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## Main Question or Discussion Point

This is a spin-off from an old thread. While that thread (in spite of its title) is about hoops, this thread will be about rods.

I plan to write this in several sections, as time permits. The first first section will be a recap of what "hyperelasticity" is about, following sections are planned to develop this into a complete Lagrangian model of a one dimensional, hyperelastic, relativistic rod.

The hyperelastic model in relativity is discussed in some of the following links:

Greg Egan's webpage

Relatistic elastodynamics, Wernig Piilchler (a PHD thesis also linked on Greg Egan's webpage)

http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3Agr-qc%2F0403073 [Broken] a paper by R Beig

The basic idea of hyperelasticity is this. Suppose we have an elastic rod, and we assume that the rod has the following characteristics:

1) It obeys Hooke's law, so that if the rod is stretched in length by some factor s (i.e. if the rod is stretched from one foot to two feet, the stretch factor is 2) that the force is always proportional to the stretch factor s, even for large values of stretch.

We will further assume that the rod does not change in width as it is stretched. This assumption is mainly for convenience, one could introduce Poisson's ratio if needed.

2) Energy is conserved, and evenly distributed throughout the rod. This assumption allows us to compute the energy density and stress-energy tensor of the stretched rod, quantities that will be needed later. In fact, as we will see, the key quantity needed is the proper energy density of the rod, i.e. the energy density in the rod as seen by an observer co-moving with some small section of the rod.

This calculation is carried out on Greg Egan's webpage, but a brief review might be helpful:

Consider a unit length of the rod, with the Hooke's law constant of the spring being k.

If it is stretched from unit length to some length s, the total energy in the rod goes from its initial rest energy [itex]E_0[/itex] to:

[tex]E = E_0 + \int_{S=1}^{S=s} -k \left( S - 1 \right) dS = E_0 + \frac{k}{2} \left( s - 1 \right)^2 [/tex]

because the total energy in the spring is equal to its initial energy plus the work done in stretching it.

We now wish to re-write this in terms of energy density, rather than energy. Since the original rod had a unit length, its energy density is equal to its total energy

[tex]\rho_0 = E_0[/tex]

The energy density in the stretched rod is just:

[tex]\rho = \frac{\rho_0}{s} + \frac{k}{2} \left( s + \frac{1}{s} - 2 \right) \hspace{1.5 in} (1) [/tex]

We divide the previous result by s, because that is the length of the stretched rod, so that the density in the stretched rod is the total energy / stretched length. We have also taken advantage of the fact that [itex]E_0 = \rho_0[/itex] in writing this expression.

These results follow from our assumption that the energy is evenly distributed throughout the spring.

The details of the dynamics of the actual energy transport are not modeled, it is simply assumed that the energy is always evenly distributed through the rod. It is important to restrict k to a small enough value so that the rod is not "too rigid". If the rod is too rigid, the speed of sound through the rod will become greater than 'c', i.e. there is no physical way our

This concludes the introductory section, next up will be the topic of an overview of Lagrangians, Lagrangian densities, and the Lagrangian of a Newtonian elastic 1d rod.

I plan to write this in several sections, as time permits. The first first section will be a recap of what "hyperelasticity" is about, following sections are planned to develop this into a complete Lagrangian model of a one dimensional, hyperelastic, relativistic rod.

The hyperelastic model in relativity is discussed in some of the following links:

Greg Egan's webpage

Relatistic elastodynamics, Wernig Piilchler (a PHD thesis also linked on Greg Egan's webpage)

http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3Agr-qc%2F0403073 [Broken] a paper by R Beig

The basic idea of hyperelasticity is this. Suppose we have an elastic rod, and we assume that the rod has the following characteristics:

1) It obeys Hooke's law, so that if the rod is stretched in length by some factor s (i.e. if the rod is stretched from one foot to two feet, the stretch factor is 2) that the force is always proportional to the stretch factor s, even for large values of stretch.

We will further assume that the rod does not change in width as it is stretched. This assumption is mainly for convenience, one could introduce Poisson's ratio if needed.

2) Energy is conserved, and evenly distributed throughout the rod. This assumption allows us to compute the energy density and stress-energy tensor of the stretched rod, quantities that will be needed later. In fact, as we will see, the key quantity needed is the proper energy density of the rod, i.e. the energy density in the rod as seen by an observer co-moving with some small section of the rod.

This calculation is carried out on Greg Egan's webpage, but a brief review might be helpful:

Consider a unit length of the rod, with the Hooke's law constant of the spring being k.

If it is stretched from unit length to some length s, the total energy in the rod goes from its initial rest energy [itex]E_0[/itex] to:

[tex]E = E_0 + \int_{S=1}^{S=s} -k \left( S - 1 \right) dS = E_0 + \frac{k}{2} \left( s - 1 \right)^2 [/tex]

because the total energy in the spring is equal to its initial energy plus the work done in stretching it.

We now wish to re-write this in terms of energy density, rather than energy. Since the original rod had a unit length, its energy density is equal to its total energy

[tex]\rho_0 = E_0[/tex]

The energy density in the stretched rod is just:

[tex]\rho = \frac{\rho_0}{s} + \frac{k}{2} \left( s + \frac{1}{s} - 2 \right) \hspace{1.5 in} (1) [/tex]

We divide the previous result by s, because that is the length of the stretched rod, so that the density in the stretched rod is the total energy / stretched length. We have also taken advantage of the fact that [itex]E_0 = \rho_0[/itex] in writing this expression.

These results follow from our assumption that the energy is evenly distributed throughout the spring.

The details of the dynamics of the actual energy transport are not modeled, it is simply assumed that the energy is always evenly distributed through the rod. It is important to restrict k to a small enough value so that the rod is not "too rigid". If the rod is too rigid, the speed of sound through the rod will become greater than 'c', i.e. there is no physical way our

**assumption**of uniform energy density can actually be maintained. At some future point after the Lagrangian has been developed, this issue will be revisited. I'll mention the important result here, though - there is a maximum stretch factor s possible for any given spring constant k that limits the speed of energy transport to below that of light.This concludes the introductory section, next up will be the topic of an overview of Lagrangians, Lagrangian densities, and the Lagrangian of a Newtonian elastic 1d rod.

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