Project: Writing $1$D Finite Element Programs

Part $1$:

Consider an anterior crucial ligament $($ACL$)$ with a constant cross$-$sectional area of $\backslash ($ A$=\backslash )\backslash (100\backslash$mathrm$\{$~mm$\}^\{2\}\backslash )$ and length $\backslash ($ L$=20\backslash$mathrm$\{$~mm$\}\backslash ).$ The leftmost first segment $(5$ mm $)$ of the tissue is bone and has a Young's modulus of $1$ GPa $.$ The middle segment $(2$ mm $)$ has a Young's modulus that varies linearly from $1$ GPa to $150$ MPa along the length. The remaining segment $(13$ mm $)$ is pure ligament and has a modulus of $150$ MPa $.$ The left end of the tissue is fixed $($no displacement$).$ The right end of the tissue is subjected to a load of $500$ N $.$ This problem is govemed by the following differential equation:

$\backslash [$

$\backslash$frac$\{$d$\}\{$d x$\}\backslash$left$($A E$($x$)\backslash$frac$\{$d u$\}\{$d x$\}\backslash$right$)+$f$($x$)=0$

$\backslash ]$

The body force, $\backslash (\backslash$mathrm$\{$f$\}(\backslash$mathrm$\{$x$\})\backslash ),$ will be zero for this problem. You have designed two replacements for the ACL $.$ One is a biomaterial structure where the leftmost segment $(5$ mm $)$ has a modulus of $900$ MPa and the rest of the structure has a modulus of $100$ MPa $.$ The second is a functionally graded biomaterial that has the same gradient properties as the normal ACL.

In this project, you will write a finite element program in MATLAB using the Galerkin procedure that plots the displacement, strain, and stress along the length of your two materials. Perform a convergence study for both scenarios to determine the element size and element order that is most appropriate. Discuss the similarities and differences between the bi$-$material model results and the functionally graded material.