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Lemaitre_ch2.pdf
CHAPTER
Elasticity and
Viscoelasticity
2
CHAPTER
2.1
Introduction to Elasticity
and Viscoelasticity
JEAN LEMAITRE
Universit!e Paris 6, LMT-Cachan, 61 avenue du Pr!esident Wilson, 94235 Cachan Cedex, France
For all solid materials there is a domain in stress space in which strains are
reversible due to small relative movements of atoms. For many materials like
metals, ceramics, concrete, wood and polymers, in a small range of strains, the
hypotheses of isotropy and linearity are good enough for many engineering
purposes. Then the classical Hooke’s law of elasticity applies. It can be derived from a quadratic form of the state potential, depending on two
parameters characteristics of each material: the Young’s modulus E and
the Poisson’s ratio n.
1
AijklðE;nÞ sij skl
c* ¼
ð1Þ
2r
eij ¼ r
@c * 1 þ n
n
sij skk dij
¼
E
E
@sij
ð2Þ
E and n are identified from tensile tests either in statics or dynamics. A great
deal of accuracy is needed in the measurement of the longitudinal and
transverse strains (de 106 in absolute value).
When structural calculations are performed under the approximation of
plane stress (thin sheets) or plane strain (thick sheets), it is convenient to
write these conditions in the constitutive equation.
Plane stress ðs33 ¼ s13 ¼ s23 ¼ 0Þ:
3
1
n
07
E
2
3 6
3
72
6E
e11
7 s11
6
7
6
1
6
6
7
7
4 e22 5 ¼ 6
07
74 s22 5
6
E
7
6
e12
7 s12
6
5
4
1þn
Sym
E
2
Handbook of Materials Behavior Models
Copyright # 2001 by Academic Press. All rights of reproduction in any form reserved.
ð3Þ
71
72
Lemaitre
Plane strain ðe33 ¼ e13 ¼ e23 ¼ 0Þ:
2
3 2
l þ 2m
s11
6
7 6
4 s22 5 ¼ 4
s12
l
l þ 2m
Sym
with
8
>
>
>l ¼
<
>
>
>
:
32
3
0
e11
7
76
0 54 e22 5
2m
e12
nE
ð1 þ nÞð1 2nÞ
ð4Þ
E
m¼
2ð1 þ nÞ
For orthotropic materials having three planes of symmetry, nine
independent parameters are needed: three tension moduli E1 ; E2 ; E3
in the orthotropic directions, three shear moduli G12 ; G23 ; G31 , and
three contraction ratios n12 ; n23 ; n31 . In the frame of orthotropy:
3
3
2
2
e11
3 s11
2
1
n
n
12
13
7
7
6
6
0
0
0
7 6 E
7
6
76
E1
E1
7 6 1
7
6
6
76
7
6 e22 7 6
s
7
22 7
1
n23
7 6
6
6
0
0
0 7
7 6
7
6
6
7
7 6
7
6
E2
E2
76
7 6
7
6
6
76
7
6 e 7 6
1
7
s
33
7
6 33 7 6
6
0
0
0 76
7 6
7
6
E
76
3
7¼6
7
6
ð5Þ
76
7 6
7
6
1
76
7 6
7
6
7
0
0
6 e23 7 6
s23 7
76
2G23
7 6
7
6
76
7 6
7
6
6
7
1
7 6
7
6
6
0 7
7 6 Sym
7
6
6
7
6 e31 7 6
6
2G31
76 s31 7
7 4
7
6
5
7
7
6
6
1
5
5
4
4
2G12
e12
s12
Nonlinear elasticity in large deformations is described in Section 2.2,
with applications for porous materials in Section 2.3 and for elastomers
in Section 2.4.
Thermoelasticity takes into account the stresses and strains induced by
thermal expansion with dilatation coefficient a. For small variations of
temperature y for which the elasticity parameters may be considered
as constant:
eij ¼
1þn
n
sij skk dij þ aydij
E
E
ð6Þ
For large variations of temperature, E; n; and a will vary. In rate
formulations, such as are needed in elastoviscoplasticity, for example, the
2.1 Introduction to Elasticity and Viscoelasticity
73
derivative of E; n; and a must be considered.
1þn
n
@ 1þn
@ n
@a
’
s’ ij s’ kk dij þ aydij þ
sij skk dij þ ydij y’
e’ij ¼
E
E
@y
E
@y E
@y
ð7Þ
Viscoelasticity considers in addition a dissipative phenomenon due to
‘‘internal friction,’’ such as between molecules in polymers or between cells in
wood. Here again, isotropy, linearity, and small strains allow for simple
models. Quadratric functions for the state potential and the dissipative
potential lead to either Kelvin-Voigt or Maxwell’s models, depending upon the
partition of stress or strains in a reversible part and in an irreversible part.
They are described in detail for the one-dimensional case in Section 2.5 and
recalled here in three dimensions.
Kelvin-Voigt model:
sij ¼ lðekk þ yl e’ kk Þdij þ 2mðeij þ ym e’ij Þ
ð8Þ
Here l and m are Lame’s coefficients at steady state, and yl and ym are two
time parameters responsible for viscosity. These four coefficients may be
identified from creep tests in tension and shear.
Maxwell model:
1þn
s
n
skk
ð9Þ
s’ ij þ
s’ kk þ
dij
e’ij ¼
E
t1
E
t2
Here E and n are Young’s modulus and Poisson’s ratio at steady state, and
t1 and t2 are two other time parameters. It is a fluidlike model:
equilibrium at constant stress does not exist.
In fact, a more general way to write linear viscoelastic constitutive
models is through the functional formulation by the convolution product as
any linear system. The hereditary integral is described in detail for the
one-dimensional case, together with its use by the Laplace transform, in
Section 2.5.
Z t
n
X
dskl
p
eijðtÞ ¼
dt þ
Jijkl ðt tÞ
Jijkl ðt tÞDskl
ð10Þ
dt
o
p¼1
p
JðtÞ is the creep functions matrix, and Dskl are the eventual stress steps.
The dual formulation introduces the relaxation functions matrix RðtÞ
Z t
n
X
dekl
p
dt þ
Rijkl ðt tÞ
Rijkl Dekl
sijðtÞ ¼
ð11Þ
dt
o
p¼1
When isotropy is considered the matrix, ½ J
and ½R
each reduce to two
functions: either JðtÞ, the creep function in tension, is identified from a creep
74
Lemaitre
test at constant stress; JðtÞ ¼ eðtÞ =s and K, the second function, from the
creep function in shear. This leads to
Dsij
Dskk
ð12Þ
eij ¼ ð J þ KÞ K
dij
Dt
Dt
where stands for the convolution product and D for the distribution
derivative, taking into account the stress steps.
Or MðtÞ, the relaxation function in shear, and LðtÞ , a function deduced
from M and from a relaxation test in tension RðtÞ ¼ sðtÞ =e; LðtÞ ¼
MðR 2MÞ=ð3M RÞ
Deij
Dðekk Þ
ð13Þ
dij þ 2M sij ¼ L Dt
dt
All of this is for linear behavior. A nonlinear model is described in
Section 2.6, and interaction with damage is described in Section 2.7.
CHAPTER
2.2
Background on
Nonlinear Elasticity
R. W. OGDEN
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK
Contents
2.2.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.2.2 Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.2.3 Stress and Equilibrium . . . . . . . . . . . . . . . . . . . . . . 77
2.2.4 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.2.5 Material Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.2.6 Constrained Materials . . . . . . . . . . . . . . . . . . . . . . . 80
2.2.7 Boundary-Value Problems . . . . . . . . . . . . . . . . . . . . 82
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.2.1 VALIDITY
The theory is applicable to materials, such as rubberlike solids and certain soft
biological tissues, which are capable of undergoing large elastic deformations.
More details of the theory and its applications can be found in Beatty [1]
and Ogden [3].
2.2.2 DEFORMATION
For a continuous body, a reference configuration, denoted by Br, is identified
and @Br denotes the boundary of Br . Points in Br are labeled by their
position vectors X relative to some origin. The body is deformed quasistatically from Br so that it occupies a new configuration, denoted B, with
Handbook of Materials Behavior Models
Copyright # 2001 by Academic Press. All rights of reproduction in any form reserved.
75
76
Ogden
boundary @B. This is the current or deformed configuration of the body. The
deformation is represented by the mapping w : Br ! B, so that
x ¼ vðXÞ
X 2 Br ;
ð1Þ
where x is the position vector of the point X in B. The mapping v is called the
deformation from Br to B, and v is required to be one-to-one and to satisfy
appropriate regularity conditions. For simplicity, we consider only Cartesian
coordinate systems and let X and x, respectively, have coordinates Xa and xi ,
where a; i 2 f1; 2; 3g, so that xi ¼ wi ðXa Þ. Greek and Roman indices refer,
respectively, to Br and B, and the usual summation convention for repeated
indices is used.
The deformation gradient tensor, denoted F, is given by
F ¼ Grad x
Fia ¼ @xi [email protected]
ð2Þ
Grad being the gradient operator in Br . Local invertibility of v and its inverse
requires that
05 J det F51
ð3Þ
wherein the notation J is defined.
The deformation gradient has the (unique) polar decompositions
F ¼ RU ¼ VR
ð4Þ
where R is a proper orthogonal tensor and U, V are positive definite and
symmetric tensors. Respectively, U and V are called the right and left stretch
tensors. They may be put in the spectral forms
3
3
X
X
ð5Þ
U¼
li uðiÞ uðiÞ
V¼
li vðiÞ vðiÞ
i¼1
i¼1
where vðiÞ ¼ RuðiÞ ; i 2 f1; 2; 3g, li are the principal stretches, uðiÞ the unit
eigenvectors of U (the Lagrangian principal axes), vðiÞ those of V (the Eulerian
principal axes), and denotes the tensor product. It follows from Eq. 3 that
J ¼ l1 l2 l3 :
The right and left Cauchy-Green deformation tensors, denoted C and B,
respectively, are defined by
C ¼ FT F ¼ U2
B ¼ FFT ¼ V2
ð6Þ
2.2.3 STRESS AND EQUILIBRIUM
Let rr and r be the mass densities in Br and B, respectively. The mass
conservation equation has the form
rr ¼ rJ
ð7Þ
77
2.2 Background on Nonlinear Elasticity
The Cauchy stress tensor, denoted r, and the nominal stress tensor, denoted
S, are related by
S ¼ JF1 r
ð8Þ
The equation of equilibrium may be written in the equivalent forms
div r þ rb ¼ 0
Div S þ rr b ¼ 0
ð9Þ
where div and Div denote the divergence operators in B and Br , respectively,
and b denotes the body force per unit mass. In components, the second
equation in Eq. 9 is
@Sai
þ rr bi ¼ 0
@Xa
ð10Þ
Balance of the moments of the forces acting on the body yields simply
rT ¼ r, equivalently ST FT ¼ FS: The Lagrangian formulation based on the
use of S and Eq. 10, with X as the independent variable, is used henceforth.
