“Size Effects in Semi-brittle Materials and Gradient ...

40
PhD presentation on “Size Effects in Semi-brittle Materials and Gradient Theories with Application to Concrete” by Antonios Triantafyllou Department of Civil Engineering School of Engineering, University of Thessaly Volos, Greece Advisor: Prof. Philip C. Perdikaris Co-advisor: Prof. Antonios E. Giannakopoulos January, 2015

Transcript of “Size Effects in Semi-brittle Materials and Gradient ...

Page 1: “Size Effects in Semi-brittle Materials and Gradient ...

PhD presentation on

“Size Effects in Semi-brittle Materials and Gradient Theories with Application to Concrete”

by

Antonios Triantafyllou

Department of Civil Engineering School of Engineering, University of Thessaly

Volos, Greece

Advisor: Prof. Philip C. Perdikaris

Co-advisor: Prof. Antonios E. Giannakopoulos

January, 2015

Page 2: “Size Effects in Semi-brittle Materials and Gradient ...

INTRODUCTION

Microcracking: ‘’branching’’ and ‘’crack face bridging’’

Images reproduced from the book ‘’Fracture Processes of Concrete’’ by Van Mier

Page 3: “Size Effects in Semi-brittle Materials and Gradient ...

INTRODUCTION

Complexity of the observed behavior of microcracking

Mode I fracture

Images reproduced from the book ‘’Fracture Processes of Concrete’’ by Van Mier

Page 4: “Size Effects in Semi-brittle Materials and Gradient ...

INTRODUCTION

Mode I cracking in compression

Images reproduced from the book ‘’Fracture Processes of Concrete’’ by Van Mier

Page 5: “Size Effects in Semi-brittle Materials and Gradient ...

INTRODUCTION

Lattice models: Modeling the full details of the microstructure

most important: particle structure

Images reproduced from ‘’Fracture Processes of Concrete’’ by Van Mier

Page 6: “Size Effects in Semi-brittle Materials and Gradient ...

INTRODUCTION

Ability to predict the complex behavior of microcracking

Images reproduced from ‘’Fracture Processes of Concrete’’ by Van Mier

Page 7: “Size Effects in Semi-brittle Materials and Gradient ...

INTRODUCTION

Dipolar elasticity

Strain gradient elasticity: (1D, ν=0)Hooke’s law extended to the 2nd spatial derivative of the Cauchy strain

Non-local models:A length scale parameter is needed for modeling materials such as concrete exhibiting size effect

Stress-strain softening law:

Size effect in elasticity:

Stiffer static response than that predicted by classical elasticity if g>0

What is a physical interpretation of the internal length?

xx22

xx E)g1(

i)/(1)/(

f ii

ii

i

0

0.2

0.4

0.6

0.8

1

0.0 0.5 1.0 1.5 2.0 2.5 3.0Normalized strain ε/εt

Nor

mal

ized

stre

ss, σ

/f t

Page 8: “Size Effects in Semi-brittle Materials and Gradient ...

HOMOGENIZATION ESTIMATES OF THE INTERNAL LENGTH

The simplest case of composite material (2D case of circular inclusions)

Input parameters

Matrix and inclusion elasticity properties

Composition value, c=a/b

Effective material properties of composite material

μ = f1(μm, νm, μi, νi, c)

ν = f2(μm, νm, μi, νi, c)

Estimation of the internal length

grcl UU

Page 9: “Size Effects in Semi-brittle Materials and Gradient ...

HOMOGENIZATION ESTIMATES OF THE INTERNAL LENGTH

0

1

2

3

4

5

6

7

0.1% 1.0% 10.0% 100.0%

Composition, c

Gra

dien

t int

erna

l len

gth

to in

clus

ion

radi

us ra

tio, ℓ

mi

mi

mi

mi

mi

mi

/15/10/5/

5.2/2/

P

r

θ 1

2 P

2a

0

1

2

3

4

5

6

7

0.1% 1.0% 10.0%

Composition, c

Gra

dien

t int

erna

l len

gth

to in

clus

ion

radi

us r

atio

, ℓ/α Loading case 1

Loading Case 2

Page 10: “Size Effects in Semi-brittle Materials and Gradient ...

