Velocity of the material point

To prove our assertions about the physical significance of the rate-of-strain and vorticity tensors, we consider the relative motion of two nearby material points in the fluid P, initially at position x and Q, which is at x + Sx. We denote the velocity of the material point P as... 

Note that the time change a (A, t) of a specific point X gives a trajectory. Then the velocity of the material point is calculated by... 

Consider a system formed of material points of mass m, mf, m",.. . suppose this system in motion, and at a given instant let V, V, 7". . . be the velocities of the various points multiply the mass of each point by the square of its velocity find the sum of the products thus obtained finally, take half of this sum we form the quantity... 

Assume that the average velocity of the material movement = j the maximum velocity, which occurs when = 0. Also assume that the average velocity over the area of contact of indenter and sheared layer can be used in the calculation. At the point of contact of specimen and indenter the velocity is zero and has a maximum in the direction /2 when 0 = 0. This later value is given by... 

If the practical tensile strain is defined as e = L/Lo 1. where L and Lq are the stretched and unstretched lengths respectively, the practical tensile strain rate is (1 /Lo)dL/dt (cf. equation 60 of Chapter 3), and a constant e can be achieved by pulling the clamps apart at a constant rate. However, if the elongational strain rate is defined as the ratio of the velocity of a material point to its displacement, this quantity, denoted ci, is ( /L)dL/dt and it will remain constant only if the clamps are pulled apart at a rate which increases exponentially with time. Several instruments which accomplish this have been described. oo-io2a Qf... 

To begin with, here, we shall present the basic principles and the expressions of the classic quantities, such as the proper time and the universal velocity in the Minkowski timespace. The law of d3mamics of the material point is then stated. 

The universe velocity vector, or 4-velocity vector, of the material point M is defined as 4V = r/4M/Jr where, in matricial notation ... 

If the reference time f and the current time t coincide then the reference and current positions will also coincide and the right-hand side of Equation (3.77) can be replaced by the reference position defined as x in Equation (3.76). In a velocity field given as = u x,t ) the motion of a material point can be described... 

Selection of the off-take position is important from the standpoint of the amount of material removed. Locating the off-take in the proximity of the material stream or at points of splash will result in greater removal of materials. This positioning may be desirable as a means to control splash effects provided that the off-take velocity is kept low. 

The elastic stress curve in figure perfectly follows elastic strain [2]. This constant is the elastic modulus of the material. In this idealized example, this would be equal to Young s modulus. Here at this point of maximum stretch, the viscous stress is not a maximum, it is zero. This state is called Newton s law of viscosity, which states that, viscous stress is proportional to strain rate. Rubber has some properties of a liquid. At the point when the elastic band is fully stretched and is about to return, its velocity or strain rate is zero, and therefore its viscous stress is also zero. 

Table 9.3 shows the measured detonation velocities and densities of various types of energetic explosive materials based on the data in Refs. [9-11]. The detonation velocity at the CJ point is computed by means of Eq. (9.7). The detonation velocity increases with increasing density, as does the heat of explosion. Ammonium ni-trate(AN) is an oxidizer-rich material and its adiabatic flame temperature is low compared with that of other materials. Thus, the detonation velocity is low and hence the detonation pressure at the CJ point is low compared with that of other energetic materials. However, when AN particles are mixed with a fuel component, the detonation velocity increases. On the other hand, when HMX or RDX is mixed with a fuel component, the detonation velocity decreases because HMX and RDX are stoichiometrically balanced materials and the incorporation of fuel components decreases their adiabatic flame temperatures. 

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