Arbitrary Non-linear Materials Research Articles

Elasto-plastic large deformation analysis of multi-patch thin shells by isogeometric approach

This paper studies elasto-plastic large deformation behaviour of thin shell structures using the isogeometric computational approach with the main focus on the efficiency in modelling the multi-patches and arbitrary material formulation. In terms of modelling, we employ the bending strip method to connect the patches in the structure. The incorporation of bending strips allows to eliminate the strict demand of the C1 continuity condition, which is postulated in the Kirchhoff-Love theory for thin shell, and therefore it enables us to use the standard multi-patch structure even with C0 continuity along the patch boundaries. Furthermore, arbitrary nonlinear material models such as hyperelasticity and finite strain plasticity are embedded in the shell formulation, from which a unified thin shell formulation can be achieved. In terms of analysis, the Bézier decomposition concept is used to retain the local support of the traditional finite element. The performance of the presented approach is verified through several numerical benchmarks.

A robust non-linear mixed hybrid quadrilateral shell element

In the paper a non-linear quadrilateral shell element for the analysis of thin structures is presented. The variational formulation is based on a Hu–Washizu functional with independent displacement, stress and strain fields. The interpolation matrices for the mid-surface displacements and rotations as well as for the stress resultants and strains are specified. Restrictions on the interpolation functions concerning fulfillment of the patch test and stability are derived. The developed mixed hybrid shell element possesses the correct rank and fulfills the in-plane and bending patch test. Using Newton's method the finite element approximation of the stationary condition is iteratively solved. Our formulation can accommodate arbitrary non-linear material models for finite deformations. In the examples we present results for isotropic plasticity at finite rotations and small strains as well as bifurcation problems and post-buckling response. The essential feature of the new element is the robustness in the equilibrium iterations. It allows very large load steps in comparison to other element formulations. Copyright © 2005 John Wiley & Sons, Ltd.

Stability of fundamental nonlinear guided waves

We determine the stability of nonlinear guided waves, including solitons, nonlinear surface waves, and modes of nonlinear waveguides, by simply reading the familiar dispersion diagram without further calculation. The waveguide can be of arbitrary cross section and arbitrary profile shape and can be composed of arbitrary nonlinear materials distributed nonuniformly. This criterion is the first to our knowledge that is universal for the long-standing problem of stability of fundamental nonlinear guided waves, i.e., waves without nodes. We also show that the transverse stability of planar nonlinear guided fundamental waves demands that the material be self-defocusing in at least some portion of space, whereas planar black solitons are unstable in arbitrary self-defocusing material.

Calculations on nonlinear dielectric slab waveguides by the iterative eigenfunction expansion method

A new treatment for the behavior of both TE and TM nonlinear waves in an arbitrary nonlinear material is given. The treatment is based on a recently developed method, the iterative eigenfunction expansion method (IEEM). Compared with the results of the finite element method (FEM), this scheme has proven very successful. It is also easily extendable to other nonlinear behavior in varying geometries.

Finite—element formulation for nonlinear dielectric slab waveguides

AbstractThis paper proposes a new general‐purpose numerical method based on the finite‐element method for both TE and TM polarizations, by which the guided mode behaviors in a dielectric slab waveguide with an arbitrary nonlinear material and an arbitrary refractive index profile can be analyzed systematically. The matrix equation derived in this paper is a nonlinear matrix eigenvalue problem with the effective relative permittivity as an eigenvalue and the principal electromagnetic field as an eigenvector. The equation can be solved in a self‐consistent way by an iterative calculation with the linear solution as an initial value. In the present formulation, the biaxial anisotropy of the nonlinear permittivity tensor for the TM wave is taken into account rigorously. A uniaxial approximation which has often been used is not employed at all. From these features, the present method is considered to be a powerful analysis method for a practical nonlinear dielectric slab waveguide. Further, it can be used for investigation of the validity of various approximate methods. The present method is applied to the nonlinear TE and TM wave propagation in a symmetric three‐layered slab waveguide with a nonlinear core layer and the effectiveness of the method is confirmed.