Appendix-BasicGeometricNonlinearities.pptVIP

July 30, 1999 Basic Geometric Nonlinearities Chapter Five - APPENDIX Basic geometric nonlinearities Appendix Large Strain Theory The information contained in this appendix is not required for successful use of geometric nonlinearities in ANSYS, and is therefore normally not presented in the class lecture. It is provided as additional background information for the user who wants a deeper understanding of ANSYS large-displacement features. Basic geometric nonlinearities … Appendix Extending the Nonlinear Strain Definitions to the General 3-D Case In 2-D and 3-D, not only does the basis for length change when a component is undergoing large strain deformations, but thickness, area and volume also change. Basic geometric nonlinearities … Appendix Motion and Deformations When a body is under the influence of some external loads, it will move and deform. If we watch the motion of a point attached to the body, with its initial position {X } and its final position {x }, it will be seen to displace an amount {u }, where Basic geometric nonlinearities … Appendix Deformation Gradient The deformation gradient is a measure of how much the body has deformed and is defined as The deformation gradient [F ] contains information on the: Volume Change Rotation Shape Change due to Straining Basic geometric nonlinearities … Appendix … deformation gradient Note that by definition, the deformation gradient [F ] eliminates translations; a necessary requirement for a strain definition. When defining our strains, we also want to exclude the rotation part (since it does not contribute to straining) and isolate the shape change part. We can accomplish this by implementing the Polar Decomposition Theorem. Basic geometric nonlinearities … Appendix Polar Decomposition The deformation gradient [F ] can be separated into a rotation and a strain part using the polar decomposition theorem: [F ] = [R ] [U ] [R ] [U ] Basic geometric nonlinearities … Appendix Implementing