NASA/TM--2002-211914
Fractional-Order Viscoelasticity (FOV): Constitutive Development Using the Fractional
Alan Glenn
Calculus:
First Annual
Freed Research
Center,
Cleveland,
Ohio
Kai Diethehn Technisch.e
Un.:i.versit_it
Yury Luchko Europe University
December
2002
Braunschweig,
Via drina,
Frankfurt,
Braunschweig,
Germany
Germany
Report
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NASA/TM--2002-211914
Fractional-Order Viscoelasticity (FOV): Constitutive Development Using the Fractional
Alan
Calculus:
First Annual
Freed
Glenn
Research
Ce:nter,
CleveI.and,
Ohio
Kai Diethehn Technisch.e
Un.iversit_it
Yury Luchko Europe University
National
Via drina,
Aeronautics
Spa ce Administration
Glelm
Research
December
Braunschweig,
Center
2002
and
Frankfurt,
Braunschweig,
Germany
Germany
Report
Acknowledgments
Alan Freed would like to thank Prof. Ronald Bagley, University of Texas-San Antonio (then Col. Bagley, USAF), for encouraging him to study the fractional calculus and to use FOV in his research on polymers and soft tissues. This work was supported in part by the U.S. Army Medical Research and Material Command to the Cleveland Clinic Foundation with NASA Glenn Research Center being a subcontractor through Space Act Agreement SAA 3---445. Numerous discussions with the PI, Dr. Ivan Vesle?, and two of his research associates, Dr. Evelyn Carew and Dr. Todd Doehring, are gratefully acknowledged. Additional support was supplied by the UltraSafe Project at the NASA Glenn Research Center. Alan Freed also gratefully acknowledges the encouragement and support of: project manager, Mr. Dale Hopkins, and supervisor, Dr. Michael Meador, at the NASA Glenn Research Center.
This report contains preliminary findings, subject to revision as analysis proceeds.
The Aerospace Propulsion and Power Program at NASA Glenn Research Center sponsored this work.
Available NASA Center 71121 Standard Hanover,
for Aerospace Drive
from National
Information
Technical
Information
Service
5285 Port Royal Road Springfield, VA 22100
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Available
electronically
at http://gltrs.grcnasa.gov
Contents
1
Fractional
Calculus
1.1
Riemann-Liouville
1.2
Caputo-Type 1.2.1
2
Caputo-Type
1.4
Numerical
1D 2.1
3
3.2
3.3
Field
Expressions FDE's
.................. ...................
.......................
Approximations
4
........................
5
1.4.3
Caputo-Type
Mittag-Leffier
FDE's
Function
.......................
Analytical
Properties
1.5.2
Numerical
Algorithms
13
.......................
13
......................
FOV
15 17
Material
Functions
2.1.1
Static
2.1.2
Dynamic
............................
Experiments
19
........................
Experiments
20
......................
Mechanics
Metric
Fields
3.1.1
Dual
3.1.2
Rates
Strain
Fields
31
................................
Contravariant
3.2.3
Dilatation Rates
30
...............................
3.2.2
Fields
29
................................
Covariant
24 29
...............................
3.2.1
Stress
5 8 l0
..........................
1.5.1
1 2 3
...........................
Caputo-type Fractional Derivatives ............... Riemann-Liouville Fractional Integrals .............
3.3.1 4
Integral
Derivative
1.4.1 1.4.2
Continuum 3.1
Fractional Fractional
Integral
1.3
1.5
1
32
.............................
32
...........................
33
.............................
34
................................
35
...............................
36
Transfer
37
4.1
Kinematics
................................
37
4.2
Deformation
Fields
39
4.3 NASA/T_2002-211914
............................
4.2.1
Duals
...............................
39
4.2.2
Rates
...............................
40
Field
Transfer
of Fractional
Operators iii
.................
44
4.4
4.3.1
Derivatives
4.3.2 Strain
Integrals .............................. Fields ................................
47 48
4.4.1
Covariant-Like
49
4.4.2
Contravariant-Like
4.4.3 4.5
5
Stress
Dilational Fields
.......................... ........................
50
.............................
Conservation
4.5.2
Rates
51
Laws
52 ........................
52
...............................
Constitutive
Theories
5.1
Integrity
Bases
5.2
Elasticity
5.3
Viscoelasticity
5.4
Tangent
53 55
..............................
55
.................................
56
...............................
Operator
57
.............................
Stability
59
..............................
60
5.5
Isotropic Elasticity 5.5.1 Field transfer
5.6
Isotropic Viscoelasticity 5.6.1 Field transfer
......................... ...........................
68 69
5.7
Transversely 5.7.1 Field
Isotropic transfer
Elasticity ..................... ...........................
70 72
5.8
Transversely 5.8.1 Field
Isotropic transfer
Viscoelasticity ...........................
75 76
Finite-Strain 6.1
6.2
7
45
................................
4.5.1
5.4.1
6
............................
Bulk 7.1 7.2
............................ ...........................
61 65
..................
Experiments
Shear-Free
Extensions
79 ..........................
79
6.1.1
Kinematics
............................
79
6.1.2
Deformation
Fields
81
6.1.3
Strain
Fields
............................
82
6.1.4
Stress
Fields
............................
84
6.1.5
Special
Cases
Simple
Shear
...............................
6.2.1
Kinematics
............................
86
6.2.2
Deformation
Fields
88
6.2.3 6.2.4
Strain Stress
............................ ............................
Material
Fields Fields
........................
...........................
85 86
........................
89 92
Models
Elastic
Response
7.1.1
Theory
Viscoelastic
93 .............................
for pressure Response
93
........................
...........................
96 97
7.2.1 7.2.2
Voigt solid ............................. Kelvin solid ............................
98 98
7.2.3
Fractional-order
99
NASA/T_2002-211914
models
..................... iv
7.3 Bridgman'sExperiment .........................
73
A Table of Caputo Derivatives
105
B Automatic Integration B.1 The FundamentalStrategy ........................ B.2 Approximation of the Integral ...................... B.3 Approximation of Error Estiamtes ....................
107 107 108 108
C Table of Pad_ Approximates for Mittag-Leffler
NASA/T_2002-211914
v
Function
111
Nomenclature Numbers N
natural
N0
counting numbers, real numbers
R
numbers,
positive C
N := {1, 2, 3,...} No := {0, 1, 2,...}
real numbers,
complex
numbers,
R+ := {a E R • a > 0}
C := {x + i y • x, y E 11(;i := v/-_}
Functions set of all continuous
n
n-differentiable
functions
E.(x)
Mittag-Leffler
function
in one parameter,
E.,,(x)
Mittag-Lemer
function
in two parameters,
1Fl(a;
b; x)
Kummer
2Fl(a,
b; c; x)
Gauss
U(x - zo)
unit
5(x)
Dirac
confluent hypergeometric
step
property
and
that
generalized
function
f_oo 5(x).f(x)dx
:= 5If]
gamma
Integral
whenever
Operators operator,
n E N
Riemann-Liouville
D."
Caputo
fractional
J_
Cauchy
n-fold
J_
Riemann-Liouville
fractional
Ai
surface
normal
dA
differential
dC
reference
dH
differential
element
for height-of-separation
dS
differential
element
for distance-of-separation
NASA/T_2002-211914
:= f(O)
characterized
function
Do
Scalar
usually
function
differential
n
(the
at 0)
continuous
digamma Differential
function
distribution
continuous Euler's
a &: f_
function
function
delta
by the
r(x)
hypergeometric
(_
fractional
differential
differential
operator,
integral
operator,
operator,
c_ E ]i¢+
c_ E R+
n E N
integral
operator,
(_ E
Fields area
whose element
distance
points
in the
ith
coordinate
direction
for area-of-surface separating
vii
neighboring
planes between between
planes points
f
is
dV
differential
element
C
dilatation
f
force
fi
force in the
for volume-of-mass
in 1D
force
ith coordinate
in the
normal
direction
ith coordinate
direction
is in the jth coordinate
G
viscoelastic
G' & G"
viscoelastic storage and ith relaxation function
(or relaxation)
n th invariant J
viscoelastic
g
current
_o
gauge
loss (dynamic) basis
length
of gauge
section
length
P
hydrostatic
pressure
P
Lagrange
8
Laplace
S
magnitude time
T
absolute
multiplier
forcing
transform
an isotropic
variable
of shear temperature
W
speed work
OL, Oq
potential function representing fractal order of evolution
of sound
fractal
C
order
of evolution
viscoelastic
material
engineering
shear
dilatation,
work
in bulk response
constant strain
classic
definition
dilatation, Hencky's strain in I-D
definition
viscosity bulk modulus A
principal stretch
stretch along
ratio
fiber direction
ith principal
stretch
#
elastic
modulus
P, Pi
characteristic
retardation
characteristic
bulk
shear
O"
mass density stress in I-D
oi
ith principal
Q
reaction NASA/T_2002-211914
moduli
compliance
function
7 6
on
modulus
of an integrity
ith memory
Vs
acting
direction
ratio time
retardation
time
stress
stress viii
constraint
a surface
whose
unit
7"
shear
T, T{
characteristic
stress relaxation
characteristic
bulk
91
first normal-stress
92
second
normal-stress
co
angular
frequency
Outer
time
relaxation
time
difference difference (rad/sec)
Products
a®b
vector
outer
product
with
components
aibj where
A®B_ A_Nb
tensor
outer
product
with
components
AijBkt
symmetric tensor outer where i, j, k, e = 1, 2, 3
product
with
i, j = 1, 2, 3
where
components
i, j, k, _, = 1, 2, 3 1 (AikBjl
Body manifold,
_ E N3
B
coordinate
system
q3
particle
(a material
coordinates, Body
Vector
and
{
=
Tensor
point) (_1,
_2,
_3)
Fields
fourth-order,
contravariant,
tangent
operator
A
d_ & d_
coordinate
differences
dq5
contact
c_0
contravariant contravariant
6
mixed
force
contravariant
E
covariant
¢
contravariant
r/
arbitrary
0
tensor mixed
on differential
metric strain
tensor strain
in preferred tensor
(strain
tensor
contravariant
material
between
(strain
material
between
tensor
of arbitrary weight, stretch tensor
v
covariant
unit
7r
contravariant
stress
71"
contravariant
deviatoric
FI
contravariant
extra-stress
NASA/TM--2002-211914
area direction
tensor
covariant
Tensor
particles
tensor
arbitrary
Body
neighboring
tensor
metric
9,-1
A
acting
unit vector areal strain
idem
covariant
between
kind
and
tensor
normal
vector tensor stress tensor
Rates ix
tensor
rank
points)
material
planes)
+
AffBjk)
D
partial
derivative
D.
