Study on the Yield Behavior of Closed-Cell Foams under Multiaxial Loads Based on Different Yield Definitions

Round 1

Reviewer 1 Report

See the enclosed file.

Comments for author File: Comments.pdf

The english quality should be improved.

Author Response

Response to the comments from the editors and reviewers

Response to the comments from reviewer 1
The research topic addressed in this manuscript is of significant interest to the scientific community, and it fits the scope of the journal. Overall, the manuscript is well-organized and the content of its different sections is clearly and carefully presented. However, my main concern when reading the paper is related to the usefulness of such approach in the modeling of the mechanical behavior of real structures made of closed-cell foams. To be accepted for publication in Applied Sciences, the authors should answer the following comments:

Question 1: The numerical approach used to predict the yield loci using the different criteria should be better explained (the FE code used for the simulations, the type of the FE element, the used constitutive framework)

Answer: According to your comment, the detail description is added in the manuscript (Section 2.1, 2.5).

The Representative Volume Element (RVE) approach is often used in the literature. By using this method, classical homogenization over the micro-scale domain can be used to analyze the mechanical property at the macroscopic scale. RVE is a micro-scale model of the material used to determine the corresponding homogenized/effective properties at the macroscopic scale. For foam material, the cubic RVE model contains cell wall material and pores. A random modeling procedure that builds closed-cell foam models with various cell structures was proposed by Hu et al. [32–34]. First, briefly describe the random modeling procedure by Hu et al. mentioned in section 2.1 is shown as follows:

(1) Create a 3-dimension coordinate system with the unit of millimeter.

(2) In x, y, and z coordinates, create a solid cubic with dimensions of  from 0 mm to 14 mm.

(3) Create a set of randomly generated x, y, and z coordinates for the center of the sphere, based on a uniform distribution from -1mm to 15 mm apart.

(4) Set the radius value () based on the normal distribution function(μ and σ).

(5) By some judgments, ensure all pores are in the solid cubic (). And there are no too small pores on the surface of the solid cubic.

(6) By comparing the distance between the centers of the sphere, exclude some poor pores generated, and judge whether two pores intersect, and exclude through hole generates.

(7) When two pores intersect, the walls of the hole are randomly generated.

(8) Repeat Step 3-7 thirty times to obtain thirty candidate cells.

(9) Measure the relative density of the generating foam model. If the prescribed  is achieved, stop this modeling procedure; otherwise go back to Step 3.

The mechanical properties of foam material are influenced by bulk material and relative density. However, the research on inner structure is not confined to the Voronoi model. The Voronoi model is put to wide use in simulation, and the disturbance factor can describe the randomness of the inner structure, but there is no method to measure disturbance factor for the experiment in practice. Differ from the Voronoi model, Hu uses characteristic diameter  and characteristic shape anisotropy  defined based on the two-dimensional geometric parameters to reflect the inhomogeneity in cell size distribution and irregularity in cell shape for the 3D foam materials. When the area fraction of the pores whose diameters are larger than d in the total area of pores is 0.2, this diameter is the characteristic diameter . As characteristic diameter  increases, the bigger pores in random model are more vulnerable generated. Taking the facts into account, the range of characteristic diameter  is set as (3.11-3.79), as shown in Table 1. For characteristic shape anisotropy ,  represents all pores in ramdom model are nearly spherical pore. As characteristic shape anisotropy  increases, shape of big pores is more promiscuous. Fig 1 shows that the random models with different characteristic diameter  and characteristic shape anisotropy  And the characteristic diameter  and characteristic shape anisotropy  can be measured in experiments. The characteristic diameter  characteristic shape anisotropy  are calculated cross section of random model cubic. There are six the cross sections which are perpendicular to x,y and z axis. The Model 1 and Model 2 with =0 only have sphere cells as shown in Fig. 1(a,b). It shows that there are more large pores with the increasing of characteristic diameter . The Models 3-7 with 0 have polyhedron cells as shown in Fig. 1(c-g). It shows that the large pores are more irregular as characteristic shape anisotropy  increases. And the characteristic diameter  and characteristic shape anisotropy  can be measured in experiments. The characteristic diameter  characteristic shape anisotropy  are calculated cross section of random model cubic. There are six the cross sections which are perpendicular to x,y and z axis. Meanwhile, the researchers in [35, 36] proposed that a rectangular cube with at least five cellular pores on each edge and more than 125 cellular pores overall can be utilized to investigate mechanical properties. And the simulation results corroborate the experimental findings well. Therefore, The RVE thus has dimensions of 14 mm in height, breadth, and depth, which are sufficient to meet RVE's requirements for material size.

