Drozdov AD - Viscoelastic structures - 1998

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Chapter I 
Kinematics of Continua 
This chapter is concerned with kinematic concepts in the nonlinear mechanics of 
continua. We discuss Eulerian and Lagrangian coordinate frames, derive expressions 
for tangent and dual vectors, and introduce operators of covariant differentiation 
in curvilinear coordinates. Explicit formulas are developed for the main strain and 
deformation tensors, as well as for the volume and surface elements in an arbitrary 
configuration. Finally, we introduce corotational derivatives of objective tensors and 
discuss their properties. A more detailed exposition of these issues with the use of 
direct tensor notation, can be found, e.g., in Drozdov (1996). 
1.1 Basic Definitions and Formulas 
1.1.1 Description of Motion 
In the nonlinear mechanics of continua, two different kinds of coordinate frames 
are distinguished. The first is the Eulerian (spatial) coordinate frame, which is fixed 
and immobile in space. Points of a moving medium change their positions in space 
with respect to the Eulerian coordinates. As common practice, Cartesian coordinates 
{xl,x2, x3}, cylindrical coordinates {r, O,z}, and spherical coordinates {r, 0, th} are 
employed as Eulerian coordinates. For cylindrical coordinates 
x 2 
r -- V / (x l ) 2 -k- (x2) 2, 0 -- tan -1 ~--i-' Z -" X 3, 
and for spherical coordinates 
r = V/(xl) 2 + (x2) 2 + (x3) 2, 
x 2 0 = tan-1 V/( xl)2 + (X2)2 1 __ 
X3 , 4) = tan- x l . 
We denote unit vectors of Cartesian coordinates as ~'1, ~'2, and 6'3, unit vectors of 
cylindrical coordinates as ~r, ~0, and ~z, and unit vectors of spherical coordinates as 
2 Chapter 1. Kinematics of Continua 
~'r, 6'0 and ~6, respectively. The following formulas are fulfilled for the derivatives of 
the unit vectors for a cylindrical coordinate frame: 
Or Or Or 
o~ e r tg e. ch _ o~ e z 
o3 6 -- 6'~b, ~(1) -- 6'r, O3(]) -- 0, 
t96'r -- 0, O~6'th -- 0, 06'z -- 0 
Oz Oz Oz 
(1.1.1) 
and for a spherical coordinate frame: 
Oe.r _ O, O~e'O - - O, °3e'dP - - O, 
Or Or Or 
- - 6 '0 , - - - -6 ' r , - - 0 , 
00 00 00 
04) - 06 sin 0, &h - 06 cos 0, 0~b --(6'r sin 0 + ~0 cos 0). 
(1.1.2) 
Lagrangian (material) coordinates ~ = {~1, ~2, ~3} provide the other kind of 
coordinate frames, which are frozen into a moving medium. Position of any point with 
respect to the Lagrangian coordinates remains unchanged in time, while the frame 
moves together with the medium. As common practice, Lagrangian coordinates 
coincide with Eulerian coordinates at the initial instant, when the motion starts. 
Position of a point M with respect to an immobile spatial coordinate frame is 
determined by its radius vector ?. Two radius vectors are distinguished: the initial 
?0(~) and the current ?(t, ~), where t stands for time (see Figure 1.1.1). To set a 
motion with respect to a Lagrangian frame means to establish a law 
= ?(t, ?o) (1.1.3) 
for any point ~ and for any instant t. Introducing the displacement vector fi(t, ~), we 
write Eq. (1.1.3) as 
?(t, ~) = ?0(~) + fi(t, ~). (1.1.4) 
Some generally accepted requirements are imposed on admissible displacement 
fields: 
(i) The map ? = ?(t, ?0) is twice continuously differentiable. 
(ii) The map ? = ?(t, ~0) is globally one-to-one and it preserves orientation. 
Condition (i) is introduced for convenience and simplicity of exposition, and it may be 
violated for the analysis of crack propagation and shock waves in deformable media. 
1.1. Basic Definitions and Formulas 3 
Initial 
configuration 
Actual 
configuration 
M fi M 
A ~_-' 
O 
Figure 1.1.1: The radius vectors and the displacement vector. 
The first part of restriction (ii) means that two distinct material points cannot occupy 
the same position simultaneously, which implies that the map ~(t, ?0) is globally 
invertible. This assertion excludes such phenomena as, e.g., collapse of a cavity and 
attachment of strips. The other part of this restriction means that orientation of any 
three noncoplanar vectors does not change. 
1.1.2 Tangent Vectors 
Let ?0(~) and ?(t, ~) be the radius vectors of a point M with Lagrangian coordinates 
= {~i} in the initial and actual configurations. We fix the coordinates ~2 and ~3 
and consider a line drawn by the radius vector, when only the coordinate ~1 changes. 
This line is called the coordinate line ~ 1. Similarly, the coordinate lines ~2 and ~3 are 
introduced as shown in Figure 1.1.2. The vectors 
07 
gi -- o~i (1.1.5) 
are linearly independent, tangent to the coordinate lines ~i, and they form a basis. 
Volume V of a parallelepiped constructed on the tangent vectors gi is calculated as 
V = gl "(,~2 X g3) - g2"(,~3 X ,~1) - g3"( ,~l X g2), (1.1.6) 
where the dot denotes the inner product, and x stands for the vector product (see 
Figure 1.1.3). 
4 Chapter 1. Kinematics of Continua 
~3 
g'3 
g'2 
... ~2 . 
-- "''-.....°. o ,,, 
Figure 1.1.2: Tangent vectors for a Lagrangian coordinate frame. 
The dual vectors ~i are orthogonal to the tangent vectors gi, 
i ~i . ~ j __ ~ j , (1.1.7) 
i where 8j are the Kronecker indices 
/ {1 ° i= , 
~J : i =/= j . 
g3 
M 
Figure 1.1.3: The elementary volume. 
1.1. Basic Definitions and Formulas 5 
Vectors gi and ~i are connected by the formulas 
~1 __ g2 X g3 ~2 __ g3 X gl ~3 __ gl X g2 
V ' V ' V 
gl = Vg 2 X ~3, g2 = Vg 3 X ~1, g3 = Vg 1 X ~2. (1.1.8) 
Any vector g/can be expanded in tangent vectors gi and in dual vectors ~i, 
gt = q'g,i : qig,', (1.1.9) 
where qi are covariant components and qi are contravariant components of g/. Sum- 
mation is assumed with respect to repeating indices, which occupy alternately the 
upper and the lower position. It follows from Eqs. (1.1.7) and (1.1.9) that 
qi = ~ . ~i, qi = q" gi. (1 .1 .10) 
Let us calculate the differential of the radius vector 
°~P i d F -- --2-~; d~ = g id~ i. (1.1.11) 
Multiplying Eq. (1.1.11) by itself, we find the square of the arc element ds 
ds 2 = d? " d? = ~id~ i " $jd(; j = (gi " g j )d~id~ j = g i jd~id~ j, (1.1.12) 
where the quantities 
gij = gi " g j (1.1.13) 
are covariant components of the metric tensor. Contravariant components g'J of the 
metric tensor are elements of the matrix inverse to the metric matrix [gij]. For any 
integers i and j we have 
• i gtk gkj = ~j. 
It can be shown that 
gij = ~ i . ~j , ~i _ g i j~ j , gi = gijg, j. (1.1.14) 
Equations (1.1.14) imply that covariant and contravariant components of the metric 
tensor allow the indices of tangent vectors to be raised and lowered. 
1.1.3 The Nabla Operator 
We multiply Eq. (1.1.11) by ~J, use Eq. (1.1.7), and find that 
d~i = ~i . d r . (1.1.15) 
6 Chapter 1. Kinematics of Continua 
Differentiation of a smooth scalar function f(~) with the use of Eq. (1.1.15) yields 
Of = ~i Of 
d f = - -~ d~ i - -~ " d r . (1.1.16) 
The Hamilton operator (the nabla operator) is introduced according to the formula 
~7 -- ~i O~ 
03~i. (1.1.17) 
Combining Eqs. (1.1.16) and (1.1.17), we obtain 
df = fT f . d?. (1.1.18) 
Equations (1.1.7) and (1.1.17) imply that 
0 
o~i -- gi" ~r. (1.1.19) 
By analogy with Eq. (1.1.16), we find that for a smooth vector function g/(~) 
dO = d~' = d? • ~i 0~/ ~T (1.1.20) 
where the tensor ~r~/is called covariant derivative of the vector field ?/(~), and T 
stands for transpose. 