Caputo
fractional
J_
Riemann-Liouville
Field
derivative fractional
integral
Transfer
t
Eulerian
transfer
of field:
body
into
Cartesian
space
' to
Lagrangian Cartesian
transfer
of field:
body
into Cartesian
space
Space
S
manifold,
C
(rectangular)
Xo
place
X
reference
S E N3 Cartesian
containing
coordinate
particle
system
_3 in initial
(Lagrangian)
position
state
vector
to
to 3¢0 with
coordinates
X
=
(X1, X2, Xa) in C X
place
X
current in C
I
unit
Kinematic
containing
_
position
in current vector
state
t
to :_ with coordinates
tensor
Fields
a
acceleration
v
velocity
F
deformation
L
velocity
R
orthogonal
Eulerian
particle
(Eulerian)
Vector
a
unit
vector
vector gradient
gradient
tensor
tensor
rotation
and
tensor
Tensor
vector
Fields
in preferred
coordinate
differences
n
body-force unit-normal
vector vector
A
Almansi
A (n)
generalized
B
Finger
C
fourth-order
Ca
anisotropic
Ce
isotropic
C ea
anisotropic
C"
isotropic
f
NASA/T_2002-211914
strain
material between
tensor
(strain
anisotropic
strain
deformation
elastic elastic viscous
neighboring
between tensor
places
material of order
points) n
tensor
tangent part
direction
operator
of elastic part
operator
of viscoelastic
part part
tangent
of viscoelastic of viscoelastic
x
tangent tangent tangent
operator operator operator
x = (xl, x2, xa)
anisotropic
viscous
part
of viscoelastic
generalized
strain
tensor
tangent
operator
m
E
(")
G
of order
n
G
arbitrary contravariant-like tensor fourth-order relaxation modulus
J
arbitrary
tensor
M
arbitrary
covariant-like
M
fourth-order
T
Cauchy
memory
stress
deviatoric left stretch
Z
Signorini
unit
a_0
stress
tensor
tensor strain
tensor
spatial-gradient Tensor
function
tensor
Cauchy
V
Eulerian
tensor
(strain
operator,
between
material
planes)
0/0x
Rates vector
in preferred
D, O/Ot
partial derivative material derivative
D
rate-of-deformation
material
direction
tensor
c,r
D
upper-fractal
D
lower-fractal
rate-of-deformation rate-of-deformation
upper-convected
}
tensor
(Oldroyd)
upper-fractal
derivative
upper-fractal
integral
corotational
tensor derivative
of order of order
of order of order
a a
of a contravariant-like
tensor
a of a contravariant-like
tensor
a of a contravariant-like
(Zaremba-Jaumann)
derivative
tensor
of an arbitrary
_G G
tensor
W
M
lower-convected
(Oldroyd)
lower-fractal
derivative
lOtM
lower-fractal
integral
W
vorticity
Lagrangian
Vector
derivative
of order of order
of a covariant-like
c_ of a covariant-like a of a covariant-like
tensor and
Tensor
Fields
A
dX & dX
coordinate
N
unit-normal
C
Green
C
fourth-order
tangent
E
Green
tensor
H
arbitrary
contravariant-like
N
arbitrary
covariant-like
NASA/T_2002-211914
differences
between
neighboring
places
vector
deformation
strain
tensor operator (strain
between tensor
tensor xi
material
points)
tensor tensor
tensor
M M
G
M
Y_
P
second
Piola-Kirchhoff
deviatoric
part
p*
Lagrangian
U
right
Y
Lagrangian
Div
spatial-gradient
Lagrangian
Tensor
D
partial
derivative
Caputo
fractional
J_
tensor
Piola-Kirchhoff
stress
tensor
(strain
material
operator,
O/OX
tensor
tensor
tensor strain
between
Rates
Riemann-Liouville
NASA/T_2002-211914
of second
stress
stretch
stress
derivative fractional
integral
xii
planes)
Preface This
is the
Command
first
annual
year project
"Advanced
supported
by grant
land
Clinic Foundation, Space
The
U.S. Army
Simulation"
Surgical
order
to the
for the three
and
through
report
to which
Act Agreement
objective
of this report
viscoelastic
(FOV)
the NASA SAA
Soft Tissue
Research
Modeling
and
for Telemedicine
No. DAMDI7-01-1-0673 Glenn
Research
Center
Material
to The
Cleve-
is a subcontractor
3-445.
is to extend
material
Medical
popular
models
into
one-dimensional
their
(ID)
three-dimensional
fractional-
(3D)
equiva-
lents for finitely deforming continua, and to provide numerical algorithms for their solution. The present report is organized into seven chapters and three appendices. The
first
chapter
for computing equations
serves
fractional
(FDE's),
as an introduction
derivatives,
and
the
Mittag-Lemer
tions of FDE's) are provided. One of the oldest applications two presents the standard
to the fractional
fractional
integrals,
function
(which
of the fractional
an overview of ID FOV. FOV solid are put forth
apprears
calculus
Definitions along with
calculus.
Algorithms
fractional-order
differential
in analytic
is viscoelasticity.
using
convected) tensor fields. Three strain fields are introduced that are strain based on changes in: length of line, separation of non-intersecting and volmne of mass. Introduced here for the first time are fractal rates fields.
Body
In the fourth
chapter,
into
objective,
those
spatial
fields
useful
when
solving
The pressible
and
grangian of elastic
then
space
derives
isotropic
NASA/T_2002-211914
are frame
potential are
measures of surfaces, of arbitrary
chapter
of field
invariant.
are mapped
transfer Spatial
is derived
theories
to an integrity
considered.
into Cartesian
experiments The suite
(e.g., uniaxial and simple
in the previous by-product
(i.e.,
equations.
and transverse-isotropic
a work
materials
transferred
contitutive
body
is that fields
are
problems.
frames. The tangent modulus and viscoelastic solids.
shear-free extension lational compression)
defined A useful
by field transfer
by applying
A suite of homogeneous sented in the sixth chapter.
deriving
fields
fields.
boundary-value
incompressible and
when
the body
created
materials
body
are useful
Cartesian,
fifth chapter
viscoelastic in the
fields
Chapter
for the standard FOV fluid and formulae that are useful in their
characterization, assuming infinitesimal strains and rotations. The third chapter provides an overview of continuum mechanics
tensor
solu-
space
These in both
for the general
for elastic
basis. theories
the
Both are
Eulerian
theoretical
and com-
derived and
La-
structures
used to characterize material models is preincludes the homogeneous deformations of:
elongation, biaxial extension, pure shear, and dishear. The deformation, stress and strain fields xiii
definedin the prior chapter,alongwith their variousrates,are all quantifiedfor this suite of experiments. Chaptersseventhrough nine provideelastic and viscoelasticconstitutivemodels appropriate for 3D analysis. Chapter sevenprovidesmaterial modelsfor bulk response.Chapter eight will introduce material modelsfor isotropic elastomers,while chapternine will introducematerial modelsfor soft biologicaltissues,which aregenerally transverseisotropic;they will be completedfor the secondannualreport. Both classicalandfractional-orderviscoelasticmodelsarepresented.Includedaresolutions for the characterizationexperimentsof chaptersix. Therearethreeappendices.The first appendixtabulatesCaputofractionalderivatives for a few of the more commonmathematicalfunctions. The secondappendix outlinesanautomaticprocedurefor numericalintegrationthat is requiredby the algorithm which computesthe Mittag-Leffier function. And the third appendixprovides an efficientschemefor approximating a specificform of the Mittag-Lemer function that arisesin FOV.
