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

Introduction
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 $$P$$ that is a functional of displacement is introduced. Consider in particular case of a mass $$m$$ attached to a spring of constant $$k$$.Deformation of the system is referenced by the elongation $$x$$ of the spring with respect to equilibrium. The virtual work \index{virtual work} associated to a displacement $$dx$$ is

Quantity $$f$$ represents the constraint, here a force, and $$x$$ is the deformation. If force $$f$$ 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 $$U$$ :

In general, force $$f$$ depends on the deformation. Relation $$f=f(x)$$ 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 $$U$$ 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 $$x$$ to equilibrium position. If $$U$$ admits an expansion at $$x=0$$, in the neighbourhood of the equilibrium position $$U$$ can be approximated by:

As $$x=0$$ is an equilibrium position, we have $$dU=0$$ at $$x=0$$. That implies that $$a_1$$ is zero. Curve $$U(x)$$ at the neighbourhood of equilibrium has thus a parabolic shape (see figure figparabe

As

the strain--deformation relation becomes:

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

A calculation using virtual powers principle would have consisted in affirming: The total elastic potential energy is in general a function $$U(x_1,\dots, x_N)x_i$$ 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 $$U$$ admits a Taylor expansion:

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

one obtains

Using expression of $$U$$ provided by equation eqdevliUch yields to:

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

Differential of $$U$$ becomes:

Assuming that $$U$$ admits a Taylor expansion around the equilibrium position:

and that $$dU=0$$ 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 $$i$$ :

if one sets $$k=-b_{ii}=-b_{ii+1}$$.

Tridimensional elastic material
Consider a system $$S$$ in a state $$S_X$$ which is a deformation from the state $$S_0$$. Each particle position is referenced by a vector $$a$$ in the state $$S_0$$ and by the vector $$x$$ in the state $$S_X$$:

Vector $$X$$ represents the deformation.

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

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

with

Tensor $$u_{i,j}^s$$ is called rate of deformation tensor. It is the symmetric part of tensor $$u_{i,j}$$. It can be shown that in the frame of SPH hypothesis, the rate of deformation tensor is simply the time derivative of SPH deformation tensor:

Thus:

Function $$U$$ can thus be considered as a function $$U(\epsilon_{ij})$$. More precisely, one looks for $$U$$ that can be written:

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

We have\footnote{{{IMP/label|footdensi}}Indeed:

and from the properties of the particulaire derivative:

Now,

From the mass conservation law:

}

Thus

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

thus:

with $$b_{ijkl}=a_{ijkl}+a_{klij}$$. Identification with equation dukij, yields to the following strain--deformation relation:

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

Nematic material
A nematic material\index{nematic} is a material whose state can be defined by vector field\footnote{ State of smectic materials can be defined by a function $$u(x,y)$$. } $$n$$. This field is related to the orientation of the molecules in the material (see figure figchampnema)

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

with

The most general form of $$u_1$$ for a linear dependence on the derivatives is:

where $$K_{ij}$$ is a second order tensor depending on $$r$$. Let us consider how symmetries can simplify this last form. where $$R_{mn}$$ are orthogonal transformations (rotations). We thus have the condition:
 * Rotation invariance. Functional $$u_1$$ should be rotation invariant.

that is, tensor $$K_{ij}$$ has to be isotrope. It is known that the only second order isotrope tensor in a three dimensional space is $$\delta_{ij}$$, that is  the identity. So $$u_1$$ could always be written like:

Thus, there is no possible energy that has the form given by equation eqsansder. This yields to consider next possible term $$u_2$$. general form for $$u_2$$ is:
 * Invariance under the transformation $$n$$ maps to $$-n$$ . The energy   of distortion is independent on the sense of $$n$$, that is    $$u_1(n)=u_1(-n)$$. This implies that the constant $$k_0$$ in the previous    equation is zero.

Let us consider how symmetries can simplify this last form.
 * Invariance under the transformation $$n$$ maps to $$-n$$ . This   invariance condition is well fulfilled by $$u_2$$.
 * Rotation invariance. The rotation invariance condition implies    that:    It is known that there does not exist any third order isotrope tensor in  $$R^3$$, but there exist a third order isotrope pseudo tensor: the signature  pseudo tensor $$e_{jkl}$$ (see appendix  secformultens). This yields to the  expression:

There are thus no term $$u_2$$ in the expression of the internal energy for a nematic crystal. Using similar argumentation, it can be shown that $$u_3$$ can always be written:
 * {\bf Invariance of the energy with respect to the axis transformation     $$x\rightarrow -x$$, $$y\rightarrow -y$$, $$z\rightarrow -z$$.} The energy of    nematic crystals has this invariance property\footnote{ Cholesteric crystal doesn't verify this condition.} . Since $$e_{ijk}$$ is a pseudo-tensor it changes its signs for such  transformation.

and $$u_4$$:

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