## non-linear systems

Nonlinear analyzes are performed due to various sources of nonlinearity. Therefore, it makes sense to classify these based on the origin of the nonlinearity. When the stiffness of a part changes under operating conditions, a nonlinear analysis is required. Geometric nonlinearity occurs when the change in stiffness is solely due to changes in shape, as in large deformations. Most FEA programs provide ways to account for changes in direction, such as rotating and constant loads. A pressure vessel under very high pressure is an example of a situation where geometric non-linearity needs to be considered.

### material consideration

There is also another type of nonlinearity called material nonlinearity. This occurs when the change in stiffness is solely due to changes in material properties under operating conditions. Unlike the linear material model, where stress is proportional to strain, material nonlinearity means that stresses and strains do not increase proportionally to load change. In addition, permanent deformations can occur and the model does not return to its original shape after the load has been removed. Therefore, in nonlinear analyses, several types of nonlinearity have to be considered, since for many problems no single cause for the nonlinearity can be identified. By considering the different causes of non-linearity, more accurate and realistic results can be achieved, which is of great importance for the design of components and systems

In principle, all materials can exhibit non-linear deformation under sufficiently high loads or strains. However, some materials inherently exhibit non-linear behavior regardless of loading. Typical examples are rubber, elastomers, plastics, plasticizers or even foams. These materials are viscoelastic and exhibit non-linear stress-strain behavior. In practice, however, most materials are non-linear, since their stiffness and strength depend on the load and can change even under high loads. These include, for example, steel, concrete, wood or composite materials

### There are many simulation programs for nonlinear systems. Some of the best known are:

- ANSYS: A comprehensive simulation software that supports Finite Element Analysis (FEA) and Computational Fluid Dynamics (CFD) for a variety of applications.
- Abaqus: FEA software that supports a wide range of materials and nonlinear analyzes including geometry, material, and contact nonlinearities.
- COMSOL Multiphysics: A multiphysics simulation software that supports FEA and Partial Differential Equation (PDE) for a variety of applications including electromagnetic fields, fluid mechanics, and chemistry.
- LS-DYNA: A software for dynamic FEA analysis, which is particularly suitable for non-linear dynamic analysis and large deformations.
- Nastran: A FEA software that supports a variety of linear and nonlinear analysis including dynamic analysis and thermal analysis.

However, these software tools are not the only options available and there are many other simulation programs on the market suitable for specific applications. The selection of a suitable simulation software depends on the specific application and should be done after careful consideration

In order to simulate the different behavior of different material types under operating conditions, special techniques and material models for FEM programs were developed. The table below lists different material models and their respective areas of application to provide an overview of these techniques.

## Models for the calculation of different material types

##### Elastoplastic

**Von Mises and Tresca** are two models of strength of materials used to calculate the strength of materials under load.

The Von Mises criterion assumes that the strength of a material is determined by a so-called "equivalent stress" which is calculated from the three principal stresses acting on a material. The criterion states that a material fails under load when the equivalent stress reaches a certain critical level. The Von Mises criterion works well for materials that exhibit plastic deformation because it considers the components of stress that cause the plastic deformation.

The Tresca criterion, on the other hand, only considers the two highest principal stresses that act on a material. It states that a material will fail under stress when the difference between these two stresses reaches a certain critical level. The Tresca criterion is better suited for materials that are brittle and fail under stress cracks or fractures.

Both models have their advantages and disadvantages and are used depending on the application and type of material.

##### Hyperelastic

**Mooney Rivlin** is a material model used to describe the mechanical behavior of elastic materials, especially rubber. The model is based on the assumption that the deformation of the material is due to a change in volume and a distortion in shape.

The Mooney-Rivlin model uses a linear combination of the material's deformation energy versus two parameters called the Mooney-Rivlin coefficients. These coefficients are empirical material constants that can be determined through experimentation.

The model also takes into account the influence of temperature on material behavior using another material parameter called the temperature-dependent Mooney-Rivlin coefficient.

The Mooney-Rivlin model is well suited to describe the behavior of rubber under small to medium-sized deformations, which are found in many engineering applications. However, it is less suitable for describing very large deformations or materials exhibiting a nonlinear stress-strain relationship, such as some polymers.

##### viscoelastic

This is often used for the calculation of viscoelastic materials **Kelvin-Voigt model or the Maxwell model** used. Both models are linear models and describe the viscoelastic material behavior through a combination of elastic and viscous deformation.

The Kelvin-Voigt model consists of a spring-damper system that describes the viscoelastic properties of the material. It assumes that the deformation of the material has both an elastic and a viscous component and that the viscous component is proportional to the rate of deformation. However, the model is not able to take into account the relaxation time of the material.

The Maxwell model consists of a spring and damper connected in series. It describes the viscoelastic properties of the material through a combination of elastic deformation and relaxation. The model is able to take into account the relaxation time of the material and is therefore often the preferred model for describing viscoelastic materials.

There are other models for describing viscoelastic materials, such as the Zener model and the Standard Linear model, but the Kelvin-Voigt model and the Maxwell model are the most common.

##### Crawl

Viscoelastic material models are usually used to calculate the creep behavior of materials. A commonly used model for creep calculation is this **Norton Hoff model.**

The Norton-Hoff model describes the creep behavior of materials as a combination of an elastic and a viscous component. It is based on the assumption that the material's creep rate is proportional to stress and the exponential factor of temperature. The model uses two material parameters, the Young's modulus and the creep exponent, to describe the creep behavior of the material.

The Norton-Hoff model is well suited for describing the creep behavior of materials at high temperatures and under high loads, such as is the case in the aerospace industry or in the manufacture of ceramic components. However, it is important to note that the model is empirical and needs to be adjusted based on the specific conditions and materials for which it is used.

##### super elastic

The model most commonly used to describe the superelastic behavior of materials is the phase transition model, also known as **Martensite-Austenite model**.

The phase transition model describes the superelastic behavior of materials through the transformation between two crystal structures called martensite and austenite. The martensite phase is the deformed, lower symmetry phase while the austenite phase is the higher symmetry phase that is stable at higher temperatures. The conversion between the two phases occurs reversibly by changing the temperature or load.

In this model, the elastic properties of the two phases are described by different moduli of elasticity that are linked to each other. The transition between the two phases is described by a phase transition condition, often expressed in terms of a critical stress or strain.

The phase transition model is an empirical model and requires experimental data to determine the material parameters. However, it is often used successfully to describe and simulate the superelastic behavior of shape memory alloys such as nitinol.