Using the Timoshenko Beam Bond Model: Example Problem

Abstract

This guide provides a reference for how to successfully use the Timoshenko Beam Bond Model (TBBM) in EDEM® to model the three dimensional response of a cemented granular material under loading. A reference manual, which includes the theory of the TBBM and a description of the input parameters required to run the model, is available separately.

Full text

Using the Timoshenko Beam Bond Model:
Example Problem
Authors:
Nick J. BRO WN
John P. MORRISSEY
Jin Y. OOI
School of Engineering,
University of Edinburgh
Jian-Fei CHE N
School of Planning, Architecture and Civil Engineering,
Queen’s University, Belfast
© 2015-2017
v1.2
March 9, 2017

Contents
List of Figures ii
List of Tables ii
1 Uni-axial Compression of Cemented Cylindrical Specimen 1
2 Specimen Generation 2
3 Simulation Parameters 3
3.1 GlobalParameters ................................. 3
3.2 BondedParameters................................. 6
3.3 Non-bondedParameters .............................. 7
4 Starting Simulation 8
4.1 BondInitialisation ................................. 8
4.2 Loading ....................................... 8
5 Example Results and Analysis 9
5.1 Plotting Stress and Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5.2 AnalysingBondBreakage ............................. 12
5.3 VisualisingBreakage ................................ 13
6 References 15
Appendices 15
Appendix A Timestep Calculations 16
–i–

List of Figures
1.1 Physical specimen compared to DEM representation . . . . . . . . . . . . . . . 1
3.1 Global parameters in the preference file . . . . . . . . . . . . . . . . . . . . . 5
3.2 Geometrydynamics ................................ 6
5.1 Axial stress and broken bonds against axial strain . . . . . . . . . . . . . . . . 9
5.2 Calculation of the specimen height . . . . . . . . . . . . . . . . . . . . . . . . 10
5.3 Exporting stress-strain data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.4 Type of broken bonds vs axial strain . . . . . . . . . . . . . . . . . . . . . . . 13
5.5 Visualising bond breakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.6 Visualisingdamage ................................ 14
List of Tables
2.1 Specimen properties for the example problem . . . . . . . . . . . . . . . . . . 2
3.1 Global parameters for the example problem . . . . . . . . . . . . . . . . . . . 5
3.2 Bonded contact parameters for type A:A interactions . . . . . . . . . . . . . . 6
3.3 Particle and boundary model parameters for the reference case . . . . . . . . 7
– ii –

1. UNI-AXIAL COMPRESSION OF CEMENTED CYLINDRICAL SPECIMEN
1 UNI-AXIAL COMPRESSION OF CEMENTED CYLINDRICAL
SPECIMEN
This guide provides a reference for how to successfully use the Timoshenko Beam Bond Model
(TBBM) in EDEM®to model the three dimensional response of a cemented granular material
under loading. A reference manual, which includes the theory of the TBBM and a description
of the input parameters required to run the model, is available separately.
The numerical simulations used as the example problem in this guide mimic physical tests of
concrete cylinders under uni-axial compression as shown in Figure 1.1.
(a) Physical cylindrical specimen (b) DEM specimen
FIGURE 1.1: Physical specimen compared to DEM representation
In this example problem a particle assembly is created and bonded together to form a spec-
imen of “concrete”. After displacement loading is applied, the important bulk (macroscopic)
characteristics of the sample specimen are determined.
–1–

2. SPECIMEN GENERATION
2 SPECIMEN GENERATION
The TBBM is only compatible with spherical particles which can be generated either using
EDEM®or third party software. To obtain a stiffness and strength that is comparable to a
real bonded cementitous material, a particle assembly with a high solid fraction is suggested.
The properties of the cylindrical specimen used in this example are shown in Table 2.1.
In order for bonds to be formed between particles that are not in physical contact, EDEM®
requires the contact radius of those particles to be greater than their physical radius. For the
example problem particles of a uniform type (e.g. type "A") and poly-disperse size range are
employed. The contact radius multiplier, ηwas set as to be 1.1 times greater than the physical
radius of each particle. This can be achieved in the particle creator section of EDEM®(See
Reference Manual for details).
TABLE 2.1: Specimen properties for the example problem
Parameter Description Value
h0Initial specimen height (mm) 200
Aratio Aspect ratio – height to diameter 2:1
NPTotal number of particles 28,982
ηContact radius multiplier 1.1
rmin Minimum particle radius (mm) 1.15
rmax Maximum particle radius (mm) 2.71
nPorosity 0.37
ρpParticle density (kg.m−3) 2700
The porosity shown in Table 2.1 relates to the tightness of the particle packing achieved by
the particle generation technique. It does not directly relate to the porosity of the subject
material, which for concrete would be expected to be much lower.
NOTE 1: SIMULATION DECK
An EDEM®simulation deck which includes this particle assembly is freely available
through the DEM Solutions website. As such users can progress to Section 3to set up
this simulation.
In this example problem, particles and bonds represent the structure of the subject material
at the mesoscopic scale. Particles and bonds do not directly represent individual grains and
cement interfaces, but rather, represent the constituent parts of the subject material and their
interactive properties at the mesoscopic scale.
The numerical concrete specimen will be loaded through the displacement of platens. The di-
mensions of these plates are not critical but they should be greater than or equal in size to the
cylinder diameter and placed to form the boundaries of the specimen (see Figure 1.1b).
–2–

