From: george grinder [laserpointerz@gmail.com]
Sent: Saturday, March 15, 2008 10:35 PM
To: tonebrush@vsnl.net
Subject: Question in Continuum Mechanics
Hi Mr Ajit Jadhav
   My name is Sridhar N. I am a 2nd year graduate student in the department of Mechanical Engg. at TAMU, Kingsville. I have had a question regarding continuum mechanics for quite sometime now and i haven't quite gotten a convincing answer from my professors. The literature too seems to be a little ambiguous about this. I have been visiting the online mechanics forum imechanica.org for a while and from your posts i feel that you seem to have a sound grasp of the fundamentals. So here is my question

Consider an expression for the elastic energy density of a linear hyperelastic material which not only depends upon the strain but also on higher order gradients of displacements.
For example, consider the following elastic energy density expression in the 1-D case

E= c. (u'^2)+f1. (u''^2) +f2. (u' u''')

In the above expression, u denotes the displacement field and u' denotes the first derivative of displacement and so on. c is related to the usual elastic modulus while f1 and f2 denote new strain-gradient material properties/moduli. Now my question is in-order to derive thermodynamic restrictions on c, f1 and f2, should we look at the positive definiteness of the strain energy density or of the total energy (which is the integral of the above energy density over a cerain volume).

From Strain-Energy Density Considerations
If I consider the strain energy density and further consider a sample displacement field whose 3rd derivative vanishes, then the term f2 will vanish from the energy density expression. Then i can conclude that both c and f1 have to be positive for the energy density to be positive definite for all such displacement fields. Now, if i want to find restrictions on f2, i do the following

I write the strain energy density as the product of these matrices [u' u'' u'''] [c 0 f2/2  [u' u'' u''']^T
                                                                                                            0 f1 0
                                                                                                            f2/2 0 0]

The ^T denotes that i am taking the transpose of the row matrix to make it into a column matrix. The matrix in the middle is a 3*3 matrix containing the elastic moduli. Now, i will try to diagonalise this matrix and impose the condition that the eigenvalues of this matrix has to be real and positive, such that the energy density is always positive. From this analysis, i can figure out that there is no restriction on f2 i.e f2 can be positive or negative.

From total energy considerations

If i take the integral of this energy density, then by integration by parts i can write the energy as

Integral of ( cu'^2+ (f1-f2).u''^2)  plus some boundary terms. Now, if i ignore the boundary terms, i get the restriction that c>0 and (f1-f2)>0. However, certain combinations which satisfy f1>f2 will make the energy density negative. Also, (f1-f2)>0 can also means f1<0, which does not look correct to me.

Can you please shed some light on which restriction should be used?

Also, one more question is if the material property f1 is found to be positive in the 1-D case, can i generalize this to the 3-D case and say that the tensor  f1 should be positive definite?

I will be obliged if you just take a look at my question. Thanks  a lot!