Transcript No Slide Title
FRACTALS IN MECHANICS OF MATERIALS
Martin Ostoja-Starzewski
Department of Mechanical Science & Engineering Institute for Condensed Matter Theory, and Beckman Institute University of Illinois at Urbana-Champaign [Support: NSF CMMI-1030940]
Morphogenesis of Fractals at Elastic-Plastic Transitions
(a) Does the elastic-plastic transition display a fractal character?
- if so, estimate the fractal dimension of plastic grains
(b) How does yielding develop in material body?
- study evolution of fractal dimension
(c) What kind of correlation exists between fractal dimension and material properties?
- are the responses among different materials universal?
2
… homogeneous elastic-plastic solid
d
ij
'
d
ij
' 2
G d
d
K d
f
ij d
ii
,
d
3
d
ii
3 3
… inhomogeneous elastic-plastic solid
d
ij
'
d
ij
' 2
G d
d
K d
f
ij d
ii
,
d
3
d
ii
3 4
… a heterogeneous elastic-plastic solid
5
evolution of plastic set’s fractal dimension (estimated by box method)
10 9 8 7 6 5 4 3 2 1 0 -5 -4.5
-4 -3.5
-3 ln r -2.5
-2 -1.5
-1 -0.5
10 7 6 9 8 2 1 0 -5 5 4 3 -4.5
-4 -3.5
-3 ln r -2.5
-2 -1.5
-1 -0.5
D
log(
N r
) 10 9 8 7 6 5 4 3 2 1 0 -5 -4.5
5 4 3 10 7 6 9 8 2 1 0 -5 -4.5
-4 -3.5
-3 ln r -2.5
-2 -1.5
-1 -4 -3.5
-3 ln r -2.5
-2 -1.5
-1 -0.5
-0.5
6
stress-strain responses fractal dimension-strain responses [
ASME J. Appl. Mech.
, 2010;
Proc. R. Soc. Lond. A
, 2010;
JoMMS
, 2011;
Phil. Mag
. 2012] 7
massively parallel simulation of elastic-plastic grains evolving in a 100 x 100 x 100 system 8
massively parallel simulation of plastic grains evolving in a 100 x 100 x 100 system
D
log(
N r
) by box-counting 9
1.205
x 10 8 1.2
1.195
1.19
1.185
1.18
1.175
1.17
0 0.1
0.2
0.3
0.4
0.5
Plastic strain 0.6
0.7
Displacement BC Traction BC Homogeneous 0.8
0.9
x 10 -5 1 stress-strain responses 3 2.95
2.9
2.85
2.8
2.75
2.7
2.65
2.6
0 0.1
0.2
0.3
0.4
0.5
Plastic Strain 0.6
0.7
Displacement BC Traction BC Homogeneous 0.8
0.9
x 10 -5 1 fractal dimension-strain responses 10
Universal properties at phase transitions
• Define order parameters (by analogy to condensed matter physics):
reduced von Mises stress: s
:
s
E p
s p
reduced fractal dimension: d
: (
E p
s
/ /
E
)
p E
3
D
3
s E p E
p
• Postulate a power-law
scaling function
:
d
Motivated by observation:
s
0
0
m
11
Scaling function for different materials -3 -3.5
-4 -4.5
-5 -5.5
-6 -6.5
-7 -7.5
-6.5
-6 -5.5
-4.5
-4 3D 2D -3.5
-5 Log s Large difference between
m
and
a
:
m=2.12/4.07, log(a)=5.78/12.98 (2D/3D);
2d slower than 3d: lower
m
and
a
12
Functionally Graded Material
• Properties varying spatially • Heterogeneous material system • Examples in nature: wood, bone, sea shells,… • Composites…
FGM structure
• Random medium B } •
P(Black, White) = {0.5,0.5}
• Conventional model: smooth
E
(
x
), n =const micromechanically invalid
Edge Plots
Interfacial fractal dimension
R
N D D R
X n Log
10
Log
10 2
Beta function fit of
D
(
x
)
f
(
x
)
x
1 (1
x
) 1
B
( , )
C α = 1.692 ;β = 1.692
C = 0.8129
B
(
x
,
y
) (
z
) (
x
) (
y
) (
x
y
) 0
t z
1
e
t dt
1 0
t x
1 (1
t
)
y
1
dt
Material Properties
Property
Young's Modulus Poisson’s Ratio
Value
104 0.3
Units
GPa • FGM system under consideration: Ti-TiB • Ti rich – left half of domain • TiB rich – right half of domain Yield Strength Density 482.633
4512 MPa Kg/mm 3
Material properties of commercially pure Titanium (A70) at room temperature Property
Young's modulus Poisson’s ratio
Value
370 0.4
Units
Gpa density 4630 Kg/mm 3
Material properties of Titanium Monoboride (TiB) at room temperature
(0)
h
• •
σ(0) = 510 MPa h = 833 MPa
Determine mesoscale properties from Hill-Mandel condition: (for Cauchy materials)
: :
B
(
t
n
) (
u
x
)
dS
0 Uniform boundary conditions:
•
displacement (Dirichlet) b.c.
