טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentMtanes Eli
SubjectFinite Element Formulations Based on Cosserat Point Theory
for Nonlinear Analysis of Elastic Anisotropic
Structures
DepartmentDepartment of Civil and Environmental Engineering
Supervisor Professor Mahmood Jabareen
Full Thesis textFull thesis text - English Version


Abstract

Technological challenges related to the safety and durability of structures demand a high level of accuracy and robustness from finite element tools. Therefore, development of a finite element program would be significantly advanced if an element could be developed that was robust and accurate enough to be applied for a wide range of finite deformation problems.

The Cosserat point theory is a special continuum theory that models deformations of a small finite region like a finite element. It uses element directors for describing its kinematic quantities. The element nodal forces of the Cosserat point element (CPE) are expressed via intrinsic director couples, which are determined by derivatives of a strain energy function that characterizes the CPE’s response. This strain energy is additively decomposed into homogeneous and inhomogeneous parts. The former can be described by any three-dimensional strain energy function of elastic materials and the latter depends on constitutive coefficients that control inhomogeneous deformations. A special care must be taken in determining those coefficients as they control the CPE’s accuracy.

The current study provides a new methodology for determining the constitutive coefficients that can be applied for any three-dimensional strain energy of elastic materials and for a general element's geometry. In the proposed methodology, the constitutive coefficients are determined by integrating three-dimensional strain energy function that quadratically depends on a modified strain filed. This strain filed consists of a compatible part and enhanced part, in which the latter is designed to improve the accuracy for bending, torsion, and higher-order hourglassing.

Using the proposed methodology, a three-dimensional CPE (3D-CPE) for accurately modeling general hyperelastic orthotropic materials at finite deformations with initially distorted element shapes was developed. However, modeling very thin structures within the 3D-CPE is not preferable due to the well-known locking pathologies in shell structures. Hence, a solid-shell formulation based on the Cosserat point theory (SSCPE) is developed. Curvature-thickness locking and transverse shear locking are avoided by applying a new concept denoted by the assumed natural inhomogeneous strain (ANIS) on the SSCPE’s Green-Lagrange strain tensor. The new methodology is adopted together with the ANIS concept to obtain constitutive coefficients that ensure accuracy and elimination of all locking pathologies in the SSCPE. Finally, the SSCPE is generalized to model linear and nonlinear laminated elastic structures denoted by LSSCPE. To account for the coupling between homogeneous and inhomogeneous deformations in composite structures, the inhomogeneous strain energy function of the LSSCPE includes dependence on the homogeneous deformations. A double mapping is introduced to perform the integration through the shell layers for evaluating the volume average stiffness of the layers.

To conclude, the Cosserat point theory associated with the new methodology provides element formulations that are accurate, robust, free of locking, computational efficient and almost insensitive to mesh distortion. Specifically, three finite element formulations based on the Cosserat point theory for modeling elastic structures under finite deformations have been developed in this work. These formulations provide numerical tools based on the same theory that can be applied for modeling a wide range of engineering problems in solid-mechanics.