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Inelastic Deformation
Shinichi Hirai
Dept. Robotics, Ritsumeikan Univ.
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 1 / 32
Agenda
1 One-dimensional Inelastic Deformation
2 Multi-dimensional Inelastic Deformation
3 Finite Element Method in Inelastic Deformation
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 2 / 32
Elastic/viscoplastic/rheologicaldeformation
elastic
rheological
plastic
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 3 / 32
Maxwell model
E c
E : Young’s modulusc : viscous modulusεela: strain at elastic elementεvis: strain at viscous elementε: strainσ: stress
ε = εela + εvis
σ = Eεela, σ = c εvis
stress-strain relationship in Maxwell model:
σ +E
cσ = E ε
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 4 / 32
Maxwell modelordinary differential equation of the first order:
σ +E
cσ = E ε
stress at time t:
σ(t) =
∫ t
0
Ee−Ec (t−t ′)ε(t ′) dt ′
In general,
σ(t) =
∫ t
0
r(t − t ′) ε(t ′) dt ′
Function r(t − t ′): relaxation function
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 5 / 32
Three-element model
E
c1
c2
E : Young’s modulusc1, c2: viscous moduliεvoigt: strain at Voigt elementεvis: strain at viscous elementε: strainσ: stress
ε = εvoigt + εvis
σ = Eεvoigt + c1εvoigt, σ = c2ε
vis
stress-strain relationship in three-element model:
σ +E
c1 + c2σ =
c1c2
c1 + c2ε +
Ec2
c1 + c2ε
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 6 / 32
Three-element modelordinary differential equation of the first order:
σ +E
c1 + c2σ =
c1c2
c1 + c2ε +
Ec2
c1 + c2ε
stress at time t:
σ(t) =
∫ t
0
r(t − t ′) ε(t ′) dt ′
where
r(t − t ′) =Ec2
c1 + c2e− E
c1+c2(t−t ′)
(1 +
c1
E
d
dt
)
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 7 / 32
Isotropic deformation modelselastic deformationspecified by a constant E :
σ = Eε
2D isotropic elastic deformationspecified by two constant λ and µ (Lame’s constants):
σ = (λIλ + µIµ)ε
Iλ =
1 1 01 1 00 0 0
, Iµ =
2 0 00 2 00 0 1
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 8 / 32
1
Isotropic deformation modelsviscoelastic deformationspecified by an operator E + c d/dt:
σ =
(E + c
d
dt
)ε
2D isotropic viscoelastic deformationspecified by two operators λ and µ:
σ = (λIλ + µIµ)ε
λ = λela + λvis d
dt, µ = µela + µvis d
dt
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 9 / 32
Isotropic deformation modelsviscoplastic deformationspecified by a convolution with a relaxation function:
σ(t) =
∫ t
0
r(t − t ′) ε(t ′) dt ′
2D isotropic viscoplastic deformationspecified by two relaxation functions:
σ(t) =
∫ t
0
R(t − t ′) ε(t ′) dt ′
R(t − t ′) = rλ(t − t ′)Iλ + rµ(t − t ′)Iµ
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 10 / 32
Isotropic deformation modelsviscoplastic deformationa relaxation function:
r(t − t ′) = E exp
{−E
c(t − t ′)
}
2D isotropic viscoplastic deformationtwo relaxation functions:
rλ(t − t ′) = λela exp
{−λela
λvis(t − t ′)
}
rµ(t − t ′) = µela exp
{−µela
µvis(t − t ′)
}
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 11 / 32
Isotropic deformation modelsrheological deformationspecified by a convolution with a relaxation function:
σ(t) =
∫ t
0
r(t − t ′) ε(t ′) dt ′
2D isotropic rheological deformationspecified by two relaxation functions:
σ(t) =
∫ t
0
R(t − t ′) ε(t ′) dt ′
R(t − t ′) = r rheoλ (t − t ′)Iλ + r rheo