3. SIMULATION PARAMETERS
3 SIMULATION PARAMETERS
The particle information for the example problem is introduced into EDEM®as a new deck
with the simulation time set to zero. The TBBM should be loaded into EDEM®in three
locations:
1. As a custom particle-particle model
2. As a custom particle-geometry model
3. As a custom body force model
The preference file titled “BondedParameters.txt” must be located in the same folder as the
contact model library file. The input parameters for the example simulation can be broken
into three categories and explained independently. The categories are: Global parameters,
Bond parameter and Non-bond parameters. The parameters for each category are discussed
in the following sections.
3.1 GLOBAL PARAMETERS
There are a number of simulation parameters that need to be considered when assessing the
numerical stability of a simulation. For the example loading problem these are the time step
∆t, the bond time tbond, the global damping ιdand the loading rate Lr. Unlike the other
global parameters the time step needs to be calculated by the user rather than defined. This
can only be done when bonds have formed, as the critical time step for bonded contacts as
well as for non-bonded contacts (calculated in EDEM) needs to be considered, as shown in
Equation (3.1).
∆t=ξmin (∆tb,crit,∆tH M,crit )(3.1)
In Equation (3.1)ξis a factor that should be kept in the range of zero to one. This parameter
is similar to the one set as a percentage of the Rayleigh time step, which is normally used in
EDEM. Before running the main loading simulation with an estimated time step it is suggested
that a dummy simulation is run. This dummy simulation can be used to determine the critical
time step for bonded contacts, which is related to the stiffness of the bonds, as shown in
Equation (3.2). The critical time step is therefore determined for each bonded contact using
the smallest particle mass mp min and the largest bond stiffness component Kb max for that
contact.
∆tb,crit = 2rmp min
Kb max
(3.2)
In the dummy simulation the time step is first set to safe percentage of the Rayleigh critical
–3–

3. SIMULATION PARAMETERS
time step. The simulation is started, but the velocities and rotations of all particles in the
simulation are capped at zero, which essentially fixes the particles in place. The simulation is
allowed to continue beyond the time at which bonds are formed (see Section 4.1) at which
point the simulation is stopped. As none of the particles will have moved, the bonds will
have been formed without any disturbance to the particle assembly. At this point the value
for axial bond stiffness (which is the largest stiffness in the stiffness matrix, Kb max) and bond
length for each bond is exported for inspection.
The unknown from Equation (3.2) for each bond is the mass of the smallest particle. For-
tunately the unknown mass can be determined by exporting axial bond stiffness and bond
length for each bond. Equation (3.2) can be rearranged as Equation (3.3), with the full
rearrangement shown in Appendix A.
∆tb,crit = 2
v
u
u
u
t
ρp
4
3πLbK1
Ebπλ 3
2
K1
(3.3)
The parameters particle density ρp, Young’s modulus for the bond Eband bond radius mul-
tiplier λare all parameters that are specified by the user. The time step ∆t, can then be
calculated using Equation (3.1).
In this example problem ξwas set at a conservative 0.05 (5% of the critical timestep), which
gives a time-step for the main simulation of 1x10−7seconds. The simulation time can now be
reset to zero and the newly determined time step included.
The global damping parameter, ιd, is set by the user. As global damping is not included in
EDEM®the value for global damping is imported from the bonded particle preference file, as
shown highlighted in red in Figure 3.1, where it is the first item in the simulation properties
section. In this example problem a value of 0.5 for global damping is used.
NOTE 2: GLOBAL DAMPING USAGE
Global damping is a non-viscous damping applied to particle accelerations and is best
suited for static and quasi-static problems.
In dynamic problems, such as breakage in a crusher or impacts, where the ini-
tial acceleration of the particles is an important aspect to capture this should be set to
zero. Very small amounts may be acceptable on a per-use basis and should be checked
by the user.
The bond time is also set by the user in the preference file; it is the second item in the
simulation properties section, as shown highlighted in green in Figure 3.1. This is the time
–4–

