Conversion of explicit microplane model with boundaries to a constitutive subroutine for implicit finite element programs

Whereas various simplistic microplane models of limited applicability, defined by stress–strain curves on the microplane, can function as either explicit or implicit, the explicit-to-implicit conversion of realistic versatile microplane models for plain or fiber-reinforced concrete, shale and composites has remained a challenge for quarter century. The reason is that these realistic models use microplane stress–strain boundaries defined by inequalities. Here, we show how the conversion can be easily achieved on the microplane level and then transferred to a tangent stiffness tensor or an inelastic stiffness tensor to be used in Newton–Raphson iterations within a loading step. To ensure convergence, a minor adjustment in the M7 algorithm is introduced to achieve continuity. Power-law convergence, almost quadratic in most cases, is also demonstrated. Seven examples of crack-band finite element simulations of challenging laboratory tests document nearly identical implicit and explicit results, as well as good match of test data. Three of them, including the vertex effect in compression-torsion tests, pure Mode II shear fracture, and the “gap test” of the crack-parallel compression effect on Mode I load-deflection curve, have not been reproduced by other models before. The coding of implicit M7 subroutine, usable in, for example, UMAT of ABAQUS, is posted for a free download.