An efficient return mapping algorithm for elastoplasticity with exact closed form solution of the local constitutive problem

Abstract

Purpose - The purpose of this paper is to present an efficient return mapping algorithm for elastoplastic constitutive problems of ductile metals with an exact closed form solution of the local constitutive problem in the small strain regime. A Newton Raphson iterative method is adopted for the solution of the boundary value problem.\ud \ud Design/methodology/approach - An efficient return mapping algorithm is illustrated which is based on an elastic predictor and a plastic corrector scheme resulting in an implicit and accurate numerical integration method. Nonlinear kinematic hardening rules and linear isotropic hardening rules are used to describe the components of the hardening variables. In the adopted algorithmic approach the solution of the local constitutive equations reduces to only one straightforward nonlinear scalar equation.\ud \ud Findings - The presented algorithmic scheme naturally leads to a particularly simple form of the nonlinear scalar equation which ultimately scales down to an algebraic (polynomial) equation with a single variable. The straightforwardness of the present approach allows to find the analytical solution of the algebraic equation in a closed form. Further, the consistent tangent operator is derived as associated with the proposed algorithmic scheme and it is shown that the proposed computational procedure ensures a quadratic rate of asymptotic convergence when used with a Newton Raphson iterative method for the global solution procedure.\ud \ud Originality/value - In the present approach the solution of the algebraic nonlinear equation is found in a closed form and accordingly no iterative method is required to solve the problem of the local constitutive equations. The computational procedure ensures a quadratic rate of asymptotic convergence for the global solution procedure typical of computationally efficient solution schemes. In the paper it is shown that the proposed algorithmic scheme provides an efficient and robust computational solution procedure for elastoplasticity boundary value problems. Numerical examples and computational results are reported which illustrate the effectiveness and robustness of the adopted integration algorithm for the finite element analysis of elastoplastic structures also under elaborate loading conditions

Subdifferential‐based implicit return‐mapping operators in computational plasticity

“…CIVP satisfies thermodynamical laws and usually involves internal variables such as plastic strains or hardening parameters. Several integration schemes for the numerical solution of CIVP have been suggested, e.g., in and the references introduced therein. If the implicit or trapezoidal Euler method is used, then the incremental constitutive problem is solved using the elastic predictor/plastic corrector method.…”

Generalized radial‐return mapping algorithm for anisotropic von Mises plasticity framed in material eigenspace

“…In this case (barring other complications, such as nonlinear elasticity), the normal to the updated yield surface is identical to the normal calculated from the trial stress (predictor) value. This simplification gives rise to the well‐known radial‐return type return mapping algorithms, which can save considerable computational effort by avoiding iterative computation of the yield surface normal, ie, the flow “direction” in associative plasticity …”

“…Maudlin et al have proposed projecting the stresses and the constitutive tensor onto a five‐dimensional deviatoric space advocated by Tomé and Kocks, where the volumetric and deviatoric coupling terms of the constitutive tensor are conveniently grouped. In consideration of especially large strain increments, Becker provided an algorithm based on an approximation from a closed‐form solution, De Angelis and Taylor have also proposed a closed‐form solution type algorithm for general elastoplasticity, and Scherzinger has provided an algorithm using a line‐search method. Liu et al described a so‐called return‐free approach for integrating the constitutive equations, which makes use of particular transformation of the elastoplastic equations such that the stress lies always on the updated yield surface.…”

Computational Algorithms for Cyclic Plasticity Based on Prager's Translation Rule

“…However, a priori, the direct inclusion of a non-linear anisotropic kinematic hardening function produces unphysical loops. Therefore, finite element programs use Prager's rule exclusively in the case of linear kinematic hardening, and for non-linear kinematic hardening, which is needed to better describe the cyclic loops, they resort to other types of formulations (if available to the user), typically based on the Armstrong and Frederick rule [36], [37].…”

“…These rules have resulted in different models as the Mróz model [39], Chaboche's model [5], [13], bounding surface models [41], [12], [11], and non-linear kinematic hardening models with the addition of multiple backstresses [5], [15]. The algorithmic imple-mentation of some of these models is usually more elaborate than that of classical plasticity [42], [43], [44], [45], [46], and even though some recent efficient algorithms are available for some cases [36], [37], they do not constitute a natural extension of classical Prager's plasticity. To improve the cyclic multiaxial plastic behavior, sometimes non-proportionality parameters obtained for certain materials under certain loading paths by fitting experiments are employed, like in [47], [48], [49], [50], [51], [52], [53], [46], [54].…”