive an explicit stability criterion for convergence relating the spatial discretization and the penalizations.

Analytical Solutions for Circular Bars Subjected to Large Strain Plastic Torsion

1990 ◽  
Vol 57 (2) ◽  
pp. 298-306 ◽  
Author(s):  
K. W. Neale ◽  
S. C. Shrivastava
Keyword(s):  
Closed Form ◽  
Analytical Solutions ◽  
Large Strain ◽  
Kinematic Hardening ◽  
Flow Theory ◽  
Constitutive Laws ◽  
Isotropic Hardening ◽  
Large Strains ◽  
Inelastic Behavior ◽  
Rate Dependent

The inelastic behavior of solid circular bars twisted to arbitrarily large strains is considered. Various phenomenological constitutive laws currently employed to model finite strain inelastic behavior are shown to lead to closed-form analytical solutions for torsion. These include rate-independent elastic-plastic isotropic hardening J2 flow theory of plasticity, various kinematic hardening models of flow theory, and both hypoelastic and hyperelastic formulations of J2 deformation theory. Certain rate-dependent inelastic laws, including creep and strain-rate sensitivity models, also permit the development of closed-form solutions. The derivation of these solutions is presented as well as numerous applications to a wide variety of time-independent and rate-dependent plastic constitutive laws.

GLOBAL EXISTENCE FOR PHASE TRANSITION PROBLEMS VIA A VARIATIONAL SCHEME

2004 ◽  
Vol 01 (04) ◽  
pp. 747-768
Author(s):  
CHRISTIAN ROHDE ◽  
MAI DUC THANH
Keyword(s):  
Phase Transition ◽  
Surface Tension ◽  
Initial Data ◽  
Time Discretization ◽  
Approximate Solutions ◽  
Implicit Time ◽  
Time Step ◽  
Dynamical Phase Transition ◽  
Transition Dynamics ◽  
Variational Scheme

We construct approximate solutions of the initial value problem for dynamical phase transition problems via a variational scheme in one space dimension. First, we deal with a local model of phase transition dynamics which contains second and third order spatial derivatives modeling the effects of viscosity and surface tension. Assuming that the initial data are periodic, we prove the convergence of approximate solutions to a weak solution which satisfies the natural dissipation inequality. We note that this result still holds for non-periodic initial data. Second, we consider a model of phase transition dynamics with only Lipschitz continuous stress–strain function which contains a non-local convolution term to take account of surface tension. We also establish the existence of weak solutions. In both cases the proof relies on implicit time discretization and the analysis of a minimization problem at each time step.

Predictions of microtorsional experiments by micropolar plasticity

2005 ◽  
Vol 461 (2053) ◽  
pp. 189-205 ◽  
Author(s):  
Paschalis Grammenoudis ◽  
Charalampos Tsakmakis
Keyword(s):  
Size Effects ◽  
Bauschinger Effect ◽  
Kinematic Hardening ◽  
Circular Cylinders ◽  
Isotropic Hardening ◽  
Gradient Plasticity ◽  
Material Response ◽  
Micropolar Plasticity ◽  
Classical Plasticity ◽  
Hardening Rules

Kinematic hardening rules are employed in classical plasticity to capture the so–called Bauschinger effect. They are important when describing the material response during reloading. In the framework of thermodynamically consistent gradient plasticity theories, kinematic hardening effects were first incorporated into a micropolar plasticity model by Grammenoudis and Tsakmakis. The aim of the present paper is to investigate this model by predicting size effects in torsional loading of circular cylinders. It is shown that kinematic hardening rules compared with isotropic hardening rules, as adopted in the paper, provide more possibilities for modelling size effects in the material response, even if only monotonous loading conditions are considered.

scholarly journals Finite thermoelastoplasticity and creep under small elastic strains

2018 ◽  
Vol 24 (4) ◽  
pp. 1161-1181 ◽  
Author(s):  
Tomáš Roubíček ◽  
Ulisse Stefanelli
Keyword(s):  
Convergence Result ◽  
Large Strains ◽  
Gradient Plasticity ◽  
Fourier Law ◽  
Large Plastic Strain ◽  
Rate Dependent ◽  
Galerkin Approximations ◽  
Elastic Strains ◽  
Rigorous Mathematical Analysis ◽  
The Continuum

A mathematical model for an elastoplastic continuum subject to large strains is presented. The inelastic response is modelled within the frame of rate-dependent gradient plasticity for non-simple materials. Heat diffuses through the continuum by the Fourier law in the actual deformed configuration. Inertia makes the nonlinear problem hyperbolic. The modelling assumption of small elastic Green–Lagrange strains is combined in a thermodynamically consistent way with the possibly large displacements and large plastic strain. The model is amenable to a rigorous mathematical analysis. The existence of suitably defined weak solutions and a convergence result for Galerkin approximations is proved.

A Fast Semi-Implicit Level Set Method for Curvature Dependent Flows with an Application to Limit Cycles Extraction in Dynamical Systems

2015 ◽  
Vol 18 (1) ◽  
pp. 203-229 ◽  
Author(s):  
Guoqiao You ◽  
Shingyu Leung
Keyword(s):  
Dynamical Systems ◽  
Level Set Method ◽  
Level Set ◽  
Limit Cycles ◽  
Time Discretization ◽  
Numerical Approach ◽  
Initial Guess ◽  
Implicit Time ◽  
Time Step ◽  
Image Regularization

AbstractWe propose a new semi-implicit level set approach to a class of curvature dependent flows. The method generalizes a recent algorithm proposed for the motion by mean curvature where the interface is updated by solving the Rudin-Osher-Fatemi (ROF) model for image regularization. Our proposal is general enough so that one can easily extend and apply the method to other curvature dependent motions. Since the derivation is based on a semi-implicit time discretization, this suggests that the numerical scheme is stable even using a time-step significantly larger than that of the corresponding explicit method. As an interesting application of the numerical approach, we propose a new variational approach for extracting limit cycles in dynamical systems. The resulting algorithm can automatically detect multiple limit cycles staying inside the initial guess with no condition imposed on the number nor the location of the limit cycles. Further, we also propose in this work an Eulerian approach based on the level set method to test if the limit cycles are stable or unstable.

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