STRUCTURAL ANALYSIS USING A DIPOLAR ELASTIC TIMOSHENKO BEAM

)x(w=zu

0=yu)x(ψz=xu

x y

zz

ux

uz

q

C.G.

Timoshenko kinematics

dxdydz221U xzxz

_

xxV

xx

_

cl

Total strain energy:

dxdydzzzxx

2xx

g21U

V

xxxx

_

xzxz

_

xxxx

_

2gr

grcltot UUU

xxxx E

xzxz kG

Cauchy stresses

dxdw

xu

zu2

dxdz

xu

zxxzxz

xxx

Non-zero strains:

Term that should not be omitted

Page 11: “Size Effects in Semi-brittle Materials and Gradient ...

STRUCTURAL ANALYSIS USING A DIPOLAR ELASTIC TIMOSHENKO BEAM

L

0

2

22

L

02

22

L

0

2

22

L

0

22

22

2

22

L

02

2

2

22

2

22

2

22

2

2

2

22

tot

dxd

dxwdkAGgw

dxdEIg

dxdw

dxdg1kAGw

dxdEAg

dxd

dxwdkAGg

dxd

dxdg1EI

dx

dxd

dxwd

dxdg1kAGw

dxdEAg

dxdw

dxdg1kAG

dxd

dxdg1EI

U

4 independent BC w, w′, ψ, ψ′ (bold are the classical elasticity variables)

4 energetically conjugate quantitiesQ, Y, M, m(Y, m: double shear or bi-force and double moment or bi-moment)

L0L0

L0

L

0

L0 mwYMwQwdxqW

Page 12: “Size Effects in Semi-brittle Materials and Gradient ...

STRUCTURAL ANALYSIS USING A DIPOLAR ELASTIC TIMOSHENKO BEAM

Qdxdg1

dxMd

dxdgg

IA1 2

22

2

222

q

dxQd

dxdg1 2

22

0wQdxdg1Q

L

0

2

22

0w

dxQdgY

L

0

2

0dxQdgMg

IAM

dxdg1M

L

0

222

22

0

dxMdgm

L

0

2

Equilibrium equations

Boundary conditions

Closed-form solution

32

3242

g/x22

g/x21

/x31

/x324

2433

xgEI6

kAGcxkAG2qx

gEI24q

egcegcededcx2

dx)dc()x(w

xddededxcgEI2

kAGxgEI6

q)x( 43/x2

2/x2

12

3

23

2

8 constants to be determined from 8 B.C.’s (4 at each end)

Page 13: “Size Effects in Semi-brittle Materials and Gradient ...

STRUCTURAL ANALYSIS USING A DIPOLAR ELASTIC TIMOSHENKO BEAM

0

0,2

0,4

0,6

0,8

1

0 0,2 0,4 0,6 0,8 1g/L

Norm

aliz

ed D

efle

ctio

n

Timoshenko beam (L/h=2)

Bernoulli-Euler beam

Finite Elements

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1x/L

Norm

aliz

ed a

xial

stra

in

g/L=0

g/L=0.005

g/L=0.01

g/L=0.05

g/L=0.1

Comparison with FEM results

+ Effect of g/h

Clamped end: Boundary layer effect as in BE beam

Free end: zero axial stains

Page 14: “Size Effects in Semi-brittle Materials and Gradient ...

NON-LOCAL DAMAGE MODEL

cA21

21G λ:Bλτ:C:τGibb’s energy: τλ 2g CB )g/1( 2

c0 CCC Microcracking adds flexibility:

n

(a) (b)

(c) (d)

Opening and closing of microcracks:

PCPC ::cc

cCI

cTI

cCCC Active microcracks:

):(:c0 τCPτCε

)(:(:c0 τ)CPτCε

(positive projections)

Minimization problem solution for a given state of stress

0cTI ε 0c

CI ε

Page 15: “Size Effects in Semi-brittle Materials and Gradient ...