3. SIMULATION PARAMETERS
that the bond initialisation procedure is triggered, a description of which is included in Sec-
tion 4.1.
Global Damping
Coefficient
Bond Initialisation
Time
Bond
Parameters
FIGURE 3.1: Global parameters in the preference file
The loading rate is also set by the user; in the simulation this is achieved by applying motion
to the geometries that were created as the top and bottom plates. Instructions on how to
apply motion to geometries are well documented in the EDEM®user manual and not intro-
duced in detail here. The loading rate used in the simulation needs to be sufficiently low to
provide a static solution, whist not requiring an unreasonable computational time. The rate
for a displacement controlled machine in physical tests is usually around 0.02 mm.s−1. In
the example problem a loading rate this low would be impractical as the computational time
would be excessive. The loading rate used in the example problem will be 200 mm.s−1(each
plate displacing at 100 mm.s−1). The geometry dynamics setup is shown in Figure 3.2.
For a numerical simulation this loading rate is acceptable as a time step of 1x10−7s means
that 10,000 calculation steps need to be computed for the specimen to displace 1 mm. The
global parameters for the example problem are summarised in Table 3.1.
TABLE 3.1: Global parameters for the example problem
Parameter Description Value
∆tTime step (s) 1x10−7
tbond Bond time (s) 0.001
LrLoading rate (mm.s−1) 200
ιdGlobal damping 0.5
In addition to the global parameters described above the use of gravity in simulations is
something that should be considered. The inclusion of gravity may be influenced by the
particle generation technique used. In the example problem no gravity is considered.
–5–

3. SIMULATION PARAMETERS
(a) Top Plate Dynamic (b) Bottom Plate Dynamic
FIGURE 3.2: Geometry dynamics
3.2 BONDED PARAMETERS
The theory describing the behaviour of particles at bonded contacts is described elsewhere
in the Reference Manual. The base properties for the bonds are contained in the interaction
properties section of the TBBM preference file, as shown in Figure 3.1.
The bonded contact parameters are shown for the example problem in Table 3.2. Defini-
tions for each bonded contact parameter are described in the accompanying reference man-
ual.
TABLE 3.2: Bonded contact parameters for type A:A interactions
Parameter Description Value
EbYoung’s modulus (GPa) 35
υbPoisson’s ratio 0.20
SCMean compressive strength (MPa) 500
ζCCoefficient of variation of compressive strength 0.0
STMean tensile strength (MPa) 60
ζTCoefficient of variation of tensile strength 0.8
SsMean shear strength (MPa) 60
ζSCoefficient of variation of shear strength 0.8
λBond radius multiplier 1
–6–

3. SIMULATION PARAMETERS
3.3 NON-BONDED PARAMETERS
The Hertz-Mindlin (no-slip) contact model is used to describe the behaviour at non bonded
contacts. The non-bonded parameters for the example problem are shown in Table 3.3.
TABLE 3.3: Particle and boundary model parameters for the reference case
Parameter Description Value
EpParticle Young’s modulus (GPa) 70
υpp Particle Poisson’s ratio 0.25
epp Particle-particle coefficient of restitution 0.5
µsP Particle-particle coefficient of static friction 0.5
µrP Particle-particle coefficient of rolling friction 0.5
EgPlate Young’s modulus (GPa) 200
υgPlate Poisson’s ratio 0.30
epg Platen-particle coefficient of restitution 0.0001
µsg Platen-particle coefficient of static friction 1
µrg Platen-particle coefficient of rolling friction 0
–7–

