Finite element formulations for large strain anisotropic material with inextensible fibers

Anisotropic material with inextensible fibers introduce constraints in the mathematical formulations. This is always the case when fibers with high stiffness in a certain direction are present and a relatively weak matrix material is supporting these fibers. In numerical solution methods like the finite element method the presence of constraints - in this case associated to a possible fiber inextensibility compared to a matrix - lead to so called locking-phenomena. This can be overcome by special interpolation schemes as has been discussed extensively for volume constraints like incompressibility as well as contact constraints. For anisotropic material behaviour the most severe case is related to inextensible fibers. In this paper a mixed method is developed that can handle anisotropic materials with inextensible fibers that can be relaxed to extensible fiber behaviour. For this purpose a classical ansatz, known from the modeling of volume constraint is adopted leading stable elements that can be used in the finite strain regime.DFG/SPP/1748DFG/WR19/50-1DFG/SCHR570/23-

A quasi-incompressible and quasi-inextensible element formulation for transversely isotropic materials

The contribution presents a new finite element formulation for quasi-inextensible and quasi-incompressible finite hyperelastic behavior of transeversely isotropic materials and addresses its computational aspects. The material formulation is presented in purely Eulerian setting and based on the additive decomposition of the free energy function into isotropic and anisotropic parts, where the former is further decomposed into isochoric and volumetric parts. For the quasi-incompressible response, the Q1P0 element formulation is outlined briefly, where the pressure-type Lagrange multiplier and its conjugate enter the variational formulation as an extended set of variables. Using the similar argumentation, an extended Hu-Washizu-type mixed variational potential is introduced, where the volume averaged fiber stretch and fiber stress are additional field variables. Within this context, the resulting Euler-Lagrange equations and the element formulation resulting from the extended variational principle are derived. The numerical implementation exploits the underlying variational structure, leading to a canonical symmetric structure. The efficiency of the proposed approached is demonstrated through representative boundary value problems. The superiority of the proposed element formulation over the standard Q1 and Q1P0 element formulation is studied through convergence analyses. The proposed finite element formulation is modular and exhibits very robust performance for fiber reinforced elastomers in the inextensibility limit

A general theory for anisotropic Kirchhoff-Love shells with embedded fibers and in-plane bending

In this work we present a generalized Kirchhoff-Love shell theory that can capture anisotropy not only in stretching and out-of-plane bending, but also in in-plane bending. This setup is particularly suitable for heterogeneous and fibrous materials such as textiles, biomaterials, composites and pantographic structures. The presented theory is a direct extension of existing Kirchhoff-Love shell theory to incorporate the in-plane bending resistance of fibers. It also extends existing high gradient Kirchhoff-Love shell theory for initially straight fibers to initially curved fibers. To describe the additional kinematics of multiple fiber families, a so-called in-plane curvature tensor -- which is symmetric and of second order -- is proposed. The effective stress tensor and the in-plane and out-of-plane moment tensors are then identified from the mechanical power balance. These tensors are all second order and symmetric for general materials. The constitutive equations for hyperelastic materials are derived from different expressions of the mechanical power balance. The weak form is also presented as it is required for computational shell formulations based on rotation-free finite element discretizations.Comment: This version updates reference list and improves text editing, results unchange

Deformations and motions in compressible non-linear elasticity

The primary purpose of this study is to identify solutions for deformations and motions in compressible non-linear elasticity. The research study mainly focuses on the mathematical theory of deformations and motions of non-linearly elastic compressible hollow spheres and hollow cylinders reinforced with inextensible fibers in the radial direction. Static and dynamic solutions for both unstressed and everted cases are presented for the hollow spheres and hollow cylinders of isotropic elastic materials that are radially inextensible. Different strain energy density functions are then applied for further demonstration to the extent allowed by the analytical approach. This thesis is concerned with the mathematical theory of non-linear elasticity and no discussion of shell or membrane theories, or of numerical methods is included. The setting is purely isothermal and no reference is made to thermodynamics. Attention is confined to twice-continuously differentiable deformations; discontinuities are not considered

Parameter Identification of Fiber Orientation Models Based on Direct Fiber Simulation with Smoothed Particle Hydrodynamics

The behavior of fiber suspensions during flow is of fundamental importance to the process simulation of discontinuous fiber reinforced plastics. However, the direct simulation of flexible fibers and fluid poses a challenging two-way coupled fluid-structure interaction problem. Smoothed Particle Hydrodynamics (SPH) offers a natural way to treat such interactions. Hence, this work utilizes SPH and a bead chain model to compute a shear flow of fiber suspensions. The introduction of a novel viscous surface traction term is key to achieve full agreement with Jeffery’s equation. Careful modelling of contact interactions between fibers is introduced to model suspensions in the non-dilute regime. Finally, parameters of the Reduced-Strain Closure (RSC) orientation model are identified using ensemble averages of multiple SPH simulations implemented in PySPH and show good agreement with literature data