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Published in ''Comput. Methods Appl. Mech. Engrg.'', Vol 193, 3087-3128, 2004<br />2
doi: 10.1016/j.cma.2003.12.0563
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==Abstract==6
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The paper presents combination of Discrete Element Method (DEM) and Finite Element Method (FEM) for dynamic analysis of geomechanics problems. Combined models can employ spherical (or cylindrical in 2D) rigid elements and finite elements in the discretization of different parts of the system. The DEM is a suitable tool to model soil/rock materials while the FEM in many cases can be a better choice to model other parts of the system considered. A typical example can be an idealization of rock cutting with a tool discretized with finite elements and rock or soil samples modelled with discrete elements. The FEM presented in the paper allows large elasto-plastic deformations in the solid regions. Such problems require the use of stabilized FEM to deal with the incompressibility constraint in order to eliminate the volumetric locking defect, especially when using triangular and tetrahedra elements with equal order interpolation for the displacement and the pressure variables. In the paper a stabilization based on the Finite Calculus (FIC) approach is used. Both theoretical algorithms of DEM and stabilized FEM are implemented in an explicit dynamic code. The paper presents some details of both formulations. A combined numerical algorithm is described finally. Selected numerical results illustrate the possibilities and performance of discrete/finite element analysis in geomechanics problems.8
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==1 Introduction==10
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The discrete element method (DEM) is nowadays acknowledged to be an effective procedure for analysis of granular and rock materials under static and dynamic loads. Typically in the DEM the soil/rock masses are modelled by a collection of spheres interacting with each other. The normal and tangential forces between spheres govern the motion of the discrete continuum. Modelling of discontinuities by the DEM is quite simple as they are produced by loss of adhesive contact between the spheres. Examples of application of the DEM to geomechanics and similar problems can be found in <span id='citeF-1'></span><span id='citeF-2'></span>[[#cite-1|[1,2]]].12
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The coupling of the finite element method with the DEM is an effective approach for problems where standard continuum finite elements are adequate to model a part of the analysis domain, whereas the DEM is more adequate to treat other areas. Examples of such problems can be found in the interaction of solids and structures with granular media. The use of simple triangular and tetrahedra elements to model the “continuum” domain is very attractive due to the simplicity of their geometry in order to treat frictional contact conditions. Standard linear triangles and tetrahedra can suffer however from serious drawbacks in the presence of material nonlinearities inducing a near-incompressible behaviour as the numerical solution locks in these cases. The volumetric locking defect can be eliminated by using adequate mixed formulations with different interpolations for the displacement and pressure variables <span id='citeF-3'></span>[[#cite-3|[3]]]. The use of an equal order interpolation for these variables is very attractive from the computational point of view and this is possible if an adequate stabilized formulation is chosen.14
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In this paper a stabilization method based on the finite calculus (FIC) approach <span id='citeF-4'></span><span id='citeF-5'></span>[[#cite-4|[4,5]]] is used for deriving locking-free linear triangles and tetrahedra which can be effectively used in conjunction with the DEM.16
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One of the motivations of the present research is the simulation of the process of rock cutting, estimating at the same time the wear of the cutting tool. The main physical phenomenon considered is the interaction of the tool with a rock leading to failure of the rock and tool wear. The latter process is usually slower, although in some cases visible changes of the tool shape can be appreciated after a few minutes. The tool is assumed to be rigid in our numerical model. Two kinds of tool discretization have been used. The first one employs finite elements to represent the tool. In the second type of discretization the tool is also discretised with discrete elements. This allows to modify the shape of the tool easily by removing “worn” particles.18
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The formulation of the discrete element method has been extended for thermal and thermo-mechanical coupled problems in order to take into account temperature as one of the important factors influencing the tool wear process. Heat is generated during cutting due to friction dissipation. In the model friction between the tool and rock particles as well as friction between rock particles themselves is considered. Heat conduction through the tool and rock is analysed and the temperature distribution is obtained. Temperature of the tool surface lowers its hardness and increases wear.20
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The content of the paper is the following. In the next sections the main features of the DEM used are described. Details of different models derived to treat the frictional contact conditions are given.22
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In the second part of the paper the formulation of volumetric locking-free linear triangles and tetrahedra is presented using a FIC approach. Next an algorithm for dynamic analysis of geomechanics problems combining DEM and locking-free FEM is described. Examples of the applications of coupled DEM-FEM formulation to a number of geomechanics problems are presented.24
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==2 DISCRETE ELEMENT METHOD FORMULATION==26
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The numerical model of rock/soil media adopted in the present study is based on the spherical discrete element method (DEM) which is widely recognized as a suitable tool to model geomaterials <span id='citeF-6'></span><span id='citeF-7'></span><span id='citeF-8'></span><span id='citeF-9'></span>[[#cite-6|[6,7,8,9]]]. Formulation of spherical discrete elements (called also distinct elements) following the main assumptions of Cundall <span id='citeF-6'></span><span id='citeF-10'></span>[[#cite-6|[6,10]]] has been developed by Rojek et al. in <span id='citeF-11'></span>[[#cite-11|[11]]] and implemented in an explicit dynamic formulation. Within the DEM, it is assumed that the solid material can be represented as a collection of rigid particles (spheres or balls in 3D and discs in 2D) interacting among themselves in the normal and tangential directions. Material deformation is assumed to be concentrated at the contact points. Appropriate contact laws allow us to obtain desired macroscopic material properties. The contact law used for rock modelling takes into account cohesive bonds between rigid particles. The contact law can be seen as the formulation of the material model at the microscopic level. Cohesive bonds can be broken, thus allowing to simulate fracture of material and its propagation.28
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===2.1 Equations of motion===30
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The translational and rotational motion of rigid spherical or cylindrical elements (particles) (Fig. [[#img-1|1]]) is described by means of the standard equations of rigid body dynamics. For the <math display="inline">i</math>-th element we have32
33
<span id="eq-1"></span>34
{| class="formulaSCP" style="width: 100%; text-align: left;" 35
|-36
| 37
{| style="text-align: left; margin:auto;width: 100%;" 38
|-39
| style="text-align: center;" | <math>m_i \ddot{\textbf{u}}_i= {\mathbf{F}}_i\,, </math>40
|}41
| style="width: 5px;text-align: right;white-space: nowrap;" | (1)42
|}43
44
<span id="eq-2"></span>45
{| class="formulaSCP" style="width: 100%; text-align: left;" 46
|-47
| 48
{| style="text-align: left; margin:auto;width: 100%;" 49
|-50
| style="text-align: center;" | <math>I_i \dot{{\omega }}_i= {\mathbf{T}}_i\,, </math>51
|}52
| style="width: 5px;text-align: right;white-space: nowrap;" | (2)53
|}54
55
where <math>{u}</math>is the element centroid displacement in a fixed (inertial) coordinate frame <math>\mathbf{X}</math>, <math>\omega </math>– the angular velocity, <math display="inline">m</math> – the element mass, <math display="inline">I</math> – the moment of inertia, <math>\mathbf{F}</math>– the resultant force, and <math>\mathbf{T}</math>– the resultant moment about the central axes. Vectors <math>\mathbf{F}</math> and <math>\mathbf{T}</math> are sums of all forces and moments applied to the <math display="inline">i</math>-th element due to external load, contact interactions with neighbouring spheres and other obstacles, as well as forces resulting from damping in the system. The form of the rotational equation ([[#eq-2|2]]) is valid for spheres and cylinders (in 2D) and is simplified with respect to a general form for an arbitrary rigid body with the rotational inertial properties represented by a second order tensor. In the general case it is more convenient to describe the rotational motion with respect to a co-rotational frame <math>\mathbf{x}</math> which is embedded at each element since in this frame the tensor of inertia is constant. The tensor of inertia for a sphere or cylinder (in 2D) does not change in the fixed global coordinate system <math>\mathbf{X}</math>, and hence the rotational motion can be easily described in this system.56
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<div id='img-1'></div>58
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"59
|-60
|[[Image:Draft_Samper_102318540-motion.png|396px|Motion of a rigid particle]]61
|- style="text-align: center; font-size: 75%;"62
| colspan="1" | '''Figure 1:''' Motion of a rigid particle63
|}64
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Equations of motion ([[#eq-1|1]]) and ([[#eq-2|2]]) are integrated in time using a central difference scheme. The time integration operator for the translational motion at the <math display="inline">n</math>-th time step is as follows:66
67
<span id="eq-3"></span>68
{| class="formulaSCP" style="width: 100%; text-align: left;" 69
|-70
| 71
{| style="text-align: left; margin:auto;width: 100%;" 72
|-73
| style="text-align: center;" | <math>\ddot{{\textbf u}}_i^{n}=\frac{{{\textbf F}}_i^n}{m_i}\,, </math>74
|}75
| style="width: 5px;text-align: right;white-space: nowrap;" | (3)76
|}77
78
<span id="eq-4"></span>79
{| class="formulaSCP" style="width: 100%; text-align: left;" 80
|-81
| 82
{| style="text-align: left; margin:auto;width: 100%;" 83
|-84
| style="text-align: center;" | <math>\dot{{\textbf u}}_i^{n+1/2}=\dot{{\textbf u}}_i^{n-1/2}+\ddot{{\textbf u}}_i^{n}{\Delta t}\,, </math>85
|}86
| style="width: 5px;text-align: right;white-space: nowrap;" | (4)87
|}88
89
<span id="eq-5"></span>90
{| class="formulaSCP" style="width: 100%; text-align: left;" 91
|-92
| 93
{| style="text-align: left; margin:auto;width: 100%;" 94
|-95
| style="text-align: center;" | <math>{{\textbf u}}_i^{n+1}={{\textbf u}}_i^{n}+\dot{{\textbf u}}_i^{n+1/2}{\Delta t}\,. </math>96
|}97
| style="width: 5px;text-align: right;white-space: nowrap;" | (5)98
|}99
100
The first two steps in the integration scheme for the rotational motion are identical to those given by Eqs.([[#eq-3|3]]) and ([[#eq-4|4]]):101
102
<span id="eq-6"></span>103
{| class="formulaSCP" style="width: 100%; text-align: left;" 104
|-105
| 106
{| style="text-align: left; margin:auto;width: 100%;" 107
|-108
| style="text-align: center;" | <math>\dot{{\omega }}_i^{n}=\frac{{{T}}_i^n}{I_i}\,, </math>109
|}110
| style="width: 5px;text-align: right;white-space: nowrap;" | (6)111
|}112
113
<span id="eq-7"></span>114
{| class="formulaSCP" style="width: 100%; text-align: left;" 115
|-116
| 117
{| style="text-align: left; margin:auto;width: 100%;" 118
|-119
| style="text-align: center;" | <math>{{\omega }}_i^{n+1/2}={{\omega }}_i^{n-1/2}+\dot{{\omega }}_i^{n}{\Delta t}\,. </math>120
|}121
| style="width: 5px;text-align: right;white-space: nowrap;" | (7)122
|}123
124
The vector of incremental rotation <math display="inline">{\Delta \theta }=\{ \Delta \theta _x\,\Delta \theta _y\,\Delta \theta _z\} ^{\mathrm{T}}</math> is calculated as125
126
<span id="eq-8"></span>127
{| class="formulaSCP" style="width: 100%; text-align: left;" 128
|-129
| 130
{| style="text-align: left; margin:auto;width: 100%;" 131
|-132
| style="text-align: center;" | <math>{{\Delta \theta }}_i={{\omega }}_i^{n+1/2}{\Delta t}\,, </math>133
|}134
| style="width: 5px;text-align: right;white-space: nowrap;" | (8)135
|}136
137
Knowledge of the incremental rotation suffices to update the tangential contact forces. It is also possible to track the rotational position of particles, if necessary <span id='citeF-12'></span>[[#cite-12|[12]]]. Then the rotation matrices between the moving frames embedded in the particles and the fixed global frame must be updated incrementally using an adequate multiplicative scheme <span id='citeF-11'></span>[[#cite-11|[11]]].138
139
===2.2 Contact search algorithm===140
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Changing contact pairs of elements during the analysis process must be automatically detected. The simple approach to identify interaction pairs by checking every sphere against every other sphere would be very inefficient, as the computational time is proportional to <math display="inline">n^2</math>, where <math display="inline">n</math> is the number of elements. In our formulation the search is based on quad-tree and oct-tree structures. In this case the computation time of the contact search is proportional to <math display="inline">n\ln{n}</math>, which allows to solve large frictional contact systems.142
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===2.3 Evaluation of contact forces===144
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====2.3.1 <span id='lb-2.3.1'></span>Decomposition of the contact force====146
147
Once contact between a pair of elements has been detected, the forces occurring at the contact point are calculated. The interaction between the two interacting bodies can be represented by the contact forces <math display="inline">{\textbf F}_1</math> and <math display="inline">{\textbf F}_2</math>, which by the Newton's third law satisfy the following relation:148
149
{| class="formulaSCP" style="width: 100%; text-align: left;" 150
|-151
| 152
{| style="text-align: left; margin:auto;width: 100%;" 153
|-154
| style="text-align: center;" | <math>{\textbf F}_{1}=-{\textbf F}_2\,. </math>155
|}156
| style="width: 5px;text-align: right;white-space: nowrap;" | (9)157
|}158
159
We take <math display="inline">{\textbf F}={\textbf F}_1</math> and decompose <math>{\textbf F}</math>into the normal and tangential components, <math display="inline">{F}_n</math>and <math display="inline">{\textbf F}_T</math>, respectively (Fig. [[#img-2|2]])160
161
{| class="formulaSCP" style="width: 100%; text-align: left;" 162
|-163
| 164
{| style="text-align: left; margin:auto;width: 100%;" 165
|-166
| style="text-align: center;" | <math> {\textbf F}={\textbf F}_n+{\textbf F}_T =F_n{\textbf n}+{\textbf F}_T \,, </math>167
|}168
|}169
170
where <math>{\textbf n}</math>is the unit vector normal to the particle surface at the contact point (this implies that it lies along the line connecting the centers of the two particles) and directed outwards from the particle 1.171
172
The contact forces <math display="inline">F_n</math> and <math display="inline">{\textbf F}_T</math> are obtained using a constitutive model formulated for the contact between two rigid spheres (or discs in 2D) (Fig. [[#img-3|3]]). The contact interface in our formulation is characterized by the normal and tangential stiffness <math display="inline">k_n</math> and <math display="inline">k_T</math>, respectively, the Coulomb friction coefficient <math display="inline">\mu </math>, and the contact damping coefficient <math display="inline">c_n</math>.173
174
<div id='img-2'></div>175
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"176
|-177
|[[Image:Draft_Samper_102318540-towspheres.png|246px|Decomposition of the contact force into the normal and tangential components]]178
|- style="text-align: center; font-size: 75%;"179
| colspan="1" | '''Figure 2:''' Decomposition of the contact force into the normal and tangential components180
|}181
182
<div id='img-3'></div>183
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"184
|-185
|[[Image:Draft_Samper_102318540-contact-point.png|384px|Model of the contact interface]]186
|- style="text-align: center; font-size: 75%;"187
| colspan="1" | '''Figure 3:''' Model of the contact interface188
|}189
190
====2.3.2 <span id='lb-2.3.2'></span>Normal contact force====191
192
