GranularMicromechanicsModelforBeams,Plates,andShells
CAUCHYHEXAGONVS.STRUCTURALELEMENTGRANULARMICROMECHANICSFORCONTINUUM
ADVANTAGES
cp3 NationalScienceFoundationwww.nsf.gov
CenterforParticulateProductsandProcessesengineering.purdue.edu/CP3
PayamPoorsolhjouyandMarcial Gonzalez www.marcialgonzalez.net
• Macroscopic mechanical behavior of materials depends upon theirmicrostructure and micromechanical properties.
• Methods at various scales and with different levels of computationaldemand may be used for incorporating material microstructure andmicromechanical properties.
• In Granular Micromechanics approach, the material is envisioned as acollection of grains interacting with each other through pseudo-bonds thatcharacterize material’s macroscopic behavior.
INTRODUCTION
• In this approach, material’s behavior is derived through micro-macrokinematic identification and an appropriate inter-granular constitutiverelationship, followed by Principle of Virtual Work.
KinematicIdentification
u
pi = u
ni +
@ui
@xj(xp
j � x
nj )
�i = upi � un
i = ✏ij lj
!
"
#$
#%
Microscopicforce-law
fi =@W↵
@�↵i= Kij�
↵j PVW
W = �ij✏ij =1
V
NcX
↵=1
W↵
kle ijs
ifkd
TensorialConstitutiveEquations
KinematicAssumption
PrincipleofVirtualWork
(PVW)
MicroscopicConstitutiveLaws
PrincipleofVirtualWork
(PVW)
StaticAssumption
TensorialContinuummechanics
GranularMicromechanicsKinematicapproach
GranularMicromechanicsStaticapproach
CauchyStress
�ij =@W
@✏ij=
1
V
NcX
↵=1
@W↵
@✏ij
Gra
nula
r sys
tem
, C
ompu
tatio
nal D
eman
d
10-2 10-3 10-5 10-6 10-9 10-8 10-7 10-4 Length scale, meters
Coarse grained – Molecular or bead-spring models.
Atomic models
Continuum models
Meso-scale particle models
Many intermediate scales and structures may be conceived. Structures and their coarse graining regimen are typically ill-defined.
Granular micromechanics – micromorphic continua.
Granular Micromechanics
Grains,boundaries,andcontacts:ill-definedforcomplexmaterials
Prohibitivelylargecomputations
Ignoresmicrostructureandmicromechanicalphenomena
Stiffnesstensor
Cijkl =1
V
NcX
↵=1
hK↵
ikl↵j l
↵l
i
• Modeling materials withdifferent levels of anisotropyusing distribution functions
• Capturing induced anisotropy automatically by using nonlinear inter-granular force laws.
Transverselyisotropic
OrthotropicIsotropic
0.05 0.1
30
210
60
240
90270
120
300
150
330
180
0
ρ = 0.55
Zavg = 5.93
ForcedistributionTriaxial loading
Stiffnesstensor
Cijkl = l2⇢
ZZ hKiklj ll
i⇠(✓,�) sin ✓d⌦
-60
-40
-20
0
20
-60 -40 -20 0 20
σ 22
(MPa
)
σ11 (MPa)
Initial
Pre-loaded (0.2σy)
Pre-loaded (0.4σy)
Pre-loaded (0.6σy)
Pre-loaded (0.8σy)
(b)
0 2.5 5 7.5 100
2
4
6
8 x 107
x (1/mm)
w (r
ad/s
)
Longitudinal
Wavenumber
Frequency
0 2.5 5 7.5 100
2
4
6
8 x 107
x (1/mm)
w (r
ad/s
)
Transverse
Wavenumber
Volumetric
Deviatoric
Average longitudinal Average transverse
Antisymmetricshear
Symmetricshear
(b)
(a)
Wave number Wave number
Frequency
Path-dependentfailure
• Minimal additional computational expense:o Only looking at different directions and not following every contact
Δ𝑥Δ𝑦
Δ𝑧𝜖&&
𝜖''
𝜖((
Δ𝑥𝑏
ℎ
𝜖&&
ElementSize𝛥𝑥×𝛥𝑦×𝛥𝑧 𝛥𝑥×𝑏×ℎ
StraincomponentsConstant Varyinginheight
Constitutivelaws
𝜎./ = 𝐶./23𝜖23𝑁 = 𝐸𝐴𝜖𝑀 = 𝐸𝐼��
KIRCHHOFFPLATEELEMENT
!
