# Nonlinear problems

Nonlinear problems of mechanics, unlike the linear, distinguished by depending on a sequence of states through which the system passed from the beginning to the end of the matter. In contrast to the linear problems, it is difficult to clearly assess the solvability of the task. For nonlinear problems, only limited resources for solving are available. The most commonly used methods are numerical methods. A solution of nonlinear problems needs more experience and intuition in comparison with the solution of linear problems. Nonlinear analyzes place greater demands on the expertise of designers. The success in solving the nonlinear problem often depends on the selected computational strategy.

## Material nonlinearity

The constitutive relations of solid deformable bodies are described by the relationship between the stress tensor (forces applied on the material) and the strain tensor (describes the response of the material). The simplest relationship between these tensors is described by Hooke’s law, and it is a linear elastic model. This material model is valid only for small displacements and small deformations, its response is time-independent, and after removing load acting on the body, a deformation will disappear. On the contrary, in materials with large reversible deformation and without hysteresis, a nonlinear elastic material model, which is defined by the relationship between the (Cauchy) stress tensor and the (Almansi) strain tensor, is used. In case the material viscosity plays a significant role, a viscoelastic material model is applied. Viscoelastic materials always respond to deformation with a time-delay. Therefore, the load induces a time-dependent response, and in the case of short-term load, the material behaves elastically. If the relaxation and the creep of metals at high temperatures are considered, then it is necessary to use the nonlinear viscoplastic material. When the state of stress in the body reaches plasticity condition, the elastoplastic model is used to describe the behaviour of materials. In the area of the material hardening it is usually assumed that the material model behaves as linear, multilinear, or has the character of a general curve. The elastic deformation of material is governed by Hooke’s law and it is reversible, but after exceeding the limit stress, a part of the irreversible deformation is present in the material. To describe the plastic deformation of real metallic materials under cyclic loading, the Bauschinger effect must be taken into account. That reflects the increase of the stress required to relaxation of dislocations accumulated on the microstructural barriers, which is associated with the change of the yield surface. This is achieved by a generalization of the hardening material model, for example, with the kinematic hardening. It is known that a viscoplastic material model have to be used for the metallic materials with a body-centred lattice because the history of plastic strains is dependent on the strain rate.

## Geometrical nonlinearity

The geometrical nonlinearity is caused by large displacements and rotations, which may be accompanied by large strain. Accordingly, we distinguish two different cases of geometrical nonlinearity:

- large displacements (displacements and rotations),
- large deformation (strain), always include large displacements.

In both cases, the material may behave linearly or nonlinearly. Geometrically nonlinear problems are therefore jobs where the monitoring of the body element must take into consideration not only the shape changes, but also displacements and rotations of the element (of the body) as a solid whole. It should be noted that the displacement may no longer be infinitesimal. Displacements of individual points are then composed of displacements and rotations of a rigid body as a whole and shifting the individual points of the body due to the deformation in the new position. Also, the definitions of the stress tensor and strain tensor are different. The definition of the stress tensor is complex. Elemental internal forces at the deformed configuration may relate to elementary areas, either in the original or in the deformed configuration. Another issue is the choice of a strain tensor. The strain tensor redesigned to meet the constitutive laws must be appropriately defined to the stress tensor. An example is the second Piola-Kirchhoff stress tensor, which is energetically conjugated with Green’s strain tensor. In deriving geometric relationships for the case of linear elasticity theory, it is assumed that the strain tensor components are small (of the order of 1e-3). Then quadratic members of geometric relationships can be neglected and the engineering strain tensor (Cauchy) is used. The limits of applicability of the engineering strain tensor are usually reported at 1% of strain, (ie strain 1E-2). The formulation is suitable for engineering analysis of structural parts and structures. In the cases of forming simulation, simulation of deformation of rubber or plastic parts , may reach even hundred percent of strain. Nonlinear members then cannot be ignored and “complex” strain tensor has to be used.

## Usage

- structural mechanics,
- multiphysical simulations (i.e. fluid-structure interaction, elasto-acoustic, thermomechanics, etc.)
- shape optimization.