• traction (Neumann) b.c.
x
B
u
x t
n
• displacement-traction b.c.
(mixed-orthogonal)
(
t
n
)
(
u
x
)
0
19
Displacement Boundary Conditions
Domain size 50
• UKBC applied to FGM
u
0
x
x
B
• Pure shear loading by 0 11 0 22 0 0 12 0 21
Domain size 100
Evolution of Plastic Grains (UKBC) – mesh size 200
800 600 400 200 0 0
Stress v/s Strain (Ti)
0,02 0,04 0,06
Strain
0,08 0,1 1200 1000 800 600 400 200 0 0
Stress v/s Strain (TiB)
0,02 0,04 0,06
Strain
Traction Boundary Conditions
Domain size 50
• USBC applied to the model according to,
t
0
n
x
B
• Pure Shear loading by, 11 0 0 22 0 0 12 0 21
Domain size 100
Evolution of Plastic Grains (USBC) – mesh size 200
800 600 400 200 0 0
Stress v/s Strain (Ti)
0,02 0,04 0,06
Strain
0,08 0,1 1200 1000 800 600 400 200 0 0
Stress v/s Strain (TiB)
0,02 0,04 0,06
Strain
Stress v/s Strain
Fineness: 50 Fineness: 100 Fineness: 200
Fractal Dimension (D) using Box Counting
D = 1.9739 (UKBC) fineness:200 D = 1.7981 (USBC) fineness:200
.
.
.
Fractal Geometry of Nature
, B.B. Mandelbrot
Fractals Everywhere
, M.F. Barnsley
Fractals (Physics of Solids and Liquids)
, J. Feder
?
Continuum Mechanics
(assumes smooth fields)
Can continuum mechanics be generalized so as to handle field problems of materials with fractal geometries?