Table 1 The characteristic diameter  and characteristic shape anisotropy  of different models.

Number	Relative density
1	0.4	3.11	0
2	0.4	3.79	0
3	0.4	3.3	0.026
4	0.4	3.29	0.06
5	0.35	3.41	0.088
6	0.3	3.68	0.101
7	0.25	3.71	0.102

Compared with the random model above, we also use the Kelvin model of RVE which regularly has a Body Centered Cubic (BCC) lattice of cells to explore the mechanical property. So we can use a few cells as RVE because of the periodic and symmetric properties of the Kelvin model. The RVE is a cuboid with a height, width, and depth of  mm, as shown in Fig. 1(h).

    

(a)                                         

  (b)

(c)     

                       

       (d)

(e)                 

                   (f)

(g)                                  

      (h)

Figure 1. The random model: (a) Model 1 in Table 1, (b) Model 2 in Table 1, (c) Model 3 in Table 1, (d) Model 4 in Table 1, (e) Model 5 in Table 1, (f) Model 6 in Table 1, (g) Model 7 in Table 1, Kelvin model : (h) The RVE of Kelvin model.

…

• Mesh convergence analysis

Quadratic element has higher calculation accuracy than linear element. And Modified mesh is more suitable for large deformation. Then, the modified quadratic tetrahedron element (C3D10 M) is used to mesh the closed-cell foam model. In order to obtain more accurate simulation results under certain conditions, so we should In order to obtain more accurate simulation results under certain conditions. Table 3 shows the number of elements in random model with different element sizes.

Table 3 The number of elements in random model with different element sizes

Number	Number of element
	Size 0.6mm		Size 0.4mm		Size 0.3mm		Size0.25mm
1 2	121478 104871	305784 259658		5857658 4575283		867593 862369

 

Table 3 shows the number of elements increases sharply with reducing the element size. In order to select the element size, Fig.2 give the stress-strain response curve of three ramdom model under uniaxial compression. It shows that the curves of random with element size 0.25mm and 0.3mm are nearly consistent. So we use 0.3mm as the element size for different random models in our study.

• Model 1
• 
• (b) Model 2

Figure 2. The stress-strain curve of random model 1 with different element sizes.

 

Because of the regular cell wall in Kelvin model, we can use hexahedral element to mesh. So C3D8I is used for the simulation of regular Kelvin model. Since the Kelvin model in Fig. 1h is simple, the number of elements is smaller, so we do not discuss mesh convergence analysis.

 

Question 2: The microscopic and macroscopic behaviors should be anisotropic due to the anisotropy of the mechanical behavior of the metal matrix and of the pores. The anisotropy aspects are neglected in this study. The authors should give some comments about this point.

Answer: According to your comment, the detail description is added in the manuscript (Section 3, Paragraph 1).

The microscopic behaviors should be anisotropic due to the anisotropy of the mechanical behavior of the metal matrix and of the pores. However, in this manuscript, we use the Representative Volume Element (RVE) approach to investigate the mechanical property of the closed-cell foam material based on macroscopic scale. RVE is a micro-scale model of the material used to determine the corresponding homogenized/effective properties at the macroscopic scale. So we analyze the mechanical property of foam without regard to the anisotropic based on microscopic scale. For the anisotropic based on macroscopic scale, we also do not analyze in this manuscript. This is because that the regular and random model is isotropic. In order to verfy, we give the stress-strain response curves along x,y and z axis of RVE cubic. Fig. 3 shows that the stress-strain response curves along x axis is almost coincided with that of along y and z axis. And in the modeling procedure, the random models ensure the coordinates of mass along x,y and z axis is nearly (7,7,7). The moment of inertia along x,y and z axis also is nearly same.

Figure 3. The stress-strain curves of Model 1 along x,y and z axis under uniaxial compression loading.  

 

Question 3: The authors should explain the potential link between their current modeling and previous frameworks studying the mechanical behavior of voided materials such as the Gurson model ([1]) or the GLD model ([2], [3]). The reviewer would like to know if it is possible to derivate a macroscopic yield function (similar to the Gurson yield function) for closed-cell foams that can to into account the pores and their growth during plastic deformation

Answer:

Gurson model [1] proposed a method for calculating approximate yield loci via an

upper bound approach for porous ductile materials, as following formula:

  （Sorry,failed to input the formula, see in "Response to the cooments from reviewers 1"）

where, due to axial symmetry

And Chen and Lu [39] also proposed a similar yield function.

