The characteristics of nonlinearity



All physical processes are to some degree nonlinear – every day, the materials that surround us all exhibit either large deformations or inelastic material behavior. For example, pressing an expanded balloon to create a dimple (the stiffness progressively increases, that is the rate of deflection decreases the more squeezing force is applied), or flexing a paper-clip until it is bent (the material has undergone a permanent change in its characteristic response).

In general there are three major, and most common, sources of nonlinear structural behavior.

  • Geometric nonlinearity: Structures whose stiffness depends on the displacement they may undergo are geometrically nonlinear. This accounts for phenomena such as the stiffening of a loaded clamped plate, and buckling or ‘snap-through’ behavior in slender structures or components. Geometric nonlinearity is usually more difficult to characterize than purely material considerations. Geometric effects may be both sudden and unexpected, but without taking them into account any computational simulation may completely fail to predict the real structural behavior.
  • Material Nonlinearity: This refers to the ability for a material to exhibit a nonlinear stress-strain (constitutive) response. Elasto-plastic, crushing, and cracking are good examples, but this can also include time-dependent effects such as visco-elasticity or visco-plasticity (creep). Material nonlinearity is often characterized by a gradual weakening of the structural response as an increasing force is applied, due to some form of internal decomposition.
  • Boundary condition nonlinearity: In highly flexible components and assemblies with multiple components, progressive displacement can cause self-contact or component-to-component contact. In such boundary condition nonlinearity, the stiffness of the structure or assembly may change when two or more parts either contact or separate from initial contact – sometimes only a little, but sometimes quite a bit. Bolted connections, toothed gears and sealing or closing mechanisms are common examples.

Many structures exhibit combinations of these three main sources of nonlinearity, and the algorithms which solve nonlinear equations are generally set up to handle nonlinear effects from a variety of sources.

To find evidence of possible nonlinear behavior, look for characteristics such as permanent deformations, and any gross changes in geometry. Cracks, necking, thinning, distortions in open section beams, buckling, stress values which exceed the elastic limits of the materials, evidence of local yielding, shear bands, and temperatures above 30% of the melting temperature are all indications that nonlinear effects may play a significant role in understanding the structural behavior.

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