The normal contact force <math display="inline">F_n</math> is decomposed into the elastic part <math display="inline">F_{ne}</math> and the damping contact force <math display="inline">F_{nd}</math>193
194
<span id="eq-10"></span>195
{| class="formulaSCP" style="width: 100%; text-align: left;" 196
|-197
| 198
{| style="text-align: left; margin:auto;width: 100%;" 199
|-200
| style="text-align: center;" | <math>F_n = F_{ne} + F_{nd}\,. </math>201
|}202
| style="width: 5px;text-align: right;white-space: nowrap;" | (10)203
|}204
205
The damping part is used to decrease oscillations of the contact forces and to dissipate kinetic energy.206
207
The elastic part of the normal contact force <math display="inline">F_{ne}</math> is proportional to the normal stiffness <math display="inline">k_n</math> and to the penetration of the two particle surfaces <math display="inline">u_{rn}</math>, i.e.208
209
<span id="eq-11"></span>210
{| class="formulaSCP" style="width: 100%; text-align: left;" 211
|-212
| 213
{| style="text-align: left; margin:auto;width: 100%;" 214
|-215
| style="text-align: center;" | <math>F_{ne}=k_n u_{rn}\,. </math>216
|}217
| style="width: 5px;text-align: right;white-space: nowrap;" | (11)218
|}219
220
The penetration is calculated as221
222
{| class="formulaSCP" style="width: 100%; text-align: left;" 223
|-224
| 225
{| style="text-align: left; margin:auto;width: 100%;" 226
|-227
| style="text-align: center;" | <math>u_{rn} = d - r_1 - r_2\,, </math>228
|}229
| style="width: 5px;text-align: right;white-space: nowrap;" | (12)230
|}231
232
where <math display="inline">d</math> is the distance between the particle centres, and <math display="inline">r_1</math>, <math display="inline">r_2</math> their radiae. In the present study no cohesion is allowed, so no tensile normal contact forces are allowed and hence233
234
<span id="eq-13"></span>235
{| class="formulaSCP" style="width: 100%; text-align: left;" 236
|-237
| 238
{| style="text-align: left; margin:auto;width: 100%;" 239
|-240
| style="text-align: center;" | <math>F_{ne}\le 0\,. </math>241
|}242
| style="width: 5px;text-align: right;white-space: nowrap;" | (13)243
|}244
245
If <math display="inline">u_{rn}<0</math>, Eq. ([[#eq-11|11]]) holds, otherwise <math display="inline">F_{ne}= 0</math>.246
247
The contact damping force is assumed to be of viscous type and given by248
249
{| class="formulaSCP" style="width: 100%; text-align: left;" 250
|-251
| 252
{| style="text-align: left; margin:auto;width: 100%;" 253
|-254
| style="text-align: center;" | <math>F_{nd}=c_n v_{rn} </math>255
|}256
| style="width: 5px;text-align: right;white-space: nowrap;" | (14)257
|}258
259
where <math display="inline">v_{rn}</math> is the normal relative velocity of the centres of the two particles in contact defined by260
261
<span id="eq-15"></span>262
{| class="formulaSCP" style="width: 100%; text-align: left;" 263
|-264
| 265
{| style="text-align: left; margin:auto;width: 100%;" 266
|-267
| style="text-align: center;" | <math>{v}_{rn}=(\dot{{\textbf u}}_2-\dot{{\textbf u}}_1)\cdot {\textbf n}\,. </math>268
|}269
| style="width: 5px;text-align: right;white-space: nowrap;" | (15)270
|}271
272
The damping coefficient <math display="inline">c_n</math> can be taken as a fraction of the critical damping <math display="inline">C_{cr}</math> for the system of two rigid bodies with masses <math display="inline">m_1</math> and <math display="inline">m_2</math>, connected with a spring of stiffness <math display="inline">k_n</math> (<span id='citeF-13'></span>[[#cite-13|[13]]]) with273
274
<span id="eq-16"></span>275
{| class="formulaSCP" style="width: 100%; text-align: left;" 276
|-277
| 278
{| style="text-align: left; margin:auto;width: 100%;" 279
|-280
| style="text-align: center;" | <math>C_{cr}=2\sqrt{\frac{m_1m_2k_n}{m_1+m_2} }\,. </math>281
|}282
| style="width: 5px;text-align: right;white-space: nowrap;" | (16)283
|}284
285
====2.3.3 <span id='lb-2.3.3'></span>Tangential frictional contact====286
287
In the absence of cohesion (if the particles were not bonded at all or the initial cohesive bond has been broken) the tangential reaction <math display="inline">{\textbf F}_T</math>appears by friction opposing the relative motion at the contact point. The relative tangential velocity at the contact point <math display="inline">{\textbf v}_{rT}</math> is calculated from the following relationship288
289
<span id="eq-17"></span>290
{| class="formulaSCP" style="width: 100%; text-align: left;" 291
|-292
| 293
{| style="text-align: left; margin:auto;width: 100%;" 294
|-295
| style="text-align: center;" | <math>{\textbf v}_{rT}= {\textbf v}_{r}-{\textbf v}_{r}\cdot {\textbf n}\,, </math>296
|}297
| style="width: 5px;text-align: right;white-space: nowrap;" | (17)298
|}299
with300
{| class="formulaSCP" style="width: 100%; text-align: left;" 301
|-302
| 303
{| style="text-align: left; margin:auto;width: 100%;" 304
|-305
| style="text-align: center;" | <math>{\textbf v}_r=(\dot{{\textbf u}}_{2} +{\omega }_{2} \times {\textbf r}_{c2})- (\dot{{\textbf u}}_{1} +{\omega }_1 \times {\textbf r}_{c1})\,, </math>306
|}307
| style="width: 5px;text-align: right;white-space: nowrap;" | (18)308
|}309
310
where <math display="inline">\dot{{\textbf u}}_{1}</math>, <math display="inline">\dot{{\textbf u}}_{2}</math>, and <math display="inline">{\omega }_1</math>, <math display="inline">{\omega }_2</math> are the translational and rotational velocities of the particles, and <math display="inline">{\textbf r}_{c1}</math> and <math display="inline">{\textbf r}_{c2}</math> are the vectors connecting particle centres with contact points.311
312
<div id='img-4'></div>313
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"314
|-315
|[[Image:Draft_Samper_102318540-coulomb.png|462px|]]316
|[[Image:Draft_Samper_102318540-coulombreg.png|462px|Friction force vs. relative tangential displacement a) Coulomb law, b) regularized Coulomb law]]317
|- style="text-align: center; font-size: 75%;"318
| colspan="2" | '''Figure 4:''' Friction force vs. relative tangential displacement a) Coulomb law, b) regularized Coulomb law319
|}320
321
The relationship between the friction force <math display="inline">\| {\textbf F}_T\| </math>and the relative tangential displacement <math display="inline">u_{rT}</math> for the classical Coulomb model (for a constant normal force <math display="inline">F_n</math>) is shown in Fig. [[#img-4|4]]a. This relationship would produce non physical oscillations of the friction force in the numerical solution due to possible changes of the direction of sliding velocity. To prevent this, the Coulomb friction model must be regularized. The regularization procedure chosen involves decomposition of the tangential relative velocity into reversible and irreversible parts, <math display="inline">{\textbf v}_{rT}^{\mathrm{r}}</math> and <math display="inline">{\textbf v}_{rT}^{\mathrm{ir}}</math>, respectively as:322
323
{| class="formulaSCP" style="width: 100%; text-align: left;" 324
|-325
| 326
{| style="text-align: left; margin:auto;width: 100%;" 327
|-328
| style="text-align: center;" | <math>{\textbf v}_{rT}= {\textbf v}_{rT}^{\mathrm{r}}+{\textbf v}_{r}^{\mathrm{ir}}\,. </math>329
|}330
| style="width: 5px;text-align: right;white-space: nowrap;" | (19)331
|}332
333
This is equivalent to formulating the frictional contact as a problem analogous to that of elastoplasticity, which can be seen clearly from the friction force-tangential displacement relationship in Fig. [[#img-4|4]]b. This analogy allows us to calculate the friction force employing the standard radial return algorithm []. First a trial state is calculated334
335
<span id="eq-20"></span>336
{| class="formulaSCP" style="width: 100%; text-align: left;" 337
|-338
| 339
{| style="text-align: left; margin:auto;width: 100%;" 340
|-341
| style="text-align: center;" | <math>{\textbf F}_T^{\mathrm{trial}}={\textbf F}_T^{\mathrm{old}}-k_T {\textbf v}_{rT} \Delta t\,, </math>342
|}343
| style="width: 5px;text-align: right;white-space: nowrap;" | (20)344
|}345
346
and then the slip condition is checked347
348
{| class="formulaSCP" style="width: 100%; text-align: left;" 349
|-350
| 351
{| style="text-align: left; margin:auto;width: 100%;" 352
|-353
| style="text-align: center;" | <math>\phi ^{\mathrm{trial}}=\| {\textbf F}_T^{\mathrm{trial}}\|{-\mu}|F_n|\,. </math>354
|}355
| style="width: 5px;text-align: right;white-space: nowrap;" | (21)356
|}357
358
If <math display="inline">\phi ^{\mathrm{trial}}\le{0}</math>, we have stick contact and the friction force is assigned the trial value359
360
{| class="formulaSCP" style="width: 100%; text-align: left;" 361
|-362
| 363
{| style="text-align: left; margin:auto;width: 100%;" 364
|-365
| style="text-align: center;" | <math>{\textbf F}_T^{\mathrm{new}}={\textbf F}_T^{\mathrm{trial}}\,, </math>366
|}367
| style="width: 5px;text-align: right;white-space: nowrap;" | (22)368
|}369
370
otherwise (slip contact) a return mapping is performed giving371
372
{| class="formulaSCP" style="width: 100%; text-align: left;" 373
|-374
| 375
{| style="text-align: left; margin:auto;width: 100%;" 376
|-377
| style="text-align: center;" | <math>{\textbf F}_T^{\mathrm{new}}=\mu |F_n|\frac{{\textbf F}_T^{\mathrm{trial}}}{\| {\textbf F}_T^{\mathrm{trial}} \| }\,. </math>378
|}379
| style="width: 5px;text-align: right;white-space: nowrap;" | (23)380
|}381
382
===2.4 Background damping===383
384
A quasi-static state of equilibrium for the assembly of particles can be achieved by application of adequate damping. The contact damping previously described is a function of the relative velocity of the contacting bodies. It is sometimes necessary to apply damping for non-contacting particles to dissipate their energy. We have considered two types of such damping of viscous and non-viscous type, referred here as background damping. In both cases damping terms <math display="inline">{{F}}_i^{\mathrm{damp}}</math> and <math display="inline">{\mathbf{T}}_{i}^{\mathrm{damp}}</math> are added to equations of motion ([[#eq-1|1]]) and ([[#eq-2|2]])385
386
<span id="eq-24"></span>387
{| class="formulaSCP" style="width: 100%; text-align: left;" 388
|-389
| 390
{| style="text-align: left; margin:auto;width: 100%;" 391
|-392
| style="text-align: center;" | <math>m_i \ddot{\mathbf{u}}_i= {\mathbf{F}}_i+{\mathbf{F}}_i^{\mathrm{damp}}\,, </math>393
|}394
| style="width: 5px;text-align: right;white-space: nowrap;" | (24)395
|}396
397
<span id="eq-25"></span>398
{| class="formulaSCP" style="width: 100%; text-align: left;" 399
|-400
| 401
{| style="text-align: left; margin:auto;width: 100%;" 402
|-403
| style="text-align: center;" | <math>I_i \dot{{\omega }}_i= {\mathbf{T}}_i+{\mathbf{T}}_i^{\mathrm{damp}}\,. </math>404
|}405
| style="width: 5px;text-align: right;white-space: nowrap;" | (25)406
|}407
408
where409
410
* for viscous damping411
412
<span id="eq-26"></span>413
{| class="formulaSCP" style="width: 100%; text-align: left;" 414
|-415
| 416
{| style="text-align: left; margin:auto;width: 100%;" 417
|-418
| style="text-align: center;" | <math>419
420
{\mathbf{F}}_{i}^{\mathrm{damp}}=-\alpha ^{vt}m_{i}\dot{\mathbf{ u}}_{i}\,, </math>421
|}422
| style="width: 5px;text-align: right;white-space: nowrap;" | (26)423
|}424
425
<span id="eq-27"></span>426
{| class="formulaSCP" style="width: 100%; text-align: left;" 427
|-428
| 429
{| style="text-align: left; margin:auto;width: 100%;" 430
|-431
| style="text-align: center;" | <math>432
433
{\mathbf{T}}_{i}^{\mathrm{damp}}=-\alpha ^{vr}I_{i}{\omega }_{i}\,, </math>434
|}435
| style="width: 5px;text-align: right;white-space: nowrap;" | (27)436
|}437
438
* for non-viscous damping439
440
<span id="eq-28"></span>441
{| class="formulaSCP" style="width: 100%; text-align: left;" 442
|-443
| 444
{| style="text-align: left; margin:auto;width: 100%;" 445
|-446
| style="text-align: center;" | <math>447
448
{\mathbf{F}}_{i}^{\mathrm{damp}}=-\alpha ^{nvt}\Vert {\mathbf{F}}_{i}\Vert \frac{\dot{\mathbf{u}}_{i}}{\Vert \dot{\mathbf{u}}_{i}\Vert }\,, </math>449
|}450
| style="width: 5px;text-align: right;white-space: nowrap;" | (28)451
|}452
453
<span id="eq-29"></span>454
{| class="formulaSCP" style="width: 100%; text-align: left;" 455
|-456
| 457
{| style="text-align: left; margin:auto;width: 100%;" 458
|-459
| style="text-align: center;" | <math>460
461
{\mathbf{T}}_{i}^{\mathrm{damp}}=-\alpha ^{nvr}\Vert {\mathbf{T}}_{i}\Vert \frac{{\omega }_{i}}{\Vert {\omega }_{i}\Vert }\,. </math>462
|}463
| style="width: 5px;text-align: right;white-space: nowrap;" | (29)464
|}465
466
where <math display="inline">\alpha ^{vt}</math>, <math display="inline">\alpha ^{vr}</math>, <math display="inline">\alpha ^{nvt}</math> and <math display="inline">\alpha ^{nvr}</math> are damping constants. It can be seen from Eqs. ([[#eq-26|26]])–([[#eq-29|29]]) that both the non-viscous and viscous damping terms are opposite to the velocity and the difference lays in the evaluation of the damping force. Viscous damping is proportional to the magnitude of velocity, while non-viscous damping is proportional to the magnitude of the resultant force and the bending moment.467
468
===2.5 Numerical stability===469
470
Explicit integration in time yields high computational efficiency and it enables the solution of large models. The known disadvantage of the explicit integration scheme is its conditional numerical stability imposing the limitation on the time step <math display="inline">{\Delta t}</math>, i.e.471
472
{| class="formulaSCP" style="width: 100%; text-align: left;" 473
|-474
| 475
{| style="text-align: left; margin:auto;width: 100%;" 476
|-477
| style="text-align: center;" | <math>{\Delta t} \le {\Delta t}_{cr} </math>478
|}479
| style="width: 5px;text-align: right;white-space: nowrap;" | (30)480
|}481
482
where <math>{\Delta t}_{cr}</math> is a critical time step determined by the highest natural frequency of the system <math>\omega _{ max}</math>483
484
<span id="eq-31"></span>485
{| class="formulaSCP" style="width: 100%; text-align: left;" 486
|-487
| 488
{| style="text-align: left; margin:auto;width: 100%;" 489
|-490
| style="text-align: center;" | <math>{\Delta t}_{cr}=\frac{2}{\omega _{max}}\,. </math>491
|}492
| style="width: 5px;text-align: right;white-space: nowrap;" | (31)493
|}494
495
If damping exists, the critical time increment is given by496
497
<span id="eq-32"></span>498
{| class="formulaSCP" style="width: 100%; text-align: left;" 499
|-500
| 501
{| style="text-align: left; margin:auto;width: 100%;" 502
|-503
| style="text-align: center;" | <math>{\Delta t}_{cr}=\frac{2}{\omega _{max}}\left(\sqrt{1+\xi ^2}-\xi \right)\,, </math>504
|}505
| style="width: 5px;text-align: right;white-space: nowrap;" | (32)506
|}507
508
where <math display="inline">\xi </math> is the fraction of the critical damping corresponding to the highest frequency <math>\omega _{max}</math>. Exact calculation of the highest frequency <math>\omega _{max}</math>would require solution of the eigenvalue problem defined for the whole system of connected rigid particles. In an approximate solution procedure, an eigenvalue problem can be defined separately for every rigid particle using the linearized equations of motion509
510
<span id="eq-33"></span>511
{| class="formulaSCP" style="width: 100%; text-align: left;" 512
|-513
| 514
{| style="text-align: left; margin:auto;width: 100%;" 515
|-516
| style="text-align: center;" | <math>{\mathbf{m}}_i \ddot{\mathbf{r}}_i+ {\mathbf{k}}_i {\mathbf{r}}_i= \mathbf{0}\,, </math>517
|}518
| style="width: 5px;text-align: right;white-space: nowrap;" | (33)519
|}520
521
where522
523
<span id="eq-34"></span>524
{| class="formulaSCP" style="width: 100%; text-align: left;" 525
|-526
| 527
{| style="text-align: left; margin:auto;width: 100%;" 528
|-529
| style="text-align: center;" | <math>{\mathbf{m}}_i = \{ m_i\; m_i\; m_i\; I_i\; I_i\; I_i\} ^{\mathrm{T}}\;,\quad {\mathbf{r}}_i = \{ (u_x)_i\; (u_y)_i\; (u_z)_i\; (\theta _x)_i\; (\theta _y)_i\; (\theta _z)_i\} ^{\mathrm{T}}\,, </math>530
|}531
| style="width: 5px;text-align: right;white-space: nowrap;" | (34)532
|}533
534
and <math display="inline">{\mathbf{k}}_i</math> is the stiffness matrix accounting for the contributions from all the penalty constraints active for the <math display="inline">i</math>-th particle. Equation ([[#eq-34|34]]) defines vectors <math display="inline">{\mathbf{m}}_i</math> and <math display="inline">{\mathbf{r}}_i</math> for a spherical particle in a 3D space. For a cylindrical particle in a 2D model the respective vectors are defined as535
536
<span id="eq-35"></span>537
{| class="formulaSCP" style="width: 100%; text-align: left;" 538
|-539
| 540
{| style="text-align: left; margin:auto;width: 100%;" 541
|-542
| style="text-align: center;" | <math>{\mathbf{m}}_i = \{ m_i\; m_i\; I_i\} ^{\mathrm{T}}\;,\quad {\mathbf{r}}_i = \{ (u_x)_i\; (u_y)_i\;(\theta _z)_i\} ^{\mathrm{T}}\,. </math>543
|}544
| style="width: 5px;text-align: right;white-space: nowrap;" | (35)545
|}546
547
Equation ([[#eq-33|33]]) leads to the eigenproblem548
549
<span id="eq-36"></span>550
{| class="formulaSCP" style="width: 100%; text-align: left;" 551
|-552
| 553
{| style="text-align: left; margin:auto;width: 100%;" 554
|-555
| style="text-align: center;" | <math>{\mathbf{k}}_i {\mathbf{r}}_i= \lambda _j {\mathbf{m}}_i {\mathbf{r}}_i\,, </math>556
|}557
| style="width: 5px;text-align: right;white-space: nowrap;" | (36)558
|}559
560
