Deformation of a continuum element

[Studious_stud]

Homework Statement

The continuum element shown in Figure (a) deforms to figure (b)

I'm looking to find the positions after deformation (x coordinates) by expressing them in terms of E coordinates.

Homework Equations

The Attempt at a Solution

x1 = E1 + E2
x2 = E1 + E2
x3 = E3

The first two equations for x I need help with, I'm not sure about the angles. I've done a previous question where the element deforms similarly (pure shear) but it stays within the same quadrant. The fact that this is flipped to the opposite quadrant is confusing me.

When the element deforms but stays in the same quadrant the answer is:

x1 = E1cosa + E2sina
x2 = E1sina + E2cosa
x3 = E3

 

[paisiello2]

What are E1, E2, and E3 supposed to be?

 

[Chestermiller]

In your first figure, it doesn't look like the deformation is pure shear. Those parameters m and n look like stretches.

Chet

 

[Chestermiller]

If the position vector between material points A and B is expressed as E₁i in the original configuration of the body (figure a), and this vector stretches by an amount m to obtain the deformed configuration (figure b), what is the equation for the position vector between material points A and B in the deformed configuration?

Chet

 

[rezeew]

However, in this case, since the element is flipped to the opposite quadrant, the angles will change. Assuming that the element is still deforming under pure shear, the equations would become:

x1 = E1cosa - E2sina
x2 = E1sina + E2cosa
x3 = E3

The first equation has a negative sign in front of the E2 term because the element is now being sheared in the opposite direction, causing a negative change in the x1 coordinate. The second equation remains the same because the element is still being sheared in the same direction, causing a positive change in the x2 coordinate. The third equation remains the same because there is no change in the x3 coordinate.

It is important to note that these equations are only valid for pure shear deformation and may not accurately represent the positions after deformation for other types of deformation. Additionally, the angles used in these equations may need to be adjusted depending on the specific orientation of the element before and after deformation. More complex deformations may require more equations to accurately represent the positions after deformation.