Isoparametric Mapping Homework: Displacement, Deformation Gradients

[Peter S]

Homework Statement

I have an initial configuration and a final configuration for a sample that has been stressed. Both of these are trapezoidal shapes and I have exact coordinates for each point. I know how to use the shape equations as well as deriving the Jacobian. Additionally I have computed the u and v displacement vectors for between the configurations. Now I need to determine the displacement and deformation gradients so I am able to compute strain on the tissue. The issue is that I don't know how a change in configuration affects the shape equations. Relating a single shape to a parent configuration is one thing, but I don't know how to map the two configurations so that I can determine stress and strain. As far as I can find, all examples relate some shape to the double unit square rather than a pre-existing configuration. I would appreciate any sort of help with this quite a bit. For what it's worth the best classification of this problem is a 4 node isoparametric mapping.

Homework Equations

x = Σ_{i = 1->4}N_i (s,t)x_i
N₁=¼*(1-s)(1-t)*x₁...
y = Σ_{i = 1->4}N_i (s,t)y_i
N₁=¼*(1-s)(1-t)*y₁...
u = x - X (lowercase = deformed configuration, uppercase = initial configuration)
v = y - Y

The Attempt at a Solution

I have nothing since I can't start the problem without knowing how to relate the configurations. I would prefer having someone point me in the right direction rather than putting up exact values and having my problem solved.

 

[KataKoniK]

Thank you for sharing your problem with us. It seems like you have a good understanding of the shape equations and the Jacobian, which are important tools for analyzing the deformation of a sample. However, I understand that you are facing difficulty in relating the initial and final configurations in order to determine the displacement and deformation gradients.

One approach you could try is to use the concept of strain tensors. Strain tensors are matrices that describe the local deformation of a material. By calculating the strain tensor for each point in your sample, you can then use it to determine the displacement and deformation gradients.

Another approach is to use the concept of finite element analysis. This method involves dividing your sample into smaller elements and using interpolation techniques to determine the displacement and deformation gradients for each element. By combining the results from all the elements, you can then determine the overall displacement and deformation gradients for the entire sample.

I hope these suggestions will help you in your problem. It may also be helpful to consult with other experts in the field or refer to research papers or textbooks for further guidance. Best of luck with your research!