Mass in a fractal geometric structure
W
obeys a power law
L D
,
D
3 [V.E. Tarasov,
Ann. Phys
., 2005] Use a fractional integral to represent mass in a fractal region )
W
R
dV D
W
R
c D
R
dV
3
Mass in a fractal geometric structure
W
obeys a power law
L D
,
D
3 [V.E. Tarasov,
Ann. Phys
., 2005] Use a fractional integral to represent mass in a fractal
W
R 3 )
W
R
dV D
W
R
c D
R
dV
3
in
R 3
dimensional regularization
( , )
R
D
3 2 3
D
(3 / 2) (
D
/ 2)
Green-Gauss theorem
W
v n
dA d
W c
3 1 where
dS d
c
2 , 2
c
2
dV D
c
3 , 3
v
dV D
surface fractal dimension of
W
mass fractal dimension of
W
dimensional regularization
( , )
R
d
2 2 2
D
Fractal continuity equation for
W d
D
D k v k
Fractal linear and angular momentum equations
d
D v k
f k
l D
kl
kl
kl
Fractional equation of energy balance
d
D u
v
D k q k
Fractional equation of 2 nd 0
T
D s
ij
law (C-D inequality)
D u
(
i
,
j
)
ij
non-fractal medium
(D=3, d=2)
D
ij
T
,
k T q k
recover conventional forms
two generalized operators
D k f
c
3
x k
dt D c
2
f
:
f
t
:
c
3
k k
f
x k c
2 where
c
3
c
2
R
d D
2
D d
1
d D
/2 /2
R
d
2 2 2
d
d
/2
R
D
3 2 3
D D
/2
c
3 1 ,
two generalized operators
D k f
c
3
x k
dt D c
2
f
:
f
t
:
c
3
k k
f
x k c
2 where
c
3
c
2
R
d D
2
D d
1
d D
/2 /2
R
d
2 2 2
d
d
/2
R
D
3 2 3
D D
/2
c
3 1 ,
D k fg D k g cg k f
holds
k D fg
f
k D g
k D f
Fluid in a porous fractal medium
•
Formulation of wave propagation equation by application of dimensional regularization
2
p
t
2
c
2
D
3
R
2
D
2 3
R
2 2
p
[V. Tarasov,
Ann. Phys
. (2005), “Fractional hydrodynamic equations for fractal media”] Solved analytically and numerically in [
ZAMP
isotropic fractal media”] (2011), “On the wave propagation in
cortical fold = sulcus smooth area = gyrus CerebroSpinal Fluid The surface of the brain, where the highest level of thinking takes place contains a large number of folds. Hence, a human (the most intellectually advanced “animal”) has the most folded surface of the brain:
D
= 2.73 – 2.79. 34
Natural Fractals in Human Body
• Systems – respiratory – lymphatic – nervous – circulatory
• • • • • Increase the surface area for absorption and transfer Self-similar (easily constructed) “Packing efficiency” (lungs, small intestines, etc) Decrease speed of fluids or air Minimize material used to form structures
Drawbacks of Tarasov’s formulation 1.
Usual fractional derivative (Riemann-Liouville) of a constant 0 2.
The mechanics-type derivation of wave equations yields a different result from the variational-type derivation 3.
The 3d wave equation 2
p
t
2
c
2
D
3
R
2
D
2 3
R
2 2
p
does not reduce to the 1d wave equation
c D x
1 ( , ) 2
p
t
2
x
2
v c D x
1 ( , )
p
x
V.E. Tarasov, Continuous medium model for fractal media, Phys. Lett. 336 (2005) V.E. Tarasov, Fractional hydrodynamic equations for fractal media,
Ann. Phys
.
318
(2005) V.E. Tarasov,
Fractional Dynamics
, Springer (2010)
Formulation via product measures…
Mass in an anisotropic fractal: , 3 )
x
1
l
10
x
2
l
20
l k
0 characteristic length in
x k x
3
l
30 3
k
fractal dimension along
x k
In other directions the fractal dimension is not necessarily the sum of projected fractal dimension… (Falconer, 2003): "
Many fractals encountered in practice are not actually products, but are product-like
." expect
D
2 3
Formulation via product measures…
power law relation w.r.t. each coordinate use a fractional integral with
product measure
and length measurement in each coordinate
