Question about stresses in material due to elastic and piezoelectric

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Discussion Overview

The discussion revolves around understanding an equation related to stresses in materials, specifically in the context of piezoelectricity and its relation to Newton's second law. Participants explore the implications of the equation, the meaning of partial derivatives in this context, and the nature of stress as it relates to force and material behavior.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant seeks clarification on the partial derivative of stress with respect to coordinates and questions whether the term density*(u_i) corresponds to 'ma' in Newton's second law, wondering if it implies a unit volume of 1.
  • Another participant explains the use of Einstein summation convention in the equation, indicating that the partial derivative of the stress tensor is a sum over indices.
  • There is a discussion on the nature of stress as an average force and the implications of constant stress on material movement, particularly in fluids.
  • A participant introduces the concept of Cauchy's first law as a continuum representation of Newton's second law, discussing the divergence of the stress tensor and its relation to force.
  • Participants mention the complexity of the stress tensor in piezoelectric materials compared to simpler models like incompressible fluids or Hookean solids.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the equation and its implications, with some clarifying concepts while others remain uncertain about specific aspects. No consensus is reached on the interpretation of the equation or the implications of the partial derivative.

Contextual Notes

Participants note that piezoelectricity is an advanced topic, and the discussion includes references to various forms of the stress tensor and their applications in different material contexts. The complexity of the stress tensor in piezoelectricity is highlighted as a significant point of discussion.

overgift
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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?
 

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Welcome to PF!

overgift said:
… 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! :smile:

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)? :smile:
 


tiny-tim said:
Hi overgift! Welcome to PF! :smile:

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)? :smile:

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

[tex]\rho[/tex]*[tex]\ddot{u<sub>i</sub>}[/tex]-[tex]\partial[/tex]Tij/[tex]\partial[/tex]xj=0

uiis the volume displacement, T is stress.
 


Hi overgift! :smile:

(use ' not dots in this forum

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

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 :smile:
 


tiny-tim said:
Hi overgift! :smile:

(use ' not dots in this forum

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

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 :smile:
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?
 
overgift said:
… 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! :smile:

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. :smile:
 
tiny-tim said:
Hi overgift! :smile:

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. :smile:

Thanks a lot! Now I totally understand.
 
overgift said:
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:

[tex]\frac{D(\rho v)}{Dt}= \nabla \bullet T[/tex]

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.
 
Andy Resnick said:
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:

[tex]\frac{D(\rho v)}{Dt}= \nabla \bullet T[/tex]

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
 

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