whose eigenvalues <math display="inline">\lambda _j</math> (<math display="inline">j=1,\ldots , 6</math> in 3D case, and <math display="inline">j=1,2,3</math> for 2D case) are the squared frequencies of the free vibration frequencies:561
562
<span id="eq-37"></span>563
{| class="formulaSCP" style="width: 100%; text-align: left;" 564
|-565
| 566
{| style="text-align: left; margin:auto;width: 100%;" 567
|-568
| style="text-align: center;" | <math>\lambda _j=\omega _j^2\,. </math>569
|}570
| style="width: 5px;text-align: right;white-space: nowrap;" | (37)571
|}572
573
In a 3D problem, three of six frequencies <math display="inline">\omega _j</math> are translational, and the other three – rotational.574
575
In the algorithm implemented, a further simplification is assumed. The maximum frequency is estimated as the maximum of natural frequencies of mass–spring systems defined for all the particles with one translational and one rotational degree of freedom. The translational and rotational free vibrations are governed by the following equations:576
577
<span id="eq-38"></span>578
{| class="formulaSCP" style="width: 100%; text-align: left;" 579
|-580
| 581
{| style="text-align: left; margin:auto;width: 100%;" 582
|-583
| style="text-align: center;" | <math>m_i \ddot{u}_n + k_n u_n=0\,, </math>584
|}585
| style="width: 5px;text-align: right;white-space: nowrap;" | (38)586
|}587
588
<span id="eq-39"></span>589
{| class="formulaSCP" style="width: 100%; text-align: left;" 590
|-591
| 592
{| style="text-align: left; margin:auto;width: 100%;" 593
|-594
| style="text-align: center;" | <math>I_i \ddot{\theta } + k_\theta \theta=0\,, </math>595
|}596
| style="width: 5px;text-align: right;white-space: nowrap;" | (39)597
|}598
599
where it is assumed that the translational motion <math display="inline">u_n</math>is due to the contact interaction along the normal direction (the spring stiffness <math display="inline">k_n</math> represents the penalty stiffness in the normal direction) and the rotational stiffness is due to the contact stiffness (penalty) in the tangential direction. Given the tangential penalty <math display="inline">k_T</math>, the rotational stiffness <math display="inline">k_\theta </math> can be obtained as600
601
<span id="eq-40"></span>602
{| class="formulaSCP" style="width: 100%; text-align: left;" 603
|-604
| 605
{| style="text-align: left; margin:auto;width: 100%;" 606
|-607
| style="text-align: center;" | <math>k_{\theta }=k_T r^2\,, </math>608
|}609
| style="width: 5px;text-align: right;white-space: nowrap;" | (40)610
|}611
612
where <math display="inline">r</math> is the length of the vector connecting the mass centre to the contact point.613
614
The natural frequency of the translational vibrations is given by:615
616
<span id="eq-41"></span>617
{| class="formulaSCP" style="width: 100%; text-align: left;" 618
|-619
| 620
{| style="text-align: left; margin:auto;width: 100%;" 621
|-622
| style="text-align: center;" | <math>\omega _n=\sqrt{\frac{k_n}{m_i}}\,, </math>623
|}624
| style="width: 5px;text-align: right;white-space: nowrap;" | (41)625
|}626
627
while the rotational frequency <math display="inline">\omega _{\theta }</math> can be obtained from628
629
<span id="eq-42"></span>630
{| class="formulaSCP" style="width: 100%; text-align: left;" 631
|-632
| 633
{| style="text-align: left; margin:auto;width: 100%;" 634
|-635
| style="text-align: center;" | <math>\omega _{\theta }=\sqrt{\frac{k_{\theta }}{I_i}}\,. </math>636
|}637
| style="width: 5px;text-align: right;white-space: nowrap;" | (42)638
|}639
640
where <math display="inline">I_i</math> is the rotational inertia of a sphere641
642
{| class="formulaSCP" style="width: 100%; text-align: left;" 643
|-644
| 645
{| style="text-align: left; margin:auto;width: 100%;" 646
|-647
| style="text-align: center;" | <math>I= \frac{2}{5} mr^2 </math>648
|}649
| style="width: 5px;text-align: right;white-space: nowrap;" | (43)650
|}651
652
and <math display="inline">k_\theta </math> is given by Eq. ([[#eq-40|40]]). Substituting these expressions into Eq. ([[#eq-42|42]]) gives the rotational frequency as653
654
{| class="formulaSCP" style="width: 100%; text-align: left;" 655
|-656
| 657
{| style="text-align: left; margin:auto;width: 100%;" 658
|-659
| style="text-align: center;" | <math>\omega _{\theta }=\sqrt{\frac{5 k_{T}}{2 m_i}}\,. </math>660
|}661
| style="width: 5px;text-align: right;white-space: nowrap;" | (44)662
|}663
664
If <math display="inline">k_T=k_n</math>, the rotational frequency <math display="inline">\omega _{\theta }</math> is considerably higher than the translational frequency <math display="inline">\omega _n</math> obtained from Eq. ([[#eq-41|41]]), which results in a smaller critical time increment (see Eq. ([[#eq-31|31]])). To reduce the influence of the rotational frequencies in the value of the critical time step, the rotational inertia terms are scaled adequately. The concept of scaling rotational inertia terms is commonly used for shell elements <span id='citeF-14'></span>[[#cite-14|[14]]].665
666
==3 MICROMECHANICAL CONSTITUTIVE MODELS==667
668
===3.1 Elastic perfectly brittle model===669
670
The elastic perfectly brittle contact model is characterized by linear elastic behaviour when cohesive bonds are active. An instantaneous breakage of these bonds occurs when the interface strength is exceeded. When two particles are bonded the contact forces in both normal and tangential directions are calculated from the linear constitutive relationships:671
672
<span id="eq-45"></span>673
<span id="eq-46"></span>674
{| class="formulaSCP" style="width: 100%; text-align: left;" 675
|-676
| 677
{| style="text-align: left; margin:auto;width: 100%;" 678
|-679
| style="text-align: center;" | <math>\sigma \!=\!k_n u_n\,, </math>680
| style="width: 5px;text-align: right;white-space: nowrap;" | (45)681
|-682
| style="text-align: center;" | <math> \tau \!=\!k_t \, u_t\,, </math>683
| style="width: 5px;text-align: right;white-space: nowrap;" | (46)684
|}685
|}686
687
where <math display="inline">\sigma </math> and <math display="inline">\tau </math> are the normal and tangential contact force, respectively, <math display="inline">k_n</math> and <math display="inline">k_t</math> are the interface stiffness in the normal and tangential directions and <math display="inline">u_n</math> and <math display="inline">u_t</math> the normal and tangential relative displacements, respectively.688
689
Cohesive bonds are broken instantaneously when the interface strength is exceeded in the tangential direction by the tangential contact force or in the normal direction by the tensile contact force. The failure (decohesion) criterion is written (for 2D) as:690
691
<span id="eq-47"></span>692
<span id="eq-48"></span>693
{| class="formulaSCP" style="width: 100%; text-align: left;" 694
|-695
| 696
{| style="text-align: left; margin:auto;width: 100%;" 697
|-698
| style="text-align: center;" | <math>\sigma \!\le \!R_n\,, </math>699
| style="width: 5px;text-align: right;white-space: nowrap;" | (47)700
|-701
| style="text-align: center;" | <math> |\tau | \!\le \!R_t\,, </math>702
| style="width: 5px;text-align: right;white-space: nowrap;" | (48)703
|}704
|}705
706
where <math display="inline">R_n</math> and <math display="inline">R_t</math> are the interface strengths in the normal and tangential directions, respectively.707
708
In the absence of cohesion the normal contact force can be compressive only, i.e.709
710
<span id="eq-49"></span>711
{| class="formulaSCP" style="width: 100%; text-align: left;" 712
|-713
| 714
{| style="text-align: left; margin:auto;width: 100%;" 715
|-716
| style="text-align: center;" | <math>\sigma \le 0 </math>717
|}718
| style="width: 5px;text-align: right;white-space: nowrap;" | (49)719
|}720
721
and the (positive) tangential contact force is given by722
723
<span id="eq-50"></span>724
{| class="formulaSCP" style="width: 100%; text-align: left;" 725
|-726
| 727
{| style="text-align: left; margin:auto;width: 100%;" 728
|-729
| style="text-align: center;" | <math>\tau = \mu |\sigma | </math>730
|}731
| style="width: 5px;text-align: right;white-space: nowrap;" | (50)732
|}733
734
if <math display="inline">\sigma{<0}</math> or zero otherwise. The friction force is given by Eq. ([[#eq-50|50]]) expressing the Coulomb friction law, with <math display="inline">\mu </math> being the Coulomb friction coefficient.735
736
<div id='img-5'></div>737
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"738
|-739
|[[Image:Draft_Samper_102318540-brittlen.png|312px|Normal contact force in the elastic perfectly brittle model]]740
|- style="text-align: center; font-size: 75%;"741
| colspan="1" | '''Figure 5:''' Normal contact force in the elastic perfectly brittle model742
|}743
744
<div id='img-6'></div>745
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"746
|-747
|[[Image:Draft_Samper_102318540-brittlet.png|312px|Tangential contact force in the elastic perfectly brittle model (tensile normal contact force)]]748
|- style="text-align: center; font-size: 75%;"749
| colspan="1" | '''Figure 6:''' Tangential contact force in the elastic perfectly brittle model (tensile normal contact force)750
|}751
752
<div id='img-7'></div>753
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"754
|-755
|[[Image:Draft_Samper_102318540-brittlesurf.png|473px|Failure surface for the elastic perfectly brittle model]]756
|- style="text-align: center; font-size: 75%;"757
| colspan="1" | '''Figure 7:''' Failure surface for the elastic perfectly brittle model758
|}759
760
Contact laws for the normal and tangential directions for the elastic perfectly brittle model are shown in Figs. [[#img-5|5]] and [[#img-6|6]], respectively. The failure surface for the elastic perfectly brittle model defined by conditions ([[#eq-47|47]]) and ([[#eq-48|48]]) is shown in Fig. [[#img-7|7]].761
762
===3.2 Elasto-plastic contact model with linear softening===763
764
A contact law relates the contact force acting between two spheres to their relative displacement. The model developed is based on the analogous elasto-plastic relationships established for the normal and tangential directions:765
766
<span id="eq-51"></span>767
{| class="formulaSCP" style="width: 100%; text-align: left;" 768
|-769
| 770
{| style="text-align: left; margin:auto;width: 100%;" 771
|-772
| style="text-align: center;" | <math>\sigma _i=k_i(u_i-u_i^p)\,, \quad i=n,t </math>773
|}774
| style="width: 5px;text-align: right;white-space: nowrap;" | (51)775
|}776
777
where <math display="inline">\sigma _i</math> is the normal or tangential contact force, <math display="inline">u_i</math> the total relative displacement, <math display="inline">u_i^p</math> the plastic component of the relative displacement, <math display="inline">k_i</math> the elastic parameters which characterize the contact interface stiffness and <math display="inline">n, t</math> the normal and tangential directions.778
779
A general yield criterion has been assumed in the following form:780
781
<span id="eq-52"></span>782
{| class="formulaSCP" style="width: 100%; text-align: left;" 783
|-784
| 785
{| style="text-align: left; margin:auto;width: 100%;" 786
|-787
| style="text-align: center;" | <math>f(\sigma ,\alpha )=|\sigma | -[\sigma _Y -H \alpha ]\,, </math>788
|}789
| style="width: 5px;text-align: right;white-space: nowrap;" | (52)790
|}791
792
where <math display="inline">H</math> is the plastic softening modulus (assumed to be positive) and <math display="inline">\alpha </math> is the softening parameter represented by the plastic part of the relative displacement <math display="inline">u^p</math>.793
794
An elasto-plastic contact law with linear softening is assumed for the shear and normal tensile contact forces (Figs. [[#img-8|8]] and [[#img-9|9]]). An elastic linear law is assumed for compression.795
796
Dependence of the tangential force on the normal force is expressed by the application of the 2D failure criterion shown in Fig. [[#img-10|10]]. Yielding of contact bonds can occur under a combined tension and shear action (if the normal contact force is tensile) or due to shear only (if the normal contact force is compressive). After the contact bonds are broken due to yielding, standard frictional contact can occur between the spherical elements.797
798
''Flow rule''799
800
The plastic strain increments for both formulations (i.e. the normal and tangential motions) are calculated from the following flow rule:801
802
<span id="eq-53"></span>803
{| class="formulaSCP" style="width: 100%; text-align: left;" 804
|-805
| 806
{| style="text-align: left; margin:auto;width: 100%;" 807
|-808
| style="text-align: center;" | <math>\dot{u}^p_i=\gamma \, \mbox{sign}(\sigma _i) \,, </math>809
|}810
| style="width: 5px;text-align: right;white-space: nowrap;" | (53)811
|}812
813
where <math display="inline">\gamma </math> is the absolute value of the slip rate (<math display="inline">\gamma{>0}</math>), <math display="inline">\dot{u}^p_i</math> the derivative of the plastic relative displacement (normal or tangential) and <math display="inline">\sigma _i</math> the contact force in the normal or tangential direction.814
815
<div id='img-8'></div>816
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"817
|-818
|[[Image:Draft_Samper_102318540-elpln.png|486px|Elasto-plastic contact law with softening for the normal direction]]819
|- style="text-align: center; font-size: 75%;"820
| colspan="1" | '''Figure 8:''' Elasto-plastic contact law with softening for the normal direction821
|}822
823
<div id='img-9'></div>824
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"825
|-826
|[[Image:Draft_Samper_102318540-elplt.png|522px|Elasto-plastic contact law with softening for the tangential direction]]827
|- style="text-align: center; font-size: 75%;"828
| colspan="1" | '''Figure 9:''' Elasto-plastic contact law with softening for the tangential direction829
|}830
831
<div id='img-10'></div>832
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"833
|-834
|[[File:Draft_Samper_102318540_6786_Fig10.png]]835
|- style="text-align: center; font-size: 75%;"836
| colspan="1" | '''Figure 10:''' Yield surface for elasto-plastic model837
|}838
839
===3.3 Contact model with elastic damage===840
841
The elastic perfectly brittle model can be easily extended to account for elastic damage by assuming a softening behaviour defined by a softening modulus <math display="inline">H</math> introduced into the force-displacement relationship (Fig. [[#img-11|11]]). <div id='img-11'></div>842
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"843
|-844
|[[Image:Draft_Samper_102318540-damagen.png|486px|Normal contact force in the contact model with elastic damage]]845
|- style="text-align: center; font-size: 75%;"846
| colspan="1" | '''Figure 11:''' Normal contact force in the contact model with elastic damage847
|}848
849
The constitutive relationship for 1D elastic damage is given by:850
851
<span id="eq-54"></span>852
{| class="formulaSCP" style="width: 100%; text-align: left;" 853
|-854
| 855
{| style="text-align: left; margin:auto;width: 100%;" 856
|-857
| style="text-align: center;" | <math>\sigma =k_n^D \,u_n =(1-\omega )k_n u_n </math>858
|}859
| style="width: 5px;text-align: right;white-space: nowrap;" | (54)860
|}861
862
where <math display="inline">k_n^D </math> is the elastic damaged secant modulus and <math display="inline">\omega </math> the scalar damage variable.863
864
The scalar damage variable <math display="inline">\omega </math> is a measure of material damage. For the undamaged state <math display="inline">\omega=0</math> and for a damaged state <math display="inline">0<\omega \le 1</math>. The scalar damage variable <math display="inline">\omega </math> can be written in the following form:865
866
<span id="eq-55"></span>867
{| class="formulaSCP" style="width: 100%; text-align: left;" 868
|-869
| 870
{| style="text-align: left; margin:auto;width: 100%;" 871
|-872
| style="text-align: center;" | <math>\omega = \frac{\psi (u_n)-1}{\psi (u_n)} </math>873
|}874
| style="width: 5px;text-align: right;white-space: nowrap;" | (55)875
|}876
877
where <math display="inline">\psi (u_n)</math> is a function of total relative displacement. For linear strain softening <math display="inline">\psi (u_n)</math> is defined by878
879
<span id="eq-56"></span>880
{| class="formulaSCP" style="width: 100%; text-align: left;" 881
|-882
| 883
{| style="text-align: left; margin:auto;width: 100%;" 884
|-885
| style="text-align: center;" | <math>\psi (u_n)=\left\{ \begin{array}{lll}1 & {for} & \displaystyle u_n\le \frac{R_n}{k_n}\\[2ex] \displaystyle \frac{k^2_n u_n}{HR_n +k_n R_n-Hk_n u_n} & {for} & \displaystyle \frac{R_n}{k_n}\le u_n\le \frac{R_n}{k_n}+\frac{R_n}{H}\\[2ex] \displaystyle \infty & {for} & \displaystyle u_n\ge \frac{R_n}{k_n}+\frac{R_n}{H} \end{array} \right. </math>886
|}887
| style="width: 5px;text-align: right;white-space: nowrap;" | (56)888
|}889
890
where <math display="inline">R_n</math> is the initial tensile strength and <math display="inline">H</math> is the softening modulus (taken as positive).891
892
A similar contact force–displacement law with damage (Fig. [[#img-12|12]]) can be introduced for the tangential direction. Then the bond decohesion can occur either due to tension or shear.893
894
<div id='img-12'></div>895
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"896
|-897
|[[Image:Draft_Samper_102318540-damaget.png|546px|Tangential contact force in the contact model with elastic damage]]898