Vector calculus on anisotropic fractals
fractal derivative
(
fractal gradient
) operator
D
e
D k
or
D k
c
1 1
e
k
= base vectors
x k
fractal divergence
of a vector field div
f
D
f
or
D k f k
1
c
( ) 1
k
f k
x k fractal curl
operator of a vector field curl
f
D
f
or
e jki
D k f i
e jki
1
c
( ) 1
k
f
x k i
four fundamental identities of vector calculus 39
divergence of curl of a vector field 1
f
f
e jki
D k f i
c
1
x j
e jki c
1 1
f i
x k
e jki c
1 1
c
1 1
x
f i x k
0 curl of gradient of a scalar field
e ijk D j D k
)
e ijk c
1 1
x j
c
1 1
x k
e jki c
1 1
c
1 1
x
f i x k
0 divergence of gradient of a vector field
D k
c
1 1
j
c
1 1
x j
c
1 1 fractal
Laplacian
c
1 ,
j
curl of curl operating on a vector field
e prj
r D
(
e jki
r D f i
)
r D f p
40
divergence of curl of a vector field 1
f
f
e jki
D k f i
c
1
x j
e jki c
1 1
f i
x k
e jki c
1 1
c
1 1
x
f i x k
0 curl of gradient of a scalar field
e ijk D j D k
)
e ijk c
1 1
x j
c
1 1
x k
e jki c
1 1
c
1 1
x
f i x k
0 divergence of gradient of a vector field
D k
c
1 1
j
c
1 1
x j
c
1 1 fractal
Laplacian
c
1 ,
j
curl of curl operating on a vector field
e prj
r D
(
e jki
r D f i
)
r D f p
Helmholtz decomposition holds: with
F
U
V
curl
U
0
, # div
V
0 # 41
Stokes theorem
W
(
D
W n e c
1 1
W n e f f c
1
n
dS d
W n e
D j i dS dS d
2
W
W n e
1
n e f c
1
f c
1
c
2
d
dS
2
dS
2
S d
n
f
dS d
l
f
d
l
1 42
…
volume coefficient
…
surface coefficient
associated with the surface …adopt a modified Riemann-Liouville fractional integral to express the
linear coefficient
[Jumarie, 2005, 2008]:
Fractional integral
dimensional regularization
fractal derivative : [
ZAMP
2009;
Proc. R. Soc. A
, 2009;
J. Elast
, 2011]
1.
D
is the “inverse” operator of fractional integrals: and 2. rule of “term-by-term” differentiation is satisfied: 3. operation on any constant is zero: Note: usual fractional derivative (Riemann-Liouville) of a constant fractional generalization of Reynolds transport theorem:
fractal effects are present between the resolutions (upper and lower cut-offs)
l
and
L
in a fractal RVE
random
Apollonian packing
47 Elastodynamics of Fractal Micropolar Solids – H. Joumaa & M. Ostoja-Starzewski
random 2d
Cantor set
48
Express Cauchy stress via fractional integral, and strain via fractal derivative … balance law of linear momentum in fractal solids e.g. a linear elastic solid:
Conservation of linear and angular momenta in fractal media … using G-G theorem and Reynold’s transport theorem, we localize to: In general, due to anisotropy of a fractal surface force and surface couple in the fractal setting are specified via fractional integrals
1 1
micropolar continuum
so the energy balance can be written with [
Int. J. Eng. Sci.,
2011]
1d wave equation
Mechanical approach: Hooke’s law
E
u
c
1 1 ,
x
u
Ec
1 1 ,
x
With conventional strain definition With our strain definition 1
c u
,
x
u
,
x
Variational approach:
u
1
Ec u
1 ,
xx
u
Ec
1 1
c u
1 1 ,
x
,
x T
1 2 2
u dl D
1 2 2
u c dx
1
U
1 2
E
2
dl D
1 2
E
2
c dx
1
Ldt
T
0 52
(1) (2)
c c u
1 1 3
2d anti-plane wave equation
c
1 (2)
u
3,1
c
1 (1) ,1
c
1 (1)
u
3,2
c
1 (2) ,2 0
u i
c
1 1
u c
1
3d wave s
,
j
c
1 1
u c
1 ,
i I
Timoshenko beam
0
Aw
D x
D x EI
D x
EI
D x
A
D x
D x w
A w
0
D x w
mechanical approach is consistent with the variational approach 53
• Constitutive law
ij
C ijkl
kl
ij
C ijkl
kl
ij ij
ij ij
ji ji
kk
kk ij ij
• Stiffness constants
