What is the physical meaning of the dyadic product \vec{a} \vec{b}?

[member 428835]

hey pf!

can someone please explain physically what the dyadic product represents: $\vec{a} \vec{b}$
i know its a matrix (tensor), but that's all i know physically.
thanks!

 

[dextercioby]

I don't believe there's a physical explanation for this. It's a very handy mathematical object with uses in classical physics mostly.

 

[Chestermiller]

hey pf!

can someone please explain physically what the dyadic product represents: $\vec{a} \vec{b}$
i know its a matrix (tensor), but that's all i know physically.
thanks!

A dyad fulfills its primary function in life only when it is dotted (i.e., contracted) with another vector (or tensor). It can be dotted with another vector on its right side by dotting the right-hand member of the dyad with the subject vector, and it can be dotted with another vector on its left side by dotting the left-hand member of the dyad with the subject vector.

Thus, for right-hand dotting,

$\vec{d}=\vec{a}\vec{b}\centerdot \vec{c}=\vec{a}(\vec{b}\centerdot \vec{c})$

or, for left-hand dotting,

$\vec{d}=\vec{c}\centerdot \vec{a}\vec{b}=(\vec{c}\centerdot \vec{a})\vec{b}$

According to the first equation, the vector $\vec{d}$ is produced by dotting the dyad $\vec{a}\vec{b}$ on its right side by the vector $\vec{c}$ to yield the vector $\vec{d}$. Thus, by dotting the dyad $\vec{a}\vec{b}$ on its right side with the vector $\vec{c}$, we have accomplished the task of mapping the vector $\vec{c}$ into the new vector $\vec{a}(\vec{b}\centerdot \vec{c})$, which points in the same direction as the vector $\vec{a}$ and has a magnitude equal to $(\vec{b}\centerdot \vec{c})$ times the magnitude of $\vec{a}$. Similarly, according to the second equation, the vector $\vec{d}$ is produced by dotting the dyad $\vec{a}\vec{b}$ on its left side by the vector $\vec{c}$ to yield the vector $\vec{d}$. Thus, by dotting the dyad $\vec{a}\vec{b}$ on its left side with the vector $\vec{c}$, we have accomplished the task of mapping the vector $\vec{c}$ into the new vector $(\vec{c}\centerdot \vec{a})\vec{b}$, which points in the same direction as the vector $\vec{b}$ and has a magnitude equal to $(\vec{a}\centerdot \vec{c})$ times the magnitude of $\vec{b}$.

When we express a vector (i.e., a first order tensor) $\vec{V}$ in component form, we write:

$$\vec{V}=V_x\vec{i}_x+V_y\vec{i}_y+V_z\vec{i}_z$$

That is, a linear summation of scalar coefficients times unit vectors taken one at a time.

Any second order tensor can be expressed in component form as the linear sumation of scalar coefficients times dyads (unit vectors taken two at a time). Such a linear sum is referred to as a Dyadic. In the case of the (3D) stress tensor, for example, one can write:

$$\vec{σ}=σ_{xx}\vec{i}_x\vec{i}_x+σ_{xy}\vec{i}_x\vec{i}_y+σ_{xz}\vec{i}_x\vec{i}_z+σ_{yx}\vec{i}_y\vec{i}_x+σ_{yy}\vec{i}_y\vec{i}_y+ σ_{yz}\vec{i}_y\vec{i}_z+σ_{zx}\vec{i}_z\vec{i}_x+σ_{zy}\vec{i}_x\vec{i}_z+σ_{zz}\vec{i}_z\vec{i}_z$$

The scalar coefficients in the dyadic sum in Eqn. 8 are the components of the stress tensor. When the stress tensor is dotted with a unit normal vector to a surface ($\vec{n}=n_x\vec{i}_x+n_y\vec{i}_y+n_z\vec{i}_z$), it maps the unit normal vector $\vec{n}$ into the stress vector (traction) $\vec{S}$ acting on the surface:

$$\vec{S}=\vec{σ}\centerdot \vec{n}$$

This is called the Cauchy stress relationship. It is one important application in which use of dyadic notation simplifies things.

Chet

 

[member 428835]

thanks!