Strain-gradient elasticity and gradient-dependent plasticity with hierarchical refinement of NURBS

Higher-order strain-gradient models are relevant for engineering materials which exhibit size-dependent behaviour as observed from experiments. Typically, this class of models incorporate a length scale - related to micro-mechanical material properties - to capture size effects, remove stress singularities, or regularise an ill-posed boundary value problem resulting from localisation of deformation. The higher-order continuity requirement on shape functions can be met using NURBS discretisation, as is considered herein. However, NURBS have a tensor-product nature which makes selective refinement cumbersome. To maintain accuracy and efficiency in analysis, a finer mesh may be required, to capture a localisation band, certain geometrical features, or in regions with high gradients. This work presents strain-gradient elasticity and strain-gradient plasticity, both of second-order, with hierarchically refined NURBS. Refinement is performed based on a multi-level mesh with element-wise hierarchical basis functions interacting through an inter-level subdivision operator. This ensures a standard finite-element data structure. Suitable marking strategies have been used to select elements for refinement. The capability of the numerical schemes is demonstrated with two-dimensional examples.