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 Template:IMP/label Template:IMP/eq 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 $U$ : Template:IMP/label Template:IMP/eq 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: Template:IMP/eq 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 Template:IMP/label

In the neighbourhood of a stable equilibrium position $x_0$, the intern energy function $U$, as a function of the difference to equilibrium presents a parabolic profile.

As Template:IMP/eq the strain--deformation relation becomes: Template:IMP/eq

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: Template:IMP/eq Template:IMP/label

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 $U(x_1,\dots ,x_N)<math>ofthedifferences$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} : Template:IMP/eq If $U$ admits a Taylor expansion: Template:IMP/label Template:IMP/eq In this last equation, repeated index summing convention as been used. Defining the differential of the intern energy as: Template:IMP/eq one obtains Template:IMP/eq Using expression of $U$ provided by equation eqdevliUch yields to: Template:IMP/eq 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 ${ϵ}_i$ defined by: Template:IMP/eq Differential of $U$ becomes: Template:IMP/eq Assuming that $U$ admits a Taylor expansion around the equilibrium position: Template:IMP/eq and that $dU=0$ at equilibrium, yields to: Template:IMP/eq As the interaction occurs only between nearest neighbours: Template:IMP/eq so: Template:IMP/eq This does correspond to the expression of the force applied to mass $i$ : Template:IMP/eq 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 $S_X$: Template:IMP/eq Vector $X$ represents the deformation. Template:IMP/rem 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 $U(X)$. Template:IMP/defn At section secpuisvirtu it has been seen that the power of the admissible intern strains for the problem considered here is: Template:IMP/eq with Template:IMP/eq 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 Template:IMP/cite that in the frame of SPH hypothesis, the rate of deformation tensor is simply the time derivative of SPH deformation tensor: Template:IMP/eq Thus: Template:IMP/label Template:IMP/eq Function $U$ can thus be considered as a function $U({ϵ}_{ij})$. More precisely, one looks for $U$ that can be written: Template:IMP/eq where $e_l$ is an internal energy density with\footnote{ Function $U$ depends only on ${ϵ}_{ij}$.} whose Taylor expansion around the equilibrium position is: Template:IMP/label Template:IMP/eq We have\footnote{Template:IMP/labelIndeed: Template:IMP/eq and from the properties of the particulaire derivative: Template:IMP/eq Now, Template:IMP/eq From the mass conservation law: Template:IMP/eq } Template:IMP/label Template:IMP/eq Thus Template:IMP/eq Using expression eqrhoel of $e_l$ and assuming that $dU$ is zero at equilibrium, we have: Template:IMP/eq thus: Template:IMP/eq with $b_{ijkl}=a_{ijkl}+a_{klij}$. Identification with equation dukij, yields to the following strain--deformation relation: Template:IMP/eq it is a generalized Hooke law\index{Hooke law}. The $b_{ijkl}$'s are the elasticity coefficients. Template:IMP/rem

Nematic material

A nematic material\index{nematic} is a material Template:IMP/cite 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) Template:IMP/label

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

Let us look for an internal energy $U$ that depends on the gradients of the $n$ field: Template:IMP/eq with Template:IMP/eq The most general form of $u_1$ for a linear dependence on the derivatives is: Template:IMP/label Template:IMP/eq where $K_{ij}$ is a second order tensor depending on $r$. Let us consider how symmetries can simplify this last form.

• Rotation invariance. Functional $u_1$ should be rotation invariant.

where $R_{mn}$ are orthogonal transformations (rotations). We thus have the condition:

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:

• 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.

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:

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: Template:IMP/eq 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:

• {\bf Invariance of the energy with respect to the axis transformation $x\to -x$, $y\to -y$, $z\to -z$.} 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.

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: Template:IMP/eq and $u_4$: Template:IMP/eq Limiting the development of the density energy $u$ to second order partial derivatives of $n$ yields thus to the expression: Template:IMP/eq