Calculate Strain Components: Given Stress Components & E, Poisson Ratio

[paramathma]

My Homework question is

The stress components at a point are given by (image attached)

units are in "Mpa (Mega pascal). calculate the strain components.
Assume E= 200GPa and poisson ratio= 0.25.

please guide me.
formula to be used?

thanks,

 

[Valkarie]

The formula to use for calculating strain components from stress components is Hooke's Law. This states that strain is directly proportional to the stress applied upon an object, and is represented by the equation: Strain = Stress/Elastic Modulus. The elastic modulus (E) in this case is 200 GPa. The Poisson's Ratio (ν) is 0.25. Using these values and the given stress components, you can calculate the strain components as follows: Strain x = Stress x / E Strain y = Stress y / E Strain z = Stress z / E Shear strain xy = (Stress x - Stress y) * ν / E Shear strain yz = (Stress y - Stress z) * ν / E Shear strain zx = (Stress z - Stress x) * ν / E For example, if the stress components in the x-direction is 10 Mpa, then the strain in the x-direction would be: Strain x = 10 Mpa / 200 GPa = 0.05

 

[Mene]

I would use the formula for calculating strain components given by:

ε = (1/ E) * [σ - ν(σ1 + σ2 + σ3)]

Where:
ε - strain component
E - Young's modulus
σ - stress component
ν - Poisson ratio
σ1, σ2, σ3 - stress components in x, y, and z directions respectively

Using the values given in the question, the strain components can be calculated as follows:

εx = (1/200GPa) * [100MPa - 0.25(100MPa + 0 + 0)] = 3.75 x 10^-4
εy = (1/200GPa) * [0 - 0.25(0 + 200MPa + 0)] = -2.5 x 10^-4
εz = (1/200GPa) * [0 - 0.25(0 + 0 + 150MPa)] = -1.875 x 10^-4

Therefore, the strain components at the given point are εx = 3.75 x 10^-4, εy = -2.5 x 10^-4, and εz = -1.875 x 10^-4. It is important to note that the units of strain are dimensionless, so they do not have any units attached to them.

I hope this helps guide you in solving your homework question. Remember to always double check your calculations and use the correct units in your final answer. Good luck!