4. STARTING SIMULATION
4 STARTING SIMULATION
It is important that the specimen is in a state of equilibrium before bonding or loading.
The TBBM will automatically create a static state during the bond initialisation procedure.
Therefore, if a user is struggling to achieve equilibrium, for example due to changing the
contact model between particles, they can simply cap the velocities and rotations of the
particles to zero until after the bond initialisation time. If particles velocities are capped it is
important to un-cap them before loading.
4.1 BOND INITIALISATION
The bond initialisation procedure is triggered when the simulation time, t, exceeds the bond
time tbond for the first time. During the bond initialisation procedure, bonds will be inserted
between particles if the two particles contact radii overlap and they are allowed to bond i.e.
that there is a bond type for that interaction in the preference file. In the example problem
the generated particles are all given the type “A” so that there are bond properties defined
for them in the particle preference file.
At the end of the computational time step where bonds have been initialised a static assembly
of bonded particles with zero overlap and no contact force at the start of the loading phase
of the simulation is created. Following bonding all capped velocities and rotations should be
removed.
NOTE 3: CAPPED LIMITS
If capped limits (velocity and/or rotation) are not removed, it is likely that no bonds
in the assembly will fail during loading.
Please remember to remove all capped limits immediately after the bond initiali-
sation time before loading begins.
4.2 LOADING
For the example problem the specimen is loaded by displacing the two pieces of geometry. In
physical experiments usually one platen would move whilst the other is static. In the example
problem both plates are moved. As described in Section 3.1 the loading rate for the specimen
is 200 mm.s−1. It should be remembered that loading should not be commenced before the
bonds have been initialised. As the bond time has been set in the preference file as 0.001
seconds the loading time could be set at a time of 0.002 seconds.
–8–

5. EXAMPLE RESULTS AND ANALYSIS
5 EXAMPLE RESULTS AND ANALYSIS
Two of the most desirable properties for a sample of concrete are the ultimate compressive
strength and the secant modulus of elasticity. These properties can be determined by plotting
the stress strain curve of the cylinder under loading. This section will describe how to plot
the stress strain curve, as shown in Figure 5.1, as well as show how to plot some additional
properties of interest such as the progression of broken bonds.
FIGURE 5.1: Axial stress and broken bonds against axial strain
5.1 PLOTTING STRESS AND STRAIN
The compressive stress σis calculated using the compressive forces acting on the top and
bottom loading plates. In theory the forces acting on each plate should be equal, assuming
that the area of the top plate in contact with specimen is the same as area of the specimen
in contact with the bottom plate. However, as this cannot be guaranteed in the numerical
simulation the average force should be used when plotting the stress, which can be calculated
as:
σ=4(FT+FB)
πd2
g
(5.1)
–9–

5. EXAMPLE RESULTS AND ANALYSIS
where FTand FBare the total compressive forces acting on the top and bottom loading plates
respectively and dgis the diameter of either loading plate in contact with the specimen, which
are assumed to be equal. The forces acting on the plates, FTand FB, are determined by
exporting the magnitude of the total forces acting on the two different geometries.
The axial strain εcan be determined from the full height of the specimen, such that:
ε=(h0+h)
h0
(5.2)
where his the current height of the specimen and h0is the initial height of the specimen.
NOTE 4: SIGN CONVENTION
Note that under compressive loading the specimen experiences axial contraction and is
considered as a positive strain in this example.
The height of the specimen is defined as the maximum vertical distance between the top and
bottom surfaces of the specimen, as shown in Figure 5.2. This definition differs from simply
assessing the axial strain from the positions of the loading plates, as it takes into consideration
the maximum overlap that develops between particles and the loading plates to remove the
influence of the rough top surface of the specimen.
FIGURE 5.2: Calculation of the specimen height
This is particularly important in stiff, bonded assemblies where significant forces develop
– 10 –

5. EXAMPLE RESULTS AND ANALYSIS
over very small strains. Neglecting the maximum overlap, simulation results showed non-
linear initial loading arising from the particles coming into contact with the loading plates.
Although initial non-linear response is sometimes seen in physical tests, the degree shown in
the numerical response was unrealistic.
The specimen height his therefore calculated such that:
h=Zt−ZB+δT+δB(5.3)
Numerical stability of this equation leads to a gradual reduction in h, as this is reliant on
quasi-static conditions such that the maximum overlaps do not change significantly between
the time steps of the DEM simulation.
In summary, to plot stress against strain the following six properties must be exported from
EDEM®:
•FT= Total contact force, particles and top plate
•FB= Total contact force, particles and bottom plate
•ZT= Pos Z, top loading plate
•ZB= Pos Z, bottom loading plate
•δT= Max overlap, particles and top loading plate
•δB= Max overlap, particles and bottom loading plate
These required data queries are shown in Figure 5.3.
The stress-strain curve shown in Figure 5.1 indicates a close to linear but gradually softening
ascending branch with a loss of stiffness noted after approximately ε= 0.0008 (43% of the
strength) when 5% of the bonds have failed. This is consistent with physical experiments on
concrete where under loading up to approximately 30% to 40% of the ultimate compressive
strength. The loss of stiffness increased with an increase in the number of broken bonds,
until the peak stress was reached. This loss of stiffness was caused by the continual failure of
bonds, resulting in a reduction in the overall stiffness of the bond network.
– 11 –

