Linear Parallel Bond Model
The linear parallel bond model with inactive dashpots and a reference gap of zero corresponds with the parallelbond model of [Potyondy2004]. It is a linearbased model that can be installed at both ballball and ballfacet contacts and is referred to in commands and FISH by the name linearpbond.
Introduction
A parallel bond provides the mechanical behavior of a finitesized piece of cementlike material deposited between the two contacting pieces (similar to the epoxy cementing the glass beads shown in Figure 2). The parallelbond component acts in parallel with the linear component and establishes an elastic interaction between the pieces. The existence of a parallel bond does not preclude the possibility of slip. Parallel bonds can transmit both force and moment between the pieces.
A parallel bond can be envisioned as a set of elastic springs with constant normal and shear stiffnesses, uniformly distributed over a {rectangular in 2D; circular in 3D} crosssection lying on the contact plane and centered at the contact point. These springs act in parallel with the springs of the linear component. Relative motion at the contact, occurring after the parallel bond has been created, causes a force and moment to develop within the bond material. This force and moment act on the two contacting pieces and can be related to maximum normal and shear stresses acting within the bond material at the bond periphery. If either of these maximum stresses exceeds its corresponding bond strength, the parallel bond breaks, and the bond material is removed from the model along with its accompanying force, moment, and stiffnesses.
Behavior Summary
The linear parallel bond model provides the behavior of two interfaces: an infinitesimal, linear elastic (notension), and frictional interface that carries a force and a finitesize, linear elastic, and bonded interface that carries a force and moment (see Figure 1). The first interface is equivalent to the linear model: it does not resist relative rotation, and slip is accommodated by imposing a Coulomb limit on the shear force. The second interface is called a parallel bond, because when bonded, it acts in parallel with the first interface. When the second interface is bonded, it resists relative rotation, and its behavior is linear elastic until the strength limit is exceeded and the bond breaks, making it unbonded. When the second interface is unbonded, it carries no load. The unbonded linear parallel bond model is equivalent to the linear model.
ActivityDeletion Criteria
A contact with the linear parallel bond model is active if it is bonded or if the surface gap is less than or equal to zero. The forcedisplacement law is skipped for inactive contacts. The surface gap is shown in this figure of the linear formulation. When the reference gap is zero, the notional surfaces of the first interface coincide with the piece surfaces. The notional surfaces of the second interface are shown in Figure 2.
ForceDisplacement Law
The forcedisplacement law for the linear parallel bond model updates the contact force and moment:
where \(\mathbf{F^{l}}\) is the linear force, \(\mathbf{F^{d}}\) is the dashpot force, \(\overline{\mathbf{F}}\) is the parallelbond force, and \(\overline{\mathbf{M}}\) is the parallelbond moment. The linear and dashpot forces are updated as in the linear model, while the force and moment in the parallel bond are updated as described below.
The parallelbond force is resolved into a normal and shear force, and the parallelbond moment is resolved into a twisting and bending moment:
where \(\overline{F}_{n} > 0\) is tension. The parallelbond shear force and bending moment lie on the contact plane and are expressed in the contact plane coordinate system:
When a parallel bond is created via the bond method, an interface between two notional surfaces is established, and the parallelbond force and moment are zeroed. The parallel bond provides an elastic interaction between these two notional surfaces, and this interaction is removed when the bond breaks. Each notional surface is connected rigidly to the piece of a body as shown in Figure 2. The parallelbond surface gap is defined as the cumulative relative normal displacement of the piece surfaces:
where \(\Delta \delta _{n}\) is the relative normaldisplacement increment of Equation (12) of the “Contact Resolution” section.
The forcedisplacement law for the parallelbond force and moment consists of the following steps (see Figure 3).
Update the bond crosssectional properties:
