# Elasticity values in stiffness matrix?

Stiffness, k, is defined by $$k=\frac{Force}{deformation}$$ if you insert this into formula $$deformation = \frac{\text{Force}\times\text{ Length}}{AE}$$ you get $$k=\frac{A.E}{L}$$ where E is elasticity, A is area and L is length.

But from before I remember that we wrote just the E values in stiffness matrix. Even many people use elasticity and stiffness as if they are the same thing, which is wrong. Why do we write elasticity values in stiffness matrix, or what am I missing here?

• Can’t you approach the calculation of any behavior of a material from the elasticity approach or the stiffness approach? Then what goes in the matrix is relevant... – Solar Mike Dec 14 '20 at 7:10
• I assume you are talking about the stiffness matrix which is in e.g. in th Finite Element Method? Or are you talking about the stress strain stiffness matrix? – NMech Dec 14 '20 at 7:11
• NMech, The hookes law matrix is not the one. That is E. As I said in my question, I look for the k, not E. so yes, the FEM matrix. – user3600630 Dec 14 '20 at 7:28
• could you put some context to where "from before" you remember only E values were used? – NMech Dec 14 '20 at 7:44
• Everywhere, you can see that E is used as a synonym for stiffness. As far as putting into the matrix, I am not sure where, but I am sure that most people use E and stiffness as synonyms. That is wrong. For example even to the straight, elastic portion of stress strain graph of steel, people say, that is E, or stiffness. Why is it used interchangibly? About matrix I cannot remember now. – user3600630 Dec 16 '20 at 6:34

## 1 Answer

There are two (common) uses of the stiffness matrix. One related to Finite Element Method and a second in stress strain relationships.

## FEM

For example in the triangle element below you get the following stiffness matrix. Where the values of stiffness $$k_{ij}$$ are $$k_{ij} = \frac{A_{ij} E_{ij}}{L_{ij}}$$ ## Hooke's law in 3D Or its inverse form, which is in the form of $$\mathbf{\sigma= K \epsilon}$$ ## Relationship between the two.

There is a relationship between the two and there are some differences.

Common theme: they both try to relate a force/stress to a displacement/strain through stiffness

Difference: the relationship between stress and strain concerns a point, while the stiffness matrix refers to a structure. This has the result that in the case of Hooke's law there are stress (points in space or infinitesimally small volumes ) involved; while FEM concerns with elements with cross-section with length.

## Elasticity and Stiffness matrix

Some people indeed use the same term. That is indeed not correct.

Some probable reasons for doing so:

• Elasticity is the inverse of stiffness. So if you know one then automatically you can derive the other.
• Elasticity modulus or Young's modulus is probably a very poorly described quantity. You would expect high values of elasticity modulus to exhibit high elasticity. Most people equate high elasticity with the ability to deform. However, high modulus of elasticity means that the material does not deform easily. So IMHO a more proper name is stiffness modulus, however hardly any book uses that term.
• Finally, it make come as a surprise to you but most people are lazy (including me). When having two terms that are equivalent they will use them interchangeably to save time and effort.
• in Hooke's law matrix that you wrote, you wrote elasticity. that is not the one. I look for stiffness. So in FEM we write stiffness values , not E – user3600630 Dec 14 '20 at 7:33
• I probably did not understand your question correctly. – NMech Dec 14 '20 at 7:42