Computational process to study the wave propagation In a non-linear medium by quasi- linearization

Two objects having distinct velocities come into contact an impact can occur. The impact study i.e., in the displacement of the objects after the impact, the impact force is function of time‘t’ which is behaves similar to compression force. The impact tenure is very short so impulses must be generated subsequently high stresses are generated. In this work we are examined the wave propagation inside the object after collision and measured the object non-linear behavior in the one-dimensional case. Wave transmission is studied by means of material acoustic parameter value. The objective of this paper is to present a computational study of propagating pulsation and harmonic waves in nonlinear media using quasi-linearization and subsequently utilized the central difference scheme. This study gives focus on longitudinal, one- dimensional wave propagation. In the finite difference scheme Non-linear system is reduced to a linear system by applying quasi-linearization method. The computed results exhibit good agreement on par with the selected non-liner wave propagation.


1.Introduction
Two bodies have distinct velocities in the same direction come into contact, an impact occurs. In the impact analysis i.e., in the displacement of the bodies after the impact, impact force is a function of time 't' is acting like a density force [2]. The objective of this work is to present a computational study of propagating pulsed and harmonic waves in a nonlinear media by using a Finite difference scheme. This study aims on longitudinal, one-dimensional wave propagation. In the finite difference scheme Non-linear model is reduced to a linear system by quasi-linearization method. The numerically computed results exhibit good quality agreement with the non-liner wave equation character.

FORMULATION OF THE PROBLEM
An object of length L1 contacts another object of length ' L2'. Both the objects have the same material configuration with non-linearity. The first object has an initial velocity of V0, whereas the second one is at rest. Here Reaction force (RN ), Normal gap c(u) are always perpendicular to another. This kind of literature is available in the monuments [2] , [5], [8] & [10].
Materials by means of plastic deformation, Materials with distributed gap, linear elastic Hooke's law is usually not adequate to describe nonlinear, inelastic nature. Here we can study the class of materials whose nature can be described by the following stress-strain Where 0 H is the initial strain s=sign ( 1 H ), D is a constant, and f ( H ) and g ( H ) are functions to be evaluated.
Now in a particular case of (2) namely with no initial stress and strain is considered as Setting D = 0 , one can obtain Where E is the elastic Young's modulus and 'c' can be considered as the phase velocity. The nonlinear equation solved by applying quasi-linearization method [3]. While the process is intiated then iteration across the time-step is introduced to reduce the non-linear equation into linear.
For nonlinear materials type Clearly, when 0 J , the material is linear elastic. The parameter J indicates the amount of material nonlinearity. The parameter J defined here is identical to the acoustic nonlinear parameter. The acoustic nonlinear parameter comes in metals due to lattice non harmonicity which is usually very small in comparison to the elastic deformation of the metals. So we can study wave propagation nature for various acceptable values of b. Here we are selected in the acceptable region, i.e the values J =10000, J =5000 and J =2500 respectively.

2.Computational solution of non-linear wave equation using Quasi-linearization
Consider the non-linear wave equation with a small rearrangement Apply the quasi-linearization method [3] on the governing equation (7) we have (9) after simplification can be expressed as now iterative so that we can get the solution profiles at n=0,1,2,3,4 …… so that With the defined initial and boundary conditions in mesh point notation transformed to equations (12) to (14) respectively. An impact taking place the velocities of the two objects are varies according to the starting compression force existed at the impact point. An impact occurs a longitudinal sound wave is generated, it propagates in the region up to free end of the second object. When it went to the free end a reflection occurs. So the boundary condition at the free end is selected as negative but very small in magnitude.
The displacement in terms of length of the impact system with respective to time is showed in the Figure 1(a) -1(d)

1)
At J =10000 at the second level the displacement u(x,t) exhibits non-linearity at the middle of the position of the objects and at all other time levels , no non-linearity is observed.

2)
At J =5000 complete time levels, displacement sustains with respect to the origin except at time level 5. At end positions at time level 5 non-linearity is observed.

3)
At J =2500 all the time levels shows the displacement with disturbance at end positions (0-2 cm and 7-10 cm) and the middle position the displacement is found to constant and rises with respective to the previous time level-1.

4)
At lower and higher Acoustic values non-linearity is not observed clearly but it gives the tendency. At middle J value the non-linearity behavior is clearly observed at higher time level-5.
5) For all the acoustic parameter values displacement u is observed to be constant at 2 to 6 units distance with respective to time level.