A fully convex reformulation of the original Matsuoka–Nakai failure criterion and its implicit numerically efficient integration algorithm

Abstract

This paper presents a reformulation of the original Matsuoka–Nakai criterion for overcoming the limitations which make its use in a stress point algorithm problematic. In fact, its graphical representation in the prin- cipal stress space is not convex as it comprises more branches, plotting also in negative octants, and it does not increase monotonically as the distance of the stress point from the failure surface rises. The proposed mathematical reformulation plots as a single, convex surface, which entirely lies in the positive octant of the stress space and evaluates to a quantity which monotonically increases as the stress point moves away from the failure surface. It is an exact reproduction, and not an approximated one, of the only significant branch of the original criterion. It is also suitable for shaping in the deviatoric plane the yield and plastic potential surfaces of complex constitutive models. A very efficient numerical algorithm for the implicit integration of the proposed formulation is presented, which enables the evaluation of the stress at the end of each incre- ment by solving a single scalar equation, both for associated and non-associated plasticity. The algorithm can be easily adapted for other smooth surfaces with linear meridian section. Finally, a close expression of the consistent Jacobian matrix is given for achieving quadratic convergence in the external structural new- ton loop. It is shown that all this results in extremely fast solutions of boundary value problems

On the existence of a unique class of yield and failure criteria comprising Tresca, von Mises, Drucker–Prager, Mohr–Coulomb, Galileo–Rankine, Matsuoka–Nakai and Lade–Duncan

“…Panteghini & Lagioia [18] have in fact shown that in this case an implicit algorithm can be formulated for the integration of the constitutive law which is based on a single scalar equation, and as such is extremely efficient and results in a very short machine runtime.…”

On the compressive strength of glass microballoons-based syntactic foams

“…In particular, we choose a model allowing the use of different strengths (or yield stresses) in uniaxial tension and compression, as exhibited by thermoset polymers well below the glass transition temperature. To this purpose, we adopt the model developed by Matsuoka and Nakai (1974) in its reformulation due to Panteghini and Lagioia (2014a). Such reformulation leads to an accurate implicit numerical algorithm, aiding its efficient implementation in a umat for ABAQUS.…”

“…For instance, the analysis on the K46-30 foam, whose FE MPUC has 365514 degrees of freedom, has ended in 19 h and 41 min to complete 1010 load increments for, overall, 1,383 iterations. 9 This is a further test on the efficiency and robustness of the reformulation and algorithm proposed in Panteghini and Lagioia (2014a) to implement the Matsuoka-Nakai plasticity model, whose original formulation has been shown to lead to relevant numerical problems (Panteghini and Lagioia, 2014b).…”

The ‘ I 3 ’ generalization of the Galileo–Rankine tension criterion

“…This is a frequently occurring problem which affects many classical criteria and a source of numerical difficulties. As shown in [3], one such example is the well-known Matsuoka-Nakai [14] failure criterion, for which an exact mathematical reformulation has been recently proposed in [4], that eliminates the undesired branches and plots as a convex surface. Similarly, in the next section a mathematical approach is proposed which makes equation (2.3) convex by removing all branches other than that of the compressive octant.…”

“…As smoothness is usually considered a desirable feature (e.g. [3,4]), most yield surfaces are affected by this problem. This is also Figure 3.…”