Largest Lyapunov Exponent Parameter of Stiffened Carbon Fiber Reinforced Epoxy Composite Laminated Plate Due to Critical Buckling Load Using Average Logarithmic Divergence Approach
Round 1
Reviewer 1 Report
The findings of this paper is to suppress the nonlinear dynamics phenomenon of carbon fiber epoxy composite laminated plate using one and two stiffeners. Largest Lyapunov exponent parameter is calculated for the bending deflection due to critical buckling load at different aspects ratios and different fiber volume fractions. It can be recommended for publication if the following concerns can be addressed:
- Clearer pictures are needed in the paper, like figures 7-15
- Please, check the formula (3) and explain the symbols.
Author Response
Response to Reviewer 1 Comments
Point 1:
Clearer pictures are needed in the paper, like figures 7-15
Response 1:
Figures (7) to (15) is treated with Adobe Photoshop CS6 to increase the resolution.
Point 2:
Please, check the formula (3) and explain the symbols.
Response 2:
The symbols in equation (3) is identified as in below:
, : Normal stress of the stiffeners in the (x) and (y) directions.
, , : Shear stresses of the stiffeners in (xy), (yz), and (xz) planes.
, : Strains of the stiffeners along (x) and (y) directions.
, : Modulus of elasticity of the stiffeners along (x) and (y) directions.
and are the shear modulus of elasticity of stiffeners along (x and y) directions.
, : Shear strain of the stiffeners along (x) and (y) directions.
Note: The corrections is highlighted in blue in the new version of the manuscript.
Author Response File: 
Author Response.docx
Reviewer 2 Report
- Rewritten the abstract section with the following parts,
 Purpose
 Design/methodology/approach
 Practical implications
 Research limitations/implications
 Findings
 Originality/value
2. The article required insert literature review for previous work (publication from 2020 to 2022), then, given conclusion remark and make a comparison between presenting and previous work.
3. Figure 3 is not clear
4. How can select the buckling load by buckling machine in the experimental work??
4. Insert more details for the Lyapunov technique and show how can solve the analytical equation (Eqs. 11 to 11) by this theory??
5. The solution for the analytical equation presented in the article (Eq. 7 to 11) does not clear?? there, required to show the solution for its equations.
6. Rewriting the conclusion section with points form and give the important points that were calculated
Author Response
Response to Reviewer 2 Comments
Point 1:
Rewritten the abstract section with the following parts,
Ø Purpose
Ø Design/methodology/approach
Ø Practical implications
Ø Research limitations/implications
Ø Findings
Ø Originality/value
Response 1:
The abstract has been paraphrased.
Point 2:
The article required insert literature review for previous work (publication from 2020 to 2022), then, given conclusion remark and make a comparison between presenting and previous work.
Response 2:
The new references between the years 2020 and 2022 have added to the introduction section.
Point3:
Figure 3 is not clear
Response 3:
I update figure (3).
Point 4:
How can select the buckling load by buckling machine in the experimental work?
Response 4:
Southwell plot is a straight line between the bending deflection and in-plane compression mechanical load in which the slope reflects the critical buckling load.
Point 5:
Insert more details for the Lyapunov technique and show how can solve the analytical equation (Eqs. 11 to 11) by this theory?
Response 5
State space of Eigen value problem is used to calculate the bending deflection analytically against time using higher order shear deformation theory in the presence of boundary conditions with the aid of MatLab software. The set of data of the bending deflection against time is used in the algorithm code of average logarithmic divergence to extract the value of largest Lyapunov exponent parameter.
Point 6:
The solution for the analytical equation presented in the article (Eq. 7 to 11) does not clear?? there, required to show the solution for its equations.
Response 6:
The solution of the analytic solution is added as shown in subsections 4.1 and 4.2 for unstiffened and plate with stiffener.
Point 7:
Rewriting the conclusion section with points form and give the important points that were calculated
Response 7:
The conclusion is rewriting again.
Note: The corrections is highlighted in red in the new version of the manuscript.
Author Response File: Author Response.docx
Reviewer 3 Report
The article “Largest Lyapunov Exponent Parameter of Stiffened Carbon Fiber Reinforced Epoxy Composite Laminated Plate Due to Critical Buckling Load Using Average Logarithmic Divergence Approach” submitted for publication in the journal Mathematics deals with the calculation of the Largest Lyapunov exponent parameter for the bending deflection due to critical buckling load at different aspects ratios and different fiber volume fractions.
The main result of the present article is to suppress the nonlinear dynamics phenomenon of carbon fiber epoxy composite laminated plate using one and two stiffeners. The largest Lyapunov exponent has declined with the increase of thickness ratio in which the nonlinear dynamics of carbon fiber reinforced epoxy composite laminated plate is decreased with the increase of thickness ratio. The value of the largest Lyapunov exponent is decreased with the increase of fiber volume fractions for the plates with one and two stiffeners while the value of the largest Lyapunov exponent is decreased with the increase in the number of stiffeners. The critical buckling load is increased with the increasing number of clamped edges while the critical buckling load is decreased with the increasing number of free edges. The value of critical buckling load for anti-symmetric cross-ply is higher than the value of critical buckling load for anti-symmetric angle ply.
The reviewer r believes that the paper is relevant for the journal "Mathematics” and suggests extending this study to more complex composite materials and using dynamic homogenization.
Author Response
Response to Reviewer 3 Comments
Point 1:
The reviewer believes that the paper is relevant for the journal "Mathematics” and suggests extending this study to more complex composite materials and using dynamic homogenization.
Response 1:
Figure (16) is added to show the behaviour of dynamics characteristics of Lyapunov exponent parameter for another complex composite materials such as E-glass fiber reinforced with polyester resin.
Note: The corrections is highlighted in green in the new version of the manuscript.
Author Response File: Author Response.docx