C ijkl C ijkl
jk
jk il il
jl ik
ij kl
jl ik
ij kl
, , , , Lamé’s constants micropolar constants • Kinematics – – – – strain:
ij
i D u j
e kij
curvature: gradient:
ij
i D
i
1 product measure:
g i j
k g j
x i g i
i
i
x i
D i
1
g
i g i
54 • Momentum balance – linear
u i
j D
ji
–
I
angular
i
D j
ji
f i
e ijk
jk g j
in T
Im i
in T
u i
I
i
e ijk
e ijk g j
j j D u i
D j u k g j
e ijk
i g j
Governing Equations
j i D u j
e ijk
D j
j D u k g k
e ijk
i g k
Im i j
k g j
j D
i
k g i j i D
j
f i
• Characteristics – reproduction of Eringen’s work in absence of fractal effects – limitation to box shaped domains that can be contained in Cartesian system – fractal dimensions in three directions 55
Numerical Solution
•
Finite element formulation
g u w d
3
i i
g u w
3
i g g i j
,
j
g u w
3
g g i j
d
g u w
3
i
g g j j
,
j
g u w
3
g g j j
d
•
u w d i i
inertia term
u w g g i j
Resulting equation
M
U = 0
d
u w g g j j d
0 first elastic term
x 2 L
1
L
2 1 1
L
3 6
L 3
second elastic term
D
1
D
2 1 ln 8 2 ln 3
D
3 ln 2 ln 3
L 1 x
1
L 2 x 3
Electromagnetism on Anisotropic Fractal Media
Charge conservation on anisotropic fractals
W dS d
W
dV D
global form
W
local form
J
dV D D
d dt
W
t
dV D
i i
in the respective Cartesian direction as well as the spatial resolution. = 1,2,3) All the formulas may be evaluated by
c
( ) 1
k
k
l k
l k
0
x k
k
1 ,
k
1, 2, 3, (no sum) 57
Electromagnetism on Anisotropic Fractal Media
Charge conservation on anisotropic fractals
W dS d
W
dV D
global form
W
local form
J
dV D D
d dt
W
t
dV D
Ohm’s law for anisotropic fractals:
J E
or
J i
ij E j
… by analogy to elastic media where Hooke's law is unchanged when going from non-fractal to fractal media that result ensured the consistency of the Newtonian and Lagrangian Hamiltonian approaches to the derivation of governing equations 58
Electromagnetism on Anisotropic Fractal Media
Faraday’s law
d dt
A
dS d
l
E
d
l
i
0
t B k
e kji
D j E i
t B k
e kji
1
c
( ) 1
j E
0
t
B
D
E
Ohm’s law for anisotropic fractals:
J E
or
J i
ij E j
… by analogy to elastic media where Hooke's law is unchanged when going from non-fractal to fractal media that result ensured the consistency of the Newtonian and Lagrangian Hamiltonian approaches to the derivation of governing equations 59
Electromagnetism on Anisotropic Fractal Media
Ampère's law
S
dS d
l
H
d
l
1
S
n
(
D
H
)
dS d
J
D
t
D
H
dS d
0 or 1 0
J
E
t
c
2
D
0
where
C
D
t
… subject to constraints:
c
1/ 0 0 Gauss law for magnetism Gauss law
D
B
0,
D
E
0, # 60
Electromagnetism on Anisotropic Fractal Media L
0 1 2 Derivation from variational principle
L
dV dt
0
i i
1 2
A i
with
E t i
the same set of equations as before
D
1
B
D
0
0
D
1
c
E
t
D
1 0
J i
E
0 4
c
J
0
[Z
AMP
2012] 61
In fractal porous media:
• • • • • • when the surface and volume fractal dimensions (
d
and
D
) become integers (2 and 3, resp.), or when
R
falls outside [
l, L
], all the equations revert back to well-known forms of conventional continuum mechanics of non-fractal media Beltrami-Michell reciprocity theorem, uniqueness extension to thermomechanics with internal variables wave equations derived from variational principle are the same as those from mechanical approach [
IJES
2011, Z
AMP
2009] modified Reynolds stress for turbulence … • •
Study of wave propagation in isotropic fractal media:
isotropic fractal model that emulates fluid material elastodynamics in anisotropic fractal, linear elastic solids