THE LATTICE DISCRETE PARTICLE MODEL (LDPM) FOR
FRACTURE DYNAMICS AND RATE EFFECT IN CONCRETE:
THEORY, CALIBRATION AND APPLICATIONS
BY
GIANLUCA CUSATIS1
1NORTHWESTERN UNIVERSITY, EVANSTON, IL
COMPUTATIONAL MODELING OF CONCRETE STRUCTURES
TOKYO TECH
JULY 27, 2018
Tokyo, Japan | July 27, 2018
Strain Rate Dependence of Concrete
10-8 10-6 10-4 10-2 10 0 10 1 10 2
Creep Quasi-static
Vehicular
impacts
Plane
crash Hard impacts
Earthquakes
Blasts
Strain rate
(s-1)
Typical strain rates for various types of loading (Bischoff and Perry, 1991)
Concrete compressive and tensile strengths increases with increasing
strain rate
Concrete fracture energy also increase with strain rate but the overall
behavior is typically more brittle
2
Strain Rate Dependence of Concrete
Strain Rate Effects
Apparent
Crack Pattern
Internal heterogeneity
Multiple crack initiation
Crack Branching
Inertia
External boundary conditions
and dynamic confinement
Intrinsic
Creep
Arrhenius-type processes
Capillary and absorbed
water
Comminution/Pulverization
3
Strain Rate Dependent Formulation
F(e.
) = 1+ c1asinhe.
c0 l
æ
è
çç
ö
ø
÷÷
é
ë
êê
ù
û
úú
e
s
1
»H0(w)F(e)
×
1
H0(w)
s0(w)F(e)
×
s0(w)
e0(w)F(e)×e
0(w)
e a) b)
4
5
Rate Effect and Dynamic Increase Factor
b) a)
6
Effect of Inertia
Inertia and Crack Patterns Effects
Apparent rate-effect phenomena captured automatically
10−4
10−2
100
102
0.95
1
1.05
1.1
1.15
1.2
1.25
Strain Rate (1/s)
DIF
a)
10-4 s-1 0.05 s-1 10 s-1
b) c) d)
7
Hopkinson Bar Test - Tension
200 mm2000 mm 2000 mm
200 mmInput
920 mm 920 mm
Output
0 0.2 0.4 0.6 0.8 1 1.2−5
0
5
10
Time (ms)
Str
ess
(MP
a)
Expt
no rate
c0: 10
−5, c
1: 0.3
c0: 10
−6, c
1: 0.3
c0: 10
−6, c
1: 0.2
0 0.2 0.4 0.6 0.8 1 1.2−5
0
5
10
Time (ms)
Str
ess
(MP
a)
Expt
c0: 10
−5, c
1: 0.3
no rate
c0: 10
−6, c
1: 0.3
c0: 10
−6, c
1: 0.2
0.6 0.8 1−2
0
2
4
6
8
10
Time (ms)
Str
ess
(M
Pa)
c0: 10
−6, c
1: 0.3
c0: 10
−5, c
1: 0.3
c0: 10
−6, c
1: 0.2
no rate
d) e)
0 0.2 0.4 0.6 0.8 1 1.2−40
−20
0
20
40
Time (ms)
Str
ess
(M
Pa)
Expt
0 0.2 0.4 0.6 0.8 1 1.2−40
−20
0
20
40
Time (ms)
Str
ess
(M
Pa)
Expt
0.6 0.8 1−2
0
2
4
6
8
10
Time (ms)
Str
ess
(M
Pa)
c0: 10
−6, c
1: 0.3
c0: 10
−5, c
1: 0.3
c0: 10
−6, c
1: 0.2
no rate
0.6 0.8 1−2
0
2
4
6
8
10
Time (ms)
Str
ess
(M
Pa)
c0: 10
−6, c
1: 0.3
c0: 10
−5, c
1: 0.3
c0: 10
−6, c
1: 0.2
no rate
0.6 0.8 1−2
0
2
4
6
8
10
Time (ms)
Str
ess
(M
Pa)
c0: 10
−6, c
1: 0.3
c0: 10
−5, c
1: 0.3
c0: 10
−6, c
1: 0.2
no rate
0.6 0.8 1−2
0
2
4
6
8
10
Time (ms)
Str
ess
(M
Pa)
c0: 10
−6, c
1: 0.3
c0: 10
−5, c
1: 0.3
c0: 10
−6, c
1: 0.2
no rate
f)
TENSION
(HIGH)
TENSION
(LOW)
a)
b) c)
8
Hopkinson Bar Test - Compression
b)
a)
2000 mm 2000 mm100 mm
200 mm
100 mm500 mm 1500 mm
Input Output
0 0.2 0.4 0.6 0.8 1 1.20
−10
−20
−30
−40
−50
Time (ms)
Str
ess
(MP
a)
Expt
Fixed
HF
LF
0 0.2 0.4 0.6 0.8 1 1.20
−10
−20
−30
−40
−50
Time (ms)
Str
ess
(MP
a)
Expt
LF
HF
Fixed
c)
9
Compression with Twins Bars
Tests on standars and dam
concrete mixes
10
Small Cylindrical Specimen (Standard concrete):
