  The paper is devoted to the numerical solution of elastoplastic constitutive
initial value problems. An improved form of the implicit return-mapping scheme
for nonsmooth yield surfaces is proposed that systematically builds on a
subdifferential formulation of the flow rule. The main advantage of this
approach is that the treatment of singular points, such as apices or edges at
which the flow direction is multivalued involves only a uniquely defined set of
non-linear equations, similarly to smooth yield surfaces. This paper (PART I)
is focused on isotropic models containing: $a)$ yield surfaces with one or two
apices (singular points) laying on the hydrostatic axis; $b)$ plastic
pseudo-potentials that are independent of the Lode angle; $c)$ nonlinear
isotropic hardening (optionally). It is shown that for some models the improved
integration scheme also enables to a priori decide about a type of the return
and investigate existence, uniqueness and semismoothness of discretized
constitutive operators in implicit form. Further, the semismooth Newton method
is introduced to solve incremental boundary-value problems. The paper also
contains numerical examples related to slope stability with available Matlab
implementation.

Computational Engineering, Finance, and Science