# Hyperelastic problems - analytical solutions?

• FEAnalyst
In summary: So, once again, I am looking for a book or tutorial that will present such a model in a more user-friendly way.In summary, according to the individual, hand calculations for nonlinear materials are possible even when plasticity or creep conditions become involved. However, there may be no analytical solutions for more complex loading scenarios involving hyperelastic materials. A book or tutorial that would present a model for stress-strain relationships for hyperelastic materials might be helpful.
FEAnalyst
TL;DR Summary
Are there analytical solutions for simple solid mechanics problems (e.g. bending of a beams, torsions of shafts, internal pressure in pipes) involving hyperelastic material?
Hi,
I have recently become interested in analytical solutions of various advanced solid mechanics problems, mostly nonlinear ones. I consider simple geometries and loads (like bending of beams, torsion of shafts, or internal pressure in pipes), but for nonlinear materials. I have learned that hand calculations for such cases are possible even when plasticity or creep conditions become involved. The results are in very good agreement with FEA. However, there is still at least one major issue left that puzzles me - hyperelasticity. Are there any analytical solutions at all for such materials (considering simple geometries and loads)? If so, what kinds of problems can be solved this way - only axial tension/compression or maybe also bending, torsion etc.? And where to look for such solutions? So far I have not been able to find anything concrete, but I am missing literature where examples of this type would be presented.

I'm not sure which hyperelastic material model could be used for calculations like that but I assume that if they are possible at all then neo-Hookean material might be the right choice as it seems to be the simplest model, with only two constants involved: https://en.wikipedia.org/wiki/Neo-Hookean_solid

Thank you in advance for any help.

Hi, I never dealt directly with hyperlastic materials, but a nice book that I came across some years ago had a chapter dedicated to hyperelasticity. The book is this one:

Notes on Continuum Mechanics

It is very dense from the mathematical point of view, but it might help! It presents not only the model you mentioned, but also other ones.

Thank you for the recommendation, I was not familiar with this publication. However, what I need is a book with exemplary stress/strain calculations involving hyperelastic material models. And all the books I've found so far (including this one) only discuss the material models themselves. I am looking for a source that will be equivalent to what helped me hand calculate problems involving plasticity and creep. Those were old Polish books in a "collection of tasks" form. I don't know if such publications are available in English and if they cover hyperelasticity as well but maybe there are also some textbooks focusing on theory but featuring occasional exemplary calculations (like a typical "mechanics of materials" or "solid mechanics" guide). Maybe I should look for something more practical, like a manual for the design of hyperelastic (polymer/rubber) components (such as gaskets).

Anyway, I will be very grateful for any other recommendations. It may not be possible to solve more complex loading scenarios (bending, torsion) involving hyperelastic materials but it should be possible to solve at least simple tension/compression analytically. I guess that it's just a matter of obtaining stress-strain curve for uniaxial tension from material model's constants. The problem is that those models use description based on strain energy density instead of directly providing stress-strain relationship.

## 1. What are hyperelastic materials?

Hyperelastic materials are materials that exhibit large deformations under applied loads, but return to their original shape once the load is removed. These materials have a nonlinear stress-strain relationship, meaning that the stress applied is not directly proportional to the strain produced.

## 2. What are some examples of hyperelastic materials?

Some common examples of hyperelastic materials include rubber, silicone, and certain types of plastics. These materials are often used in applications where flexibility and resilience are important, such as in medical devices, seals, and gaskets.

## 3. What are the main challenges in solving hyperelastic problems analytically?

The main challenge in solving hyperelastic problems analytically is the complexity of the material behavior. Since hyperelastic materials have a nonlinear stress-strain relationship, their behavior cannot be described by simple equations and requires more advanced mathematical models. Additionally, the large deformations exhibited by these materials make it difficult to apply traditional analytical methods.

## 4. What are some common analytical solutions for hyperelastic problems?

Some common analytical solutions for hyperelastic problems include the Mooney-Rivlin model, the Ogden model, and the Neo-Hookean model. These models use different mathematical equations to describe the behavior of hyperelastic materials and are often used in specific applications based on the material properties and loading conditions.

## 5. How are analytical solutions for hyperelastic problems validated?

Analytical solutions for hyperelastic problems are typically validated by comparing the results to experimental data. This involves conducting tests on the hyperelastic material and measuring its behavior under different loading conditions. The results from the experiments are then compared to the analytical solutions to ensure their accuracy and reliability.

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