NON-LOCAL DAMAGE MODEL

)(TT I

cI τRC )(

CC IcI τRC Damage rules: and

Irreversible character of damage: 0

.

Internal length is assumed to be a function of damage: )(gg

0A)(:)(

ddgg

)(:)(g21::

21::

21

d c

c

I2

II TCT

τCτ

τRττRττRτ

Energy density dissipation:,

Associated damage surface: )(t2

c21

21 2

τ:ττ:τ

0d/dg 0positive definite andcIT

C cIC

CTIRCIR, , ,

Page 16: “Size Effects in Semi-brittle Materials and Gradient ...

APPLICATION TO 4-POINT BENDING

i0iii0

ii0ii0ii for ,

E1EE)D1(

i0iii0i for ,E

Definition of damage

i)/(1)/(f

ii

iiii

Assumed stress-strain law:

i0i for 0D i0ii

ii0ii for

)/(1)/(11D

i

i

Damage functions:

iii0i

iii0 i)/(1

fE

Young’s modulus and isotropy:

t

c

)/(1f)/(1f

tt0tcc

cc0ctt

c

t

4-point bending: an assumed stress-strain law is sufficient input information for damage characterization

Page 17: “Size Effects in Semi-brittle Materials and Gradient ...

APPLICATION TO 4-POINT BENDING

0dzbN2/h

2/hxx

dzzbM2/h

2/hxx

kzmxx Linear strain distribution:

Input: a given value for curvature, k

Find: εm that satisfies equilibrium (stress-strain law)

Output: M, k point on the response diagram (size independent)

Input: k-δ kinematic relation (boundary value problem)

Final output: Force vs. midspan deflection diagram (size dependent)

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20

Predicted midspan curvature, k (mm-1 x 10-6)

Pred

icte

d be

ndin

g m

omen

t ca

paci

ty, M

(kNm

)

05

101520

25303540

455055

6065

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Predicted midspan deflection, δ (mm)

Pred

icte

d ap

plie

d lo

ad, P

(kN)

Page 18: “Size Effects in Semi-brittle Materials and Gradient ...

APPLICATION TO 4-POINT BENDING

Non-local predictions:

The local M vs. k is transformed to M vs. kgrad (scaling of curvature-elasticity)

The gradient solution of the boundary value problem is used to obtain the kinematic relation, δgrad/kgrad = f(g/L)This relation is not constant but evolves with damage since g=g(D)

Internal length evolution law:nD

0egg

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20

Predicted midspan curvature, k (mm-1 x 10-6)

Pred

icte

d be

ndin

g m

omen

t ca

paci

ty, M

(kNm

)

LocalNon-local

05

101520253035404550556065

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Predicted midspan deflection, δ (mm)

Pred

icte

d ap

pied

load

, P (k

N)

LocalNon-local

Page 19: “Size Effects in Semi-brittle Materials and Gradient ...

APPLICATION TO 4-POINT BENDING

Acoustic tensor positive definite

Good agreement with AE findings:

•Initiation of softening (D>0) and therefore microcracking

•Intense localization of AE energy release rate – macro-crack formation (D>0.95)

Objective damage characterization:

(for both g=0 and g>0)

Objectivity of the model:

Page 20: “Size Effects in Semi-brittle Materials and Gradient ...

EXPERIMENTAL PROGRAM

Leather pad

Pin support

Aluminum frameDCDT 1, 2

L/3

DCDT 2DCDT 1

Midspan cross-section

h=L/3

b=h

L/3L/3

Steel plate

Applied load, P

SG

SGSGSG

SG SG

3D geometrical similar specimens

Page 21: “Size Effects in Semi-brittle Materials and Gradient ...

