How Do You Calculate Displacement Gradients and Strain Tensors from Marker Data?

[tunk]

displacement gradients?

Homework Statement

Arrays of markers created on the surface (i.e. 2-D) of a tissue are often used to quantify
tissue deformations. Find the displacement gradients and calculate the Green strain tensor,
E, and the infinitessimal strain tensor, ε. Measurement of marker positions are listed below:
(marker #, before deformation, after deformation)
1 (0, 0) (0, 0)
2 (1, 0) (1.02, 0.01)
3 (2, 0) (2.15, 0.03)
4 (0, 1) (-0.02, 1.03)
5 (1, 1) (1.01, 1.05)
6 (2, 1) (2.19, 1.15)
7 (0, 2) (-0.04, 2.06)
8 (1, 2) (1.00, 2.09)
9 (2, 2) (2.30, 2.22)

a) Generate images of the marker locations before and after deformation.
b) Using these data perform linear fits to compute the four displacement gradients, ∂u/∂x for
each of the areas defined by points 1-2-4-5, 2-3-5-6, 4-5-7-8, and 5-6-8-9.

Homework Equations

The Attempt at a Solution

at this point all i can do is part a, i have no idea what to calc for b.

 

[nilesh_pat]

A) Before deformation: (0,0) (1,0) (2,0) (0,1) (1,1) (2,1) (0,2) (1,2) (2,2)After deformation: (0,0) (1.02, 0.01) (2.15, 0.03) (-0.02, 1.03) (1.01, 1.05) (2.19, 1.15) (-0.04, 2.06) (1.00, 2.09) (2.30, 2.22)

 

[Mene]

I would first clarify the concept of displacement gradients for those who may not be familiar with it. Displacement gradients refer to the change in displacement (or position) of a point in a tissue with respect to a change in position in a different direction. In other words, it measures how much a point has moved in a specific direction.

To calculate the displacement gradients in this scenario, we can use the linear fit method. This involves fitting a line to the data points and calculating the slope of the line. The slope represents the displacement gradient for that specific area.

For example, to calculate the displacement gradient for the area defined by points 1-2-4-5, we can fit a line to the data points (0,0), (1,0), (0,1), and (1,1). The slope of this line would represent the displacement gradient for this area. Similarly, we can perform linear fits for the other areas defined by points 2-3-5-6, 4-5-7-8, and 5-6-8-9 to calculate their respective displacement gradients.

Once we have the four displacement gradients for each area, we can then use them to calculate the Green strain tensor, E, and the infinitesimal strain tensor, ε. These tensors represent the amount of deformation in the tissue and can be used to further analyze the tissue's behavior under stress or strain.

In summary, displacement gradients are an important tool for quantifying tissue deformations and can be calculated using the linear fit method. They can then be used to calculate other important tensors that can provide insights into the behavior of the tissue.