 

                                                                                            

                          (8)

                                                                               

where  is the stress tensor and  and  are the material parameters to be determined from experiments. For isotropic materials,  is dependent on the mean stress , the effective stress  and third deviatoric stress invariant .

And the total strain rate can be obtained by the associated flow rule as

                                                       

where the proportionality factor dλ is determined from the consistence condition:

 

We use the method above to give the yield point, give the yield surface determined by the yield function, as shown in the following. It shows that there is large scatter of yield points. And the yield function underestimates the yield strength near .                                              

Question 4: The use and applicability of the developed approach in the modeling of real structures made of closed-cell foams is not very clear. How we can use the results of the paper in more complex or industrial applications by the finite element method for example. The authors should better highlight the applicability of their approach for real applications

Answer: According to your comment, the detail description is added in the manuscript (Section 3.5).

“…Finally, it is presented that the yield points of all random models in Table 1 are basic superposition as shown as Fig.13. It mean that the yield point defined by  get rid of related density and inner structure influences. Compared with the complicated yield surface in Combaz et al. [15-16], we can obtain simple yield function to describe the yield. And the yield definition is also used for the continuum materials. In addition, different from yield definition such as critical plastic dissipation energy or the rate of critical plastic dissipation energy and initial energy, the definition formula (3.15) can be obtained in experiments. ”

 

 

Author Response File: Author Response.docx

Reviewer 2 Report

The work regards the definition of a formulation to evaluate the plastic behavior of aluminum foams. Overall, the manuscript is clear and well-written, but some issues require revision:

·         Briefly describe the random modeling procedure by Hu et al. mentioned in paragraph 2.

·         Discuss the reasons for choosing model parameters in Table 1. Are the values of relative density, D_ch and alpha_ch widely used?

·         To be coherent with the other subfigures in Figure 1, report the CAD model also for the Kelvin cell instead of the FE model.

·         Page 5 – regarding the reference [37], it should be better to take material properties from a paper.

·         Paragraph 2.3 – The characteristics of the FE model are not described. It is necessary to report details about the modeling strategy, type of elements, mesh convergence analysis, material properties and a figure showing the models.

·         How do the geometrical nonlinearities influence the elastic properties? The initial behavior of the FE model should be linear and then becomes nonlinear; did you find a change in the behavior during the compression analysis?

·         Equation (4), the meaning of the quantity sigma_e is not explained in the text.

·         Page 10 – It is unclear why the numerical simulations and experiments' materials differ.

·         Paragraph 3 – Report the values of density and plastic dissipation energy in a Table for greater clarity.

·         Paragraph 3 – A quantitative difference between the energy values obtained with the plasticity models and the experiment should be reported

·         The references should include more papers about the aluminum foams considering their relevance in the presented study, such as: https://doi.org/10.1016/j.compositesa.2020.105923; https://doi.org/10.1016/j.euromechsol.2021.104291; https://doi.org/10.1016/j.commatsci.2013.08.021

Author Response

Response to the comments from the editors and reviewers

Response to the comments from reviewer 2

This paper discusses about the yield behavior of aluminum alloy foam simulated by different models.  The author suggested the paper to be published after major revision. 
Question 1: Briefly describe the random modeling procedure by Hu et al. mentioned in paragraph 2.

Answer: The modeling procedure is briefly shown as follows:

(1) Create a 3-dimension coordinate system with the unit of millimeter.

(2) In x, y, and z coordinates, create a solid cubic with dimensions of  from 0 mm to 14 mm.

(3) Create a set of randomly generated x, y, and z coordinates for the center of the sphere, based on a uniform distribution from -1mm to 15 mm apart.

(4) Set the radius value ( ) based on the normal distribution function(μ and σ).

(5) By some judgments, ensure all pores are in the solid cubic ( ). And there are no too small pores on the surface of the solid cubic.

(6) By comparing the distance between the centers of the sphere, exclude some poor pores generated, and judge whether two pores intersect, and exclude through hole generates.

(7) When two pores intersect, the walls of the hole are randomly generated.

(8) Repeat Step 3-7 thirty times to obtain thirty candidate cells.

(9) Measure the relative density of the generating foam model. If the prescribed  is achieved, stop this modeling procedure; otherwise go back to Step 3.

 

Question 2 :Discuss the reasons for choosing model parameters in Table 1. Are the values of relative density,  and  widely used?

Answer: According to your comment, the detail description is added in the manuscript (Section 2.1, Paragraph 1).

