How can you use tensors in theory of elasticity? I am interested in physics and Icame across the usage of tensors in elasticity. How do you do that? How can tensors be useful?
 A: Tensors pop up in elasticity for any number of perfectly justifiable reasons, which include:

*

*The main physical objects of interest in elasticity are tensors (stress, strain, strain rate, etc.)

*Because we study the connection between quantities that are vectors (traction, displacement, etc.), tensors appear because they are general linear maps between vector quantities.

To expand on Argument 2, think of it this way; in elasticity, we almost always have a traction field $\mathbf{T}$ that is acting on a material, causing some displacement field $\mathbf{x}$. This means we have a mapping that takes some traction vector field and turns it into a displacement vector field:
$$\mathbf{x} = f(\mathbf{T}),\ f: \mathbb{R}^n\rightarrow\mathbb{R}^n$$
Now, we know this mapping needs to satisfy some transformation rules if we mess around with the coordinate systems of the vectors involved. For example, if I rotate to a new coordinate system where now $\{x,y,z\} \rightarrow \{-y,x,z\}$, I know that my mapping needs to "process" that change accordingly to not spout out nonsense results.
If I then also assume that my mapping is linear, then this mapping is by definition a tensor; tensors are defined as linear maps that satisfy transformation rules precisely such that the example above holds. As a result, any time you have linear maps between vectors that are physically meaningful, you will always find tensors.