MATERIALS

Mix

Quantities (Kg/m3)Dry density

(Kg/m3)Slump (cm)

Air-content (d)

(%)Cement (a) Aggregates (b) Water (w/c ratio) Additives (c)

CM 450 1350 293 (0.65) 3.6 2100 - e 2.5

LC 208 1980 162 (0.78) 1.6 2335 25 3.0

NC 276 2080 176 (0.64) 1.5 2365 10 2.5

MC1 448 1720 204 (0.45) 4.0 2410 22.4 2.0

MC2 447 1640 207 (0.46) 6.0 2440 15.6 2.0

Sieve opening

(mm)

% passing

LC NC MC1 MC2 CM

32 100 100 100 100 -

16 85.8 84.1 80.6 78.7 -

8 70.7 67.8 60.0 57.7 -

4 62.7 59.7 49.6 49.1 -

2 45.4 43.3 35.7 35.5 -

1 29.6 28.2 23.3 23.2 100

0.5 - - - - 30

0.25 12.9 12.3 10.2 10.1 -

0.075 8.0 7.6 6.3 6.3 -

Four concrete mixes dmax=32 mm

Low-strength (LC)

Normal-strength (NC)

Medium-strength (MC1 and MC2)

Cement mortar dmax=1 mm

Page 22: “Size Effects in Semi-brittle Materials and Gradient ...

CLASSICAL ELASTICITY PROPERTIES

10

15

20

25

30

35

40

45

12 16 20 24 28 32 36 40 44

Compressive strength, fc (MPa)

Youn

g's

mod

ulus

, E (G

Pa)

Uniaxial CompressionSplitting tension (vertical SGs of Fig.1)Splitting tesnion (horizontal SGs of Fig. 1)

0.12

0.14

0.16

0.18

0.20

0.22

0.24

0.26

0.28

12 16 20 24 28 32 36 40 44Compressive strength, fc (MPa)

Pois

son'

s ra

tio, ν

Uniaxial Compression Splitting tension

Mix LC NC MC1 MC2 CM

E, GPa 25.0 30.70 32.7 34.0 22.3

ν 0.2

Classical elasticity properties used in the analysis

Page 23: “Size Effects in Semi-brittle Materials and Gradient ...

INTERNAL LENGTH AND SIZE EFFECT IN ELASTICITY

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

50 100 150 200 250

Beam height, h (mm)

Stiff

ness

ratio

, Kex

p/Kcl

g=1 mmg=5 mmDef lection dataCurv ature data

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

50 100 150 200 250

Beam height, h (mm)St

iffne

ss ra

tio, K

exp/K

cl

g=10 mmg=15 mmg=20 mmDef lection dataCurv ature data

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

50 100 150 200 250

Beam height, h (mm)

Stiff

ness

ratio

, Kex

p/Kcl

g=5 mmg=10 mmg=15 mDelf ection dataCurv ature data

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

50 100 150 200 250Beam height, h (mm)

Stiff

ness

ratio

, Kex

p/Kcl

g=1 mmg=5 mmg=10 mDef lection dataCurv ature data

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

50 100 150 200 250

Beam height, h (mm)

Stiff

ness

ratio

, Kex

p/Kcl

g=10 mmg=15 mmg=20 mmDef lection dataCurv ature data

CM

LC NC

MC1

MC2 Elasticity: P=K δ & P=S k

Size effect in elasticity

if: Kexp/Kcl > 1

Sexp/Scl>1

Page 24: “Size Effects in Semi-brittle Materials and Gradient ...

INTERNAL LENGTH AND MICROSTRUCTURE

Concrete contains different aggregate sizes in different volume fractions

Fracture surfaces and fractured aggregates

0

5

10

15

20

25

20 22 24 26 28 30 32 34 36Young's modulus, E (GPa)

Gra

dien

t int

erna

l len

gth,

g (m

m) All data

Curvature dataDeflection data

1

4

21

20

5

62

3 78

10

19

20 1

1817

51

44

43

50

12

11139 14

15

16

42

373625

2423

26

28 2729

3031 32

3334

353938

4047

41

46 45

4849

21

22 Cas

ting

dire

ctio

n

compressive fiber

tensile fiber

Average inclusion size ranges between

10 to 20 mm

Page 25: “Size Effects in Semi-brittle Materials and Gradient ...