(in Section 2.1, Paragraph 1)

 

“…The mechanical properties of foam material are influenced by bulk material and relative density. However, the research on inner structure is not confined to the Voronoi model. The Voronoi model is put to wide use in simulation, and the disturbance factor can describe the randomness of the inner structure, but there is no method to measure disturbance factor for the experiment in practice. Differ from the Voronoi model, Hu uses characteristic diameter  and characteristic shape anisotropy  defined based on the two-dimensional geometric parameters to reflect the inhomogeneity in cell size distribution and irregularity in cell shape for the 3D foam materials. When the area fraction of the pores whose diameters are larger than d in the total area of pores is 0.2, this diameter is the characteristic diameter . As characteristic diameter  increases, the bigger pores in random model are more vulnerable generated. Taking the facts into account, the range of characteristic diameter  is set as (3.11-3.79), as shown in Table 1. For characteristic shape anisotropy ,  represents all pores in ramdom model are nearly spherical pore. As characteristic shape anisotropy  increases, shape of big pores is more promiscuous. Fig 1 shows that the random models with different characteristic diameter  and characteristic shape anisotropy  And the characteristic diameter  and characteristic shape anisotropy  can be measured in experiments. The characteristic diameter  characteristic shape anisotropy  are calculated cross section of random model cubic. There are six the cross sections which are perpendicular to x,y and z axis. The Model 1 and Model 2 with =0 only have sphere cells as shown in Fig. 1(a,b). It shows that there are more large pores with the increasing of characteristic diameter . The Models 3-7 with 0 have polyhedron cells as shown in Fig. 1(c-g). It shows that the large pores are more irregular as characteristic shape anisotropy  increases. And the characteristic diameter  and characteristic shape anisotropy  can be measured in experiments. The characteristic diameter  characteristic shape anisotropy  are calculated cross section of random model cubic. There are six the cross sections which are perpendicular to x,y and z axis.”

 

(a)          

                            (b)

(c)       

       

                   (d)

(e)                 

                    (f)

(g)                        

                 (h)

Figure 1. The random model: (a) Model 1 in Table 1, (b) Model 2 in Table 1, (c) Model 3 in Table 1, (d) Model 4 in Table 1, (e) Model 5 in Table 1, (f) Model 6 in Table 1, (g) Model 7 in Table 1, Kelvin model : (h) The RVE of Kelvin model.

Question 3 : To be coherent with the other subfigures in Figure 1, report the CAD model also for the Kelvin cell instead of the FE model.

Answer: According to your comment, the manuscript has been carefully checked. Figure 1 in the manuscript has been revised.

Question 4 :Page 5 – regarding the reference [37], it should be better to take material properties from a paper.

Answer: According to your comment, the detail description is added in the manuscript (Section 2.2, Paragraph 1).

(in Section 2.2, Paragraph 1)

We choose the aluminum alloy as the cell wall material because many researchers [2-5, 7, 10-16, 18] investigated the mechanical property of aluminum alloy via experiments in past years. So, in this work, aluminum alloy is selected as the bulk material of the foam model. As shown in Table 2, according to Su et al. [38], the constitutive relation of the Al alloy in the plastic region can be written as Formula (1),

 

 

Question 5 :Paragraph 2.3 – The characteristics of the FE model are not described. It is necessary to report details about the modeling strategy, type of elements, mesh convergence analysis, material properties and a figure showing the models.

Answer: According to your comment, detail description about the modeling strategy and the figure of the models are introduced in section 2.1.

 

the type of elements is introduced in section 2.3:

Quadratic element has higher calculation accuracy than linear element. Then, the modified quadratic tetrahedron element (C3D10 M) is used to mesh the closed-cell foam model.

mesh convergence analysis:

“

• Mesh convergence analysis

Quadratic element has higher calculation accuracy than linear element. And Modified mesh is more suitable for large deformation. Then, the modified quadratic tetrahedron element (C3D10 M) is used to mesh the closed-cell foam model. In order to obtain more accurate simulation results under certain conditions, so we should In order to obtain more accurate simulation results under certain conditions. Table 3 shows the number of elements in random model with different element sizes.

Table 3 The number of elements in random model with different element sizes

Number	Number of element
	Size 0.6mm		Size 0.4mm		Size 0.3mm		Size0.25mm
1 2	121478 104871	305784 259658		5857658 4575283		867593 862369

 

Table 3 shows the number of elements increases sharply with reducing the element size. In order to select the element size, Fig.2 give the stress-strain response curve of three ramdom model under uniaxial compression. It shows that the curves of random with element size 0.25mm and 0.3mm are nearly consistent. So we use 0.3mm as the element size for different random models in our study.