2.2.4 ELASTICITY
The constitutive equation of an elastic material is given in the equivalent
forms
S ¼ HðFÞ ¼
@W
ðFÞ
@F
r ¼ GðFÞ J1 FHðFÞ
ð11Þ
where H is a tensor-valued function, defined on the space of deformation
gradients F, W is a scalar function of F and the symmetric tensor-valued
function G is defined by the latter equation in Eq. 11. In general, the form of
H depends on the choice of reference configuration and it is referred to as the
response function of the material relative to Br associated with S. For a given
Br , therefore, the stress in B at a (material) point X depends only on the
deformation gradient at X. A material whose constitutive law has the form of
Eq. 11 is generally referred to as a hyperelastic material and W is called a
strain-energy function (or stored-energy function). In components, (11)1 has
the form Sai ¼ @[email protected] , which provides the convention for ordering of the
indices in the partial derivative with respect to F.
If W and the stress vanish in Br , so that
WðIÞ ¼ 0
@W
ðIÞ ¼ O
@F
ð12Þ
where I is the identity and O the zero tensor, then Br is called a natural
configuration.
78
Ogden
Suppose that a rigid-body deformation x * ¼ Qx þ c is superimposed on
the deformation x ¼ vðXÞ, where Q and c are constants, Q being a rotation
tensor and c a translation vector. The resulting deformation gradient, F * say,
is given by F * ¼ QF: The elastic stored energy is required to be independent
of superimposed rigid deformations, and it follows that
WðQFÞ ¼ WðFÞ
ð13Þ
for all rotations Q. A strain-energy function satisfying this requirement is said
to be objective.
Use of the polar decomposition (Eq. 4) and the choice Q ¼ RT in Eq. 13
shows that WðFÞ ¼ WðUÞ: Thus, W depends on F only through the stretch
tensor U and may therefore be defined on the class of positive definite
symmetric tensors. We write
@W
ð14Þ
T¼
@U
for the (symmetric) Biot stress tensor, which is related to S by
T ¼ ðSR þ RT ST Þ=2.
2.2.5 MATERIAL SYMMETRY
Let F and F0 be the deformation gradients in B relative to two different
reference configurations, Br and B0r respectively. In general, the response of
the material relative to B0r differs from that relative to Br , and we denote by W
and W 0 the strain-energy functions relative to Br and B0r . Now let P ¼ Grad X0
be the deformation gradient of B0r relative to Br , where X0 is the position
vector of a point in B0r . Then F ¼ F0 P: For specific P we may have W 0 ¼ W,
and then
WðF0 PÞ ¼ WðF0 Þ
ð15Þ
for all deformation gradients F0 . The set of tensors P for which Eq. 15 holds
forms a multiplicative group, called the symmetry group of the material relative
to Br . This group characterizes the physical symmetry properties of
the material.
For isotropic elastic materials, for which the symmetry group is the proper
orthogonal group, we have
WðFQÞ ¼ WðFÞ
ð16Þ
for all rotations Q. Since the Q’s appearing in Eqs. 13 and 16 are independent,
the combination of these two equations yields
WðQUQT Þ ¼ WðUÞ
ð17Þ
79
2.2 Background on Nonlinear Elasticity
for all rotations Q. Equation 17 states that W is an isotropic function of U. It
follows from the spectral decomposition (Eq. 5) that W depends on U only
through the principal stretches l1 ; l2 , and l3 and is symmetric in these
stretches.
For an isotropic elastic material, r is coaxial with V, and we may write
r ¼ a0 I þ a1 B þ a2 B2
ð18Þ
where a0 ; a1 , and a2 are scalar invariants of B (and hence of V) given by
@W
1=2 @W
1=2 @W
1=2 @W
a1 ¼ 2I3
þ I1
ð19Þ
a0 ¼ 2I3
a2 ¼ 2I3
@I3
@I1
@I2
@I2
and W is now regarded as a function of I1 ; I2 , and I3 , the principal invariants
of B defined by
I1 ¼ trðBÞ ¼ l21 þ l22 þ l23 ;
ð20Þ
I2 ¼ 12 ½I21 tr ðB2 Þ
¼ l22 l23 þ l23 l21 þ l21 l22
ð21Þ
I3 ¼ det B ¼ l21 l22 l23
ð22Þ
Another consequence of isotropy is that S and r have the decompositions
S¼
3
X
ti uðiÞ vðiÞ
i¼1
r¼
3
X
si vðiÞ vðiÞ
ð23Þ
i¼1
where si ; i 2 f1; 2; 3g are the principal Cauchy stresses and ti the principal
Biot stresses, connected by
ti ¼
@W
¼ Jl1
i si
@li
ð24Þ
Let the unit vector M be a preferred direction in the reference configuration
of the material, i.e., a direction for which the material response is indifferent
to arbitrary rotations about the direction and to replacement of M by M.
Such a material can be characterized by a strain energy which depends on F
and the tensor M M [2, 4, 5] Thus, we write WðF; M MÞ. The required
symmetry (transverse isotropy) reduces W to dependence on the five invariants
I1 ; I2 ; I3 ; I4 ¼ M ðCMÞ
I5 ¼ M ðC2 MÞ
ð25Þ
where I1 ; I2 ; and I3 are defined in Eqs. (20)–(22). The resulting nominal
stress tensor is given by
S ¼ 2W1 FT þ 2W2 ðI1 I CÞFT þ 2I3 W3 F1 þ 2W4 M FM
þ 2W5 ðM FCM þ CM FMÞ
where Wi ¼ @[email protected] ; i ¼ 1; . . . ; 5.
ð26Þ
80
Ogden
When there are two families of fibers corresponding to two preferred
directions in the reference configuration, M and M0 say, then, in addition to
Eq. 25, the strain energy depends on the invariants
I6 ¼ M0 ðCM0 Þ
I7 ¼ M0 ðC2 M0 Þ
I8 ¼ M ðCM0 Þ
ð27Þ
0
and also on M M (which does not depend on the deformation); see Spencer
[4, 5] for details. The nominal stress tensor can be calculated in a similar
way to Eq. 26.
2.2.6 CONSTRAINED MATERIALS
An internal constraint, given in the form CðFÞ ¼ 0, must be satisfied
for all possible deformation gradients F, where C is a scalar function. Two
commonly used constraints are incompressibility and inextensibility, for
which, respectively,
CðFÞ ¼ detF 1
CðFÞ ¼ M ðFT FMÞ 1
ð28Þ
where the unit vector M is the direction of inextensibility in Br . Since any
constraint is unaffected by a superimposed rigid deformation, C must be an
objective scalar function, so that CðQFÞ ¼ CðFÞ for all rotations Q.
Any stress normal to the hypersurface CðFÞ ¼ 0 in the (nine-dimensional)
space of deformation gradients does no work in any (virtual) incremental
deformation compatible with the constraint. The stress is therefore
determined by the constitutive law (11)1 only to within an additive
contribution parallel to the normal. Thus, for a constrained material, the
stress-deformation relation (11)1 is replaced by
S ¼ HðFÞ þ q
@C @W
@C
¼
þq
@F
@F
@F
ð29Þ
where q is an arbitrary (Lagrange) multiplier. The term in q is referred to as
the constraint stress since it arises from the constraint and is not otherwise
derivable from the material properties.
For incompressibility and inextensibility we have
@W
@W
ð30Þ
S¼
þ qF1
þ 2qM FM
S¼
@F
@F
respectively. For an incompressible material the Biot and Cauchy stresses are
given by
@W
ð31Þ
pU1
T¼
det U ¼ 1
@U
2.2 Background on Nonlinear Elasticity
81
and
@W
ð32Þ
pI
det F ¼ 1
@F
where q has been replaced by p, which is called an arbitrary hydrostatic
pressure. The term in a0 in Eq. 18 is absorbed into p, and I3 ¼ 1 in the
remaining terms in Eq. 18. For an incompressible isotropic material the
principal components of Eqs. 31 and 32 yield
@W
@W
ð33Þ
ti ¼
pl1
s i ¼ li
p
i
@li
@li
r¼F
respectively, subject to l1 l2 l3 ¼ 1.
For an incompressible transversely isotropic material with preferred
direction M, the dependence on I3 is omitted and the Cauchy stress tensor
is given by
r ¼ pI þ 2W1 B þ 2W2 ðI1 B B2 Þ þ 2W4FM FM
þ 2W5 ðFM BFM þ BFM FMÞ
ð34Þ
For a material with two preferred directions, M and M0 , the Cauchy stress
tensor for an incompressible material is
r ¼ pI þ 2W1 B þ 2W2 ðI1 B B2 Þ þ 2W4 FM FM
þ 2W5 ðFM BFM þ BFM FMÞ þ 2W6 FM0 FM0
þ 2W7 ðFM0 BFM0 þ BFM0 FM0 Þ
þ W8 ðFM FM0 þ FM0 FMÞ
ð35Þ
where the notation Wi ¼ @[email protected] now applies for i ¼ 1; 2; 4; . . . ; 8.
2.2.7 BOUNDARY-VALUE PROBLEMS
The equilibrium equation (second part of Eq. 9), the stress-deformation
relation (Eq. 11), and the deformation gradient (Eq. 2) coupled with Eq. 1 are
combined to give
@W
Div
F ¼ Grad x
x ¼ vðXÞ
X 2 Br
ð36Þ
þ rr b ¼ 0
@F
Typical boundary conditions in nonlinear elasticity are
x ¼ nðXÞ on @Bxr
ð37Þ
ST N ¼ sðF; XÞ on @Btr
ð38Þ
where n and s are specified functions, N is the unit outward normal to @Br ,
82
Ogden
and @Bxr and @Btr are complementary parts of @B. In general, s may depend
on the deformation through F. For a dead-load traction s is independent of F.
For a hydrostatic pressure boundary condition, Eq. 38 has the form
s ¼ JPFT N
on @Btr
ð39Þ
Equations 36–38 constitute the basic boundary-value problem in nonlinear elasticity.
In components, the equilibrium equation in Eq. 36 is written
Aaibj
@ 2 xj
þ rr bi ¼ 0
@Xa @Xb
ð40Þ
for i 2 f1; 2; 3g, where the coefficients Aaibj are defined by
Aaibj ¼ Abjai ¼
@2W
@Fia @Fjb
ð41Þ
When coupled with suitable boundary conditions, Eq. 41 forms a system of
quasi-linear partial differential equations for xi ¼ wi ðXa Þ. The coefficients
Aaibj are, in general, nonlinear functions of the components of the
deformation gradient.
For incompressible materials the corresponding equations are obtained by
substituting the first part of Eq. 30 into the second part of Eq. 9 to give
Aaibj
@ 2 xj
@p
þ rr b i ¼ 0
@Xa @Xb @xi
detð@xi @Xa Þ ¼ 1
ð42Þ
where the coefficients are again given by Eq. 41.
In order to solve a boundary-value problem, a specific form of W needs to
be given. The form of W chosen will depend on the particular material
considered and on mathematical requirements relating to the properties of
the equations, an example of which is the strong ellipticity condition.
Equations 40 are said to be strongly elliptic if the inequality
Aaibj mi mj Na Nb > 0
ð43Þ
holds for all nonzero vectors m and N. Note that Eq. 43 is independent
of any boundary conditions. For an incompressible material, the
strong ellipticity condition associated with Eq. 42 again has the form of
Eq. 43, but the incompressibility constraint now imposes the restriction
m ðFT NÞ ¼ 0 on m and N.