(5)\[\begin{split}\begin{array}{c} \overline{R} = \overline{\lambda} \left\{ \begin{array}{rl} \min\left( R^{(1)}, R^{(2)} \right), & \mbox{ballball} \\ R^{(1)}, & \mbox{ballfacet} \end{array} \right. ,\quad \overline{A} = \left\{ \begin{array}{rl} 2 \bar{R} t, & \mbox{2D $(t = 1)$} \\ \pi \bar{R}^2, & \mbox{3D} \end{array} \right. \\[3mm] \overline{I} = \left\{ \begin{array}{rl} \tfrac{2}{3} t \bar{R}^3, & \mbox{2D $(t = 1)$} \\ \tfrac{1}{4} \pi \bar{R}^4, & \mbox{3D} \end{array} \right. ,\quad \overline{J} = \left\{ \begin{array}{rl} 0, & \mbox{2D} \\ \tfrac{1}{2} \pi \bar{R}^4, & \mbox{3D} \end{array} \right. \end{array}\end{split}\]where \(\overline{A}\) is the crosssectional area, \(\overline{I}\) is the moment of inertia of the parallel bond cross section (about the line passing through \(\mathbf{x_{c}}\) and in the direction of \(\overline{\mathbf{M}}_\mathbf{b}\)), and \(\overline{J}\) is the polar moment of inertia of the parallel bond cross section (about the line passing through \(\mathbf{x_{c}}\) and in the direction of \(\hat{\mathbf{n}}_\mathbf{c}\)). The bond cross section is {rectangular in 2D; circular in 3D}.
Update \(\overline{F}_n\):
(6)\[\overline{F}_{n} :=\overline{F}_{n} +\overline{k}_{n} {\kern 1pt} \overline{A}\, \Delta \delta _{n}\]where \(\Delta \delta _{n}\) is the relative normaldisplacement increment of Equation (12) of the “Contact Resolution” section.
Update \(\overline{\mathbf{F}}_\mathbf{s}\):
(7)\[\overline{\mathbf{F}}_\mathbf{s} :=\overline{\mathbf{F}}_\mathbf{s} \overline{k}_{s} {\kern 1pt} \overline{A}{\kern 1pt} \Delta \pmb{δ} _\mathbf{s}\]where \(\Delta \pmb{δ}_\mathbf{s}\) is the relative sheardisplacement increment of Equation (12) of the “Contact Resolution” section.
Update \(\overline{M}_{t}\):
(8)\[\begin{split} \overline{M}_t := \left\{ \begin{array}{rl} 0, & \mbox{2D} \\ \overline{M}_t  \overline{k}_s \overline{J} \Delta \theta_t, & \mbox{3D} \end{array} \right.\end{split}\]where \(\Delta \theta _{t}\) is the relative twistrotation increment of Equation (12) of the “Contact Resolution” section. The torsional stiffness \((k_{t})\) of a twisted elastic circular shaft of length \(L\) loaded at its ends by equal and opposite twisting moments is given by [Crandall1978] (Eq. 6.10):
(9)\[M_{t} =k_{t} \theta _{t} ,\quad k_{t} =\frac{GJ}{L}\]where \(G\) is the shear modulus and \(J\) is the polar moment of inertia of the crosssectional area about the axis of the shaft. The momenttwist relation of Equation (8) is obtained by setting \(k_{t} =\overline{k}_{s} {\kern 1pt} \overline{J}\).
Update \(\overline{\mathbf{M}}_\mathbf{b}\):
(10)\[\overline{\mathbf{M}}_\mathbf{b} := \overline{\mathbf{M}}_\mathbf{b}  \overline{k}_{n} {\kern 1pt} \overline{I}{\kern 1pt} \Delta \pmb{θ}_\mathbf{b}\]where \(\Delta \pmb{θ}_\mathbf{b}\) is the relative bendrotation increment of Equation (12) of the “Contact Resolution” section. The bending stiffness \(\left(k_{b} \right)\) of an elastic symmetrical beam of length \(L\) loaded at its ends by equal and opposite bending moments is given by [Crandall1978] (Eq. 7.14):
(11)\[M_{b} =k_{b} \theta _{b} ,\quad k_{b} =\frac{EI}{L}\]where \(E\) is the Young’s modulus and \(I\) is the moment of inertia of the crosssectional area about the neutral axis of the beam. The momenttwist relation of Equation (11) is obtained by setting \(k_{b} =\overline{k}_{n} {\kern 1pt} \overline{I}\).
Update the maximum normal (\(\overline{\sigma}\), \(\overline{\sigma} > 0\) is tension) and shear stresses at the parallelbond periphery:
(12)\[\begin{split}\begin{array}{l} \overline{\sigma} = \displaystyle\frac{\overline{F}_n}{\overline{A}} + \overline{\beta} \frac{\left\ \overline{\mathbf{M}}_\mathbf{b} \right\ \overline{R}}{\overline{I}} ,\quad \overline{\tau} = \frac{\left\ \overline{\mathbf{F}}_\mathbf{s} \right\}{\overline{A}} + \left\{ \begin{array}{rl} 0, & \mbox{2D} \\ \overline{\beta} \frac{\left \overline{M}_t \right \overline{R}}{\overline{J}}, & \mbox{3D} \end{array} \right. \\[1mm] \mbox{with }\quad \overline{\beta} \in [0, 1]. \end{array}\end{split}\]The momentcontribution factor \(\left( \overline{\beta } \right)\) is discussed in [Potyondy2011].
Enforce the strength limits (see Figure 4). If the tensilestrength limit is exceeded \(\left(\overline{\sigma }>\overline{\sigma }_{c} \right)\), then break the bond in tension:
(13)\[\overline{B}=1,\quad \left\{\overline{F}_{n} ,\overline{F}_{ss} ,\overline{F}_{st} ,\overline{M}_{t} ,\overline{M}_{bs} ,\overline{M}_{bt} \right\}=0.\]If the bond has not broken in tension, then enforce the shearstrength limit. The shear strength \(\overline{\tau}_{c} = \overline{c}  \sigma \tan \overline{\phi}\), where \(\sigma = \overline{F}_{n} / \overline{A}\) is the average normal stress acting on the parallel bond cross section. If the shearstrength limit is exceeded \((\overline{\tau} > \overline{\tau}_{c})\), then break the bond in shear:
(14)\[\overline{B}=2,\quad \left\{\overline{F}_{n} ,\overline{F}_{ss} ,\overline{F}_{st} ,\overline{M}_{t} ,\overline{M}_{bs} ,\overline{M}_{bt} \right\}=0.\]If the bond has broken, then the bond_break callback event is triggered.
Energy Partitions
The linear parallel bond model provides four energy partitions:
strain energy, \(E_{k}\), stored in the linear springs;
slip energy, \(E_{\mu }\), defined as the total energy dissipated by frictional slip;
dashpot energy, \(E_{\beta }\), defined as the total energy dissipated by the dashpots; and
bond strain energy, \(\overline{E}_k\), stored in the parallelbond springs.
Keyword 
Symbol 
Description 
Range 
Accumulated 

