A numeric-symbolic approach to the problem of localization of plastic flow

A method combining numerical and symbolic computations is presented for deciding whether or not a mechanical state given numerically in terms of its fourth-order tenser of moduli satisfies the localization condition, and more generally, any constitutive restriction on the tenser of moduli having a polynomial form. The idea underlying the method consists in posing the problem not as an optimization problem (minimizing the determinant of the acoustic tenser with respect to direction), but as a polynomial inequality. The polynomiality of the problem is then fully taken advantage of by using a powerful algorithm to solve polynomial inequalities: (a simplified version of) Collins' cylindrical algebraic decomposition algorithm. The method was implemented using the computer algebra system MATHEMATICA, which provides accuracy-controlled symbolic and numerical computations and many built-in functions to handle polynomials. These features make it possible to determine the onset of localization more accurately and more reliably than with the usual optimization-based approaches, which are not guaranteed to converge to the global minimum. The potential of the method is illustrated by addressing loss of ellipticity and loss of strong ellipticity in the case of Gurson's porous material.