Sun Yuan 2024a - Revision history

Waqas999 at 05:41, 14 February 2025

2025-02-14T05:41:52Z

Rimni at 14:13, 19 June 2024

2024-06-19T14:13:42Z

Rimni at 14:13, 19 June 2024

2024-06-19T14:13:10Z

Rimni at 14:12, 19 June 2024

2024-06-19T14:12:02Z

Rimni at 13:45, 19 June 2024

2024-06-19T13:45:34Z

Rimni: /* 4. Numerical Simulation of Damage and Failure of Tunnel Surrounding Rock */

2024-06-19T13:34:22Z

‎4. Numerical Simulation of Damage and Failure of Tunnel Surrounding Rock

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Rimni at 12:18, 19 June 2024

2024-06-19T12:18:33Z

Rimni at 12:17, 19 June 2024

2024-06-19T12:17:24Z

Rimni: /* 3.2 Numerical simulation of Brazilian splitting */

2024-06-19T12:15:39Z

‎3.2 Numerical simulation of Brazilian splitting

Rimni at 12:14, 19 June 2024

2024-06-19T12:14:31Z

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@@ Line 454: / Line 454: @@
 ==Acknowledgement==
-This research was supported by National Natural Science Foundation of China (grant number: 52379114).
+This research was supported by National Natural Science Foundation of China (grant number: 52379114) and Research Fund Project of Nanjing University of Aeronautics and Astronautics Jincheng College (grand number: XJ202207).
 ==References==

@@ Line 192: / Line 192: @@
 Where <math display="inline"> n </math> takes the integer part of the calculation result of Eq. (7), that is, the value range is <math display="inline">1, 2, \ldots , 20</math>. The effective elastic modulus <math display="inline">{E}^{\ast }</math> of the damage element is obtained from Eq. (4) based on the value of <math display="inline"> n </math>.
-In the linear elastic state, the expression for the strain energy density of the element is：
+In the linear elastic state, the expression for the strain energy density of the element is:
 {| class="formulaSCP" style="width: 100%; text-align: center;"

@@ Line 177: / Line 177: @@
 where <math display="inline">{\sigma }_{x}^{i}</math>, <math display="inline">{\sigma }_{y}^{i}</math>, <math display="inline">{\sigma }_{z}^{i}</math>, <math display="inline">{\tau }_{xy}^{i}</math>, <math display="inline">{\tau }_{xz}^{i}</math>, <math display="inline">{\tau }_{yz}^{i}</math> is the stress of the element at step <math display="inline"> i </math>, and <math display="inline">{\sigma }_{x}^{i-1}</math>, <math display="inline">{\sigma }_{y}^{i-1}</math>, <math display="inline">{\sigma }_{z}^{i-1}</math>, <math display="inline">{\tau }_{xy}^{i-1}</math>, <math display="inline">{\tau }_{xz}^{i-1}</math>, <math display="inline">{\tau }_{yz}^{i-1}</math> is the stress at the previous step. The strain is expressed in the same manner. If <math display="inline">i=0</math>, then the strain energy density of the element is <math display="inline">(dW/dV)=</math><math>0</math>.
-When the strain energy density of an element <math display="inline">(dW/dV)>(dW/dV{)}_{u}</math>, the element begins to damage, and the degree of damage <math display="inline"> n </math> is calculated by the following equation:
+When the strain energy density of an element <math display="inline">(dW/dV)>(dW/dV{)}_{u}</math>, the element begins to damage, and the degree of damage <math display="inline"> n </math> is calculated as follows:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 243: / Line 243: @@
 | style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 3'''. Calculation flow of numerical simulation
 |}
 ===3.2 Numerical simulation of Brazilian splitting===

@@ Line 141: / Line 141: @@
 where <math display="inline"> D </math> is the damage variable, <math display="inline"> E </math> is the elastic modulus of the non-damaged material, and <math display="inline"> {E}^{\ast }(n) </math>  is the elastic modulus after damage.
-Generally, it can be seen from the above equation (5) that the value range of the damage variable <math display="inline"> D </math> is 0-1. Since the damage variable <math display="inline"> D </math> has a monotonically increasing relationship with the value <math display="inline"> n </math> of the elastic modulus reduction, <math display="inline"> n </math> (<math display="inline">0\leq n\leq 20</math>) is used to represent the damage degree of the element in the following in order to represent the damage state of the element in a more concise and intuitive way.
+Generally, it can be seen from Eq. (5) that the value range of the damage variable <math display="inline"> D </math> is 0-1. Since the damage variable <math display="inline"> D </math> has a monotonically increasing relationship with the value <math display="inline"> n </math> of the elastic modulus reduction, <math display="inline"> n </math> (<math display="inline">0\leq n\leq 20</math>) is used to represent the damage degree of the element in the following in order to represent the damage state of the element in a more concise and intuitive way.
 In summary, the criterion of rock damage and failure based on the strain-energy-density theory is as follows:

