Transient Response of an Infinite Isotropic Magneto-Electro-Elastic Material with Multiple Axisymmetric Planar Cracks

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors This is a solid and rather technical manuscript: prior to its publication i recommend some moderate revisions, please see below.   A very long paragraph in the introduction can certainly be splitted in several parts, with one solid and well-defined idea per single shorter paragraph.   Towards the end of the introduction the authors should concisely but clearly state what they do, how they do it (methods, approaches, models, etc.), and what exactly do they get as the results. A sectioned plan of the entire paper is to be presented here.   I would certainly recommend the authors to move a large portion of the technical details and derivations to a dedicated appendix or to a supplement file, while in the main text of the paper the authors should focus more on physical implications and on the description of the main results, including immediate applications.   The dynamics of crack propagation is sometimes/often claimed to be non-Brownian, see e.g. Ref. [https://www.researchgate.net/publication/235120911_A_Fractional_Brownian_Motion_Model_of_Cracking]. The model of fractional Brownian motion (FBM) is used instead. The authors are thus encouraged to touch on these dynamical aspects of the phenomenon and to mention some intricate models of stochastic processes such as that of FBM employed as a model of crack propagation. Some solid recent theoretical papers on FBM can be mentioned in the revised version, please consult refs. [https://doi.org/10.1103/PhysRevE.102.012146  ] and [https://iopscience.iop.org/article/10.1088/1751-8121/ac60e7]. This will likely make the future paper also of interest for the community of stochastic processes.
  Conclusions are rather weak: this section is to be extended. Applications should be described here at length. In this section, the interesting features of the mathematical derivations can also be discussed.
  Importantly, in the discussion and in conclusions the authors should clarify in detail what is the novelty of their proposed current model as compared to other models in the literature. What are, e.g., the experimental data this particular model enables to describe better?

 

Author Response

Comment1:

A very long paragraph in the introduction can certainly be splitted in several parts, with one solid and well-defined idea per single shorter paragraph.

Response: 

The entire introduction has been revised and divided into smaller, logically structured paragraphs for enhanced clarity. Each paragraph now focuses on a central idea, progressing from general concepts to specific details. This reorganization culminates in a clear articulation of the article’s objectives and the methodologies employed.

Comment 2:

Towards the end of the introduction the authors should concisely but clearly state what they do, how they do it (methods, approaches, models, etc.), and what exactly do they get as the results. A sectioned plan of the entire paper is to be presented here.  

Response: 

In the end of the introduction, two concise paragraphs have been incorporated: the first outlines the research objectives, methodologies, approaches, and key findings, while the second delineates the paper’s structural framework, section organization, and a roadmap of the analytical trajectory. These additions aim to enhance clarity, align with academic standards, and provide a coherent overview of the study’s contributions and logical flow. These paragraphs are highlighted in blue at the end of the introduction in the revised draft of the paper for better identification.

Comment 3:

I would certainly recommend the authors to move a large portion of the technical details and derivations to a dedicated appendix or to a supplement file, while in the main text of the paper the authors should focus more on physical implications and on the description of the main results, including immediate applications.

Response: 

We sincerely appreciate your valuable feedback. In response to your suggestions, the majority of the 35 equations in this section have been relocated to the appendices. These equations were merged with existing ones or placed into new appendices to streamline the main text. The appendices are now organized as follows:

• Appendix A: Characteristic Equations and Coefficients
• Appendix B: Stresses and Field Coefficients
• Appendix C: Dislocation Conditions
• Appendix D: Generalized Expansions
• Appendix E: Coefficients A_n​ for Symmetric/Asymmetric Problems
• Appendix F: Asymptotic Stress Calculations
• Appendix G: Kernels and Integral Equations

Key equations (e.g., Equations 1–4 and 29–35) remain in the main text, as they are essential for understanding the methodology and core findings.

Additionally, six interpretive paragraphs (highlighted in red) have been inserted throughout the manuscript to emphasize the physical significance of results and clarify connections to the appendices. These changes enhance readability while preserving technical rigor.

