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A low velocity impact between a rigid sphere and transversely isotropic strain-hardening plate supported by a rigid substrate is generalized to the concept of noninteger derivatives order. A brief history of fractional derivatives order is presented. The fractional derivatives order adopted is in Caputo sense. The new equation is solved via the analytical technique, the Homotopy decomposition method (HDM). The technique is described and the numerical simulations are presented. Since it is very important to accurately predict the contact force and its time history, the three stages of the indentation process, including (1) the elastic indentation, (2) the plastic indentation, and (3) the elastic unloading stages, are investigated.

The concept of noninteger order derivative has been intensively applied in many fields. It is worth nothing that the standard mathematical models of integer-order derivatives, including nonlinear models, do not work adequately in many cases. In the recent years, fractional calculus has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics, notably control theory, signal image processing, and groundwater problems; an excellent literature of this can be found in [

However, there exist a quite number of these fractional derivative definitions in the literature which range from Riemann-Liouville to Jumarie [

There are many physical situations in which a thin plate made of strain-hardening materials resting on a rigid substrate is impacted by a rigid indenter. For example, such a phenomenon may be caused by the impact of hailstones, run way debris, or small stones on the panels of a vehicle or aircraft [

In this paper, approximated solutions for the generalized version of a low velocity impact between a rigid sphere and transversely isotropic strain-hardening plate supported by a rigid substrate will be obtained via the relatively new analytical method HDM.

The remaining of this paper is structured as follows: in Section

There exists a vast literature on different definitions of fractional derivatives. The most popular ones are the Riemann-Liouville and the Caputo derivatives. For Caputo, we have

For the case of Riemann-Liouville we have the following definition:

Guy Jumarie proposed a simple alternative definition to the Riemann-Liouville derivative:

For the case of Weyl we have the following definition:

With the Erdelyi-Kober type we have the following definition:

Here

With Hadamard type, we have the following definition:

With Riesz type, we have the following definition:

We will not mention the Grunward-Letnikov type here because it is in series form [

In 1998, Davison and Essex [

In an article published by Coimbra [

It is very important to point out that all these fractional derivative order definitions have their advantages and disadvantages; here we will include Caputo, variational order, Riemann-Liouville Jumarie, and Weyl [

With the Jumarie definition which is actually the modified Riemann-Liouville fractional derivative, an arbitrary continuous function needs not to be differentiable; the fractional derivative of a constant is equal to zero and more importantly it removes singularity at the origin for all functions for which

With the Riemann-Liouville fractional derivative, an arbitrary function needs not to be continuous at the origin and it needs not to be differentiable.

One of the great advantages of the Caputo fractional derivative is that it allows traditional initial and boundary conditions to be included in the formulation of the problem [

It is customary in groundwater investigations to choose a point on the centreline of the pumped borehole as a reference for the observations and therefore neither the drawdown nor its derivatives will vanish at the origin, as required [

Although these fractional order derivatives display great advantages, however, they are not applicable in all the situations. We will begin with the Liouville-Riemann type.

The Riemann-Liouville derivative has certain disadvantages when trying to model real-world phenomena with fractional differential equations [

Caputo’s derivative demands higher conditions of regularity for differentiability: to compute the fractional derivative of a function in the Caputo sense, we must first calculate its derivative. Caputo derivatives are defined only for differentiable functions while functions that have no first-order derivative might have fractional derivatives of all orders less than one in the Riemann-Liouville sense.

With the Jumarie fractional derivative, if the function is not continuous at the origin, the fractional derivative will not exist, for instance, what will be the fractional derivative of

Variational order differential operator cannot easily be handled analytically. Numerical approach is some time needs to deal with the problem under investigation.

Although Weyl fractional derivative found its place in groundwater investigation, it is still displaying a significant disadvantage; because the integral defining these Weyl derivatives is improper, greater restrictions must be placed on a function [

To illustrate the basic idea of this method, we consider a general nonlinear nonhomogeneous fractional partial differential equation with initial conditions of the following form:

Subject to the initial condition

We obtain

In the homotopy decomposition method, the basic assumption is that the solutions can be written as a power series in

and the nonlinear term can be decomposed as

The homotopy decomposition method is obtained by the graceful coupling of homotopy technique with Abel integral and is given by

Comparing the terms of same powers of

In this section, the analytical technique described in Section

The governing equation under investigation here is given as follows:

Subject to the initial conditions

Here,

Now following the description of the HDM, we arrive at the following equation:

Comparing the terms of the same power of

Integrating the above we obtain the following solutions:

In the same manner one can obtain the rest of the components. But in this case, few terms were computed and the asymptotic solution is given by

Equation (

The contact force in the elastic indentation phase may be interpreted in terms of the indentation value [

Figures

Approximate solution (

Approximate solution (

Approximate solution of the contact force in the elastic indentation phase (

Approximate solution of the contact force in the elastic indentation phase (

Surface showing the approximate solution of the governing differential equation in the elastic indentation phase equation (

Approximate solution of the contact force in the elastic indentation phase equation (

The governing equation under investigation here is given as follows.

Subject to the initial conditions

Here,

Such that (

Employing the HDM, we obtain the following integral equations:

Integrating the above we arrived at the following:

Using the package Mathematica, in the same manner one can obtain the rest of the components. But in this case, few terms were computed and the asymptotic solution is given by

The governing equation of motion of the indenter mass in the unloading phase under investigation here is given as follows:

Subject to the initial conditions

Initial conditions of this phase may be obtained from solutions of the previous stage at the time of the maximum indentation. The velocity of the indenter at the time instant that it attains its maximum indentation is zero. Therefore, time of the maximum indentation may be determined by differentiating (

For simplicity let:

Thus (

Following carefully the steps involved in the HDM we obtain the following integral equations:

Integrating the above we arrive at the following series solutions:

Using the package Mathematica, in the same manner one can obtain the rest of the components. But in this case, few terms were computed and the asymptotic solution is given by

Low velocity impact between a rigid sphere and a transversely isotropic strain-hardening plate supported by a rigid substrate was extended to the concept of noninteger derivatives. The governing equations of the elastic indentation were obtained by Yigit and Christoforou [

A brief history of the fractional derivative orders was presented. Advantages and disadvantages of each definition were presented. The new equations were solved approximately using the relatively new analytical technique, the homotopy decomposition methods. The numerical simulations showed that the approximate solutions are continuous and increase functions of the fractional derivative orders. The method used to derive approximate solution is very efficient, easier to implement, and less time consuming. The HDM is a promising method for solving nonlinear fractional partial differential equations.

The authors declare that they have no conflict interests.

A. Atangana and A. Ahmed made the first draft and N. Bıldık corrected and improved the final version. All the authors read and approved the final draft.

The authors would like to thank the referee for some valuable comments and helpful suggestions.