Mixed isogeometric collocation for geometrically exact 3D beams with elasto-visco-plastic material behavior and softening effects

A geometrically nonlinear, shear-deformable 3D beam formulation with inelastic material behavior and its numerical discretization by a mixed isogeometric collocation method are presented. In particular, the constitutive model captures elasto-visco-plasticity with damage/softening from Mullin's effect, which applies to the modeling of metallic and polymeric materials, e.g., in additive manufacturing applications and meta-materials. The inelastic material behavior is formulated in terms of thermodynamically consistent internal variables for viscoelastic and plastic strains and isotropic and kinematic hardening variables, as well as accompanying evolution equations. A mixed isogeometric collocation method is applied for the discretization of the strong form of the quasi-static nonlinear differential equations. Thus, the displacements of the centerline curve, the cross-section orientations, and the stress resultants (forces and moments) are discretized as B-spline or NURBS curves. The internal variables are defined only locally at the collocation points, and an implicit return-mapping algorithm is employed for their time discretization. The method is verified in comparison to 1D examples as well as reference results for 3D beams. Furthermore, its applicability to the simulation of beam lattice structures subject to large deformations and instabilities is demonstrated.