Bulk powder behaviour under flow and compactionjae1001.user.srcf.net/teaching/materials/M2_PartIII/M2_L... · 2006-10-27 · 3.20.4 Using Mohr’s circle for stress analysis zFor

Dr James Elliott

Bulk powder behaviour under flow and compaction

[continued…]

PARTICLE TECHNOLOGYPart III Materials SciencePart IIB Chemical Engineering

MSM III M2/CET IIB – Lecture 8

27/10/2006

Handouts will available online from http://www.msm.cam.ac.uk/Teaching/PtIII/M2/

Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.

3.20.4 Using Mohr’s circle for stress analysis

For situations where τzx = τzy = 0, we can use Mohr’s circle construction for 3D systemsWe have three nested Mohr’s circles, each representing rotation of the system about one of the three principal axes

[1] R.M. Nedderman, “Statics and kinematics of granular materials” CUP (1992)

Intersections of circles and σ-axis give the principal stresses σ1(major), σ2 (intermediate) and σ3 (minor)

We usually ignore the intermediate stress and associated Mohr’s circles


3.21.1 Mohr-Coulomb failure analysis

In 3.13.2, 3.14.1 and 3.15, we introduced the Coulomb frictional yielding criterion for smooth, cohesive particles and an expression for the angle of internal friction ϕ

The expression above gives the maximum shear stress that can be supported across a given plane in the powder before yield occurs, known as Coulomb yield criterionIn fact, Coulomb criterion does not concern direction of slip, only its magnitude: tan cτ = σ φ +

0w w

n

0

τµ µ tanφ

tan τ for slippage

F Wp

≤ = +

⇒ τ = σ φ + [τ0 called cohesivity, c]



We can plot this yield locus on the Mohr circle diagram

O

S

φc

τ

σ

90 − φ

IYL

IYL

S'

A

(i) If Coulomb line lies above the Mohr circle, then no slip plane is formed → material is in static equilibrium

(ii) If Coulomb line just touches circle (as shown on right) then the material is about to slip along a plane defined by S or S’ (a situation referred to as incipient slip)

(iii) If Coulomb line cuts circle, then τ > σ tanϕ + c, which can’t happen!

IYL stands for internal yield locus, to distinguish from slip at walls



Example application to a powder column between walls

E

C B F

A

D

Consider a material supported by two parallel smooth walls, as shown on left

The material will be filled with two families of incipient slip planes. If the left hand wall were to break at A, the incipient slip plane AB would become an actual slip plane and the triangular wedge of material ABC would slip downwards and to the left.

If the right hand wall were to break at D, the incipient slip plane DE would become an actual slip plane and the trapezium DECF would slip downwards and to the right.


3.22.1 Rankine states

The Rankine states describe the form of failure that will happen when the stresses in a granular material reach the incipient yield value on some planeConsider the stresses set up in a semi-infinite soil due to self-weight:

A

A

B

B

hy

x

σ yy

The shear stress τxy on the plane AA is zero by symmetry, and similarly on BB

Force balance at depth h gives:

Hence can label the a co-ordinates of a point Y on the Mohr’s circle at (ρgh,0)

yy ghσ = ρ



We want to find σxx (where τxy = 0)This plane is denoted by points X on the Mohr’s circle (MC)If X is at X1, the material MC lies entirely within the Coulomb lines, and the material is stable

Y (ρgh,0)X

X

X

max

min

1

τ

σ

IYL If X is at Xmin, then MC touches the Coulomb line, and the material is in a state of incipient slip.

X cannot lie to the left of Xminotherwise, the circle would cut the Coulomb line, which is forbidden.

Similarly X cannot lie to right of Xmax

Thus X lies between Xmin and Xmax.



We cannot find the position of X, just range in which it liesXmin represents stresses on a smooth retaining wall that is just about to fail, the material slipping downwards and outwards – this is said to be the Active StateXmax represents case of a smooth bulldozer just about to push the material inwards, the material is failing inwardsand upwards – this is said to be the Passive State

Active

Passive

Stress

Displacement of wallInwards Outwards

Actual

Idealised rigid-plastic



Resolving stresses for the active stateSince the angle ASO is a right angle, the radius of the circle, R, is given by R = p sin φ

KA is Rankine’s coefficient of Active Earth Pressure

σ

c

O A

S

R

p

X Y

c cot φ

φ

IYLτ

( )A

cot 1 sincot 1 sin

xx Ac

Kgh c

σ + ⋅ φ − φ⇒ = ≡

ρ + ⋅ φ + φ

( )ρg cot 1 sin cot h p R c p c = + − φ = + φ − φ

At point Y:

( ) cotxx Ap R cσ = − − ⋅ φ

At point X:



For active state:

Similarly, in the passive state:

KP is Rankine’s coefficient of Passive Earth PressureThe Rankine coefficients are sometimes written in the following form, where κ = –1 for Active State, and +1 for the Passive State 1 sin

1 sinK + κ φ

≡− κ φ

( ) 1 sin 2cos1 sin 1 sinxx A

gh c⎡ ⎤ ⎡ ⎤− φ φσ = ×ρ − ×⎢ ⎥ ⎢ ⎥+ φ + φ⎣ ⎦ ⎣ ⎦

( ) 1 sin 2cos1 sin 1 sinxx P

gh c⎡ ⎤ ⎡ ⎤+ φ φσ = ×ρ + ×⎢ ⎥ ⎢ ⎥− φ − φ⎣ ⎦ ⎣ ⎦

P1 sin1 sin

K + φ≡

− φ


3.23 Angle of repose

The angle of repose is the maximum angle of slope that can be supported in a pile of a granular materialFor cohesionless powders, the angle of repose is just the internal angle of friction (see lecture 7, 3.15)However, for powders with cohesion forces, the angle of repose depends on the height of the slope, and is not a constant for a given material

h

X

α

θ

φ

PossibleSlip Plane

O

A

( )1 cos22cos sin

cgh

− θ − φ>

ρ φ θ

Stable if:


