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

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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


\delta W=f.dx

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 :


\delta W=-dU

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 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


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.



the strain--deformation relation becomes:


Oscillators chains[edit]

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 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> of the differences 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 U admits a Taylor expansion:


U(x_1=0,\dots x_N=0)=a+a_i x_i+a_{ij}x_ix_j+O(x^2)

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

dU=f_i dx_i

one obtains

f_i=\frac{\partial U}{\partial x_i}.

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:

b_{ij}=0 \mbox{ si  } i\neq j\pm 1



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[edit]

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.


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

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).

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


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

P_i=\int K_{ij}^s u_{i,j}^s d\tau



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 [ph:fluid:Germain80] that in the frame of SPH hypothesis, the rate of deformation tensor is simply the time derivative of SPH deformation tensor:




dU=-\int K_{ij}^s d\epsilon_{ij}d\tau.

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

U=\int \rho e_l d\tau

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:


\rho e_l=a+a_{ij}\epsilon_{ij}+a_{ijkl}\epsilon_{ij}\epsilon_{kl}

We have\footnote{



\frac{d}{dt}U=\frac{d}{dt}\int \rho e_l d\tau

and from the properties of the particulaire derivative:

\frac{d}{dt}\int \rho e_l d\tau=\int\frac{d}{dt}( \rho e_l d\tau)


\frac{d}{dt} (\rho e_ld\tau)=e_l\frac{d}{dt} (\rho d\tau) + \rho

From the mass conservation law:

\frac{d}{dt} \rho=0



\frac{dU}{dt}=\int \rho (\frac{d}{dt}e_l) d\tau


dU=\int \rho de_l d\tau

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

dU=\int \rho  [a_{ijkl}\epsilon_{ij}d\epsilon_{kl}+


dU=\int \rho  b_{ijkl}\epsilon_{kl}d\epsilon_{ij}d\tau

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.

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


Nematic material[edit]

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 u(x,y). } n. This field is related to the orientation of the molecules in the material (see figure figchampnema)


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:

U=\int u d\tau


u=u_1(\partial_in_j)+u_2(n_i\partial_jn_k)+ u_3(\partial_i
n_j\partial_kn_l)+ u_4(n_pn_q\partial_i n_j\partial_kn_l)+\dots

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


u_1(\partial_in_j)= K_{ij} \partial_i n_j

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.

  u_1(\partial_i n_j)=u_1(R_{ik}R_{jl}\partial_k n_l)

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


that is, tensor K_{ij} has to be isotrope. It is know 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:

  u_1=k_0\mbox{ div } n

  • 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:


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:

  u_2=k_1e_{ijk}n_k\partial_in_j=k_1n\mbox{ rot } n.

  • {\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.

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:

u_3=K_1 (\mbox{ div } n)^2

and u_4:

u_4=K_2 (n.\mbox{ rot } n)^2+K_3(n \wedge \mbox{ rot } n)^2

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

u=K_1 (\mbox{ div } n)^2+K_2 (n.\mbox{ rot } n)^2+K_3(n \wedge \mbox{ rot } n)^2