U is the fluid velocity, K is the
elastic constant for Winkler foundation and L is the length of the nanotube.
In its simplest ("equal
elastic constant") form, the functional can be written
With elastic tensile loading ([F.sub.max] < 47.71 kN), the relative nonlinear coefficients nearly keeps unchanged; thus, the
elastic constant changes very little and increases rapidly when the tensile loading exceeds 47.71 kN at the position of 2, 3, and 4.
In the equations above, [x.sub.j] are Cartesian spatial coordinates, V is the volume of solution domain bounded by the surface S, [[sigma].sub.ij] is the stress tensor, [n.sub.j] is the outward unit normal to the surface S, [f.sub.1] is the volume force, [C.sub.ijkl] is the
elastic constant tensor components, [[epsilon].sub.kl] is the strain tensor, and [u.sub.k] represents the point displacement.
Since it is our aim to simulate also rigid bodies using particle systems, it is of fundamental importance to be sure that the simulation will behave appropriately in a large interval of possible values for the
elastic constant. For this reason, it has been necessary to balance damping, to define the most appropriate refreshing rate for the simulator, and to select the best possible numeric integrator, that is to say, the one that guarantees the most correct behavior while, at the same time, presenting good performances in terms of calculation time (we are dealing with real-time systems).
The values for the Lame constants and the third-order
elastic constants are approximated on the basis of the experimental data [13]; as a result the linear
elastic constant is [[alpha].sub.0] = 400 GPa and the nonlinear
elastic constant [[beta].sub.0] = -1000 GPa.
This allows us to differentiate between for example [[micro].sub.em], the electromagnetic permeability of free space, and [[micro].sub.0], the Lame
elastic constant for the shear modulus of the spacetime continuum.
Oda, "High-temperature
elastic constant data on minerals relevant to geophysics," Reviews of Geophysics, vol.
R is
elastic constant corresponding to fluid phase.
The elastic components, as previously mentioned, can be modeled as springs of
elastic constant E, given [sigma] = Ez is the formula: where o is the stress, E is the elastic modulus of the material and ?is the strain that occurs under the given stress, similar to Hooke's Law.
elastic modulus - (physics) the ratio of the applied stress to the change in shape of an elastic body