1.1.4 Deformation Gradient 
We now differentiate Eq. (1.1.4) with respect to ~i and find
that 
07 07o Off Off 
gi -- o~i o~i + ~ = goi + a~ i , (1.1.21) 
where g0i and gi are tangent vectors in the initial and actual configurations. Equations 
(1.1.19) and (1.1.21) imply that 
gi -- g0i + g0i" V0 ~ -- g0i" (I + ~0 ~) = (I + ~r0uT)'g0i, 
goi : gi -- gi " ~ ~l : gi " ( I -- ~ ~l) : ( I -- v ~lT) " gi, (1.1.22) 
where i is the unit tensor. 
Denote by ~(~) and ~,i(t, ~) the dual vectors, and by ~'0r(~) and ~r?0(t, ~) the 
deformation gradients 
- 0? 
VO ~ -- g ,~-~ -- ~tO~i, ~7~0 = ~i ¢~0 • -~ = ~t~O i. (1.1.23) 
1.1. Basic Definitions and Formulas 
The tensors ~'o? and ~r ?o are "gradients" of the map ?(t, ?o), which characterize it in 
a small vicinity of any point. In particular, if a map ?(t, ?0) preserves orientation, then 
det ~'07 > 0. (1.1.24) 
Let us discuss properties of the deformation gradients. It follows from Eqs. 
(1.1.23) that 
and 
~'o?o = ~' ? = i (1.1.25) 
¢0 ~T -- ~i~io, V E~ -= ~Oi~ i. 
Substitution of expressions (1.1.22) into Eqs. (1.1.23) yields 
Vo~ = i + Vo~, V ~o = i - ~ ~. 
Multiplying Eqs. (1.1.23) and using Eq. (1.1.7), we find that 
Vor" Vro = i. 
It follows from this equality that 
(1.1.26) 
(1.1.27) 
~r0? = ~r ?o 1. (1.1.28) 
We multiply the first equality in Eq. (1.1.23) by ~'0i, the other equality by ~,~, and use 
Eq. (1.1.7). As a result, we obtain 
g0i" ~707 -- gi, ~rr0. g0 -- ~i. (1.1.29) 
It follows from Eqs. (1.1.17), (1.1.28), and (1.1.29) that 
~7 =~i O3 i ¢9 ~-~ = ¢ r0" g0 ~ = ~7 r0" ¢0, ~7 0 = ~7 ~O 1 . ~7 = VO P . ~7. (1.1.30) 
Let us consider a vector d Po = ~oid~ i in the initial configuration and its image 
d ? = ~id,~ i in the actual configuration. Equations (1.1.20) and (1.1.30) imply that 
dP = dPo" VoP = Vo~ T" d~o, dPo = d? . V~o = VPff" dP. (1.1.31) 
1.1.5 Deformation Tensors and Strain Tensors 
Denote by dso and ds the arc elements in the initial and actual configurations 
ds 2 = d?o " d?o, ds 2 = d? . d? . (1.1.32) 
Substitution of expressions (1.1.31) into the second formula (1.1.32) yields 
ds 2 = d? • d? = d?o" ~ro? • ~'0 ?T • d?o = d?o" ~" d?o, (1.1.33) 
Chapter 1. Kinematics of Continua 
where 
~' = ~'o~ • ~7o?r (1 .1 .34) 
is the Cauchy deformation tensor. It follows from Eqs. (1.1.27) and (1.1.34) that 
where 
= ~? + 2~o(fi) + V0fi" Vofi r, (1.1.35) 
~'o = V?o" V?~ (1.1.38) 
is the Almansi deformation tensor. According to Eqs. (1.1.27) and (1.1.38), 
~o = i - 2~(fi) + V ft. V fir, (1.1.39) 
where 
1 
~(fi) = ~(~,fi + ¢fir) (1.1.40) 
is the second (Swainger) infinitesimal strain tensor. 
It follows from Eqs. (1.1.33) and (1.1.37) that the Cauchy and Almansi deforma- 
tion tensors indicate changes in the arc element for transition from the initial to actual 
configuration. Substitution of expressions (1.1.23) into Eqs. (1.1.34) and (1.1.38) 
yields 
- / - j ,, • .. 
g, = g i jgogo , go = goijg, tg, J (1.1.41) 
Multiplying the deformation gradients ~'o? and V ?o, we may construct four 
symmetrical tensors. The Finger deformation tensor is determined as 
P = Vo ?r" ~'o? = i + 2~o(fi) + Vofi r" ~7ofi, (1.1.42) 
and the Piola deformation tensor equals 
F0 = ¢~" ~'~0 = i - 2~(fi) + ~,fir. eft. (1.1.43) 
It follows from Eqs. (1.1.28), (1.1.34), (1.1.38), (1.1.42), and (1.1.43) that 
P = go 1, /~0 = ~-1. (1.1.44) 
1 
&o(~) - ~(Vo~ + Vo~ T) (1.1.36) 
is the first (Cauchy) infinitesimal strain tensor. 
To obtain a reciprocal deformation tensor, we substitute expressions (1.1.31 ) into 
the first formula in Eq. (1.1.32) and find that 
ds~ = dPo" dPo = d~. ~r~o-~'?~. dP = dP" ~'o" dP, (1.1.37) 
where 
1.1. Basic Def in i t ions and Formulas 9 
Substitution of expressions (1.1.23) into Eqs. (1.1.42) and (1.1.43) implies that 
p i j - - " " 
= go gigj, FO -- g'Jgoigoj. 
It follows from Eqs. (1.1.34), (1.1.38), (1.1.42), and (1.1.43) that 
Ik(P) = Ik(~), Ik(P0) = I~(~0), (1.1.45) 
where I~ (k = 1, 2, 3) stands for the principal invariant of a tensor. 
Other deformation tensors can be presented as functions of the Cauchy and 
Finger tensors. For example, the Hencky deformation tensor is defined as 
1 
/ : /= ~ lnF (1.1.46) 
[see Fitzgerald (1980).] In general, to construct the tensor/t we should find the eigen- 
values and eigenvectors of the tensor F, which requires cumbersome calculations. 
It follows from Eqs. (1.1.33) and (1.1.37) that 
ds 2 - ds 2 = d ?o " ~ " d ro - d ?o " i . d ~o = 2d?o" ¢~" d?o, 
ds 2 - ds 2 = d~ • i" d? - d? "g0" d~ = 2d? • A" d~, (1.1.47) 
where 
C-- ~(g--I) : E0(U) + ~~70~" ~70uT (1.1.48) 
is the Cauchy strain tensor and 
^ 1 1 
A = ~( i - ~o) = ~(fi) - ~ ' f i " ~fir (1.1.49) 
is the Almansi strain tensor. Substitution of expressions (1.1.41) into Eqs. (1.1.48) 
and (1.1.49) implies that 
1 - i - j ^ 1 • 
= -~(gij -- goij)gogo, A = -~(gij - goij)g, ig, J, 
which means that the Cauchy and Almansi strain tensors have the same covariant 
components, but in different bases. Obviously, their contravariant and mixed compo- 
nents may differ from each other. 