Linear Group: 


\(E_{k}\) 
strain energy 
\([0.0,+\infty)\) 
NO 

\(E_{\mu}\) 
total energy dissipated by slip 
\((\infty,0.0]\) 
YES 
Dashpot Group: 


\(E_{\beta}\) 
total energy dissipated by dashpots 
\((\infty,0.0]\) 
YES 
ParallelBond Group: 


\(\overline{E}_k\) 
bond strain energy 
\([0.0,+\infty)\) 
NO 
The strain energy, slip energy, and dashpot energy are updated as for the linear model (see this section). The slip energy remains equal to zero until the bond breaks. The bond strain energy is updated via:
Properties
The properties defined by the linear parallel bond model are listed in the table below for a concise reference; see the “Contact Properties” section for a description of the information in the table columns. The mapping from the surface inheritable properties to the contact model properties is also discussed below.
Keyword 
Symbol 
Description 
Type 
Range 
Default 
Modifiable 
Inheritable 

linearpbond 
Model name 

Linear Group: 

kn 
\(k_n\) 
Normal stiffness [force/length] 
FLT 
\([0.0,+\infty)\) 
0.0 
YES 
YES 
ks 
\(k_s\) 
Shear stiffness [force/length] 
FLT 
\([0.0,+\infty)\) 
0.0 
YES 
YES 
fric 
\(\mu\) 
Friction coefficient [] 
FLT 
\([0.0,+\infty)\) 
0.0 
YES 
YES 
rgap 
\(g_r\) 
Reference gap [length] 
FLT 
\(\mathbb{R}\) 
0.0 
YES 
NO 
lin_mode 
\(M_l\) 
Normalforce update mode [] 
INT 
\(\{0,1\}\) 
0 
YES 
NO 
\(\;\;\;\;\;\;\begin{cases} \mbox{0: update is absolute} \\ \mbox{1: update is incremental} \end{cases}\) 

