“size effects in semi-brittle materials and gradient
TRANSCRIPT
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
INTRODUCTION
Microcracking: ‘’branching’’ and ‘’crack face bridging’’
Images reproduced from the book ‘’Fracture Processes of Concrete’’ by Van Mier
INTRODUCTION
Complexity of the observed behavior of microcracking
Mode I fracture
Images reproduced from the book ‘’Fracture Processes of Concrete’’ by Van Mier
INTRODUCTION
Mode I cracking in compression
Images reproduced from the book ‘’Fracture Processes of Concrete’’ by Van Mier
INTRODUCTION
Lattice models: Modeling the full details of the microstructure
most important: particle structure
Images reproduced from ‘’Fracture Processes of Concrete’’ by Van Mier
INTRODUCTION
Ability to predict the complex behavior of microcracking
Images reproduced from ‘’Fracture Processes of Concrete’’ by Van Mier
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
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
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
/α
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
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
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
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)
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
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
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, , ,
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
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)
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
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:
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
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
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
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
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
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
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
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)
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
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
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]
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,
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
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
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
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
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 ?
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
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
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)
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