Similar to Eqs. (1.1.48) and (1.1.49), we define the Finger strain tensor 
^ 1 
EF = ~(I3(g0)/2" -- I) (1.1.50) 
and the Piola strain tensor 
1 ^ I3(~)P0) /~F0 = ~( I - . (1.1.51) 
10 Chapter 1. Kinematics of Cont inua 
Several constitutive equations for viscoelastic media employ the so-called differ- 
ence histories of strains [see, e.g., Coleman and Noll (1961).] The difference history 
of the Cauchy strain Cd(t , T) equals 
Cd(t , "r) = C( t ) - C( t - "r), (1.1.52) 
where t~(t) is the Cauchy strain tensor. It follows from Eqs. (1.1.47) and (1.1.52) that 
the difference history of the Cauchy strain characterizes changes in the arc element 
for transition from the actual configuration at instant t - ~- to the actual configuration 
at instant t 
ds2 ( t ) - ds2 ( t - ~-) = 2d?0-Cd(t, ~') " d ?o . 
The terminology used in the nonlinear mechanics has not yet been fixed. 
The deformation gradient is also called the distortion tensor. The Cauchy strain 
tensor is also called the Cauchy-Green strain tensor and the Green strain tensor. The 
Cauchy deformation tensor is also called the left Cauchy tensor, whereas the Almansi 
deformation tensor is called the Green tensor, the right Cauchy tensor, and the Euler 
strain tensor. 
1.1.6 Stretch Tensors 
According to the polar decomposition theorem, any nonsingular tensor can be pre- 
sented as a product of a symmetrical positive definite tensor and an orthogonal tensor. 
Applying this assertion to the deformation gradient V0?, we arrive at the left polar 
decomposition formula 
~ro? = O'l" O, (1.1.53) 
where tit is a symmetrical positive definite left stretch tensor, C]~ = ~]l, and O is an 
orthogonal rotation tensor, 0 T = 0 -1. Substitution of expression (1.1.53) into Eq. 
(1.1.34) implies that 
= Ol " O" O- ' " Ol = 
It follows from this equality that 
~Jl = ~1/2. (1.1.54) 
Another important relation is derived by using the right polar decomposition of 
the deformation gradient 
Vo? = O. ~-/r, (1.1.55) 
where Clr is a symmetrical positive definite right stretch tensor, Of = Clr, and O is 
an orthogonal rotation tensor, O r = 0 -1. It follows from Eqs. (1.1.42) and (1.1.55) 
1.1. Basic Definitions and Formulas 11 
that 
~)r r = p 1/2. (1 .1 .56) 
The eigenvalues Vl, v2, v3 of the stretch tensors UI and Or coincide. These eigenvalues 
are called principal stretches. Equation (1.1.56) implies that 
I1(/~) v 2 + v 2 + v 2, /2(/~) v2v 2 + v2v 2 + v2v 2, I 3 (F )= v 2 2 2 = = V~ V 3 .
(1.1.57) 
Other deformation tensors can be also expressed in terms of the left and right 
stretch tensors. For example, substituting expression (1.1.56) into Eq. (1.1.46), we 
obtain the formula for the Hencky deformation tensor 
/~ = In Ur. (1.1.58) 
1.1.7 Relative Deformat ion Tensors 
The deformation tensors describe transformations from the initial (at instant t = 0) to 
the actual (at the current instant t) configuration. For the analysis of the viscoelastic 
behavior, it is convenient to use relative deformation tensors, which characterize the 
entire history of deformations in the interval [0, t]. 
Let us consider transition from the actual configuration at instant ~" to the 
actual configuration at instant t -> ~-. The corresponding deformation gradients 
fTr?(t) = ~7~?o" fTo?(t) = g i (T )~, i ( t ) , 
(Ttr(~') = (Ttro" (7or( ' r)= gi(t)~oi(r) (1.1.59) 
are called relative deformation gradients. It follows from Eqs. (1.1.59) that for any 
0_<,r_<t <~, 
~r~?(t) = (~r0?(r))-I. ~r0?(t ). (1.1.60) 
Relative deformation tensors are introduced with the use of Eqs. (1.1.34), (1.1.38), 
(1.1.42), (1.1.43), and (1.1.59). For example, the Cauchy and Finger tensors are 
determined as follows: 
~°(t, 1-) = ~'TP(t) • ~rTpT(t) = ~'¢P0 " ~(t) • ~'TPff, 
Pc( t , r) = fTr?T(t) • fTr?(t) = fTO?r(t) " ~-l(r)" ~'0?(t). (1.1.61) 
Similar formulas can also be obtained for the relative strains tensors, for example, 
for the relative Cauchy strain tensor 
1 
C'<> (t, r) = ~ [~<> (t, r) - i] (1.1.62) 
12 Chapter 1. Kinematics of Continua 
and for the relative Almansi strain tensor 
1 -1] A<>(t, ~-) = ~[i - (F<>(t, ~')) . (1.1.63) 
It follows from Eqs. (1.1.61) that for any t -> 0, 
~<> (t, t) = i, F<> (t, t) = L (1.1.64) 
Equations (1.1.44) show that only two deformation tensors (e.g., the Cauchy and 
Finger tensors) are independent, and the other two tensors are inverse to them. 
Equations (1.1.61) imply that for any 0 -< ~" - t < ~, 
F°(t, 1-) = [~°(~-,t)]-l, (1.1.65) 
which means that only one relative deformation tensor is independent, and the others 
may be expressed in terms of it. 
1.1.8 Rigid Motion 
A motion is called rigid if for any instant t - 0 the distance between two arbitrary 
points ~1 and ~2 remains unchanged 
]P(t, ~1) - ?(t, ~2)1 = I r0 (~ l ) - P0(~2)l. 
For any rigid motion, there are a constant vector k~, a vector function R* (t) and an 
orthogonal tensor function O(t) such that for any point ~ and for any instant t, 
?(t, ~) = R*(t) + [?0(~) - k~]-0(t). (1.1.66) 
Differentiation of Eq. (1.1.66) with respect to ~i with the use of Eq. (1.1.5) yields 
gi(t) = goi " O(t). 
Combining this equality with Eqs. (1.1.23) and (1.1.27), we obtain 
C7o~ = O, v ~o = 0 T, CTor~ = 0 - i, ~ r~ = i - O r. 
Substitution of these expressions into Eqs. (1.1.34), (1.1.36), (1.1.39), (1.1.40), 
(1.1.42), (1.1.43), (1.1.48) to (1.1.51) implies that 
~,=b =k=Po =i, ~ = ~=b~ =b~0 =o, 
~0= ~(o+0T) - - i#0 , ~=i - -~(0+0T)#0. (1.1.67) 
It follows from Eqs. (1.1.67) that the strain tensors ~, 6",/~F, and/~Fo adequately 
describe deformation of continua, since they vanish for any rigid motion. The in- 
finitesimal strain tensors ~0 and ~ do not vanish for rigid motions with nonzero 
rotations. 
1.1. Basic Definitions and Formulas 13 
1.1.9 Genera l ized Strain Tensors 
Four strain tensors A, ~7, EF, and EF0 are expressed in terms of the deformation 
tensors ~,, ~'0, F, and F0. It is natural to generalize this approach and to treat any 
admissible tensor function Z of the deformation tensors as a generalized measure of 
strains. Several generalized strain tensors are introduced by Seth (1964). Conditions 
of admissibility for generalized strain tensors are formulated by Hill (1968): 
1. Generalized strain tensors L coincide for two motions which differ from each 
other by a rigid motion. 
2. Generalized strain tensors Z vanish for the identical transformation from the initial 
to the actual configuration, L(0) = 0. 
3. L is a tensor of the second rank. 
4. Z is an isotropic invertible tensor function of either the Cauchy deformation tensor 
~, or the Finger deformation tensor F. 
5. The derivative of a generalized strain tensor Z with respect to the strain tensor 
vanishes in the initial configuration, Lele=0 = 0. 
6. For infinitesimal strains, a generalized strain tensor Z coincides with the first and 
second infinitesimal strain tensors. 
^ 
7. The eigenvalues of a generalized strain tensor L are positive, provided the corre- 
sponding principal stretches are greater than unity. 
Two generalized strain tensors are widely used in applications: 
1. The Eulerian m-tensor of strains 
1 (Din~2 ~), m 4: 0, 
e(em ) = m 
1 
= ln~,, m = O, 
2 ~ 
2. The Lagrangian m-tensor of strains 
E(L m) = 1 j _ p-m/2) , 
~lnP , 
m4:0 , 
m--O. 
