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MCE 561 Computational Methods in Solid Mechanics Nonlinear Issues

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MCE 561 Computational Methods in Solid Mechanics

Nonlinear Issues

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.

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

s

e

e

W

(j)

(i)

L

Example Nonlinear Problems Material NonlinearityNonlinear Stress-Strain Behavior

Elastic/Plastic Stress-Strain Behavior

This behavior leads to an FEA formulation with a stiffness response that depends on the deformation

Under Large Deformation Truss Has a Different Geometry Thus Implying a New Stiffness Response

Undeformed Configuration

Example Nonlinear Problems Large DeformationFinite Deformation Lagrangian Strain-Displacement Law

Large Deflection Beam Bending

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

Nonlinear FEA ExampleTemperature Dependent Conductivity

Hence Nonlinearity in Both Stiffness Matrix and Loading Vector

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

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 MethodMethod 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

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