3.24 Powder behaviour under flow

When powders flow, their components are subjected to internal forces that can produce either mixing or segregation of particles of different shape, size, density or mechanical/surface propertiesWe need to define carefully what we mean by the terms mixed and segregated, with respect to powder blendsThe consequences of segregation are of great industrial importance, since poorly mixed powders can be disastrous for engineering applications or even lethal in pharmaceutical formulationsThe precise mechanisms of segregations in powders are also of academic interest, and still an open question


3.25 Mixing and segregation of powders

Must be precise about what we mean by these terms, in order that we can be clear whether system is segregated

Usually aim for distributive mixing when blending dry powders or low viscosity liquids.

Aim for dispersive mixing when compounding rubbers or high viscosity liquids.

[1] D.H. Morton-Jones, “Polymer Processing” Chapman & Hall (1989)


3.26.1 The ‘Brazil nut’ effect

This is a classical anecdotal phenomenon in vibrated or poured powders containing a particle (an intruder) or particles of different shape, size and density, and can be demonstrated easily in a wide range of material systemsThe physical mechanisms underlying the effect are multifarious and rather complicated, with dependencies not just on particle properties, but also wall friction and ambient air pressureThe general consensus seems to be that the action of wall friction creates a circulating flow of particles that drives larger or less dense particles to the top or edges of the container, but this can be modified by the presence of interstitial air in the powder


3.26.2 The ‘Brazil nut’ effect

The Jaeger group in University of Chicago have studied particle flows in an air-driven Brazil nut effect using MRI imaging techniquesThe following video demonstrates the phenomenon, along with the circulation flows of the powder matrix, and it also shows that the time taken for intruder particle to rise to the top of bed can be influenced by interstitial air

MRI imaging of air-driven Brazil nut effect

[1] http://jfi.uchicago.edu/%7Ejaeger/granular2/videos/grcvideo1.avi


3.27.1 Dilatancy in flowing powders

Dilatancy is defined as a change in bulk volume of a particulate suspension undergoing shear strainIt is required by the change in pore volume needed for the particles to move past each other in the flow.

rearrangement

plastic flow fragmentation

A

B,DC

E

F

ln p

ε

λ

maxo plnλεε −=


3.27.2 Dilatancy in flowing powders

Mixtures that are closely packed will tend to expand (or dilate), and those which are loosely packed will tend to contact, until a steady state with a particular critical density is reachedThe critical density depends on the shape and size of the particles in the mixture, as does the static packing densityThe two are intimately related, and in principle it should be possible to connect them by measuring changes in bulk volume as a function of applied torqueMany people have tried to construct a measure of powder ‘viscosity’ similar to that of a liquid, but such attempts fail unless initial state is ‘preconditioned’


3.28 Characterisation of dilatancy

Torque

Vb

timeτc τs

∆V

∆T

Vb[static]

Vb[roll]

T[roll]

T[crit]


3.29.1 Jenike split-box and shear cell apparatus

These consist of a split-box or an annular trough filled with the powder material under investigationA lid is fitted and subjected to a consolidating normal load from which the consolidating stress σc can be calculated

In shear cell, the top plate is rotated at about 1 revolution per hour until a steady shear stress τc is obtainedDuring this process the material compacts, more or less uniformly, and reaches a density ρc in equilibrium with the stress state (σc,τc)

N → σ

τ

N → σ τ


3.29.2 Jenike shear cell apparatus

The procedure needs to be repeated many times to give a set of yield loci, one for each density

Often the yield loci are nearly straight and lines of type:

Mohr-Coulomb

can be fitted where µ and c are functions of the density.

τ

σ

cτ = µσ +


3.30.1 Origin and nature of “force chains”

Transmission of stresses through real particle assembly is not homogeneous, but is concentrated in preferred paths due to irregularity in particle packing

[1] H.M. Jaeger, Physics World, December edition (2005).


3.30.2 Origin and nature of “force chains”

Stresses are transmitted through these force chains so that some particles experience loads much greater than the average value of W, while others may experience no force at allAs the load (and effective stress) increases, heavily loaded particles will deform or break and the stress chain structure will incorporate more particlesThe existence of stress chains means that continuum-based models can only offer approximate descriptions for processes which happen on a particle-particle scale


Summary of Lecture 8

Mohr’s circle allows straightforward diagonalisation of the stress tensor for 2D cases, allowing the major and minorprincipal stresses to be found easilyCombining Mohr’s circle with Coulomb yield criterion gives a description of powders as Mohr-Coulomb materialsMohr-Coulomb failure analysis method can be used to analyze a variety of different situations involving powdersWe looked at the phenomenon of segregation in flowing powders, and methods to characterise the dilatancy of these materials undergoing shearThis is a good way to define ‘flowability’ and can be used to measure friction angle and cohesivity in real powders via the Jenike annular shear cell

Documents

Bulk powder behaviour under flow and compactionjae1001.user.srcf.net/teaching/materials/M2_PartIII/M2_L... · 2006-10-27 · 3.20.4 Using Mohr’s circle for stress analysis zFor