# Usage of First Order Elastic Constants in Soft Body Equations

• I
• doenn1616
In summary, the author is trying to find the first order elasticity constants and is confused because the symmetry relations Eijmn equal Ejinm make Ejinm look like it is fourth order instead of first order. The author finds the first order elasticity constants using symmetry and writes a summary of the material.

#### doenn1616

TL;DR Summary
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:

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:

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

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.

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,

C2132 = C1232 = C1223 and Eijmn also equals Ejinm.

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..!

## 1. What are first order elastic constants?

First order elastic constants are parameters that describe the linear relationship between stress and strain in a material. They are used to calculate the elastic modulus, Poisson's ratio, and shear modulus of a material.

## 2. How are first order elastic constants used in soft body equations?

First order elastic constants are used in soft body equations to model the behavior of soft materials, such as rubber or biological tissues. They are incorporated into the equations to describe the material's response to external forces and deformations.

## 3. What are the limitations of using first order elastic constants in soft body equations?

One limitation is that first order elastic constants only describe the linear relationship between stress and strain, and may not accurately capture the behavior of highly deformable materials. Additionally, these constants may vary depending on the type of deformation or loading conditions, making it difficult to accurately model complex soft body behaviors.

## 4. How are first order elastic constants measured?

First order elastic constants can be measured through various experimental techniques, such as tensile or compression tests. These tests apply known forces and measure the resulting strains in the material, allowing for the calculation of the elastic constants.

## 5. Can first order elastic constants be used for all types of materials?

No, first order elastic constants are only applicable to materials that exhibit linear elastic behavior. Materials that exhibit non-linear or viscoelastic behavior may require higher order elastic constants or other parameters to accurately model their behavior.