Yield and plastic potential surfaces are often affected by problems related to convexity. One such problem is encountered when the yield surface that bounds the elastic domain is itself convex; however, convexity is lost when the surface expands to pass through stress points outside the current elastic domain. In this paper, a technique is proposed, which effectively corrects this problem by providing linear homothetic expansion with respect to the centre of the yield surface.A very compact implicit integration scheme is also presented, which is of general applicability for isotropic constitutive models, provided that their yield and plastic potential functions are based on a separate mathematical definition of the meridional and deviatoric sections and that stress invariants are adopted as mechanical quantities. The elastic predictor-plastic corrector algorithm is based on the solution of a system of 2 equations in 2 unknowns only. This further reduces to a single equation and unknown in the case of yield and plastic potential surfaces with a linear meridional section. The effectiveness of the proposed convexification technique and the efficiency and stability of the integration scheme are investigated by running numerical analyses of a notoriously demanding boundary value problem.
In this paper, it is mathematically demonstrated that classical yield and failure criteria such as Tresca, von Mises, Drucker-Prager, Mohr-Coulomb, MatsuokaNakai and Lade-Duncan are all defined by the same equation. This can be seen as one of the three solutions of a cubic equation of the principal stresses and suggests that all such criteria belong to a more general class of non-convex formulations which also comprises a recent generalization of the Galileo-Rankine criterion. The derived equation is always convex and can also provide a smooth approximation of continuity of at least class C 2 of the original Tresca and Mohr-Coulomb criteria. It is therefore free from all the limitations which restrain the use of some of them in numerical analyses. The mathematical structure of the presented equation is based on a separate definition of the meridional and deviatoric sections of the graphical representation of the criteria. This enables the use of an efficient implicit integration algorithm which results in a very short machine runtime even when demanding boundary value problems are analysed.
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
We propose a plastic potential for higher-order (HO) phenomenological strain gradient plasticity (SGP), predicting reliable size-dependent response for general loading histories. By constructing the free energy density as a sum of quadratic plastic strain gradient contributions that each transitions into linear terms at different threshold values, we show that we can predict the expected micron-scale behaviour, including increase of strain hardening and strengthening-like behaviour with diminishing size. Furthermore, the anomalous behaviour predicted by most HO theories under non-proportional loading is avoided. Though we demonstrate our findings on the basis of Gurtin (Gurtin 2004
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