REFERENCES
1. Beatty, M. F. (1987). Topics in finite elasticity: Hyperelasticity of rubber, elastomers and
biological tissues } with examples. Appl. Mech. Rev. 40; 1699–1734.
2.2 Background on Nonlinear Elasticity
83
2. Holzapfel, G. A. (2000). Nonlinear Solid Mechanics. Chichester: Wiley.
3. Ogden, R. W. (1997). Non-linear Elastic Deformations. New York: Dover Publications.
4. Spencer, A. J. M. (1972). Deformations of Fibre-Reinforced Materials. Oxford: Oxford University
Press.
5. Spencer, A. J. M. (1984). Constitutive theory for strongly anisotropic solids. In Continuum
Theory of the Mechanics of Fibre-Reinforced Composites, CISM Courses and Lectures No. 282,
pp. 1–32, Spencer, A. J. M., ed., Wien: Springer-Verlag.
CHAPTER
2.3
Elasticity of Porous
Materials
N. D. CRISTESCU
231 Aerospace Building, University of Florida, Gainesville, Florida
Contents
2.3.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.3.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.3.3 Identification of the Parameters . . . . . . . . . . . . . . 85
2.3.4 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.3.1 VALIDITY
The methods used to determine the elasticity of porous materials and/or
particulate materials as geomaterials or powderlike materials are distinct from
those used with, say, metals. The reason is that such materials possess pores
and=or microcracks. For various stress states these may either open or closed,
thus influencing the values of the elastic parameters. Also, the stress-strain
curves for such materials are strongly loading-rate-dependent, starting from
the smallest applied stresses, and creep (generally any time-dependent
phenomena) is exhibited from the smallest applied stresses (see Fig. 2.3.1 for
schist, showing three uniaxial stress-strain curves for three loading rates and a
creep curve [1]). Thus information concerning the magnitude of the elastic
parameters cannot be obtained:
from the initial slope of the stress-strain curves, since these are loadingrate-dependent;
by the often used ‘‘chord’’ procedure, obviously;
from the unloading slopes, since significant hysteresis loops are
generally present.
84
Handbook of Materials Behavior Models
Copyright # 2001 by Academic Press. All rights of reproduction in any form reserved.
2.3 Elasticity of Porous Materials
85
FIGURE 2.3.1 Uniaxial stress-strain curves for schist for various loading rates, showing time
influence on the entire stress-strain curves and failure (stars mark the failure points).
2.3.2 FORMULATION
The elasticity of such materials can be expressed as ‘‘instantaneous
response’’ by
1
1 1 ’
T’
ð1Þ
ðtrTÞ1
þ
D¼
3K 2G 3
2G
%
where G and K are the elastic parameters that are not constant, D is the strain
rate tensor, T is the stress tensor, tr( ) is the trace operator, and 1 is the unit
tensor. Besides the elastic properties described by Eq. 1, some other
mechanical properties can be described by additional terms to be added to
Eq. 1. For isotropic geomaterials the elastic parameters are expected to
depend on stress invariants and, perhaps, on some damage parameters, since
during loading some pores and microcracks may close or open, thus
influencing the elastic parameters.
2.3.3 IDENTIFICATION OF THE PARAMETERS
The elastic parameters can be determined experimentally by two procedures.
With the dynamic procedure, one is determining the travel time of the two
86
Cristescu
elastic (seismic) extended longitudinal and transverse waves, which
are traveling in the body. If both these waves are recorded, then the
instantaneous response is of the form of Eq. 1. The elastic parameters are
obtained from
4 2
2
K ¼ r vp vS
ð2Þ
G ¼ rv2S
3
where vS is the velocity of propagation of the shearing waves, vp the velocity of
the longitudinal waves, and r the density.
The static procedure takes into account that the constitutive equations for
geomaterials are strongly time-dependent. Thus, in triaxial tests performed
under constant confining pressure s, after loading up to a desired stress state t
(octahedral shearing stress), one is keeping the stress constant for a certain
time period tc [2, 3]. During this time period the rock is creeping. When the
strain rates recorded during creep become small enough, one is performing an
unloading–reloading cycle (see Fig. 2.3.2). From the slopes
1
1 1
1
1 1
ð3Þ
þ
þ
3G 9K
6G 9K
of these unloading–reloading curves one can determine the elastic parameters.
For each geomaterial, if the time tc is chosen so that the subsequent unloading
is performed in a comparatively much shorter time interval, no significant
interference between creep and unloading phenomena will take place. An
example for schist is shown in Figure 2.3.3, obtained in a triaxial test with five
unloading–reloading cycles.
FIGURE 2.3.2 Static procedures to determine the elastic parameters from partial unloading
processes preceded by short-term creep.
87
2.3 Elasticity of Porous Materials
FIGURE 2.3.3 Stress-strain curves obtained in triaxial tests on shale; the unloadings follow a
period of creep of several minutes.
If only a partial unloading is performed (one third or even one quarter of
the total stress, and sometimes even less), the unloading and reloading follow
quite closely straight lines that practically coincide. If a hysteresis loop is still
recorded, it means that the time tc was chosen too short. The reason for
performing only a partial unloading is that the specimen is quite ‘‘thick’’ and
as such the stress state in the specimen is not really uniaxial. During complete
unloading, additional phenomena due to the ‘‘thickness’’ of the specimen
will be involved, including, e.g., kinematic hardening in the opposite
direction, etc.
Similar results can be obtained if, instead of keeping the stress constant,
one is keeping the axial strain constant for some time period during which the
axial stress is relaxing. Afterwards, when the stress rate becomes relatively
small, an unloading–reloading is applied to determine of the elastic
parameters. This procedure is easy to apply mainly for particulate materials
(sand, soils, etc.) when standard (Karman) three-axial testing devices are used
and the elastic parameters follow from
K¼
1
Dt
3 De1 þ 2De2
G¼
1
Dt
2 De1 De2
ð4Þ
where D is the variation of stress and elastic strains during the unloading–
reloading cycle. The same method is used to determine the bulk modulus K in
hydrostatic tests when the formula to be used is
K¼
Ds
Dev
ð5Þ
with s the mean stress and ev the volumetric strain.
Generally, K is increasing with s and reaching an asymptotic constant value
when s is increasing very much and all pores and microcracks are closed
88
Cristescu
under this high pressure. The variation of the elastic parameters with t is
more involved: when t increases but is still under the compressibility–
dilatancy boundary, the elastic parameters are increasing. For higher values,
above this boundary, the elastic parameters are decreasing. Thus their
variation is related to the variation of irreversible volumetric strain, which, in
turn, is describing the evolution of the pores and microcracks existing in the
geomaterial. That is why the compressibility–dilatancy boundary plays the
role of reference configuration for the values of the elastic parameters so long
as the loading path (increasing s and=or t) is in the compressibility domain,
the elastic parameters are increasing, whereas if the loading path is in the
dilatancy domain (increasing under constant s), the elastic parameters are
decreasing. If stress is kept constant and strain is varying by creep, in the
compressibility domain volumetric creep produces a closing of pores and
microcracks and thus the elastic parameters increase, and vice versa in the
dilatancy domain. Thus, for each value of s the maximum values of the elastic
parameters are reached on the compressibility–dilatancy boundary.
2.3.4 EXAMPLES
As an example, for rock salt in uniaxial stress tests, the variation of the elastic
moduli G and K with the axial stress s1 is shown in Figure 2.3.4 [4]. The
variation of G and K is very similar to that of the irreversible volumetric
FIGURE 2.3.4 Variation of the elastic parameters K and G and of irreversible volumetric strain
in monotonic uniaxial tests.
2.3 Elasticity of Porous Materials
89
FIGURE 2.3.5 Variation in time of the elastic parameters and of irreversible volumetric strain in
uniaxial creep tests.
strain eIV . If stress is increased in steps, and if after each increase the stress in
kept constant for two days, the elastic parameters are varying during
volumetric creep, as shown in Figure 2.3.5. Here D is the ratio of the applied
stress and the strength in uniaxial compression sc ¼ 17:88 MPa. Again, a
similarity with the variation of eIV is quite evident. Figure 2.3.6 shows for a
different kind of rock salt the variation of the elastic velocities vP and vS in
true triaxial tests under confining pressure pc ¼ 5 MPa (data by Popp,
Schultze, and Kern [5]). Again, these velocities increase in the compressibility
domain, reach their maxima on the compressibility–dilatancy boundary, and
then decrease in the dilatancy domain.
For shale, and the conventional (Karman) triaxial tests shown in
Figure 2.3.3, the values of E and G for the five unloading–reloading cycles
shown are: E ¼ 9:9, 24.7, 29.0, 26.3, and 22.3 GPa, respectively, while
G ¼ 4:4, 10.7, 12.1, 10.4, and 8.5 GPa.
For granite, the variation of K with s is given as [2]
8
>
< K0 K1 1 s ; if s s0
s0
KðsÞ :¼
ð6Þ
>
: K ; if s s
0
0
with K0 ¼ 59 GPa, K1 ¼ 48 GPa, and s0 ¼ 0:344 GPa, the limit pressure when
all pores are expected to be closed.
90
Cristescu
FIGURE 2.3.6 The maximum of vs takes place at the compressibility–dilatancy boundary
(figures and hachured strip); changes of vp and vs as a function of strain (’e ¼ 105 s1,
pc ¼ 5 Mpa, T ¼ 308 C), showing that the maxima are at the onset of dilatancy (after
Reference [4]).
The same formula for alumina powder is
s
KðsÞ :¼ K pa exp b
pa
1
ð7Þ
with K1 ¼ 1 107 kPa the constant value toward which the bulk modulus
tends at high pressures, a ¼ 107 , b ¼ 1:2 104 , and pa ¼ 1 kPa. Also for
alumina powder we have
EðsÞ :¼ E1 pa b expðdsÞ
ð8Þ
with E1 ¼ 7 105 kPa, b ¼ 6:95 105 , and d ¼ 0:002.
For the shale shown in Figure 2.3.3, the variation of K with s for
0 s 45 MPa is
KðsÞ :¼ 0:78s2 þ 65:32s 369
ð9Þ
REFERENCES
1. Cristescu, N. (1986). Damage and failure of viscoplastic rock-like materials. Int. J. Plasticity
2 (2): 189–204.
2. Cristescu, N. (1989). Rock Rheology, Kluver Academic Publishing.
3. Cristescu, N. D., and Hunsche, U. (1998). Time Effects in Rock Mechanics, Wiley.
4. Ani, M., and Cristescu N. D. (2000). The effect of volumetric strain on elastic parameters for
rock salt. Mechanics of Cohesive-Frictional Materials 5 (2): 113–124.
5. Popp, T., Schultze, O., and Kern, H. (
). Permeation and development of dilatancy and
permeability in rock salt, in The Mechanical Behavior of Salt (5th Conference on Mechanical
Behavior of Salt), Cristescu, N. D., and Hardy, Jr., H. Reginald, eds., Trans Tech Publ.,
Clausthal-Zellerfeld.