|- style="text-align: center; font-size: 75%;"899
| colspan="1" | '''Figure 12:''' Tangential contact force in the contact model with elastic damage900
|}901
902
An optional failure criterion has been implemented with a failure criterion based on the tensile contact force only. This criterion models better fracture of brittle materials, as the macroscopic failure is due to brittle rupture of atomic bonds in tension. This microscopic mechanism explains the macroscopic strain softening behaviour both under compression and tension states.903
904
After the contact bonds are broken due to damage (<math display="inline">\omega=0</math>) standard frictional contact can occur between the spherical elements.905
906
===3.4 Contact model with friction, wear and heat generation===907
908
Frictional contact, wear and heat generation must be considered in order to accurately model tool-rock/soil interaction. Cohesion is not included into the formulation of this model. Hence, the normal contact forces can be compressive only. The normal contact force <math display="inline">\sigma </math> is calculated from the linear relationships:909
910
<span id="eq-57"></span>911
{| class="formulaSCP" style="width: 100%; text-align: left;" 912
|-913
| 914
{| style="text-align: left; margin:auto;width: 100%;" 915
|-916
| style="text-align: center;" | <math>\sigma = k_n u_n </math>917
|}918
| style="width: 5px;text-align: right;white-space: nowrap;" | (57)919
|}920
921
where <math display="inline">u_n\le 0</math> is the normal relative displacement (penetration of one sphere against another). The frictional contact force is evaluated using the regularized Coulomb law922
923
<span id="eq-58"></span>924
{| class="formulaSCP" style="width: 100%; text-align: left;" 925
|-926
| 927
{| style="text-align: left; margin:auto;width: 100%;" 928
|-929
| style="text-align: center;" | <math>\tau = \mu |\sigma | </math>930
|}931
| style="width: 5px;text-align: right;white-space: nowrap;" | (58)932
|}933
934
where <math display="inline">\mu </math> is the Coulomb friction coefficient. The frictional dissipation rate <math display="inline">\dot{D}</math> is calculated as935
936
<span id="eq-59"></span>937
{| class="formulaSCP" style="width: 100%; text-align: left;" 938
|-939
| 940
{| style="text-align: left; margin:auto;width: 100%;" 941
|-942
| style="text-align: center;" | <math>\dot{D} = \tau v_t </math>943
|}944
| style="width: 5px;text-align: right;white-space: nowrap;" | (59)945
|}946
947
where <math display="inline">v_t</math> is the relative tangential velocity. The frictional dissipation is used to calculate the wear rate <math display="inline">\dot{w}</math> and the heat generated by friction <math display="inline">Q_{gen}</math>. This is computed by948
949
<span id="eq-60"></span>950
{| class="formulaSCP" style="width: 100%; text-align: left;" 951
|-952
| 953
{| style="text-align: left; margin:auto;width: 100%;" 954
|-955
| style="text-align: center;" | <math>Q_{gen} = \chi \dot{D} </math>956
|}957
| style="width: 5px;text-align: right;white-space: nowrap;" | (60)958
|}959
960
where <math display="inline">\chi </math> is the part of the friction work converted to heat (<math display="inline">\chi \le 1</math>). Heat generation through frictional dissipation is assumed to be absorbed equally by the two particles in contact.961
962
Wear rate is calculated using the Archard law <span id='citeF-15'></span>[[#cite-15|[15]]], which assumes that the wear rate <math display="inline">\dot{w}</math> is proportional to the contact pressure <math display="inline">\sigma </math> and to the slip velocity <math display="inline">v_T</math>963
964
<span id="eq-61"></span>965
{| class="formulaSCP" style="width: 100%; text-align: left;" 966
|-967
| 968
{| style="text-align: left; margin:auto;width: 100%;" 969
|-970
| style="text-align: center;" | <math>\dot{w}=k \frac{\sigma v_T}{H} </math>971
|}972
| style="width: 5px;text-align: right;white-space: nowrap;" | (61)973
|}974
975
with <math display="inline">H</math> being the hardness of worn surface and <math display="inline">k</math> being a dimensionless parameter. The Archard law was derived originally for adhesive wear, the same form of law, however, can be obtained for abrasive wear <span id='citeF-16'></span>[[#cite-16|[16]]]. Values of adhesive and abrasive wear constants <math display="inline">k</math> for different combinations of materials can be found in <span id='citeF-16'></span>[[#cite-16|[16]]]. It is commonly accepted that wear is related to friction and thus friction coefficients are often introduced into Eq. ([[#eq-61|61]]). If the Coulomb friction law ([[#eq-58|58]]) holds, the Archard wear law can be written in the following equivalent form:976
977
<span id="eq-62"></span>978
{| class="formulaSCP" style="width: 100%; text-align: left;" 979
|-980
| 981
{| style="text-align: left; margin:auto;width: 100%;" 982
|-983
| style="text-align: center;" | <math>\dot{w}=\bar{k} \frac{\tau v_T}{H}= \bar{k} \frac{\dot{D}}{H} </math>984
|}985
| style="width: 5px;text-align: right;white-space: nowrap;" | (62)986
|}987
988
in which the wear rate is related to frictional dissipation rate <math display="inline">\dot{D}</math> and <math display="inline">\bar{k}=k/\mu </math>.989
990
Influence of temperature on wear can be taken into account by adaptation of the law of Archard given by Eq. ([[#eq-61|61]]). Thermal effects can be captured approximately by taking the hardness <math display="inline">H</math> as a function of the temperature <math display="inline">T</math>991
992
<span id="eq-63"></span>993
{| class="formulaSCP" style="width: 100%; text-align: left;" 994
|-995
| 996
{| style="text-align: left; margin:auto;width: 100%;" 997
|-998
| style="text-align: center;" | <math>H=H(T) </math>999
|}1000
| style="width: 5px;text-align: right;white-space: nowrap;" | (63)1001
|}1002
1003
Then the Archard law ([[#eq-61|61]]) takes the following form:1004
1005
<span id="eq-64"></span>1006
{| class="formulaSCP" style="width: 100%; text-align: left;" 1007
|-1008
| 1009
{| style="text-align: left; margin:auto;width: 100%;" 1010
|-1011
| style="text-align: center;" | <math>\dot{w}=k \frac{\sigma v_T}{H(T)} </math>1012
|}1013
| style="width: 5px;text-align: right;white-space: nowrap;" | (64)1014
|}1015
1016
Computation of temperature can be performed by solving thermal conduction problem on the discrete domain <span id='citeF-17'></span>[[#cite-17|[17]]].1017
1018
==4 FORMULATION OF QUASI-INCOMPRESSIBLE SOLID FINITE ELEMENTS==1019
1020
Many continuum finite elements exhibit so called “volumetric locking” in the analysis of incompressible or quasi-incompressible problems. Situations of this type are usual in the structural analysis of rubber materials, some geomechanical problems and most bulk metal forming processes. Volumetric locking is an undesirable effect leading to incorrect numerical results <span id='citeF-3'></span>[[#cite-3|[3]]].1021
1022
Volumetric locking is present in all low order solid elements based on the standard displacement formulation. The use of a mixed formulation or a selective integration technique eliminates the volumetric locking in many elements. These methods however, fail in some elements such as linear triangles and tetrahedra, due to lack of satisfaction of the Babuska-Brezzi conditions <span id='citeF-3'></span><span id='citeF-18'></span><span id='citeF-19'></span>[[#cite-3|[3,18,19]]] or alternatively the mixed patch test <span id='citeF-3'></span><span id='citeF-20'></span><span id='citeF-21'></span>[[#cite-3|[3,20,21]]] not being passed.1023
1024
Considerable efforts have been made in recent years to develop linear triangles and tetrahedra producing correct (stable) results under incompressible situations. Brezzi and Pitkäranta <span id='citeF-22'></span>[[#cite-22|[22]]] proposed to extend the equation for the volumetric strain rate constraint for Stokes flows by adding a laplacian of pressure term. A similar method was derived for quasi-incompressible solids by Zienkiewicz and Taylor <span id='citeF-3'></span>[[#cite-3|[3]]]. Zienkiewicz ''et al.'' <span id='citeF-23'></span>[[#cite-23|[23]]] have proposed a stabilization technique which eliminates volumetric locking in incompressible solids based on a mixed formulation and a Characteristic Based Split (CBS) algorithm initially developed for fluids <span id='citeF-24'></span><span id='citeF-25'></span><span id='citeF-26'></span>[[#cite-24|[24,25,26]]] where a split of the pressure is introduced when solving the transient dynamic equations in time. Extensions of the CBS algorithm to solve bulk metal forming problems have been recently reported by Rojek ''et al.'' <span id='citeF-24'></span>[[#cite-24|[24]]]. Other methods to overcome volumetric locking are based on mixed displacement (or velocity)-pressure formulations using the Galerkin-Least-Square (GLS) method <span id='citeF-25'></span>[[#cite-25|[25]]], average nodal pressure and average nodal deformation <span id='citeF-26'></span><span id='citeF-27'></span>[[#cite-26|[26,27]]] techniques, and Sub-Grid Scale (SGS) methods <span id='citeF-28'></span><span id='citeF-29'></span><span id='citeF-30'></span><span id='citeF-31'></span>[[#cite-28|[28,29,30,31]]].1025
1026
In this paper volumetric locking is avoided by using a finite calculus (FIC) formulation. The basis of the FIC method is the satisfaction of the equations of balance of momentum and the constitutive equation relating the pressure with the volumetric strain in a domain of finite size. The modified governing equations contain additional terms from standard infinitesimal theory. These terms introduce the necessary stability in the discretized equations to overcome the volumetric locking problem.1027
1028
The FIC approach has been successfully used to derive stabilized finite element and meshless methods for a wide range of advective-diffusive and fluid flow problems <span id='citeF-32'></span><span id='citeF-33'></span><span id='citeF-34'></span><span id='citeF-35'></span><span id='citeF-36'></span><span id='citeF-37'></span><span id='citeF-38'></span><span id='citeF-39'></span>[[#cite-32|[32,33,34,35,36,37,38,39]]]. The same ideas were applied in <span id='citeF-4'></span><span id='citeF-5'></span>[[#cite-4|[4,5]]] to derive a stabilized formulation for quasi-incompressible and incompressible solids allowing the use of linear triangles and tetrahedra.1029
1030
In the next section, the basis of the FIC method are given for static quasi-incompressible solid mechanics problems. The stabilized dynamic formulation for linear triangles and tetrahedra is presented and both semi-implicit and explicit monolithic solution schemes are described.1031
1032
===4.1 Equilibrium equations===1033
1034
The equilibrium equations for an elastic solid are written using the FIC technique as <span id='citeF-32'></span>[[#cite-32|[32]]]1035
1036
<span id="eq-65"></span>1037
{| class="formulaSCP" style="width: 100%; text-align: left;" 1038
|-1039
| 1040
{| style="text-align: left; margin:auto;width: 100%;" 1041
|-1042
| style="text-align: center;" | <math>{r}_{i}- \underline{\frac{{h}_k}{2}{\partial {r}_{i} \over \partial {x}_k}}\frac{{h}_k}{2}{\partial {r}_{i} \over \partial {x}_k}=0 \quad \mbox{in} \;\,\Omega \qquad k=1,n_d </math>1043
|}1044
| style="width: 5px;text-align: right;white-space: nowrap;" | (65)1045
|}1046
1047
where <math display="inline">n_d</math> is the number of space dimensions of the problems (i.e. <math display="inline">n_d =3</math> for 3D)1048
1049
<span id="eq-66"></span>1050
{| class="formulaSCP" style="width: 100%; text-align: left;" 1051
|-1052
| 1053
{| style="text-align: left; margin:auto;width: 100%;" 1054
|-1055
| style="text-align: center;" | <math>{r}_{i}: ={\partial {\sigma }_{ij} \over \partial {x}_j}+{b}_i </math>1056
|}1057
| style="width: 5px;text-align: right;white-space: nowrap;" | (66)1058
|}1059
1060
In ([[#eq-65|65]]) and ([[#eq-66|66]]) <math>{\sigma }_{ij}</math>and <math>{b}_i</math>are the stresses and the body forces, respectively and <math>{h}_k</math>are characteristic length distances of an arbitrary prismatic domain where equilibrium of forces is considered.1061
1062
Equations ([[#eq-65|65]]) and ([[#eq-66|66]]) are completed with the boundary conditions on the displacements <math display="inline">u_i</math>1063
1064
<span id="eq-67"></span>1065
{| class="formulaSCP" style="width: 100%; text-align: left;" 1066
|-1067
| 1068
{| style="text-align: left; margin:auto;width: 100%;" 1069
|-1070
| style="text-align: center;" | <math>{u}_i-\bar{u}_i=0\quad \mbox{on} \;\,\Gamma _u </math>1071
|}1072
| style="width: 5px;text-align: right;white-space: nowrap;" | (67)1073
|}1074
1075
and the equilibrium of surface tractions1076
1077
<span id="eq-68"></span>1078
{| class="formulaSCP" style="width: 100%; text-align: left;" 1079
|-1080
| 1081
{| style="text-align: left; margin:auto;width: 100%;" 1082
|-1083
| style="text-align: center;" | <math>{\sigma }_{ij}{n}_j- \bar{t}_i- \underline{\frac{1}{2}{h}_k{n}_k{r}_{i}}\frac{1}{2}{h}_k{n}_k{r}_{i} =0\quad \mbox{on} \;\,\Gamma _t </math>1084
|}1085
| style="width: 5px;text-align: right;white-space: nowrap;" | (68)1086
|}1087
1088
In the above <math>\bar{u}_i</math>and <math>\bar{t}_i</math>are prescribed displacements and tractions over the boundaries <math display="inline">\Gamma _u</math>and <math display="inline">\Gamma _t</math>, respectively, <math display="inline">n_i</math> are the components of the unit normal vector and <math display="inline">h_k</math> are again the characteristic lengths.1089
1090
The underlined terms in Eqs. ([[#eq-65|65]]) and ([[#eq-68|68]]) are a consequence of expressing the equilibrium of body forces and surface tractions in domains of finite size and retaining higher order terms than those usually accepted in the infinitesimal theory <span id='citeF-32'></span>[[#cite-32|[32]]].1091
1092
These terms are essential to derive a stabilized finite element formulation as described below.1093
1094
====4.1.1 Constitutive equations====1095
1096
The stresses are split into deviatoric and volumetric (pressure) parts1097
1098
<span id="eq-69"></span>1099
{| class="formulaSCP" style="width: 100%; text-align: left;" 1100
|-1101
| 1102
{| style="text-align: left; margin:auto;width: 100%;" 1103
|-1104
| style="text-align: center;" | <math>{\sigma }_{ij}= s_{ij} +p{\delta }_{ij} </math>1105
|}1106
| style="width: 5px;text-align: right;white-space: nowrap;" | (69)1107
|}1108
1109
where <math>{\delta }_{ij}</math>is the Kronnecker delta function. The linear elastic constitutive equations for the deviatoric stresses <math display="inline">s_{ij}</math> are written as1110
1111
<span id="eq-70"></span>1112
{| class="formulaSCP" style="width: 100%; text-align: left;" 1113
|-1114
| 1115
{| style="text-align: left; margin:auto;width: 100%;" 1116
|-1117
| style="text-align: center;" | <math>s_{ij}\!\!=\!\! 2G \left({\varepsilon }_{ij}-\frac{1}{3}{\varepsilon }_{v}{\delta }_{ij}\right) </math>1118
|}1119
| style="width: 5px;text-align: right;white-space: nowrap;" | (70)1120
|}1121
1122
where <math display="inline">G</math>is the shear modulus,1123
1124
{| class="formulaSCP" style="width: 100%; text-align: left;" 1125
|-1126
| 1127
{| style="text-align: left; margin:auto;width: 100%;" 1128
|-1129
| style="text-align: center;" | <math>{\varepsilon }_{ij}= \frac{1}{2} \left( {\partial {u}_i \over \partial {x}_j}+ {\partial {u}_j \over \partial x_i} \right) \quad \mbox{and} \quad {\varepsilon }_{v}={\varepsilon }_{ii}\,. </math>1130
|}1131
| style="width: 5px;text-align: right;white-space: nowrap;" | (71)1132
|}1133
1134
The constitutive equation for the pressure <math display="inline">p</math> can be written for an arbitrary domain of finite size using the FIC formulation as [30]1135
1136
<span id="eq-72"></span>1137
{| class="formulaSCP" style="width: 100%; text-align: left;" 1138
|-1139
| 1140
{| style="text-align: left; margin:auto;width: 100%;" 1141
|-1142
| style="text-align: center;" | <math>\left({1\over K} p - {\varepsilon }_{v}\right)- \frac{{h}_k}{2}{\partial \over \partial {x}_k} \left({1\over K} p - {\varepsilon }_{v}\right)=0\, ,\quad k=1,2,3\,\,\, \hbox{for }3D </math>1143
|}1144
| style="width: 5px;text-align: right;white-space: nowrap;" | (72)1145
|}1146
1147
where <math display="inline">K</math> is the bulk modulus of the material.1148
1149
Note that for <math display="inline">h_k \to 0</math> the standard relationship between the pressure and the volumetric strain of the infinitesimal theory <math display="inline">(p=K\varepsilon _v )</math> is found.1150
1151
For an incompressible material <math display="inline">K\to \infty </math> and Eq. ([[#eq-72|72]]) yields1152
1153
<span id="eq-73"></span>1154
{| class="formulaSCP" style="width: 100%; text-align: left;" 1155
|-1156
| 1157
{| style="text-align: left; margin:auto;width: 100%;" 1158
|-1159
| style="text-align: center;" | <math>\varepsilon _v - \frac{{h}_k}{2}{\partial {\varepsilon }_{v} \over \partial {x}_k}=0 </math>1160
|}1161
| style="width: 5px;text-align: right;white-space: nowrap;" | (73)1162
|}1163
1164
Eq. ([[#eq-73|73]]) expresses the limit incompressible behaviour of the solid. This equation is typical in incompressible fluid dynamic problems and there arises from the mass continuity conditions <span id='citeF-32'></span><span id='citeF-33'></span>[[#cite-32|[32,33]]].1165
