m
p
representing small volume
m
p
/ρ
p
. In generalized interpolation material point method (GIMP), any function can be
expanded in a particle basis using:
)
(
)
(
x
f
x
f
P
P
P
χ
∑
=
(3.11a)
where
ρ
f
is property or volume function
f
on the particle and
( )
x
P
χ
is a particle shape
function. For example, density is expanded as
)
(
)
(
x
V
m
x
P
P
P
P
χ
ρ
∑
=
(3.11b)
P
P
P
P
V
m
f
=
=
ρ
(3.11c)
The particle characteristic (shape function) is required to integrate to particle values in
the initial configuration and undeformed state as
P
V
P
V
dV
x
=
∫
)
(
χ
. The force that is
distributed over the volume of particles P is then:
)
(
)
(
x
V
P
x
dt
v
d
V
m
a
P
P
P
P
P
P
P
P
P
χ
χ
ρ
∑
∑
=
=
(3.11d)
where
P
P
is momentum of particle
p
and equal to mass times velocity. The internal stress
is then:
)
(
x
P
P
P
χ
σ
σ
∑
=
,
0
≠
P
χ
only near particles
p
, otherwise it is 0. At the start of the
time step, all the information on the particles is projected to the grid (such as mass,
velocity, internal force, etc.). Equation (3.10) with particle expansions becomes
∑∫
∑∫
∑
∫
∑∫
∇
+
⋅
=
⋅
+
⋅
+
⋅
P V
P
P
P V
P
P
P
P
P
S
P V
P
P
P
dV
u
x
dV
u
x
V
u
F
dS
u
T
dV
u
b
x
V
m
δ
χ
σ
δ
χ
ρ
δ
δ
δ
χ
:
)
(
~
~
)
(
)
(
The next step is to distribute
P
F
over one particle according to the
)
(
x
P
χ
function.
78
dV
x
V
F
F
P
P
P
P
∫
=
)
(
χ
(3.12a)
The sum of the forces that are distributed over entire particle
p
is then:
∑ ∫
∑
⋅
=
⋅
P
V
P
P
P
P
P
u
x
V
F
u
d
F
δ
χ
)
(
(3.12b)
The next step is to expand the virtual displacement in terms of grid based shape functions
)
(
x
N
u
u
i
i
i
∑
=
δ
δ
(3.13a)
leading to
∑∑
∑∑∫
⋅
=
⋅
i
P
i
i
P
P
i
i
P
i
V
P
P
u
x
N
x
V
b
m
x
N
u
d
b
x
V
m
δ
χ
χ
)
(
)
(
1
)
(
)
(
(3.13b)
∑∫
∫
⋅
=
⋅
i
S
i
S
dS
u
x
N
T
dS
u
T
δ
δ
)
(
(3.13c)
∑∑
∫
∑
⋅
=
⋅
i
P
i
V
i
P
P
P
P
P
u
dv
x
N
x
V
F
u
d
F
δ
ψ
)
(
)
(
1
(3.13d)
∑∑
∫
∑∫
⋅
=
⋅
i
P
i
i
P
P
P V
P
P
P
u
d
dv
x
N
x
V
dV
u
x
V
)
(
)
(
1
)
(
χ
ρ
δ
χ
ρ
(3.13e)
∑∑
∫
∑∫
⋅
∇
=
∇
i
P
i
i
P
P
P
P
P V
P
P
u
dV
x
N
x
V
V
dV
u
x
δ
χ
σ
δ
χ
σ
)
(
)
(
1
~
~
:
)
(
~
~
(3.13f)
By theorem of virtual work,
i
u
δ
is arbitrary. Thus equating all terms in summations over i
gives:
∑
∑
∫
∑
∑
+
+
+
∇
⋅
=
P
P
iP
P
S
i
P
iP
P
iP
P
P
P
iP
P
S
F
dS
x
N
T
S
b
m
S
V
S
)
(
~
~
σ
ρ
(3.14a)
which can be written as:
ext
i
s
i
b
i
i
i
f
f
f
f
P
+
+
+
=
int
(3.14b)
∑
=
iP
P
i
S
P
ρ
(3.14c)
where
∑
∑
∇
⋅
−
=
∇
−
=
P
iP
s
P
P
P
iP
P
P
i
S
m
S
V
f
)
(
int
~
~
~
~
σ
σ
(3.14d)
79
∑
−
=
P
iP
P
b
i
S
b
m
f
(3.14e)
∫
∫
⋅
=
=
S
i
S
S
i
S
i
dx
x
N
n
dx
x
N
T
f
)
(
ˆ
~
~
)
(
)
(
σ
ρ
(3.14f)
and
∑
=
P
iP
P
ext
i
S
F
f
(3.14g)
Here the rate change of momentum on the grid is denoted by
i
P
, the “internal force” due
to stress is denoted by
int
i
f
, and the forces due to body forces, surface tractions and
external load are denoted by
b
i
f
,
s
i
f
,
ext
i
f
respectively.
