Principal Stress and Strain on a Fracture

[ticalcman]

Homework Statement

Find the expressions for principal stresses in both plane strain and plane stress conditions

sigma(x) = (K/(sqrt(2*pi*r)))*cos(theta/2)*(1-sin(theta/2)*sin(3*theta/2))
sigma(y) = (K/(sqrt(2*pi*r)))*cos(theta/2)*(1+sin(theta/2)*sin(3*theta/2))
T(xy) = (K/(sqrt(2*pi*r)))*cos(theta/2)*sin(theta/2)*cos(3*theta/2))

Since the problem asks for expressions, no variables are given values.

Homework Equations

Mohrs Circle for Principal Stress
Principal Strain

The Attempt at a Solution

Should I take derivatives of (x), (y), and (xy) and then use that to solve for the angle theta? All I know is I'm supposed to look for sigmas x, y, and xy (not sure if my teacher meant x', y' and xy') and then sigmas 1 and 2. Ideas?

 

[KataKoniK]

As a fellow scientist, I would suggest starting by reviewing the equations for Mohr's circle for principal stress and principal strain. These equations involve the angle theta, which is the angle between the x-axis and the plane in which the stresses are acting. So, to find the expressions for the principal stresses in both plane strain and plane stress conditions, we need to first determine the angle theta for each case.

For plane strain conditions, theta = 0, so the expressions for principal stresses become:

sigma(1) = (K/(sqrt(2*pi*r)))*cos(0/2)*(1-sin(0/2)*sin(3*0/2)) = K/sqrt(2*pi*r)
sigma(2) = (K/(sqrt(2*pi*r)))*cos(0/2)*(1+sin(0/2)*sin(3*0/2)) = K/sqrt(2*pi*r)

For plane stress conditions, theta = 45 degrees, so the expressions for principal stresses become:

sigma(1) = (K/(sqrt(2*pi*r)))*cos(45/2)*(1-sin(45/2)*sin(3*45/2)) = K/2*pi*r
sigma(2) = (K/(sqrt(2*pi*r)))*cos(45/2)*(1+sin(45/2)*sin(3*45/2)) = K*sqrt(2)/sqrt(pi*r)

To find the expression for T(xy), we can use the equation T = (sigma(1) - sigma(2))/2 * sin(2*theta). Plugging in the values we just found for sigma(1) and sigma(2), we get:

T(xy) = (K/sqrt(2*pi*r) - K/2*pi*r)/2 * sin(2*45) = K/(2*sqrt(pi*r))

I hope this helps! Remember to always double check your equations and make sure they make sense intuitively. Good luck with your calculations!

 

[Mene]

it is important to first clarify any confusion or ambiguity in the problem statement. In this case, it would be helpful to ask for clarification on the variables and terms used in the expressions for the principal stresses. Additionally, it would be beneficial to know the context or application of these expressions, as it may influence the approach to solving the problem.

Assuming that the problem is asking for the expressions for the principal stresses in both plane strain and plane stress conditions, the Mohr's circle can be used to determine the principal stresses. In plane strain conditions, the principal stresses are given by sigma1 = K/(sqrt(2*pi*r)) and sigma2 = -K/(sqrt(2*pi*r)), where K is the maximum stress and r is the radius of the Mohr's circle.

In plane stress conditions, the principal stresses are given by sigma1 = K/(sqrt(2*pi*r)) and sigma2 = 0, where K is the maximum stress and r is the radius of the Mohr's circle.

To find the expressions for the principal strains, the equations for strain can be used, which are related to stress through Hooke's law. In plane strain conditions, the principal strains are given by epsilon1 = sigma1/E and epsilon2 = sigma2/E, where E is the elastic modulus.

In plane stress conditions, the principal strains are given by epsilon1 = sigma1/E and epsilon2 = 0.

As for the expressions involving theta, it is not clear how they relate to the principal stresses and strains. Further clarification or context is needed in order to provide a solution.