• Model 1  
• 
• (b) Model 2

Figure 2. The stress-strain curve of random model 1 with different element sizes.

Because of the regular cell wall in Kelvin model, we can use hexahedral element to mesh. So C3D8I is used for the simulation of regular Kelvin model. Since the Kelvin model in Fig. 1h is simple, the number of elements is smaller, so we do not discuss mesh convergence analysis.”

 

Question 6 :How do the geometrical nonlinearities influence the elastic properties? The initial behavior of the FE model should be linear and then becomes nonlinear; did you find a change in the behavior during the compression analysis?

Answer: According to your comment, the detail description is added in the manuscript (Section 3, Paragraph 1).

(in Section 3, Paragraph 1)

“ Fig. 3 shows that the stress-strain response curve of foam material under uniaxial state. Alougth foam is non-continuum, it also has a change from linear to becomes nonlinear. When the loading is small, the stress-strain response is linear. With the increasing of loading, the micro part of foam begin to yield that makes increase amplitude of stress decrease with increasing of strain. And when more part of foam yield, the stress-strain response curve of foam material becomes nonlinear.”

 

Question 7 :Equation (4), the meaning of the quantity sigma_e is not explained in the text.

Answer:  is Mises equivalent stress.

Question 8 :Page 10 – It is unclear why the numerical simulations and experiments' materials differ.

Answer: According to your comment, the manuscript has been carefully checked. In the manuscript, we use the bulk material as shown in Table 2.

Compared with the bulk material in Su et al.[37], the constitutive relation of the Al alloy is nearly same with the bulk material in the manuscript. And we cannot guarantee that the numerical simulations and experiments' materials are not identical. So the comparison of different aluminum alloys as cell wall materials can lead to mean little. To solve this problem, we give a comparison of dimensionless yield surfaces. To give a normalized yield surface, the uniaxial compression yield stress  is used to nondimensionalize the yield points ( , ) in the -  space. The uniaxial compression yield stress  is obtained with the method shown as point B in Fig. 3. The normalized yield points ( , ) for each model are plotted in Fig. 5-8,11.

 

Question 9 : Paragraph 3 – Report the values of density and plastic dissipation energy in a Table for greater clarity.

Answer: According to your comment, Table 5 shows that the relationship between density and plastic dissipation energy per unit volume.

Table 5 shows the plastic dissipation energy per unit volume for different models with different related densities and inner structure. It is presented that the plastic dissipation energy per unit volume at yield point is not subject solely to the influence of related density of foam material. However, inner structure also effect the plastic dissipation energy per unit volume. And the plastic dissipation energy per unit volume at yield point of the regular Kelvin model is bigger that that of random model.

Table 5 The relationship of related density and plastic dissipation energy per unit volume

Related density		plastic dissipation energy per unit volume ( )
	Kelvin	Model 1	Model 2	Model 3	Model 4	Model 5	Model 6	Model 7
0.43	60.6	57.3	48.1	39.7	31.5	21.4	20.1	15.7
0.2	22.8
0.4
0.35
0.3
0.25

 

Question 10 : Paragraph 3 – A quantitative difference between the energy values obtained with the plasticity models and the experiment should be reported

Answer: According to your comment, we analyze the relationship between the energy values obtained with the plasticity models and the experiment under uniaxial loading for different random models in Table 1.

The following Fig. 13 give the the energy values obtained with the plasticity models and the experiment. It shows that the the energy value obtained by the experiment is bigger than that obtained with the plasticity models. And with the increasing of relate density, the energy values increase. However, the energy values is also dependent on characteristic diameter  and characteristic shape anisotropy . Because of the characteristic diameter  with small difference, we only discuss the effect the characteristic shape anisotropy  on the difference energy values. Fig.14 gives the effect of the characteristic shape anisotropy . And this effect can be fitted by quadratic formula. It shows that the difference between the energy values obtained with the plasticity models and the experiment descrease as characteristic shape anisotropy  increases.

Figure 13: The energy values obtained with the plasticity models and the experiment of RVE models in Table 1 under uniaxial loading.

Figure 14: The difference value between the energy values obtained with the plasticity models and the experiment of RVE models with 0.4 relate density under uniaxial loading.

 

 

 

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

The revised version adequately addressed the comments from the reviewers.

Reviewer 2 Report

The paper has been revised according to my suggestions.

In my opinion, it can be accepted for publication on Applied Sciences.