CHAPTER
2.4
Elastomer Models
R. W. OGDEN
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK
Contents
2.4.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.4.3 Description of the Model. . . . . . . . . . . . . . . . . . . . . 93
2.4.4 Identification of Parameters . . . . . . . . . . . . . . . . . . 93
2.4.5 How to Use It . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2.4.6 Table of Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . 94
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2.4.1 VALIDITY
Many rubberlike solids can be treated as isotropic and incompressible elastic
materials to a high degree of approximation. Here describe the mechanical
properties of such solids through the use of an isotropic elastic strain-energy
function in the context of finite deformations. For general background on
finite elasticity, we refer to Ogden [5].
2.4.2 BACKGROUND
Locally, the finite deformation of a material can be described in terms of the
three principal stretches, denoted by l1 ; l2 ; and l3 . For an incompressible
material these satisfy the constraint
l1 l2 l3 ¼ 1
ð1Þ
The material is isotropic relative to an unstressed undeformed (natural)
configuration, and its elastic properties are characterized in terms of a
Handbook of Materials Behavior Models
Copyright # 2001 by Academic Press. All rights of reproduction in any form reserved.
91
92
Ogden
strain-energy function Wðl1 ; l2 ; l3 Þ per unit volume, where W depends
symmetrically on the stretches subject to Eq. 1.
The principal Cauchy stresses associated with this deformation are
given by
si ¼ li
@W
p;
@li
ð2Þ
i 2 f1; 2; 3g
where p is an arbitrary hydrostatic pressure (Lagrange multiplier). By
regarding two of the stretches as independent and treating the strain energy as
1
# 1 ; l2 Þ ¼ Wðl1 ; l2 ; l1
a function of these through the definition Wðl
1 l2 Þ,
we obtain
s1 s3 ¼ l1
#
@W
@l1
s2 s3 ¼ l2
#
@W
@l2
ð3Þ
For consistency with the classical theory, we must have
#
Wð1;
1Þ ¼ 0;
#
#
#
@2W
@W
@2W
ð1; 1Þ ¼ 2m;
ð1; 1Þ ¼ 0;
ð1; 1Þ ¼ 4m;
2
@la
@l1 @l2
@la
ð4Þ
a 2 f1; 2g
where m is the shear modulus in the natural configuration. The equations in
Eq. 3 are unaffected by superposition of an arbitrary hydrostatic stress. Thus,
# and hence those of W, it suffices to set
in determining the characteristics of W,
s3 ¼ 0 in Eq. 3, so that
s1 ¼ l1
#
@W
@l1
s2 ¼ l2
#
@W
@l2
ð5Þ
Biaxial experiments in which l1 ; l2 and s1 ; s2 are measured then provide
# Biaxial deformation of a thin sheet where the
data for the determination of W.
deformation corresponds effectively to a state of plane stress, or the combined
extension and inflation of a thin-walled (membranelike) tube with closed
ends provide suitable tests. In the latter case the governing equations are
written
1
P * ¼ l1
1 l2
#
@W
@l2
F* ¼
# 1
#
@W
@W
l2 l1
1
@l1 2
@l2
ð6Þ
where P * ¼ PR=H, P is the inflating pressure, H the undeformed membrane
thickness, and R the corresponding radius of the tube, while F * ¼ F=2pRH,
with F the axial force on the membrane (note that the pressure contributes to
the total load on the ends of the tube). Here l1 is the axial stretch and l2 the
azimuthal stretch in the membrane.
93
2.4 Elastomer Models
2.4.3 DESCRIPTION OF THE MODEL
A specific model which fits very well the available data on various rubbers is
that defined by
N
X
ð7Þ
mn ðla1n þ la2n þ la3n 3Þ=an
W¼
n¼1
where mn and an are material constants and N is a positive integer, which for
many practical purposes may be taken as 2 or 3 [3]. For consistency with
Eq. 4 we must have
N
X
ð8Þ
mn an ¼ 2m
n¼1
and in practice it is usual to take mn an > 0 for each n ¼ 1; . . . ; N.
In respect of Eq. 7, the equations in Eq. 3 become
N
N
X
X
s1 s3 ¼
mn ðla1n la3n Þ
s2 s3 ¼
mn ðla2n la3n Þ
n¼1
ð9Þ
n¼1
2.4.4 IDENTIFICATION OF PARAMETERS
Biaxial experiments with s3 ¼ 0 indicate that the shapes of the
curves of s1 s2 plotted against l1 are essentially independent of l2 for
many rubbers. Thus the shape may be determined by the pure shear test with
l2 ¼ 1, so that
N
N
X
X
s1 s2 ¼
mn ðla1n 1Þ
s2 ¼
mn ðla3n 1Þ
ð10Þ
n¼1
n¼1
for l1 1; l3 1. The shift factor to be added to the first equation in Eq. 10
when l2 differs from 1 is
N
X
ð11Þ
mn ð1 la2n Þ
n¼1
Information on both the shape and shift obtained from experiments at fixed
l2 then suffice to determine the material parameters, as described in detail in
References [3] or [4].
Data from the extension and inflation of a tube can be studied on this basis
by considering the combination of equations in Eq. 6 in the form
s1 s 2 ¼ l 1
#
#
@W
@W
1
l2
¼ l1 F * l22 l1 P *
@l1
@l2
2
ð12Þ
94
Ogden
2.4.5 HOW TO USE IT
The strain-energy function is incorporated in many commercial Finite
Element (FE) software packages, such as ABAQUS and MARC, and can be
used in terms of principal stretches and principal stresses in the FE solution of
boundary-value problems.
2.4.6 TABLE OF PARAMETERS
Values of the parameters corresponding to a three-term form of Eq. 7 are now
given in respect of two different but representative vulcanized natural
rubbers. The first is the material used by Jones and Treloar [2]:
a1 ¼ 1:3; a2 ¼ 4:0; a3 ¼ 2:0;
m1 ¼ 0:69; m2 ¼ 0:01; m3 ¼ 0:0122 Nmm2
The second is the material used by James et al. [1], the material constants
having been obtained by Treloar and Riding [6]:
a1 ¼ 0:707; a2 ¼ 2:9; a3 ¼ 2:62;
m1 ¼ 0:941; m2 ¼ 0:093; m3 ¼ 0:0029 Nmm2
For detailed descriptions of the rubbers concerned, reference should be made
to these papers.
REFERENCES
1. James, A. G., Green, A., and Simpson, G. M. (1975). Strain energy functions of rubber.
I. Characterization of gum vulcanizates. J. Appl. Polym. Sci. 19: 2033–2058.
2. Jones, D. F., and Treloar, L. R. G. (1975). The properties of rubber in pure homogeneous strain.
J. Phys. D: Appl. Phys. 8: 1285–1304.
3. Ogden, R. W. (1982). Elastic deformations of rubberlike solids, in Mechanics of Solids
(Rodney Hill 60th Anniversary Volume) pp. 499–537, Hopkins, H. G., and Sevell, M. J., eds.,
Pergamon Press.
4. Ogden, R. W. (1986). Recent advances in the phenomenological theory of rubber elasticity.
Rubber Chem. Technol. 59: 361–383.
5. Ogden, R. W. (1997). Non-Linear Elastic Deformations, Dover Publications.
6. Treloar, L. R. G., and Riding, G. (1979). A non-Gaussian theory for rubber in biaxial strain.
I. Mechanical properties. Proc. R. Soc. Lond. A369: 261–280.
CHAPTER
2.5
Background on
Viscoelasticity
KOZO IKEGAMI
Tokyo Denki University, Kanda-Nishikicho 2-2, Chiyodaku, Tokyo 101-8457, Japan
Contents
2.5.1
2.5.2
2.5.3
2.5.4
2.5.5
2.5.6
Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Mechanical Models . . . . . . . . . . . . . . . . . . . . . . . . 95
Static Viscoelastic Deformation. . . . . . . . . . . . . . . 98
Dynamic Viscoelastic Deformation . . . . . . . . . 100
Hereditary Integral . . . . . . . . . . . . . . . . . . . . . . . . 102
Viscoelastic Constitutive Equation by the
Laplace Transformation . . . . . . . . . . . . . . . . . . . . 103
2.5.7 Correspondence Principle . . . . . . . . . . . . . . . . . . 104
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.5.1 VALIDITY
Fundamental deformation of materials is classified into three types: elastic,
plastic, and viscous deformations. Polymetric material shows time-dependent
properties even at room temperature. Deformation of metallic materials is also
time-dependent at high temperature. The theory of viscoelasticity can be
applied to represent elastic and viscous deformations exhibiting timedependent properties. This paper offers an outline of the linear theory
of viscoelasticity.
2.5.2 MECHANICAL MODELS
Spring and dashpot elements as shown in Figure 2.5.1 are used to represent
elastic and viscous deformation, respectively, within the framework of the
Handbook of Materials Behavior Models
Copyright # 2001 by Academic Press. All rights of reproduction in any form reserved.
95
96
Ikegami
FIGURE 2.5.1 Mechanical model of viscoelasticity.
linear theory of viscoelasticity. The constitutive equations between stress s
and stress e of the spring and dashpot are, respectively, as follows:
de
ð1Þ
s ¼ ke
s¼Z
dt
where the notations k and Z are elastic and viscous constants, respectively.
Stress of spring elements is linearly related with strain. Stress of dashpot
elements is related with strain differentiated by time t, and the constitutive
relation is time-dependent.
Linear viscoelastic deformation is represented by the constitutive equations
combining spring and dashpot elements. For example, the constitutive
equations of series model of spring and dashpot shown in Figure 2.5.2 is
as follows:
Z ds
de
ð2Þ
sþ
¼Z
k dt
dt
This is called the Maxwell model. The constitutive equation of the parallel
model of spring and dashpot elements shown in Figure 2.5.3 is as follows:
s ¼ ke þ Z
This is called the Voigt or Kelvin model.
de
dt
ð3Þ
97
2.5 Background on Viscoelasticity
FIGURE 2.5.2 Maxwell model.
There are many variations of constitutive equations giving linear
viscoelastic deformation by using different numbers of spring and dashpot
elements. Their constitutive equations are generally represented by the
following ordinary differential equation:
p0 s þ p1
ds
d2 s
dn s
þ p2 2 þ . . . þ pn n
dt
dt
dt
¼ q0 e þ q1
de
d2 e
dn e
þ q2 2 þ . . . þ qn n
dt
dt
dt
ð4Þ
The coefficients p and q of Eq. 4 give the characteristic properties of linear
viscoelastic deformation and take different values according to the number of
spring and dashpot elements of the viscoelastic mechanical model.
98
Ikegami
FIGURE 2.5.3 Voigt (Kelvin) model.
2.5.3 STATIC VISCOELASTIC DEFORMATION
There are two functions representing static viscoelastic deformation; one is
creep compliance, and another is the relaxation modulus. Creep compliance
is defined by strain variations under constant unit stress. This is obtained by
solving Eqs. 2 or 3 for step input of unit stress. For the Maxwell model and
the Voigt model, their creep compliances are represented, respectively, by
the following expressions. For the Maxwell model, the creep compliance is
t 1 1 t
þ1
e þ ¼
ð5Þ
Z k k t
where tM ¼ Z=k, and this is denoted as relaxation time. For the Voigt model,
the creep compliance is
1
kt
1
t
e ¼ 1 exp ¼ 1 exp ð6Þ
k
Z
k
tk
where tK ¼ Z=k, and this is denoted as retardation time.