NASA/T_2002-211914
xiv
Chapter
1
Fractional numerical
1.1
Calculus: methods
Riemann-Liouville
In the classical n-fold
calculus
integration
an Abel
of Newton
and
function
y(x)
of the
(power
law)
Jny(x)
Fractional Leibniz, into
Integral
Cauchy
a single
reduced convolution
:=
"'"
y(x0) dxo..,
integers,
and
possessing
_
1
dx,_-2 dx,_-i (1.1)
-(n-1)!f, J'_ is the
integral
of an
kernel,
1
where
the calculation
n-fold
noN,
integral
N+ is the
set
continued
Cauchy's
result
continuous
gamma
function
operator
with
of positive
reals.
by replacing F(n),
d_y(x)
noting
= y(x),
Liouville
xe ,
N is the set of positive
and
Riemann*
the
discrete
factorial
(n -
that
(n-1)Y
= F(n),
thereby
analytically
1)! with
Euler's
producing
[67,
Eqn. AI JaY(X) where
ja is the
report
we take
positive finite real. need it to be real. A brief
(x - x') 1-a y(x')
Riemann-Liouville
JaJ_y(x) = Jf_J'_y(x) the fractional calculus, In this
:=
history
integral
operator
dx',
a,x
of order
e 1[¢+,
a, which
(1.2) commutes
(i.e.,
-- Ja+_y(x) V a, _ _ ][_+). Equation (1.2) is the cornerstone of although it may vary in its assignment of limits of integration. the
lower
Actually, of the
limit
to be zero
a can be complex
development
and
the
[102], but
of fractional
upper for our
calculus
can
limit
to be some
purposes
we only
be found
in Ross
[100] and Miller and Ross [78, Chp. 1]. A survey of many emerging applications of the fractional calculus in areas of science and engineering can be found in the recent text
by Podtubny
[86, Chp.
10].
*Riemann's pioneering work in the field of fractional calculus was done during his student years, but published posthumous--forty-four years after Liouville first published in the field [100]. NASA/TM
2002-211914
1
1.2
Caputo-Type
From
this
single
Fractional
definition
for fractional
initions for fractional differentiation that we choose to use, which requires I(_]-times
differentiable
Derivative
in the
integration
several
def-
(cf. e.g., [86, 102]). The special operator the dependent variable y to be continuous
D_ and
independent
D,ay(x)
one
variable
can construct
x, is defined
by
:= yFal-aD[Cdy(x),
(1.3)
such that lim
n,_y(x)----
Day(x)
for
n E N,
(1.4)
_--+n-
with
D°y(x)
greater The
than
operator
to call
-- y(x),
where
(or equal
to)
Caputo
Caputo
classical
differential
operator a table
operator
function
a -+ n_
operator
to use this A presents functions.
differential
ceiling
where
differential
D,_(y that
is the
and
D n, n E N, is the
D,_ the
the amoung the first properties, t Appendix common mathematical The
ral a,
a,
the
c_ goes
operator.
of order
smallest
integer
to n from
below.
It is accepted
after
in applications and of Caputo derivatives
is a linear
+ z)(x)
giving
means
Caputo
practice
[12] who
to study for some
was
some of its of the more
operator
= D_y(x)
(1.5a)
+ D,_z(x)
commutes D_D_,y(x)
if y(x)
is sufficiently
= D_,D_y(x)=
smooth,
and D_c
it possesses = 0
The more common Riemann-Liouville not commute [86, pg. 74]; furthermore, Ross
D_+Zy(x)
is a function
of x!
[100] attributes
the fractional mathematics
calculus has historically and physics communities.
V c_,/_ E R+
the desirable
for any constant
property
(1.5b) that
c.
(1.5c)
fractional derivative D R, although D'_c = D [_] J['_l-'_c = cx-'_/F(1 this had
startling
fact
as the
a difficult
time
being
main
linear, need - o0, which reason
embraced
why
by the
factually, Liouville introduced the operator in his historic first paper on the topic [67, ¶6, Eqn. B]. Still, nothing in Liouville's works suggests that he ever saw any difference between D,_ = J[_]-aD [_] and D _ -- D [_] j[al-a, D _ being his accepted definition [67, first formula on pg. 10]--the RiemannLiouville differential operator of order a. Liouville freely interchanged the order of integration and differentiation, because the class of problems that he was interested in happened to be a class where such an interchange is legal, and he made only a few terse remarks about the general requirements on the class of functions for which his fractional calculus works [74]. The accepted naming of the operator D,a after Caputo therefore seems warrented. Rabotnov [90, pg. 129] introduced this same differential operator into the Russian viscoelastic literature a year before Caputo's paper was published. Regardless of this fact, operator D,_ is commonly named after Caputo in the current liturature. NASA/T_2002-211914
2
The Riemann-Liouvilleintegral operator J_ D.a are inverse
operators
D'_J'_y(x)
= y(x)
in the
sense
and
J"D_.y(x)
and
the
Caputo
differential
operator
that L_J xk = y(x)
- E
_/Y_k+)'
c_ C R+,
(1.6)
k=O
with
y_k+) := Dky(O+)
than
c_. The
•
like formulae, A word
n-fold
.
n
calculus•
n
= y(x)
Fractional Leibniz
function
differential
and
J
n
n
y(0 +) F(1-a)
the
= y(x) Caputo
is given z(x)
the
largest
of integer -
do not satisfy
whenever rule
giving
operators
D y(x)
derivatives
product
× z)(x)-
floor
and
For example,
0 < c_ < 1, the
D,_(y
integral
D J y(x)
of caution•
of classical that
vm..
L_J is the
' where
classic
n--1
integer order
xk
less
satisfy
(k)
_--]k--0 _ Yo+, n E N.
the Leibniz derivative
product
rule
is restricted
so
by
- z(0 +) x"
×
(1•7)
+
(J'-°yl(x) k=l
where,
unlike
the
Leibniz
product
rule
coefficients (k) = _(o-1)(a-2)...(a-k+l)k, become zero whenever k > a because extent). Liouville
A similar fractional
1.2.1
Integral
The Caputo
derivative
This
D_y(x)
(with (o) = 1, a E _ a _ N (i.e., the binomial
the
binomial
and k E N) do not sum is now of infinite rule
of the
Riemann-
Expressions (1.3) can be expressed
1
the weak singularity
observed.
derivatives,
infinite sum exists for the Leibniz product derivative (cf. Podlubny [86, pp. 91-97]).
D,_y(x)-F([a]_a) where
for integer-order
singularity
notation
f0 (x_x,)O,_L_,j(Dr"ly)(x')dx 1 caused
by the Abel kernel
can be removed
through
JO+ ( xrOl-o,,(rol)
1 -_) raq
= F(I+
in more explicit
-t-
fox
',
as the integral
_,xER,,_,
(1.8a)
of the integral
operator
an integration
by parts
(x --
is readily
')
x')r°'l-a(D:t+r':"ly)(x')dx
(1.8b) provided
that
the dependent
in the independent (1.8b)
the power-law
in (1.8a) The
the kernel
kernel
fractional
NASA/T_2002-211914
but
variable
in order derivatives,
y is continuous
x over the interval is bounded
is singular
two representations
calculations, such
variable
over the entire
at the upper of (1.8a)
to obtain we found
and
limit
and
(1.85)
a numerical it even 3
(l+[c_])-times
of differentiation
differentiable
(integration)
interval
of integration;
[0, x]. In whereas,
of integration. are quite
useful
scheme
for the
more
helpful
for pen-and-paper approximation
to look
at yet
another
of
representation
that
seems
to have
been
introduced
into
this
context
by Elliott
[30];
namely, D*_y(x)
F(-a)1
-
fo x (x-
This representation
can also be obtained
by parts,
the
is that
but the
with
function
roles
disadvantage
is that
thus
to interpret
we have
is cumbersome problem brief
singularity
of such
reader
differential initial and
equations boundary
the
using
the
the
in the
to [20, 30[ and
method The
instead
is now strong
but,
makes
E N+.
interchanged.
integrand
calculations
following
the
advantage
here The
rather
as we shall
models,
the subject
of this report,
pages.
(possibly
see below,
= f(x,y(x)),
inhomogeneous)
solution
A typical
feature
to (1.9)
to specify
additional
case
of Caputo
FDE's,
at the initial not physical; experiment,
details,
a a
the
therein.
are systems
of fractional-order
(1.9a)
conditions
k = 0, 1,...,
[aJ,
(1.9b)
equations
conditions these
additional
fractional
derivatives
X E N+. It turns f of the right-hand
[21].
in order
(both
classical
to produce
conditions
tions listed in (1.9b), which are akin to those familiar to us. In contrast, for Riemann-Liouville certain
it is not
c_,x E R+,
initial
does exist
of differential
need
constitute
and This
We provide
For more
cited
and whose solution is sought over an interval [0, X], say, where out that under some very weak conditions placed on the function a unique
weak,
integral.
