ME 3255 Final Project
Designing a graphene pressure sensor
Submission link (https://goo.gl/forms/g4zKUjMY9TidE43E2)
In this final project, you and your team mates will design a graphene pressure sensor. Graphene is a single-layer of carbon atoms arranged in a hexagonal lattice. Graphene films have a thickness of 0.3 nm, Young's modulus of 1 TPa, and Poisson's ratio of 0.31. The film has no bending stiffness so the shape of the membrane under a given pressure, P, with a pretension per unit length, T is given by:
(1)
This equation assumes you know both the applied pressure and tension in the membrane, but the tension will be a function of the applied pressure, assuming there is no initial pretension. We will use an energy balance to solve the nonlinear relation between tension and pressure.
(2)
(3)
Where w(x,y) is the displacement of the membrane at a given point, (x,y), and the integral is over the entire of area of the membrane, A. We can rewrite this exact integral as a summation over the elements, as such.
(4)
Where, 
is the average w of the four nodes in the element, and
the summation goes from 1 to the number of elements, nel.
In this design problem, we will solve for the tension and deflection of a square membrane, as seen in figure 2.
Figure 2. Square membrane design with sides of 10 um. 2A demonstrates the node and element numbering for 9 interior nodes and 16 elements. 2B shows element 6 and its nodal displacments along with the aveerage displacement as the red point. 2C shows a cross-section of the force-balance for the square membrane window and 2D shows an isometric view of the membrane displacement solution and force-balance.
a. Create a central finite difference approximation of the gradient with 3-by-3 interior nodes of w
for the given membrane solution in terms of P and T. [w]=membrane_solution3(T,P); The
output w should be a vector, but the solution represents a 2D data set w(x,y).
b. Solve for w given a pressure, P=0.001 MPa and tension, T=0.006 uN/um. Plot the result with
surf(X,Y,W) where X, Y, and W are the x-, y-, and z-coordinates of each point on the
membrane from 0-10um.
c. Create a general central finite difference approximation of the gradient with
n-by-n interior nodes of w
for the given membrane solution in terms of P and T. [w]=membrane_solution(T,P,n); The
output w should be a vector, but the solution represents a 2D data set w(x,y).
d. Solve for w given a pressure, P=0.001 MPa and tension, T=0.006 uN/um with 10 interior
nodes. Plot the result with surf(X,Y,W) where X, Y, and W are the x-, y-, and
z-coordinates of each point on the membrane from 0-10um. Include the graph in your README.
e. Create a function SE_diff that calculates the difference in strain energy (right hand side Eq.
4) and work done by pressure (left hand side Eq. 4) for n-by-n elements.
[pw_se,w]=SE_diff(T,P,n)
Use the solution from part c to calculate w, then do a numerical integral over the elements to calculate work done and strain energy.
f. Use a root-finding method to calculate the tension in the membrane given a pressure, P=0.001 MPa, and n=[20:5:40] interior nodes.
Show that the error in tension is decreasing with a table:
|number of nodes|Tension (uN/um)| rel. error|
|---|---|---|
|3 |0.059 |n/a|
|20|0.0618|4.5%|
|25|0.0616|0.3%|
|30|0.0614|0.3%|
|40|0.0611|0.3%|
g. Plot the Pressure vs maximum deflection (P (y-axis) vs max(w) (x-axis)) for P=linspace(0.001,0.01,10). Use a root-finding method to determine tension, T, at each pressure. Use a cubic best-fit to find A, where, P(x)=A*dw^3. State how many interior nodes were used for the graph. Plot the data and best-fit curve in your README.
h.[Bonus 5 pts] Show that the constant A is converging as the number of nodes is increased (Similar table to f).
i.[Bonus 10 pts] If the square membrane sides are always equal, but have a tolerance of 0.1%, what should the depth of the sensor be if 2.5% of the sensors won't hit the bottom given a maximum pressure of 0.01 MPa.