(a) In the process of fragmentation, clusters split up into parts of a certain fraction. (b) The reverse dynamics, 'fractional percolation. In contrast to ordinary ...
Fragmentation & Pulverization
H O. Ghaffari With W.A.Griffith &Troy J. Barber
Dec 2016
Fragmentation :a wide range in nature
(a) In the process of fragmentation, clusters split up into parts of a certain fraction. (b) The reverse dynamics, ‘fractional percolation. In contrast to ordinary aggregation processes, in fractional percolation the coalescence of clusters that substantially differ in size is systematically suppressed.
“Pulverization of chromosomes in micronuclei may also be one explanation for ‘chromothripsis’ in cancer and developmental disorders, where isolated chromosomes or chromosome arms undergo massive local DNA breakage and rearrangement”.
Qin, Z., Pugno, N. M., & Buehler, M. J. (2014). Mechanics of fragmentation of crocodile skin and other thin films. Nature’s Scientific reports, 4.
Theories of Fragmentations
(Key-Schultz,2011)
Fragmentation :a wide range in nature
(PRL,2006)
Onset of Reflection of Incident Signal
Reflected signals in Incident bar
Fragmentation :a wide range in nature
(PRL,2006)
(epl,2008)
Fragmentation Topological Vortex Defects in Ferroelectric
Dependence of the defect density nv on cooling rate (Nature Physics ,2015) (PRL,2010-PNAS 2009)
(PRX,2014) -leveling off : saturated min.fragment’s size -suppression of defect formation : controlling fragmentation
Fragmentation :a wide range in nature
(PRL,2010-PNAS 2009)
Coloring the domains(fragments) : a new method to characterize the fragments
The famous four-color theorem states that all regions of every two-dimensional map can be colored with only four colors in a manner that no two adjacent regions have the same color (Apple-Haken theory,1976)
Heavily crumpled surfaces analogous with heavily fragmented solids
Ridge networks : fracture networks
Ridge networks formed in flattened crumpling sheets of size L=8 1, 16 2, and 35 3 cm of Albanene-1 paper. B
PHYSICAL REVIEW E 74, 061602 2006
crumpled surfaces are analogous to Spin field undulation Useful analogy of crumpling with Heisenberg ferromagnetism : the direction of growth or decay is the surface normals (i.e., spin field), and a crumpled surface is similar to a Heisenberg paramagnet. Therefore, destroying long-range order of surface normals are in analogy with spin waves.
Kantor, Y. and Nelson, D.R., 1987. Crumpling transition in polymerized membranes. Physical review letters, 58(26), p.2774.
fragmentation to pulverization Transition : coloring domains approach Transition to Fragmentation Transition to Pulverization
Finite domains Pulverized state == max. Fragmentation
Finite domains
I
I
2d system II (direct transition)
Heavily fragmented
Stress Pulse(Destructive)
III
III Partially /fragmented pulverized (mixed state)
Impulsive loading ?
Stress (i.e., Creep/plastic deformation)
1d system:
Phase diagram of Solid-Granules Transition (s-g transition) U
Fragmented regime is broken symmetry
M
initial state: is symmetric state
I
U
Finite domains
I
M
II (direct transition) Pulverized state: is another ordered state Stress Pulse(Destructive)
III
III Partially pulverized or fragmented/mixed state
Pulverized state ==
max. Fragmentation = Anti-ferromagnetic state 2d system
1d system:
Imaginary state in s-g transition ... ... ... ... ... ... ... ...
Ordered sate
U
Finite domains
M
initial state: is symmetric state
I
U
I real situation in s-g transition
M
II (direct transition)
Pulverized state: is another ordered state
Stress Pulse(Destructive)
III
III Partially pulverized or fragmented/mixed state
Phase diagram of Solid-Granules Transition Possible existence of Tipple point in fragmentation-pulverization transition
J 0
J 0
J 0
II
Energy
U
I J 0
M
U
I
J 0
M
Minimum size of a fragment is a unit cell in lattice structure
Another possibility of Phase diagram of Solid-Granules Transition :modulated state and shape of domains modulated state
.. ..
modulated state II
Energy
.. .. U
I M
U
I M
U
M
modulated state U
M
Fragmentation mediated granular flow : example soft rocks such as sandstone
Crystalline Solid
Fragmented Solid Critical point I
Floating solid
Critical point II
Frozen impurities in these types of Models : Spin glasses
Stress (or impulse rate)
Fragmentation to Pulverization : Solid-granules transition
(Aben et al,2016)
Evolution of strain energy along the loading path (dashed line) and its changes under small strain perturbations around points with markers (out of scale). The flattening of the parabolas around markers 3 and 4 represents material degradation due to damage increase compared with undamaged material (markers 1 and 2). (Lyakhovsky,1 Yehuda Ben-Zion,2016).
