Introduction to Mathematical Physics/Continuous approximation/Virtual powers principle

Principle statement

Momentum conservation has been introduced by using averages over particles of quantities associated to those particles. Distant forces have been modelized by force densities $f_{i}$, internal strains by a second order tensor $\tau _{ij}$,\dots This point of view is directly related to the Newton's law of motion. The dual point of view is presented here: strains are described by the means of movement they permit ([#References|references])). This way corresponds to our day to day experience

• to know if a wallet is heavy, one lifts it up.
• to appreciate the tension of a string, one moves it aside from its equilibrium position.
• pushing a car can tell us if the brake is on.

Strains are now evaluated by their effects coming from a displacement or deformation. This point of view is interesting because it allows to defines strains when they are bad defined in the first point of view, like for frictions or binding strains. Freedom in modelization is kept very large because the modelizer can always choose the size of the virtual movements to be allowed. let us precise those ideas in stating the principle.

Principle:

Virtual power of acceleration quantities is equal to the sum of the virtual powers of all strains applied to the system, external strains, as well as internal strains:

$\int \rho \gamma _{i}{\hat {u}}_{i}d\tau =P^{int}+P_{dist}^{ext}+P_{contact}^{ext}$

where $P^{int}$ represents power of internal strains, $P_{dist}^{ext}$, distant external strains, $P_{contact}^{ext}$ contact external strains.

At section sepripuiva it is shown how a partial differential equation system can be reduced to a variational system: this can be used to show that Newton's law of motion and virtual powers principle are dual forms of a same physical law.

Powers are defined by giving spaces $A$ and $E$ where $A$ is the affine space attached to $E$:

${\begin{array}{llll}P:&L(A,E)&\longrightarrow &R\\&u(r)&\longrightarrow &P(u(r))\end{array}}$

At section seccasflu we will consider an example that shows the power of the virtual powers point of view.

sepripuiva

Virtual powers and local equation

A connection between local formulation (partial derivative equation or PDE) and virtual powers principle (variational form of the PDE problem considered) is presented on an example. Consider the problem:

Problem:

Find $u$ such that:

$\partial _{j}\sigma _{i,j}(u)+f_{i}=0{\mbox{ in }}\Omega$

$u_{i}=0{\mbox{ on }}\Gamma _{1}$

$\sigma _{i,j}(u)n_{j}=0{\mbox{ on }}\Gamma _{2}$

with $\Gamma _{1}\cup \Gamma _{2}=\partial \Omega$.

Let us introduce the bilinear form:

$a(u,v)=\int _{\Omega }\sigma _{i,j}(u)\epsilon _{i,j}(v)dx$

and the linear form:

$L(v)=\int _{\Omega }f_{i}v_{i}dx+\int _{\Gamma _{1}}g_{i}v_{i}dx$

it can be shown that there exist a space $V$ such that there exist a unique solution $u$ of

$\forall v\in V,a(u,v)=L(v)$

$a(u,v)$ represents the deformation's work of the elastic solid \index{virtual power} \index{elasticity} corresponding to virtual displacement $v$ from position $u$. $L(v)$ represents the work of the external forces for the virtual displacement $v$. The virtual powers principle can thus be considered as a consequence of the great conservation laws: \begin{prin}Virtual powers principle (static case): Actual displacement $u$ is the displacement cinematically admissible such that the deformation's work of the elastic solid corresponding to the virtual displacement $v$ is equal to the work of the external forces, for any virtual displacement $v$ cinematically admissible. \end{prin} Moreover, as $a(.,.)$ is symmetrical, solution $u$ is also the minimum of

$J(v)={\frac {1}{2}}\int _{\Omega }\sigma _{i,j}(v)\epsilon _{i,j}(v)dx-(\int _{\Omega }f_{i}v_{i}dx+\int _{\Gamma _{1}}g_{i}v_{i}dx)$

$J(v)$ is the potential energy of the deformed solid, ${\frac {1}{2}}\int _{\Omega }\sigma _{i,j}(v)\epsilon _{i,j}(v)dx$ is the deformation energy. $-(\int _{\Omega }f_{i}v_{i}dx+\int _{\Gamma _{1}}g_{i}v_{i}dx)$ is the potential energy of the external forces. This result ([#References|references]) can be stated as follows: \begin{prin} The actual displacement $u$ is the displacement among all the admissible displacement $v$ that minimizes the potential energy $J(v)$. \end{prin}

seccasflu

Case of fluids

Consider for instance a fluid ([#References|references]). Assume that the power of the internal strains can be described by integral:

$P^{int}=\int (K_{i}{\hat {u}}_{i}+K_{ij}{\hat {u}}_{i,j})dx$

where $u_{i,j}$ designs the derivative of $u_{i}$ with respect to coordinate $j$. The proposed theory is called a first gradient theory.

Remark:

The step of the expression of the power as a function of the speed field $u$ is the key step for modelization. A large freedom is left to the modelizator. Powers being scalars, they can be obtained by contraction of tensors (see appendix chaptens) using the speed vector field $u_{i}$ as well as its derivatives. method to obtain intern energies in generalized elasticity is similar (see section secelastigene).

Denoting $a$ and $s$ the antisymmetric and symmetric [art of the considered tensors yields to:\index{tensor} :

$P^{int}=\int (K_{i}{\hat {u}}_{i}+K_{ij}^{s}{\hat {u}}_{i,j}^{s}+K_{ij}^{a}{\hat {u}}_{i,j}^{a})dx$

where it has been noted that cross products of symmetric and antisymmetric tensors are zero\footnote{ That is: $(a_{ij}+a_{ji})(b_{ij}-b_{ji})=0$. } . Choosing the uniformly translating reference frame, it can be shown that term $K_{i}$ has to be zero:

$K_{i}=0$

Antisymmetric tensor is zero because movement is rigidifying:

$K_{ij}^{a}=0.$

Finally, the expression of the internal strains is:\index{strains}

$P^{int}=\int (K_{ij}^{s}{\hat {u}}_{i,j}^{s})dx$

$K_{ij}^{s}$ is called strain tensor since it describes the internal deformation strains. The external strains power is modelized by:

$P_{dist}^{ext}=\int (f_{i}{\hat {u}}_{i}+F_{ij}{\hat {u}}_{i,j})dx$

Symmetric part of $F_{ij}$ can be interpreted as the volumic double--force density and its antisymmetrical part as volumic couple density. Contact strains are modelized by:

$P_{contact}^{ext}=\int (T_{i}{\hat {u}}_{i}+T_{ij}{\hat {u}}_{i,j})dx$

Finally the PDE problem to solve is:

$f_{i}+\tau _{ij,j}=\rho \gamma _{i}{\mbox{ in }}V$

where $\tau _{ij}=K_{ij}^{s}-F_{ij}$

$T_{i}=\tau _{ij}n_{j},{\mbox{ sur }}\partial V$

$T_{ij}=0,{\mbox{ sur }}\partial V$

Stress-deformation tensor

The next step is to modelize the internal strains. that is to explicit the dependence of tensors $K_{ij}$ as functions of $u_{i}$. This problem is treated at chapter parenergint. Let us give here two examples of approach of this problem.

Example:

For a perfect gas, pressure force work on a system of volume $V$ is:

$\delta W=-pdV.$

The state equation (deduced from a microscopic theory)

$pV=nk_{B}T$

is used to bind the strain $p$ to the deformation $dV$.

Example:

The elasticity theory (see chapter parenergint) allows to bind the strain tensor $\tau _{ij}$ to the deformation tensor $\epsilon _{ij}$