emod 
\(E^*\) 
Effective modulus [force/area] 
FLT 
\([0.0,+\infty)\) 
0.0 
NO 
N/A 
kratio 
\(\kappa^*\) 
Normaltoshear stiffness ratio [] 
FLT 
\([0.0,+\infty)^*\) 
0.0\(^*\) 
NO 
N/A 
\(\kappa^* \equiv \frac{k_n}{k_s}\) 

lin_slip 
\(s\) 
Slip state [] 
BOOL 
{false,true} 
false 
NO 
N/A 
\(\;\;\;\;\;\;\begin{cases} \mbox{true: slipping} \\ \mbox{false: not slipping} \end{cases}\) 

lin_force 
\(\mathbf{F^l}\) 
Linear force (contact plane coord. system) 
VEC 
\(\mathbb{R}^3\) 
\(\mathbf{0}\) 
YES 
NO 
\(\left( F_n^l,F_{ss}^l,F_{st}^l \right) \quad \left(\mbox{2D model: } F_{ss}^l \equiv 0 \right)\) 

user_area 
\(A\) 
Constant area [length*length] 
FLT 
\((0.0,+\infty)\) 
0.0 
YES 
NO 
Dashpot Group: 

dp_nratio 
\(\beta_n\) 
Normal critical damping ratio [] 
FLT 
\([0.0,1.0]\) 
0.0 
YES 
NO 
dp_sratio 
\(\beta_s\) 
Shear critical damping ratio [] 
FLT 
\([0.0,1.0]\) 
0.0 
YES 
NO 
dp_mode 
\(M_d\) 
Dashpot mode [] 
INT 
{0,1,2,3} 
0 
YES 
NO 
\(\;\;\;\;\;\;\begin{cases} \mbox{0: full normal & full shear} \\ \mbox{1: notension normal & full shear} \\ \mbox{2: full normal & slipcut shear} \\ \mbox{3: notension normal & slipcut shear} \end{cases}\) 

dp_force 
\(\mathbf{F^d}\) 
Dashpot force (contact plane coord. system) 
VEC 
\(\mathbb{R}^3\) 
\(\mathbf{0}\) 
NO 
NO 
\(\left( F_n^d,F_{ss}^d,F_{st}^d \right) \quad \left(\mbox{2D model: } F_{ss}^d \equiv 0 \right)\) 

ParallelBond Group: 

pb_rmul 
\(\overline{\lambda}\) 
Radius multiplier [] 
FLT 
\((0.0,+\infty)\) 
1.0 
YES 
NO 
pb_kn 
\(\overline{k}_n\) 
Normal stiffness [stress/disp.] 
FLT 
\([0.0,+\infty)\) 
0.0 
YES 
NO 
pb_ks 
\(\overline{k}_s\) 
Shear stiffness [stress/disp.] 
FLT 
\([0.0,+\infty)\) 
0.0 
YES 
NO 
pb_mcf 
\(\overline{\beta}\) 
Momentcontribution factor [] 
FLT 
\([0.0,1]\) 
1.0 
YES 
NO 
pb_ten 
\(\overline{\sigma}_c\) 
Tensile strength [stress] 
FLT 
\([0.0,+\infty)\) 
0.0 
YES 
NO 
pb_coh 
\(\overline{c}\) 
Cohesion [stress] 
FLT 
\([0.0,+\infty)\) 
0.0 
YES 
NO 
pb_fa 
\(\overline{\phi}\) 
Friction angle [degrees] 
FLT 
\([0.0,90.0)\) 
0.0 
YES 
NO 
pb_state 
\(\overline{B}\) 
Bond state 
INT 
{0,1,2,3} 
0 
NO 
NO 
\(\;\;\;\;\;\;\begin{cases} \mbox{0: unbonded} \\ \mbox{1: unbonded & broke in tension} \\ \mbox{2: unbonded & broke in shear} \\ \mbox{3: bonded} \end{cases}\) 

pb_radius 
\(\overline{R}\) 
Bond radius [length] 
FLT 
\((0.0,+\infty)\) 
N/A 
NO (set via \(\overline{\lambda}\)) 
NO 
pb_emod 
\(\overline{E}^{*}\) 
Bond effective modulus [stress] 
FLT 
\([0.0,+\infty)\) 
0.0 
NO 
N/A 
pb_kratio 
\(\overline{\kappa }^{*}\) 
Bond normaltoshear stiffness ratio 
FLT 
\([0.0,+\infty)^{*}\) 
0.0 
NO 
N/A 
\(\overline{\kappa}^* \equiv \frac{\overline{k}_n}{\overline{k}_s}\) 

