Thermoelastic Diffusion Multicomponent Half-Space under the Effect of Surface and Bulk Unsteady Perturbations

Round 1

Reviewer 1 Report

1-mention the paper that derived the equations number 1

2-what is the possible value for N?

3-comparison with numerical method is required to improve your work.

Author Response

The authors thank the Reviewer for careful acquaintance with the article. About the made remarks we report the following:

1. Links added

2. The “N” can be random natural number. In the calculation examples we take N=2.

3. Of course, the authors are aware of the use of finite difference methods, finite elements, etc. to solve such problems. However, it should be noted that the numerical methods, for all its undoubted effectiveness, have a number of disadvantages. For example, they tend to accumulate errors and are limited for forecasting. It is necessary for their verification to have some reference solution, which reliability is absolute. Most often, for such problems solution Laplace and Fourier integral transformations based on numerical and analytical methods are used. In this case, the Durbin method is mainly used to reverse the Laplace transform. It allows to express the Mellin integral through the Fourier transform. The comparison of the results obtained by the authors is made with these works. References to relevant works were in the article initial version under the numbers [8,9,10]. Also, in the second computational example, a sentence was added indicating the coincide of the proposed algorithm with an analytical solution for a purely elastic problem.

Reviewer 2 Report

1.    Authors must specify a reference for equations (2).

2.    The affirmation after Eq. (7) is not motivated.

3.    Details about the solution (20) for Eq. (19) are needed.

4.    What is the motivation for the funtions  f_i and F_i,  before the Section 9?

5.    A very great number of notions and results are "borrowed" from different already published paper. As such, I think the authors need to emphasize more clearly the contribution of the manuscript from a scientific point of view.

6.    Some editing "glitches" need to be corrected.

7.    Punctuations are used randomly. Insert comma or full stop after each and every equation accordingly.

8.    References are not uniformly written. In some references the name of the  journal  is written in full and in others it is abbreviated.

9.    I think, the authors  must strengthen the References section with some articles that use the same techniques,  to make the techniques used more plausible, for instance: The effect of a dipolar structure on the Holder stability in Green-Naghdi thermoelasticity,  Contin Mech Thermodyn, 29(6), 1365-1374, 2017; Cesaro means in thermoelasticity of dipolar bodies,  Acta Mech, 122(1-4), 155-168, 1997

Author Response

The authors thank the Reviewer for careful acquaintance with the article materials. About the made remarks we report the following:

1. Link added

2. The solution of the problem is proposed to search in the form of Green’s functions. Due to superposition the problem is divided into the separately construction of surface and bulk Green’s functions. Reference added.

3. The solution of systems (15) and (21) is described through determinants in more detail. Polynomials $ P_{ik} $  for a two-component half-space are also written.

4. The title of section 8 is a typo. Its true name is “The calculation example for the pulse-periodic processes”. The functions of surface and bulk perturbations were chosen empirically, based on sources [21,22]. Links to sources added in the text.

5. Of course, the authors use "standard" terminology, which has been established for several decades. However, it should be noted that the problem considered in this article has not been solved by anyone before. The well-known scientific sources do not consider the formulation and methods of analytical solution for mechanical diffusion problems taking into account cross-diffusion effects. This fact determines the scientific novelty of the work. It is stated in the introduction (line 54-63). Additional information about scientific novelty is also added in the introduction and conclusion.

6. Typos corrected. English language corrected.

7. Punctuation corrected. The final decision on this question is left to the journal editors.

8. References corrected and supplemented

9. The theme of these works lies somewhat away from the presented thermoelastic diffusion problem. In the “Thermalodynamic Thermoelasticity, Contin Mech Thermodyn, 29 (6), 1365-1374, 2017” and “Cesaro means in thermoelasticity of the dipolar bodies, Acta Mech, 122 (1-4), 155-168, 1997” it is a question of thermoelasticity problems. At the same time, in the article presented by us, multicomponent diffusion processes are also considered. However, the article “Thermalodynamic Thermoelasticity, Contin Mech Thermodyn, 29 (6), 1365-1374, 2017” is interesting because of the thermal perturbation finite speed study in it. For this reason, we consider it expedient to include it in the list of publications. Reference added.

Thank You very much for accurate remarks!