"#
u1(x, y, z) = u(x, y)� z
@w
@x
u2(x, y, z) = v(x, y)� z
@w
@y
u3(x, y, z) = w(x, y)
Displacement Straintensor
✏ij
= u(i,j) =
2
4✏xx
+ zxx
✏xy
+ zxy
0✏xy
+ zxy
✏yy
+ zyy
00 0 0
3
5
Constitutiverelationships
8>>>>>><
>>>>>>:
N1
N2
M1
M2
V12
Q12
9>>>>>>=
>>>>>>;
=Eh
1� ⌫2
2
6666664
1 ⌫ 0 0 0 0⌫ 1 0 0 0 00 0 h2/12 h2⌫/12 0 00 0 h2⌫/12 h2/12 0 00 0 0 0 1�⌫
2 0
0 0 0 0 0 h2
121�⌫2
3
7777775
8>>>>>><
>>>>>>:
✏11✏2211
22
�12212
9>>>>>>=
>>>>>>;
PARTICLE-BINDERMATERIALSYSTEM
SHELLS
FUNCTIONALLYGRADEDMATERIALS(FGM)
0 0.002 0.004 0.006 0.008 0.0175 [1=m]
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Norm
al fo
rce a
nd M
omen
t
NM
0 0.002 0.004 0.006 0.008 0.0175 [1=m]
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Norm
al fo
rce a
nd M
omen
t
NM
NormalForce,N
[KN]
BendingMoment,M
[KN.m
]
Bending strain, [1/m]
Homogeneous
0 0.002 0.004 0.006 0.008 0.0175 [1=m]
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Nor
mal
forc
e and
Mom
ent
NM
0 0.002 0.004 0.006 0.008 0.0175 [1=m]
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Nor
mal
forc
e and
Mom
ent
NM
NormalForce,N
[KN]
BendingMoment,M
[KN.m
]
Bending strain, [1/m]
FGM
Microscopicforce law
Compression
0 0.5 1 1.575 [1=m]
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
Nor
mal
forc
e an
d M
omen
t
NM
Bending
0 0.5 1 1.575 [1=m]
-14
-12
-10
-8
-6
-4
-2
0
2
<11
[MPa
]
�11
CYCLICLOADING
Bend
ing
Axial
Constantgrainsize FGM
Straincomponents✏1 =
R1
R1 � z(✏1 + z1)
✏2 =R2
R2 � z(✏2 + z2)
�12 =R1
R1 � z(✏12 + z12) +
R2
R2 � z(✏21 + z21)
�23 = �13 = 0
In a spherical shell,constitutive laws willbe identical to thatof a Kirchhoff plate.
!
"
#
$
%
Coupledconstitutiverelationship
8>>>>>><
>>>>>>:
N1
N2
M1
M2
V12
Q12
9>>>>>>=
>>>>>>;
=Ch
1� ⌫2
2
66666664
kn kn⌫�knh
h2
12�knh
h2
12 ⌫ 0 0
kn⌫ kn�knh
h2
12 ⌫�knh
h2
12 0 0�knh
h2
12�knh
h2
12 ⌫ knh2/12 knh2⌫/12 0 0�knh
h2
12 ⌫�knh
h2
12 knh2⌫/12 knh2/12 0 0
0 0 0 0 kn1�⌫2
�knh
h2
121�⌫2
0 0 0 0 �knh
h2
121�⌫2 kn
h2
121�⌫2
3
77777775
8>>>>>><
>>>>>>:
✏k11✏k22k11
k22
�k12
2k12
9>>>>>>=
>>>>>>;
[1]
[2]
REFERENCES1- Misra, Anil, and Poorsolhjouy, Payam, Acta Mechanica 227, no. 5 (2016): 1393.2- Poorsolhjouy, Payam, and Misra, Anil, Intl. J. of Solids and Structures 108 (2017): 139-152.
-20 -15 -10 -5 070 [%]
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Nor
mal
forc
e an
d M
omen
t
NM
-1.5 -1 -0.5 0 0.5 1 1.575 [1=m]
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
Nor
mal
forc
e an
d M
omen
t
N
M
-1.5-1
-0.50
0.51
1.575
[1=m]
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05 0
0.05
Normal force and Moment
2 4 6 8 10 12 14 16 18
#10
4
-1.5 -1 -0.5 0 0.5 1 1.575 [1=m]
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
Nor
mal
forc
e an
d M
omen
t
N
M
Loadingprogress
-25 -20 -15 -10 -5 070 [%]
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
Nor
mal
forc
e an
d M
omen
t
N
M
-25 -20 -15 -10 -5 070 [%]
-1
-0.8
-0.6
-0.4
-0.2
0
Nor
mal
forc
e an
d M
omen
t
N
M
