Semi-implicit finite strain constitutive integration and mixed strain/stress control based on intermediate configurations

• Rotations are calculated at the end of each time step and therefore do not require linearization.
• Frame-invariance during time step iterations is ensured by relative Green–Lagrange strains.
• Plasticity: von-Mises, Hill and Barlat 91 yield functions are used.
• Anisotropic hyperelasticity, the approach by Bonet and Burton [17] for transversely isotropic Kirchhoff/Saint–Venant.
• Replacement of the complementarity condition by the Chen–Mangasarian function.

A new semi-implicit stress integration algorithm for finite strain plasticity (compatible with hyperelasticity) is introduced. Its most distinctive feature is the use of different parameterizations of equilibrium and reference configurations. Rotation terms (nonlinear trigonometric functions) are integrated explicitly and correspond to a change in the reference configuration. In contrast, relative Green–Lagrange strains (which are quadratic in terms of displacements) represent the equilibrium configuration implicitly. In addition, the adequacy of several objective stress rates in the semi-implicit context is studied. We parametrize both reference and equilibrium configurations, in contrast with the so-called objective stress integration algorithms which use coinciding configurations. A single constitutive framework provides quantities needed by common discretization schemes. This is computationally convenient and robust, as all elements only need to provide pre-established quantities irrespectively of the constitutive model. In this work, mixed strain/stress control is used, as well as our smoothing algorithm for the complementarity condition. Exceptional time-step robustness is achieved in elasto-plastic problems: often fewer than one-tenth of the typical number of time increments can be used with a quantifiable effect in accuracy. The proposed algorithm is general: all hyperelastic models and all classical elasto-plastic models can be employed. Plane-stress, Shell and 3D examples are used to illustrate the new algorithm. Both isotropic and anisotropic behavior is presented in elasto-plastic and hyperelastic examples.

Figure 1: Pinched cylinder with Hill yield criterion (full constitutive system with stress condition). The anisotropic direction I is shown, as well as a top perspective of the (half-)thickness contour plots.Figure optionsDownload as PowerPoint slide