Talk:Strain for scientists and engineers

I'm still confused

And I am the author of this page!

None of the Wikipedia pages and nothing I can find on the internet seem sufficient in getting from to Cauchy's Infinitesimal strain theory (which I find simple and beautiful) to Large Strain Theory. After consulting with the Bard Chatbot, I came to the conclusion that we need a Wikiversity article. The title might be:

• Large Lagrangian strain theory

What follows is my question and Bard's response. I asked the question because if a theory is (1) exact and (2) already worked out by someone else, then I'm confident I can work it out and create an improvement over what I see on the internet.--Guy vandegrift (discuss • contribs) 02:22, 25 January 2024 (UTC)

Is the Lagrangian strain tensor valid for arbitrarily large strain?

The validity of the Lagrangian strain tensor (also known as the Green-Lagrange strain tensor) for arbitrarily large strain depends on the context and the specific application. Here's a more nuanced answer:

Yes, the Lagrangian strain tensor is theoretically valid for arbitrarily large strains. It accurately captures the deformation of a material particle regardless of the magnitude of the strain. This is because it is defined in the reference (undeformed) configuration of the material, where the geometry is still well-defined.

However, there are practical limitations to using the Lagrangian strain tensor for very large strains:

• Numerical issues: When implemented in numerical simulations, the Lagrangian strain tensor can lead to instabilities and convergence problems for extremely large strains, especially with complex loading conditions. This is because the calculations involve higher-order terms in the deformation gradient, which can amplify small numerical errors.
• Loss of physical meaning: While mathematically valid, the Lagrangian strain tensor may not be the most relevant measure of strain for very large deformations. For example, it does not directly reflect the elongation or compression of material fibers, which can be a crucial factor in some applications.

Alternative strain measures for large strains:

• Engineering strain: This is a simpler measure based on the relative change in length, often used for small strains. It can be inaccurate for large deformations due to its linear nature.
• Logarithmic strain (Hencky strain): This measure is based on the natural logarithm of the stretch ratio and is more suitable for large strains compared to engineering strain. However, it can become infinite for certain types of deformations.
• Almansi strain: This measure is defined in the current (deformed) configuration and may be easier to interpret for some applications at large strains. However, it is not symmetric and can lead to difficulties in certain calculations.

Choosing the right strain measure:

The choice of strain measure for large deformations depends on the specific context and the desired level of accuracy. Ultimately, the goal is to capture the essential features of the deformation in a way that is both mathematically sound and physically meaningful for the specific application.

In conclusion, while the Lagrangian strain tensor is theoretically valid for arbitrarily large strains, its practical applicability depends on the specific situation and the existence of alternative measures that might be more suitable for large deformation scenarios. BardChatBot (not really) (discuss • contribs) 02:20, 25 January 2024 (UTC)