(FDE's) that need to be solved in accordance with appropriate conditions. A FDE of the Caputo type has the form
y_k+)= Dky(O+),
side,
than
do this job.
references
of integration
of its derivative. finite-part
computer
(1.8c)
FDE's
D._y(x) satisfying
a,x
as a Hadamard-type
that
Caputo-Type material
(1.8a)
of the kernel
an algorithm
is referred
Fractional
in the
this integral
an algorithm
description
1.3
the
from
two factors
appears
in pen-and-paper
to devise
interested
of the
y itself
1 a+l y(x')dx', x')
(and/or
and
a unique
are just
the
fractional) solution.
static
is the For the
initial
condi-
of classical ODE's, and are therefore FDE's, these additional conditions integrals)
of the
unknown
solution
point x = 0 [57], which are functions of x! These initial conditions are furthermore, it is not clear how such quantities are to be measured from say, so that
they
can be appropriately
assigned
in an analysis,
t If for no
*We explicitly note, however, the very recent paper of Podlubny [87] who attempts to give highly interesting geometrical and physical interpretations for fractional derivatives of both the Riemann-Liouville and Caputo types. These interpretations are deeply related to the questions: What precisely is time? Is it absolute or not? And can it be measured correctly and accurately, and if so, how? Thus, we are still a long way from a full understanding of the geometric and physical nature of a fractional derivative, let alone from an idea of how we can measure it in an experiment, but our mental picture of what fractional derivatives and integrals 'look like' continues to improve. NASA/T_2002-211914
4
other reason,the need to solveFDE's is justification enoughfor choosingCaputo's definition (i.e., D, _ -- Jr_I-_D[_I) for fractional differentiation over the more commonly used (at least in mathematical (viz., D _ : DF_Ijr_I-_).
1.4
Numerical
1.4.1
ordinary
definition
of Liouville
and
Riemann
Approximations
Caputo-type
Unlike
analysis)
Fractional
derivatives,
hereditary functionals rithm for computing
which
Derivatives are
point
possessing a total Caputo derivatives
functionals,
fractional
derivatives
are
memory of past states. A numerical algohas been derived by Diethelm [20] l and is
listed in Alg. 1.1. Validity of its Richardson extrapolation scheme for 1 < c_ < 2, or one similar to it, has to date not been proven, or disproven. Here Yn denotes y(xn), while YN represents y(X) where [0, X] is the interval of integration (fractional differentiation) with 0 < xn < X. This algorithm was arrived at by approximating the integral
(1.8c)
Similar
algorithms
general
procedure
The
with
derived
and Podlubny The extent of a fractional illustrated for the backward
method,
ranges
fractional
become
is often
derivative
used
(and
restricting
six cases
(cf., e.g., with
Oldham
and
Spanier
to appear
with
in this
exhibits
figure.
h = X/8,
by a like backward-difference weights
being
the
[82, §8.2]
for approximating
integrals).
This operator
plotted
difference
the
approximate
of rememberance of past states exhibited by the derivative is manifest, for example, in its weights
in Fig. 1.1.
0 < c_ < 2. by using
needed. to numerically
7]) and it was the first algorithm
a fading
If Dy(X)
then
the
scheme, zero.
then
It is evident
memory: were
effective
be a0,s = 1 and al,s = -1 with all remaining by the line segments in this figure. Similarly, remaining
thereby
of c_ can be constructed
in Ref. [20], if they algorithm
[86, Chp.
derivatives
product
to larger
Grfinwald-Letnikov
Riemann-Liouville fractional
a trapezoidal
applicable
0.001
of quadrature
by
a
would
being zero, as represented were to be approximated
a0,s = 1, al,s = -2 from the data
< las,sl < 0.01
to be approximated
weights
weights if D2y(X)
hereditary nature of quadrature, as
and
presented
a2,s = 1 with
in Fig. 1.1 that
all the
weights of quadrature an,s for approximating D_y(X) are compatible with those for the first- and second-order backward differences, and that fractional quadratures have additional
contributions
that
monotonically
diminish
with
increasing
from node n -- 2 fading all the way back to the origin at node that a truncation scheme may be able to be used to enhance for some
classes
of functions,
but
nodal
number
n : N. This suggests algorithmic efficiency
not all.
§Apparently this algorithm first appeared in the PhD thesis of Chern [15], unbeknownst to us (KD) at the time of writing Ref. [20]. Chern used this algorithm to differentiate a Kelvin-Voigt, fractional-order, viscoelastic, material model in a finite element code. He did not address stability or uniqueness of solution issues; he did not compute error estimates; and he did not utilize an extrapolation scheme to enhance solution accuracy. NASA/T_2002-211914
5
Algorithm
1.1 Computation
For interval
[0, X] with
grid
h_r(2-a)
D,_y(Z) using
{x_, = nh:
_n=0
= D_, yg(h)
the quadrature
of a Caputo
fractional
n = 0, 1,2,...,N}
(
an,y
derivative
YN-n
--
0
where
weights
(derived
from
a trapezoidal
if2_no fl 0.
the mappings
symmetric
J_tt
reproduce
1 F(a)
J_H--
F(a)
of (4.25),
tensor
it follows
tt maps
the
into Cartesian
(t - t') 1-_ tt(t')
F(a)
that
Riemann-Liouville space
integral
in the Eulerian
frame
of a as
dt'
(t-t')
1-_F-T(t''t)'M(t')'F-l(t''t)dt'
(4.29a)
t
I _LM =:= =F_T. __
(F__)
f_
1 1-_ FT. M(t').Ft, (t - t')
dt' )
. F -1
= =F •OroN).=F -1, while
the
Riemann-Liouville
F(a)
integral
of a symmetric
contravariant
tensor
rI maps
as
(t - t') 1-_ _7(t') dt'
:= F-_1
_rG t
= F.
_t I (t-t') 1 1-_F(t''t)
( fl
'
(t - t') '-a
"G(t')'FT(t"t)dt'
F}'.
G(t').
(4.29b)
F} T dt'
)
•F T
=F= (J°H) for all a > 0.[]
4.4 Strain
Strain
Fields
is not a unique
Cartesian,
strain
separation
between
NASA/T_2002-211914
fields
concept that
material
in finite-deformation
relate points,
to changes
while the second 48
analysis.
in length-of-line. pertains
Here The
we present first pertains
to a separation
two, to a
between
material
planes.
A third
material point. constructs.
These
measure
concepts
of strain
accounts
are presented
for changes
in both
their
in the
Eulerian
volume
and
of a
Lagrangian
As it turns out, the strain fields used in the constitutive theories of Chp. 5 are different from the classic strain fields that are discussed in this chapter. Be that as it may, seeing
the
strain
how
arrive
how the spatial
4.4.1
fields
they
presented from
strain
fields
in this
chapter
transfer
should
of Chp.
5 are arrived
are of historical
significance,
aid the
understanding
reader's
and of
at.
Covariant-Like
The covariant
strain
In particular,
in an Eulerian
tensor
= where field,
field
A(:E; to, t) is the
of (3.8) maps transfer
:= 1
into spatial
_ := l(t[2] strain
1
tensor,
E :=
__
between
are well known.
a pair
B=-I),
while
where _E(:E0; to, t) is the popular Green [46] strain The-Almansi and Green strains are symmetric distance-of-separation
fields that
of field,
_
Almansi
strain
1
in a Lagrangian
transfer
,
(C -/)
tensor. fields that
of neighboring
(4.30a) of
(4.30b)
measure
material
a change
in the
points.
Rates From
(4.22a),
the
time
rate-of-change
1D De__= _
of strain
_
_e_maps
into
A = = D, = DE= ½ DC=
Cartesian
space
as
(4.31) FT. D" F,_
V
where
A is the
lower-convected
rate
of Almansi
strain,
and
DE
is the
Green
strain
V
rate,
both
of which
of the corotational
are well-known derivative
via
results.
Formula
(4.16a)
A_ = D can be rewritten
in terms
as
&=D-D.A-A.D, thereby producing Zaremba-Jaumann Fractional into Cartesian
order: space
a quasi-linear derivative. The
equation
fractal,
according
for
covariant,
to (4.23a)
(4.32) the
evolution
strain-rate
tensor,
of A in terms
DT_e, of (3.11)
of its
maps
as
_L
_A
= D, aL
C=F 51Da. =
D_E= __
NASA/T_2002-211914
f:o 1
1
(t-t')
49
(4.33)
T " D.F FT.
_ __
D(t,) __
. Ft, dt ,, __
where
_tA_
Green, 0 tl > 0.
