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A quick question about Finite Strain Formulation for Solid Mechanics Module

Posted Oct 5, 2021, 2:30 p.m. EDT Structural & Acoustics, Geometry, Structural Mechanics 2 Replies

Yueming Song
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Hi all,

I am curious about this governing equation when geometric nonlinearity is turned on in solid mechanics module. It looks like the basic idea is calcuating the Cauchy Stress first based on C's and elastic strains and then correlate to Second Piola- Kirchhoff Stress by considering volume changes and inelastic deformation.

I am also wondering if I want to model particle intercalation induced stress and strain with big volumetric change, is it a good path to follow by turning on geometric nonlinearity and introducing external stress and strain?

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2 Replies Last Post Oct 19, 2021, 10:40 a.m. EDT
Henrik Sönnerlind COMSOL Employee

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Posted: 3 years ago Oct 14, 2021, 3:42 p.m. EDT

The basis of the geometrically nonlinear formulation is what often is called a Total Lagrangian formulation. The strain measure that is used is the Green-Lagrange strain.

The inner part of the expression you have posted, is the tensor multiplication of the linear constitutive tensor and the elastic part of the Green-Lagrange strain. In the absence of inelastic strains, that would be the final 2nd Piola-Kirchhoff stress.

With inelastic deformations, things get trickier. By multiplicatively removing the inelastic part of the deformation, the elastic problem can be seen as computed in an intermediate configuration. To get from the undeformed to the final state, we do two things: First the inelastic stretching, followed by the elastic stretching. So the stresses computed by the expression above are of 2nd Piola-Kirchhoff type, but not in the correct configuration. Hence the extra pull-back operation.

If you are going to use large inelastic strains, caused by intercalation, I would suggest that you use geometric nonlinearity, and then add an External Strain node. Set Strain input to Deformation gradient, inverse. Then enter a diagonal tensor, where each element is (current_volume/original_volume)^(-1/3)

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Henrik Sönnerlind
COMSOL
The basis of the geometrically nonlinear formulation is what often is called a *Total Lagrangian* formulation. The strain measure that is used is the Green-Lagrange strain. The inner part of the expression you have posted, C:\varepsilon_{el} is the tensor multiplication of the linear constitutive tensor and the elastic part of the Green-Lagrange strain. In the absence of inelastic strains, that would be the final 2nd Piola-Kirchhoff stress. With inelastic deformations, things get trickier. By multiplicatively removing the inelastic part of the deformation, the elastic problem can be seen as computed in an intermediate configuration. To get from the undeformed to the final state, we do two things: First the inelastic stretching, followed by the elastic stretching. So the stresses computed by the expression above are of 2nd Piola-Kirchhoff type, but not in the correct configuration. Hence the extra pull-back operation. If you are going to use large inelastic strains, caused by intercalation, I would suggest that you use geometric nonlinearity, and then add an *External Strain* node. Set *Strain input* to *Deformation gradient, inverse*. Then enter a diagonal tensor, where each element is (current_volume/original_volume)^(-1/3)
Yueming Song
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Posted: 3 years ago Oct 19, 2021, 10:40 a.m. EDT

Thank you Henrik! Now everything makes sense to me.

Thank you Henrik! Now everything makes sense to me.

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