Creep deformations of the Maxwell and Voigt models are illustrated in
Figures 2.5.4 and 2.5.5, respectively. Creep strain of the Maxwell model
2.5 Background on Viscoelasticity
FIGURE 2.5.4 Creep compliance of the Maxwell model.
FIGURE 2.5.5 Creep compliance of the Voigt model.
99
100
Ikegami
increases linearly with respect to time duration. The Voigt model exhibits
saturated creep strain for a long time.
The relaxation modulus is defined by stress variations under constant unit
strain. This is obtained by solving Eqs. 2 or 3 for step input of unit strain. For
the Maxwell and Voigt models, their relaxation moduli are represented by the
following expressions, respectively. For the Maxwell model,
kt
t
s ¼ k exp ¼ k exp ð7Þ
Z
tM
For the Voigt model,
s¼k
ð8Þ
Relaxation behaviors of the Maxwell and Voigt models are illustrated in
Figures 2.5.6 and 2.5.7, respectively. Applied stress is relaxed by Maxwell
model, but stress relaxation dose not appear in Voigt model.
2.5.4 DYNAMIC VISCOELASTIC DEFORMATION
The characteristic properties of dynamic viscoelastic deformation are
represented by the dynamic response for cyclically changing stress or strain.
FIGURE 2.5.6 Relaxation modulus of the Maxwell model.
2.5 Background on Viscoelasticity
101
FIGURE 2.5.7 Relaxation modulus of the Voigt model.
The viscoelastic effect causes delayed phase phenomena between input and
output responses. Viscoelastic responses for changing stress or strain are
defined by complex compliance or modulus, respectively. The dynamic
viscoelastic responses are represented by a complex function due to the phase
difference between input and output.
Complex compliance J of the Maxwell model is obtained by calculating
changing strain for cyclically changing stress with unit amplitude. Substituting changing complex stress s ¼ expðiotÞ, where i is an imaginary unit and
o
is the frequency of changing stress, into Eq. 2, complex compliance J is
obtained as follows:
J ¼
1
1
1
1
i
¼ i
¼ J0 iJ00
k
oZ k
kotM
ð9Þ
where the real part J0 ¼ 1=k is denoted as storage compliance, and the
imaginary part J00 ¼ 1=kotM is denoted as loss compliance.
The complex modulus Y of the Maxwell model is similarly obtained by
calculating the complex changing strain for the complex changing strain
102
Ikegami
e ¼ expðiotÞ as follows:
Y ¼ k
ðotM Þ2
1 þ ðotM Þ2
þ ik
otM
1 þ ðotM Þ2
¼ Y 0 þ iY 00
ð10Þ
where Y 0 ¼ kððotM Þ2 =ð1 þ ðotM Þ2 ÞÞ and Y 00 ¼ kðotM =ð1 þ ðotM Þ2 ÞÞ. The
notations Y 0 and Y 00 are denoted as dynamic modulus and dynamic loss,
respectively. The phase difference d between input strain and output stress is
given by
tan d ¼
Y 00
1
¼
Y 0 otM
ð11Þ
This is called mechanical loss.
Similarly, the complex compliance and the modulus of the Voigt model are
able to be obtained. The complex compliance is
"
#
"
#
1
1
1
otK
J ¼
i
¼ J0 iJ00
ð12Þ
k 1 þ ðotK Þ2
k 1 þ ðotK Þ2
"
#
"
#
1
1
1
otK
00
where J ¼
and J ¼
k 1 þ ðotK Þ2
k 1 þ ðotK Þ2
0
The complex modulus is
Y ¼ k þ iotK ¼ Y 0 þ iY 00
ð13Þ
where Y 0 ¼ k and Y 00 ¼ kotK .
2.5.5 HEREDITARY INTEGRAL
The hereditary integral offers a method of calculating strain or stress variation
for arbitrary input of stress or strain. The method of calculating strain
for stress history is explained by using creep compliance as illustrated in
Figure 2.5.8. An arbitrary stress history is divided into incremental constant
stress history ds0 Strain variation induced by each incremental stress history
is obtained by creep compliance with the constant stress values. In
Figure 2.5.8 the strain induced by stress history for t0 5t is represented by
the following integral:
Z t
ds0
ð14Þ
Jðt t0 Þ 0 dt0
eðtÞ ¼ s0 JðtÞ þ
dt
0
103
2.5 Background on Viscoelasticity
FIGURE 2.5.8 Hereditary integral.
This equation is transformed by partially integrating as follows:
Z t
dJðt t0 Þ 0
dt
sðt0 Þ
eðtÞ ¼ sðtÞJð0Þ þ
dðt t0 Þ
0
Similarly, stress variation for arbitrary strain history becomes
Z t
ds0
Yðt t0 Þ 0 dt0
sðtÞ ¼ e0 YðtÞ þ
dt
0
Partial integration of Eq. & gives the following equation:
Z t
dYðt t0 Þ 0
dt
sðtÞ ¼ eðtÞYð0Þ þ
sðt0 Þ
dðt t0 Þ
0
ð15Þ
ð16Þ
ð17Þ
Integrals in Eqs. 14 to 17 are called hereditary integrals.
2.5.6 VISCOELASTIC CONSTITUTIVE EQUATION
BY THE LAPLACE TRANSFORMATION
The constitutive equation of viscoelastic deformation is the ordinary
differential equation as given by Eq. 4. That is,
n
X
k¼0
pk
m
dk s X
dk e
¼
q
k
dtk
dtk
k¼0
ð18Þ
104
Ikegami
This equation is written by using differential operators P and Q,
Ps ¼ Qe
where P ¼
n
P
k
pk
k¼0
m
P
ð19Þ
k
d
d
and Q ¼
qk k .
dtk
dt
k¼0
Equation (1?) is represented by the Laplace transformation as follows.
n
n
X
X
pk sk s% ¼
qk sk e%
ð20Þ
k¼0
k¼0
where s% and e% are transformed stress and strain, and s is the variable of
the Laplace transformation. Equation 20 is written by using the Laplace
% as follows:
transformed operators of time derivatives P% and Q
%
Q
ð21Þ
s% ¼ e%
P%
n
m
P
P
% ¼
pk sk and Q
q k sk .
where P% ¼
k¼0
k¼0
% P%
Comparing Eq. 21 with Hooke’s law in one dimension, the coefficient Q=
corresponds to Young’s modulus of linear elastic deformation. This fact
implies that linear viscoelastic deformation is transformed into elastic
deformation in the Laplace transformed state.
2.5.7 CORRESPONDENCE PRINCIPLE
In the previous section, viscoelastic deformation in the one-dimensional state
was able to be represented by elastic deformation through the Laplace
transformation. This can apply to three-dimensional viscoelastic deformation.
The constitutive relations of linear viscoelastic deformation are divided into
the relations between hydrostatic pressure and dilatation, and between
deviatoric stress and strain.
The relation between hydrostatic pressure and dilatation is represented by
m
n
X
dk s0ij X
dk eii
p0k k ¼
q00k k
ð22Þ
dt
dt
k¼0
k¼0
P00 sii ¼ Q00 eii
ð23Þ
n
P
dk
dk
and Q00 ¼
q00k k . In Eq. 22 hydrostatic pressure is (1/3)
k
dt
dt
k¼0
k¼0
sii and dilatation is eii .
where P00
m
P
p00k
105
2.5 Background on Viscoelasticity
The relation between deviatoric stress and strain is represented by
m
X
p0k
k¼0
dk s0ij
dtk
¼
n
X
q0k
k¼0
P0 s0ij ¼ Q0 e0ij
dk e0ij
dtk
ð24Þ
ð25Þ
n
P
dk
dk
and Q0 ¼
q0k k . In Eq. 24 deviatoric stress and strain
k
dt
dt
k0
k¼0
are s0ij and e0ij , respectively.
The Laplace transformations of Eqs. 22 and 24 are written, respectively, as
follows:
where P0 ¼
m
P
p0k
% 00 e%ii
P% 00 s% ii ¼ Q
ð26Þ
% 00 ¼ Q
% sðsÞ, and
where P% 00 ¼ P% 00 ðsÞ and Q
00
% 0 e% 0ij
P% 0 s% 0ij ¼ Q
ð27Þ
%0 ¼ Q
% 0 ðsÞ.
where P% 0 ¼ P% 0 ðsÞ and Q
The linear elastic constitutive relations between hydrostatic pressure and
dilatation and between deviatoric stress and strain are represented as follows:
sii ¼ 3Keii
ð28Þ
s0ij ¼ 2Ge0ii
ð29Þ
Comparing Eq. 17 with Eq. 19, and Eq. 18 with Eq. 20, the transformed
viscoelastic operators correspond to elastic constants as follows:
3K ¼
% 00
Q
P% 00
ð30Þ
2G ¼
%0
Q
P% 0
ð31Þ
where K and G are volumetric coefficient and shear modulus, respectively.
For isotropic elastic deformation, volumetric coefficient K and shear
modulus G are connected with Young’s modulus E and Poisson’s ratio n as
follows:
E
2ð1 þ nÞ
ð32Þ
E
3ð1 2nÞ
ð33Þ
G¼
K¼
106
Ikegami
Using Eqs. 30–33, Young’s modulus E and Poisson’s ratio are connected with
the Laplace transformed coefficient of linear viscoelastic deformation
as follows:
% 00
% 0Q
3Q
E ¼ 0 00
ð34Þ
%
%0
2P% Q þ P% 00 Q
n¼
% 00 P% 00 Q
%0
P% 0 Q
0
00
00
% þ P% Q
%0
2P% Q
ð35Þ
Linear viscoelastic deformation corresponds to linear elastic deformation
through Eqs. 30–31 and Eqs. 34–35. This is called the correspondence
principle between linear viscoelastic deformation and linear elastic deformation. The linear viscoelastic problem is the transformed linear elastic problem
in the Laplace transformed state. Therefore, the linear viscoelastic problem is
able to be solved as a linear elastic problem in the Laplace transformed state,
and then the elastic constants of solved solutions are replaced with the
Laplace transformed operator of Eqs. 30–31 and Eqs. 34–35 by using
the correspondence principle. The solutions replaced the elastic constants
become the solution of the linear viscoelastic problem by inversing the
Laplace transformation.
REFERENCES
1.
2.
3.
4.
5.
6.
Bland, D. R. (1960). Theory of Linear Viscoelasticity, Pergamon Press.
Ferry, J. D. (1960). Viscoelastic Properties of Polymers, John Wiley & Sons.
Reiner, M. (1960). Deformation, Strain and Flow, H. K. Lewis & Co.
Flluege, W. (1967). Viscoelasticity, Blaisdell Publishing Company.
Christensen, R. M. (1971). Theory of Viscoelasticity: An Introduction, Academic Press.
Drozdov, A. D. (1998). Mechanics of Viscoelastic Solids, John Wiley & Sons.