In classic
viscoelasticity,
the
fourth-rank
material
functions
Gijkl
and
Mijk_ account for material anisotropy, when present. The first of the three formulations listed above requires the strain to be continuous and differentiable over time. The second and third formulations are less restrictive in that they only require strain to be continuous over time. The last two formulations differ in the moduli of their elastic terms, and they also differ in the states that define strain in their viscoelastic (integral) terms. It is the third expression of these three equivalent expressions that we choose to analytically continue from the infinitesimal into the finite. In all three
of these
stress responds. control variable displacements for the cause Using
Because to which
classical
dW
NASA/T_2002-211914
as our
- like viscoelastic
respond,
guide,
and
solids that
9Jt.(t qS.(t)
produce
= EN(
is the controlled
variable
to which
the theory is linear, it can also be written so that stress is the strain responds. But in our end application (finite elements),
viscoelasticity
a class of K-BKZ
as such,
strain
which
motivates
adopting
the
selecting
hypothesis
strain of Kaye
et al. [7] (which they applied to viscoelastic liquids) wherein strain gradient of strain energy, as in elasticity theory, we therefore con-
Qo -_-__ = 260 _ and
formulations,
are assigned to which forces and stress for the effect.
[56] and Bernstein is replaced by the sider
classic
OI_
a work
¢3_(t) O_ OI.
O'y(t)
increment
O_'(t) Oi_(O,t)
+
obey
the constitutive
O_ OI.(t',t) - t') OI_ O'_,(t)
hypothesis
dt'
,
(5.2) of
+ fot 9:;t.(t58
t') O_20 Oil 0I_(t',t) O'),(t)
dt' )
" d')'(t),
(5.4)
where the viscoelasticstrain-energydensity W(_; with
invariants
such
that
strain
I,(_t,,
0_(_;
t, t)/O_
are one in the
The with
units
task
is relegated
mass
density.
and
present,
fact that
:= 0¢5_(t from
energy,
the memory
strain
0_/0_),
tive
(viz.,
theory
whereas
strain
constitutive
rate
like the
tissues). Equation
are positive
that
II = -r/s
(based
solids
which
the
that
elastic
response
for the
that
strain-energy
5.4
i
Commercial (or modulus) NASA/T_2002-211914
derivative
(i.e., deriva-
K-BKZ
- like for our
of the solvent
place
infinity
with
is particularly
some and
viscoelastic
soft biological
is slightly
[71], which
different
from
is
tr D__7-1 = 0,
no restrictions _r,
are usually considered to be incomthe body stress ,r in its construction. present
but
solid.
Finally,
in the fluid theory, the
on the extent on what
relaxation
topic,
in a viscoelastic
of viscoelasticity,
loose constraints
density
solids,
fluids
in a viscoelastic
zero to minus
places
density)
O0
is present
this is an important
Tangent
on
monotonic,
foundatation
,, O_a30I_(t', t) ffJtn(t - t )oi_ O_'(t) dr',
response
from
in this chapter
elasticity,
and
of the
elastomers
for viscoelastic
have a unique reference state. In contrast with the classic formulae Thermodynamics
selection
elastic
to us (viz.,
use for viscoelastic
N f_ D_7 -1 + 200 Z j_
is moved
presented
of stress
with a second-order
strain-energy
are predominantly
is applicable
one would
the viscous
integration not
on an elastic
rls represents solvent viscosity. Fluids and as such, the extra stress II replaces
Furthermore,
our
This
t = 0, as the reference with the hypothesis of
a first-order
convolves
motivated
of interest
n--_-
wherein pressible,
with
anisotropic
has units
in (5.4) as the phenomenological
materials
(5.4),
construct
function
specify
solids.
of construction
for modeling
attributes,
convolves
the relaxation
presented
of viscoelastic
attractive
function
02_!lJ/Ot 0_,), that
structure
This type
the
[18, 19]. the memory
along
is positive,
viscoelasticity.
which
and that the state of integration, t', replaces the initial state, state in the invariants of the integrand, are both in agreement a fading memory It is because
which
do not
_,
defining
and dimensionless,
classical
0_/0_,
functions,
two states
-t')/Ot',
decreasing,
is different
the
IN" °43;t', t) is constrained
t' C [0, t].
is positive
-t')
is monotone of strain
when
t') with
t'), which _n(t
which
to the gradient The
_'t, := _(_;
¢3n(t-
function,
time,
when
can be no strain
where
function, memory
of reciprocal
characteristics,
and
t', t) = _]3(I1,/2,..., _3; t t, t), n = 1, 2,..., N,
XM-2:
= 0 (there
same),
n th relaxation
its associated
has
_', X1, X2,-..,
_5_, and
it lies outside
of either
the scope
the
lower
because
constitutive
are admissible memory
fluid replaces
strain
limit
of
fluids
do
theories or rotation.
functional
forms
9Yt_ functions.
Like
of this report.
Operator
finite-element for user-defined
packages
often require
constitutive 59
a user to supply
equations
so that
a tangent
its solver
operator
can construct
an optimal tensors, and
stiffness
matrix.
to then
element
map
it into the
tangent
operator
_E:=2 -_e is the
coordinate
_(q3;
finite-element
frame
programs
do not employ
this fourth-rank
of Cartesian
mandates
sothat
tensor
tensor
space
for later
contravariant
tensor
d(_)=_(ld_=_E'dE, _
of (3.8).
This
operator
__ _jike = _jlk
= _jitk
In component
constitutive
form,
equation
body
in the use
body,
in finite
n=l
an additional
components
N_l
__
__
=
symmetry
of
the
_ symmetries
because
stress
as _jke tangent
_
(in _ and
(0__n
N
strain
13.
02fn 03" +-0I_ 02_ 03"
_e
becomes
tensor
In most
(5.6a) outer
material
includingthose of this report, 02fCO/OIm OIn = 0 for all m _ n, thereby the above modulus so that it becomes
=4QoE
a body-
= 20(e_ 7tij)/Oykl. modulus
The
_klij.
system
by
=
o2fllJ OIm OI._ 0±m0zoo3"®b-4/' : :
_°°°°_ jke
xliJ_]ke in coodrinate
defined
(5.5) has
(5.5) reads of (5.3),
N (Of$1J 02I. 0 03+
--
r/ has
0, t) is a fourth-rank
B) of _/jkl
fields.
For the elastic
r/®
required
0'e_],[_ 03"
strain
system
are symmetric
which
though
for us to construct
analysis.
The
where
Even
it is still advantageous
product models,
simplifying
OI. OI_) 03' ®_/"
(5.6b)
In keeping with the assumption that there are no cross-coupling terms arising between the various invariants in the functions selected to represent strain-energy density, it follows that
= 4Qo EN --
Q
_5.(t) O_ 0[.
for the viscoelastic
5.4.1
in the
OIn
OIn
constitutive elastic
fot
equation terms;
02_
of (5.4).
whereas,
Some
I_ =
Ofn
OIn
)
care is required
I_(t',t)
here becuase
in viscoelastic
(integral)
Stability
For an elastic
or viscoelastic
and as such, satisfies sufficient that
solid to be stable
a condition
of strong
_iJklxiYjXky
NASA/T_2002-211914
- t ,.02YO )_ O'y 03" dt ' O2in
(5.7)
_ . 02f_
I_ = I_(O,t) terms.
03" 02fn03" + fo t 9Jt.(t
rill
requires
ellipticity e >
60
O,
V
that
where X__, y.
it be Hadamard it becomes
necessary
stable, and (5.8a)
A more useful is that
condition
(in accordance de:
This
is a thermodynamic
postulate, The
which
E" de=
d(-_)
condition
is not
necessary
with 5.5), which -d__e> 0,
for stability,
a physical
and
to be useful
[92], Leonov
conditions
Dissipative has
Kwon
stability,
no role to play
5.5
to the
[651 and
An
isotropic
when
elastic
being
det(v-l-_0) invertible three
where = (0/00)
(and
solid
for strong
solid
the
one field
analysis
Of note analysis
complex
that attempts have that are of ample
are the
address
works
of Renardy
viscoelastic
liquids.
of viscoelastic
of viscoelastic
be formulated
liquids
[65],
solids.
is the
_o:"
in terms
fields; inverse
2 > 0, guarentees
that
perspective,
of the
variable
other.
and
network
a constitutive
theory
The
conservation
of rubber described
fields
too).
and
of mass,
consequently, As such,
In what
only
follows,
is comprised of sums of like in doing so, we average their
to choose
V_0, or vice versa.
equation
and
invertible, that and
tensor A := _o15
are independent.
is no reason
_-1.
mixed,
consider
V__-:'_0 is non-singular,
in (5.1a & 5.1b)
there
of two,
in particular,
_ is non-singular
invariants
molecular
[53]) produces
[28] stability
are extremely
It is for this reason for strong ellipticity
for a stability
can
able _o:" _ over deformation perspective,
Drucker
ellipticity
the existance of an integrity basis from these two deformation tensors,
a mechanics
so-called
[61], all of which
by body-tensor
of the ten possible
From
(5.8b)
Elasticity
therefore,
we will postulate invariants taken characteristics.
V d__e.
the
developer.
