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

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

6. Example Nonlinear Problems Contact Boundary Conditionspc w
No Contact No Contact Force
Initial Contact Leads to New Boundary Condition With Contact Force
Evolving Contact Boundary Condition Changing With Deformation; i.e. w and pc Depend on Deformation and Load

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