µ (t − t ′)IµShinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 12 / 32
Isotropic deformation modelsrheological deformationa relaxation function:
r(t − t ′) =Ec2
c1 + c2e− E
c1+c2(t−t ′)
(1 +
c1
E
d
dt
)
2D isotropic rheological deformationtwo relaxation functions:
r rheoλ (t − t ′) =
λelaλvis2
λvis1 + λvis
2
exp
{− λela
λvis1 + λvis
2
(t − t ′)
}(1 +
λvis1
λela
d
dt
)
r rheoµ (t − t ′) =
µelaµvis2
µvis1 + µvis
2
exp
{− µela
µvis1 + µvis
2
(t − t ′)
}(1 +
µvis1
µela
d
dt
)
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 13 / 32
Nodal elastic forcesstress-strain relationship
σ = (λIλ + µIµ)ε
a set of elastic forces applied to nodal points:
elastic force = −(λJλ + µJµ)uN
from stress-strain relationship to nodal force setreplacing Iλ by Jλ, Iµ by Jµ, and ε by uN in thestress-strain relationship yields the elastic force set
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 14 / 32
Nodal viscoelastic forcesstress-strain relationship
σ = (λelaIλ + µelaIµ)ε + (λvisIλ + µvisIµ)ε
replacing Iλ by Jλ, Iµ by Jµ, and ε by uN in thestress-strain relationship yields a viscoelastic force set
⇓a set of viscoelastic forces applied to nodal points:
viscoelastic force = − Jλ(λelauN + λvisuN)
− Jµ(µelauN + µvisuN)
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 15 / 32
Nodal viscoplastic forcesstress-strain relationship
σ(t) = Iλ
∫ t
0
rλ(t − t ′) ε(t ′)dt ′ + Iµ
∫ t
0
rµ(t − t ′) ε(t ′)dt ′
replacing Iλ by Jλ, Iµ by Jµ, and ε by uN in thestress-strain relationship yields a viscoplastic force set
⇓a set of viscoplastic forces applied to nodal points
viscoplastic force = − Jλ
∫ t
0
rλ(t − t ′) uN(t ′) dt ′
− Jµ
∫ t
0
rµ(t − t ′) uN(t ′) dt ′
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 16 / 32
2
Nodal viscoplastic forcesintroduce
fλ =
∫ t
0
λela exp
{−λela
λvis(t − t ′)
}uN(t ′) dt ′
fµ =
∫ t
0
µela exp
{−µela
µvis(t − t ′)
}uN(t ′) dt ′
Vectors fλ and fµ have dimension of force/length
viscoplastic force = −Jλfλ − Jµfµ
fλ = −λela
λvisfλ + λelauN = −λela
λvisfλ + λelavN
fµ = −µela
µvisfµ + µelauN = −µela
µvisfµ + µelavN
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 17 / 32
Nodal viscoplastic forces
ordinary differential equation of the first order:
σ +E
cσ = E ε
⇓
fλ = −λela
λvisfλ + λelauN = −λela
λvisfλ + λelavN
fµ = −µela
µvisfµ + µelauN = −µela
µvisfµ + µelavN
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 18 / 32
Nodal rheological forcesstress-strain relationship
σ(t) = Iλ
∫ t
0
r rheoλ (t−t ′)ε(t ′)dt ′+Iµ
∫ t
0
r rheoµ (t−t ′)ε(t ′)dt ′
replacing Iλ by Jλ, Iµ by Jµ, and ε by uN
⇓
rheological force = − Jλ
∫ t
0
r rheoλ (t − t ′) uN(t ′) dt ′
− Jµ
∫ t
0
r rheoµ (t − t ′) uN(t ′) dt ′
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 19 / 32
Nodal rheological forcesintroduce
fλ =
∫ t
0
λelaλvis2
λvis1 + λvis
2
e− λela
λvis1
+λvis2
(t−t ′)(uN +
λvis1
λelauN
)(t ′) dt ′
fµ =
∫ t
0
µelaµvis2
µvis1 + µvis
2
e− µela
µvis1
+µvis2
(t−t ′)(uN +
µvis1
µelauN
)(t ′) dt ′
Vectors fλ and fµ have dimension of force/length
rheological force = −Jλfλ − Jµfµ
fλ = − λela
λvis1 + λvis
2
fλ +λelaλvis
2
λvis1 + λvis
2
vN +λvis
1 λvis2
λvis1 + λvis
2
vN
fµ = − µela
µvis1 + µvis
2
fµ +µelaµvis
2
µvis1 + µvis
2
vN +µvis
1 µvis2
µvis1 + µvis
2
vN
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 20 / 32
Nodal rheological forces