pb_shear 
\(\overline{\tau }_{c}\) 
Bond shear strength [stress] 
FLT 
\([0.0,+\infty)\) 
0.0 
NO (set via \(\overline{c}\) and \(\overline{\sigma}_c\)) 
N/A 
pb_sigma 
\(\overline{\sigma }\) 
Normal stress at bond periphery [stress] 
FLT 
\([0.0,+\infty)\) 
0.0 
NO 
N/A 
pb_tau 
\(\overline{\tau }\) 
Shear stress at bond periphery [stress] 
FLT 
\([0.0,+\infty)\) 
0.0 
NO 
N/A 
pb_force 
\(\overline{\mathbf{F}}\) 
Parallelbond force (contact plane coord. system) 
VEC 
\(\mathbb{R}^3\) 
\(\mathbf{0}\) 
YES 
NO 
\(\left( \overline{F}_n,\overline{F}_{ss},\overline{F}_{st} \right) \quad \left(\mbox{2D model: } \overline{F}_{ss} \equiv 0 \right)\) 

pb_moment 
\(\overline{\mathbf{M}}\) 
Parallelbond moment (contact plane coord. system) 
VEC 
\(\mathbb{R}^3\) 
\(\mathbf{0}\) 
YES 
NO 
\(\left( \overline{M}_t,\overline{M}_{bs},\overline{M}_{bt} \right) \quad \left(\mbox{2D model: } \overline{M}_{t} \equiv \overline{M}_{bt} \equiv 0 \right)\) 

\(^*\) By convention, \(\kappa^*\) and \(\overline{\kappa }^{*}\) equal zero if either corresponding normal or shear stiffness is zero. 
Note
Modifying the contact model force will not alter forces accumulated to the bodies. Therefore, any change to \(\mathbf{F^l}\), \(\overline{\mathbf{F}}\), or \(\overline{\mathbf{M}}\) may only be effective during the next forcedisplacement calculation. When \(M_l = 0\), the normal component of the linear force is automatically overridden during the next forcedisplacement calculation.
Surface Property Inheritance
The linear stiffnesses, \(k_n\) and \(k_s\), and the friction coefficient, \(\mu\), may be inherited from the contacting pieces. See this section from the linear formulation for details on property inheritance.
Methods
Method 
Arguments 
Symbol 
Type 
Range 
Default 
Description 


Linear Group: 

area  Set user_area to the area 

deformability  Set deformability 

emod 
\(E^*\) 
FLT 
\([0.0,+\infty)\) 
N/A 
Effective modulus 

kratio 
\(\kappa^*\) 
FLT 
\([0.0,+\infty)^*\) 
N/A 
Normaltoshear stiffness ratio 

ParallelBond Group: 

pb_deformability 
Set bond deformability 

emod 
\(\overline{E}^*\) 
FLT 
\([0.0,+\infty)\) 
N/A 
Bond effective modulus 

kratio 
\(\overline{\kappa}^*\) 
FLT 
\([0.0,+\infty)^*\) 
N/A 
Bond normaltoshear stiffness ratio, \(\overline{\kappa}^* \equiv \frac{\overline{k}_n}{\overline{k}_s}\) 

bond 
Bond the contact if \(g_c \in G\) 

gap 
\(G\) 
VEC2 
\(\mathbb{R}^2\) 
\((\infty,0]\) 
Gap interval 

unbond 
Unbond the contact if \(g_c \in G\) 