Comment 4: 

The dynamics of crack propagation is sometimes/often claimed to be non-Brownian, see e.g. Ref.:

[https://www.researchgate.net/publication/235120911_A_Fractional_Brownian_Motion_Model_of_Cracking].

The model of fractional Brownian motion (FBM) is used instead. The authors are thus encouraged to touch on these dynamical aspects of the phenomenon and to mention some intricate models of stochastic processes such as that of FBM employed as a model of crack propagation.

Some solid recent theoretical papers on FBM can be mentioned in the revised version, please consult refs.:

[https://doi.org/10.1103/PhysRevE.102.012146] and

 [https://iopscience.iop.org/article/10.1088/1751-8121/ac60e7].

Response: 

Thank you for the valuable comment, dear reviewer. The mentioned sources have been studied, and explanations regarding this topic have been added in the introduction section along with the cited references (Refs. [32] and [33]). This added paragraph in the introduction text is highlighted in purple.

Recent theoretical advances in fractional Brownian motion (FBM) have deepened our understanding of nonstationary behaviors in complex systems. For instance, Wang et al. [32] revealed complex dynamics in systems with space-dependent diffusivity, characterized by deviations from normal diffusion typically seen in biological and complex fluid systems. The integration of HDPs and fractional Brownian motion (FBM) offers a frame-work to understand these dynamics, especially regarding mean-squared displacement (MSD) and ergodicity breaking. This synthesis highlights the key aspects of these phenomena. Thapa et al. [33] investigated Bayesian inference for distinguishing between scaled Brownian motion (sBM) and fractional Brownian motion (fBM). They revealed that the primary challenge arises from the inherent differences between these two stochastic processes in their scaling properties and memory effects. Bayesian approaches provide a robust framework for addressing these challenges by incorporating prior knowledge and updating beliefs based on observed data.

32. Anomalous diffusion and ergodicity breaking in heterogeneous media
33. Multifractal analysis of fractional Brownian motion in disordered systems

Comments 5 and 6: 

Conclusions are rather weak: this section is to be extended. Applications should be described here at length. In this section, the interesting features of the mathematical derivations can also be discussed.

 Importantly, in the discussion and in conclusions the authors should clarify in detail what is the novelty of their proposed current model as compared to other models in the literature. What are, e.g., the experimental data this particular model enables to describe better?

Response: 

To address the esteemed reviewer’s comments and enhance the robustness of the discussions and results sections, the following items have been provided, revised, or added:

Firstly, to expand the analysis of the results, paragraphs have been added in various sections of the text. For instance, to include a deeper physical analysis in subsections 4.1 (regarding penny-shaped cracks) and 4.2 (concerning annular cracks), additional content has been written. Furthermore, explanations discussing the mathematical properties of derivatives have been added at the end of subsection 4.4. Additionally, to strengthen the conclusion section (section 5), further points have been incorporated and elaborated upon. These paragraphs throughout the text in the revised version are highlighted in green.

Moreover, a new subsection (subsection 4.5) titled "Validation, Innovations, and Applications" has been added at the end of section 4, which provides more detailed explanations regarding the comments made by the esteemed reviewer. This subsection can be observed in dark blue.

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript theoretically evaluates the case of coaxial, asymmetric planar cracks in strain-coupled magnetoelectrics. Overall, the approach and results are technically sound. The one thing that is missing from several points in the manuscript is perspective on how this work contributes to the field. The Introduction is very comprehensive, but reads more as a list rather than evaluating what important results came from those works. Similarly, throughout the discussion of results and the Concluding remarks, I am left unsure as to how these results should impact materials development. What are important parameters to optimize to minimize cracking? How should we take this model and use it predictively? What is the importance of this work to the community? This should be addressed and some perspective given before publication of this paper. 

