MCE 561 Computational Methods in Solid Mechanics Nonlinear Issues

1. MCE 561 Computational Methods in Solid Mechanics Nonlinear Issues

2. Nonlinear FEA Many problems of engineering interest involve nonlinear behavior. Such behavior commonly arises from the following three sources: • Nonlinear Material Behavior • This is one of the most common forms of nonlinearity, and would include nonlinear elastic, plastic, and viscoelastic behavior. For thermal problems, a temperature dependent thermal conductivity will produce nonlinear equations. • Large Deformation Theory (Geometric Nonlinearity) • If a continuum body under study undergoes large finite deformations, the strain-displacement relations will become nonlinear. Also for structural mechanics problems under large deformations, the stiffness will change with deformation thus making the problem nonlinear. Buckling problems are also nonlinear. • Nonlinear Boundary or Initial Conditions • Problems involving contact mechanics normally include a boundary condition that depends on the deformation thereby producing a nonlinear formulation. Thermal problems involving melting or freezing (phase change) also include such nonlinear boundary conditions.

3. Features of Nonlinear FEA Problems • While Linear Problems Always Have a Unique Solution, Nonlinear Problems May Not • Iterative/Incremental Solution Methods Commonly Used on Nonlinear Problems May Not Always Converge or They May Converge To The Wrong Solution • The Solution To Nonlinear Problems May Be Sensitive To Initial and/or Boundary Conditions • In General Superposition and Scalability Will Not Apply To Nonlinear Problems

4. s s e e W (j) (i) L Example Nonlinear Problems Material Nonlinearity Nonlinear Stress-Strain Behavior Elastic/Plastic Stress-Strain Behavior This behavior leads to an FEA formulation with a stiffness response that depends on the deformation

5. Simple Truss Under Large Deformation Truss Has a Different Geometry Thus Implying a New Stiffness Response Undeformed Configuration Example Nonlinear Problems Large Deformation Finite Deformation Lagrangian Strain-Displacement Law Large Deflection Beam Bending

6. Example Nonlinear Problems Contact Boundary Conditions pc w No ContactNo Contact Force Initial ContactLeads to New Boundary Condition With Contact Force Evolving ContactBoundary Condition Changing With Deformation; i.e. w and pcDepend on Deformation and Load

7. Nonlinear FEA ExampleTemperature Dependent Conductivity Hence Nonlinearity in Both Stiffness Matrix and Loading Vector

8. Solution Techniques for Nonlinear Problems • Since Direct Inversion of the Stiffness Matrix Is Impossible, Other Methods Must Be Used To Solve Nonlinear Problems • Incremental or Stepwise Procedures • Iterative or Newton Methods • Mixed Step-Iterative Techniques

9. Ku Ku F F Solution ToK(u)u=F Solution ToK(u)u=F u0 u1 u2 u u3 u1 u2 u0 u Concave Ku-u Relation - Divergence Convex Ku-u Relation - Convergence Direct Iteration Method Method is based on making successive approximations to solution using the previous value of u to determine K(u) Therefore nonlinear solution methods may result in no converged solution