Large Cylindrical Specimen (Standard concrete):
0 5 10−10
−5
0
5
10
15
20
Str
ess
[MP
a]
Time [ms]
Signal at Section 1
Experimental Data
Simulation Results
0 5 10−10
−5
0
5
10
15
20
Str
ess
[MP
a]
Time [ms]
Signal at Section 2
Experimental Data
Simulation Results
0 5 10−10
−5
0
5
10
15
20
Str
ess
[MP
a]
Time [ms]
Signal at Section 3
Experimental Data
Simulation Results
0 5 10−10
−5
0
5
10
15
20
Str
ess
[MP
a]
Time [ms]
Signal at Section 1
Experimental Data
Simulation Results
0 5 10−10
−5
0
5
10
15
20
Str
ess
[MP
a]
Time [ms]
Signal at Section 2
Experimental Data
Simulation Results
0 5 10−10
−5
0
5
10
15
20
Str
ess
[MP
a]
Time [ms]
Signal at Section 3
Experimental Data
Simulation Results
Compression with Twins Bars, Cont
11
Compression with Twins Bars, Cont
12
Dynamic Concrete Tension Test
200
mm
200
mm
25
mm
50
mm
64
mm 5
mm
18
mm
Reaction
Loading
Reaction
Loading
Reaction
Loading
a) b) c)
35 mm/s 1400 mm/s 4300 mm/s
35 mm/s 1400 mm/s 4300 mm/s
13
Concrete Ball Impact Test
0 100 200 300 4000
1
2
3
4
Fra
gm
ent
Mas
s (k
g)
Strain rate (1/s)
a) b) c)
meridian cracks
crushed cone
Forc
e (N
)
Displacement (mm)0 0.5 1 1.5 2 2.5 3
0
2
4
6
8x 10
4
Numerical
Experimental
0 100 200 300 4000
1
2
3
4
Fra
gm
ent
Mas
s (k
g)
Strain rate (1/s)
a) b) c)
meridian cracks
crushed cone
Forc
e (N
)
Displacement (mm)0 0.5 1 1.5 2 2.5 3
0
2
4
6
8x 10
4
Numerical
Experimental
Strain rate: 51 s-1 140 s-1 260 s-1
meridian cracks secondary cracks
place of cone of fines meridian cracks
place of ring tensile stress secondary cracks
c)
14
Concrete Ball Impact Test
1.1E-5 s-1 140 s-1 353 s-1
15
Penetration of UHPC Panels
12 in (304.8 mm)
12 in
2 in
(50.8 mm)
2.5 in
(63.5 mm)
3 in
(76.2 mm)
A B C
Side view Front view
Top view Back view
(FSP) projectile:
4340-H steel
Yield strength = 930 MPa
Diameter = 12.5 mm
Length = 14.8 mm
Smith J et. al.. “Discrete Modeling of Ultra High Performance Concrete with Application to Projectile Penetration” 65 (2014) 13 – 32, Int J of Imp Eng.16
0 0.02 0.04 0.06 0.08 0.10
25
50
75
100
125
150
175
200
Per
fora
tio
n D
epth
(%
)
Time (ms)
Damage Evolution
0 (ms) 0.01 0.02 0.04 0.06
0.08 0.1
0.03 0.05
0.0
7
0.09
Crack Opening (mm)
0.30.05
Progression of projectile penetration for CORTUF-Plain size Aa)
b)
0 0.02 0.04 0.06 0.08 0.1500
1000
1500
2000
2500
3000
Vel
oci
ty (
m/s
)
Time (ms)
Smith J et. al.. “Discrete Modeling of Ultra High Performance Concrete with Application to Projectile Penetration” 65 (2014) 13 – 32, Int J of Imp Eng.17
Effect of Fiber Content
Crack
Opening
(mm)
0.05
0.3
RSC UHPC UHPC - 3% UHPC - 5%
Scabbing
Fixed
boundary
Scabbing
Free
boundary
Side view
Fixed
boundary
Smith J et. al.. “Discrete Modeling of Ultra High Performance Concrete with Application to Projectile Penetration” 65 (2014) 13 – 32, Int J of Imp Eng.18
Penetration of Regular Strength Concrete
• Experimental data (Hanchak et al. 1992 )
relevant to impact of steel projectiles against
lightly reinforced concrete slabs
• Projectile of mass m=0.5 kg and diameter d =
25.4 mm
• Slab 610 x 610 x 178 mm
