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 straindisplacement 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
Example Nonlinear Problems
Material Nonlinearity
s
s
e
e
Nonlinear Stress-Strain Behavior
Pi   AE
ui
du i
dx
(i)
Elastic/Plastic Stress-Strain Behavior
uj
W
L
(j)
P j   AE
This behavior leads to an FEA formulation with a
stiffness response that depends on the deformation
[ K ( u )]{ d }  { F }
du
dx
j
Example Nonlinear Problems
Large Deformation
Simple Truss
Under Large Deformation Truss Has
a Different Geometry Thus Implying
a New Stiffness Response
Undeformed Configuration
Large Deflection Beam Bending
2
 du
d 
1  dw   

 
   q  0
 EA 
dx 
2  dx   
 dx


d
dx
2
2
2

d w
 EI
2
dx

Finite Deformation Lagrangian
Strain-Displacement Law
2

d 
dw  du
1  dw   
 
 
   f  0

 EA
dx
dx
dx
2
dx

  



[ K ( u )]{ d }  { F }
u
1  u 
Ex 
 

x
2  x 
2
Example Nonlinear Problems
Contact Boundary Conditions
pc
w
No Contact
No Contact Force
Initial Contact
Leads to New
Boundary Condition
With Contact Force
[ K ( u )]{ d }  { F ( u )}
Evolving Contact
Boundary Condition
Changing With
Deformation; i.e.
w and pc Depend on
Deformation and Load
Nonlinear FEA Example
Temperature Dependent Conductivity
Law q   k ( T )
Fourier Conduction
dT
dx
[ K ( T )]{ T }  { F ( T )}
[ K ( T )] 
{ F ( T )} 

T
W

W
k ( T )[ B ] [ B ]dx
[  ] Qdx  [  ] k ( T )
T
T
dT
dx
Hence Nonlinearity in Both Stiffness
Matrix and Loading Vector
h
0
Solution Techniques for Nonlinear Problems
[ K ( u )]{ d }  { F }
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
Direct Iteration Method
Method is based on making successive approximations to
solution using the previous value of u to determine K(u)
{u
r 1
1
}  [ K ( u )] { F }
r
Ku
Ku
F
F
Solution To
K(u)u=F
u0 u1 u2
u
Convex Ku-u Relation - Convergence
Solution To
K(u)u=F
u2 u0
u1 u3
u
Concave Ku-u Relation - Divergence
Therefore nonlinear solution methods may result in no converged solution