Usage of First Order Elastic Constants in Soft Body Equations

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Discussion Overview

The discussion revolves around the usage of first order elastic constants in soft body equations, particularly focusing on the indexing of these constants and their application in constitutive equations. Participants explore the relationship between different orders of elasticity and the implications for their specific equations.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant seeks clarification on how to properly index first order elasticity constants for their soft body equations, referencing specific materials and matrices.
  • Another participant notes the symmetry relations in elasticity constants, indicating that certain constants can be related through a 6x6 matrix due to the symmetric nature of stress and strain.
  • A distinction is made between “first order” as linear in a parameter and “fourth order” as referring to the rank of the elasticity tensor, which has four indices.
  • Further discussion confirms the symmetry of the constants and the equivalence of certain indices, while expressing the need for thorough verification of these symmetries in the context of ongoing simulations.

Areas of Agreement / Disagreement

Participants express agreement on the symmetry relations of the elasticity constants and the distinction between different orders of elasticity. However, there remains some uncertainty regarding the implications of these distinctions for the participant's specific equations and whether they are confined to lower order constants.

Contextual Notes

Participants mention the complexity of the relationships between different orders of elasticity and the potential confusion arising from the terminology used in the literature. There is also a reference to the limitations of the material data available, which is presented in a 6x6 matrix format.

Who May Find This Useful

This discussion may be useful for researchers and practitioners working with soft body physics, elasticity theory, and those involved in computational simulations of material behavior under various conditions.

doenn1616
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TL;DR
Trying to properly index elastic constants in elasticity equations.
Hi, I have some soft body equations that require first order elasticity constants. Just trying to figure out the proper indexing.

From Finite Elements of Nonlinear Continua by J.T. Oden, the elastic constants I am trying to obtain are the first order, circled below:

1640307271734.png

My particular constitutive equation doesn't need the zeroeth order, just first (circled in blue) or higher. I have a quite lengthy tutorial from Nasa on elastic constants called An In-Depth Tutorial on Constitutive Equations for Elastic Anisotropic Materials by Michael Nemeth, which includes:
1640307140508.png

So, say I need the constant at index 2,1,3,2. I see that in the full matrix only, circled below in blue:

1640307750435.png


I see there is symmetry, but can't figure out yet how to get this one. From the Materials Project Database, I have some elasticities of a material
Nb4CoSi:

"elastic_tensor": [
[
311.33514638650246,
144.45092552856926,
126.17558149507941,
0.0,
-0.11034746666666635,
0.0
],
[
144.45092552856926,
311.3204320131957,
126.16885826858503,
0.0,
-0.11216067833333321,
0.0
],
[
126.17558149507941,
126.16885826858503,
332.18500448217554,
0.0,
-0.10754095333333334,
0.0
],
[
0.0,
0.0,
0.0,
98.91818763333335,
0.0,
0.0
],
[
-0.11034746666666635,
-0.11216067833333321,
-0.10754095333333334,
0.0,
98.92097952333339,
0.0
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
103.33913232000003
]
],

Which is only a 6x6 matrix. Is there something simple I can do to use these constants? Am I confined to a lower order? Confusing since Eijmn looks fourth order when Oden says first. Am I close or is there a lot more to do?

Regards.
 
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In view of the symmetry relations C_{2132} = C_{1223}. Since \sigma_{ij} and \epsilon_{ij} are symmetric they each have only six independent entries, which can be related to each other by a 6x6 matrix.
 
doenn1616 said:
Confusing since Eijmn looks fourth order when Oden says first.

Distinguish
“first order” as “one factor of \gamma in the expansion” (“linear in \gamma”)
from
“fourth order” or “fourth rank” as “four indices” in that E-tensor (as a multilinear mapping of four vectors to the reals).
 
Yeah, yeah, you're totally right.. where is my mind at!

So,
1640633379432.png

C2132 = C1232 = C1223 and Eijmn also equals Ejinm.
1640633965516.png

Since Ejinm wasn't specified, just wanted to double check. And sorry for the delay, want to be thorough in proving these symmetries check out, but my mind is super full trying to simulate this and some fluid equations graphically, kind of digesting this slow this week. Take care..!