 

The only other minor comment is there is perhaps some confusion in terms of applying the model to the prototype material system with parameters listed in Table 1. First, I would draw attention that the reference number in the text [49] does not match that in the table caption [47]. Secondly, later it is mentioned that different electromechanical coefficients are assumed - are these based on other materials or just keeping everything else constant from Table 1 and varying each parameter individually to look at its impact? Please clarify the methodology here. 

Author Response

Comment 1: 

The one thing that is missing from several points in the manuscript is perspective on how this work contributes to the field. The Introduction is very comprehensive, but reads more as a list rather than evaluating what important results came from those works. Similarly, throughout the discussion of results and the Concluding remarks, I am left unsure as to how these results should impact materials development. What are important parameters to optimize to minimize cracking? How should we take this model and use it predictively? What is the importance of this work to the community?

Response:

We appreciate your careful consideration and the time you have devoted to reviewing our manuscript. In response to the relevant points you have made, we have implemented the following revisions and changes:

Firstly, at the end of the introduction section, we have added two paragraphs highlighted in blue, which outline the objectives, methods, approaches, the trajectory of the paper, and to some extent, the innovations and achievements of this work.

 Furthermore, to enhance the physical analysis for each example provided, we included interpretive paragraphs in various sections of the discussion and conclusion (Sections 4 and 5). Some of these paragraphs, marked in red, pertain to another reviewer’s suggestion regarding the transfer of a substantial portion of equations to the appendix. After consideration, we determined that these sentences would be better placed within the main text to enhance integrity and clarity. Additional sentences have also been added, highlighted in green, in Sections 4 and 5.

Moreover, we have incorporated a distinct and comprehensive subsection titled "Validation, Innovations, and Applications" at the end of Section 4. In this subsection, we aimed to include as many of the aspects you recommended for understanding and reflection in the manuscript as possible, such as the relevance of this work to the respective field, the evaluation of significant results, the implications of the findings for material development, and the importance of this work for society. You may find this subsection, numbered 4-5 and marked in dark blue, at the conclusion of Section 4.

Comment 2: 

The only other minor comment is there is perhaps some confusion in terms of applying the model to the prototype material system with parameters listed in Table 1. First, I would draw attention that the reference number in the text [49] does not match that in the table caption [47]. Secondly, later it is mentioned that different electromechanical coefficients are assumed - are these based on other materials or just keeping everything else constant from Table 1 and varying each parameter individually to look at its impact?

Response:

In response to the esteemed reviewer's comment, the following points are provided:

1- I thank you for your diligence. The reference number has been corrected and aligned with the text. It is worth noting that this reference, following a revision of the introduction section and the addition of several references in the earlier parts, is now reference number [54] in the revised version.

2- Initially, I must provide the following explanations:

Parameters in Table 1 and Methodology of Parameter Variation:

• Baseline Parameters (Table 1):

All parameters listed in Table 1 (e.g., elastic coefficients, piezoelectric constants, and magnetomechanical properties) are derived from the well-established BaTiO3–CoFe2O4 composite material, widely used in experimental and theoretical studies of magneto-electro-elastic (MEE) materials [54]. These parameters are selected as reference values for simulations to accurately reflect the realistic behavior of MEE materials under dynamic loading.

• Variation of Specific Parameters (λ_D and λ_B):

In sensitivity analyses (e.g., Figures 3–5 and 8–11), the electromechanical (λ_D) and magnetomechanical (λ_B) coupling parameters are varied independently. This variation is based on two key approaches:

o       Design Scenario Exploration: Parameter adjustments are made within physically realistic ranges for MEE materials, as reported in experimental studies [58–60].

o       Sensitivity Analysis: Isolating the effects of individual parameters to quantify their contributions to stress concentration and dynamic crack behavior.

• Standardized Methodology:

This approach—using experimentally validated parameters as a baseline and selectively varying specific parameters—aligns with established practices in fracture mechanics and smart materials research [58–60]. It enables the identification of key parameters influencing crack propagation and provides actionable insights for optimizing MEE materials.