Fracture mechanics aspects of Solid-Granules Transition
Rice 1975; Bhat etal.2012
Gd (2 surface
G 2 surface wInelastic
G K I2 “Toughening” term (s) due to inelastic regime in a main/major crack tip due to micro-cracks ;
Perfect-Elastic Inducing domains during pressure-quench (P-impulse)
G 2 surface G K I2
0
surface
wInelastic )d
Fracture mechanics aspects of Solid-Granules Transition initial state: is symmetric state
R=1
The maximum number of fragmentation is obtained by the pattern of (d) which is in analogy with pulverization. In terms of spin models, the ideal pulverization resembles antiferromagnetic: an ordered state.
Anti- ferromagnet : Ising model with metropolis algorithm
Fragmentation in 1D system
Background noise Control Parameter
Rise time of the pulse
VGinzburg Landau ( ) 2
4
Formation of kinks in a 1-D Landau-Ginzburg system VGinzburg Landau ( ) 2
4
driven by white noise.
with real
Overdamped Gross-Pitaevskii evolution with
t / Q and:
c 2 noise 2
2
2
(Laguna-Zurek, 1997)
very fast stress loadings are ‘‘all impulse,’’ and—according to Dynamic transition model —the number of kinks should saturate.
(Laguna-Zurek, 1997)
G-L framework and its connection with Phase field model
t k ( xi , t ) 2 ki
2
g (ki )( i 2 c2 ); g (ki )
ui ui b .[ g ( k ) (1 t )ui ] i t 2 t 2
ki ; i ui
Approach II: Define the characteristics of the “domains” with their phase using velocity vectors A thin-film
vi
strain-control ring
Rock Sample
Prior to fragmentation/ pulverization Onset of the impact of the fragments on the film ;
We assume similar size of the fragments but different velocity vectors
Exaggerated shape of the film in 2+1D
The shape of the contour serves as the order function
ˆ
vibration modes of the “contour” v.s stress pulse
(
d )* dt
Colliding and Impact formulation Hertezian’s Contact
vi vi
Elastic Impact
R2
1 1 R R1
Velocity of approach
Aluminum foil /pressure film as the thin sheet
Control Ring+ pressure thin film
A thin-film Longitudinal strain-control ring Rock Sample
i
Control Ring+ pressure thin film
b)
d) WG-6
WG-5
WG-2
WG-4
WG-1
Increasing ramp-rate
Ghaffari-Griffith-Baraber , in revision
Damage parameter from Patterns o pressure films
dD(t ) dl D(t ) 2 / 3 D1/0 3 dt dt Rate of Growth of domain walls
Bulk damage
D(t ) 1
A (t ) w
Atotal
(t ) t
dD( ) dl D( ) 2 / 3 D1/0 3 d d
Slope of D-e proportional to growth of domains
Damage parameter from Patterns o pressure films
D(t ) 0 L
D(t )
3 0.43 L 7
(t ) 1
(t ) 1 Therefore,
1
A (t ) w
Atotal
D
D(t )
1 L
(t ) 1
Damage Parameter
Order Parameter
BET (surface area ) vs. patterns of thin films
Why sort of correlation?
fractional BET on Prescaled film
a)
b)
c)
vn n
vi
vs s
max
max
max
max Sconatct
Imprinted surface area
S real 4 ( ) 2 2 BETreal
Approximated contact surface area
2 r... r 2
4 ( ) 1 1 2 4 r ( )3 3 2 2 2
n ....
n 2
BETmapped
r r
(vapp. ) 2
(
3 ( ) ) 4 r 2 r2
2
max (
15mV 2 app
16
1/ 2
E
*
)2 / 5 ,
Patterns of film hold both information of formed fracture surfaces and kinetic energy
Inferring Normal and Tangential colliding velocity of exerted fragments : Estimating Kinetic Energy term
15mV 2 app
max ( )2 / 5 , 1/ 2 * 16 E P(r , )
3 2
4 K (r , ) 2 (r , ) ( 1) 2 2
With assuming a max. allowed indentation depth and having calibrated pressure films ,we can estimate an interval of colliding normal velocity
Approximation of fragment distribution
Measured Distribution
Estimated Distribution
Ghaffari-Griffith-Baraber , in revision
Inferring Normal and Tangential colliding velocity of exerted fragments : vn n vs s
In current calculation vs 0 But any shear component on an exerted particle adds a second terms to kinetic energy . If the heat energy is due to frictional forces between fragments ,then we can assume a shear (rotational) component is added to net force (and then velocity vector ) which results an oblique impact of particles. The corresponding shear penetration s might be estimated using estimated frictional heat:
s
NW f
n
Here the bar sign is average over all over fragments and N is the number of fragments. In current calculation:
MC simulation using a well defined distribution over fragments with having mean value of Vs from dissipated energy and BET analysis
Ek
1 N 1 N 2 2 m ( v v ) mi vn2 i n s 2 2 i 1 i 1
i ei
A “phase”, therefore, can be extracted for each fragment
i
Spectrum of energy for fragmented and pulverized states (many-events spectrum) . gaped states are eliminated in an ideal pulverized state.
Ghaffari-Griffith-Baraber , in revision
Phase-transition from Intact state (I-state) to the pure pulverized state (P-state). Here, we assumed the system moves on a rigid energy landscape. In reality, the energy landscape is not rigid.
