77Chapter 3: Nonlinearity and Analysis TypesNonlinear Effects and Formulations

Nonlinear Effects and FormulationsThere are three sources of nonlinearity in structural analysis: material, geometric, and nonlinear boundary conditions.

Material NonlinearityRecall that linear analysis assumes a linear relationship between stress and strain. Material nonlinearity results from the nonlinear relationship between stresses and strains. In addition, large strain can influence the material behavior.

Considerable progress has been made in attempts to derive the continuum or macroscopic behavior of materials from microscopic backgrounds, but, up to now, commonly accepted constitutive laws are phenomenological. Difficulty in obtaining experimental data is usually a stumbling block in accurately simulating material behavior. A plethora of models exist for more commonly available materials like elastomers and metals. Material models of considerable practical importance are: composites, viscoplastics, creep, soils, concrete, powder, and foams. Figure 3-1 shows representation of the elastoplastic, elasto-viscoplasticity, and creep.

Examples of material nonlinearities include metal plasticity, materials such as soils and concrete, or rubbery materials (where the stress-strain relationship is nonlinear elastic). Various plasticity theories such as von Mises or Tresca (for metals), and Mohr-Coulomb or Drucker-Prager (for frictional materials such as soils or concrete) can be selected by you. Three choices for the definition of subsequent yield surfaces are available in SOL 400. They are isotropic hardening, kinematic hardening, or combined isotropic and kinematic hardening. With such generality, most plastic material behavior, with or without the Bauschinger effect, can be modeled.

Figure 3-1 Material Nonlinearity

Elasto-Plastic Behavior Elasto-Viscoplastic Behavior

tCreep Behavior

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78 Nonlinear User�s GuideNonlinear Effects and Formulations

Geometric NonlinearityGeometrically nonlinear problems involve large displacements; �large� means that the displacements invalidate the small displacement assumptions inherent in the equations of linear analysis. For example, consider a classical thin plate subject to a lateral load; if the deflection of the plate�s midplane is anything close to the thickness of the plate, then the displacement is considered large and a linear analysis is not applicable.

Geometric nonlinearity results from the nonlinear relationship between strains and displacements on the one hand and the nonlinear relation between stresses and forces on the other hand. If the stress measure is conjugate to the strain measure, both sources of nonlinearity have the same form. This type of nonlinearity is mathematically well defined, but often difficult to treat numerically. Three important types of geometric nonlinearity occur:

1. Problems where large rotation occur.2. The analysis of buckling and snap-through problems (see Figure 3-2 and Figure 3-3).3. Large strain problems such as manufacturing, crash, and impact problems. In such problems, due to large strain

kinematics, the mathematical separation into geometric and material nonlinearity is not unique.

Figure 3-2 Buckling

Figure 3-3 Snap-Through

Nonlinear Boundary ConditionsBoundary conditions and/or loads can also cause nonlinearity. These loads can be conservative, as in the case of a centrifugal force field (see Figure 3-4); they can also be non conservative, as in the case of a follower force on a cantilever beam (see Figure 3-5). Also, such a follower force can be locally nonconservative, but represent a

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79Chapter 3: Nonlinearity and Analysis TypesNonlinear Effects and Formulations

conservative loading system when integrated over the structure. A pressurized cylinder (see Figure 3-6) is an example of this.

Figure 3-4 Centrifugal Load Problem (Conservative)

Figure 3-5 Follower Force Problem (Non conservative)

Figure 3-6 Pressurized Cylinder (Globally Conservative)

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80 Nonlinear User�s GuideNonlinear Effects and Formulations

Contact and friction problems lead to nonlinear boundary conditions. This type of nonlinearity manifests itself including assembly modeling, metal forming, gears, interference of mechanical components, pneumatic tire contact, and crash (see Figure 3-7). Loads on a structure cause nonlinearity if they vary with the displacements of the structure. If there is a change in constraints due to contact during loading, the problem may be classified as a boundary nonlinear problem and would require the use of BCTABLE/ BCTABLE1, BCONECT, BCONPRG, BCONPRP, BCBODY, or BSURF bulk data entry options. CGAP elements would have been used in the traditional nonlinear sequences of SOL 106 or 129; however, the use of GAP elements is strongly discouraged in SOL 400.

Figure 3-7 Contact and Friction Problem

Geometric NonlinearitiesGeometric nonlinearity leads to two types of phenomena: change in structural behavior and loss of structural stability.

There are two natural classes of large deformation problems: the large displacement, small strain problem and the large displacement, large strain problem. For the large displacement, small strain problem, changes in the stress-strain law can be neglected, but the contributions from the nonlinear terms in the strain displacement relations cannot be neglected. For the large displacement, large strain problem, the constitutive relation must be defined in the correct frame of reference and is transformed from this frame of reference to the one in which the equilibrium equations are written.

The kinematics of deformation can be described by the following approaches:

Lagrangian FormulationIn the Lagrangian method, the finite element mesh is attached to the material and moves through space along with the material. In this case, there is no difficulty in establishing stress or strain histories at a particular material point and the treatment of free surfaces is natural and straightforward.

The Lagrangian approach also naturally describes the deformation of structural elements; that is, shells and beams.

Shortcomings of the Lagrangian method are that flow problems are difficult to model and that the mesh distortion is as severe as the deformation of the object.

The Lagrangian approach can be classified in two categories: the total Lagrangian method and the updated Lagrangian method. In the total Lagrangian approach, the equilibrium is expressed with the original undeformed state as the reference; in the updated Lagrangian approach, the current configuration acts as the reference state. The kinematics of