CHAPTER
2.6
A Nonlinear Viscoelastic
Model Based on
Fluctuating Modes
RACHID RAHOUADJ
AND
CHRISTIAN CUNAT
LEMTA, UMR CNRS 7563, ENSEM INPL 2, avenue de la For#et-de-Haye, 54500 Vandoeuvre-l"esNancy, France
Contents
2.6.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
2.6.2 Background of the DNLR . . . . . . . . . . . . . . . 108
2.6.2.1 Thermodynamics of Irreversible
Processes and Constitutive Laws . . . 108
2.6.2.2 Kinetics and Complementary
Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
2.6.2.3 Constitutive Equations of
the DNLR . . . . . . . . . . . . . . . . . . . . . . . . 112
2.6.3 Description of the Model in the Case
of Mechanical Solicitations . . . . . . . . . . . . . . 113
2.6.4 Identification of the Parameters . . . . . . . . . 113
2.6.5 How to Use It . . . . . . . . . . . . . . . . . . . . . . . . . . 115
2.6.6 Table of Parameters. . . . . . . . . . . . . . . . . . . . . 115
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
2.6.1 VALIDITY
We will formulate a viscoelastic modeling for polymers in the temperature
range of glass transition. This physical modeling may be applied using integral
or differential forms. Its fundamental basis comes from a generalization of the
Gibbs relation, and leads to a formulation of constitutive laws involving
control and internal thermodynamic variables. The latter must traduce
Handbook of Materials Behavior Models
Copyright # 2001 by Academic Press. All rights of reproduction in any form reserved.
107
108
Rahouadj and Cunat
different microstructural rearrangements. In practice, both modal analysis
and fluctuation theory are well adapted to the study of the irreversible
transformations.
Such a general formulation also permits us to consider various
nonlinearities as functions of material specificities and applied perturbations.
To clarify the present modeling, called ‘‘the distribution of nonlinear
relaxations’’ (DNLR), we will consider the viscoelastic behavior in the simple
case of small applied perturbations near the thermodynamic equilibrium. In
addition, we will focus our attention upon the nonlinearities induced by
temperature and frequency perturbations.
2.6.2 BACKGROUND OF THE DNLR
2.6.2.1 THERMODYNAMICS OF IRREVERSIBLE
PROCESSES AND CONSTITUTIVE LAWS
As mentioned, the present irreversible thermodynamics are based on a
generalization of the fundamental Gibbs equation to systems evolving outside
equilibrium. Note that Coleman and Gurtin [1], have also applied this
postulate in the framework of rational thermodynamics. At first, a set of
internal variables (generalized vector denoted z) is introduced to describe the
microstructural state. The generalized Gibbs relation combines the two laws
of thermodynamics into a single one, i.e., the internal energy potential:
e ¼ eðs; e; n; . . . ; zÞ
ð1Þ
which depends on overall state variables, including the specific entropy, s.
Furthermore, with the positivity of the entropy production being always
respected, one obtains for open systems:
n
X
dDi s
T
¼ Tss ¼ Js : rT Jk : rmk þ A z’ 0
ð2Þ
dt
k¼1
where the nonequilibrium thermodynamic forces may be separated into two
groups: (i) the gradient ones, such as the gradient of temperature gradient
rT, and the gradient of generalized chemical potential rmk ; and (ii) The
generalized forces A, or affinities as defined by De Donder [2] for chemical
reactions, which characterize the nonequilibrium state of a uniform medium.
The vectors Js , Jk , and z’ correspond to the dual, fluxes, or ratetype variables.
To simplify the formulation of the constitutive laws, we will now consider
the behavior of a uniform representative volume element (RVE without any
2.6 A Nonlinear Viscoelastic Model Based on Fluctuating Modes
109
gradient), thus:
Tss ¼ A z’ 0
ð3Þ
The equilibrium or relaxed state (denoted by the index r) is currently
described by a suitable thermodynamic potential (cr ) obtained via the
Legendre transformation of Eq. 1 with respect to the control or state variable
(g). In this particular state, the set of internal variables is completely
governed by (g):
cr ¼ cr ðg; zr ðgÞÞ ¼ cr ðgÞ
ð4Þ
Our first hypothesis [3] states that it is always possible to define a
thermodynamic potential c only as a function of g and z, even for systems
outside equilibrium:
ð5Þ
c ¼ cðg; zÞ
Then, we assume that the constitutive equations may be obtained as functions
of the first partial derivatives of this potential with respect to the dual
variables, and depend consequently on both control and internal variables;
i.e., b ¼ bðg; zÞ and A ¼ Aðg; zÞ. In fact, this description is consistent with the
principle of equipresence, as postulated in rational thermodynamics. Therefore, the thermodynamic potential becomes in a differential form:
q
r
X
X
dck ¼
bm dgm Aj dzj
ð6Þ
m¼1
j¼1
Thus the time evolution of the global response, b, obeys a nonlinear
differential equation involving both the applied perturbation g and the
internal variable z (generalized vector):
ð7aÞ
b’ ¼ au : g’ þ b : z’
A’ ¼ t b : g’ g : z’
ð7bÞ
This differential system resumes in a general and condensed form the
announced constitutive relationships. The symmetrical matrix au ¼ @2 [email protected]@g
is the matrix of Tisza, and the symmetrical matrix g ¼ @2 [email protected]@z traduces the
interaction between the dissipation processes [3]. The rectangular matrix
b ¼ @2 [email protected]@g expresses the coupling effect between the state variables and
the dissipation variables.
In other respects, the equilibrium state classically imposes the thermo’ ¼ 0. From Eq. 7b
dynamic forces and their rate to be zero; i.e., A ¼ 0 and A
we find, for any equilibrium state, that the internal variables’ evolution results
directly from the variation of the control variables:
z’ r ¼ g1 : t b : g’
ð8Þ
110
Rahouadj and Cunat
According to Eqs. 7b and 8, the evolution of the generalized force becomes
’ ¼ g : ð’z z’ r Þ
ð9Þ
A
and its time integration for transformation near equilibrium leads to the
simple linear relationship
A ¼ gðz zr Þ
ð10Þ
where g is assumed to be constant.
2.6.2.2 KINETICS AND COMPLEMENTARY LAWS
To solve the preceding three equations (7a–b, 10), with the unknown variables being b, z, zr , and A, one has to get further information about the kinetic
relations between the nonequilibrium driving forces A and their fluxes z’ .
2.6.2.2.1 First-Order Nonlinear Kinetics and Relaxation Times
We know that the kinetic relations are not submitted to the same
thermodynamic constraints as the constitutive ones. Thus we shall consider
for simplicity an affine relation between fluxes and forces. Note that this wellknown modeling, early established by Onsager, Casimir, Meixner, de Donder,
De Groot, and Mazur, is only valid in the vicinity of equilibrium:
z’ ¼ L A
ð11Þ
z’ ¼ L g ðz zr Þ ¼ t1 :ðz zr Þ
ð12Þ
and hence, with Eq. 10:
According to this nonlinear kinetics, Meixner [4] has judiciously suggested a
base change in which the relaxation time operator t is diagonal. Here, we
consider this base, which also represents a normal base for the dissipation
modes. In what follows, the relaxation spectrum will be explicitly defined on
this normal base. To extend this kinetic modeling to nonequilibrium
transformations, which is the object of the nonlinear TIP, we also suggest
referring to Eq. 12 but with variable relaxation times. Indeed, each relaxation
time is inversely proportional to the jump frequency, u, and to the probability
þ;r
pj ¼ expðDFþ;r
j =RTÞ of overcoming a free energy barrier, DFj . It follows
that the relaxation time of the process j may be written:
trj ¼ 1=u expðDFþ;r
j =RTÞ
ð13Þ
where the symbol (þ) denotes the activated state, and the index (r) refers to
the activation barrier of the REV near the equilibrium.
2.6 A Nonlinear Viscoelastic Model Based on Fluctuating Modes
111
The reference jump frequency, u0 ¼ kB T=h, has been estimated from
Guggenheim’s theory, which considers elementary movements of translation
at the atomic level. The parameters h, kB , and r represent the constants of
Plank, Boltzmann, and of the perfect gas, respectively, and T is the absolute
temperature. It seems natural to assume that the frequency of the microscopic
rearrangements is mainly governed by the applied perturbation rate, g’ ,
through a shift function að’gÞ:
u ¼ u0 =að’gÞ
ð14Þ
Assuming now that the variation of the activation energy for each process is
governed by the evolution of the overall set of internal variables leads us to
the following approximation of first order:
þ;r
þ Kz :ðz zr Þ
DFþ
j ¼ DFj
ð15Þ
In the particular case of a viscoelastic behavior, this variation of the free
energy becomes negligible. The temperature dependence obviously intervenes
into the basic definition of the activation energy as
¼ DEþ;r T DSþ;r
DFþ;r
j
j
ð16Þ
where the internal energy DEþ;r is supposed to be the same for all processes. It
follows that we may define another important shift function, noted aðTÞ,
which accounts for the effect of temperature. According to the Arrhenius
approximation, DEþ;r being quasi-constant, this shift function verifies the
following relation:
ln aðT; Tref Þ ¼ DEþ;r ð1=T 1=Tref Þ
ð17Þ
where Tref is a reference temperature. For many polymers near the glass
transition, this last shift function obeys the WLF empiric law developed by
William, Landel, and Ferry [5]:
lnðaT Þ ¼ c1 ðT Tref Þ=½c2 þ ðT Tref Þ
ð18Þ
In summary, the relaxation times can be generally expressed as
tj ðTÞ ¼ trj ðTref ÞaðT; Tref Þ að’gÞ aðz; zr Þ
ð19Þ
r
and the shift function aðz; z Þ becomes negligible in viscoelasticity.
2.6.2.2.2 Form of the Relaxation Spectrum near the Equilibrium
We now examine the distribution of the relaxation modes evolving during the
solicitation. In fact, this applied solicitation, g, induces a state of fluctuations
which may be approximately compared to the corresponding equilibrium one.
According to prigogine [6], these fluctuations obey the equipartition of the
entropy production. Therefore, we can deduce the expected distribution in
112
Rahouadj and Cunat
the vicinity of equilibrium as
n
qffiffiffiffi
X
p0j ¼ B trj with
p0j ¼ 1
and
B¼1
X
n qffiffiffiffi
trj
j¼1
ð20Þ
j¼1
where trj is the relaxation time of the process j, p0j its relative weight in the
overall spectrum, and n the number of dissipation processes [3].
As a first approximation, the continuous spectrum defined by Eq. 20 may
be described with only two parameters: the longest relaxation time
corresponding to the fundamental mode, and the spectrum width. Note
that a regular numerical discretization of the relaxation time scale using
a sufficiently high number n of dissipation modes, e.g., 30, gives a
sufficient accuracy.
2.6.2.3 CONSTITUTIVE EQUATIONS
OF THE
DNLR
Combining Eqs. 7a and 12 gives, whatever the chosen kinetics,
b’ ¼ au : g’ b ðz zr Þ: t1 ¼ au : g’ ð#a a# r Þ:t1
b
z
ð21aÞ
To simplify the notation, tb will be denoted t. In a similar form and after
introducing each process contribution in the base defined above, one has
n
n b p0 br
X
X
jm
j m
b’ m ¼
aump g’ p ð21bÞ
t
j
p¼1
j¼1
where the indices u and r denote the instantaneous and the relaxed
values, respectively.
Now we shall examine the dynamic response due to sinusoidally
varying perturbations gn ¼ g0 expðiotÞ, where o is the applied frequency,
and i2 ¼ 1, i.e., g’ n ¼ iogn . The response is obtained by integrating the
above differential relationship. Evidently, the main problem encountered
in the numerical integration consists in using a time step that must
be consistent with the applied frequency and the shortest time of
relaxation. Furthermore, a convenient possibility for very small perturbations is to assume that the corresponding response is periodic and out
of phase:
ð22Þ
b ¼ b0 expðiot þ jÞ and b’ ¼ iob
n
n
n
where j is the phase angle. In fact, such relations are representative of various
physical properties as shown by Kramers [7] and Kronig [8].
