Analytical solution for thermal stresses in a rectangular plate

[Sargam]

Hello everybody,

I am solving a 2D problem of thermal stresses in a rectangular plate in which temperature is changing only y direction. Plate has fixed displacement conditions. Could anyone help me to find out analytical solution of thermal stresses for my problem? Does anyone suggest me where I can find analytical solution?

Thanks in advance.


Sargam

 

[Enthalpy]

What are "fixed displacement conditions"?

Solutions could be something like a parabola or a cosine in the transverse direction, multiplied by a tanh in the expansion direction. Try a few ones.

 

[Sargam]

Hi,

Thanks for the reply. I have mistakenly mention fixed displacement boundary condition but actually all surfaces of rectangle are traction free (free surfaces). Temperature change is parabolic on y-axis.

Could you please let me know where I can get analytical solution for this problem?

 

[Enthalpy]

If temperature is uniform along the X axis and all edges are free to move, stress is zero everywhere, whatever the profile is along the Y axis.

In the more general case, try a product of cos(x) and cos(y) with different harmonic periods - but a particular polynomial solution can look good.

 

[AlephZero]

If temperature is uniform along the X axis and all edges are free to move, stress is zero everywhere, whatever the profile is along the Y axis.

I think I can see how you reached that (wrong) conclusion, but what about non-zero shear strains in the plate, and the stresses they create?

The direct stress component normal to a free edge must be zero, but the other stress components along a free edge need not be zero. Axial tension in a rod is a simple (non-thermal) example.

 

[Enthalpy]

Right!

It must be something like tanh(x) times a polynomial of y.