Cho
pertinent
described
B := _-:_i0,*
model
and
in the stability
Isotropic
but not sufficient,
law.
sufficient
and awkward to verify for any given model. been made to obtain sufficient conditions generality
is necessary
the
deformation
However,
elasticity soley
from
(i.e., the in terms
vari-
a physics
neo-Hookean of _o1._
[72].
This contrasts with the popular phenomenological model of Mooney [80] that utilizes both deformation fields in its description, and whose rigorous derivation from statistical mechanics continues to elude researchers. Rivlin
[93, 94, 95, 96, 97, 98] and his students,
were amoung
the first to derive
constitutive
Saunders
equations
from an integrity
strain isotropic elasticity, and to perform multi-dimensional admissible functional forms for the strain-energy density. *As is the norm in general tensor _'_o)is achieved through
a contraction
[99] and Gent
[39, 40, 41],
basis
for finite-
experiments to seek out Reiner [91] was the first to
analysis, the trace of a tensor field (in this case, 7_o: and with the metric 7__:= 7__(_,t), t > 0, or, when appropriate,
its inverse 7_-1 := _-:(_, t). It is because of this fact that gradients (i.e., c9_tJ/O"l.=.) can arise from a strain-energy density W = _(I_;i = 1, 2,..., N) that may otherwise be considered as resulting from non-conservative sources, (e.g., one can introduce D__ as a state variable). This is one very important reason why we prefer using body-tensor fields for the purpose of deriving constitutive theories. NASA/T_2002-211914
61
actually deriveconstitutive equationsfrom an integrity basis. Todaythey are called Reinerfluids. Flory [34]proposeda multiplicative decompositionof the deformationfield (defining _ := (det_)-I/3_ so that det_ = I) as a way to uncoupledeviatoricand hydrostatic responses, We choose not
which has certain advantages to make such a decomposition
variants we have chosen as is typically the case, introducing
such
Adding
a decomposition
respective
each anchored
have gradients and therefore,
invariants
to a different
held by these
two measures
of invariants
as our integrity
from a computational perspective in this body of work because
that are strain fields, not deformation fields it is not certain if the added complexity of
is warrented
at this time.
from the two deformation
state,
[105]. the in-
averages
the effect
for deformation
and
tensors,
of reference
leads
_,oI-_ , and _,-i. To, state
us to consider
that
is tacitly
the following
set
basis:
) ,
(5.9a)
:: /det( o whose
gradients,
t
(5.9b)
tHere we have used the following results to derive the gradients OIn/O"¢ and 02I_,/O"y O"i: 0"y -- = fi [] _,
with components in B of
O_ -1
OYkl
i)ykt _ __-1[]
,l-i,
with
components
in B of
07ij
1
_
L_.i£J
i j
_(¢_
and
noting
NASA/T_2002-211914
that
62
I_oX._l= I_o_lI_l- i_ll•
are symmetric
strain
fields,
and
o2h/o o ==
0213/0.70.7=
in the
coefficients they
Z°)+
I_
moduli
on these
with
Idet(z°l"
det .7o 1- .7
At first
invariants
the definition
scale their
Each strain measure these strains is additive
in (5.9b) vanishes and anti-symmetric
the
definition tensor
desirable
property for strain,
fields
Utilizing
preferable
which
first
the gradients
of using
equation the
(especially
variables.
and
measures
their defThe
so that
limit.
The
or/2, has to do with by the trace, t
an asymmetric
tension/compression
theory
= lnA= when
expressed
1 • 7)
mixed
in (5.9b)
obtained
for an isotropic deformation
for viscoelastic
From
the
identity
from the postulated
elastic
tensors
liquids)
from
the
identity
solid derived
7_o-I-
to use
and
the
7_ and
__,-I.
mixed
stretch
=A-I=_=
with
Hencky's
in terms
[49]
of body-
_
----
0A-
-
of (5.9a),
the work potential variables,
it may
be
tensors
,
that
• OA" .A-I
07
A- A = 7_01. 7_, it follows
02
invariants
from
"To as state
=
A -1 • A_ -- 5, it follows
07 likewise,
strain
in the small-strain
in finite-strain
OA -i
and
invariants
in the reference state; however, not one of in its dependence upon state; yet they
of producing
is _1 ln(_o
associated strain
in/3, but not in/1 that are not shared
appeared
_--A := ____1. _/_ = as state
J
[36].
the constitutive *Instead
property that
these
(5.9c)
/
® Z-1
.7-1 [].7-1
glance,
of infinitesimal
can be used directly of the determinant
response--a
,
complex but, as we shall soon discover, these constitutive formulae in the Eulerian frame.
fact that a square root some unique properties
all possess
_--)) Z-1
I_ _/)_ _
-
of (5.6).
to be somewhat relatively simple
imposed
coincide
I
.7-1. "To" .7-1 + .7.),-1..70" .7-1. "70" .7-1 [] 7=-1) ['
det .7-1..7o
tangent
appear produce
derivatives,
-1
4 (v/det(_-l" + !
gradients initions
second
I
+ .7-1 [] ,7-1..70"
appear
whose
A-l,
---that
• 0A A+A
_ =7oN5.==
Like relations exist in the case of time derivatives. Only when a solution can be found for 0A/0% assuming that a solution does indeed exist, will it be possible to construct a theory using stretches instead of deformations as the state variables. NASA/T_2002-211914
63
in (5.2) has a stress
response
of
(
1
a_0- a__0= 1
(5.25a)
__o'_ • _o = A2 _=> to g_o. C • _a_o=A2
**It was our desire to construct meaningful, anisotropic, finite-strain measures that led us to choose invariants that are themselves a sum of like invariants constructed from fields with opposing variance (e.g., one constituent invariant arises from the contravariant form of a state variable, while the other arises from the covariant form for that same state variable). Constructing meaningful, isotropic, strain measures is not a difficult task; there are many admissible choices. However, there are few admissible choices to select from when attempting to construct meaningful, anisotropic, strain measures. Equation (5.23b) presents one such pair of acceptable strain fields. NASA/T_2002-211914
72
so that
in an Eulerian
transfer
of field one
gets
t 2o _:::_ A a_ = 1 _=_ a • B-1 -_a_= A-2
_o.,To._o
(5.25b)
, 2 _=:_ t a .a = l
20"q"C_o=A and
therefore
a = A-1F-
a_0, or equivalently,
£0 = A F_-I" a_• Material
curves
defined
by the trajectory of a0 are indicative of fiber direction. After a deformation F, these fibers have a new direction of a, and they have stretched by a factor of A. Vectors a_0 and a are the same vector fields that were utilized by Spencer [107, pg. 13]. To the invariants 14
which
=I
of (5.11 or 5.13) we add
are different
Affiliated
(C-q-C
_ao"
-1)
_ao
(gotten
and
invariants !A2
of (5.26)
a®a
-B
of Eqn.
used by Spencer:
are the two,
-1.(aQa
_ao.__C-_aoand a_o-_C2._ao.
anisotropic,
Eulerian,
strain
((-_ --
which
result
from
-)
B -1.
:
-
((a@a)-B
a transfer
of the
these
anisotropic
strain
fields
fields
(5.27a) -)=_))
-1
A2 = a0. C. a_o allows
fields
.B -1
±A 2 (((a®a).B+B.(a@a :=1o - _- = A(1)::4A (2)
5.23a) (5.26)
15=lJ_£0-(g2-i-C-2)-ao,
in form from the invariants
with the
by a field transfer
-I-B
)
-1-
in (5.23b). and
}
(a@a)).B
-1)
Recalling
that
a = A-l_F._a_0,
to be recast
(5.27b)
as
A= (_)= _ ((_.._o)®(B_. __._-0)+ (_.g-._o)®(g"_-o) . (5.27c) _(_)= _((_. ao)®(_-a0)- (E_-a0)®(E_"ao)) } _ (_=-T. ao) ®(_._-1. ET. a0)_ (8=-1. E_.._0)®(g-_'-_o)) These the
are still Eulerian
material
vector
_F- a0 is tangent t-he plane
ao instead
fields,
of isotropy, response
(mapped
that
variant
line of anisotropy,
as illustrated transversely
it is just
of its spatial
to the material
A compressible, has a stress
strain
they
a. In the vector
in Fig. 5.1, and these
vectors
isotropic, from
Eqn.
elastic
solid whose
in the
incompressible T + go/=
where These
the are
NASA/T_2002-211914
vector
F -T. a0 is normal
to
need
not be coaxial.
invariants
are so defined
_
,
(5.28a)
case becomes
2Qo (_,
chapters
state,
of
5.24) of
1E (1) +_-E
(2) + _,4A_(1)
Lagrange multiplier p forces a constraint the formulations for transverse-isotropic
the remaining
in terms
deformed
while
{_T=2eo(_,lE(1)+_-E(2)+_,3eI+_,4A(1)+_,sA(2) )_ = ,.,.= = that
are represented
of this report. 73
+ !_, 5 A(2))
(5.28b)
for incompressiblity, det F = 1. elasticity that we investigate in
X v-T" a0 _.-1
Normal,
Tangent,
_
Materiall
Figure
Tangent
5.1:
Kinematics
Directionj Fib/_r
_
Plane
_
of a transverse-isotropic
material.
operator
To the
isotropic,
elastic,
tangent
and
F
then
elastic,
tangent
operator
mapping
operator
(obtained
the
listed
from
resultant
into
in (5.18),
substituting
Cartesian
one must
add
5.23b
& 5.23c
Eqns. space),
which
is given
an anisotropic, into
Eqn.
