# 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... Dec 14, 2020 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, 2020 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. Dec 14, 2020 at 7:28
• could you put some context to where "from before" you remember only E values were used?
– NMech
Dec 14, 2020 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. Dec 16, 2020 at 6:34

Let's try this way:

$$k = \dfrac{F}{\Delta}$$, and

$$\Delta = \dfrac{FL}{EA}$$. Rearrange the terms, it becomes

$$\dfrac{F}{\Delta} = \dfrac{EA}{L} = k$$

Note the terms $$F$$ (internal force/reaction of a member in an assembly) and $$\Delta$$ are both "unknowns" before solving the problem. The member stiffness is therefore characterized by the terms on the right-hand side, which are all "knowns" that define the member. Agreed?!

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 Dec 14, 2020 at 7:33
• I probably did not understand your question correctly.
– NMech
Dec 14, 2020 at 7:42

The spring constant, $$k$$ and the elastic modulus, $$E$$ both are an indicator of stiffness but differ in what phenomenon they are indicating.

In material science, the variation in $$E$$ indicates the stiffness of different materials, which is easily understood that within the elastic range, a stiffer material will require higher stress to produce a strain that equal to the strain of a lesser stiffness material. Thus, the larger the elastic modulus, the stiffer the material is true.

On the other hand, the spring constant $$k$$ indicates the stiffness of a structural element that having dimensional parameters, $$A, L$$ & $$I$$. If we hold $$A$$ & $$L$$ to be unit in a parameter study, then the equation $$k = AE/L$$ becomes $$k = E$$. Note that while the moment of inertia, $$I$$, which has no force/stress term, can be considered as a dimensional parameter, however, in practice, it is often used to indicate one cross-section is stiffer than the other because of the larger $$I$$ of the former. Hope the explanations make sense.