Determine the strain rate for a material fiber

[LeFerret]

Homework Statement

Determine the Strain Rate for a Material Fiber in the direction of the surface normal.

The Velocity Field is
V=((4y-3x)i+(5x+3y)j) ft/s

http://puu.sh/9hQ7Q/2bda80620f.jpg is the picture

which describes a steady, planar flow

where i and j are unit vectors.

Homework Equations

n * (n * ∇) V

where n is the unit normal, and * are dot products.

The Attempt at a Solution

I know that the solution to this itself is very simple, it is just math, my biggest issue though is how do I find the unit normal vector? I have no idea where to begin, any hints would be greatly appreciated!

 

[Chestermiller]

In your figure, the z axis lies within the plane, and the point 1,1,1 lies within the plane. Do you know how to determine a unit normal to this plane?

Also, the rate of deformation tensor is 1/2 the velocity gradient tensor and its transpose, not just the velocity gradient tensor.

Chet

 

[LeFerret]

In your figure, the z axis lies within the plane, and the point 1,1,1 lies within the plane. Do you know how to determine a unit normal to this plane?

Also, the rate of deformation tensor is 1/2 the velocity gradient tensor and its transpose, not just the velocity gradient tensor.

Chet

The only thing I can think of is a cross product of two vectors.
If I defined two vectors from the origin to the coordinates (0,0,1) and (1,1,0) and crossed them, this would give me -i+j
however the solution is sqrt(2)/2(-i+j) and I'm not sure where that common factor is coming from

 

[Chestermiller]

The only thing I can think of is a cross product of two vectors.
If I defined two vectors from the origin to the coordinates (0,0,1) and (1,1,0) and crossed them, this would give me -i+j
however the solution is sqrt(2)/2(-i+j) and I'm not sure where that common factor is coming from

(1,1,0) is not a unit vector. Divide it by its magnitude, and you will see where the sqrt(2)/2 came from. Another way to get the unit vector normal to the plane is just to draw a diagram of the intersection of the plane with the x-y plane.

Chet

 

[LeFerret]

(1,1,0) is not a unit vector. Divide it by its magnitude, and you will see where the sqrt(2)/2 came from. Another way to get the unit vector normal to the plane is just to draw a diagram of the intersection of the plane with the x-y plane.

Chet

In general, when given something like, this how would I know which vectors to use for my cross product?

 

[Chestermiller]

In general, when given something like, this how would I know which vectors to use for my cross product?

Any two convenient in-plane unit vectors will do the trick. But often, it's easier to draw a diagram with a unit normal to the plane, and resolve it into components in the coordinate directions.

Chet

 

[LeFerret]

Any two convenient in-plane unit vectors will do the trick. But often, it's easier to draw a diagram with a unit normal to the plane, and resolve it into components in the coordinate directions.

Chet

Ah I see, thank you.