# Question about stresses in material due to elastic and piezoelectric

1. Oct 1, 2008

I'm learning piezoelectricity right now and got an equation I can't understand. It writes the newton's sencond law for the stresses in materials due to elastic and piezoelectric contribution.

The equation is in the attachment.

In this equation I'd like to ask what is the partial derivative between stress and the coordiante? And if density*(u_i)" means 'ma' in newton's sencond law, does it mean the volume is an unit which equals to 1? Or there is another explaination?

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2. Oct 1, 2008

### tiny-tim

Welcome to PF!

It will be hours before the attachment is approved.

Can you type the equation for us (use the X2 and X2 tags above the reply field for SUP and SUB)?

3. Oct 1, 2008

Re: Welcome to PF!

$$\rho$$*$$\ddot{ui}$$-$$\partial$$Tij/$$\partial$$xj=0

uiis the volume displacement, T is stress.

4. Oct 1, 2008

### tiny-tim

Re: Welcome to PF!

(use ' not dots in this forum

and you needed "\partial T" rather than "\partialT" in your first attempt )

Let's see … rho ui'' = ∂Tij/∂xj

Are you asking what ∂Tij/∂xj is?

It's using the Einstein summation convention … you add all the possible values of i …

∂Tij/∂xj = ∂Ti1/∂x1 + ∂Ti2/∂x2 + ∂Ti3/∂x3

5. Oct 1, 2008

Re: Welcome to PF!

Thank you again for your help! ∂Tij/∂xj = ∂Ti1/∂x1 + ∂Ti2/∂x2 + ∂Ti3/∂x3 this I can understand, what I can't understand is stress is already the average amount of force,doesn't it already fit F=ma? so why take partial derivative, what does this mean?

6. Oct 1, 2008

### tiny-tim

Yes, stress is a sort of average of force …

but if the force is the same everywhere, nothing will move …

suppose it's a fluid, with Tij purely diagonal, so that T11, for example, is the pressure in the x1 direction.

Then T1j/∂xj = T11/∂x1, which is the pressure gradient in that direction, and u1'' will be 0, not T11, if T11 is constant.

7. Oct 1, 2008

Thanks a lot! Now I totally understand.

8. Oct 1, 2008

### Andy Resnick

Tiny-tim did a nice job stepping you through the notation. I'd just like to add some conceptual information- piezoelectricity is a quite advanced topic to try and introduce many of these fundamental concepts.

Newton's second law, F = ma, when written in terms of a continuum is known as Cauchy's first law:

$$\frac{D(\rho v)}{Dt}= \nabla \bullet T$$

Where D/Dt is the total derivative, and T the stress tensor. The physics comes in when writing down the form of the stress tensor. The divergence of the stress tensor is associated with 'Force'. Simple forms of the stress tensor can be written down for incompressible Newtonian fluids, Hookean elastic solids, linear combinations of the two (viscoelastic materials), etc. etc. and is known as "constitutive relations" or constitutive equations.

For piezoelectricity, the stress tensor is considerably more complex than that for an isotropic incompressible fluid, but the concept is the same as above.

9. Oct 1, 2008