(1.1.68) 
(1.1.69) 
The following formulas for the fractional powers of the Cauchy and Finger tensors 
are valid: 
3 ~2 _ (I1 -- v2)g + 13 Vk21 
~m/2 = ~ V~ 2 4 2 
k=l Ulc -- ll v[c + 13 Vk 2 
3 p2 (I1 2 " m -- -v~)F+I3vE2 i 
pm/2 = ~ Vi~ 2 4 2 
k=l V~: -- Il V ~ + 13 Vk 2 
(1.1.70) 
14 Chapter 1. Kinematics of Continua 
where vk are the principal stretches and Ik are the principal invariants of the Cauchy 
and Finger deformation tensors [see Morman (1986)]. 
1.1.10 Volume Deformation 
It follows from Eq. (1.1.41) that 
,, - i - j k j - i - 
g = gijgogo = gikgo gogo j, 
which implies that 
[ koj ] det[gik] _ g (1.1.71) I3(g') -- det ~ = det gikg -- det[gik] det [golj] -- det[g0 kj] go' 
where go = det[g0 ik] and g = det[gik]. 
Volume V of a parallelepiped erected on the tangent vectors gi is calculated using 
the formula for the triple product of vectors and Eq. (1.1.6) 
. [glxl g lx 2 glx3] 
V= det |g2xl g2x2 g2x3 , (1.1.72) 
I_g3x 1 g3x 2 g3x 3 
where gixJ are projections of the tangent vectors gi on Cartesian axes x j. Multiplying 
Eq. (1.1.72) by itself, we obtain 
r ax ,ix] r, xl , xl 3xl] 
V 2 = det [g2x 1 g2x 2 g2x 3 [glx 2 g2x 2 g3x 2 
[.g3x~ g3x2 g3x3 I_g lx 3 gzx 3 g3x 3 
[gl" gl gl'g2 gl" g3] [gll g12 g13] 
-- det g2 gl g2"g2 g2"g3 = det /gEl g22 g23 
g3 gl g3 "g2 g3 "g3 [.g31 g32 g33 
It follows from this equality that 
=g. 
V = ~. (1.1.73) 
Denote by dVo and dV the volume elements (volumes of the elementary paral- 
lelepipeds erected on the tangent vectors g0i and gi with sides d~ i) in the initial and 
actual configurations. It follows from Eq. (1.1.73) that 
dVo = v~d~ 1 d~ 2 d~ 3, dV = v~d~ 1 d~ 2 d~ 3. (1.1.74) 
Substitution of expression (1.1.71) into Eqs. (1.1.74) with the use of Eq. (1.1.45) 
implies that 
dv _ U - - 1175 
dVo go 
1.1. Basic Definitions and Formulas 15 
A medium is called incompressible provided its volume element remains unchanged 
under deformation. According to Eq. (1.1.75), the incompressibility condition reads 
I3(F) = 1. (1.1.76) 
1.1.11 Deformation of the Surface Element 
Let us consider a rectangular surface element in the initial configuration erected on 
vectors d ?d and d ?ff (see Figure 1.1.4). Denote by h0 the unit normal vector to the 
surface element and by dSo its area 
ho dSo = d ?d × d ?d I. (1.1.77) 
In the actual configuration, vectors d ?d, d ?if, and ho are transformed into d 7/, d ?", 
and h, and Eq. (1.1.77) is written as 
hdS = d? I × d? n. (1.1.78) 
The surface elements in the initial and actual configurations obey the equality [see, 
e.g., Drozdov and Kolmanovskii (1994)], 
(1.1.79) 
We multiply Eq. (1.1.79) by itself, use the condition h • h = 1, and
obtain 
dS 2 = g ~o" ¢ ~ " ¢ ~o" nodS~. 
go 
This equality together with Eqs. (1.1.43) and (1.1.44) implies that 
dS 
dSo 
• h0) l /2 . (1.1.80) 
hdS 
d? n 
y dS 
d? I 
.// 
Figure 1.1.4: The surface element. 
16 Chapter 1. Kinematics of Continua 
To derive another expression for this ratio, we present Eq. (1.1.79) as 
hodSo = ~r0? • hdS = n. fTo~ rdS. (1.1.81) 
We multiply Eq. (1.1.81) by itself, use Eqs. (1.1.42), (1.1.44), and the condition 
h0 • h0 = 1, and find that 
dS (h. F. n) 1/2 -- (n. go 1. ~)1/2 (1.1.82) 
Equations (1.1.47), (1.1.80), and (1.1.82) demonstrate the geometrical meaning 
of deformation tensors. The Cauchy tensor ~, and the Almansi tensor ~0 determine 
changes in the arc element, whereas the Finger tensor P and the Piola tensor F0 
characterize changes in the area element for transition from the initial to the actual 
configuration. 
1.1.12 Objective Tensors 
Let us consider two motions of a continuum. At the current instant t, the radius-vector 
of a point ~ equals 
= R(t, ~) (1.1.83) 
in the first motion, and equals 
F = R'(t, ~) (1.1.84) 
in the other motion. We choose an arbitrary point P as a pole and denote by/~0 and 
k~ radius vectors of the point P in the first and second motions, respectively (see 
Figure 1.1.5). The relative radius vectors (with respect to the pole) are denoted as/~ 
and/~/: 
= R0 + R, R '= k~ + R'. (1.1.85) 
The motions R and R~ differ from each other by a rigid motion, provided that 
for any point ~ and for any instant t 
]/~(t)] = ] R(t)[. (1.1.86) 
Equality (1.1.86) implies that there exists an orthogonal tensor function t) = O(t) 
such that for any point ~ and for any instant t, 
k ~ =/~. 0. (1.1.87) 
Substitution of expression (1.1.87) into Eq. (1.1.85) yields 
Rt(t, ~) = k~(t) + [R(t, ~) - R0(t)]" O(t). (1.1.88) 
1.1. Basic Definitions and Formulas 17 
79 
M M 
79 
O 
Figure 1.1.5: Kinematics of two rigid motions. 
Equation (1.1.88) determines a rigid motion superimposed on the motion 
(1.1.83). We differentiate Eq. (1.1.88) with respect to ~i to obtain the formula for the 
tangent vectors 
~(t , ~) = gi(t, ~) " O(t) = 0 T (t) " gi(t, ~). (1.1.89) 
The dual vectors ~i and ~i! satisfy the equality 
~it(t ' ~) = ~i(t , ~) " O(t) = 0 T( t ) " ~i(t , ~). (1.1.90) 
A vector field 
77 = qi gi, ~, = qi ,~, (1.1.91) 
frozen into a deformable medium, is called objective (indifferent with respect to rigid 
motions) if 
qit = qi. (1.1.92) 
It follows from Eqs. (1.1.89) and (1.1.92) that an objective vector satisfies the condi- 
tion 
~! = qi(~)T " gi) -- oT " q = q" O. (1.1.93) 
By analogy with Eq. (1.1.92), a tensor field of the second rank 
~_~ = QiJ~i~j, Ot i j l - I - I = Q gig j, (1.1.94) 
18 Chapter 1. Kinematics of Continua 
frozen into a deformable medium, is called objective (indifferent with respect to rigid 
motions) if 
Qijt = Qij. (1.1.95) 
Using Eq. (1.1.89), formula (1.1.95) can be also presented in the invariant form 
01= aiJ (OT . ~i) (~j . O) "- 0 T" O" O. (1.1.96) 
It follows from Eq. (1.1.96) that the product P • (~ of two objective tensors P and (~ 
is an objective tensor as well. 
