Negative Stiffness, Incompressibility, and Strain Localisation in Particulate Materials

Round 1

Reviewer 1 Report

In the paper two mechanisms of strain localizations in particulate material under compression are considered. The strain localization phenomena continue to be one of the most interesting features of homogeneous particulate material behavior under stresses and attract attentions of researchers. In the article, perfect introduction contains the necessary references. It was shown in the reviewed paper, that when not all the bonds between particles of the material are broken, a kind of rotating particle clusters are formed, and the apparent negative stiffness is induced. Another mechanism of strain localization considered in the paper is the emergence of a state of incremental incompressibility due to dilatancy. Both suggested mechanisms were studied numerically.

The paper is very interesting and contains valuable results.

I have some miner remarks:

1. There are several errors of the type “reference source not found”, at least in the manuscript version that I got (lines 76, 77, 117, 149, 211), I marked them by yellow.
2. It is not enough clear from the text, why do authors apply doble couple shear forces (see Figure 1), to avoid rotation?
3. Line 96 “thesample” instead of “the sample”.
4. Why do you use such a strange number of particles – 3786 in numerical calculations?
5. Lines 177 and 205 – I suppose it should be “2D” and “3D” instead of “D”

I recommend the paper for publishing.

Comments for author File: Comments.pdf

Author Response

I have some miner remarks:

1. There are several errors of the type “reference source not found”, at least in the manuscript version that I got (lines 76, 77, 117, 149, 211), I marked them by yellow.
2. It is not enough clear from the text, why do authors apply doble couple shear forces (see Figure 1), to avoid rotation?
3. Line 96 “thesample” instead of “the sample”.
4. Why do you use such a strange number of particles – 3786 in numerical calculations?
5. Lines 177 and 205 – I suppose it should be “2D” and “3D” instead of “D”

Response

Thank you for noticing the errors, items 1, 3 and 5. They are rectified.

Item 2. We added explanation to the caption to Figure 1. It is now reads: “Partially detached cluster of particles within matrix. Two pairs of forces are shown in order to emphasise that moment equilibrium should hold.”

Item 4. The 2D sample was generated by randomly placing particles with radii between D_min and D_max until the target porosity was reached. Therefore there is no control over how many particles will be generated to form the virtual sample. We added the following explanation to the para before Table 1; it now reads: “In order to illustrate the existence of rotating clusters a 2D discrete element simulation of uniaxial compression test on a sample of particulate material was conducted. The virtual sample was created by randomly placing particles with radii between D_min and D_max=D_rD_min, where D_r is the maximum to minimum disc diameter ratio, until the target porosity was reached. This was performed in 4 steps: (1) the initial un-bonded compact virtual sample was built within 4 frictionless walls; (2) floaters that is the particles with no contact with other particles were deleted; (3) The particle bonds where installed leading to the particles being “glued” together; (4) the initial frictionless walls were removed and the sample was relaxed by performing the modelling so that the average displacement becomes less than 10^-6 of the average disc radius. This procedure produced a sample containing 3786 particles. The built-in linear parallel bond model is used as the contact bond model. This bond model represents bonds between particles that are able to break one-by-one eventually separating particle clusters and leaving them free to rotate. It is assumed that the bond breakage takes place only when the force acting in it reaches the tensile strength of the bond. Parameters of the virtual sample have been are shown in Tables 1 and 2.”

Reviewer 2 Report

Manuscript concerns DEM modelling of uniaxial compression test of assembly of spherical particles connected by BPM contact model. Authors considered rotations of clusters during axial loading as a source of dilatation and the negative stiffness. Manuscript introduce some new information. It could be recommended for publication after some revision following particular comments:

1. Lack of detailed information how the sample of material was prepared. When the BPM was introduced?

2. Lack of sensitivity analyses to material parameters.

3. It is not clearly indicated what kind of uniaxial compression test was modeled: confined or unconfined. If it was unconfined uniaxial compression test the observed decrease of loading force vs. axial strain could be simply considered as exhaustion of the material strength. The stress-strain plot would be more informative than the stress-time plot.

4. Plenty of minor editorial errors, for example: Table 1. Heading. Line 159 – change of the letter font. Line 177. “4.1. D simulations” ??? Line 205. „4.2. D simulations” ??? References: Lack of spaces: Line 342. “DevicesJ.” Line 374. „characteristicsPhys.” Abbreviations of journal names should be unified.

Author Response

We are grateful to the reviewer for their comments.

 

Manuscript concerns DEM modelling of uniaxial compression test of assembly of spherical particles connected by BPM contact model. Authors considered rotations of clusters during axial loading as a source of dilatation and the negative stiffness. Manuscript introduce some new information. It could be recommended for publication after some revision following particular comments:

1. Lack of detailed information how the sample of material was prepared. When the BPM was introduced?

Response

It is clarified at the end of the para before Table 1. It reads: “The built-in linear parallel bond model is used as the contact bond model. This bond model represents bonds between particles that are able to break one-by-one eventually separating particle clusters and leaving them free to rotate. It is assumed that the bond breakage takes place only when the force acting in it reaches the tensile strength of the bond. Parameters of the virtual sample have been are shown in Tables 1 and 2.”

 

2. Lack of sensitivity analyses to material parameters.

Response

The model parameters were chosen in such a way that the sample macro properties (Young’s modulus, Poisson’s ratio) match typical values the rock properties taken from the literature. However the aim of the modelling was only to illustrate how the incremental incompressibility can produce shear strain localisation.

 

3. It is not clearly indicated what kind of uniaxial compression test was modeled: confined or unconfined. If it was unconfined uniaxial compression test the observed decrease of loading force vs. axial strain could be simply considered as exhaustion of the material strength. The stress-strain plot would be more informative than the stress-time plot.

Response

It is unconfined compressive test. The clarification was added to the first para of Section 4.1. Yes, it is exhaustion of the material strength, but it can lead to splitting or shear failure or spalling. We point out at dilatancy-induced incremental incompressibility as a mechanism of shear failure. We added clarification in the first and third para of Conclusions.

4. Plenty of minor editorial errors, for example: Table 1. Heading. Line 159 – change of the letter font. Line 177. “4.1. D simulations” ??? Line 205. „4.2. D simulations” ??? References: Lack of spaces: Line 342. “DevicesJ.” Line 374. „characteristicsPhys.” Abbreviations of journal names should be unified.

Response

Thank you. These were corrected.

Round 2

Reviewer 2 Report

Authors improved the manuscript following particular remarks. Therefore, I can recommend it for publication.