GENERALISED HOMOGENISATION PROCEDURES FOR GRANULAR AND LAYERED MATERIALS

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GENERALISED HOMOGENISATION PROCEDURES FOR
GRANULAR AND LAYERED MATERIALS
Elena Pasternak Technische Universität Clausthal,
Germany University of Western Australia and
Hans Mühlhaus University of Queensland, Australia
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Continuous vs. discrete materials
DltltL
r, E, n, ...
Equivalent continuum
L
Interpretation s, e, u, j ...
Granulate materials
V
Layered materials
D
3
Non-standard continua
4
Plan

• Homogenisation techniques in the Cosserat theory
  modelling of granular materials
• Layered materials with sliding layers. Cosserat
  Continuum model

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Homogenisation techniques for granulate materials

• Simple 1D granular structures. Spherical
  particles connected by translational and
  rotational springs
• Homogenisation by differential expansion
• Homogenisation by integral transformation
  Kunins method
• An application Vertical duct
• Non-local continuum for structures with random
  distribution of stiffness
• Cosserat Continuum model for granular medium
• Wave propagation in 1D structures
• Localisation in elastic gouge

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1D structure. Balls connected by translational
and rotational springs
a
j2i
k


r
x1
k
j
ra/2
u
3i
x3
x1
x2
x3
Energy density
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Homogenisation by differential expansion
Discrete energy density
xia
Taylor expansion up to second order terms
Deformation measures
Constitutive equations
- deformations
- curvature
Cosserat continuum
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Cosserat solution

• Cosserat, xja, ra/2, a1
• Exact

Coincide
Stresses
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Homogenisation by integral transformation
Kunins method
The structure of a material is supposed to be
periodical, a - the parameter of microstructure
Trigonometrical interpolation of discrete
functions
- Kunin's d - function
Kunin's homogenization gives the exact solution
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Non-local model
Stresses
Moment (couple) stresses
11
Kernels
5
C(x)
4
3
d(x)
2
C(xa)-2 C(x) C(x-a)
1
0
1
x/a
10
5
0
5
10
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Solution

• Kunin, xja, ra/2, a1
• Exact

Coincide
Coincide
Stresses
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An application Vertical duct
j2i
k


x1
k
j
u3i
r
g
x3
B.c.
0ltxltL
zero moment and constant volume force
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Discrete, Kunin and Cosserat
0.5
x
0
0.2
0.4
0.6
1
0.8
Kunin
0.5
1
Cosserat
Discrete
Configuration before ... and after
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Reference sphere and contacting spheres
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Equations of motion of the reference sphere
- the displacement vector
- rotation vector
m - mass of the sphere
r - density
N - number of spheres in a representative volume V
and
- resultant force and moment acting on the
reference sphere coming from all neighbouring
spheres
- lever-arm of the force
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Integral representation of the equations of motion
Mühlhaus and Oka (1996)
- a probable number of contacts in the element dn
centred at
- for isotropic and independent of the position
distribution of contacts
k - coordination number (the average number of
contacts per grain)
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Averaged equations of motion
- force and moment acting on the sphere at the
point of contact at
and
In the continuum description - translational and
rotational degrees of freedom of each point of
the continuum
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Homogenised equations of motion
Introducing the classical Cosserat continuum
deformation measures (eg, Nowacki, 1970)
strains
curvature twists
constitutive relationship for stress and moment
stress tensors
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Lamé coefficients
Non-standard moduli
Cosserat moduli
Bending stiffness
Cosserat shear modulus
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Localisation in elastic gouge
(Pasternak, Mühlhaus, Dyskin, 2002)
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Constitutive equations
s11(2ml)g11lg22
s21(ma)g21(m-a)g12
s22lg11(2ml)g22
s12(ma)g12(m-a)g21
m13Bk13
m23Bk23
u20
Ll1/l
l1(B/4m)1/2
l2(B/4a)1/2
l(l12l22)1/2
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Displacement gradient u1,2
The displacement gradient u1,2 at the middle
layer of the gouge (x20) as a function of
microstructural parameters L, l
1 - (L0.5, l0.5), 2-(L0.7, l0.5), 3-(L0.9,
l0.5), 4-(L1, l1), 5-standard (conventional)
elastic solution (L0, l0).
Bltlt(ma)l2
l2?0, l1?l
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Normalised moment stress
1 - (L0.1, l0.1), 2-(L0.5, l0.1), 3-(L0.9,
l0.1), 4-(L0.9, l0.5), 5-(L1, l1)
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Finite deformations. Kinematics
The rates of the Cosserat deformation measures
vi- velocity
Wc - Cosserat spin
wic- Cosserat rotation rate
Objectivity
x(X,t) Q(t) x(X,t) c(t), t t - a
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Constitutive equations
stresses
moment stresses
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Wave propagation and dispersion relationship. 1D
case
Propagation of harmonic waves
q3(x1)M2(x1)0
Normalisation a1
x - the wave number
vp - the phase velocity
Two types of waves exist rotational-shear and
shear-rotational
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Square of phase velocity
Shear-rotational wave
Rotational-shear wave
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Granulate materials

• Homogenisation the governing equations by
  differential expansion leads to a Cosserat
  Continuum, while by integral transformation - to
  a non-local Cosserat Continuum with oscillating
  kernels. Randomisation of the microstructure does
  not remove the oscillations
• Cosserat Continuum describes the behaviour of
  granulates with reasonable accuracy being a long
  wave asymptotic approximation to the exact model
• Non-local and discrete solution for 1D granulates
  coincide
• The non-local model for a vertical duct shows
  boundary effects concentrated over the distances
  smaller than the particle size
• Homogenisation of equations of motion of separate
  particles based on averaging over all possible
  contacts leads to a Cosserat Continuum model for
  granular medium

