MMC_Lecture 7_ Integration Algorithms for Generalised Elasto-plasticity

Computational Solid Mechanics

MCC - Master in Computational Mechanics


2 MCC - Master in Computational Mechanics

Contents

Lecture 1: Physical Motivation

Lecture 2: 1D Mathematical model

Lecture 3: Computational aspects of 1D elasto-plasticity

Lecture 4:Classical model of E- P

Lecture 5:Computational aspects of elasto - plasticity

Lecture 6:Plane strain Von-Mises elasto - plastic model

Lecture 7:Continuum model for plane strain and 3-D Von-Mises elasto-plasticity

Lecture 8:Integration algorithms for generalised elasto-plasticity

Lecture 9: Generalisations and applications of plasticity

Lecture 10: 1-D Mathematical model large strain elastoplasticity

Lecture 11: Computational aspects of 1-D large strain elasto-plasticity



Integration algorithms for generalised elasto-plasticity

Layout of Lecture Incremental Problem Generalised Elasto-Plasticity - A Remainder.

-Numerical Discretisation. Backward Euler. Stress Integration Algorithm. - Operator Split. Trial Elastic State. - Plastic Corrector. Iterative Newton-Raphson Procedure - Geometric Interpretation - Closest Point Projection.

Example: Stress Integration Algorithm for the Barlat Anisotropic Yield Criterion.




G.1 The Incremental Problem of Generalised Elasto-Plasticity

By applying the generalised midpoint algorithm to the evolution

problem of rate-independent plasticity and restricting attention to

the backward Euler scheme ( = 1), the incremental version of the

elasto-plastic evolution problem is obtained in the form




G.1 The Incremental Problem of Generalised Elasto-Plasticity The associative elasto-plastic flow with isotropic linear hardening will be considered, with corresponding discrete forms where the standard notation is used. By standard arguments, the updated variables at must also satisfy a discrete version of the Kuhn-Tucker conditions in the form




G.2 Elastic Predictor In a standard manner a global solution of the BVP at supplies incremental strain (and follows immediatly) at the local (Gauss point) level. The problem consists in updating in a manner which is consistent with the model (G.1)-(G.3). Structure of the incremental elasto-plastic problem (G.1)-(G.3) allows solution to be pursued in two distinct steps which have a compelling physical interpretation




G.2 Elastic Predictor By freezing the plastic flow during the time increment Is obtained in the form




G.2 Elastic Predictor

The above state constitutes the so-called elastic predictor. In general, unless step is purely elastic, this state is physically inadmissible and requires further correction in the second phase of the algorithm. The discrete Kuhn-Tucker conditions provide information which is governing further algorithmic steps. The main result may be summarised in the form




G.3 Plastic Corrector. The objective is to obtain a solution which in turn denes which in turn defines In the case of generalised elasto-plasticity this solution is not available in the explicit form. It is necessary to solve locally (at Gauss point level) the following nonlinear system of algebraic equations for variables




G.3 Plastic Corrector. The solution is typically obtained by employing the Newton-Raphson iterative procedure. BOX 15 Elastic Predictor Algorithm for Generalised Rate-Independent Isotropic Hardening Elasto-Plasticity.




G.3 Plastic Corrector. BOX 16 Plastic Corrector Algorithm for Generalised Rate-Independent Isotropic Hardening Plasticity




G.3 Plastic Corrector.

Figure G.1: Illustration of the elastic predictor -

plastic corrector algorithm for generalised elasto-

plasticity: Return mapping of the trial stress trial

to the current yield surface

Owing to its geometric interpretation, this

algorithm is also known as the closest point

projection algorithm.




G.3 Plastic Corrector. BOX 17 Summary of Stress Integration Procedure for Generalised Rate-Independent Elasto- Plasticity




G.4 Barlat Anisotropic Yield Criterion. The Barlat yield criterion is formulated by F. Barlat in 1989. to accurately model the anisotropic plastic behaviour of textured polycrystalline sheets. The mathematical representation of this criterion for the plane stress conditions takes the form where a; h; p and M are material constants and is the yield stress from a uniaxial tension test.




G.4 Barlat Anisotropic Yield Criterion. For a given value of the exponent M material constants a; h; p can be evaluated using R values, i.e. plastic strain ratios of the in-plane strains to the thickness strain obtained from uniaxial tension tests in three different directions. The resulting set of functions span the set of yield surfaces which include the standard von Mises and Tresca yield surfaces for M=2 and respectively. Also the Hill orthotropic yield criterion is included as a special case.




Figure G.2: Representation of the Barlat yield criterion: (a) Isotropic yield surfaces for several values of material constant M and for shear stress . (b) Isotropic and anisotropic yield surfaces for

G.4 Barlat Anisotropic Yield Criterion.

Questions ?



MMC_Lecture 7_ Integration Algorithms for Generalised Elasto-plasticity

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