Question about stresses in material due to elastic and piezoelectric

[overgift]

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 explanation?

 

[tiny-tim]

Welcome to PF!

… 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 explanation?

Hi overgift! 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)?

 

[overgift]

Hi overgift! 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)?

Hi thank you for your kind reply. The equation writes:

$$\rho$$*$$\ddot{u_i}$$-$$\partial$$T_ij/$$\partial$$x_j=0

u_iis the volume displacement, T is stress.

 

[tiny-tim]

Hi overgift!

(use ' not dots in this forum

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

Let's see … rho u_i'' = ∂T_ij/∂x_j …

Are you asking what ∂T_ij/∂x_j is?

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

∂T_ij/∂x_j = ∂T_i1/∂x₁ + ∂T_i2/∂x₂ + ∂T_i3/∂x₃

 

[overgift]

Hi overgift!

(use ' not dots in this forum

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

Let's see … rho u_i'' = ∂T_ij/∂x_j …

Are you asking what ∂T_ij/∂x_j is?

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

∂T_ij/∂x_j = ∂T_i1/∂x₁ + ∂T_i2/∂x₂ + ∂T_i3/∂x₃

Thank you again for your help! ∂T_ij/∂x_j = ∂T_i1/∂x₁ + ∂T_i2/∂x₂ + ∂T_i3/∂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?

 

[tiny-tim]

… 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?

Hi overgift!

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 T_ij purely diagonal, so that T₁₁, for example, is the pressure in the x₁ direction.

Then T_1j/∂x_j = T₁₁/∂x₁, which is the pressure gradient in that direction, and u₁'' will be 0, not T₁₁, if T₁₁ is constant.

 

[overgift]

Hi overgift!

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 T_ij purely diagonal, so that T₁₁, for example, is the pressure in the x₁ direction.

Then T_1j/∂x_j = T₁₁/∂x₁, which is the pressure gradient in that direction, and u₁'' will be 0, not T₁₁, if T₁₁ is constant.

Thanks a lot! Now I totally understand.

 

[Andy Resnick]

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 explanation?

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.

 

[overgift]

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.

Thank you for your post. It really helps a lot