Local strain energy density for a plate subjected to in-plane linear load

In summary, Mattia is seeking help with a problem related to their Master thesis. They are looking for the strain energy density for a circle on a square plate under specific boundary conditions. Mattia has provided a formula, but it does not match the solution obtained through FEM. They request help in deriving the equation and comparing it to their own. Later, they realize their mistake and provide the correct formula.
  • #1
mp87
2
0
Dear all,


I would like to know from you the solution about this problem (which is not a homework, but a topic of my Master thesis!): I need the strain energy density related to a circle of radius r0 centered in an arbitrary point of a square plate, under the boundary conditions described in the attached picture (on the right edge a linear stress is applied, which ranges from 0 to S). The value I obtained, under the plain strain hypothesis, is:

[itex]
SED=\frac{1-\nu^2}{2\,E}\,\left(\frac{S}{h}\right)^2\,\left[\frac{r_0^2}{4}+(h+y_c)^2\right][/itex]

where h is the edge of the square plate and yc the y coordinate of the center. The fact is that it doesn't match with the FEM solution (which is surely right, since it was obtained by my Supervisor :)).

Can you please derive the equation and compare it with mine?


Thanks for your help!


Mattia
 

Attachments

  • plate.pdf
    5.2 KB · Views: 232
Engineering news on Phys.org
  • #2
Oops, I found the mistake! It was hidden inside the choose of the reference system :P

Here's the correct formula:

[itex]
SED=\frac{1-\nu^2}{2\,E}\,\left(\frac{S}{h}\right)^2\,\left[\frac{r_0^2}{4}+(h-y_c)^2\right][/itex]





Mattia
 

1. What is local strain energy density?

Local strain energy density is a measure of the deformation or strain that a specific material experiences in a small region, such as a plate, when subjected to a load. It represents the amount of energy stored in the material due to the deformation.

2. How is local strain energy density calculated?

The local strain energy density for a plate subjected to in-plane linear load can be calculated using the formula W = 1/2 * σ * ε, where W is the local strain energy density, σ is the stress acting on the plate, and ε is the strain or deformation of the plate.

3. What is the significance of local strain energy density in a plate?

The local strain energy density in a plate is an important factor in determining the strength and stability of the plate. It can help engineers and scientists understand the behavior of the material under different loading conditions and make design decisions accordingly.

4. How does the local strain energy density change with different loading conditions?

The local strain energy density will vary depending on the type and magnitude of the load applied to the plate. It will also change with the material properties of the plate, such as its stiffness and elasticity. Generally, a higher load or stiffer material will result in a higher local strain energy density.

5. Can local strain energy density be used to predict failure in a plate?

Yes, local strain energy density can be used as an indicator of potential failure in a plate. If the local strain energy density exceeds the material's ultimate strength, it can lead to failure in the form of cracks or fractures. Therefore, engineers can use this information to design plates that can withstand the expected loading conditions without failure.

Similar threads

  • Mechanical Engineering
Replies
2
Views
839
  • Quantum Physics
Replies
3
Views
299
  • Introductory Physics Homework Help
Replies
5
Views
404
  • Set Theory, Logic, Probability, Statistics
Replies
9
Views
2K
  • Introductory Physics Homework Help
Replies
1
Views
683
Replies
7
Views
2K
Replies
1
Views
526
  • Electromagnetism
Replies
1
Views
672
  • Classical Physics
Replies
4
Views
2K
Replies
3
Views
1K
Back
Top