Introduction to Mathematical Physics/Energy in continuous media/Generalized elasticity

IntroductionEdit

In this section, the concept of elastic energy is presented. \index{elasticity} The notion of elastic energy allows to deduce easily "strains--deformations" relations.\index{strain--deformation relation} So, in modelization of matter by virtual powers method \index{virtual powers} a power   that is a functional of displacement is introduced. Consider in particular case of a mass   attached to a spring of constant  .Deformation of the system is referenced by the elongation   of the spring with respect to equilibrium. The virtual work \index{virtual work} associated to a displacement   is

deltWfdx

 

Quantity   represents the constraint , here a force, and   is the deformation. If force   is conservative, then it is known that the elementary work (provided by the exterior) is the total differential of a potential energy function or internal energy   :

eqdeltaWdU

 

In general, force   depends on the deformation. Relation   is thus a constraint--deformation relation .


The most natural way to find the strain-deformation relation is the following. One looks for the expression of   as a function of the deformations using the physics of the problem and symmetries. In the particular case of an oscillator, the internal energy has to depend only on the distance   to equilibrium position. If   admits an expansion at  , in the neighbourhood of the equilibrium position   can be approximated by:

 

As   is an equilibrium position, we have   at  . That implies that   is zero. Curve   at the neighbourhood of equilibrium has thus a parabolic shape (see figure figparabe

figparabe

 
In the neighbourhood of a stable equilibrium position  , the intern energy function  , as a function of the difference to equilibrium presents a parabolic profile.

As

 

the strain--deformation relation becomes:

 

Oscillators chainsEdit

Consider a unidimensional chain of   oscillators coupled by springs of constant  . this system is represented at figure figchaineosc. Each oscillator is referenced by its difference position   with respect to equilibrium position. A calculation using the Newton's law of motion implies:

 

figchaineosc

 
A coupled oscillator chain is a toy example for studying elasticity.

A calculation using virtual powers principle would have consisted in affirming: The total elastic potential energy is in general a function  x_i</math> to the equilibrium positions. This differential is total since force is conservative\footnote{ This assumption is the most difficult to prove in the theories on elasticity as it will be shown at next section} . So, at equilibrium: \index{equilibrium} :

 

If   admits a Taylor expansion:

eqdevliUch

 

In this last equation, repeated index summing convention as been used. Defining the differential of the intern energy as:

 

one obtains

 

Using expression of   provided by equation eqdevliUch yields to:

 

But here, as the interaction occurs only between nearest neighbours, variables   are not the right thermodynamical variables. let us choose as thermodynamical variables the variables   defined by:

 

Differential of   becomes:

 

Assuming that   admits a Taylor expansion around the equilibrium position:

 

and that   at equilibrium, yields to:

 

As the interaction occurs only between nearest neighbours:

 

so:

 

This does correspond to the expression of the force applied to mass   :

 

if one sets  .

secmaterelast

Tridimensional elastic materialEdit

Consider a system   in a state   which is a deformation from the state  . Each particle position is referenced by a vector   in the state   and by the vector   in the state  :

 

Vector   represents the deformation.

Remark:

Such a model allows to describe for instance fluids and solids.

Consider the case where   is always "small". Such an hypothesis is called small perturbations hypothesis (SPH). The intern energy is looked as a function  .

Definition: The deformation tensor SPH is the symmetric part of the tensor gradient of  .

 

At section secpuisvirtu it has been seen that the power of the admissible intern strains for the problem considered here is:

 

with

 

Tensor   is called rate of deformation tensor. It is the symmetric part of tensor  . It can be shown [ph:fluid:Germain80] that in the frame of SPH hypothesis, the rate of deformation tensor is simply the time derivative of SPH deformation tensor:

 

Thus:

dukij

 

Function   can thus be considered as a function  . More precisely, one looks for   that can be written:

 

where   is an internal energy density with\footnote{ Function   depends only on  .} whose Taylor expansion around the equilibrium position is:

eqrhoel

 

We have\footnote{

footdensi

Indeed:

 

and from the properties of the particulaire derivative:

 

Now,

 

From the mass conservation law:

 

}

eqdudt

 

Thus

 

Using expression eqrhoel of   and assuming that   is zero at equilibrium, we have:

 

thus:

 

with  . Identification with equation dukij, yields to the following strain--deformation relation:

 

it is a generalized Hooke law\index{Hooke law}. The  's are the elasticity coefficients.

Remark: Calculation of the footnote footdensi show that calculations done at previous section secchampdslamat should deal with volumic energy densities.

secenernema

Nematic materialEdit

A nematic material\index{nematic} is a material [ph:liqcr:DeGennes74] whose state can be defined by vector field\footnote{ State of smectic materials can be defined by a function  . }  . This field is related to the orientation of the molecules in the material (see figure figchampnema)

figchampnema

 
Each molecule orientation in the nematic material can be described by a vector  . In a continuous model, this yields to a vector field  . Internal energy of the nematic is a function of the vector field   and its partial derivatives.

Let us look for an internal energy   that depends on the gradients of the   field:

 

with

 

The most general form of   for a linear dependence on the derivatives is:

eqsansder

 

where   is a second order tensor depending on  . Let us consider how symmetries can simplify this last form.

  • Rotation invariance. Functional   should be rotation invariant.
 

where   are orthogonal transformations (rotations). We thus have the condition:

 

that is, tensor   has to be isotrope. It is known that the only second order isotrope tensor in a three dimensional space is  , that is the identity. So   could always be written like:

 
  • Invariance under the transformation   maps to   . The energy of distortion is independent on the sense of  , that is  . This implies that the constant   in the previous equation is zero.

Thus, there is no possible energy that has the form given by equation eqsansder. This yields to consider next possible term  . general form for   is:

 

Let us consider how symmetries can simplify this last form.

  • Invariance under the transformation   maps to   . This invariance condition is well fulfilled by  .
  • Rotation invariance. The rotation invariance condition implies that:
     
    It is known that there does not exist any third order isotrope tensor in  , but there exist a third order isotrope pseudo tensor: the signature pseudo tensor   (see appendix secformultens). This yields to the expression:
 
  • {\bf Invariance of the energy with respect to the axis transformation  ,  ,  .} The energy of nematic crystals has this invariance property\footnote{ Cholesteric crystal doesn't verify this condition.} . Since   is a pseudo-tensor it changes its signs for such transformation.

There are thus no term   in the expression of the internal energy for a nematic crystal. Using similar argumentation, it can be shown that   can always be written:

 

and  :

 

Limiting the development of the density energy   to second order partial derivatives of   yields thus to the expression: