A question on elasticity from Kittel's Solid State Physics book

hagopbul · Jul 23, 2023

Hello All :

i am reviewing charlse kettle book introduction to solid state physics and came across an equation in the 3ed chapter which is a bit unclear
hope that physics forums members can clear it more if that possible

the equation 1st mentioned in the 3ed chapter page number 84
it is about the elastic energy density U :

##U =(1/2)\sum_{\lambda=1}^{6}\sum_{\mu=1}^{6}[C_{\lambda\mu}e_{\lambda}e_{\mu}]##

then the writer start deriving the above equation which i didnt understand , could any one explain it especially that the above equation is consist of constants (c,e)?

Best
Hagop

Frabjous · Jul 23, 2023

hagopbul said:

TL;DR Summary: reviewing the introduction to solid state physics by charles kettle and in the 3ed chapter there is this equation that is little unclear

Hello All :

i am reviewing charlse kettle book introduction to solid state physics and came across an equation in the 3ed chapter which is a bit unclear
hope that physics forums members can clear it more if that possible

the equation 1st mentioned in the 3ed chapter page number 84
it is about the elastic energy density U :

##U =(1/2)\sum_{\lambda=1}^{6}\sum_{\mu=1}^{6}[C_{\lambda\mu}e_{\lambda}e_{\mu}]##

then the writer start deriving the above equation which i didnt understand , could any one explain it especially that the above equation is consist of constants (c,e)?

Best
Hagop

C_λμ are the elastic moduli (constants)
e_i are strains (not constants)

it is essentially the spring energy (½kx²) for elastic materials

hagopbul · Jul 23, 2023

thanks but if they are constant shouldnt the derivative of C and e be = 0 in the book e also constant
you will read in the above pages coefficients

" it is usual to work with coefficients e rather than epsilon"

the writer give the impression that he get e from deriving a 1st degree equation

Frabjous · Jul 23, 2023

I do not have Kittel in front of me. So I am responding blind.

Derivative with respect to what?

There is no reason for the derivative of strain to be zero.
C_ij is commonly a constant. I believe the equation you wrote requires it to be, but I might be wrong.

hagopbul · Jul 23, 2023

tomorrow i will take a screenshot of the pages and paste them here excuse me in is near 2 am here

Frabjous · Jul 24, 2023

I took a look at Kittel. He is trying to motivate waves in cubic crystals. I do not think it is a particularly insightful motivation.

Vanadium 50 · Jul 24, 2023

It is Kittel and not Kettle. (Unless Ma and Pa Kettle wrote a textbook)

Drakkith · Jul 24, 2023

Vanadium 50 said:

It is Kittel and not Kettle. (Unless Ma and Pa Kettle wrote a textbook)

I've changed the title.

bsmile · Jul 25, 2023

My two cents, mathematically, it's basically an energy expression expanded in terms of six degree of freedoms to the 2nd order of geometry changes (those e_i). Since the solid is stable, thus 1st term vanishes, only the quadratic term is left. The derivative of (free) energy w.r.t geometry changes renders the constant coefficient tensor. One can then write in the compact fancy form of repeated summation over 6 indices.

Frabjous · Jul 25, 2023

Drakkith said:

I've changed the title.

I do not believe that Kittle is better than Kettel.

Drakkith · Jul 25, 2023

Frabjous said:

I do not believe that Kittle is better than Kettel.

Hah! Touché. Title changed again. Or is it titel?

hagopbul · Jul 26, 2023

thanks for changing the title it was 2 am , and going to post the pages in few hours

A question on elasticity from Kittel's Solid State Physics book

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