Oñate et al 2017a - Revision history

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@@ Line 646: / Line 646: @@
 The enhanced local DEM constitutive model can be used in conjunction with any constitutive law in solid mechanics for predicting the linear and non linear behavior of a solid. This opens the door for predicting with the DEM the linear and non linear response of solids and structures of any material (including metallic materias) <span id='citeF-3'></span>[[#cite-3|[3]]], which is unusual for standard DEM models.
-==4 interaction of discrete particles and solids modelled with the FEM==
+==3. Interaction of discrete particles and solids modelled with the FEM==
 The interaction of discrete particles and solids can be studied by coupling the potential of the DEM and FEM approaches. Isolated particles can be modeled with the DEM, while solids are better modeled using the FEM. Indeed, complex objects can also be modeled using a collection of DEM particles, which allows one to use the standard frictional contact algorithms between particles and rigid/deformable objects  developed for the DEM. Clearly, for rigid objects only the DEM particles discretizing the boundary of the object need to be accounted for.
@@ Line 730: / Line 730: @@
 |}
-===4.1 A FEM-DEM procedure for multifracture analysis of solids===
+===3.1 A FEM-DEM procedure for multifracture analysis of solids===
 The DEM is a flexible method to simulate granular and non-continuum media, in particular the propagation of initial cracks. On the other hand, these contact properties are defined at the micro scale, while the material properties usually refer to experimental results in the macro scale. The step between both scales is not easy and requires a calibration task. The FEM otherwise is based in a continuum formulation involving the macro properties of the material. The FEM allows one to establish failure criteria compatible with the equilibrium equations in continuum mechanics, which makes it consistent and easy to apply for different materials.
@@ Line 787: / Line 787: @@
 A relevant point in the FEM-DEM approach described above is its computational cost. Most of the cost in a DEM simulation is due to the contact searching algorithms. In the FEM-DEM technique presented the number of discrete elements is in general much lower than the number of nodes because the fractured area is typically smaller than the whole calculation domain. Hence, the computational time consumed by the contact searching algorithms is much lower than in a standard DEM solution.
-===4.2 Examples of application of the FEM-DEM procedure===
+===3.2 Examples of application of the FEM-DEM procedure===
 ====Four-point bending beam====

@@ Line 485: / Line 485: @@
 |}
-===3.4 Enhanced local constitutive model for the DEM that accurately reproduces the behavior of a continnum===
+===2.4 Enhanced local constitutive model for the DEM that accurately reproduces the behavior of a continnum===
 The standard DEM technique is known to suffer from sensitivity to the packaging pattern and density of the particles when applied to the linear elastic analysis of a continuum. Standard features of a continuum domain, such as the Poisson's ratio effect, are difficult to predict with the DEM in a consistent manner.

