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**I have deliberately allowed slider variable ranges that can result in a break up of
the solid. It that happens, the learner just needs to refresh the page, which in Windows means hitting the F5 key.**
Here I compute the longitudinal and shear modulii of the 2D solid.
For the longitudinal modulus (Young's Modulus `E`) I apply force to the right hand
end and anchor the left end to cause about 1% compression and extension.
Of course this is analogous to placing a weight of mass, `M`, on a vertical spring and letting it go.
The weight exhibits vertical oscillations between the
starting height and the displacement `dy=(Mg)/k` where `k` is the spring constant.
To cause these oscillations to relax, I had to
multiply the disc velocities by a drag coefficient which is adjustable by the learner.
For the shear modulus (`G`) I apply force to the top row and anchor the bottom row and cause about 0.1 radian of angle change.
In this simulation the bonding is done with a Lennard-Jones (LJ) potential.
LJ potentials have extremely large repelling forces when the particles get too close and longer
range attractive forces when the particles are farther apart.
The LJ potential is of the form

`V(r)=4V_(LJ)[(sigma/r)^(2b)-(sigma/r)^b]`

where `V_(LJ)` is the potential minimum, `sigma` is a length scaling parameter, `r` is the particle separation, and `b` is an integer often chosen to be 6. The larger the value of `b`, the more abrupt is the force change with distance,`r`. The solid is initially set up so that all the particles are separated by a distance that causes the nearest neighbor (NN) forces to be zero. The force between a site and its second nearest neighbor (SNN) is not initially zero but is small anyway. An obvious result of that set up is that alternate rows of particles have to be staggered by 1/2 the spacing in the horizontal direction and the vertical spacing of rows is `sqrt(3)/2` times the horizontal spacing of particles in any column so that the 2D solid has 6 fold symmetry. SNN would be at just the `sqrt(2)` times the NN distance. With b=6 that would mean that the force due to the SNN would be 1/8 that of the NN. With the 6 fold symmetry SNN distance is twice NN distance so the SNN force is just 1/64 that of the NN force,`G=E/(2(1+nu))`

The shear modulus (SM) calculation has difficulty for the 6 fold symmetry case because that symmetry exhibits very little NN distance change during shear and thus has very minimal restoring force. For that reason the SM is estimated from a combination of E and Poissons ratio, `nu`. During the longitudinal modulus strain, the length of the array is increased and there is a corresponding decrease in the height of the array. These result in a change in total volume (Area in 2D) change. From the fractional change in volume and the longitudinal strain, the Poisson ratio is computed. The SM uses the Poisson ratio for calculation of the longitudinal modulus, LM

`SM=(LM)/(2(1+nu)`

From the LM and SM and the mass density, the longitudinal and shear wave speeds, c, can be computed using the equation

`c=sqrt(LM)/rho)`

where `rho` is density measured in kilograms per `m^3`. These agree quite well with those computed in a companion page Speed of Sound in a 2D Numerically Modeled Solid Lattice Plots of the strain Vs time as well as the change of short range spacing are shown below the graphic.