Inventor Nastran - What are the different types of nonlinear behaviour?

In this blog, you will learn about the different types of nonlinear analysis with examples taken from Inventor Nastran and explanations of phenomena causing nonlinear behaviours.

Nonlinear analysis generically falls into the following three categories.

Geometric nonlinearity - Where a component experiences large deformations and, as a result, can cause the component to experience nonlinear behaviour. A typical example is a fishing rod. 

Material nonlinearity - When the component goes beyond the yield limit, the stress/strain relationship becomes nonlinear as the material starts to deform permanently.

Contact - Includes the effect of two components coming into contact; that is, they can experience an abrupt change in stiffness, resulting in localised material deformation at the region of contact.

While many practical problems can be solved using linear analysis, some or all its inherent assumptions may not be valid:

  • Displacements and rotations may become large enough that equilibrium equations must be written for the deformed rather than the original configuration. Large rotations can cause pressure loads to change in direction and to change in magnitude if there is a change in the area to which they are applied.
  • Elastic materials may become plastic, or the material may not have a linear stress-strain relation at any stress level.
  • Part of the structure may lose stiffness because of buckling or material failure.
  • Adjacent parts may make or break contact with the contact area as the loads change.

Geometric Nonlinearity

The geometric nonlinearity becomes a concern when the part(s) deform such that the small displacement assumptions are no longer valid. The significant displacement effects are a collection of different nonlinear properties, such as:

  1. Large deflections.
  2. Stress stiffening/softening.
  3. Snap-thru.
  4. Large strain.

Large Deflections

When your components or assemblies experience more than 10 degrees rotations, you should consider nonlinear analysis. This is because linear analysis assumes a small displacement theory in which sin(θ) ≈ (θ).

Stress Stiffening

Stress stiffening (also known as geometric stiffening) only affects thin structures where the bending stiffness is minimal compared to the axial stiffness. For instance, could you consider the following plate subjected to a load? The structure is fixed around the perimeter. This thin-walled structure will undergo significant stress stiffening as the part transitions from reacting to the load in bending to reacting to the load in-plane.

The images below taken from Inventor Nastran show two plate results; the first image results from a nonlinear analysis (peak deflection 3.321mm). The second image results from a linear analysis (peak deflection is 26.03mm).

 

Stress stiffening effects are caused by tensile stresses, which result from larger displacements, not by the displacements themselves. The actual displacement in the model is not a clear indication of the degree of nonlinearity, nor is the tensile stress magnitude. A similar tension in one geometry or load orientation may result in significantly less stress stiffening than in another.

Snap-thru and Buckling

Other common geometric nonlinear situations involve snap-thru and buckling problems, often referred to as bi-stable or multi-stable systems. Many snap-thru problems behave nearly linearly until the point where a small amount of additional load causes a large amount of deflection, where a second stable position is reached. Capturing this snap-thru is a complicated numerical problem.

Large Strain

Large strains are typically associated with large displacements causing permanent deformation as stresses above yield have been exceeded. Cold heading, rubber seal compression, and metal forming are good examples of large strain examples.

 

Material Nonlinearity

When components experience stress above yield, the linear analysis results are not valid. In these cases, we need to define the stress and strain behaviour of materials above yield to get an accurate behaviour. However, most materials and even metals have some amount of ductility. This ductility allows hot spots to locally yield, thus reducing the stresses compared to what a linear analysis would predict.

The metal bracket from the image below shows very different stress distributions between linear and nonlinear materials. The right image contains linear analysis results and shows peak stresses well above yield. The nonlinear material analysis on the left shows a different contour due to the stress redistribution. The Peak plastic strain was 5.7% in the nonlinear material analysis.

Boundary Condition Nonlinearity

The most common boundary nonlinearities are:

  1. Follower forces.
  2. Contact

Contact conditions model the interaction of two separate parts. Boundary conditions such as separation contacts are generally considered nonlinear, as the contact allows separation and sliding between components. This type of contact is typically used in bolted connections where the bolts hold two plates, and the plates are allowed to slide and separate depending on the extent of the loading conditions.  Another example is in impact-type analyses, as illustrated below.

Follower Forces

This nonlinear effect means the force's direction moves with the part's deformation or movement. This can be best demonstrated with the cantilevered strip shown below, which is loaded with a force of 100N, and three analyses are performed with different large displacement settings.

The first image shows the unrealistic "growth" that occurs when large displacement effects are turned off (LGDISP=OFF). The second image shows the results of large displacements turned on but follower forces turned off (LGDISP=2). The final image uses large displacement effects with follower forces and is the most accurate (LGDISP=ON).

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