• Concrete Young Modulus 20000 MPa
• Concrete Strength f’c = 48 MPa
• Impact velocity from 300 m/s to 1000 m/s
19
Full Meso-Scale Simulations
1,229,348 LDPM tets208,967 nodes
1,253,802 dofs
Steel reinforcement
diameter = 0.569 cm
spacing = 7.62 cm
20
Full Meso-Scale Simulations: Results
Severe Damage Nonlinear
Behavior
Front Face
Front Face
Scabbing
Initiation
21
Comparison with Experiments
Col VS VR,num VR,exp
Blu 1058 947 947
Red 749 611 615
Gre 606 444 449
Yel 434 244 214
Mag 381 173 136
Cya 360 132 67
Bla 350 0 0
Grey 301 0 0
22
Animation: Ballistic Limit (~350 m/s)
Projectile Velocity vs. Time
23
Penetration: Multiple Hits
Case # 1: Centered Hits Case # 2: Offset Hits
P1P2 P1
P2
24
Multiple Hits: Projectile Velocity
P1P1
P2P1P2
P1P2
P1
P2
P2
25
s
18 in
18 in
R6 in
D
2 in
W
Blast Simulations: Geometry 1
26
Simulated Tests
Test
No.
Rebar spacing s
(mm)
W C4
(kg)
R (m) D (mm)
1 50.8 0.454 0.183 152
2 25.4 0.454 0.183 152
3 50.8 0.454 0.183 152
4 25.4 0.454 0.183 152
5 50.8 0.227 0.152 229
Compressive Strength=26.7 MPa
Experimental Data from “Explosive fragmentation of dividing walls”,
Report ARLCD-CR-81018;
Blast-reflected pressures computed using US Army, US Navy, US Air
Force, 1990. “Structures to resist the effects of accidental explosions”.
Technical report TM5-1300, NAVFAC P-397, AFR 88-22 and Hyde, D.W.,
1992. “CONWEP, Conventional Weapons Effects Program.” Technical
report, US Army Engineer Waterways Experiment Station, Vicksburg, MS.
27
Results: Test 1
28
Animation Test 1
29
Results: Test 3
30
Animation Test 3
31
Blue = test 1; Red = test 2; Green = test 3;
Pink = test 4; Cyan = test 5.
Test5
100
101
102
103
100
101
102
103
Fra
gm
en
ts w
ith
m >
M
M [g]
100
101
102
103
100
101
102
103
Fra
gm
en
ts w
ith
m >
M
M [g]
Fragment Distributions
Experimental Numerical
32
33
MPI Parallelization
Domains are visualized using
exploded views and different colors
MPI = Message Passing Interface
Parallel Simulation of Perforation
• Panel is model using 3.17 M LDPM tet element
• One geometric tet element requires 40 bytes of
memory; one LDPM element requires over 5 Kbytes
of memory• For this problem, recursive
bisection employs tet
centers as points
34
35
Parallel Simulation of Perforation, Cont.
Number of cores
Cycle
s/m
inute
Number of cores
Cycle
s/m
inute
Adjusted for size
43% speed up
from 64 to 128
processors
Whole model does not fit in the
memory of a single compute
node
36
Parallel Simulation of Perforation, Cont.
37
Impact of a steel cylindrical rod against a quasi-brittle
brick
The objective is to study fragmentation processes
Various velocities and masses of the cylinder are
considered
Parametric Study of Fragmentation
Brick Mass = 2.75 KgM = 0.72 Kg
38
No. of Fragments Increases with velocity
V=400 in/s
V=800 in/s
V=1200 in/s
V=1600 in/s
39
No. of Fragments Increases with velocity
0 0.5 1 1.50
20
40
60
80
100
Cu
mu
lati
ve m
ass
fra
cti
on
[%
]
Fragments mass [kg]
1600 in/s
1200 in/s
1000 in/s
800 in/s
400 in/s
40
Fragment Analysis