1.1.13 Ve loc i ty Vector and Its Grad ient 
The velocity vector ~ is defined as the derivative of the radius vector ~ with respect 
to time 
0 = at(t,~). (1.1.97) 
Applying the Hamilton operator to the velocity vector ~, we obtain the velocity 
gradient ~7 ~. This tensor is expanded into the symmetrical and skew-symmetrical 
components 
~r~ = b - t', (1.1.98) 
which are called the rate-of-strain tensor 
1 
D = -~(f7 9 T + f7 ~), (1.1.99) 
and the vorticity tensor 
1 
~" = ~(~79T -- ~'9). (1.1.100) 
The rate-of-strain tensor [ is connected with the relative Finger deformation tensor 
F<>(t, r) by the formula [see, e.g., Drozdov (1996)], 
10 
D(t) = -~ F<>(t, T) . (1.1.101) 
T=t 
Denote by ~ and ~/the velocity vectors corresponding to the motions (1.1.83) 
and (1.1.84) 
- a t ' v = O---t-" (1.1.102) 
We transform the latter formula in Eq. (1.1.102) with the use of Eq. (1.1.88) 
0/ d/~ ( d/~o~ dO 
- + ~ • 0 + (R - /~o)" (1.1 103) 
dt \ - - -~- / -dt" " 
1.1. Basic Definitions and Formulas 19 
For any orthogonal tensor O, 
dO dO T 
- -O . • O. (1.1.104) 
dt dt 
It follows from Eqs. (1.1.88), (1.1.103), and (1.1.104) that 
dko~ dO T 
_ , _ dk~ + ~- "0 - (R - [~o) "O" dt "0 dt -Yi-) 
(d ido) , , dO T 
_ ddt [~ + fp - - -~ "0 - ( -k~)" dt "O" (1. .1 105) 
The skew-symmetrical tensor 
fi = dOT • 0 (1.1.106) 
dt 
is called spin tensor. According to Eqs. (1.1.104) and (1.1.106), tensors 0 and 0 T 
satisfy the differential equations 
dO 0 ~, dOT 
. . . . f t . 0 T. (1.1.107) 
dt dt 
It follows from Eqs. (1.1.105) and (1.1.106) that 
( dko) + 
dt -~- J 
( dko) 
- + a - • 0 + 1~. (R ' - / ?~) . (1.1 108) 
dt --d~ j 
To calculate the covariant derivative of the vector 9 ~, we use Eqs. (1.1.5), (1.1.7), 
(1.1.89), (1.1.90), and (1.1.108) 
fT ' TJ ' = g- i t a ~l l - O T " g, i a ~) t -- O T " g, i ( O -~ - -~ "0 - - a ~t~ t ~ i . f i 
: OT . (~, i ~)~) " ) 0 T " - -~ " 0 -- ~t~,[. f i __ • (V~) " O -- g, tg, i • O" f i ) 
= O T. fT~. 0 - 0 T ' i " (9. fi = 0 T. fT~. O- ft. (1.1.109) 
It follows from Eqs. (1.1.108) and (1.1.109) that the velocity vector 9 and its gradient 
~ are not objective. 
We calculate the rate-of-strain tensor b and the vorticity tensor Y according to 
Eqs. (1.1.99), (1.1.100), and (1.1.109) and find that 
D' -- O r . D. O, Y' - 0 T . ~" . 0 + 19,. (1.1.110) 
20 Chapter 1. Kinematics of Continua 
It follows from Eq. (1.1.110) that the rate-of-strain tensor/3 is objective, whereas the 
vorticity tensor t" is not indifferent with respect to rigid motions. Resolving the latter 
equation in Eq. (1.1.110) with respect to the spin tensor ~, we obtain 
h= ? ' -b r . ? .b . (1.1.111) 
1.1.14 Corotational Derivatives 
In the constitutive theory for inelastic media, corotational derivatives serve as analogs 
of material time derivatives (which are not indifferent with respect to rigid motions). 
Corotational derivatives are defined as linear operators, which transform an objective 
tensor into an objective tensor of the same range. 
The Jaumann Derivative Let gt(t, {~) be an objective vector field that satisfies Eq. 
(1.1.93). The derivative of ?:/with respect to time at fixed Lagrangian coordinates {~ 
is called the material derivative. Differentiation of Eq. (1.1.93) with the use of Eq. 
(1.1.107) yields 
= ~ dO a?::/ ,, aq' aq .o+q . . . . O -q .b . f~ . 
at at dt at 
Substitution of expressions (1.1.93) and (1.1.111) into this equality implies that 
- 0- 0. 0 o) 
at at 
= a~.O_q .O.? ,+q.? .O 
at 
= a~.o -O, . ? ,+q.? .O . 
at 
It follows from this equality that 
~?/~+?/~.~,~= (a?/ ) 
a-7 -a-7 + 0. ? .0, 
which means that the vector 
Z/~ = aZ/ + ~/" ~, (1.1.112) 
at 
is indifferent with respect to rigid motions. The vector g/<> is called the Jaumann 
derivative of an objective vector g/. 
Employing similar reasoning, the Jaumann derivative may be defined for an 
objective tensor of the second rank (~. Using Eqs. (1.1.96), (1.1.107), and (1.1.111), 
1.1. Basic Definitions and Formulas 21 
and repeating the preceeding calculations, we find that 
aO.' d b ~ aO_. d b 
• 0_" b + b ~ • ~. b + b ~" (2" 
at dt at dt 
= h. b ~. O. b + b ~. aa~. b - U . O b. fi 
at 
= (? ' - U . ? . b)" b ~" (2" b + b ~" a/_OO, b 
at
- b ~ " O_ " b" (?' - b ~ " ? . b) 
= ?"O_ ' -U '? 'O_ 'O+O ~. aO. .b 
at 
-O'.~'+b~.O.~.b. 
Therefore 
aQ~-Y~'Q~+Q~'~'~=br'at ( aO- - ~ " O" + . t ' ) . b, 
which implies that the Jaumann derivative of an objective tensor 
0< > _ aQ + Q" }" - ~'" 0 (1.1.113) 
at 
is indifferent with respect to superposed rigid motions. 
The Oldroyd Derivatives It follows from Eqs. (1.1.98) to (1.1.100) that 
~" = b - ~7 ~, t" = -D + ~7 ~T. (1.1.114) 
Let ~/be an objective vector field with the Jaumann derivative (1.1.112). Substitution 
of the first expression (1.1.114) into Eq. (1.1.112) yields 
q+ = a~ +~.b-q .#9 = qa +q.b , 
at 
(1.1.115) 
where 
F/A _ OF:/ _ Z/. ~'~. (1.1.116) 
at 
It follows from Eq. (1.1.115) that qA is an objective tensor, since ~/, D, and ?/<> are 
objective vectors and tensors. It is called contravariant (upper) convected derivative 
of a vector q [see Oldroyd (1950)]. 
22 Chapter 1. Kinematics of Continua 
Let us now consider an objective tensor field (~ with the Jaumann derivative 
(1.1.113). Substitution of the first expression (1.1.114) into Eq. (1.1.113) yields 
0o _ 0(~ + (~. b - D -0 - (~" ~7~ + ¢~. (~. (1.1.117) 
0t 
It follows from Eq. (1.1.99) that 
V0 = 2 /3 - V0 r. 
Substituting this expression into Eq. (1.1.117), we obtain 
0_<>_ OQ 
ot 
+O.b-O.eo+b.O. -eor .o 
=0 A +O.b+b.O, (1.1.118) 
where the objective tensor 
^ 
OA _ 0Q _ Q" V0 - V0 T. 0 (1.1.119) 
Ot 
is the contravariant (upper) convected derivative of an objective tensor Q. 
Using the latter expression in Eq. (1.1.114) and repeating similar transformations, 
we introduce the covariant (lower) convected derivatives of an objective vector g:/and 
an objective tensor (~ [see Oldroyd (1950)], 
g/v = Og/ + ~/" ¢~,r, ~)v _ 00 + 0" l~'vr + ~'0" Q. (1.1.120) 
0t 0t 
Spriggs et al. (1966) introduced the generalized corotational derivative of an 
objective tensor Q of the second rank 
0 D = 0 ° + al(Q" b + D. (2) + a2(Q"/3)i + a3Ii(0)D, (1.1.121) 
where ak (k = 1, 2, 3) are arbitrary constants. 