5. EXAMPLE RESULTS AND ANALYSIS
(a) Platen Total Force (b) Platen Position
(c) Maximum Normal Over-
lap
FIGURE 5.3: Exporting Stress-Strain Data - EDEM data queries during export
5.2 ANALYSING BOND BREAKAGE
To determine how damage propagates in the specimen the failure mode of bonds over time
can be plotted. In Figure 5.4 the number of broken bonds as a percentage of total bonds in
the specimen has been plotted against axial strain. Axial strain can be determined as shown
in Section 5.1.
In the TBBM, bonds fail because either their compressive strength, tensile strength or shear
strength is exceeded by the respective stresses. To plot broken bonds the failure mode can be
exported from EDEM®from each time step. The TBBM is implemented in EDEM®as a custom
contact model. As such, when using the EDEM®Analyst the user should ensure to look at the
properties of interest under the “contacts” category rather than the “bond” category.
To determine the number of bonds that have broken through each failure mode export the
property failure mode, where 0,1 and 2 refer to the failure mode in compression, tension and
shear respectively. As can be seen in Figure 5.4 the majority of bonds in the specimen failed
because their tensile strength was exceeded. The rate of bond failure increased as the loading
continued towards the peak failure and after the peak stress, the rate began to decrease in
the softening regime.
– 12 –

5. EXAMPLE RESULTS AND ANALYSIS
FIGURE 5.4: Type of broken bonds vs axial strain
5.3 VISUALISING BREAKAGE
Sections 5.1 and 5.2 have shown how to extract prominent results such as peak failure
strength or peak strain from the bonded assembly. However, the many custom contact prop-
erties are also available for visualisation within EDEM analyst.
Figures 5.5 and 5.6 use EDEM’s capability to slice through the assembly to examine the
internal structure of your assembly. Both figures plot results at three different stage of loading
in Figure 5.1 - the peak strain, the post peak value at approximately ε= 0.0035, and at a much
larger strain before the sample collapses, which is not included in that figure.
In Figure 5.5 the custom contact property of "Bonded" (See Reference Manual for details)
is plotted to show the evolution of bond failure in the sample. Intact bonds are shown in
yellow, while non-bonded contacts are shown as black. Non-bonded contacts can be either
broken bonds where the particles are now in contact or contacts between particles and other
elements that were never bonded.
At the peak strain no clear shear pattern is observed from the broken contacts, but as the
sample continues to be loaded, a clear shear band develops post peak as the number of
broken bonds continues to increase. Finally, large cracking and separation is observed at
large strains that are significantly past the peak.
– 13 –

5. EXAMPLE RESULTS AND ANALYSIS
(a) At Peak (b) Post-Peak (c) Before collapse
FIGURE 5.5: Visualising bond breakage - Broken bonds at various stages of loading
In Figure 5.6 the custom particle property of "Damage" (See Reference Manual for details)
is plotted for the same three time-steps. Damage is the ratio of broken bond to the initial
number of bonds a particle had. A ratio of 1 means all of the original bonds that particle
formed are now broken. A similar pattern is observed for the sliced assembly, with the ad-
vantage that the particles can still be visualised once the contact no longer exists following
breakage.
(a) At Peak (b) Post-Peak (c) Before collapse
FIGURE 5.6: Visualising damage - Damage ratio at various stages of loading
– 14 –

6. REFERENCES
6 References
– 15 –

A. TIMESTEP CALCULATIONS
A TIMESTEP CALCULATIONS
Calculations to determine the mass of the smallest particle in a bonded contact:
K1=EbAb
Lb
(A.1)
Ab=πr2
bλ(A.2)
Therefore, by rearranging Equation (A.2) and substituting for Abin Equation (A.1):
rb=rLbK1
Ebπλ (A.3)
The mass of a particle can be calculated as:
mp=ρp
4
3πr3
b(A.4)
By substituting Equation (A.3) into Equation (A.4):
mp=ρp
4
3πLbK1
Ebπλ 3
2
(A.5)
– 16 –