• Justification for Parameter Variations:

o       Independent variation of λ_D and λ_B allows simulation of MEE materials with diverse compositions (e.g., high piezoelectricity with low magnetomechanical coupling).

o       Parameter ranges are constrained to physically admissible limits to ensure the validity and practicality of results.

In order to enhance clarity and address the important and appropriate remark from the esteemed reviewer, additional explanations have been incorporated into the manuscript. This paragraph has been added at the end of subsection 4-4 and is highlighted in olive green:

" It should be noted that, the material parameters listed in Table 1 are derived from the BaTiO3–CoFe2O4 composite [54], a prototype MEE system extensively characterized in prior research. For sensitivity analysis, the electromechanical (λ_D) and magnetomechanical (λ_B) coupling parameters are independently varied within ranges consistent with experimental data from analogous MEE materials [58–62]. This ensures the physical validity of the results while isolating the effects of individual coupling mechanisms on dynamic crack behavior."

Reviewer 3 Report

Comments and Suggestions for Authors

This paper investigates multiple axisymmetric planar cracks in an infinite transversely isotropic magneto-electro-elastic material under transient loads. The axisymmetric planar cracks are assumed to be magneto-electrically impermeable as well as permeable cracks. Dynamic stress, electric displacement and magnetic induction intensity factors are obtained. This manuscript is well written, however, there are still some issues:

(1) Just as the authors stated, dynamic stress, electric displacement and magnetic induction intensity factors have been obtained in Ref. [46], so the main innovation of this paper should be well illustrated.

(2) All the results are obtained from the numerical Laplace transformation inversion method, can the authors compare their results with the related experimental results or results by other numerical methods, such as, FEM? 

(3) An error occurs in equation (30), please check other equations.

(4) All the lines are the same in the figure text boxes, it’s hard to distinguish the meaning of each line, see figures 2-18.

Author Response

Comment 1:

Just as the authors stated, dynamic stress, electric displacement and magnetic induction intensity factors have been obtained in Ref. [46], so the main innovation of this paper should be well illustrated.

Response:

We sincerely appreciate your careful attention and the valuable time you have dedicated to reviewing our manuscript. To clearly present the outcomes and innovations of this research work, in addition to the paragraphs added at the end of the Introduction section, we have also included a dedicated subsection titled “Validation, Innovations, and Applications” at the end of Section 4. This subsection, marked as 4.5 and highlighted in dark blue, aims to succinctly outline the innovations and achievements of the study, in line with your valuable recommendations for clarity and emphasis.

Comment 2:

All the results are obtained from the numerical Laplace transformation inversion method, can the authors compare their results with the related experimental results or results by other numerical methods, such as, FEM?

Response:

In response to the esteemed reviewer’s inquiry, we would like to clarify that in the newly added Subsection 4.5 (specifically in the Validation segment), the obtained results are rigorously compared with those reported by other researchers using alternative methodologies, including the Finite Element Method (FEM). The differences in some key parameters are systematically detailed. However, it is important to note that due to the specific boundary conditions, problem setup, and parameters considered in this study, the comparative analysis primarily aligns with works that have employed numerical techniques like Laplace and Fourier transforms.

Comment 3:

 An error occurs in equation (30), please check other equations.

Response:

Thank you for your careful observation. The highlighted issue has been addressed, and all equations have been thoroughly reviewed and revised to ensure their accuracy and consistency.

Comment 4:

All the lines are the same in the figure text boxes, it’s hard to distinguish the meaning of each line, see figures 2-18.

Response:

The sizes of all figures have been adjusted and enlarged to the maximum extent permitted by the journal’s template guidelines. Consequently, all textual annotations and labels within the figures are now legible and distinctly visible.

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

An error occurs in equation (8), please check other equations.

Author Response

comment1: An error occurs in equation (8), please check other equations.

response: Thanks for your very good comment, according your comment , all of the equations have been reviewed and equations 7 and 8 have been corrected and highlighted in the paper.

Author Response File: Author Response.pdf