The coefficients of the matrices of Tisza, au and ar , and the relaxation
times, tj , may be dependent on temperature and=or frequency. In uniaxial
2.6 A Nonlinear Viscoelastic Model Based on Fluctuating Modes
113
tests of mechanical damping, these Tisza’s coefficients correspond to the
storage and loss modulus E0 (or G0 ) and E00 (or G00 ), respectively.
2.6.3 DESCRIPTION OF THE MODEL IN THE
CASE OF MECHANICAL SOLICITATIONS
We consider a mechanical solicitation under an imposed strain e. Here, the
perturbation g and the response b are respectively denoted e and s. According
’ may be finally written
to Eqs. 19 and 21b, the stress rate response, s,
n
n
X
X
sj p0j ar : e
s’ ¼
p0j au : e’ ð23Þ
að’eÞ aðe; er Þ aðT; Tref Þtj ðTref Þ
j¼1
j¼1
As an example, for a pure shear stress this becomes
n
n
X
X
sj 12 p0j Gr e12
s’ 12 ¼
p0j Gu e’ 12 að’eÞ aðe; er Þ aðT; Tref ÞtGj ðTref Þ
j¼1
j¼1
ð24Þ
In the case of sinusoidally varying deformation, the complex modulus is
given by
G ðoÞ ¼ Gu þ ðGr Gu Þ
n
X
j¼1
p0j
1
1 þ iotGj
It follows that its real and imaginary components are, respectively,
n
X
1
p0j
G0 ðoÞ ¼ Gu þ ðGr Gu Þ
1 þ o2 ðtGj Þ2
j¼1
G00 ðoÞ ¼ ðGr Gu Þ
n
X
j¼1
p0j
otj
1 þ o2 ðtGj Þ2
ð25Þ
ð26Þ
ð27Þ
2.6.4 IDENTIFICATION OF THE PARAMETERS
The crucial problem in vibration experiments concerns the accurate
determination of the viscoelastic parameters over a broad range of frequency.
Generally, to avoid this difficulty one has recourse to the appropriate principle
of equivalence between temperature and frequency, assuming implicitly
identical microstructural states. A detailed analysis of the literature has
brought us to a narrow comparison of the empirical model of Havriliak and
114
Rahouadj and Cunat
Negami (HN) [9] with the DNLR. The HN approach appears to be
successful for a wide variety of polymers; it combines the advantages
of the previous modeling of Cole and Cole [10] and of Davidson
and Cole [11]. For pure shear stress the response given by this HN
approach is
G ¼ GuHN þ ðGrHN GuHN Þ
1
½1 þ ðiotHN Þa b
ð28Þ
where GuHN ; GrHN ; a; and b are empirical parameters. Thus the real and
imaginary components are, respectively,
G0 ¼ GuHN þ ðGrHN GuHN Þ
G00 ¼ ðGrHN GuHN Þ
cosðbyÞ
½1 þ
2oa taHN cosðap=2Þ
þ o2a t2a b=2
sinðbyÞ
½1 þ
2oa taHN cosðap=2Þ
þ o2a t2a b=2
ð29Þ
ð30Þ
The function y is defined by
y ¼ tan1
oa taHN sinðap=2Þ
1 þ oa taHN cosðap=2Þ
ð31Þ
Eqs. 28 to 30 are respectively compared to Eqs. 25 to 27 in order to establish a
correspondence between the relaxation times of the two models:
logðtGr
j Þ ¼ logðtHN Þ þ jL=n þ Y
ð32Þ
where Y, L, and n are a scale parameter, the number of decades of the
spectrum, and the number of processes, respectively. A precise empirical
connection is obtained by identifying the shift function for the time scale
with the relation
tanðbyÞ Gr
G
Gr
Gr
tj ¼ að’gÞtj ¼ aðoÞtj ¼
ð33Þ
tj
otHN
This involves a progressive evolution of the difference of modulus as a
function of the applied frequency:
ðGr Gu Þ ¼ ðGrHN GuHN ÞfG
ð34Þ
The function fG is given by
fG ¼ cosðbyÞ
ð1 þ tan2 ðbyÞÞ
b=2
½1 þ 2oa taHN cosðap=2Þ þ o2a t2a
HN ð35Þ
2.6 A Nonlinear Viscoelastic Model Based on Fluctuating Modes
115
2.6.5 HOW TO USE IT
In practice, knowledge of the only empirical parameters of HN’s modeling
(and=or Cole and Cole’s and Davidson and Cole’s) permits us, in the
framework of the DNLR, to account for a large variety of loading histories.
2.6.6 TABLE OF PARAMETERS
As a typical example given by Hartmann et al. [12], we consider the case of a
polymer whose chemical composition is 1PTMG2000=3MIDI=2DMPD*. The
master curve is plotted at 298 K in Figure 2.6.1. The spectrum is discretized
FIGURE 2.6.1 Theoretical simulation of the moduli for PTMG ( J).*
FIGURE 2.6.2 Theoretical simulations of the shift function aðoÞ and of fG for PTMG.*
* PTMG: poly (tetramethylene ether) glycol; MIDI: 4,40 -diphenylmethane diisocyanate; DMPD:
2,2-dimethyl-1, 3-propanediol with a density of 1.074 g=cm3, and glass transition Tg ¼ 408C.
116
Rahouadj and Cunat
with L ¼ 6, a scale parameter Y equal to 5.6, and 50 relaxation times. The
parameters
GrHN ¼ 2:14 MPa,
GuHN ¼ Gu ¼ 1859 MPa,
tHN ¼
7
1.649 10 s, a ¼ 0:5709 and b ¼ 0:0363 allow us to calculate the shift
function aðoÞ and the function fG which is necessary to estimate the difference
between the relaxed and nonrelaxed modulus, taking into account the
experimental conditions. Figure 2.6.1 illustrates the calculated viscoelastic
response, which is superposed to HN’s one. The function fG and the shift
function aðoÞ illustrate the nonlinearities introduced in the DNLR modeling
(Fig. 2.6.2).
REFERENCES
1. Coleman, B. D., and Gurtin, M. (1967). J. Chem. Phys. 47 (2): 597.
2. De Donder, T. (1920). Lecon de thermodynamique et de chimie physique, Paris: Gauthiers,
Villars.
3. Cunat, C. (1996). Rev. Gçn. Therm. 35: 680–685.
4. Meixner, J. Z. (1949). Naturforsch., Vol. 4a, p. 504.
5. William, M. L., Landel, R. F., and Ferry, J. D. (1955). The temperature dependence of
relaxation mechanisms in amorphous polymers and other glass-forming liquids. J. Amer.
Chem. Soc. 77: 3701.
6. Prigogine, I. (1968). Introduction a" la thermodynamique des processus irr!eversibles, Paris:
Dunod.
7. Kramers, H. A. (1927). Atti. Congr. dei Fisici, Como, 545.
8. Kronig, R. (1926). J. Opt. Soc. Amer. 12: 547.
9. Havriliak, S., and Negami, S. (1966). J. Polym. Sci., Part C, No. 14, ed. R. F. Boyer, 99.
10. Cole, K. S., and Cole, R. H. (1941). J. Chem. Phys. 9: 341.
11. Davidson, D. W., and Cole, R. H. (1950). J. Chem. Phys. 18: 1417.
12. Hartmann, B., Lee, G. F., and Lee, J. D. (1994). J. Acoust. Soc. Amer. 95 (1).
CHAPTER
2.7
Linear Viscoelasticity
with Damage
R. A. SCHAPERY
Department of Aerospace Engineering and Engineering Mechanics, University of Texas,
Austin, Texas
Contents
2.7.1
2.7.2
2.7.3
2.7.4
Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Description of the Model. . . . . . . . . . . . . . . . . . .
Identification of the Material Functions
and Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.5 How to Use It . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117
118
119
121
123
123
2.7.1 VALIDITY
This paper describes a homogenized constitutive model for viscoelastic
materials with constant or growing distributed damage. Included are threedimensional constitutive equations and equations of evolution for damage
parameters (internal state variables, ISVs) which are measures of damage.
Anisotropy may exist without damage or may develop as a result of
damage. For time-independent damage, the specific model covered here is
that for a linearly viscoelastic, thermorheologically simple material in which
all hereditary effects are expressed through a convolution integral with one
creep or relaxation function of reduced time; nonlinear effects of transient
crack face contact and friction are excluded. More general cases that account
for intrinsic nonlinear viscoelastic and viscoplastic effects as well as
thermorheologically complex behavior and multiple relaxation functions are
published elsewhere [10].
Handbook of Materials Behavior Models
Copyright # 2001 by Academic Press. All rights of reproduction in any form reserved.
117
118
Schapery
2.7.2 BACKGROUND
As background to the model with time-dependent damage, consider first the
constitutive equation with constant damage, in which e and s represent
the strain and stress tensors, respectively,
e ¼ fSdsg þ eT
ð1Þ
where S is a fully symmetric, fourth order creep compliance tensor and eT is
the strain tensor due to temperature and moisture (and other absorbed
substances which affect the strains). The braces are abbreviated notation for a
linear hereditary integral. Although the most general form could be used,
allowing for general aging effects, for notational simplicity we shall use the
familiar form for thermorheologically simple materials,
Z t
Z x
@g
0 @g 0
ð2Þ
f fdgg ¼
f ðx x Þ 0 dt ¼
f ðx x0 Þ 0 dx0
@t
@x
o
o
where it is assumed f ¼ g ¼ o for t5o and
Z t
x
dt00 =aT ½Tðt00 Þ
x0 ¼ xðt0 Þ
ð3Þ
o
Also, aT ðTÞ is the temperature-dependent shift factor. If the temperature is
constant in time, then x x0 ¼ ðt t0 Þ=aT : Physical aging [12] may be taken
into account by introducing explicit time dependence in aT ; i.e., use
aT ¼ aT ðT; t00 Þ in Eq. 3. The effect of plasticizers, such as moisture, may also
be included in aT : When Eq. 2 is used with Eq. 1, f and g are components of
the creep compliance and stress tensors, respectively.
In certain important cases, the creep compliance components are
proportional to one function of time,
S ¼ kD
ð4Þ
where k is a constant, dimensionless tensor and D ¼ DðxÞ is a creep
compliance (taken here to be that obtained under a uniaxial stress state).
Isotropic materials with a constant Poisson’s ratio satisfy Eq. 4. If such a
material has mechanically rigid reinforcements and=or holes (of any shape), it
is easily shown by dimensional analysis that its homogenized constitutive
equation satisfies Eq. 4; in this case the stress and strain tensors in Eq. 1
should be interpreted as volume-averaged quantities [2]. The Poisson’s ratio
for polymers at temperatures which are not close to their glass-transition
temperature, Tg , is nearly constant; except at time or rate extremes, somewhat
above Tg Poisson’s ratio is essentially one half, while below Tg it is commonly
in the range 0.35–0.40 [5].
