Integration schemes for von-Mises plasticity models based on exponential maps: numerical investigations and theoretical considerations

Abstract

SUMMARYWe consider three different exponential map algorithms for associative von-Mises plasticity with linear isotropic and kinematic hardening. The first scheme is based on a different formulation of the time continuous plasticity model, which automatically grants the yield consistency of the method in the numerical solution. The second one is the quadratically accurate but non-yield consistent method already proposed in Auricchio and Beirão da Veiga (Int. J. Numer. Meth. Engng 2003; 56: 1375-1396). The third method is an improved version of the second one, in which the yield consistency condition is enforced a posteriori. We also compare the performance of the three methods with the classical radial return map algorithm. We develop extensive numerical tests which clearly show the main advantages and disadvantages of the three methods.

A novel ‘optimal’ exponential‐based integration algorithm for von‐Mises plasticity with linear hardening: Theoretical analysis on yield consistency, accuracy, convergence and numerical investigations

“…This innovative statement of the problem represents a generalization of the formulation presented in References [2,10] and thus constitutes an extension of the ones proposed in References [1,11,12]. Such a formulation allows to rewrite the system in the formẊ = AX (11) which is the starting point for the numerical scheme developed in Section 4.…”

“…The first choice for R in (40) corresponds to the ESC scheme already proposed in Reference [2], the second one leads to the new ESC 2 scheme which is the object of the present paper. While the reasonings for the first choice for R are evident ('forward integration' scheme), the second choice for R comes from considerations regarding the improved numerical properties of the new exponential-based algorithm, discussed from the analytical standpoint in Section 5.…”

On the integration schemes for Drucker–Prager's elastoplastic models based on exponential maps

“…Note that using EIM(II) with practical time steps gives a very precise result, which in practical engineering problems may be referred to as a near-exact approach. In order to better investigate the rate of convergence of these methods, here, the total error can be defined as [7] …”

“…Then they developed a new numerical scheme by employing an exponential map, exp(A n t), as an approximation to the above system. Artioli et al [7] enhanced this method to obtain a fully consistent algorithm. Finally, further improvements were made to this scheme and consistent methods with a second-order accuracy were developed [8,9].…”

“…Otherwise, similar to all explicit integration schemes, a partially plastic step should be divided into elastic and plastic portions and then contact stress could be determined, e.g. References [6][7][8]. To accomplish this task, a scalar parameter, r , that divides e into an elastic step, r e, and a plastic step, (1 − r ) e, is used.…”