Nonlinear effects in structures occur mainly due to nonlinear materials and large deformations (rotations and/or strains). Contact problems exhibit nonlinear effects due to changes in boundary conditions. Primary operations for nonlinear elements are updating element coordinates and applying loads for large displacements, and the internal relaxation iteration for material nonlinearity.

The primary solution operations are gradual load or time increments, iterations with convergence tests for acceptable equilibrium error and stiffness matrix updates.

Nonlinear characteristics and General Recommendations

Modeling for nonlinear analysis should follow the guidelines for good modeling practice pertaining to linear analysis, which are summarized as follows:

• The analyst should have some insight into the behavior of the structure to be modelled; otherwise, a simple model should be the starting point.
• Discretization should be based on the anticipated stress gradient, i.e., a finer mesh in the area of stress concentrations.
• If there is a change in constraints due to contact during loading, the problem may be classified as a boundary nonlinear problem.

The materials nonlinearity is characterized by material properties. However, the material nonlinear effects may or may not be significant depending on the magnitude and duration of the loading.

The geometric nonlinearity is characterized by large rotations which usually cause large displacements. Intuitively, geometric nonlinear effects should be significant if the deformed shape of the structure appears distinctive from the original geometry without amplifying the displacements. There is no distinct limit for large displacements because geometric nonlinear effects are related to the dimensions of the structure and the boundary conditions. The key to this issue is to know where the loading point is in the load-deflection curve of the critical area.

Some nonlinear problems, such as plastic deformation, creep deformation and contact problems with frictional forces are path-dependent; others, such as hyperelastic or purely geometric nonlinear, are path dependent. An incremental process may not be required for a purely geometric nonlinear problem. For computational efficiency, however, the analysis of such a problem does frequently need incremental solutions.

Notes:

- Localize the nonlinear region for computational efficiency. If unsure, perform a linear analysis prior to the nonlinear analysis.

- The nonlinear region usually requires a finer mesh. Use a finer mesh if severe element distortions or stress concentrations are anticipated.

Large element net deformations should be avoided. In areas of the structure where large total deformations are expected, the mesh must be fine enough to keep the element net deformations small. The element net rotation should not exceed 20 degrees and the element should not be stretched by more than 10%. If stretches exceed 20%, it is recommended to use hyperelastic elements if applicable.

Material nonlinearity

Material nonlinearity is an inherent property of any engineering material. Material nonlinear effects may be classified into many categories such as plasticity, nonlinear elasticity, hyperelasticity, creep and viscoelasticity. Material nonlinearity becomes an issue in your analysis if the stress-strain relationship is nonlinear. Constitutive relations characterize the macroscopic behavior of the material, which stems from its microscopic constitution. All engineering materials are inherently nonlinear. However, it isn't feasible to characterize a nonlinear material by a single set of equations for the entire range of environmental conditions, such as loading, temperature, and rate of deformation.

The linear elastic (Hookean) material is the simplest case of all. As the strain exceeds the infinitesimal range, the material exhibits nonlinear behavior:

• Nonlinear elastic behavior if it is recoverable.
• Plastic behavior if it is irrecoverable.

Some types of nonlinear materials are:

• Nonlinear elastic: both the loading and unloading occur along the same path. In general, it is usefull to predict multiaxial stress state for materials for which only the tension and compression stress-strain data is available. The stress-strain data in tension may be different from that in compression.
• Hyperelastic: used to analyze elastomers up to a large strain. Elastomers regain their original shape once the force is removed (unless damage occurs). If the strains are small, the results of the hyperelastic elements should compare with those of the linear elastic elements. The hyperelastic elements are fully nonlinear finite deformation elements, including large strain and large rotation.
• Elastic-plastic.
• Visco-elastic: a viscoelastic analysis lets you analyze situations in which the strain rate has a significant effect on the material response (creep).
• Temperature dependent.

Geometric nonlinearity

Geometric nonlinearities are manifested in problems involving large rotations and large deformation. The geometric nonlinearity becomes discernible when the structure is subjected to large displacement and rotation. Geometric nonlinear effects are prominent in two different aspects: geometric stiffening due to initial displacements and stresses, and follower forces due to a change in loads as a function of displacements. The characteristics of a geometrically nonlinear problem are:

• Follower forces due to large rotations.
• Geometric stiffening due to initial stress effect (as a result of large rotations).
• Large strains due to large deformation.

Geometric nonlinear effects should be significant if the deformed shape of the structure appears distinctive from the original geometry when you inspect it visually. In geometric nonlinear analysis, the elements may undergo large total displacements and rotations but the net deformation of each element has to remain small, therefore these elements are called "small strain" elements. For linear material and infinitesimal deformation, a nonlinear analysis with small strain elements gives results which are identical to a lineal analysis.

A more rigorous and quantitative definition for the large displacements can be derived from the plate theory of Kirchhoff and Love, which states that the small deflection theory is valid for a maximum deflection of less than 20% of the plate thickness or 2% of the small span length.

Stiffening of a membrane, stiffness in a pendulum or snap-through of an arch are geometric nonlinear effects. The motion of a pendulum under gravity is caused by geometric (differential) stiffness. Follower forces are manifested when the applied loads are displacement dependent, such as pressure load and thermal load applied on the surface that rotates. Centrifugal force is another example of follower forces.

Large strain effects are pronounced in metal forming, rubber and elastomer applications. In such applications, the strains exceed 100%. Finite strain formulation is required to treat the problems in this category.

The previous text is a summary of the document: Basic Nonlinear Analysis User's guide. Siemens, 2014.