@@ Line 12: / Line 12: @@
 In this chapter we present first recent advances in the DEM for non linear analysis of cohesive and non cohesive materials. Then a method for coupling the DEM and FEM procedures and for studying the interaction of physical particles and deformable solids es explained. In the last part of the chapter we present an approach for modeling multifracture situations in a solid by starting with the FEM analysis of the continuum domain in the standard manner. Discrete elements at the element nodes are progressively introduced as damage on the center of the element sides exceeds a certain value. Examples of this FEM-DEM technique are presented for a number of structural problems involving single fracture and multifracture situations such as the failure analysis of concrete samples and beams under external loads, a shale rock domain under a pulse load and  blasting situations in a tunnel front and a granite rock specimen.
-==3 a local DEM model==
+==2. A local DEM model==
-===3.1 DEM model overview===
+===2.1 DEM model overview===
 The DEM  was initially developed by Cundall et al. <span id='citeF-5'></span>[[#cite-5|[5]]] in the 1970's. It is based in the interaction of discrete elements (also called particles) &#8211; typically cylinders (in 2D) and spheres (in 3D) &#8211; to simulate the behavior of  continuum and discontinuum domains <span id='citeF-2'></span><span id='citeF-3'></span><span id='citeF-6'></span><span id='citeF-7'></span><span id='citeF-12'></span><span id='citeF-14'></span><span id='citeF-15'></span><span id='citeF-21'></span><span id='citeF-24'></span><span id='citeF-26'></span><span id='citeF-31'></span>[[#cite-2|[2,3,6,7,12,14,15,21,24,26,31]]]. This interaction is governed by a set of kinematic equations involving the forces acting over the discrete elements and the displacements, velocities and accelerations of the particles. The forces acting over a discrete element are related to the stresses and strains according to a constitutive model. In our work we use the local constitutive model for the DEM  for cohesive and non-cohesive materials proposed by Oñate et al. <span id='citeF-21'></span>[[#cite-21|[21]]]. In the following a brief description of this model is presented.
@@ Line 40: / Line 40: @@
 |}
-====3.1.1 Kinematic equations and integration scheme====
+====2.1.1 Kinematic equations and integration scheme====
 The translation and rotation of the  particles in the DEM is governed by the standard dynamics equations for rigid bodies,
@@ Line 98: / Line 98: @@
 where <math display="inline">\xi </math> is a fraction of the critical damping <span id='citeF-5'></span><span id='citeF-22'></span>[[#cite-5|[5,22]]].
-====3.1.2 Forces acting over the discrete element====
+====2.1.2 Forces acting over the discrete element====
 The interaction forces at the contact interface between two particles ''i'' and ''j'' (<math display="inline">{F}^{ij}</math>) are obtained from the normal (<math display="inline">{F}_{n}^{ij}</math>) and tangential (<math display="inline">{F}_{s}^{ij}</math>) components (Figure [[#img-2|2]]b).
@@ Line 251: / Line 251: @@
 The local DEM constitutive model described above holds for cohesive and non-cohesive materials, this latter as a particular case of the former, when the bonds between the particles are assumed to be initially broken. More details can be found in <span id='citeF-21'></span>[[#cite-21|[21]]].
-===3.2 Normal and shear failure===
+===2.2 Normal and shear failure===
 Cohesive bonds at a contact interface are assumed to start breaking when the interface strength is exceeded in the normal direction by the tensile  contact force, ''or'' in the tangential direction by the shear force. The ''uncoupled'' failure (decohesion) criterion for the normal and tangential directions at the contact interface between particles <math display="inline">i</math> and <math display="inline">j</math> is written as
@@ Line 337: / Line 337: @@
 |}
-===3.3 Elasto-plastic model for compression forces===
+===2.3 Elasto-plastic model for compression forces===
 The compressive stress-strain behaviour in the normal direction at the contact interface for frictional cohesive materials, such as cement, rock and concrete, is typically governed by an initial elastic law followed by a non-linear constitutive equation that varies for each material. The compressive normal stress increases under  linear elastic conditions until it reaches the  limit normal compressive  stress <math display="inline">\sigma _{n_c}^l</math>. This is defined as the axial stress level where the experimental curve relating the axial stress and the axial strain   starts to deviate from the linear elastic behaviour. After this point the material is assumed to yield under ''elastic-plastic'' conditions.

@@ Line 4: / Line 4: @@
 In this chapter we present recent advances in the Discrete Element Method (DEM) and in the coupling of the DEM with the Finite Element Method (FEM) for solving a variety of problems in non linear solid mechanics involving damage, plasticity and multifracture situations.
-==2 INTRODUCTION==
+==1. Introduction==
 The Discrete Element Method (DEM) is a popular technique for analysis of the mechanics of granular matter, as well as for modeling multifracture situations in frictional materials such as concrete and geomaterials.

@@ Line 12: / Line 12: @@
 In this chapter we present first recent advances in the DEM for non linear analysis of cohesive and non cohesive materials. Then a method for coupling the DEM and FEM procedures and for studying the interaction of physical particles and deformable solids es explained. In the last part of the chapter we present an approach for modeling multifracture situations in a solid by starting with the FEM analysis of the continuum domain in the standard manner. Discrete elements at the element nodes are progressively introduced as damage on the center of the element sides exceeds a certain value. Examples of this FEM-DEM technique are presented for a number of structural problems involving single fracture and multifracture situations such as the failure analysis of concrete samples and beams under external loads, a shale rock domain under a pulse load and  blasting situations in a tunnel front and a granite rock specimen.
-==3 A LOCAL DEM MODEL==
+==3 a local DEM model==
 ===3.1 DEM model overview===

← Older revision	Revision as of 11:20, 26 October 2017
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 This chapter has shown the possibility of the DEM for linear and non linear analysis of cohesive materials and structures, as well as the advantages of coupling the FEM and DEM techniques for studying the interaction of particles with structures and the prediction of complex multifracture situations in solids.
-==ACKNOWLEDGEMENTS==
+==Acknowledgements==
 This research was partially funded by the ICEBREAKER project of the European Research Council. Support for this work was also provided from the Office for Naval Research Global (ONRG) of the US Navy through the NICE-SHIP project. We also acknowledge the financial support of the CERCA programme of the Generalitat de Catalunya.

@@ Line 936: / Line 936: @@
 |}
-==CONCLUDING REMARKS==
+==Concluding remarks==
 This chapter has shown the possibility of the DEM for linear and non linear analysis of cohesive materials and structures, as well as the advantages of coupling the FEM and DEM techniques for studying the interaction of particles with structures and the prediction of complex multifracture situations in solids.