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The small particle size also implies that the disturbed flow can be found in the limit of very small Reynolds number, leading to a drag force given by Stokes' drag. Unsteadiness of the flow relative to the particle results in force contributions by added mass and the Basset force. The BBO equation states:
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
The original Langevin equation [1] [2] describes Brownian motion, the apparently random movement of a particle in a fluid due to collisions with the molecules of the fluid, = + (). Here, v {\displaystyle \mathbf {v} } is the velocity of the particle, λ {\displaystyle \lambda } is its damping coefficient, and m {\displaystyle m} is its mass.
where denotes the surface normal direction of the solid structure, and the imaginary particle density is calculated through the equation of state. Accordingly, the interaction forces f a F : p {\displaystyle \mathbf {f} _{a}^{F:p}} and f a F : v {\displaystyle \mathbf {f} _{a}^{F:v}} acting on a solid particle a {\displaystyle a} are given by
This is called Abel's integral equation and allows us to compute the total time required for a particle to fall along a given curve (for which / would be easy to calculate). But Abel's mechanical problem requires the converse – given T ( y 0 ) {\displaystyle T(y_{0})\,} , we wish to find f ( y ) = d ℓ / d y {\displaystyle f(y)={d\ell }/{dy ...
The curve represents a plot of equation with p 1, v 1, c 0, and s known. If p 1 = 0, the curve will intersect the specific volume axis at the point v 1. Hugoniot elastic limit in the p-v plane for a shock in an elastic-plastic material. For shocks in solids, a closed form expression such as equation cannot be
The equation is a nonlinear integro-differential equation, and the unknown function in the equation is a probability density function in six-dimensional space of a particle position and momentum. The problem of existence and uniqueness of solutions is still not fully resolved, but some recent results are quite promising. [3] [4]
MPM can be implemented to solve either quasi-static or dynamic equations of motion, depending on the type of problem that is to be modeled. Several versions of MPM include Generalized Interpolation Material Point Method [33];Convected Particle Domain Interpolation Method; [34] Convected Particle Least Squares Interpolation Method. [35]