^ 
For any tensor Q, the Jaumann and the Oldroyd corotational derivatives are 
particular cases of the generalized corotational derivative (1.1.121). The Jaumann 
derivative corresponds to the case al = a2 = a3 = 0; the upper Oldroyd derivative 
corresponds to the case al = -1 , a2 = a3 -- 0; and the lower Oldroyd derivative 
corresponds to the case a l = 1, a2 = a3 = 0. 
1.1.15 The R iv l in -Er i cksen Tensors 
The Oldroyd covariant derivatives of the rate-of-strain tensor b were introduced by 
Oldroyd (1950) and used by Rivlin and Ericksen (1955) to construct constitutive 
models for viscoelastic fluids. The tensors, known as the Rivlin-Ericksen tensors, are 
Bibliography 23 
determined by the formulas 
A0 = i, ~ = 2b = Vo + Vo T, 
^ aAn 
an+l = An v - + An" ~,oT + ~,~. ,21n (n = 1,2 .. . . ). (1.1.122) 
Ot 
Instead of the Oldroyd covariant derivatives, other corotational derivatives may 
be used. For example, employing the Oldroyd contravariant derivatives, we arrive at 
the White-Metzner tensors [see White and Metzner (1963)], 
/~0 = -L /~1 = 2D = ~r ~ + ~ oT, 
^ a/~n Bn+l =/Tn A -- --/Tn" ~'~ -- ~roT. /Tn (n = 1,2 .. . . ). (1.1.123) 
at 
The Rivlin-Ericksen tensors (1.1.122) were used by Coleman et al. (1966), the White- 
Metzner tensors (1.1.123) were employed by Astarita and Marrucci (1974) and Huil- 
gol (1979) to develop constitutive equations in the nonlinear mechanics of continua. 
Bibliography 
[1] Astarita, G. and Marrucci, G. (1974). Principles of Non-Newtonian Fluid Me- 
chanics. McGraw-Hill, London. 
[2] Coleman, B. D., Markowitz, H., and Noll, W. (1966). Viscometric Flows of 
Non-Newtonian Fluids. Springer-Verlag, New York. 
[3] Coleman, B. D. and Noll, W. (1961). Foundations of linear viscoelasticity. Rev. 
Modem Phys. 33, 239-249. 
[4] Drozdov, A. D. (1996). Finite Elasticity and Viscoelasticity. World Scientific, 
Singapore. 
[5] Drozdov, A. D. and Kolmanovskii, V.B. (1994). Stability in Viscoelasticity. 
North-Holland, Amsterdam. 
[6] Fitzgerald, J. E. (1980). Tensorial Hencky measure of strain and strain rate for 
finite deformations. J. Appl. Phys. 51, 5111-5115. 
[7] Hill, R. (1968). On constitutive inequalities for simple materials. J. Mech. Phys. 
Solids 16, 229-242. 
[8] Huilgol, R. R. (1979). Viscoelastic fluid theories based on the left Cauchy-Green 
tensor history. Rheol. Acta 18,451-455. 
[9] Morman, K. N. (1986). The generalized strain measure with application to 
nonhomogeneous deformations in rubber-like solids. Trans. ASME J. Appl. 
Mech. 53,726-728. 
24 Chapter 1. Kinematics of Continua 
[10] Oldroyd, J. G. (1950). On the formulation of rheological equations of state. 
Proc. Roy. Soc. London A200, 523-541. 
[ 11 ] Rivlin, R. S. and Ericksen, J. L. (1955). Stress-deformation relations for isotropic 
materials. J. Rational Mech. Anal. 4, 323-425. 
[12] Seth, B. R. (1964). Generalized strain measure with applications to physical 
problems. In Second-Order Effects in Elasticity, Plasticity and Fluid Dynamics. 
(M. Reiner, and D. Abir, eds.), pp. 162-172. Pergamon Press, Oxford. 
[13] Spriggs, T. W., Huppler, J. D., and Bird, R. B. (1966). An experimental appraisal 
of viscoelastic models. Trans. Soc. Rheol. 10, 191-213. 
[14] White, J. L. and Metzner, A. B. (1963). Development of constitutive equations 
for polymeric melts and solutions. J. Appl. Polym. Sci. 7, 1867-1889. 
Drozdov AD - Viscoelastic structures - 1998/Chapter-2-Constitutive-Models-in-Linear-Viscoelasticity_1998_Viscoelastic-Structures.pdf
Chapter 2 
Constitutive Models in 
Linear Viscoelasticity 
This chapter is concerned with constitutive relations for linear viscoelastic media 
with infinitesimal strains. The viscoelastic behavior is typical of a number of mate- 
rials that are extremely important for applications: polymers and plastics [see, e.g., 
Aklonis et al. (1972), Bicerano (1993), Ferry (1980), Struik (1978), Tschoegl (1989), 
Vinogradov and Malkin (1980), and Ward (1971)], composites [see, e.g., Alberola 
et al. (1995)], metals and alloys at elevated temperatures [see, e.g., Skrzypek (1993) 
and Szczepinski (1990)], concrete [see, e.g., Bazant (1988) and Neville et al. (1983)], 
soils [see, e.g., Adeyeri et al. (1970)], road construction materials [see, e.g., Mac- 
carrone and Tiu (1988) and Vinogradov et al. (1977)], building materials [see, e.g., 
Papo (1988)], biological tissues [see, e.g., Deligianni et al. (1994)], food-stuffs [see, 
e.g., Robert and Sherman (1988) and Struik (1980)], etc. 
Section 2.1 deals with linear differential and fractional differential models. In 
section 2.2, we discuss integral models and introduce the concept of adaptive links. 
Section 2.3 is concerned with properties of creep and relaxation measures. Basic ther- 
modynamic potentials for an aging viscoelastic medium are derived in Section 2.4. A 
new constitutive models for an aging viscoelastic material is developed in Section 2.5. 
2.1 Differential Constitutive Models 
In this section, a brief survey is provided of differential and fractional differential 
models in linear viscoelasticity. For simplicity, we confine ourselves to uniaxial 
loading. 
25 
26 Chapter 2. Constitutive Models in Linear Viscoelasticity 
2.1.1 Differential Constitutive Models 
We begin with differential constitutive models, where the stresses and strains at the 
current instant t are connected by linear differential equations with constant coeffi- 
cients. Any differential model for the viscoelastic behavior reflects some rheological 
model, where elastic elements (springs) and viscous
elements (dashpots) are con- 
nected in series and in parallel. 
A linear spring obeys Hooke's law 
o- = EE, (2.1.1) 
where o" is the stress, E is the strain, and E is Young's modulus. A linear dashpot 
obeys Newton's law 
dE 
~r = r/~-, (2.1.2) 
where r/is the Newtonian viscosity. 
To construct more complicated rheological models, the basic elements are con- 
nected in series or in parallel. To derive differential equations for a rheological model, 
the following rules are employed: 
• For elements connected in parallel, their strains coincide, and the total stress equals 
a sum of stresses in individual elements. 
• For elements connected in series, their stresses coincide, and the total strain equals 
a sum of strains in individual elements. 
The Maxwell model consists of a spring and a dashpot connected in series (see 
Figure 2.1.1). The constitutive equation of the Maxwell model reads 
dE 1 do- o- 
- + --. (2.1.3) 
dt E dt rl 
I I 
i 
> 
< -> 
Figure 2.1.1: The Maxwell model. 
2.1. Differential Constitutive Models 27 
% iII 
Figure 2.1.2: The Kelvin-Voigt model. 
The Kelvin-Voigt model consists of a spring and a dashpot connected in parallel 
(see Figure 2.1.2). This model obeys the constitutive relation 
dE 
cr = EE + r/-w. (2.1.4) 
at 
To generalize these models, we combine springs and dashpots, as well as the 
Maxwell and Kelvin-Voigt elements, in parallel and in series. Several Maxwell 
elements connected in parallel provide the generalized Maxwell model (the Maxwell- 
Weichert model). Several Kelvin-Voigt elements connected in series provide the 
generalized Voigt model. A model consisting of a Maxwell element in parallel with 
a spring is called the standard viscoelastic solid (the Zener model) (see Figure 2.1.3). 