gap 
\(G\) 
VEC2 
\(\mathbb{R}^2\) 
\((\infty,0]\) 
Gap interval 

\(^*\) By convention, setting \(\kappa^*\) and \(\overline{\kappa }^{*}\) equal to zero sets the corresponding shear stiffness to zero but does not modify the corresponding normal stiffness. 
Area
Set the user_area property via the current contact area. This operation means that the contact area stays constant and is fixed independent of changes to the piece sizes/geometries. In order for the stiffnesses to be recomputed accounting for this area, one should subsequently call the deformabilty and/or pb_deformability methods.
Deformability
See this section from the linear formulation for details on this method.
pb_deformability
The deformability provided by the first interface can be specified with the deformability method. The additional deformability provided by the parallel bond can be specified with the pb_deformability method, which sets
The first term in this expression is obtained by equating the normal stiffness to the axial stiffness of the volume of material shown in this figure of the linear model formulation.
Bond
Create a parallel bond by bonding the second interface if the contact gap between the pieces is within the bondinggap interval. If no gap is specified, then the second interface is bonded if the pieces overlap. A single value can be specified with the gap keyword corresponding to the maximum gap. One can ensure the existence of contacts between all pieces with a contact gap less than a specified bonding gap \((g_b)\) by specifying \(g_b\) with the proximity in the contact cmat default
command of the Contact Model Assignment Table (CMAT). If the second interface becomes bonded, then the parallel bond force, moment and surface gap are zeroed.
Unbond
Remove a parallel bond by unbonding the second interface if the contact gap between the pieces is within the bondinggap interval. If no gap is specified, then the second interface is unbonded if the pieces overlap. A single value can be specified with the gap keyword corresponding to the maximum gap. If the second interface becomes unbonded, then the bond state becomes unbonded \((\overline{B}=0)\). The parallel bond force and moment are unaffected and will be updated during the next cycle.
Callback Events
Event 
Array Slot 
Value Type 
Range 
Description 

contact_activated 
Contact has become active 

1 
C_PNT 
N/A 
Contact pointer 

Linear Group: 

slip_change 
Slip state has changed 

1 
C_PNT 
N/A 
Contact pointer 

2 
INT 
{0,1} 
Slip change mode 

\(\;\;\;\;\;\;\begin{cases} \mbox{0: slip has initiated} \\ \mbox{1: slip has ended} \end{cases}\) 

ParallelBond Group 

bond_break 
Bond has broken 

1 
C_PNT 
N/A 
Contact pointer 

2 
INT 
{1,2} 
Failure mode 

\(\;\;\;\;\;\;\begin{cases} \mbox{1: failed in tension} \\ \mbox{2: failed in shear} \end{cases}\) 

3 
FLT 
\([0.0,+\infty)\) 
Failure strength [force] (\(\overline{\sigma }_{c}\) or \(\overline{\tau }_{c}\), according to the failure mode) 

4 
FLT 
\([0.0,+\infty)\) 
Bond strain energy \(\overline{E}_k\) at onset of failure 
Usage and Verification Examples
The tutorial example “Generating a Bonded Assembly” demonstrates the use of the linear parallel bond model.
The verification problem “TipLoaded Cantilever Beam” uses the linear parallel bond contact model to model a cantilever beam and compares its mechanical response to load against analytical solutions.
The example “Rock Testing” compares the response of a parallel bonded material under typical laboratory testing conditions (unconfined compression and directtension tests) with that of flatjointed material.
The tutorial example “Inclusions in a Matrix” also uses the linear parallel bond model. Its purpose is to illustrate how material regions with different mechanical properties may be created.
The tutorial examples “Creation of a Synthetic Rock Mass (SRM) Specimen” and “Slip on a Fault” illustrate how interfaces may be inserted in a parallel bonded material and modeled using the SmoothJoint Model.
Model Summary
An alphabetical list of the linear parallel bond contact model methods is given here. An alphabetical list of the linear parallel bond contact model properties is given here.
References
Crandall, S.H., N.C. Dahl and T.J. Lardner. “An Introduction to the Mechanics of Solids, Second Edition,” New York: McGrawHill Book Company (1978).
Holt, R.M., J. Kjølass, I. Larsen, L. Li, A. Gotusso Pillitteri and E.F. Sønstebø. “Comparison Between Controlled Laboratory Experiments and Discrete Particle Simulations of the Mechanical Behavior of Rock,” Int. J. Rock Mech. Min. Sci., 42, 985995.
Potyondy, D.O. “ParallelBond Refinements to Match Macroproperties of Hard Rock,” in Continuum and Distinct Element Numerical Modeling in Geomechanics 2011 (Proceedings of Second International FLAC/DEM Symposium, Melbourne, Australia, 14–16 February 2011), pp. 459–465. D. Sainsbury, R. Hart, C. Detournay and M. Nelson, Eds. ISBN 9780976757726. Minneapolis: Itasca International.
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