INELASTICITY: PEAK LOAD

Mix

Uniaxial stress-strain law parameters Young’s modulusCompression Tension

fc(a) εc ε0c

(b) bc(b) ft

(c)/ fsp(a) bt

(c) ε0t(d) εt

(d) E (a)

(MPa) (x10-6) (x10-6) - - - (x10-6) (x10-6) (GPa)

LC 15.9 1500 250 1.643 0.80 4.5 70 110 25.0

NC 20.5 1500 270 1.707 0.85 5.0 65 100 30.7

MC1 34.7 1500 430 3.360 0.88 6.0 80 120 32.7

MC2 38.0 1500 450 3.890 0.90 6.5 80 110 34.0

CM (e) 32.4 1800 600 5.183 0.95 10 130 130 22.4

0

200

400

600

800

1000

10 15 20 25 30 35 40 45Compressive strength, fc (MPa)

Cyl

inde

r mid

heig

ht c

ompr

essi

vest

rain

(μs)

at a

(%) x

f c

a=55%

a=40%

0

30

60

90

120

150

180

210

2.0 2.5 3.0 3.5 4.0Splitting strength, fsp (MPa)

Stra

ins

at 0

.5 x

f sp (μs

)

Tensile strain Compressive strain

Page 26: “Size Effects in Semi-brittle Materials and Gradient ...

INELASTICITY: PEAK LOAD

0

10

20

30

40

50

60

50 100 150 200 250Size, h (mm)

Peak

load

(kN

)MeasuredPredicted

0

10

20

30

40

50

60

50 100 150 200 250Size, h (mm)Pe

ak lo

ad (k

N)

MeasuredPredicted

0

10

20

30

40

50

60

50 100 150 200 250Size, h (mm)

Peak

load

(kN

)

MeasuredPredicted

0

10

20

30

40

50

60

50 100 150 200 250Size, h (mm)

Peak

load

(kN

)

MeasuredPredicted

Model predictions correspond to a size independent flexural strength

Scatter of predicted values corresponds to a +/- 5% deviation of the assumed material’s tensile strength

LC NC

MC1 MC2

Page 27: “Size Effects in Semi-brittle Materials and Gradient ...

INELASTICITY: SOFTENING BRANCH

0

10

60

20

30

40

50

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Midspan deflection (mm)

App

lied

Load

(kN

)

Non-local model predictions

size S1size S2 size S3

LC mix: D9.0e17g

0

10

60

20

30

40

50

size S1size S2 size S3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Midspan deflection (mm)

App

lied

Load

(kN

) Non-local model predictions

MC2 mix: D2e8g (mm) (mm)

Page 28: “Size Effects in Semi-brittle Materials and Gradient ...

INELASTICITY: UNLOADING PATH

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0Normalized midspan deformation

after unloading

Nor

mal

ized

load

at u

nloa

ding

0pl K)D1/(P

0

2

4

6

8

10

12

14

16

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50Midspan deflection (mm)

App

lied

load

(kN

)

Experimental results

Non-local prediction

0

6

12

18

24

30

36

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50Midspan deflection (mm)

App

lied

load

(kN

)

Experimental results

Non-local prediction

0

10

20

30

40

50

60

70

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50Midspan deflection (mm)

App

lied

load

(kN

)

Experimental results

Non-local prediction

MC2 mix

Page 29: “Size Effects in Semi-brittle Materials and Gradient ...

EVOLUTION LAW OF GRADIENT INTERNAL LENGTH

Non-local parameters

Mix

LC NC MC1 MC2

g0 (mm) 17 15 12.5 8.0

n 0.90 1.30 1.65 2.00

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1Damage, D

Nor

mal

ized

inte

rnal

leng

th, g

(D)/g

0

n=0.90n=1.30n=1.65n=2.00

0

10

60

20

30

40

50

Local pred.