Its constitutive equation reads 
1 ( 
+-or = 1 + - - +- -e . (2.1.5) 
E~ at ~ ~ dt 
Here E1 is Young's modulus of the spring and E2 and r/are Young's modulus and the 
Newtonian viscosity of the Maxwell element. 
Figure 2.1.3: The standard viscoelastic solid. 
28 Chapter 2. Constitutive Models in Linear Viscoelasticity 
The differential constitutive equation (2.1.5) with the zero initial conditions is 
equivalent to the Volterra integral constitutive equation 
[ /o t or(t) = E e(t) + Qo(t - T)E(r) d . (2.1.6) 
Here 
E=EI +E2 
is Young's modulus and 
t 
Qo(t) = -x [1 - exp ( -T ) I 
is the relaxation measure with the parameters 
(2.1.7) 
E2 ~/ 
X = T - 
El +E2' E2 
Combining basic rheological elements, an arbitrary number of differential con- 
stitutive models can be designed. Some of these models are equivalent to each other, 
the others are independent [see Giesekus (1994)]. The constitutive equation of a 
rheological model with an arbitrary number of springs and dashpots reads 
do" dno " dE dmE 
ao~r + al dt + "'" + an- -~ -- boe + bl--d- [ + "'" + bm dt--- W, (2.1.8) 
where m and n are positive integers and ak and bl are adjustable parameters. 
2.1.2 Fractional Differential Models 
Fractional differential models (that are "transient" between differential and integral 
models) obey linear differential equations with constant coefficients and fractional 
derivatives. A fractional derivative is a Volterra operator with the Abel kernel 
t o~ 
J~ (t) = F(1 + a)" (2.1.9) 
Here a is a real parameter, and F(z) is the Euler gamma-function of a complex 
variable z 
fo c° F(z) = t z- 1 exp(- t) dt. (2.1.10) 
Fractional differential constitutive models in linear viscoelasticity were proposed by 
Bagley and Torvik (1983a,b, 1986) and Rogers (1983), and were discussed in detail 
by Bagley (1987), Ffiedfich (1991a,b), Glockle and Nonnenmacher (1991, 1994), 
Heymans and Bauwens (1994), Koeller (1984), Nonnenmacher and Glockle (1991), 
2.1. Differential Constitutive Models 29 
Suarez and Shokooh (1995), and Tschoegl (1989). For a detailed exposition of the 
theory of fractional derivatives see, e.g., Miller and Ross (1994), Oldham and Spanier 
(1974), and Srivastava and Menocha (1984). 
Fractional differential models 
• Describe adequately the creep and relaxation processes in polymeric materials 
by means of simple relationships between stresses and strains with relatively 
small number of adjustable parameters [see, e.g., Bagley and Torvik (1983a) and 
Heymans and Bauwens (1994)]. 
• Reflect correctly the effect of vibrational frequency on the viscoelastic response 
in damping materials [see, e.g., Bagley and Torvik (1983b), Nashif et al. (1985), 
and Suarez and Shokooh (1995)]. 
• Lead to the non-Debay relaxation behavior [see, e.g., Glockle and Nonnenmacher 
(1993), Ngai (1987), and Nonnenmacher and Glockle (1991)]. 
• Take into account stochastic (micro-Brownian) motion of chain molecules at the 
microlevel in the phenomenological description of viscoelastic media [see Bagley 
and Torvik (1983a)], and provide a basis for application of fractal rheological 
models in viscoelasticity [see Heymans and Bauwens (1994)]. 
• Allow some interesting phenomena to be analyzed arising in the theory of wave 
propagation in media described by hyperbolic-parabolic equations [see e.g., Choi 
and MacCamy (1989) and Hrusa and Renardy (1985)]. 
Let f ( t ) be a sufficiently smooth function that vanishes in the interval ( -~ , 0] 
and that is integrable in [0, T] for an arbitrary T > 0. The primitive Fa(t) for the 
function f ( t ) equals 
f0 t Fl(t) = f ( s )ds . 
The second primitive Fz(t) of f ( t ) is calculated as 
/0 t /0 • /0 t F2(t) = d~" f ( s ) ds = (t - s ) f (s ) ds. 
Similarly, the nth primitive of f ( t ) reads 
1 /0t F,(t ) = (n - 1)! (t - - S ) n - 1 f ( s ) ds, (2.1.11) 
where n! = 1.2 . • • n. Since F(n) = (n - 1)! for any positive integer n, Eq. (2.1.11) 
is presented as 
fo 
t 
Fn(t) = Jn-1 (t - s ) f (s ) ds. 
30 Chapter 2. Constitutive Models in Linear Viscoelasticity 
According to Eq. (2.1.11), the fractional operator 
f0 t F~(t) = Ja - l ( t - s ) f ( s )ds (2.1.12) 
is reduced to the standard operator of integration for a positive integer c~. It is 
convenient to rewrite Eq. (2.1.12) as 
/0 t /0 F~(t) = J~- l ( s ) f ( t - s)ds = J~- l ( s ) f ( t - s)ds. (2.1.13) 
Equation (2.1.12) determines the fractional operator F~(t) for an arbitrary a > 0. 
For a E (0, 1), the special notation is used [see, e.g., Glockle and Nonnenmacher 
(1991, 1994) and VanArsdale (1985)], 
D-~ f ( t ) = F~(t) = J~- l ( t - s ) f (s )ds . (2.1.14) 
To define the function F~ (t) for an arbitrary negative c~, we employ the formula 
d n 
dtnJ~+n(t) = J~(t), (2.1.15) 
which is satisfied for any real a and for any positive integer n. Let us consider the 
functional 
la(th) = Ja_l(S)dp(s)ds. (2.1.16) 
For ct > 0, the integral in Eq. (2.1.16) converges for any continuous function th(t) 
such that 
4~(0) = 0, 14~(t)l dt < ~. 
To define the functional I,~(~b) for c~ E ( -n , -n + 1), we assume that ~b(t) has n 
continuous derivatives which vanish at t = 0. In this case, Eq. (2.1.16) is integrated 
n times by parts with the use of Eq. (2.1.15) to obtain 
I~(dp) = J~-l(S)dp(s)ds 
/o = J~(s)4)(s)lo - J~(s s) ds 
ds (s) (s) ds 
~_. * * , 
o/0 "~ d~d~ = ( -1 ) ~ J~+n- l (S ) - -~(s )as . (2.1.17) 
as ,~ 
2.1. Differential Constitutive Models 31 
Since the integral in the right side of Eq. (2.1.17) converges, the functional I~(th) can 
be defined for a E ( -n , -n + 1) according to formula (2.1.17). Returning to the 
function f it) , which vanishes at t = 0 with its derivatives, we set [see Eq. (2.1.13)] 
fo fo' F~(t) = Ja+n-l(S)f(n)(t -- s)ds = Ja+n-l(S)f(n)(t -- s)ds,
(2.1.18) 
provided a E ( -n , -n + 1). Here n is a positive integer and 
dnf 
f(n)(t) = -~( t ) . 
The function F~ (t) is defined now for any real a except for nonpositive integers. 
To define it on the whole real axis, we set 
F-n(t) = lim Fa(t). 
t2g.--.* - - n - - O 
Substitution of expressions (2.1.9) and (2.1.18) into this equality implies that 
f0 t F-n(t) = lim Ja+n(S)f (n+ l)(t - s) ds a--,-n-O 
lim foo t sa+n f(n+l)(t _ S) ds 
a---*-n-0 F(1 + c~ + n) 
fo t f(n+ 1)(t s) ds f(n)(t). (2.1.19) 
It follows from Eq. (2.1.19) that for any positive integer n, function F-n(t) determines 
the nth derivative of f(t). 
This concept may be extended to an arbitrary negative real c~. In particular, the 
fractional derivative of the order a ~ [0, 1) is defined as follows: 
daf 
f{a}(t) = d---~(t) = F_a(t) 
fO t df _ ds = J_~(s)-d- i (t S) 
/o' = J_~( t - s) (s) ds 
1_ a) fOO t °~ d fd t = (t - s)- -c(s)ds. 