1166
By combining Eqs. ([[#eq-65|65]]), ([[#eq-66|66]]), ([[#eq-69|69]]), ([[#eq-70|70]]) and ([[#eq-73|73]]) a mixed displacement–pressure formulation can be written as1167
1168
<span id="eq-74"></span>1169
{| class="formulaSCP" style="width: 100%; text-align: left;" 1170
|-1171
| 1172
{| style="text-align: left; margin:auto;width: 100%;" 1173
|-1174
| style="text-align: center;" | <math>{\partial s_{ij} \over \partial {x}_j}+{\partial p \over \partial x_i}+{b}_i-\frac{{h}_k}{2} {\partial r_i \over \partial {x}_k}=0 </math>1175
|}1176
| style="width: 5px;text-align: right;white-space: nowrap;" | (74)1177
|}1178
1179
<span id="eq-75"></span>1180
{| class="formulaSCP" style="width: 100%; text-align: left;" 1181
|-1182
| 1183
{| style="text-align: left; margin:auto;width: 100%;" 1184
|-1185
| style="text-align: center;" | <math>\left(\frac{p}{K}-{\varepsilon }_{v}\right)-\frac{{h}_k}{2}{\partial \over \partial {x}_k}\left(\frac{p}{K}-{\varepsilon }_{v}\right)=0 </math>1186
|}1187
| style="width: 5px;text-align: right;white-space: nowrap;" | (75)1188
|}1189
1190
Substituting Eq. ([[#eq-70|70]]) into ([[#eq-74|74]]) leads after some algebra to1191
1192
<span id="eq-76"></span>1193
{| class="formulaSCP" style="width: 100%; text-align: left;" 1194
|-1195
| 1196
{| style="text-align: left; margin:auto;width: 100%;" 1197
|-1198
| style="text-align: center;" | <math>{\partial {\varepsilon }_{v} \over \partial {x}_i}= \frac{3}{2G}\left[\hat{r}_{i}- \frac{{h}_k}{2}{\partial {r}_{i} \over \partial {x}_k} \right] </math>1199
|}1200
| style="width: 5px;text-align: right;white-space: nowrap;" | (76)1201
|}1202
1203
where <math>{r}_{i}</math>is defined by Eq. ([[#eq-66|66]]) and1204
1205
<span id="eq-77"></span>1206
{| class="formulaSCP" style="width: 100%; text-align: left;" 1207
|-1208
| 1209
{| style="text-align: left; margin:auto;width: 100%;" 1210
|-1211
| style="text-align: center;" | <math>\hat{r}_{i}=\frac{\partial }{\partial x_j}(2G \varepsilon _{ij})+{\partial p \over \partial x_i} + b_i </math>1212
|}1213
| style="width: 5px;text-align: right;white-space: nowrap;" | (77)1214
|}1215
1216
Substituting Eq. ([[#eq-76|76]]) into ([[#eq-75|75]]) gives after some simplification1217
1218
<span id="eq-78"></span>1219
{| class="formulaSCP" style="width: 100%; text-align: left;" 1220
|-1221
| 1222
{| style="text-align: left; margin:auto;width: 100%;" 1223
|-1224
| style="text-align: center;" | <math>\left(\frac{p}{K}-{\varepsilon }_{v}\right)- {h_i\over 2} \left({1\over K} {\partial p \over \partial x_i} - {3\over 2G}\hat r_i \right)- \left({3h_i\over 8}{{h}_k\over G}{\partial r_i \over \partial {x}_k}\right)=0 </math>1225
|}1226
| style="width: 5px;text-align: right;white-space: nowrap;" | (78)1227
|}1228
1229
<span id="eq-79"></span>1230
{| class="formulaSCP" style="width: 100%; text-align: left;" 1231
|-1232
| 1233
{| style="text-align: left; margin:auto;width: 100%;" 1234
|-1235
| style="text-align: center;" | <math>{p\over K} - {\varepsilon }_{v}- \sum \limits _{i=1}^{n_d} \tau _i {\partial r_i \over \partial x_i}=0 </math>1236
|}1237
| style="width: 5px;text-align: right;white-space: nowrap;" | (79)1238
|}1239
1240
where1241
1242
{| class="formulaSCP" style="width: 100%; text-align: left;" 1243
|-1244
| 1245
{| style="text-align: left; margin:auto;width: 100%;" 1246
|-1247
| style="text-align: center;" | <math>\tau _i = {3h_i^2\over 8G} </math>1248
|}1249
| style="width: 5px;text-align: right;white-space: nowrap;" | (80)1250
|}1251
1252
The coefficients <math display="inline">\tau _{i} </math> in Eq. ([[#eq-79|79]]) are also referred to as ''intrinsic time'' parameters per unit mass (their dimensions are <math display="inline">{t^2 m^3\over Kg}</math> where <math display="inline">t</math> is the time).1253
1254
===4.2 Non linear transient dynamic analysis===1255
1256
The static formulation can be readily extended for the transient dynamic case accounting for geometrical and material non linear effects. Indeed in many situations of this kind, material quasi-incompressibility develops in specific zones of the solid due to the accumulation of plastic strains. It is well known that in these cases the use of equal order interpolations for displacements and pressure leads to locking solutions unless some precautions are taken.1257
1258
The transient FIC equilibrium equations can be written in an identical form to eq. ([[#eq-65|65]]) (neglecting time stabilization terms <span id='citeF-32'></span><span id='citeF-37'></span><span id='citeF-38'></span>[[#cite-32|[32,37,38]]]) with1259
1260
<span id="eq-81"></span>1261
{| class="formulaSCP" style="width: 100%; text-align: left;" 1262
|-1263
| 1264
{| style="text-align: left; margin:auto;width: 100%;" 1265
|-1266
| style="text-align: center;" | <math>r_i:= - \rho \frac{\partial ^2{u_i}}{\partial{t}^2} + {\partial \sigma _{ij} \over \partial x_j}+ b_i </math>1267
|}1268
| style="width: 5px;text-align: right;white-space: nowrap;" | (81)1269
|}1270
1271
where <math display="inline">\rho </math> is the density.1272
1273
Eq. ([[#eq-81|81]]) is completed with the constitutive equations for the deviatoric stresses (eq. ([[#eq-70|70]])) and the pressure (eq. ([[#eq-72|72]])), as well with the boundary conditions ([[#eq-67|67]]) and ([[#eq-68|68]]) and the initial conditions for <math display="inline">t=0</math>.1274
1275
Following the arguments of the static case, the stabilized constitutive equation for the pressure can be expressed in terms of the residuals of the momentum equations by an expression identical to eq. ([[#eq-79|79]]). This equation is now written in an incremental form more suitable for non linear transient analysis.1276
1277
We note that the stabilization terms are not larger needed in the numerical (FEM) solution equilibrium equations which can be asted in the standard form. This simply implies neglecting the terms involving <math display="inline">h_i</math> in Eq. ([[#eq-74|74]]). The stabilized FIC form of the pressure constitutive equation (Eq. ([[#eq-79|79]])) is however mandatory in order to avoid volumetric locking.1278
1279
The set of stabilized equations to be solved are now:1280
1281
'''''Equilibrium'''''1282
1283
<span id="eq-82"></span>1284
{| class="formulaSCP" style="width: 100%; text-align: left;" 1285
|-1286
| 1287
{| style="text-align: left; margin:auto;width: 100%;" 1288
|-1289
| style="text-align: center;" | <math>r_i =0 </math>1290
|}1291
| style="width: 5px;text-align: right;white-space: nowrap;" | (82)1292
|}1293
1294
'''''Pressure constitutive equation'''''1295
1296
<span id="eq-83"></span>1297
{| class="formulaSCP" style="width: 100%; text-align: left;" 1298
|-1299
| 1300
{| style="text-align: left; margin:auto;width: 100%;" 1301
|-1302
| style="text-align: center;" | <math>{\Delta p\over K} -{\partial (\Delta u_i) \over \partial x_i} - \sum \limits _{i=1}^{n_d} \tau _i {\partial r_i \over \partial x_i}=0 </math>1303
|}1304
| style="width: 5px;text-align: right;white-space: nowrap;" | (83)1305
|}1306
1307
where <math display="inline">\Delta p = p^{n+1}-p^n</math> and <math display="inline">\Delta u_i = u_i^{n+1}-u_i^n</math> are the increments of pressure and displacements, respectively. As usual <math display="inline">(\cdot )^n</math> denotes values at time <math display="inline">t_n</math>.1308
1309
In the derivation of Eq. ([[#eq-83|83]]) we have accepted that <math display="inline">\Delta r_i =r_i^{n+1} \equiv r_i</math> as the infinitesimal equilibrium equations are assumed to be satisfied at time <math display="inline">t_n</math> (and hence <math display="inline">r_i^n =0</math>).1310
1311
The weighted residual form of the FIC governing equations ([[#eq-82|82]]), ([[#eq-68|68]]) and ([[#eq-83|83]]) is1312
1313
'''''Equilibrium''''1314
1315
<span id="eq-84"></span>1316
{| class="formulaSCP" style="width: 100%; text-align: left;" 1317
|-1318
| 1319
{| style="text-align: left; margin:auto;width: 100%;" 1320
|-1321
| style="text-align: center;" | <math>\int _\Omega \!\delta u_i \left[-\rho \frac{\partial ^2{u_i}}{\partial{t}^2}+ {\partial \sigma _{ij} \over \partial x_j}+b_i\right] d\Omega + \!\!\int _{\Gamma _t}\! \delta u_i \left[\sigma _{ij}n_j - \bar t_i \right]d\Gamma =0 </math>1322
|}1323
| style="width: 5px;text-align: right;white-space: nowrap;" | (84)1324
|}1325
1326
'''''Pressure constitutive equation'''''1327
1328
<span id="eq-85"></span>1329
{| class="formulaSCP" style="width: 100%; text-align: left;" 1330
|-1331
| 1332
{| style="text-align: left; margin:auto;width: 100%;" 1333
|-1334
| style="text-align: center;" | <math>\int _\Omega q \left({p\over K} - {\varepsilon }_{v}\right)d\Omega - \int _\Omega q \left(\sum \limits _{i=1}^{n_d} \tau _i {\partial {r}_{i} \over \partial x_i}\right)\, d\Omega =0 </math>1335
|}1336
| style="width: 5px;text-align: right;white-space: nowrap;" | (85)1337
|}1338
1339
where <math display="inline">\delta u_i </math> and <math display="inline">q</math> are arbitrary test functions representing virtual displacements and virtual pressure fields, respectively.1340
1341
Integrating by parts, the terms involving <math display="inline">s_{ij}</math> and <math display="inline">p</math> in Eq. ([[#eq-84|84]]) and the term involving <math display="inline">r_i</math> in Eq. ([[#eq-85|85]]) and neglecting the space derivatives of the intrinsic time parameters leads to1342
1343
<span id="eq-86"></span>1344
{| class="formulaSCP" style="width: 100%; text-align: left;" 1345
|-1346
| 1347
{| style="text-align: left; margin:auto;width: 100%;" 1348
|-1349
| style="text-align: center;" | <math>\int _{\Omega }\delta u_i \rho \frac{\partial ^2{u_i}}{\partial{t}^2}d\Omega + \int _{\Omega }\!\delta {\varepsilon }_{ij}\sigma _{ij}\,d\Omega{-} \int _{\Omega }\delta {u}_i{b}_i\, d\Omega - \int _{\Gamma _t}\delta {u}_i\bar{t}_i\,d\Omega =0 </math>1350
|}1351
| style="width: 5px;text-align: right;white-space: nowrap;" | (86)1352
|}1353
1354
<span id="eq-87"></span>1355
{| class="formulaSCP" style="width: 100%; text-align: left;" 1356
|-1357
| 1358
{| style="text-align: left; margin:auto;width: 100%;" 1359
|-1360
| style="text-align: center;" | <math>\int _\Omega q \left({p\over K} - {\varepsilon }_{v}\right)d\Omega + \int _\Omega \left(\sum \limits _{i=1}^{n_d} {\partial q \over \partial x_i} \tau _{i} r_i \right)\,d\Omega =0 </math>1361
|}1362
| style="width: 5px;text-align: right;white-space: nowrap;" | (87)1363
|}1364
1365
The residual <math display="inline">r_i</math> is split now as1366
1367
{| class="formulaSCP" style="width: 100%; text-align: left;" 1368
|-1369
| 1370
{| style="text-align: left; margin:auto;width: 100%;" 1371
|-1372
| style="text-align: center;" | <math>r_i = \pi _i + {\partial p \over \partial x_i} </math>1373
|}1374
| style="width: 5px;text-align: right;white-space: nowrap;" | (88)1375
|}1376
1377
where1378
1379
{| class="formulaSCP" style="width: 100%; text-align: left;" 1380
|-1381
| 1382
{| style="text-align: left; margin:auto;width: 100%;" 1383
|-1384
| style="text-align: center;" | <math>\pi _i = - \frac{\partial ^2{u_i}}{\partial{t}^2} + {\partial s_{ij} \over \partial x_i}+b_i </math>1385
|}1386
| style="width: 5px;text-align: right;white-space: nowrap;" | (89)1387
|}1388
1389
Note that <math display="inline">\pi _i</math> is the part of <math display="inline">r_i</math> not containing the pressure gradient and may be interpreted as the negative of a projection of the pressure gradient. In a discrete setting the terms <math display="inline">\pi _i</math> can be considered belonging to a ''sub-scale'' space orthogonal to that of the pressure gradient terms.1390
1391
In the infinitesimal limit <math display="inline">r_i=0</math> and <math display="inline">{\partial p \over \partial x_i} + \pi _i=0</math>. This limit relationship between <math display="inline">{\partial p \over \partial x_i}</math> and <math display="inline">\pi _i</math> can be weakly enforced by means of a weighted residual form.1392
1393
The final set of integral equations is therefore1394
1395
<span id="eq-90"></span>1396
{| class="formulaSCP" style="width: 100%; text-align: left;" 1397
|-1398
| 1399
{| style="text-align: left; margin:auto;width: 100%;" 1400
|-1401
| style="text-align: center;" | <math>\int _{\Omega }\delta u_i \rho \frac{\partial ^2{u_i}}{\partial{t}^2}\,d\Omega + \int _{\Omega }\delta {\varepsilon }_{ij}{\sigma }_{ij}\,d\Omega{-} \int _{\Omega }\delta u_i b_i \,d\Omega - \int _{\Gamma _t}\delta {u}_i\bar{t}_i \,d\Gamma _t=0 </math>1402
|}1403
| style="width: 5px;text-align: right;white-space: nowrap;" | (90)1404
|}1405
1406
<span id="eq-91"></span>1407
{| class="formulaSCP" style="width: 100%; text-align: left;" 1408
|-1409
| 1410
{| style="text-align: left; margin:auto;width: 100%;" 1411
|-1412
| style="text-align: center;" | <math>\int _{\Omega }q \left({\Delta p\over K} - {\partial (\Delta u_i) \over \partial x_i} \right)\,d\Omega + \int _\Omega \left[ \sum \limits _{i=1}^{n_d}{\partial q \over \partial x_i} \tau _{i}\left({\partial p \over \partial x_i}+\pi _i \right)\right]d\Omega =0 </math>1413
|}1414
| style="width: 5px;text-align: right;white-space: nowrap;" | (91)1415
|}1416
1417
<span id="eq-92"></span>1418
{| class="formulaSCP" style="width: 100%; text-align: left;" 1419
|-1420
| 1421
{| style="text-align: left; margin:auto;width: 100%;" 1422
|-1423
| style="text-align: center;" | <math>\int _{\Omega }\left[\sum \limits _{i=1}^{n_d} w_i \tau _{i} \left({\partial p \over \partial x_i}+\pi _i \right)\right]d\Omega =0 </math>1424
|}1425
| style="width: 5px;text-align: right;white-space: nowrap;" | (92)1426
|}1427
1428
where the <math display="inline">\tau _i</math> coefficients are introduced in Eq. ([[#eq-92|92]]) for convenience.1429
1430
We will choose <math display="inline">C^0</math>continuous linear interpolations of the displacements, the pressure and the pressure gradient projection <math display="inline">\pi _i</math> over three-node triangles (2D) and four-node tetrahedra (3D) <span id='citeF-3'></span>[[#cite-3|[3]]]. The linear interpolations are written as1431
1432
<span id="eq-93"></span>1433
{| class="formulaSCP" style="width: 100%; text-align: left;" 1434
|-1435
| 1436
{| style="text-align: left; margin:auto;width: 100%;" 1437
|-1438
| style="text-align: center;" | <math>{u}_i=\sum _{j=1}^{n} N_j \bar{u}_i^j \quad , \quad p=\sum _{j=1}^{n} N_j \bar{p}^j \quad , \quad \pi _i =\sum _{j=1}^{n} N_j \bar \pi ^j_i </math>1439
|}1440
| style="width: 5px;text-align: right;white-space: nowrap;" | (93)1441
|}1442
1443
where <math display="inline">n=3(4)</math> for 2D(3D) problems and <math display="inline">(\bar{\cdot })</math> denotes nodal variables. As usual <math display="inline">N_j</math> are the linear shape functions <span id='citeF-3'></span>[[#cite-3|[3]]]. The nodal variables are a function of the time <math display="inline">t</math>. Substituting the approximations ([[#eq-93|93]]) into eqs. ([[#eq-90|90]])–([[#eq-92|92]]) gives the following system of discretized equations1444
1445
<span id="eq-94"></span>1446
{| class="formulaSCP" style="width: 100%; text-align: left;" 1447
|-1448
| 1449
{| style="text-align: left; margin:auto;width: 100%;" 1450
|-1451
| style="text-align: center;" | <math>\mathbf{M} \ddot{\bar \mathbf{u}} + \mathbf{g} - \mathbf{f}=\mathbf{0} </math>1452
|}1453
| style="width: 5px;text-align: right;white-space: nowrap;" | (94)1454
|}1455
1456
<span id="eq-95"></span>1457
{| class="formulaSCP" style="width: 100%; text-align: left;" 1458
|-1459
| 1460
{| style="text-align: left; margin:auto;width: 100%;" 1461
|-1462
| style="text-align: center;" | <math>\mathbf{G}^T \Delta \bar \mathbf{u} -\mathbf{C} \Delta \bar \mathbf{p}- \mathbf{L} \bar \mathbf{p} - \mathbf{Q}\bar {\boldsymbol \pi }= {0} </math>1463
|}1464
| style="width: 5px;text-align: right;white-space: nowrap;" | (95)1465
|}1466
1467
<span id="eq-96"></span>1468
{| class="formulaSCP" style="width: 100%; text-align: left;" 1469
|-1470
| 1471
{| style="text-align: left; margin:auto;width: 100%;" 1472
|-1473
| style="text-align: center;" | <math>\mathbf{Q}^T \bar \mathbf{p}+ \bar \mathbf{C} \bar {\boldsymbol \pi }= \mathbf{0} </math>1474
|}1475
| style="width: 5px;text-align: right;white-space: nowrap;" | (96)1476
|}1477
1478
where <math display="inline">\ddot{\bar \mathbf{u}}</math> is the nodal acceleration vector. The expression of the different matrices in vectors appearing in Eqs. ([[#eq-94|94]])–([[#eq-96|96]]) is presented in the Appendix.1479
1480
A four steps semi-implicit time integration algorithm can be derived from eqs. ([[#eq-94|94]])–([[#eq-96|96]]) as follows1481
1482
step1483
1484
'''Step 1'''. Compute the nodal velocities <math display="inline">\dot{\bar \mathbf{u}}^{n+1/2}</math>1485
1486
<span id="eq-97"></span>1487
{| class="formulaSCP" style="width: 100%; text-align: left;" 1488
|-1489
| 1490
{| style="text-align: left; margin:auto;width: 100%;" 1491
|-1492
| style="text-align: center;" | <math>1493
1494
\dot{\bar \mathbf{u}}^{n+1/2} = \dot{\bar \mathbf{u}}^{n-1/2}+\Delta t \mathbf{M}^{-1}_d (\mathbf{f}^n - \mathbf{g}^n) </math>1495
|}1496
| style="width: 5px;text-align: right;white-space: nowrap;" | (97)1497
|}1498
1499
'''Step 2'''. Compute the nodal displacements <math display="inline">\bar \mathbf{u}^{n+1}</math>1500
1501
<span id="eq-98"></span>1502
{| class="formulaSCP" style="width: 100%; text-align: left;" 1503
|-1504
| 1505
{| style="text-align: left; margin:auto;width: 100%;" 1506
|-1507
