Linear vs. non-linear 2

[Tunalover]

Folks-
How can you tell if a structural problem is non-linear? When you exceed the yield point of an aluminum test specimen in a tension test the specimen goes into plastic deformation. Is non-linear? What kind of linearities are there in solid mechanics? What makes a problem non-linear? It's OK to talk with math!
Thank in advance!



Tunalover

 

[dfish67]

Geometry and Material properties can be non-linear. These are the two biggies in simple stress analysis. One can consider none, or both of these during analysis. Frankly, a majority of problems fit nicely into linear analysis in both an analytical and engineering (practical) sense.

Geometry- large deflections cause secondary effects such as P-Delta (a moment caused by deflection).

Material- Plastic deformation and Cracking (concrete) are nonlinear effects that, in a lot of instances, are considered failures so analysis beyond these effects is not required (cracking in concrete would be an exception as it is often expected).

If you want to go beyond yield (you did in your question) and deflections are large (your call) use non-linear analysis.

 

[TheRaptor]

Tunalover,

Linearity is a simple mathematical approximation to simplify real time problems. In linear analysis, the deflections and rotations are very small, stresses are proportional to strain, material is elastic, the equations are written for the initial structual configuration, loads maintain their original directions as the structure deforms, and the global equations (KU=P) are solved in a single step.

No real time problem will satisfy all these conditions.

There are basically 3 kinds of non-linearities. The 1st two are geometric and material non-linearities.Geometric non-linearity is caused due to large deflections in the structure and material non-linearity is where the material properties are dependant on the strain. I need not go into the details of these as they have already been well explained by dfish67.

The 3rd kind is contact non-linearity. This is the case with crash analysis.

Buckling is inherently non-linear.

 

[btrueblood]

In dynamic simulations, coulomb (sliding) friction creates non-linear responses.

 

[prost]

Linearity is just an assumption we make to simplify the modeling--in general to describe any physics, we write very complicated, nonlinear differential equations. "Linearity" then comes from our efforts to simplify those complicated equations in order to create a workable engineering model. Every problem is nonlinear, it matters only the degree. We don't model everything with nonlinear kinematics and/or materials, because it would be too expensive. Therefore we try to run linear models whenever possible. However, the assumptions built into the linearity must be considered with ever model. You should incorporate testing of any relevant assumptions into your interpretation of the results. The answer to the question "what should I calculate to test those assumptions?" is a very complicated one. Consider a nice smooth structure, no cracks it. Say all you were interested in was testing a material for nonlinearities. You have to have some idea what the material's constitutive relation (or material curve, stress vs. strain) is before the analysis. You would generally complete the analysis, first assuming the material indeed behaved linearly. Now test the linearity assumption. A metal (nonlinearity in composites and polymers should evaluated differently) is assumed to be nonlinear with the maximum stress exceeds the yield stress (even the choice of stress is not constant. What stress? Some use von Mises stress, others use max. shear, it depends strongly on accepted industry practice). Say you like von Mises. If the von Mises stress anywhere exceed's the material's yield stress, then the material is considered to behave nonlinearly. Testing the assumption--now set up the material in your analysis so that it is elasto-plastic (elastic perfectly plastic, Ramberg Osgood, etc.). Run the analysis again, this time make sure it does the nonlinear material analysis (some FE software runs the linear analysis first, then the nonlinear analysis starts with this linear solution to complete the nonlinear analysis). Check the max. von Mises stress again--if the max. von Mises stress in this nonlinear run was less than one percent smaller than the von Mises stress calculated in the linear solution, then you can say the linear solution was adequate because the nonlinearities (in this case, plasticity) was small.

 

[btrueblood]

Nicely wrapped, prost. Ought to make it a FAQ.

 

[gurmeet2003]

Tunalover,

To decide if you should include non-linear affects (material, goemetry etc.) first run a linear analysis. Then look at the results. Usually you can tell if a linear analysis will be enough or not. If stresses exceed yield in area of concern, a non-linear material may be required. If deformations are too large, a non-linear geometry effect may be required.

Gurmeet