Types of Non-Linearity
Nonlinearity
is natural in physical problems. In fact, the linear assumptions we make are
only valid in special circumstances and usually involve some measure of
“smallness”, for example, small strains, small displacements, small
rotations, small changes in temperature, and so on.
We
use linear approximations not because they are more correct but because
· Linear solutions are easier to compute.
· The computational cost is smaller.
· Solutions can be superposed on each other.
A nonlinear analysis is needed if the loading on a structure causes significant changes in stiffness. Typical reasons for stiffness to change significant are:
⁻ Contact
between two bodies
⁻
Strains
beyond the elastic limit (plasticity)
⁻
Large
deflections, such as with a loaded fishing rod
However, linear analysis is not adequate and
nonlinear analysis is necessary when
·
Designing
high performance components.
·
Establishing
the causes of failure.
·
Simulating
true material behavior.
·
Trying to
gain a better understanding of physical phenomena.
Modern analysis software makes it
possible to obtain solutions to nonlinear problems. However, experienced skill
is required to determine their validity and these analyses can easily be
inappropriate. Care should be taken to specify appropriate model and solution
parameters. Understanding the problem, the role played by these parameters and
a planned and logical approach will do much to ensure a successful solution.
Generally, linear analysis means that
the relationship between the applied force and the response to this force (Ex:
displacement) is linear in nature. We know that this displacement can be
obtained by inverting the stiffness matrix and then multiplying it with the
force vectors. In other words, this means that the stiffness matrix does
not change during a linear analysis.
This can be further emphasized by
taking the example of a spring with linear and nonlinear characteristics.
As the stiffness is dependent on
the displacement, which keeps changing, the initial stiffness matrix cannot be
used without continually updating and inverting it during the course of the
analysis. This is the reason why nonlinear analysis takes more time to solve
compared to a linear FEM analysis.
Mostly,
nonlinear analysis depends on the need of the goals i.e., for fatigue, dynamic
and so on. A cost-benefit analysis is usually necessary before embarking on the
nonlinear analysis of a problem.
There
are three major types of non-linearity:
1. Material (plasticity, creep,
viscoplasticity/viscoelasticity)
2. Geometric (large deformation, large strains)
3. Boundary
These may occur singly or in combination.
Material
Non-linearity:
Nonlinear
stress-strain relationships are a common cause of nonlinear structural
behaviour. Material
nonlinearity involves the nonlinear behavior of a material based on a current
deformation, deformation history, rate of deformation, temperature, pressure,
and so on. Examples of nonlinear material models are large strain (visco)
elasto-plasticity and hyper elasticity (rubber and plastic materials) and other
types.
Many factors
can influence a material’s stress-strain properties, such as:
- Load
history (as in elastoplastic response) - Environmental
conditions (such as temperature) - The amount
of time that a load is applied (as in creep response)
Geometric Non-linearity
Geometric nonlinearity arises when the
changes in the model’s geometry are very high during the course of deformation.
This happens when there are
1. Large
deformations
2. Large
rotations
3. Both
4. Initial
loading conditions before the start of the analysis
Ø If a structure experiences large deformations, its changing geometric
configuration can cause the structure to respond nonlinearly
Ø Geometric nonlinearity is characterized by “large”
displacements and/or rotations
Ø In linear geometric analysis, the deformations and rotations are smaller
like within 5 % as generic rule, but in case of nonlinear geometric analysis,
displacement and rotations are large. Small changes in magnitude of force for
nonlinear geometry, can change convergence behavior considerably. However the
geometric nonlinearity is not only due to large deformation/rotation but varies
based loading and situation too.
Ø An example would be the fishing
rod
It is difficult to predict the case of
geometric nonlinearity, as it needs lot of experiments and judge the case by
looking the scenario. In FEA software geometry nonlinear is solved simply by
keeping nonlinear zone on.
In analyses involving geometric
nonlinearity, changes in geometry as the structure deforms are considered in
formulating the constitutive and equilibrium equations. Many engineering
applications such as metal forming, tire analysis, and medical device analysis
require the use of large deformation analysis based on geometric nonlinearity.
Small deformation analysis based on geometric nonlinearity is required for some
applications, like analysis involving cables, arches and shells.
Contact
non-linearity:
Ø When two separate surfaces touch each other such that they become
mutually tangent, they are said to be in Contact. In contact
nonlinearity abrupt change in stiffness may occur when bodies come into or out
of contact each other. This type on nonlinearity is used to simulate the gap
between two parts.
Ø In the common physical sense, surfaces that are in contact have below
characteristics:
ü They do not interpenetrate
ü They can transmit compressive normal forces and tangential friction
forces
ü They often do not transmit tensile normal forces
ü They are therefore free to separate and move away from each other
Nonlinear
FEA Issues:
- Achieving convergence: Obtaining
convergence is biggest challenge in nonlinear analysis.
·
Trial and error is sometimes required.
·
Complex problems might require more load
increments, and many iterations at the each load step to achieve the
convergence.
- Balancing expenses versus accuracy
·
FEA involves expenses (Solution time, disk
and memory requirements)
·
More detail and a finer mesh generally lead
to a more accurate solution, but require more time and system resources.
·
Nonlinear analysis need additional iteration
affects both accuracy and expenses.
- Verification
·
Difficult to verify the FEA results due to
increased complexity of nonlinear behaviour.
·
The sensitivity studies (mesh convergence,
increased mesh density, reduced load increments, varying other model
parameters) become more expensive.
- Results of the nonlinear analysis cannot
be scaled. - The structural behavior can be markedly
non-proportional to the applied load.
