Papers by Keyword: Eulerian Formulation

Abstract: The present paper investigates possible improvements to the performances of beam finite element with curvature correction where large slopes as well as large displacement are involved. The displacement field of the beam element is based on simple strain functions satisfying the requirement of exact representation of curvature. The truss element is introduced with the current cross sectional and the current length instead of initial area and initial length in large displacement solution. The finite element method is used in conjunction with linearised incrementation and the Newton-Raphson iterative technique. The two basic formulations to problem involving geometric non-linear, Eulerian and Lagrangian are also discussed. The present elements offer significant advantages over existing stiffness-based elements. Consequently, fewer elements are needed to yield results of comparable accuracy. This is demonstrated with the analysis of several simple example structures by comparing the results to those of stiffness based elements and analytical solution.

Abstract: Using two-dimensional Eulerian formulations coupling viscoplastic flow and heat transfer, the behaviors of aluminum alloys and stainless steel during FSW were overviewed. The plastic behaviors of the materials are complicated and the flow stresses are depending on deformation rate, temperature and deformation histories. Constitutive equations considering both strain hardening from accumulation of crystal defects and softening from recovery or recrystallization were used to model the materials. Strain hardening is incorporated with a strength that evolves with deformation rate and temperature along streamlines in the flow field. Strength evolutions have a Voce-like saturation limit because of the severe plastic deformation during FSW process. The model equations for kinematic and temperature were solved using the standard finite element method. The evolution equation for the strength is integrated along streamlines. The strength and temperature distribution vary with process conditions and constitutive equations. Stainless steel and AA6061 have different strengthening mechanisms. Modified constitutive equations were applied to reflect microstructural features of each material.

Abstract: Friction stir welding (FSW) process of aluminum alloys was investigated using a two-dimensional Eulerian formulation coupling viscoplastic flow and heat transfer and strain hardening. The thermal equation for the temperature was modified to stabilize temperature distribution using a Petrov-Galerkin method. The evolution equation for strength was calculated using a streamline integration method. Predicted strength was compared with experiments. Based on crystal plasticity, texture evolution was predicted during FSW of AA6061.

Abstract: Texture evolution during friction stir welding of stainless steel was investigated using both predictions by crystal plasticity and EBSD measurements. Two- and three-dimensional Eulerian formulations are used to model friction stir welding. Plane strain deformation is assumed in a two-dimensional model, and an initial uniform texture changes into a torsion texture with monoclinic sample symmetry after deformation. Around the tool pin, the texture strengthens, weakens and restrengthens repeatedly. It is found from a simple circular streamline model that the relative magnitudes of the deformation rate and spin along the streamlines decide textural stability. In order to consider more complicated material behaviors, such as movement along the thickness direction due to a threaded tool pin and a tool shoulder, a three-dimensional Eulerian formulation is also implemented. Materials starting under the tool shoulder travel down to the bottom, producing the longest material streamlines. Those material points are predicted to have stronger texture components than others. EBSD results are compared with the predictions.