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07 Oct 2022

# Linear Analysis and Non-Linear Analysis in FEA

Skill-Lync

In finite-element analysis (FEA) when the applied load conditions (force, heat, etc.,) hold a linear relation with the response (displacement, temperature, etc.,) obtained from a system it is called linear analysis. This type of analysis always obeys Hooke’s law (stress is directly proportional to the strain). In reality, these conditions will be applicable for a material until it reaches the elastic limit.

## What is Linear Analysis?

The most typical common method used in FEA is linear analysis. In

, the relationship between applied forces and displacements is linear. In actuality, this is applicable to structural issues where stresses stay within the material's linear elastic range. The stiffness matrix of the model is constant in linear static analysis, and the solving time is comparatively quick in comparison to the non-linear analysis of the same model. As a result, before doing a complete nonlinear analysis, the linear static analysis is frequently utilised for an initial estimate.

The materials that obey Hooke’s law are called linear elastic Materials. Elastic materials return to their shape after the unloading. The property of linear analysis is good enough to predict its physics exactly. There are few materials which don't obey Hookes’s law even in the elastic region. Such kinds of materials cannot be analysed under linear conditions. A few examples are elastomers and rubber.

## The Hypothesis For Linear Analysis

• Displacement will be very small
• The load being applied is constant
• Material is considered an elastic material.

## How does the Solver solve the Linear Analysis Problem?

From the stress-strain curve, it can be known whether the structure will fail or break after the yield point is reached. But when the solver took this as a Linear Analysis problem, the structure won’t fail. We need to interpret the stress plot by comparing the maximum stress value to the yield point of that material.

## What is Nonlinear Analysis?

When the applied load conditions (force, heat, etc.,) hold a non-linear relation with the response (displacement, temperature, etc.,) obtained from a system called Non-linear Analysis. In Nonlinear analysis, the stiffness matrix keeps on changing throughout the analysis. The non-linearity could be caused by the geometry, material and also the contacts being applied. Based on this, nonlinearity is split into three types

1. Geometric nonlinearity
2. Material nonlinearity
3. Contact nonlinearity

## Geometric Nonlinearity

When a structure experiences higher deformation the stiffness matrix will change and this is characterized by geometric nonlinearity.

## Material Nonlinearity

Material nonlinearity is characterized by various parameters which affect the material properties. It is based on the deformation, temperature, pressure, strain, rate of deformation and so on. These parameters will either follow elasticity or plasticity. For example, if the temperature has an effect on the material, both the mechanical and thermal properties should be considered through thermo-elasticity or thermo-plasticity.

## Contact Nonlinearity

Contact nonlinearity is characterized by the change in contacts during the analysis because of the applied load.

## Processing the Non-Linear Analysis

As the stiffness matrix keeps changing throughout the analysis, it needs to be calculated throughout the analysis. Loads are applied as a load step during the non-linear analysis. For each load step, the residual load is calculated which is the difference between the applied load and the internal load developed. The structure becomes equilibrium when the residual load is zero. Achieving zero residual loads in Non-linear analysis is not so easy and a negligible load condition is considered an equilibrium condition. The stiffness matrix is calculated at each load step iteratively to achieve the equilibrium condition. This method of calculating the stiffness matrix is called the Newton-Raphson method.

The below table shows the difference between Linear and Non-Linear Analysis.

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