| style="text-align: center;" | <math>1508
1509
\bar \mathbf{u}^{n+1} = \bar\mathbf{u}^n + \Delta t \dot{\bar \mathbf{u}}^{n+1/2} </math>1510
|}1511
| style="width: 5px;text-align: right;white-space: nowrap;" | (98)1512
|}1513
1514
'''Step 3'''. Compute the nodal pressures <math display="inline">\bar \mathbf{p}^{n+1}</math>1515
1516
<span id="eq-99"></span>1517
{| class="formulaSCP" style="width: 100%; text-align: left;" 1518
|-1519
| 1520
{| style="text-align: left; margin:auto;width: 100%;" 1521
|-1522
| style="text-align: center;" | <math>1523
1524
\bar \mathbf{p}^{n+1}=[ \mathbf{C} + \mathbf{L}]^{-1} [\Delta t \mathbf{G}^T \dot{\bar \mathbf{u}}^{n+1/2}+ \mathbf{C} \bar \mathbf{p}^n- \mathbf{Q} \bar {\boldsymbol \pi }^n] </math>1525
|}1526
| style="width: 5px;text-align: right;white-space: nowrap;" | (99)1527
|}1528
1529
'''Step 4'''. Compute the nodal projected pressure gradients <math display="inline">\bar {\boldsymbol \pi }^{n+1}</math>1530
1531
<span id="eq-100"></span>1532
{| class="formulaSCP" style="width: 100%; text-align: left;" 1533
|-1534
| 1535
{| style="text-align: left; margin:auto;width: 100%;" 1536
|-1537
| style="text-align: center;" | <math>1538
1539
\bar {\boldsymbol \pi }^{n+1}=- \bar \mathbf{C}_d^{-1} \mathbf{Q}^T \bar \mathbf{p}^{n+1} </math>1540
|}1541
| style="width: 5px;text-align: right;white-space: nowrap;" | (100)1542
|}1543
1544
In above, all matrices are evaluated at <math display="inline">t^{n+1}</math>, <math display="inline">(\cdot )_d =\hbox{diag}\, (\cdot )</math> and1545
1546
{| class="formulaSCP" style="width: 100%; text-align: left;" 1547
|-1548
| 1549
{| style="text-align: left; margin:auto;width: 100%;" 1550
|-1551
| style="text-align: center;" | <math>\mathbf{g}^n =\int _{\Omega ^e} [\mathbf{B}^T {\boldsymbol \sigma }]^n\,d\Omega </math>1552
|}1553
| style="width: 5px;text-align: right;white-space: nowrap;" | (101)1554
|}1555
1556
where the stresses <math display="inline">{\boldsymbol \sigma }^n</math> are obtained by consistent integration of the adequate (non linear) constitutive law [33].1557
1558
Note that steps 1, 2 and 4 are fully explicit as a diagonal form of matrices <math display="inline">\mathbf{C}</math> and <math display="inline">\bar\mathbf{C}</math> has been chosen. The solution of step 3 with a diagonal form for <math display="inline">\mathbf{C}</math> still requires the inverse of a Laplacian matrix. This can be an inexpensive process using an iterative equation solution method (e.g. a preconditioned conjugate gradient method).1559
1560
A ''three steps'' approach can be obtained by evaluating the projected pressure gradient variables <math display="inline">\bar {\boldsymbol \pi }^{n+1}</math> at <math display="inline">t_{n+1}</math> in a fully implicit form in Eq. ([[#eq-99|99]]). Eliminating then <math display="inline">\bar {\boldsymbol \pi }^{n+1}</math> from the fourth step using Eq. ([[#eq-100|100]]) and substituting this expression into Eq. ([[#eq-99|99]]) leads to1561
1562
{| class="formulaSCP" style="width: 100%; text-align: left;" 1563
|-1564
| 1565
{| style="text-align: left; margin:auto;width: 100%;" 1566
|-1567
| style="text-align: center;" | <math>\bar\mathbf{p}^{n+1}= [\mathbf{C} + \mathbf{L}-\mathbf{S}]^{-1} [\Delta t \mathbf{G}^T \dot{\bar \mathbf{u}}^{n+1/2} + \mathbf{C} \bar\mathbf{p}^n ] </math>1568
|}1569
| style="width: 5px;text-align: right;white-space: nowrap;" | (102)1570
|}1571
1572
where1573
1574
{| class="formulaSCP" style="width: 100%; text-align: left;" 1575
|-1576
| 1577
{| style="text-align: left; margin:auto;width: 100%;" 1578
|-1579
| style="text-align: center;" | <math>\mathbf{S} = \mathbf{Q} \bar \mathbf{C}_d^{-1} \mathbf{Q}^T </math>1580
|}1581
| style="width: 5px;text-align: right;white-space: nowrap;" | (103)1582
|}1583
1584
Recall that for the full incompressible case <math display="inline">K=\infty </math> and <math display="inline">\mathbf{C}=0 </math> in all above equations.1585
1586
The critical time step <math display="inline">\Delta t</math> is taken as that of the standard explicit dynamic scheme [28].1587
1588
'''''Explicit algorithm'''''1589
1590
A fully explicit algorithm can be obtained by computing <math display="inline">\bar \mathbf{p}^{n+1}</math> from step 3 in Eq. ([[#eq-99|99]]) as follows1591
1592
<span id="eq-104"></span>1593
{| class="formulaSCP" style="width: 100%; text-align: left;" 1594
|-1595
| 1596
{| style="text-align: left; margin:auto;width: 100%;" 1597
|-1598
| style="text-align: center;" | <math>\bar\mathbf{p}^{n+1}=\mathbf{C}_d^{-1}[\Delta t \mathbf{G}^T \dot{\bar \mathbf{u}}^{n+1/2} + (\mathbf{C}_d+\mathbf{L}) \bar\mathbf{p}^n - \mathbf{Q} {\boldsymbol \sigma }^n] </math>1599
|}1600
| style="width: 5px;text-align: right;white-space: nowrap;" | (104)1601
|}1602
1603
Obviously, solution of eq. ([[#eq-104|104]]) breaks down for <math display="inline">K=\infty </math> as <math display="inline">{C}=0 </math> in this case. Therefore, the explicit algorithm is not applicable in the full incompressible limit. The explicit form can however be used with success in problems where quasi-incompressible regions exist adjacent to standard “compressible” zones.1604
1605
==5 DEM–FEM CONTACT ALGORITHM==1606
1607
Use of combined DEM/FEM models involves treatment of contact between spherical discrete elements and boundary of continuum subdomains discretized with finite elements as shown in Fig. [[#img-13|13]]. Similarly as in case of contact between two spheres (Fig. [[#img-2|2]]) the contact force between the sphere and external edge <math display="inline">\mathbf{F}</math> of a finite element is decomposed into normal and tangential components, <math display="inline">\mathbf{F}_n</math> and <math display="inline">\mathbf{ F}_T</math>. In a general case the contact model between discrete elements and finite element edges can include cohesion, damping, friction, wear, heat generation and exchange.1608
1609
<div id='img-13'></div>1610
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"1611
|-1612
|[[Image:Draft_Samper_102318540-fesphere.png|270px|Contact between a sphere and finite element edge]]1613
|- style="text-align: center; font-size: 75%;"1614
| colspan="1" | '''Figure 13:''' Contact between a sphere and finite element edge1615
|}1616
1617
Similarly to the relationship ([[#eq-10|10]]) the normal contact force <math display="inline">F_n</math> is decomposed into the elastic part <math display="inline">F_{ne}</math> and the damping contact force <math display="inline">F_{nd}</math>1618
1619
{| class="formulaSCP" style="width: 100%; text-align: left;" 1620
|-1621
| 1622
{| style="text-align: left; margin:auto;width: 100%;" 1623
|-1624
| style="text-align: center;" | <math>F_n = F_{ne} + F_{nd}\,. </math>1625
|}1626
| style="width: 5px;text-align: right;white-space: nowrap;" | (105)1627
|}1628
1629
The elastic part of the normal contact force <math display="inline">F_{ne}</math> is proportional to the normal stiffness <math display="inline">k_n</math> and to the penetration of the two particle surfaces <math display="inline">u_{rn}</math>1630
1631
<span id="eq-106"></span>1632
{| class="formulaSCP" style="width: 100%; text-align: left;" 1633
|-1634
| 1635
{| style="text-align: left; margin:auto;width: 100%;" 1636
|-1637
| style="text-align: center;" | <math>F_{ne}=k_n u_{rn}\,. </math>1638
|}1639
| style="width: 5px;text-align: right;white-space: nowrap;" | (106)1640
|}1641
1642
The penetration is calculated here as1643
1644
{| class="formulaSCP" style="width: 100%; text-align: left;" 1645
|-1646
| 1647
{| style="text-align: left; margin:auto;width: 100%;" 1648
|-1649
| style="text-align: center;" | <math>u_{rn} = d - r\,, </math>1650
|}1651
| style="width: 5px;text-align: right;white-space: nowrap;" | (107)1652
|}1653
1654
where <math display="inline">d</math> is the distance of the particle centre <math display="inline">C</math> and <math display="inline">r</math> is the particle radius.1655
1656
The contact damping force is assumed to be of viscous type1657
1658
{| class="formulaSCP" style="width: 100%; text-align: left;" 1659
|-1660
| 1661
{| style="text-align: left; margin:auto;width: 100%;" 1662
|-1663
| style="text-align: center;" | <math>F_{nd}=c_n v_{rn} </math>1664
|}1665
| style="width: 5px;text-align: right;white-space: nowrap;" | (108)1666
|}1667
1668
where <math display="inline">c_n</math> is the damping coefficient and <math display="inline">v_{rn}</math> is the normal component of the relative velocity at the contact point. The relative velocity at the contact point is calculated as1669
1670
{| class="formulaSCP" style="width: 100%; text-align: left;" 1671
|-1672
| 1673
{| style="text-align: left; margin:auto;width: 100%;" 1674
|-1675
| style="text-align: center;" | <math>\mathbf{v}_{r}= ({\mathbf{v}}_{C} +{\omega }_{C} \times \mathbf{r})- ({\mathbf{v}}_{1}H_{1} + {\mathbf{v}}_{2}H_{2}) \,, </math>1676
|}1677
| style="width: 5px;text-align: right;white-space: nowrap;" | (109)1678
|}1679
1680
where <math display="inline">{\mathbf{v}}_{C} +{\omega }_{C} \times \mathbf{r}</math> is the discrete element velocity at the contact point and <math display="inline">{\mathbf{v}}_{1}H_{1} + {\mathbf{v}}_{2}H_{2}</math> is the velocity of the finite element edge at the contact point expressed in terms of the nodal velocities, <math display="inline">{\mathbf{v}}_{1}</math> and <math display="inline">{\mathbf{v}}_{2}</math>, and respective shape functions <math display="inline">H_{1}</math> and <math display="inline">H_{2}</math>. The relative velocity is decomposed into the normal and tangential velocity using the formulae ([[#eq-15|15]]) and ([[#eq-17|17]]).1681
1682
The cohesive contact force is calculated using the elastic-perfectly brittle model described in Section [[#3.1 Elastic perfectly brittle model|3.1]]. In the absence of cohesion the frictional contact reaction is calculated using the regularized Coulomb friction model described in section [[#lb-2.3.3|2.3.3]].1683
1684
==6 GENERAL DISCRETE ELEMENT-FINITE ELEMENT DYNAMIC FORMULATION==1685
1686
The general algorithm for the transient dynamic problem involving discrete elements and (stabilized) finite elements includes the following steps.1687
1688
====Step1. Compute the nodal velocities====1689
1690
'''Discrete elements'''1691
1692
<span id="eq-110"></span>1693
{| class="formulaSCP" style="width: 100%; text-align: left;" 1694
|-1695
| 1696
{| style="text-align: left; margin:auto;width: 100%;" 1697
|-1698
| style="text-align: center;" | <math>\dot{\mathbf{u}}_i^{n+1/2}=\dot{\mathbf{u}}_i^{n-1/2}+\ddot{\mathbf{u}}_i^{n}{\Delta t}\,, </math>1699
|}1700
| style="width: 5px;text-align: right;white-space: nowrap;" | (110)1701
|}1702
1703
with1704
1705
<span id="eq-111"></span>1706
{| class="formulaSCP" style="width: 100%; text-align: left;" 1707
|-1708
| 1709
{| style="text-align: left; margin:auto;width: 100%;" 1710
|-1711
| style="text-align: center;" | <math>\ddot{\mathbf{u}}_i^{n}=\frac{{\mathbf{F}}_i^n}{m_i}\,, </math>1712
|}1713
| style="width: 5px;text-align: right;white-space: nowrap;" | (111)1714
|}1715
1716
<span id="eq-112"></span>1717
{| class="formulaSCP" style="width: 100%; text-align: left;" 1718
|-1719
| 1720
{| style="text-align: left; margin:auto;width: 100%;" 1721
|-1722
| style="text-align: center;" | <math>{{\omega }}_i^{n+1/2}={{\omega }}_i^{n-1/2}+\dot{{\omega }}_i^{n}{\Delta t}\,. </math>1723
|}1724
| style="width: 5px;text-align: right;white-space: nowrap;" | (112)1725
|}1726
1727
with1728
1729
<span id="eq-113"></span>1730
{| class="formulaSCP" style="width: 100%; text-align: left;" 1731
|-1732
| 1733
{| style="text-align: left; margin:auto;width: 100%;" 1734
|-1735
| style="text-align: center;" | <math>\dot{{\omega }}_i^{n}=\frac{{\mathbf{T}}_i^n}{I_i}\,, </math>1736
|}1737
| style="width: 5px;text-align: right;white-space: nowrap;" | (113)1738
|}1739
1740
'''Finite elements'''1741
1742
<span id="eq-114"></span>1743
{| class="formulaSCP" style="width: 100%; text-align: left;" 1744
|-1745
| 1746
{| style="text-align: left; margin:auto;width: 100%;" 1747
|-1748
| style="text-align: center;" | <math>\dot{\bar \mathbf{u}}^{n+1/2} = \dot{\bar \mathbf{u}}^{n-1/2}+\Delta t \mathbf{M}^{-1}_d (\mathbf{f}^n - \mathbf{g}^n) </math>1749
|}1750
| style="width: 5px;text-align: right;white-space: nowrap;" | (114)1751
|}1752
1753
====Step2. Compute the nodal displacements====1754
1755
'''Discrete elements'''1756
1757
<span id="eq-115"></span>1758
{| class="formulaSCP" style="width: 100%; text-align: left;" 1759
|-1760
| 1761
{| style="text-align: left; margin:auto;width: 100%;" 1762
|-1763
| style="text-align: center;" | <math>{\mathbf{u}}_i^{n+1}={\mathbf{u}}_i^{n}+\dot{\mathbf{u}}_i^{n+1/2}{\Delta t}\,. </math>1764
|}1765
| style="width: 5px;text-align: right;white-space: nowrap;" | (115)1766
|}1767
1768
<span id="eq-116"></span>1769
{| class="formulaSCP" style="width: 100%; text-align: left;" 1770
|-1771
| 1772
{| style="text-align: left; margin:auto;width: 100%;" 1773
|-1774
| style="text-align: center;" | <math>{{\Delta \theta }}_i={{\omega }}_i^{n+1/2}{\Delta t}\,, </math>1775
|}1776
| style="width: 5px;text-align: right;white-space: nowrap;" | (116)1777
|}1778
1779
'''Finite elements'''1780
1781
<span id="eq-117"></span>1782
{| class="formulaSCP" style="width: 100%; text-align: left;" 1783
|-1784
| 1785
{| style="text-align: left; margin:auto;width: 100%;" 1786
|-1787
| style="text-align: center;" | <math>\bar \mathbf{u}^{n+1} = \bar\mathbf{u}^n + \Delta t \dot{\bar \mathbf{u}}^{n+1/2} </math>1788
|}1789
| style="width: 5px;text-align: right;white-space: nowrap;" | (117)1790
|}1791
1792
====Step3. Compute the nodal pressures====1793
1794
'''Finite elements'''1795
1796
{| class="formulaSCP" style="width: 100%; text-align: left;" 1797
|-1798
| 1799
{| style="text-align: left; margin:auto;width: 100%;" 1800
|-1801
| style="text-align: center;" | <math>\bar\mathbf{p}^{n+1}= [\mathbf{C} + \mathbf{L}-\mathbf{S}]^{-1} [\Delta t \mathbf{G}^T \dot{\bar \mathbf{u}}^{n+1/2} + \mathbf{C} \bar\mathbf{p}^n ] </math>1802
|}1803
| style="width: 5px;text-align: right;white-space: nowrap;" | (118)1804
|}1805
1806
====Step4. Update the nodal coordinates====1807
1808
'''Discrete and finite elements'''1809
1810
{| class="formulaSCP" style="width: 100%; text-align: left;" 1811
|-1812
| 1813
{| style="text-align: left; margin:auto;width: 100%;" 1814
|-1815
| style="text-align: center;" | <math>\mathbf{X}_i^{n+1} = \mathbf{X}_i^{n}+\Delta \bar \mathbf{u}_i </math>1816
|}1817
| style="width: 5px;text-align: right;white-space: nowrap;" | (119)1818
|}1819
1820
{| class="formulaSCP" style="width: 100%; text-align: left;" 1821
|-1822
| 1823
{| style="text-align: left; margin:auto;width: 100%;" 1824
|-1825
| style="text-align: center;" | <math>\Delta \bar \mathbf{u}_i =\bar \mathbf{u}_i^{n+1} - \bar \mathbf{u}_i^n </math>1826
|}1827
| style="width: 5px;text-align: right;white-space: nowrap;" | (120)1828
|}1829
1830
====Step 5. Check frictional contact forces====1831
1832
====Step 6. Update residual force vector====1833
1834
====Go to step 1====1835
1836
==7 EXAMPLES==1837
1838
===7.1 Uniaxial compression and tension tests of a rock material===1839
1840
Mechanical properties of rock materials are determined from laboratory tests such as compression, triaxial and tension tests. Figure [[#img-14|14]] shows a cube specimen of a sandstone rock after unconfined compression test. The parameters of numerical discrete element model can be obtained carrying out simulations of basic laboratory tests. Such analyses allow us to determine microscopic constitutive parameters for a material sample modelled with discrete elements yielding required macroscopic properties. The most important macroscopic properties are Young modulus, Poisson ratio and compressive and tensile strengths. Microscopic parameters are in turn all the constitutive model parameters governing the frictional contact interaction between a pair of particles. Here elastic perfectly brittle constitutive model will be used with the contact stiffness in normal and tangential direction, interface tensile and shear strength and Coulomb friction coefficient being the microscopic parameters.1841
1842
1843
<div id='img-14'></div>1844
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"1845
|-1846
|[[Image:Draft_Samper_102318540-ucs-lab.png|462px|Rock specimen after laboratory unconfined compression test]]1847
|- style="text-align: center; font-size: 75%;"1848
| colspan="1" | '''Figure 14:''' Rock specimen after laboratory unconfined compression test1849
|}1850
1851
1852
Figure [[#img-15|15]] presents the material sample prepared for testing. A material sample of <math display="inline">109\times 109</math> mm is represented by an assembly of randomly compacted 2100 discs of radii 1–1.5 mm. It is shown in <span id='citeF-40'></span>[[#cite-40|[40]]] that preparing a well-connected densely packed irregular assembly of particles is the key to successful simulation with discrete elements. Compaction of the particle assembly shown in Fig. [[#img-15|15]] is characterized by a porosity of <math display="inline">13</math>%.1853
1854
1855
<div id='img-15'></div>1856
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"1857
|-1858
|[[Image:Draft_Samper_102318540-uni-comp-geo.png|360px|Set-up of the numerical model of uncofined compression test]]1859
|- style="text-align: center; font-size: 75%;"1860