ordinary differential equation of the first order:
σ +E
c1 + c2σ =
c1c2
c1 + c2ε +
Ec2
c1 + c2ε
⇓
fλ = − λela
λvis1 + λvis
2
fλ +λelaλvis
2
λvis1 + λvis
2
vN +λvis
1 λvis2
λvis1 + λvis
2
vN
fµ = − µela
µvis1 + µvis
2
fµ +µelaµvis
2
µvis1 + µvis
2
vN +µvis
1 µvis2
µvis1 + µvis
2
vN
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 21 / 32
FE formulation of viscoplastic deformation
a set of equations of elastic deformation:
−KuN + fext + AλA − MuN = 0
elastic force
⇓a set of equations of viscoplastic deformation:
−Jλfλ − Jµfµ + fext + AλA − MuN = 0
viscoplastic force
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 22 / 32
FE formulation of viscoplastic deformation
uN = vN[M −A
−AT
] [vN
λA
]=
[−Jλfλ − Jµfµ + fext
AT(2αvN + α2uN)
]
fλ = −λela
λvisfλ + λelavN
fµ = −µela
µvisfµ + µelavN
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 23 / 32
FE formulation of rheological deformation
a set of equations of elastic deformation:
−KuN + fext + AλA − MuN = 0
elastic force
⇓a set of equations of rheological deformation:
−Jλfλ − Jµfµ + fext + AλA − MuN = 0
rheological force
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 24 / 32
3
FE formulation of rheological deformation
uN = vN[M −A
−AT
] [vN
λA
]=
[−Jλfλ − Jµfµ + fext
AT(2αvN + α2uN)
]
fλ = − λela
λvis1 + λvis
2
fλ +λelaλvis
2
λvis1 + λvis
2
vN +λvis
1 λvis2
λvis1 + λvis
2
vN
fµ = − µela
µvis1 + µvis
2
fµ +µelaµvis
2
µvis1 + µvis
2
vN +µvis
1 µvis2
µvis1 + µvis
2
vN
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 25 / 32
Example (Sample Program)
simulation movie
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 26 / 32
Example (Sample Program)
simulation movie
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 27 / 32
Summary
one-dimensional inelastic deformationMaxwell model for viscoplastic deformation
three-element model for rheological deformation
2D/3D inelastic deformationisotropic deformation models
formulating nodel force sets
finite element formulation
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 28 / 32
Simulating Inelastic DeformationReport #7 due date : Jan. 29 (Fri)Simulate the deformation of a rectangular inelasticobject shown in the figure.
P2P3 is fixed to the floor.P1 and P4 may slide on the floor.[ 0, tpush ] push P14P15 downward[ tpush, thold ] keep P14P15
[ thold , tend ] release P14P15
Use appropriate values of geometrical and physicalparameters of the object.
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 29 / 32
Simulating Inelastic Deformation
P5
P3
P6
P2P1
P7
P4
P9 P10 P11
P13 P14 P15
P8
P12
P16
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 30 / 32
AppendixLet us solve the following ordinary differential equation:
x + ax = u(t)
Assuming x(0) = 0, Laplace transform of the aboveequation yields
sX − aX = U
Thus, we have
X (s) =1
s − aU ,
implying that x(t) is the convolution of eat and u(t).Consequently,
x(t) =
∫ t
0
ea(t−τ)u(τ) dτ
Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 31 / 32
AppendixDifferentiating
x(t) =
∫ t
0
ea(t−τ)u(τ) dτ = eat
∫ t
0
e−aτu(τ) dτ
with respect to t, we have
x = aeat
∫ t
0
e−aτu(τ) dτ + eat · e−atu(t)
= a
∫ t
0
ea(t−τ)u(τ) dτ + u(t)
= ax + u(t),
which coincides with the ordinary differential equation.Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Inelastic Deformation 32 / 32
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