The functions
iP
S
and
iP
S
∇
are particle shape, and gradient shape functions
respectively. Note that both are implicit functions of grid vertex position.
∫
=
S
i
P
P
iP
dV
x
N
x
V
S
)
(
)
(
1
χ
(3.14h)
The interpolation function is formed by gradients of shape functions by:
∫
∇
=
∇
S
i
P
P
iP
dV
x
N
x
V
S
)
(
)
(
1
χ
(3.14i)
When MPM was first introduced,
)
(
x
P
χ
was effectively equal to
)
(
x
V
P
δ
where
)
(
x
δ
is Dirac delta function. This assumption leads to
)
(
P
i
iP
x
N
S
=
and
)
(
p
i
iP
x
N
S
∇
=
∇
This approached could result in numerical noise when particles cross element
boundaries in the grid. The noise was caused by discontinuities in
)
(
p
i
x
N
∇
. The problem
is greatly minimized in GIMP by choosing
)
(
x
P
χ
=1 over a volume
V
P
around the
particle and 0 elsewhere. The resulting
iP
S
∇
, unlike
)
(
p
i
x
N
∇
, has no discontinuities.
80
Appendix 3.2: Calculated mechanical property from elastic modulus
The experiments on double lap shear specimens gave results for interfacial
properties (
D
t
). For modeling of OSB, we also needed orthotropic material properties of
the strands. The strands, however, are too small for most tests except axial modulus. For
modeling we estimated all other properties by ratios from measured modulus derived
from solid wood properties. The results are described here.
The hybrid polar strands (OSB type) were cut into small sizes to prepare for the
double lap shear tests. MOE was measured for each individual strand. During DLS tests,
over 1000 individual strands were loaded in tension. The results for these MOE parallel
to grain were E
L
=9936±935MPa.
Hybrid poplar is a combination of Western and Eastern poplar. From the Wood
Handbook (Table 4-1), solid yellow poplar properties have the ratios E
T
/E
L
= 0.043,
E
R
/E
L
= 0.092, G
LR
/E
L
= 0.075, G
LT
/E
L
= 0.069, G
RT
/E
L
= 0.011. We therefore estimated
hybrid poplar strand properties using E
T
= 9936*0.043 ± 935*0.043 = 427 ± 40MPa, E
R
=
9936*0.092 ± 935*0.092 = 914 ± 86MPa, G
LR
= 9936*0.075 ± 935*0.075 = 745 ± 6Mpa,
G
LT
= 9936*0.069 ± 935*0.069 = 686 ± 64MPa, G
RT
= 9936*0.011 ± 935*0.011 = 109 ±
10MPa. These results are shown in Table 2.1 as well. The Poisson’s ratio were assumed
to be the same as yellow poplar or µ
LR
=0.31, µ
LT
=0.39, µ
RT
=0.70, µ
TR
=0.33, µ
RL
=0.03,
and µ
TL
=0.02.
For orthotropic materials, such as wood, Poisson’s ratios are different in each
direction (x, y, and z). Additionally, pairs of Poison ratios are related by:
Ej
E
ji
i
ij
µ
µ
=
,
i
≠
j
i, j =L, R, T
(3.15)
after substituting L,R, and T we have:
R
RL
L
LR
E
E
µ
µ
=
,
T
TL
L
LT
E
E
µ
µ
=
,
T
TR
R
RT
E
E
µ
µ
=
. The results
of Poisson’s ratio are calculated using the relationship from equation 3.15. The Poisson’s
ratio of hybrid poplar is show in Table 3.3. µ
TR
is the highest and µ
TL
is the lowest. From
Table 3.3, the trend of elastic properties values from the experiment is consistent with
others such as the example from Katz et al (2008) that E
L
is the largest and E
T
is the
81
smallest. The COV (coefficient of variation) and standard deviation were similar (but
better) to Wood Handbook (United State Department of Agriculture, 1999). The literature
value of COV is 25 for tension parallel to grain, whereas we obtained 16 for our results.
Similarly, the mechanical properties of VTC strands were calculated from strand
MOE and the results are given in Table 2.2. Due to densification effects in VTC, all
stiffnesses were higher. The Poisson ratios were assumed to be the same.
Table 3.3: Elastic properties of yellow poplar and hybrid poplar.
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