| colspan="1" | '''Figure 15:''' Set-up of the numerical model of uncofined compression test1861
|}1862
1863
1864
The macroscopic response of a square material sample subjected to uniaxial compression has been studied. The loading has been introduced under kinematic control by prescribing the motion of rigid plates pressing on the top and bottom of the sample. The deformation in the <math display="inline">x</math> direction was free. The velocity of the wall displacement was 1 mm/s which was found to be sufficiently low to obtain quasi-static loading.1865
1866
The purpose of the present study was to determine the microscopic parameters for sandstone with the following macroscopic properties: Young modulus <math display="inline">E=14</math> GPa, Poisson ratio <math display="inline">\nu=0.3</math> and unconfined compressive strength <math display="inline">\sigma _{c}=60</math> MPa. The stress-strain relationship shown in Fig. [[#img-16|16]] satisfying these requirements has been obtained with the following set of micromechanical parameters: contact stiffness in the normal and tangential directions <math display="inline">k_n=k_T=20</math> GPa, Coulomb friction coefficient <math display="inline">\mu=0.839</math> and cohesive bond strengths in the normal and tangential direction, <math display="inline">R_n=0.1</math> MN/m and <math display="inline">R_T=1</math> MN/m, respectively. The failure mode of the specimen obtained in the simulation is shown in Fig. [[#img-17|17]]. Particles with broken cohesive bonds are marked with different colour. Comparison of Figs. [[#img-14|14]] and [[#img-17|17]] show that numerical analysis yields a failure mode similar to that observed in experiments.1867
1868
<div id='img-16'></div>1869
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"1870
|-1871
|[[Image:Draft_Samper_102318540-com-brittle.png|450px|Stress–strain relationship for uniaxial unconfined compression test]]1872
|- style="text-align: center; font-size: 75%;"1873
| colspan="1" | '''Figure 16:''' Stress–strain relationship for uniaxial unconfined compression test1874
|}1875
1876
<div id='img-17'></div>1877
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"1878
|-1879
|[[Image:Draft_Samper_102318540-com-fail.png|390px|Failure mode obtained in the simulation of unconfined compression test]]1880
|- style="text-align: center; font-size: 75%;"1881
| colspan="1" | '''Figure 17:''' Failure mode obtained in the simulation of unconfined compression test1882
|}1883
1884
Tensile strength of rocks is obtained experimentally from indirect tension (Brasilian) test. In the numerical procedure we checked the tensile strength of the particle assembly carrying out simulations of direct tension test. The stress–strain relationship obtained in this simulation is plotted in Fig. [[#img-18|18]]. The elastic modulus for tension and tensile strength obtained from the plots are the following: <math display="inline">E_t=11.5</math> GPa, <math display="inline">\sigma _{t}=12.7</math> MPa. It can be noted that the curves both for compression and tension are typical for materials with brittle failure. The failure mode obtained in the analysis of a direct tension test is shown in Fig. [[#img-19|19]].1885
1886
<div id='img-18'></div>1887
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"1888
|-1889
|[[Image:Draft_Samper_102318540-ten-brittle.png|450px|Stress–strain relationship for uniaxial direct tension test]]1890
|- style="text-align: center; font-size: 75%;"1891
| colspan="1" | '''Figure 18:''' Stress–strain relationship for uniaxial direct tension test1892
|}1893
1894
<div id='img-19'></div>1895
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"1896
|-1897
|[[Image:Draft_Samper_102318540-ten-bb.png|354px|Failure mode obtained in the simulation of direct tension test]]1898
|- style="text-align: center; font-size: 75%;"1899
| colspan="1" | '''Figure 19:''' Failure mode obtained in the simulation of direct tension test1900
|}1901
1902
===7.2 Simulation of rock cutting===1903
1904
A material sample representing sandstone with parameters determined in the previous section has been used in the simulation of a rock cutting process. Two different solutions with the tool discretized with discrete and finite elements have been studied. The combined FEM/DEM model is shown in Fig. [[#img-20|20]]. The material sample <math display="inline">109\times 109</math> mm is represented by an assembly of randomly compacted 2100 discs of radii 1–1.5 mm. Contact interface between particles representing the rock is characterized with the stiffness in the normal and tangential directions <math display="inline">k_n=k_T=20</math> GPa, the cohesive bond strengths <math display="inline">R_n=0.1</math> MN/m, <math display="inline">R_T=1</math> MN/m, and the friction coefficient <math display="inline">\mu=0.839</math>. The tool has been modelled with 6800 linear triangles. The following elasto-plastic parameters have been assumed for the tool material: Young modulus <math display="inline">E=2\cdot 10^5</math> MPa, Poisson's ratio <math display="inline">\nu=0.3</math>, yield stress <math display="inline">\sigma _Y=600</math> MPa and hardening modulus <math display="inline">300</math> MPa. The following parameters have been assumed for the tool-rock interface: contact stiffness modulus <math display="inline">k_n=20</math> GPa, Coulomb friction coefficient <math display="inline">\mu = 0.839</math>.1905
1906
<div id='img-20'></div>1907
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"1908
|-1909
|[[Image:Draft_Samper_102318540-cut-fem12-ini.png|600px|Rock cutting problem. 2100 discrete elements are used to model the rock whereas 6800 linear triangles are used to discretize the tool.]]1910
|- style="text-align: center; font-size: 75%;"1911
| colspan="1" | '''Figure 20:''' Rock cutting problem. 2100 discrete elements are used to model the rock whereas 6800 linear triangles are used to discretize the tool.1912
|}1913
1914
1915
Cutting process has been carried out with prescribed horizontal velocity of the tool of 4 m/s. The cutting is shown in Fig. [[#img-21|21]]. In this figure we can also see the failure mode. Particles with broken bonds are coloured in blue. It can be clearly seen the formation of a chip during cutting which is typical for brittle materials.1916
1917
Variation of the cutting force in the initial phase of cutting is shown in Fig. [[#img-22|22]]. The Mises stress and efffective plastic strain distributions in the tool in the initial phase of cutting are shown in Figs. [[#img-23|23]] and [[#img-24|24]], respectively.1918
1919
<div id='img-21'></div>1920
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"1921
|-1922
|[[Image:Draft_Samper_102318540-cut-fem12-bb-0015.png|468px|]]1923
|[[Image:Draft_Samper_102318540-cut-fem12-bb-0025.png|450px|]]1924
|-1925
|[[Image:Draft_Samper_102318540-cut-fem12-bb-0065.png|426px|]]1926
|[[Image:Draft_Samper_102318540-cut-fem12-bb-012.png|378px|Rock cutting process analysed using the FEM/DEM model. The failure mode]]1927
|- style="text-align: center; font-size: 75%;"1928
| colspan="2" | '''Figure 21:''' Rock cutting process analysed using the FEM/DEM model. The failure mode1929
|}1930
1931
<div id='img-22'></div>1932
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"1933
|-1934
|[[Image:Draft_Samper_102318540-cut-fem-tcx.png|450px|Variation of the cutting force in the initial phase of cutting]]1935
|- style="text-align: center; font-size: 75%;"1936
| colspan="1" | '''Figure 22:''' Variation of the cutting force in the initial phase of cutting1937
|}1938
1939
<div id='img-23'></div>1940
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"1941
|-1942
|[[Image:Draft_Samper_102318540-cut-fem12ns-vms-0009.png|582px|Rock cutting process analysed using the FEM/DEM model. The Mises stresses in the tool in the initial phase of cutting]]1943
|- style="text-align: center; font-size: 75%;"1944
| colspan="1" | '''Figure 23:''' Rock cutting process analysed using the FEM/DEM model. The Mises stresses in the tool in the initial phase of cutting1945
|}1946
1947
<div id='img-24'></div>1948
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"1949
|-1950
|[[Image:Draft_Samper_102318540-cut-fem12ns-eps-0009.png|582px|Rock cutting process analysed using the FEM/DEM model. The effective plastic strain in the tool in the initial phase of cutting]]1951
|- style="text-align: center; font-size: 75%;"1952
| colspan="1" | '''Figure 24:''' Rock cutting process analysed using the FEM/DEM model. The effective plastic strain in the tool in the initial phase of cutting1953
|}1954
1955
The model of rock cutting process presented above has been used to demonstrate the capabilities of the developed algorithm for wear estimation. Next the tool has been discretized with 4500 discrete elements (Fig. [[#img-25|25]]). This allows us to easily modify the shape of the tool. The tool shape is changed by eliminating particles if the accumulated wear exceeds the particle diameter. Particles are eliminated automatically and the analysis is continued with the modified shape of the tool.1956
1957
<div id='img-25'></div>1958
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"1959
|-1960
|[[Image:Draft_Samper_102318540-w2d-ini.png|408px|Rock cutting process analyzed with the tool discretized with 4500 discrete elements]]1961
|- style="text-align: center; font-size: 75%;"1962
| colspan="1" | '''Figure 25:''' Rock cutting process analyzed with the tool discretized with 4500 discrete elements1963
|}1964
1965
The required material constants <math display="inline">k</math> and <math display="inline">H</math> necessary for wear evaluation have been obtained in laboratory tests. The average Brinell hardness of 400 has been obtained for the tool steel and a value of the Archard wear constant <math display="inline">k=0.0025</math> has been taken in the wear simulation.1966
1967
Evolution of the tool shape is obtained in a number of consecutive simulations of work cycles. The tool shape is modified continuously during the analysis. After finishing one work cycle the tool with a new shape is again used in the simulation of next work cycle. The whole procedure can be repeated as needed to reproduce the real working conditions of the tool. In this manner the evolution of change of tool shape in wear process during its whole service life can be obtained.1968
1969
Tool shapes at different stages of wear are shown in Fig. [[#img-26|26]]. As much as 40% of the tool mass has been lost due to abrasive wear. Figure [[#img-27|27]] presents the tool mass loss as a fraction of the initial tool mass versus analysis time. The wear curve obtained in the analysis agrees quite well with experimental curves obtained in field tests.1970
1971
<div id='img-26'></div>1972
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"1973
|-1974
|[[Image:Draft_Samper_102318540-tshape-a.png|264px|]]1975
|[[Image:Draft_Samper_102318540-tshape-b.png|390px|]]1976
|-1977
|[[Image:Draft_Samper_102318540-tshape-c.png|378px|]]1978
|[[Image:Draft_Samper_102318540-tshape-d.png|378px|Evolution of the cutting tool shape due to wear during the rock cutting process]]1979
|- style="text-align: center; font-size: 75%;"1980
| colspan="2" | '''Figure 26:''' Evolution of the cutting tool shape due to wear during the rock cutting process1981
|}1982
1983
<div id='img-27'></div>1984
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"1985
|-1986
|[[Image:Draft_Samper_102318540-tmass-exp.png|600px|Mass loss fraction in the tool versus analysis time –- comparison of experimental and simulation wear curves]]1987
|- style="text-align: center; font-size: 75%;"1988
| colspan="1" | '''Figure 27:''' Mass loss fraction in the tool versus analysis time –- comparison of experimental and simulation wear curves1989
|}1990
1991
===7.3 Strip punch test===1992
1993
The strip punch test investigated experimentally by Dede <span id='citeF-41'></span>[[#cite-41|[41]]] and studied numerically using a FE model by Klerck <span id='citeF-42'></span>[[#cite-42|[42]]] is analysed here using the DEM model. Prismatic quartzite specimens of dimensions <math display="inline">80\times 80\times 80</math> mm are crushed by a punch load applied in the strip 10 mm wide on the top side along the symmetry line as shown in Fig. [[#img-28|28]]. A confining pressure of 4.5 MPa was applied to the specimen sides.1994
1995
<div id='img-28'></div>1996
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"1997
|-1998
|[[Image:Draft_Samper_102318540-dede.png|528px|Strip punch geometry and loading configuration]]1999
|- style="text-align: center; font-size: 75%;"2000
| colspan="1" | '''Figure 28:''' Strip punch geometry and loading configuration2001
|}2002
2003
2004
The experimental fracture pattern of the specimen is shown in Fig. [[#img-29|29]] and the experimental axial stress versus the specimen axial strain is plotted in Fig. [[#img-30|30]]. <div id='img-29'></div>2005
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"2006
|-2007
|[[Image:Draft_Samper_102318540-strip-exp.png|450px|Experimental fracture pattern, after <span id='citeF-41'></span>[[#cite-41|[41]]]]]2008
|- style="text-align: center; font-size: 75%;"2009
| colspan="1" | '''Figure 29:''' Experimental fracture pattern, after <span id='citeF-41'></span>[[#cite-41|[41]]]2010
|}2011
<div id='img-30'></div>2012
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"2013
|-2014
|[[Image:Draft_Samper_102318540-strip-forcex.png|600px|Experimental axial stress vs. axial strain plot, after <span id='citeF-41'></span>[[#cite-41|[41]]]]]2015
|- style="text-align: center; font-size: 75%;"2016
| colspan="1" | '''Figure 30:''' Experimental axial stress vs. axial strain plot, after <span id='citeF-41'></span>[[#cite-41|[41]]]2017
|}2018
2019
2020
The 2D discrete element model used in our analysis is shown in Fig. [[#img-31|31]]. The square rock specimen <math display="inline">80\times 80</math> mm is represented by an assembly of randomly compacted 2100 discs of radii 0.8–1.2 mm. Sliding conditions are prescribed at the bottom specimen side and confining pressure is introduced by means of rigid plates transmitting a confining load. Axial load is introduced by a rigid punch under a prescribed displacement.2021
2022
<div id='img-31'></div>2023
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"2024
|-2025
|[[Image:Draft_Samper_102318540-strip-ini.png|432px|2D discrete element model]]2026
|- style="text-align: center; font-size: 75%;"2027
| colspan="1" | '''Figure 31:''' 2D discrete element model2028
|}2029
2030
2031
Required microscopic parameters for the DEM model corresponding to the macroscopic properties of quartzite rock given in <span id='citeF-41'></span>[[#cite-41|[41]]] are: Young's modulus <math display="inline">E=68</math> GPA, compressive strength <math display="inline">200</math> MPa and tensile strength <math display="inline">27</math> MPa. Those values have been obtained using the procedure presented in example [[#7.1 Uniaxial compression and tension tests of a rock material|7.1]] for the following microscopic parameters: the stiffness in the normal and tangential directions <math display="inline">k_n=k_T=100</math> GPa, the cohesive bond strengths <math display="inline">R_n=0.25</math> MN/m, <math display="inline">R_T=2.5</math> MN/m, and the friction coefficient <math display="inline">\mu=0.839</math>.2032
2033
The numerical failure pattern is shown in Fig. [[#img-32|32]]. The cracking pattern obtained is similar to experimental results shown in Fig. [[#img-29|29]]. The triangular crushed zone below the punch and the axial cleavage crack are reproduced in the numerical simulation.2034
2035
Quite a good agreement can also be observed between experimental (Fig. [[#img-30|30]]) and numerical (Fig. [[#img-33|33]]) critical values of axial stresses.2036
2037
<div id='img-32'></div>2038
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"2039
|-2040
|[[Image:Draft_Samper_102318540-strip-punch2-bb-035.png|420px|]]2041
|[[Image:Draft_Samper_102318540-strip-punch2-bb-040.png|426px|]]2042
|-2043
|[[Image:Draft_Samper_102318540-strip-punch2-bb-050.png|414px|]]2044
|[[Image:Draft_Samper_102318540-strip-punch2-bb-1.png|432px|Numerical failure pattern, after <span id='citeF-41'></span>[[#cite-41|[41]]]]]2045
|- style="text-align: center; font-size: 75%;"2046
| colspan="2" | '''Figure 32:''' Numerical failure pattern, after <span id='citeF-41'></span>[[#cite-41|[41]]]2047
|}2048
2049
<div id='img-33'></div>2050
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"2051
|-2052
|[[Image:Draft_Samper_102318540-strip-punch2-f.png|450px|Numerical axial stress vs. axial strain plot]]2053
|- style="text-align: center; font-size: 75%;"2054
| colspan="1" | '''Figure 33:''' Numerical axial stress vs. axial strain plot2055
|}2056
2057
===7.4 Impact of projectile against rock plate===2058
2059
The impact of the projectile against a thick rock plate as defined in Fig. [[#img-34|34]] has been studied in order to illustrate possibilities of combined FEM/DEM modelling for this kind of problems.2060
2061
<div id='img-34'></div>2062
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"2063
|-2064
|[[Image:Draft_Samper_102318540-rock-impact.png|276px|Impact of projectile – definition of geometry]]2065
|- style="text-align: center; font-size: 75%;"2066
| colspan="1" | '''Figure 34:''' Impact of projectile – definition of geometry2067
|}2068
2069