5.6,
by
/ C a
_--- 4kO0/1_,4z_2(t[_
(_m_-1"
(a@a)-B
-1)
--_ (_-1.
(a@a).B-1)
[_t)
+ 1__,5 a_(_=[](a®a) + (a®a)[]g -I-/[_
(_-1.
((a®a)
"B-1
-I- _-1.
(a®a)).
_-1)
(5.29) -t-
(B
-1.
((a®a)._-1___
B-1.
(a®n))-B
-1)
_]t
(c q-_,44
Again, exists
(A
additional
(1)
®A
NASA/T_2002-211914
q-_,55
components
a cross-product
strain-energy
(1,)
density
(A
will
dependence that
is being
(2,
need
®A(2)))
to
between
.
be the
considered. 74
included invariants
into
this
in the
expression
function
if there
representing
5.8
Transversely
Isotropic
In accordance
with
of the
- like constitutive
K-BKZ
coelastic material given
our
elastic
theory
Viscoelasticity for tansverse
hypothesis
isotropy
(5.24),
for viscoelasticity
(5.4)
an application produces
a vis-
constitutive relation governing the stress response of a transversely isotropic that has five constituent parts, one associated with each invariant, and is
by
__:2_o +
¢_l(t)_(ot)_ I (%;1 9Jh(t-t'l_,l(t',t)_("/,,l
_0 _
+ _2(t)_I_,2(0, +
:
o'o • 0,-1)
-- ,._--I.
_ _-1 . __
--I
t)_ (0'o1" "y "'Yo 1 -- _-1"
/0
i
_2(t-t')_,2(t',t)g
2t- _)3(t)_,3(O,t)1
(v/det(%01"
"_t'"
,7-1)dt,
0_0" 0'-1" _0" _ -1)
-i t' ""Y'"_t' -i _ =_-i. "It" .y-i"_t""_ 2
-
Cdet(_/-1.
-1
)
dt'
_=0))_ -1
+ _ootO23(t_t,)_,3(t,,t)½(v/det(v_t,l.v__)_
v/det(o,_i.O,r))d
t, %-1
+ _(t) _ _(o,t) _((__o®__o)- _=-1. _=o. (_o®__o).2o._=-_) +
£
"4- e5(t
gYq(t-t')fJ2J,4(t',t)l((_e®_r)-2-1"2t,'(_t,®_t')'Tt"'Y-1)
) _l, 5(O,t)
-_-%O1"%
+
i
((CI_ 0 ®
" (____0®____0)
_.-_-0)" _"
--
%--1"
_t01
_'0"
-- %--1.
_y--1"
%0"
dr' %0"
(_-._-0 ®
_-_-0) "'_0" ,_f--l.,.)tO.,_t--1
(O_0®C_0)"20"%
-1)
gJts(t-t')fgOs(t',t)
× 1 ((___, ®___,) •__._j_- Z-_--2,.(m,®_,) .-.-- - ¢ _ .
O_
t >_ 0, t' E [0,t],
1 < ¢J(t--
_
predicts
on the positive
s (> 0) denotes
the
(7.11c)
real line whenever
rubbery
(quasi-static)
the glassy (dynamic) bulk modulus; time; and > (> 7) is the characteristic,
in (7.11a)
and the
integral
equations
¢ (> 0) bulk,
of (7.9) with mate-
rial functions (7.11b &: 7.11c) are equivalent formulations. Whether one chooses the differential or integral form of a particular model for use in analysis is largely a matter of personal
taste
that
may
be swayed
by factors
like:
the
being solved, and the solution (e.g., numerical) methods The limited data that are available for characterizing that
the
bulk response
example,
the
bulk
of a material
viscoelastic
is not as viscoelastic
response
(with exists
is predicted
modulus
way
for experimentally
happens
to be _(>/_). evaluating
to travel
both
v8 and
bulk
modulus,
materials.
test, like what Bridgman used. The from their ratio >/_ by performing, experiment
used previously Measuring where
to establish wave speeds there
warrented
when
of waves:
logitudinal
is typically
this approach (bulk)
Fractional-order
It is a straightforward
bulk modulus,
in dilational
matter
_. to
compression
quantify
the
is applied
to solids,
and
transverse
fact that there (i.e., dynamic)
_(_/_), from
the
same
a dilational
bulk
present. because
compression
modulus
However, solids
fixture
works
well for
additional
care is
propagate
two types
(shear).
models to extend
(7.10 & 7.11) to fractional
order.
flexibility afforded by fractional models over their classical counterparts, comes time to correlate with experimental data, is justification enough consider such extensions. NASA/T_2002-211914
a viable
in isotropic
time constants > and ¢ can then for example, an additional stress
dynamic
of wave
For
for >/_
O is therefore
s, can be obtained
using
one type
response.
of vs = V/-_-/¢Q
of the effective
dynamic
compression be extracted relaxation
The static
it. suggest
a ratio
at a speed
Measuring the
problem
that is less than 10; whereas, for of dynamic to quasi-static moduli
compressible
7.2.3
[77] has
0 denoting mass density). This is a direct consequence a finite, inhomogeneous, elastic, initial condition whose
bulk
fluids,
as its shear
of polyisobutylene
(i.e., the ratio of dynamic to quasi-static moduli) the same material, the shear response has a ratio (e.g., p/T) that exceeds 104 [33]. For this material model, sound
boundary-value
available for solving bulk viscoelasticity
99
The
greater
when it for us to
Fractional Voigt solid The simplestfractional-orderviscoelasticsolid that onecanconsideris the fractional Voigt solid introducedby Caputo [121 which, for dilatation, takes on the form p(t) and
satisfies
(_ that
a homogeneous
controls
the rate
= -g(1
initial
+ _aD_)A(t),
condition.
of evolution.
This
Equation
(7.12a)
model
(7.12a)
- t') = 1 + fia r(1
an additional
has a relaxation
1 _(t
has
while
_5 is weakly
9)I is strongly
The
fractional
- (_) (t - t') a'
(7.125)
The
Kelvin
simplest,
representing Caputo
and
ation
with
speed
differential is the
[13] which,
limit
(7.12c)
of integration
on physical due
to the
-_(1
equation standard
as it applies
initial
that FOV
to bulk
grounds,
weak
t' = t,
because
it too
present
in
is physically solid
of (2.2)
response,
condition.
This
admissible
for
introduced
by
is given
model
by
Z_O+, has a bulk
(7.13a) relax-
of [75]
_a
memory
Ea
(
-((t-t')/¢)
a
)
l___(t-t')_
¢ > 0, and 0 < _ < 1; however, because -E_,o(-((t-t')/¢)a)/(t-t function becomes E,,l(X)
is the
exponential
unbounded
') _ (t-t') at time
Mittag-Leffier
function
function
e _ whenever
in two parameters,
a and
_-1 as t' --+ t, it is apparent that this memory t exhibiting a weak singularity. Here E_(x) -in one
parameter,
a = 1, while
_, (cf. §1.5).
E_,z(x)
Again,
c_, which is the
reduces
_ (> 0) is the
rubbery
constant
extracted
from
accounting the
Experimental
same data
for the
fractal
order
set of experiments
[103] suggests
of evolution.
used
that
the
to quantify
bulk time
function
bulk modulus;
now _(fi/¢)a (> _) represents the glassy bulk modulus; parameters the characteristic, bulk, relaxation and retardation times, respectively; material
to the
Mittag-Leffler
"Y and fi are still and _ is a new
Their
values
can be
the
constants
of (7.11).
constants
_ and
fi are much
smaller than their counterparts T and p for shear (bulk transients fade faster), and that the fractal order for bulk evolution (_ is smaller than the fractal order for shear evolution
a (bulk
behavior
is less rate
(0, 1]. In other words, the response is more fluid-like.
bulk
sensitive),
both
tends
to be more
response
being
bound
to the
solid-like,
while
interval the
shear
Surface plots of y = Ea,l(-x a) and y = -E_,o(-X_)/x, of _ and _ in (7.13), respectively, are presented in Figs.
which are representative 7.2 & 7.3. In both plots
0.001
< x in the
< a
[a]
0 if j E No and j < Fa], orj_Nandj> LaJ.