119
2.7 Linear Viscoelasticity with Damage
Equations 1 and 4 give
e ¼ fDdðksÞg þ eT
ð5Þ
s ¼ kI fEdeg kI fEdeT g
ð6Þ
The inverse is
where kI ¼ k
for t > o,
1
and E ¼ EðxÞ is the uniaxial relaxation modulus in which,
fDdEg ¼ fEdDg ¼ 1
ð7Þ
In relating solutions of elastic and viscoelastic boundary value problems,
and for later use with growing damage, it is helpful to introduce the
dimensionless quantities
1
1
1
eR fEdeg
eRT fEdeT g
uR fEdug
ð8Þ
ER
ER
ER
where ER is an arbitrary constant with dimensions of modulus, called the
reference modulus; also, eR and eRT are so-called pseudo-strains and uR is
the pseudo-displacement. Equation 6 becomes
s ¼ CeR CeRT
ð9Þ
where C ER kI is like an elastic modulus tensor; its elements are called
pseudo-moduli. Equation 9 reduces to that for an elastic material by taking
E ¼ ER ; it reduces to the constitutive equation for a viscous material if E is
proportional to a Dirac delta function of x. The inverse of Eq. 9 gives the
pseudo-strain eR in terms of stress,
# þ eR
ð10Þ
eR ¼ Ss
T
where S# ¼ C1 ¼ k=ER : The physical strain is given in Eq. 5.
2.7.3 DESCRIPTION OF THE MODEL
The correspondence principle (CPII in Schapery [4, 8]) that relates elastic and
viscoelastic solutions shows that Eqs. 1–10 remain valid, under assumption
Eq. 4, with damage growth when the damage consists of cracks whose faces
are either unloaded or have loading that is proportional to the external loads.
With growing damage k; C, and S# are time-dependent because they are
functions of one or more damage-related ISVs; the strain eT may also depend
on damage. The fourth-order tensor k must remain inside the convolution
integral in Eq. 5, just as shown. This position is required by the
correspondence principle. The elastic-like Eqs. 9 and 10 come from Eq. 5,
and thus have the appropriate form with growing damage. However, with
120
Schapery
healing of cracks, pseudo-stresses replace pseudo-strains because k must
appear outside the convolution integral in Eq. 5 [8].
The damage evolution equations are based on viscoelastic crack growth
equations or, in a more general context, on nonequilibrium thermodynamic
equations. Specifically, let W R and WCR denote pseudo-strain energy density
and pseudo-complementary strain energy density, respectively,
1
ð11Þ
W R ¼ CðeR eRT ÞðeR eRT Þ F
2
1#
þ eRT s þ F
WCR ¼ Sss
2
ð12Þ
WCR ¼ W R þ seR
ð13Þ
so that
and
@W R
@WCR
R
ð14Þ
e
¼
@eR
@s
The function F is a function of damage and physical variables that cause
residual stresses such as temperature and moisture.
For later use in Section 2.7.4, assume the damage is fully defined by a set of
scalar ISVs, Sp (p ¼1, 2, . . . P) instead of tensor ISVs. Thermodynamic forces,
which are like energy release rates, are introduced,
s¼
fp @W R
@Sp
ð15Þ
or
fp @WCR
@Sp
ð16Þ
where the equality of these derivatives follows directly from the total
differential of Eq. 13.
Although more general forms could be used, the evolution equations for
S’p dSp =dx are assumed in the form
ð17Þ
S’p ¼ S’p ðSq ; fp Þ
in which S’p may depend on one or more Sq (q ¼ 1, . . . P), but on only one
force fp . The entropy production rate due to damage is non-negative if
X
fp S’p O
ð18Þ
p
thus satisfying the Second Law of Thermodynamics. It is assumed that when
j fp j is less than some threshold value, then S’p ¼ O.
2.7 Linear Viscoelasticity with Damage
121
Observe that even when the stress vanishes, there may be damage growth
due to F. According to Eqs. 12 and 16,
fp ¼
@WCR 1 @ S#
@eR
@F
¼
ss þ T s þ
2 @Sp
@Sp
@Sp
@Sp
ð19Þ
which does not vanish when r ¼ o, unless @[email protected] ¼ 0.
The use of tensor ISVs is discussed and compared with scalar ISVs by
Schapery [10]. The equations in this section are equally valid for tensor and
scalar ISVs.
2.7.4 IDENTIFICATION OF THE MATERIAL
FUNCTIONS AND PARAMETERS
The model outlined above is based on thermorheologically simple behavior
in that reduced time is used throughout, including damage evolution
(Eq. 17). In studies of particle-reinforced rubber [4], this simplicity
was found, implying that even the microcrack growth rate behavior
was affected by temperature only through viscoelastic behavior of the
rubber. If the damage growth is affected differently by temperature (and
plasticizers), then explicit dependence may be introduced in the rate
(Eq. 17). In the discussion that follows, complete thermorheological
simplicity is assumed.
The behavior of particle-reinforced rubber and asphalt concrete has been
characterized using a power law when fp > o,
ð20Þ
S’p ¼ ð fp Þap
where ap is a positive constant. (For the rubber composite two ISVs, with
a1 ¼ 4:5 and a2 ¼ 6, were used for uniaxial and multiaxial behavior, whereas
for asphalt one ISV, with a ¼ 2:5, was used for uniaxial behavior.) A
coefficient depending on Sp may be included in Eq. 20; but it does not really
generalize the equation because a simple change of the variable Sp may be
used to eliminate the coefficient.
Only an outline of the identification process is given here, but details are
provided by Park et al. [3] for uniaxial behavior and by Park and Schapery [4]
and Ha and Schapery [1] for multiaxial behavior. Schapery and Sicking
[11] and Schapery [9] discuss the model’s use for fiber composites. The effects
of eT and F are neglected here.
(a) The first step is to obtain the linear viscoelastic relaxation modulus
EðxÞ and shift factor aT for the undamaged state. This may be done
122
Schapery
using any standard method, such as uniaxial constant strain rate tests
at a series of rates and temperatures. Alternatively, for example, uniaxial
creep tests may be used to find DðxÞ, after which EðxÞ is derived
from Eq. 7.
(b) Constant strain rate (or stress rate) tests to failure at a series of
rates or temperatures may be conveniently used to obtain the additional data
needed for identification of the model. (However, depending on the
complexity of the material and intended use of the model, unloading and
reloading tests may be needed [7].) Constant strain rate tests often are
preferred over constant stress rate tests because meaningful post-stress peak
behavior (prior to significant strain localization) may be found from the
former tests.
# where
For isothermal, constant strain rate, R, tests, the input is Rt ¼ Rx;
R# ¼ RaT and x ¼ t=aT . Inasmuch as the model does not depend on
temperature when reduced time is used, all stress vs. reduced time response
# regardless of temperature.
curves depend on only one input parameter R,
Thus, one may obtain a complete identification of the model from a series of
tests over a range of R# using one temperature and different rates or one rate
and different temperatures; both types of tests may be needed in practice for R#
to cover a sufficiently broad range. One should, however, conduct at least
a small number of both types of tests to check the thermorheologically
simple assumption.
(c) Convert all experimental values of displacements and strains
from step (b) tests to pseudo-quantities using Eq. 8. This removes intrinsic
viscoelastic effects, thus enabling all subsequent identification steps to be
those for a linear elastic material with rate-dependent damage. If controlled
strain (stress) tests are used, then one would employ W R ðWCR Þ in the
identification. However, mixed variables may be input test parameters, such
as constant strain rate tests of specimens in a test chamber at a series of
specified pressures [4]. In this case it is convenient to use mixed pseudoenergy functions in terms of strain and stress variables. Appropriate
energy functions may be easily constructed using methods based on linear
elasticity theory.
(d) The procedure for finding the exponent a and pseudo Young’s modulus
in terms of one damage parameter is given by Park et al. [3]. After this, the
remaining pseudo-moduli or compliances may be found in terms of one or
more ISVs, as described by Park and Schapery [4] using constant strain rate
tests of bar specimens under several confining pressures. The material
employed by them was initially isotropic, but it became transversely isotropic
as a result of damage. Identification of the full set of five pseudo-moduli and
the pseudo-strain energy function, as functions of two ISVs, is detailed by Ha
and Schapery [1].
2.7 Linear Viscoelasticity with Damage
123
2.7.5 HOW TO USE IT
Implementation of user-defined constitutive relations based on this model in a
finite element analysis is described by Ha and Schapery [1]. Included are
comparisons between theory and experiment for overall load-displacement
behavior and for local strain distributions. The model employed assumes the
material is locally transversely isotropic with the axis of isotropy assumed
parallel to the local maximum principal stress direction, accounting for prior
stress history at each point. A procedure is proposed by Schapery [10] that
enables use of the same model when transverse isotropy is lost due to rotation
of the local maximum principal stress direction.
REFERENCES
1. Ha, K., and Schapery, R. A. (1998). A three-dimensional viscoelastic constitutive model for
particulate composites with growing damage and its experimental validation. International
Journal of Solids and Structures 35: 3497–3517.
2. Hashin, Z. (1983). Analysis of composite materials } a survey. Journal of Applied Mechanics
105: 481–505.
3. Park, S. W., Kim, Y. R., and Schapery, R. A. (1996). A viscoelastic continuum damage
model and its application to uniaxial behavior of asphalt concrete. Mechanics of Materials
24: 241–255.
4. Park, S. W., and Schapery, R. A. (1997). A viscoelastic constitutive model for particulate
composites with growing damage. International Journal of Solids and Structures 34: 931–947.
5. Schapery, R. A. (1974). Viscoelastic behavior and analysis of composite materials, in
Mechanics of Composite Materials, pp. 85–168, vol. 2, Sendeckyi, G. P., ed., New York:
Academic.
6. Schapery, R. A. (1981). On viscoelastic deformation and failure behavior of composite
materials with distributed flaws, in 1981 Advances in Aerospace Structures and Materials,
pp. 5–20, Wang, S. S., and Renton, W. J., eds., ASME, AD-01.
7. Schapery, R. A. (1982). Models for damage growth and fracture in nonlinear viscoelastic
particulate composites, in: Proc. Ninth U.S. National Congress of Applied Mechanics, Book No.
H00228, pp. 237–245, Pao, Y. H., ed., New York: ASME.
8. Schapery, R. A. (1984). Correspondence principles and a generalized J integral for large
deformation and fracture analysis of viscoelastic media, in: International Journal of Fracture
25: 195–223.
9. Schapery, R. A. (1997). Constitutive equations for special linear viscoelastic composites with
growing damage, in Advances in Fracture Research, pp. 3019–3027, Karihaloo, B. L., Mai, Y.W., Ripley, M. I., and Ritchie, R. O., eds., Pergamon.
10. Schapery, R. A. (1999). Nonlinear viscoelastic and viscoplastic constitutive equations with
growing damage. International Journal of Fracture 97: 33–66.
11. Schapery, R. A., and Sicking, D. L. (1995). On nonlinear constitutive equations for elastic and
viscoelastic composites with growing damage, in Mechanical Behavior of Materials, pp. 45–76,
Bakker, A., ed., Delft: Delft University Press.
12. Struik, L. C. E. (1978). Physical Aging in Amorphous Polymers and Other Materials,
Amsterdam: Elsevier.
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