Computational and Applied Mathematics seminar

Abstrakt finns enbart på engelska: Two approaches to static equilibrium of fiber network microstructures will be presented. Orientation-based averaging of network microstructures with non-affine kinematics associates the ensemble of fibers at a given material point with a unit sphere of initial orientations (called also microsphere or representative orientation space). A special micro-macro relation between these microscopic deformations and the macroscopic deformation gradient is derived by the concept of maximal advance paths in the network. The averaged microscopic free energy of the fibers is minimized subjected to this kinematic constraint. The derived model can be applied to networks of flexible polymer chains (as found in elastomers), semiflexible or stiff filaments (elastin, collagen and other biopolymers and soft tissues). The model has been recently extended to composite networks consisting of two or more distinguished components. Sliding of inextensible fibers in discrete network microstructure of nonwoven materials is defined as a nonsmooth convex optimization problem. The rate-independent sliding and the quasi-static equilibrium loading of such network structure are constituted within the theory of standard dissipative systems. A minimum principle for incremental potential is formulated with respect to the displacement-based variables as a second-order cone programming (SOCP) problem. A pure complementary energy principle is derived as the dual formulation in terms of stress-like variables. Both SOCP problems can be solved numerically by interior-point methods.