+ 1)
c > 0 and
cj-[_]-I
r(j - [aJ)I'([a]
= x j lnx
the
to distinguish
= (x + c)_ for arbitrary =
along
for positive
(D_f)(z)=
2. Let f(x)
cut
we only need
case the power series analytically into the
-a
for some
j E ]R. Then
[a] -j;
+ 1)zF_l-%Fl(l'
[a] -a+
1;-x/c).
j > LaJ. Then
L_J (Da, f)(x)
=
E(-l)[a]-k+l(_)_ k=o
r(j + 1)
+ F(j--4. Let
f(x)
= exp(jx)
a + 1)
for some
f(x)
= sinjx
for some
xk-_(¢(j
- LaJ) - ¢(j - a + 1) + lnz).
j E N. Then
D_ (,f)(x) 5. Let
k r(jl-_(J-- a L°_J) + 1) x'-_"
[
= j['_lxF'_l-'_El,[,_l_,_+x(jx). j E JR. Here
again
we have
two cases:
JF li(-1)r°V2xr l- [1F1(1; - a + 1;ijx) 2r(fal - a + 1) - 1Fl(1; (D_f)(x)=
jral(-1)L_Jl2xFc_l-a 2r(r l - + 1)
Fal - a + 1;-ijx)]
[1Fl(1;
+ 1FI(1; 6. Finally we consider f(x) we obtain two cases:
= cosjx
[a]-a+
if Fal is even,
1;ijx)
pal - a + 1;-ijx)]
if [a 1 is odd.
with some j E R. As in the previous
example,
jr:l(_l) rOv2xr:l-: 2r([c_l -a+
1) [_Ft(1; [c_l-a+ + 1FI(1;
(Dyf)(x)
=
-a
+ 1;-ijx)]
if [a]
is even,
if [a]
is odd.
jF_li(_l)tOj/2xF_l__ 2r(F_l-_+
1) -
NASA/T_2002-211914
[a]
1;ijx)
[1Fl(1; 1FI(1;
106
Fal-a+
1;ijx)
[(_l - a + 1;-ijx)]
is
Appendix
B
Automatic In §1.5.2 complete possible
Integration
we had found the evaluation strategy
explained
for the solution
in QUADPACK
accepted
standard
domain,
software packages. for example, from
two integrals numerically function. In this appendix
integrals.
routines
Essentially
introduced
for efficient
and reliable
but they can also be found
book
in §1.5.2,
Quadrature,
Adaptive,
this routine
follows
it turns
General a top-down
much
out that
purpose) structure
(i.e.,
on the
details,
information
we first
algorithms.
commercially
the routine
is the method
are the generally
quadrature
in many
in order to we outline a
we will follow the ideas
in that
The source code is written in FORTRAN77 the URL http://www.netlib.org/quadpack.
the form encountered
giving
of these
The
[84].
when looking
are in the public
without
the need to calculate of the Mittag-Leffler
and
may be obtained, For integrands of
DQAG
(Double
of choice. explain
and at a later
They
distributed
Our the stage
precision
description
general
of
strategy
we fill in those
details).
B.1 The
The
Fundamental
fundamental
idea of automatic
integrand
function
and
define
integral
in question)
the
to satify
the
integration
the bounds
user's
for the
and
or absolute error that he/she return either an approximation accurate
Strategy is the following.
interval
a desired
accuracy
is willing to accept). for the correct value requirements
The
of integration
user supplies
(i.e., the
(i.e., a bound
the
data
for the
that
relative
The routine is then supposed to of the integral that is sufficiently
or an error
flag
if it fails to find such
an
approximation. Typically integration where the in Alg.
the
to achieve
this
goal by sub-dividing
the
interval
of
B.1.
In practice,
should
try
in an adaptive way, thus concentrating more quadrature nodes in areas integrand is difficult to approximate. This leads to a structure indicated
for example, memory,
algorithms
one
may
too many
or too much return
NASA/T_2002-211914
an error
also
terminate
sub-divisions computing
have
the
WHILE
taken
place
time has been consumed.
flag. 107
loop
in this
thereby
using
algorithm
when;
up all available
In such cases the algorithm
Algorithm B.1 Automatic Integration GIVEN two realnumbersa and b, a function and
some tolerance
Calculate
an approximation
error
Q[f]
for f: f(x)dx
and
a list L of the
approximations
obtained
by L := { ([a, b], Q[f], _)} WHILE error tolerance is not satified the
interval
and remove Bisect
from
it from
L with the
so far
DO
the largest
error
approximations
of while
RETURN
B.2
and
error
estimates
for the two newly
the sum of the
B.1 we have
how the
indicated
obtained
approximations,
loop approximations
Approximation
In Alg.
estimate
list;
subintervals; Add these subintervals, their corresponding and their error estimates to the list L
said
_ for the
this interval;
Calculate
END
an estimate
S(x)dx- Q[S]
Initialize
Take
f : [a, b] -+ C,
e > 0 THEN
left
for all the intervals.
of the Integral open
approximation
some
of the
of the integral
how the required
error
to these two questions. The basic idea behind That is, we calculate of the form
obtained
the
estimates
solution
two different
key details. itself
can be found.
is the
concept
approximations
Specifically,
will be performed,
and
We now turn
we have
not
we have
not
our attention
of Gauss-Kronrod
for the integral,
both
integration. of which
are
n
Za.S(x.,), 5=1
with
suitably
begin
with
chosen a first
nodes, which is the value of the integral 2nl
-
1.
This
values
of n,
aj,n, and
approximation
that
(uniquely whenever
determined) the integrand
formula
has
been
xj,_
is just
(j
=
a Gauss
1,...,n).
In particular,
quadrature
formula
we
with
nl
quadrature formula that gives the exact is a polynomial of degree not exceeding
thoroughly
investigated.
For a recent
survey
we
refer to Ref. [10] and the references cited therein. Specifically, there is no quadrature formula with the same number of nodes that is exact for a larger class of polynomials. Moreover, method
B.3 From
as stated is a very
in [10], both
good
Approximation rather
general
considerations,
can only give an approximation, NASA/T_2002-211914
theoretical
and
practical
evidence
suggest
that
this
one.
of Error it is known
Estimates that
one, single,
but not both an approximation 108
quadrature
and an error
formula estimate.
Therefore,
to derive
quadrature
formula.
quadrature
formula
also
an error
The
estimate,
heuristic
is much
the two approximations the two approximations,
it is necessary
argument
more
is as follows.
accurate
than
to introduce Assuming
the first,
then
the
In view
selecting
again,
nl nodes nodes,
of the
a formula
formula
would
quality
of the
with n2 nodes,
but
the second
difference
between
will be a good approximation for the error of the cruder of and therefore, a rather conservative (and thus quite reliable)
upper bound for the error of the finer of the two approximations. a second quadrature formula that is significantly better than the formula.
a second
that
this
have
and so almost
would
at most none
Gauss
where
be very one
of the
formula,
n2 > nl.
this
node
with
gathered
be achieved
one could
because
in common
information
can only
In principle,
uneconomical
Hence, we need first (Gauss-type)
a Gauss
a Gauss
by
use a Gauss formula
formula
with
with
so far (i.e., the function
n2
values
f (xj,,_ 1), whose calculation is typically the most computationally expensive part of the algorithm) could be reused. To overcome this difficulty, Kronrod [59, 60] suggested to construct a formula that is nowadays called the Kronrod extension of the Gauss formula
or, shortly,
formula
with
the
following
the
Gauss-Kronrod
nl nodes;
it uses
formula.
• The remaining are determined mials
of degree
(cf. the
For our purposes, the
15 points integral
not
exceeding
survey
suggestions
made with
with
high accuracy
method
form
function without
mentioned
above
and
to use the tabulated there, the
K mentioned
properties
for routine
with
in Alg.
using
Kronrod
Gauss
according
of the
n2 nodes
to
xj,._, of
and
references
effort.
NAG
the
cited
Gaussian
extension
for
D01AKF).
109
This
are
therein).
method
with
computing
the
part).
For the
other
P, these formulm using the 30-point
Library
weights
given in [84]. In particular,
1.4 (the monotonic
extension. in the
nodes
In view of the
this gives us an approximation
computational
Kronrod integrals
the
values
we propose 31-point
of the integrand,
too much
its 61-point
for oscillatory
NASA/T_2002-211914
a subset
is given in [29]. of the required
algorithm, with the oscillatory integrand function to follow the oscillations properly; thus, we propose suggested
on the
3nl + 1. formulas calculation
papers
it is sufficient
in combination
nice smoothness
together
is based
is constructed
nodes and the weights aj,n2 of the of the Gauss-Kronrod scheme in such a way that the resulting method is exact for all polyno-
A recent survey on Gauss-Kronrod Algorithms for the concrete
following
and
criteria:
• The nl nodes xj, m of the Gauss the Gauss-Kronrod method.
available
His formula
n2 = 2nl + 1 nodes
is essentially [81] (see the
with quite
integral
in that
may be unable Gauss method also
the
method
documentation
Appendix Table
C
of Pad
Approximates
Mittag-Leffier This
appendix
Function
presents
a scheme
for fast
E_(-x_), suitable
for finite
elements.
(1.16)
is used
Pad_ approximate specifically,
computations
0
