Computational modelling and data-driven homogenisation of knitted membranes

Abstract

Knitting is an e ective technique for producing complex three-dimensional surfaces owing to the inherent exibility of interlooped yarns and recent advances in manufacturing providing better control of local stitch patterns. Fully yarn-level modelling of large-scale knitted membranes is not feasible. Therefore, we use a two-scale homogenisation approach and model the membrane as a Kirchho -Love shell on the macroscale and as Euler-Bernoulli rods on the microscale. The governing equations for both the shell and the rod are discretised with cubic B-spline basis functions. For homogenisation we consider only the in-plane response of the membrane. The solution of the nonlinear microscale problem requires a signi cant amount of time due to the large deformations and the enforcement of contact constraints, rendering conventional online computational homogenisation approaches infeasible. To sidestep this problem, we use a pre-trained statistical Gaussian Process Regression (GPR) model to map the macroscale deformations to macroscale stresses. During the o ine learning phase, the GPR model is trained by solving the microscale problem for a su ciently rich set of deformation states obtained by either uniform or Sobol sampling. The trained GPR model encodes the nonlinearities and anisotropies present in the microscale and serves as a material model for the membrane response of the macroscale shell. The bending response can be chosen in dependence of the mesh size to penalise the ne out-of-plane wrinkling of the membrane. After verifying and validating the di erent components of the proposed approach, we introduce several examples involving membranes subjected to tension and shear to demonstrate its versatility and good performance.