Non-local pred.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Midspan deflection (mm)

App

lied

Load

(kN

)

Model predictions

Size S3Experimentalresults

• Progressively stiffer response if

dg/dD>0 (size effect in inelasticity)

• Cauchy stiffness decreases with damage: dK/dD = -K0 <0

But, K0gr>K0cl

n increases as brittleness increases in the mixes

Page 30: “Size Effects in Semi-brittle Materials and Gradient ...

STRAIN GAGE MEASUREMENTS

SG0,2

SG1,3

SG4

SG5

2 cm

2 cm

0

2.5

5

7.5

10

12.5

15

0.00 0.40 0.80 1.20 1.60 2.00Midspan curvature, k (1/mm)

App

lied

load

, P (k

N)

SG4,5SG2,3SG0,1S=7.636 Nm

For g>0, Sexp>Scl or kexp<kcl

(P=Sk)

0

50

100

150

200

250

0 25 50 75 100 125Measured strain εx (μs)

Mea

sure

d st

rain

IεyI

(μs)

SG1,4

SG2,30.00

0.50

1.00

1.50

2.00

2.50

0 50 100 150 200Measured strain εx (μs)

Appl

ied

stre

ss, σ

(MPa

)

SG1

SG2

0.00

0.50

1.00

1.50

2.00

2.50

0 80 160 240 320

Measured strain IεyI (μs)

Appl

ied

stre

ss, σ

(MPa

)

SG3

SG4

[ν] [Εt] [Εc]

[g0]

Page 31: “Size Effects in Semi-brittle Materials and Gradient ...

STRAIN GAGE MEASUREMENTS

300 mm

SG (z=80 mm)

300 mm

xz

0

0.2

0.4

0.6

0.8

1

-150-125-100-75-50-25025Axial strain (x10-6)

Nor

mal

ized

load

, P/P

peak

LC-S3-01

LC-S3-02

-100

-80

-60

-40

-20

0

20

40

60

80

100

-5 -4 -3 -2 -1 0 1 2 3

Axial stress (MPa) z (m

m)

LC-S3-01LC-S3-02

Damage characterization

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700

Axial compressive strain (x10-6)

Nor

mal

ized

load

, P/P

peak

MC2-S3

Plastic strains: verification of progressively stiffer response in inelasticity

)g(E)(D1),( xx,2

xx,

Page 32: “Size Effects in Semi-brittle Materials and Gradient ...

SIZE EFFECTS

Size effect in elasticity (g>0)

Size effect in inelasticity (g=g(D) with dg/dD>0)

Size effect on strength

Sources/Explanations of size effect on strength

1. Fracture mechanics size effect

2. Statistical

3. Stress redistribution at meso-scale (Lattice models)

4. Multi-fractal size effect

1. Diffusion phenomena

2. Hydration heat – microcracking during cooling

3. Wall effect due to inhomogeneous boundary layer

Page 33: “Size Effects in Semi-brittle Materials and Gradient ...

SIZE EFFECT ON FLEXURAL STRENGTH

0

1

2

3

4

5

6

7

50 100 150 200 250Size, h (mm)

Flex

ural

str

engt

h, σ

Ν (M

Pa) Measured

Predicted

0

1

2

3

4

5

6

7

50 100 150 200 250Size, h (mm)

Flex

ural

str

engt

h, σ

Ν (M

Pa) Measured

Predicted

0

1

2

3

4

5

6

7

50 100 150 200 250Size, h (mm)

Flex

ural

str

engt

h, σ

Ν (M

Pa) Measured

Predicted

0

1

2

3

4

5

6

7

50 100 150 200 250Size, h (mm)

Flex

ural

str

engt

h, σ

Ν (M

Pa) Measured

Predicted

0

1

2

3

4

5

6

7

50 100 150 200 250Size, h (mm)

Flex

ural

str

engt

h, σ

Ν (M

Pa) Measured

Predicted

8/1N h-

. Statistical size effect

Observed size effect on flexural strength cannot be attributed to statistical sources

CMMC2

MC1NCLC

Page 34: “Size Effects in Semi-brittle Materials and Gradient ...

