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