The projectile has been discretized with either equal order linear triangular elements using the FIC formulation previously described, and also with quadrilateral Q1/P0 elements based on a mixed formulation <span id='citeF-3'></span>[[#cite-3|[3]]]. Projectile has been assumed to be made of steel with the following properties: Young's modulus <math display="inline">E=2\cdot 10^5</math> MPa, Poisson's ratio <math display="inline">\nu=0.3</math>, yield stress <math display="inline">\sigma _Y=400</math> MPA and isotropic linear hardening modulus <math display="inline">H=300</math> MPA. The macroscopic parameters of rock has been assumed the same as in the previous example (section [[#7.3 Strip punch test|7.3]]). Again the same collection of discrete elements has been used.2070
2071
Fracturing of the plate under the impact as well as projectile deformation is shown in Figs. [[#img-35|35]] and [[#img-36|36]] for triangular and quadrilateral projectile discretizations, respectively. Results obtained with quadrilateral elements are considered as the reference ones for the solution obtained with the FIC formulation. The agreement observed between Figs. [[#img-35|35]] and [[#img-36|36]] confirms correctness of the FIC algorithm developed by the authors. Advantages of this algorithm can be observed especially in 3D models with more complicated geometry where meshing with tetrahedra is easier than with hexahedra. Effective plastic strains obtained with triangles and quadrilaterals also coincide as it can be seen in Fig. [[#img-37|37]].2072
2073
Non-symmetry observed in the deformation of the projectile can be attributed in the non-symmetry of the discrete elements in the zone of impact which initiates global buckling of the projectile.2074
2075
<div id='img-35'></div>2076
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"2077
|-2078
|[[Image:Draft_Samper_102318540-rock-bara-ps3t-c2-ini.png|258px|]]2079
|[[Image:Draft_Samper_102318540-rock-bara-ps3t-c2-bb-024.png|258px|]]2080
|[[Image:Draft_Samper_102318540-rock-bara-ps3t-c2-bb-040.png|264px|]]2081
|-2082
|[[Image:Draft_Samper_102318540-rock-bara-ps3t-c2-bb-056.png|270px|]]2083
|[[Image:Draft_Samper_102318540-rock-bara-ps3t-c2-bb-080.png|282px|]]2084
|[[Image:Draft_Samper_102318540-rock-bara-ps3t-c2-bb-112a.png|288px|Impact of projectile – numerical results obtained using linear triangular elements based on FIC formulation]]2085
|- style="text-align: center; font-size: 75%;"2086
| colspan="3" | '''Figure 35:''' Impact of projectile – numerical results obtained using linear triangular elements based on FIC formulation2087
|}2088
2089
<div id='img-36'></div>2090
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"2091
|-2092
|[[Image:Draft_Samper_102318540-rock-bara-ps3-c2-ini.png|252px|]]2093
|[[Image:Draft_Samper_102318540-rock-bara-ps3-c2-bb-024.png|258px|]]2094
|[[Image:Draft_Samper_102318540-rock-bara-ps3-c2-bb-040.png|264px|]]2095
|-2096
|[[Image:Draft_Samper_102318540-rock-bara-ps3-c2-bb-056.png|276px|]]2097
|[[Image:Draft_Samper_102318540-rock-bara-ps3-c2-bb-080.png|282px|]]2098
|[[Image:Draft_Samper_102318540-rock-bara-ps3-c2-bb-112.png|288px|Impact of projectile – numerical results obtained using mixed bilinear Q1/P0 quadrilateral elements]]2099
|- style="text-align: center; font-size: 75%;"2100
| colspan="3" | '''Figure 36:''' Impact of projectile – numerical results obtained using mixed bilinear Q1/P0 quadrilateral elements2101
|}2102
2103
<div id='img-37'></div>2104
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"2105
|-2106
|[[Image:Draft_Samper_102318540-rock-bara-ps3t-c2-eps-112.png|360px|]]2107
|[[Image:Draft_Samper_102318540-rock-bara-ps3-c2-eps-112.png|384px|Impact of projectile – effective plastic strain distribution; comparison of numerical results obtained with FIC-based linear triangles with those obtained using mixed bilinear Q1/P0 quadrilateral elements]]2108
|- style="text-align: center; font-size: 75%;"2109
| colspan="2" | '''Figure 37:''' Impact of projectile – effective plastic strain distribution; comparison of numerical results obtained with FIC-based linear triangles with those obtained using mixed bilinear Q1/P0 quadrilateral elements2110
|}2111
2112
===7.5 Study of pipe ovalization===2113
2114
Interaction of soil and pipe leading to pipe ovalization has been studied. This example demonstrates another possibility of combined FEM/DEM modelling in geomechanics.2115
2116
<div id='img-38'></div>2117
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"2118
|-2119
|[[Image:Draft_Samper_102318540-pipeoval.png|452px|Pipe ovalization – geometry definition]]2120
|- style="text-align: center; font-size: 75%;"2121
| colspan="1" | '''Figure 38:''' Pipe ovalization – geometry definition2122
|}2123
2124
2125
The definition of the problem geometry is shown in Fig. [[#img-38|38]]. A pipe of 1m diameter and 1 cm thickness is buried 1 m under the surface of a soil. Due to symmetry only half of the geometry is modelled. The surface load <math display="inline">Q</math> increases linearly as2126
2127
{| class="formulaSCP" style="width: 100%; text-align: left;" 2128
|-2129
| 2130
{| style="text-align: left; margin:auto;width: 100%;" 2131
|-2132
| style="text-align: center;" | <math> Q =k \bar{Q} </math>2133
|}2134
|}2135
2136
where <math display="inline">k</math> is load factor and <math display="inline">\bar{Q}</math> is the constant reference load defined by a set of uniformly distributed concentrated forces <math display="inline">F_i</math>2137
2138
{| class="formulaSCP" style="width: 100%; text-align: left;" 2139
|-2140
| 2141
{| style="text-align: left; margin:auto;width: 100%;" 2142
|-2143
| style="text-align: center;" | <math> \bar{Q}=\{ F_i\} , i=1,12 </math>2144
|}2145
|}2146
2147
The following values of reference forces have been assumed:2148
2149
{| class="formulaSCP" style="width: 100%; text-align: left;" 2150
|-2151
| 2152
{| style="text-align: left; margin:auto;width: 100%;" 2153
|-2154
| style="text-align: center;" | <math> F_1=200 \mbox{N}, \quad F_2=400 \mbox{N}, \quad F_3=400 \mbox{N}, \quad F_4=300 \mbox{N}, \quad F_5=200 \mbox{N}, \quad </math>2155
|}2156
|}2157
2158
{| class="formulaSCP" style="width: 100%; text-align: left;" 2159
|-2160
| 2161
{| style="text-align: left; margin:auto;width: 100%;" 2162
|-2163
| style="text-align: center;" | <math> F_6=100 \mbox{N}, \quad F_7=50 \mbox{N}, \quad F_8=5 \mbox{N}, \quad F_9=2 \mbox{N}, \quad F_{10}=1 \mbox{N}, \quad F_{11}=1 \mbox{N}. </math>2164
|}2165
|}2166
2167
The load has been transmitted to soil by a kind of diaphragm with hinges in the places of force application.2168
2169
The following material properties for the pipe have been assumed: Young's modulus <math display="inline">E=2\cdot 10^5</math> MPa, Poisson's coefficient <math display="inline">\nu=0.3</math>, density <math display="inline">\rho=7800</math> kg/m<math display="inline">^3</math>, yield stress <math display="inline">\sigma _Y = 300</math> MPa, isotropic hardening modulus <math display="inline">H=300</math> MPa. The pipe has been discretized with 42 plane strain triangular elements based on the FIC formulation.2170
2171
The soil specimen has been modelled by 15 000 cylindrical discrete elements of diameters in the range from 2 cm to 3 cm. The following microscopic parameters in the soil model have been used: density <math display="inline">\rho=2800</math> kg/m<math display="inline">^3</math>, the stiffness in the normal and tangential directions <math display="inline">k_n=k_T=200</math> GPa, Coulomb friction coefficient <math display="inline">\mu=0.8</math>.2172
2173
The pipe–soil interface has been defined by the stiffness in the normal and tangential directions <math display="inline">k_n=k_T=20</math> GPa, Coulomb friction coefficient <math display="inline">\mu=0.3</math>.2174
2175
Numerical results for the deformed pipe shape are shown in Fig. [[#img-39|39]]. The increasing soil pressure leads to ovalization of the pipe. Due to large soil pressure the pipe instability is observed. Pipe material undergoes elasto-plastic deformation. Pipe cross-sections with effective plastic strain distribution at different load level are shown in Fig. [[#img-40|40]].2176
2177
Pipe cross-sections deformation can be described using an ovalization factor parameter <math display="inline">f_{ov}</math>, defined in terms of the maximum and minimum pipe diameters, <math display="inline">D_{max}</math> and <math display="inline">D_{min}</math>, respectively as2178
2179
{| class="formulaSCP" style="width: 100%; text-align: left;" 2180
|-2181
| 2182
{| style="text-align: left; margin:auto;width: 100%;" 2183
|-2184
| style="text-align: center;" | <math> f_{ov}= \frac{D_{max}-D_{min}}{D_{min}+D_{max}}\cdot 100% </math>2185
|}2186
|}2187
2188
Variation of the ovalization factor for different load levels is shown in Fig. [[#img-41|41]]. Numerical results agree with those given in <span id='citeF-43'></span>[[#cite-43|[43]]].2189
2190
2191
<div id='img-39'></div>2192
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"2193
|-2194
|[[Image:Draft_Samper_102318540-pipeovaln3s-wd-geo-0.png|480px|]]2195
|[[Image:Draft_Samper_102318540-pipeovaln3s-wd-geo-02.png|480px|]]2196
|-2197
|[[Image:Draft_Samper_102318540-pipeovaln3s-wd-geo-04.png|480px|]]2198
|[[Image:Draft_Samper_102318540-pipeovaln3s-wd-geo-06.png|480px|Pipe ovalization – deformed configuration at different load levels]]2199
|- style="text-align: center; font-size: 75%;"2200
| colspan="2" | '''Figure 39:''' Pipe ovalization – deformed configuration at different load levels2201
|}2202
2203
<div id='img-40'></div>2204
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"2205
|-2206
|[[Image:Draft_Samper_102318540-pipeovaln3s-wd-eps-0.png|180px|]]2207
|[[Image:Draft_Samper_102318540-pipeovaln3s-wd-eps-02.png|230px|]]2208
|-2209
|[[Image:Draft_Samper_102318540-pipeovaln3s-wd-eps-04.png|270px|]]2210
|[[Image:Draft_Samper_102318540-pipeovaln3s-wd-eps-06.png|306px|Pipe ovalization – deformed configuration]]2211
|- style="text-align: center; font-size: 75%;"2212
| colspan="2" | '''Figure 40:''' Pipe ovalization – deformed configuration2213
|}2214
2215
<div id='img-41'></div>2216
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"2217
|-2218
|[[Image:Draft_Samper_102318540-p3s-ovf.png|450px|Pipe ovalization factor]]2219
|- style="text-align: center; font-size: 75%;"2220
| colspan="1" | '''Figure 41:''' Pipe ovalization factor2221
|}2222
2223
==Conclusions==2224
2225
The combination of spherical rigid elements and finite elements is a promising approach for simulation of geomechanical problems involving fracture, penetration and wear. The transient dynamic formulation presented allows to model the motion of discrete and finite elements using a unified algorithm. Frictional contact effects between the rigid spheres and between the spheres and finite elements can be simply accounted for. FIC stabilization procedure allows to use low order triangles and tetrahedra elements with equal order interpolation of the displacement and pressure variables free of the volumetric locking problem. This allows us to apply the presented formulation to problems where large plastic deformation occur in the continuum part of the domain. The discrete formulation can be used to model tools in rock cutting operations. This allows to reproduce tool wear simply by removing the worn particles from the tool surface, thus reproducing the material loss in a realistic manner. Further applications of this approach to the prediction of wear of rock cutting process can be found in <span id='citeF-17'></span><span id='citeF-44'></span>[[#cite-17|[17,44]]].2226
2227
==References==2228
2229
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'''[[#citeF-35|[35]]]''' E. O<math display="inline">\tilde{\mbox{n}}</math>ate, J. García, and S. Idelhson. An alpha-adaptive approach for stabilized finite element solution of advective-diffusive problems. In P. Ladeveze and J.T. Oden, editors, ''New Adv. in Adaptive Comp. Meth. in Mech.'' Elsevier, 1998.2333
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'''[[#citeF-36|[36]]]''' E. O<math display="inline">\tilde{\mbox{n}}</math>ate and M. Manzan. A general procedure for deriving stabilized space-time finite element methods for advective-diffusive problems. ''Int. J. Num. Meth. Fluids.'', 31:203 – 221, 1999.2336
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'''[[#citeF-37|[37]]]''' E. O<math display="inline">\tilde{\mbox{n}}</math>ate and J. García. A finite element method for fluid–structure interaction with surface waves using a finite calculus formulation. ''Comput. Meth. Appl. Mech. Eng.'', 191:635–660, 2001.2339
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'''[[#citeF-38|[38]]]''' J. García and E. O<math display="inline">\tilde{\mbox{n}}</math>ate. An unstructured finite element solver for ship hydrodynamic problems. ''J. Appl. Mech.'', 2002.2342
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'''[[#citeF-39|[39]]]''' E. O<math display="inline">\tilde{\mbox{n}}</math>ate, C. Sacco, and S. Idelhson. A finite point method for incompressible flow problems. ''Computing and Visualization in Science'', 3:67–75, 2000.2345
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'''[[#citeF-40|[40]]]''' H. Huang. ''Discrete Element Modeling of Tool-Rock Interaction''. PhD thesis, University of Minnesota, 1999.2348
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'''[[#citeF-43|[43]]]''' http://geosim.engr.mun.ca/pipesoil.htm.2357
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'''[[#citeF-44|[44]]]''' E. Oñate, C. Recarey, F. Zarate, J. Miquel, and J. Rojek. Characterization of micro-macro parameters in discrete element formulation. (to be published).2360
2361
==APPENDIX A==2362
2363
===A.1 Matrices and vectors in FEM equations of motion using the FIC formulation===2364
2365
In FEM equations of motion using the FIC formulation ([[#eq-94|94]])–([[#eq-96|96]]) the following vectors and matrices have been used:2366
2367
{| class="formulaSCP" style="width: 100%; text-align: left;" 2368
|-2369
| 2370
{| style="text-align: left; margin:auto;width: 100%;" 2371
|-2372
| style="text-align: center;" | <math>{\bf M}_{ij}=\int_{\Omega^e} \rho {\bf N}_i {\bf N}_j\,d\Omega\quad ,\quad {\bf N}_i = N_I {\bf I}_3 </math>2373
|}2374
| style="width: 5px;text-align: right;white-space: nowrap;" | (A.1)2375
|}2376
2377
is the mass matrix2378
2379
<span id="eq-2"></span>2380
{| class="formulaSCP" style="width: 100%; text-align: left;" 2381
|-2382
| 2383
{| style="text-align: left; margin:auto;width: 100%;" 2384
|-2385
| style="text-align: center;" | <math>{\bf g}= \int_\Omega {\bf B}^T {\boldsymbol\sigma} d\Omega </math>2386
|}2387
| style="width: 5px;text-align: right;white-space: nowrap;" | (A.2)2388
|}2389
2390
is the internal nodal force vector and the rest of matrices and vectors are2391
2392
{| class="formulaSCP" style="width: 100%; text-align: left;" 2393
|-2394
| 2395
{| style="text-align: left; margin:auto;width: 100%;" 2396
|-2397
| style="text-align: center;" | <math>{\bf G}_{ij} = \int_{\Omega^e} ({\boldsymbol\nabla} N_i) N_j \, d\Omega a </math>2398
|-2399
| style="text-align: center;" | <math> L_{ij}= \int _{\Omega ^e} {\boldsymbol \nabla }^T N_i [\tau ] {\boldsymbol \nabla } N_j \, d\Omega \qquad ,\quad C_{ij} = \int _{\Omega ^e} {1\over K} N_i N_j \, d\Omega </math>2400
|-2401
| style="text-align: center;" | <math> {\bf C} = \left[\begin{matrix} \bar {\bf C}^1 & {\bf 0}&{\bf 0}\\2402
{\bf 0}&\bar {\bf C}^2 & {\bf 0}\\ {\bf 0}&{\bf 0}&\bar {\bf C}^3\end{matrix}\right]\qquad ,\quad \bar C_{ij}^k = \int _{\Omega ^e} \tau _k N_i N_j \, d\Omega </math>2403
| style="width: 5px;text-align: right;white-space: nowrap;" | (A.3)2404
|-2405
| style="text-align: center;" | <math> {\bf Q} = [{\bf Q}^1, {\bf Q}^2, {\bf Q}^3]\quad , \quad Q_{ij}^k = \int _{\Omega ^e} \tau _k {\partial N_i \over \partial x_k} N_j d\Omega </math>2406
|-2407
| style="text-align: center;" | <math> {\bf f}_i = \int_{\Omega^e} N_i {\bf b} \, d\Omega +2408
\int_\Gamma N_i \bar {\bf t} \, d\Gamma \quad , \quad i,j=1, n_d </math>2409
|}2410
|}2411
2412
In above <math display="inline">{\bf b}=[b_1,b_2,b_3]^T</math> and <math display="inline">\bar{\bf t} = [\bar t_1, \bar t_2,\bar t_3]^T</math>,2413
2414
{| class="formulaSCP" style="width: 100%; text-align: left;" 2415
|-2416
| 2417
{| style="text-align: left; margin:auto;width: 100%;" 2418
|-2419
| style="text-align: center;" | <math>{\boldsymbol \nabla } = \left\{\begin{matrix} {\partial \over \partial x_1}\\ {\partial \over \partial x_2}\\ {\partial \over \partial x_3}\\\end{matrix}\right\}\quad , \quad [\tau ] =\left[\begin{matrix} \tau _{1} & & {\bf 0}\\ & \tau _{2} &\\{\bf 0}&&\tau _{3}\\\end{matrix}\right]\quad , \quad {I}_3=\left[\begin{matrix}1&0&0\\ 0&1&0\\ 0&0&1\\\end{matrix} \right] </math>2420
|}2421
| style="width: 5px;text-align: right;white-space: nowrap;" | (A.4)2422
|}2423
2424
<math display="inline">{\bf B}</math> is the standard infinitesimal strain matrix and <math display="inline">{\bf D}_d</math> is the deviatoric constitutive matrix <span id='citeF-3'></span>[[#cite-3|[3]]]. Note that the expression of <math display="inline">{\bf g}</math> of eq. ([[#eq-2|A.2]]) is adequate for non linear structural analysis.2425
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Published on 01/01/2004
DOI: 10.1016/j.cma.2003.12.056
Licence: CC BY-NC-SA license
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