SIZE EFFECT ON FLEXURAL STRENGTH

.

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0.01 0.1 1 10Nomalized size, h/l1

Nor

mal

ized

mod

ulus

of r

uptu

re f

r/ft

LC NC MC1 MC2

1

1.05

1.1

1.15

1.2

1.25

1.3

0 0.5 1 1.5

Normalized size, Dcyl/l1

Nor

mal

ized

str

engt

h, f s

p/ft

LC NC MC1 MC2

Limited scale range , 1 : 1.5 : 2,

for examining size effect on strength

Page 35: “Size Effects in Semi-brittle Materials and Gradient ...

CONCLUDING REMARKS

1. Experimental quantification of the internal length assumed by simplified dipolar elasticity

6. Further experimental work is needed

2. Concrete can be seen as a model material for studying size effects in elasticity

4. Experimental verification of key assumptions

5. Use of SG’s and continuous strain gradient models

3. Physical correlation between the internal length and material’s microstructure

7. Size effect on strength

Page 36: “Size Effects in Semi-brittle Materials and Gradient ...

FUTURE WORK

Introducing size effect on strength in the proposed modelfor the case of concrete

• Refinement of the modelInclude effect of distributed damage to the Timoshenko model

•Size dependent damage characterization (Van Vliet & Van Mier)

• Damage characterization not based on Cauchy strains alone: D(ε,ε,x)

• Strain gradient lattice models predicting size effect on strength ?

Page 37: “Size Effects in Semi-brittle Materials and Gradient ...

FUTURE WORK

Fiber Reinforced Concrete (FRC)

0

5

10

15

20

25

30

35

40

45

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

Midspan deflection (mm)

Appl

ied

load

(kN)

Very ductile softening response – bt parameter of stress-strain in the range of 1.5

For FRC, n→0. Therefore, g(D)=g0 (the thermodynamic limit of dg/dD=0 is recovered)

0

3

5

8

10

13

15

18

20

20 22 24 26 28 30 32 34 36

Young's modulus (GPa)

Gra

dien

t int

erna

l len

gth,

g (m

m)

4-point bending experiments

g0 of FRC = 8 mm

g0 of matrix = 4 mm

same order of magnitude as predicted by the simplified 2D case

FRC

Page 38: “Size Effects in Semi-brittle Materials and Gradient ...

FUTURE WORK

Micro-inertia length and dynamic loading:

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7 8 9 10kr

Dim

ensi

onle

ss p

hase

vel

ocity

, c/c s

Classic, (+) sign Classic, (-) signGradient, g/H=0.1, h/g=0.9 Gradient, g/H=0.1, h/g=0.9Gradient, g/H=0.1, h/g=1.2 Gradient, g/H=0.1, h/g=1.2Gradient, g/H=0.1, h/g=2 Gradient, g/H=0.1, h/g=2

1

0

t

ttot 0dtKWUHamilton’s

principle:

I2

222

2

22

2

222 m

dxdh

IAh1Q

dxdg1

dxMd

dxdgg

IA1

wmdxdh1q

dxQd

dxdg1 A2

22

2

22

2 Eqs. of motion and 2 gradient length: stiffness related g and inertia related h

Page 39: “Size Effects in Semi-brittle Materials and Gradient ...

ACKNOWLEDGMENTS

Laboratory of “Reinforced Concrete Technology and Structures”(Director: Prof. P. C. Perdikaris)

&Laboratory for “Strength of Materials and Micromechanics”

(Director: Prof. A. E. Giannakopoulos)

Page 40: “Size Effects in Semi-brittle Materials and Gradient ...

PhD presentation on

“Size Effects in Semi-brittle Materials and Gradient Theories with Application to Concrete”

By

Antonios Triantafyllou

Department of Civil Engineering School of Engineering, University of Thessaly

Volos, Greece

Advisor: Prof. Philip C. Perdikaris

Co-advisor: Prof. Antonios E. Giannakopoulos