# Comparison of implicit and explicit numerical integration schemes for a bounding surface soil model without elastic range

## FG Grundbau und Bodenmechanik

Standard integration schemes for rate constitutive equations were designed within the classical theory of plasticity. Consequently, they rely on the assumption that a yield criterion defines a range of purely elastic material behaviour. Many constitutive models for non-cohesive soils discard that assumption. They account for inelastic deformations without employing a yield criterion. This simplifies the formulation of the models but raises questions concerning their numerical implementation. In particular, it is not fully clear which algorithmic method is most appropriate for the numerical integration of constitutive models without elastic range. To investigate this, two different stress point algorithms for a critical state bounding surface model for sands were developed and implemented. The explicit method employs substepping to automatically control the local error. The implicit method updates stresses and state variables through a local Newton iteration, the Jacobian of which is computed by numerical differentiation. The two algorithms were compared by means of calculations at integration point level as well as with respect to a boundary value problem. The results show that for a given level of accuracy, the explicit update procedure is significantly more efficient than the implicit one. This holds regardless of initial state parameters and input strain increment magnitudes.