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• Two types of waves exist simultaneously
  shear-rotational and rotational-shear waves. For
  long waves the shear component dominates the
  shear-rotational wave, while the rotational
  component dominates the rotational-shear wave.
• There is a range of parameters L and l, for which
  the displacement gradient is highly
  non-homogeneous and displays localisation. The
  maximum values of the moment stress and the
  antisymmetric shear stress are attained at the
  gouge boundaries, while the maximum
  (localisation) of the symmetric stress is
  achieved at the middle layer of the gouge.

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Layered materials with sliding layers. Cosserat
continuum model

• Stress and failure localization associated with
  sliding in layered materials
• Cosserat continuum model of layered material
• Dislocations and disclinations in the Cosserat
  continuum
• Large-scale asymptotics
• Bending crack - a new fracture mode
• Path-independent integrals
• Wave propagation in layered materials

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Layered materials. Cosserat model
2-D
Cosserat Deformation measures
Degrees of Freedom
u,v-translational
Wz-rotational
Meaning of the Cosserat rotation
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Stresses
(Adhikary Dyskin, 1998)
Equilibrium equations
Interpretation of the moment stresses
A
y
ks
Non-symmetry
x
z
myz 0, mxz is the bending moment per unit
area, A
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Hookes law (Constitutive equations)
where A11, A12, A22 are the conventional
effective elastic moduli and B, B2 are the
Cosserat moduli
(Mühlhaus, 1993 Adhikary, Mühlhaus and Dyskin,
1998)
Continuum stays anisotropic
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Disclination
Cosserat continuum
Layered material
y
bW1
x
z
y
Wz1/2 u 0 sxy 0
Boundary-value problem
Wz1/2 u 0 sxy 0
x
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Large-scale asymptotics
Layered material
Cosserat continuum
y
y
by1
bltltL
x
x
For fixed (x, y)
Mathematically
Thus, we must use far-field asymptotics
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Large-scale asymptotics for disclination
Asymptotics
Stress localisation
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Bending crack
(Dyskin and Pasternak, 1999)
Bending crack is the surface of discontinuity in
the Cosserat rotation.
Cosserat continuum
Layered material
y
cut
(free of moments)
x
Given
Stresses are continuous through the crack surface
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Bending crack. Stress concentration
y
Stress concentration at the crack tip
x
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Propagation of bending crack
A post-failure photograph of the silica glass
slope model (SGM-02) (after Adhikary and Dyskin,
1998)
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m-criterion
Cosserat continuum
Layered material
y
b
x
Criterion of bending crack propagation
smax
mxz(b)
m-criterion
Fracture smax st
st is the local tensile strength
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Path independent (conservation) integrals.
P-integral
Cosserat continuum
Layered material
mxz0
b
mxz m(DWz)
(process zone)
Energy variation
lzone- process zone length
?xmam
Ti sjinj
Mi mjinj
DWz is the maximum rotation discontinuity
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Bending crack model of brittle fracture
mxz0
a
Moment fracture toughness
mxz0M?(y-a)
to start kink propagation
aminb1
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Wave propagation in the layered material
Three types of waves exist longitudinal, shear
and rotational
fxfymz0
Harmonic waves
x and h - wave numbers in x and y (parallel and
perpendicular to the layering) directions
vvpx, vpy - phase velocity
w xvpx hvpy - frequency
kx, h
- wave vector
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Phase velocities Directions perpendicular and
parallel to layering
Perpendicular to the layering
Parallel to the layering
(x0)
(h0)
h
x
vvpx, 0 ?? kx, 0
v0, vpy ?? k0, h
First solution
w2(x,0)0
w2(0,h)0
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Wave amplitudes Directions perpendicular and
parallel to layering
Perpendicular to the layering
Parallel to the layering
(x0)
(h0)
h
B20
no shear wave
B10
no shear wave
B30
no rotational wave
B30
no rotational wave
B1?0
x
B2?0
only longitudinal wave
only longitudinal wave
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A possible mechanism of impact failure
(Halliday et al, 1997)
Impact action is transmitted through the layered
material by longitudinal wave.
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Layered materials

• Moment stresses create a fracture that develops
  as a progressive rupture of layers forming a
  narrow zone propagating as a "bending crack
• The concentration of bending moments at the tip
  of the bending crack is singular being however
  weaker than the stress singularity at the tip of
  a conventional crack
• In Cosserat continuum, 3 additional (rotational)
  DOF produce 3 additional crack modes (Modes
  IV-VI)
• The phase velocity and the wave vector are
  collinear only in the directions parallel and
  perpendicular to the layering
• Only longitudinal waves exist in the directions
  perpendicular and parallel to the layering with
  both the rotational and shear waves vanishing
• Impact action is transmitted through the layered
  material by longitudinal wave. This can induce
  fracture propagating normal to layers

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Conclusions

• There exist heterogeneous materials which
  mechanical behaviour is strongly affected by
  rotational DOFs.
• Rotational DOFs can cause apparent strain
  localisation which can be passed for localisation
  caused by non-linearity.
• Rotational DOFs lead to new types of waves.
• Rotational DOFs lead to new fracture modes,
  bending crack being an example.