1. Basic Geometric Nonlinearities Chapter Five - APPENDIX

2. Training Manual October 15, 2001 Inventory # 001565 5a-2 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.

3. Training Manual October 15, 2001 Inventory # 001565 5a-3 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. A A0A0 P

4. Training Manual October 15, 2001 Inventory # 001565 5a-4 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 X Y

5. Training Manual October 15, 2001 Inventory # 001565 5a-5 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

6. Training Manual October 15, 2001 Inventory # 001565 5a-6 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.

7. Training Manual October 15, 2001 Inventory # 001565 5a-7 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 ] = rotation matrix containing information about the amount and direction that the material point rotated as a rigid body = stretch matrix containing the information about the straining of the body at the material point

8. Training Manual October 15, 2001 Inventory # 001565 5a-8 Basic geometric nonlinearities … Appendix Implementing [U] in the Definitions of Strain By knowing the stretch matrix [U ], the 3-D generalizations for the 1-D logarithmic strain  l and the 1-D Green-Lagrange strain  G can be constructed.

9. Training Manual October 15, 2001 Inventory # 001565 5a-9 Basic geometric nonlinearities … Appendix The Hencky Strain The log (Hencky) strain is computed by: where [  H ] is the strain tensor expressed in matrix form. [  H ] is, in this case, the 3-D equivalent to the 1-D logarithmic, or true strain  l.

10. Training Manual October 15, 2001 Inventory # 001565 5a-10 Basic geometric nonlinearities … Appendix Green-Lagrange Strain In 3-D, the Green-Lagrange strain can be computed directly from the stretch matrix [U ] as shown below: This measure directly neglects the rotation matrix [R ] from the strain field evaluation. [  G ] can be re-written in terms of the gradient of the displacement field as shown below: The first two terms are the linear small strain terms and the last term is the nonlinear contribution to the strain measure.

11. Training Manual October 15, 2001 Inventory # 001565 5a-11 Basic geometric nonlinearities … Appendix Extending the Nonlinear Stress Definitions to the General 3-D Case As in the 1-D case, there are conjugate stress measures which can be defined for each of the nonlinear strain definitions in 2-D and 3-D.

12. Training Manual October 15, 2001 Inventory # 001565 5a-12 Basic geometric nonlinearities … Appendix The Cauchy Stress The Cauchy or true stress tensor [  ] (written here in matrix form) gives the current force in the deformed configuration per unit deformed area. If we let then, in 3-D, the Cauchy stress tensor [  ] relates {dP } to {dA } as The Cauchy stress is a readily interpretable physical quantity. = vector defining components of an elemental area in the deformed body = the corresponding elemental force acting on

13. Training Manual October 15, 2001 Inventory # 001565 5a-13 Basic geometric nonlinearities … Appendix The 2 nd Piola-Kirchhoff Stress Let represent a force component derived from the transformation Also let The 2 nd Piola-Kirchhoff stress tensor [S ] relates and as –[S ] is a symmetric stress tensor which is frequently used in finite-strain elasticity formulations and is the conjugate stress tensor to the Green-Lagrange strain [  G ]. [S ] is a non-physical stress tensor (a pseudo stress tensor). [S ] does not lend itself to direct physical interpretation. = vector defining an elemental area in the undeformed body where

14. Training Manual October 15, 2001 Inventory # 001565 5a-14 Basic geometric nonlinearities … Appendix The Relationship Between [  ] and [S ] The physical Cauchy [  ] stress can be related directly to the non-physical 2 nd Piola-Kirchhoff pseudo stress [S ] via the expression 

15. Training Manual October 15, 2001 Inventory # 001565 5a-15 Basic geometric nonlinearities … Appendix The Fully Consistent Nonlinear Tangent Stiffness Matrix It is a well known fact that in most nonlinear structural problems, the use of a consistent nonlinear tangent stiffness matrix can rapidly increase the rate of convergence of a Newton-Raphson based solution process. A consistent, or full tangent stiffness matrix normally leads to a quadratic convergence rate in the iterative solution process.

16. Training Manual October 15, 2001 Inventory # 001565 5a-16 Basic geometric nonlinearities … Appendix What is a Consistent Nonlinear Stiffness Matrix? The consistent nonlinear stiffness matrix is obtained by taking the derivative of the discretized finite element equations and is a function of both the element internal force vector and the element applied load vector.

17. Training Manual October 15, 2001 Inventory # 001565 5a-17 Basic geometric nonlinearities … Appendix The Discretized Nonlinear Static Finite Element Equations The discretized nonlinear static finite element equations which are solved can be characterized at the element level by: where = number of total elements = element internal force vector in the element coordinate system = transformation matrix transferring into the global coordinate system = applied load vector at the element level in the global coordinate system

18. Training Manual October 15, 2001 Inventory # 001565 5a-18 Basic geometric nonlinearities … Appendix The Element Internal Force Vector The element internal force vector is given by where = element strain - nodal displacement matrix = element stress vector = element volume In terms of the internal force definition given above, the discretized nonlinear finite element equations (force balance) can be re-written as

19. Training Manual October 15, 2001 Inventory # 001565 5a-19 Basic geometric nonlinearities … Appendix Developing the Incremental Nonlinear Stiffness Matrix The consistent nonlinear stiffness matrix is obtained by taking the derivative of the discretized finite element equations as shown below where and where

20. Training Manual October 15, 2001 Inventory # 001565 5a-20 Basic geometric nonlinearities … Appendix The Consistent Nonlinear Stiffness Matrix = the main tangent matrix = the initial stress matrix which incorporates the effects of stress-stiffening = the initial displacement-rotation matrix which includes the effect of changing geometry in the stiffness relation = the initial load matrix which includes the effect of changing load orientation (follower forces) in the stiffness relation