@@ Line 458: / Line 458: @@
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+[14] Liu X.S., Ning J.G., Tan Y.L., Gu Q.H. Damage constitutive model based on energy dissipation for intact rock subjected to cyclic loading. International Journal of Rock Mechanics and Mining Sciences, 85:27-32. 2016.
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← Older revision		Revision as of 12:18, 19 June 2024
Line 269:		Line 269:


−	The Mohr-Coulomb constitutive model in FLAC3D software is used for numerical calculation. The physical and mechanical characteristics of the model are as follow: the modulus of elasticity <math display="inline">E=3</math>GPa, the Poisson's ratio <math display="inline">v=0.37</math>, the cohesion <math display="inline">c=12</math>MPa, the friction angle <math display="inline">\varphi =50\mbox{°}</math>, the density <math display="inline">\rho =1800\mbox{kg/{m}}^{3}</math>, the ultimate stress <math display="inline">{\sigma }_{u}=2</math>MPa, the ultimate strain <math display="inline">{\epsilon }_{u}=1.2\times 10^{-4}</math>, and the strain in case of fracture <math display="inline">{\epsilon }_{f}=1.2\times 10^{-3}</math>.	+	The Mohr-Coulomb constitutive model in FLAC3D software is used for numerical calculation. The physical and mechanical characteristics of the model are as follow: the modulus of elasticity <math display="inline">E=3</math>GPa, the Poisson's ratio <math display="inline">v=0.37</math>, the cohesion <math display="inline">c=12</math>MPa, the friction angle <math display="inline">\varphi =50\mbox{°}</math>, the density <math display="inline">\rho =1800\mbox{kg/m}^{3}</math>, the ultimate stress <math display="inline">{\sigma }_{u}=2</math>MPa, the ultimate strain <math display="inline">{\epsilon }_{u}=1.2\times 10^{-4}</math>, and the strain in case of fracture <math display="inline">{\epsilon }_{f}=1.2\times 10^{-3}</math>.

	The failure process of Brazilian splitting numerical simulation is shown in [[#img-6\|Figure 6]]. Under the influence of the load, the two ends of the disc began to fail due to stress concentration, then the failure range continued to extend inside the specimen. Ultimately, hole-through fracture occurred along the axis of the disc.		The failure process of Brazilian splitting numerical simulation is shown in [[#img-6\|Figure 6]]. Under the influence of the load, the two ends of the disc began to fail due to stress concentration, then the failure range continued to extend inside the specimen. Ultimately, hole-through fracture occurred along the axis of the disc.

@@ Line 263: / Line 263: @@
 {| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
 |-style="background:white;"
-|style="text-align: center;padding:10px;"| [[File:Draft_Sun_139268125-image5.png|centre|236x236px]]
+|style="text-align: center;padding:10px;"| [[File:Draft_Sun_139268125-image5.png|centre|266x266px]]
 |-
 | style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 5'''. Numerical model
@@ Line 269: / Line 269: @@
-The Mohr-Coulomb constitutive model in FLAC3D software is used for numerical calculation. The physical and mechanical characteristics of the model are as follow: the modulus of elasticity <math display="inline">E=3</math>GPa, the Poisson's ratio <math display="inline">v=0.37</math>, the cohesion <math display="inline">c=12</math>MPa, the friction angle <math display="inline">\varphi =50\mbox{°}</math>, the density <math display="inline">\rho =1800\mbox{kg/{m}}^{3}</math>, the ultimate stress <math display="inline">{\sigma }_{u}=2</math>MPa, the ultimate strain <math display="inline">{\epsilon }_{u}=1.2\times 1{0}^{-4}</math>, and the strain in case of fracture <math display="inline">{\epsilon }_{f}=1.2\times 1{0}^{-3}</math>.
+The Mohr-Coulomb constitutive model in FLAC3D software is used for numerical calculation. The physical and mechanical characteristics of the model are as follow: the modulus of elasticity <math display="inline">E=3</math>GPa, the Poisson's ratio <math display="inline">v=0.37</math>, the cohesion <math display="inline">c=12</math>MPa, the friction angle <math display="inline">\varphi =50\mbox{°}</math>, the density <math display="inline">\rho =1800\mbox{kg/{m}}^{3}</math>, the ultimate stress <math display="inline">{\sigma }_{u}=2</math>MPa, the ultimate strain <math display="inline">{\epsilon }_{u}=1.2\times 10^{-4}</math>, and the strain in case of fracture <math display="inline">{\epsilon }_{f}=1.2\times 10^{-3}</math>.
 The failure process of Brazilian splitting numerical simulation is shown in [[#img-6|Figure 6]]. Under the influence of the load, the two ends of the disc began to fail due to stress concentration, then the failure range continued to extend inside the specimen. Ultimately, hole-through fracture occurred along the axis of the disc.