F(1 
(2.1.20) 
It is of interest to establish a correspondence between the fractional derivative 
f{~}(t) and the fractional operator D -~ f(t). For this purpose, we integrate expression 
(2.1.14) by parts and use the initial condition f(0) = 0. As a result, we find that 
32 Chapter 2. Constitutive Models in Linear Viscoelasticity 
1 fOt f(s) 
D-(1 -~) f ( t ) = F(1 - a) (t - s) ~ ds 
1 [ 
(1 - a )F (1 - a) - ( t - s)l-af(s) /0 t ] s=t + (t -- S) 1-a df s=0 -ji (s) & 
1 /0 t (1 - a)F(1 - a ) (t -- S) 1-a (s)ds. (2.1.21) 
Differentiation of Eq. (2.1.21) with respect to time implies that 
d (1 1 fot ad f 
dt D- -~) f ( t ) = F(1 - a ) ( t - s ) - -d-~(s)ds. 
Finally, combining this equality with Eq. (2.1.20), we obtain 
d~f d (1 
dt ~ (t) - -~V- -~) f ( t ) . (2.1.22) 
Equation (2.1.2) determines a Newtonian dashpot, where the stress is propor- 
tional to the first derivative of the strain. A natural generalization of this rheological 
element is a fractional dashpot with the constitutive equation 
daE 
or = rl dt ~ , (2.1.23) 
which is characterized by two material parameters: a E (0, 1) and ~. The limiting 
cases correspond to the Hookean spring (a = 0) and to the Newtonian dashpot 
(a = 1). 
Using the fractional dashpot, we can construct analogs of the Maxwell model 
(2.1.3): 
d~or E d~e 
+ -or = E~ (2.1.24) 
dt ~ rl dt ~ ' 
of the Kelvin-Voigt model (2.1.4): 
daE 
or = Ee + ~ dt ~ , (2.1.25) 
and of the Zener model (2.1.5): 
d '~or ~- 1 E d'~e E1 
- --or = + ~e. (2.1.26) 
dt ~ T dt ~ T 
Equation (2.1.26) is easily generalized by introducing derivatives of different frac- 
tional orders. For example, the following equation may be proposed for the standard 
viscoelastic solid: 
d~ tr 
dt a 
1 d/3a E1 
+ ~or= Ed- ~ + Te, (2.1.27) 
2.1. Differential Constitutive Models 33 
where a,/3, E, El, and T are adjustable parameters. However, not all extensions of 
the Zener model are thermodynamically admissible [see a discussion of this question 
in Friedrich (1991a)]. 
Models with fractional springs and dashpots permit experimental data in dy- 
namic tests to be predicted adequately for a number of polymeric materials. To 
demonstrate fair agreement between results provided by the Zener model (2.1.26) 
and observations, we consider steady uniaxial oscillations of a viscoelastic specimen. 
In accordance with Burton (1983), to derive an equation for steady oscillations, we 
have to replace zero as the lower limit of integration in Eq. (2.1.20) by -~. As a 
result, we obtain the constitutive equation 
1 [ ; (t - ~d° + or(t) 
F(1 - a ) J _ s ) -~ dt (S)ds o~ T 
/: E (t - s) -'~ de E1 e(t) = - : - ( s )ds + ~ . (2.1.28) F(1 og) o~ t i t - - T I 
As common practice, we seek solutions of Eq. (2.1.28) in the form 
o-(t) = o0 exp(~wt), E(t) = E0 exp(w~t), (2.1.29) 
where o'0 and e0 are the amplitudes of oscillations to be found, ~o is the frequency of 
oscillations, and ~ = ~ 1. Substituting expressions (2.1.29) into Eq. (2.1.28) and 
introducing the new variable ~- = t - s, we find that 
E /0 1 too ~'-'~ exp(-~to~')d~" + o0 F(1 - a) 
= eo F(1 - a) ~'-'~ exp( - ~or) d~- + . 
Calculation of the integrals with the use of Eq. (2.1.10) implies that 
E*(w) - oo _ E1 + ET( rw) '~ (2.1.30) 
~0 1 + T(~w) '~ ' 
where 
E*(og) = E'(w) + tE"(w) (2.1.31) 
is the complex Young's modulus. Combining Eqs. (2.1.30) and (2.1.31), we obtain 
expressions for the storage modulus E'(oJ) and the loss modulus E'(o~) 
[El + Eo9 a cos(Tra/2)][1 + to '~ cos(ara/2)] + Eto 2a sin2(ara/2) 
E'(to) = 
E'(to) = 
[1 + to a cos('n'a/2)] 2 + o) 2c~ sin2(Tra/2) 
(E - E1)o9 c~ sin(ara/2) 
[1 + to c~ cos('n'a/2)] 2 + ~2c~ sin2(,n.c~/2) • 
(2.1.32) 
34 Chapter 2. Constitutive Models in Linear Viscoelasticity 
4.0 
logE ~ 
logE" 
1.0 
0 0 0 
I I I I I I I I I 
-2.0 log to 7.0 
Figure 2.1.4: The storage modulus E ~ (MPa) and the loss modulus E" (MPa) versus 
frequency to (Hz) of steady oscillations for poly(methyl methacrylate) (PM MA). 
Circles show experimental data obtained by Rogers (1983): unfilled circles: E'; filled 
circles: E'. Solid lines show prediction of the fractional Zener model (2.1.26) with 
c~ = 0.1946, E = 6354.0 MPa, E1 = 2130.0 MPa, and T = 0.8234 sec ~. 
Experimental data for PMMA and the dynamic moduli calculated according to 
Eq. (2.1.32) are plotted in Figure 2.1.4, which demonstrates fair agreement between 
experimental data in dynamic tests and their theoretical prediction. 
Calculation of the material response in static tests requires a more sophisticated 
analysis, since even for the simplest programs of loading (creep, relaxation, recovery, 
etc.) the behavior of the model (2.1.26) is expressed in terms of special functions 
(either the generalized Mittag-Leffler functions or the Wright functions). 
2.2 Integral Constitutive Models 
In this section, some integral constitutive models are discussed for linear viscoelastic 
media. We begin with Boltzmann's superposition principle and derive integral equa- 
2.2. Integral Constitutive Models 35 
tions for the creep and relaxation measures of aging viscoelastic media. Afterward, 
the concept of adaptive links is introduced, and a balance law is developed for the 
number of links. Finally, constitutive equations for uniaxial deformation are extended 
to three-dimensional loading. 
2.2.1 Boltzmann's Superposition Principle 
Let us consider a specimen in the form of a rectilinear rod, which is in its natural 
(stress-free) state. At the initial instant t = 0, tensile forces are applied to the ends 
of the rod. Boltzmann's superposition principle states that the stress cr at the current 
instant t depends on the entire history of strains e in the interval [0, t]. Assuming this 
functional to be linear and applying Riesz's theorem, we find that 
/0 o(t) = X(t, ~')de(~'), (2.2.1) 
where X(t, T) is a function integrable in ~" for any fixed t >- 0. Equation (2.2.1) provides 
the general presentation of the stress-strain dependence in linear viscoelasticity. 
We suppose that the stress o- and the strain e are sufficiently smooth functions 
of time that satisfy the conditions 
o-(0) = 0, e(0) = 0. (2.2.2) 
Integration of Eq. (2.2.1) by parts with the use of Eq. (2.2.2) implies that 
foo t OX or(t) = X(t, t)e(t) - -~r (t, r)e(r) dr. 
It is convenient to present the relaxation function X(t, r) in the form 
X(t, r) = E(r) + Q(t, r), 
(2.2.3) 
(2.2.4) 
where 
E(I") = X(~', 1") (2.2.5) 
is the current Young's modulus, and 
Q(t, r) = x(t , r) - x ( r , r) (2.2.6) 
is the relaxation measure. It follows from Eq. (2.2.6) that for any t ->
0 
Q(t, t) = O. (2.2.7) 
The relaxation kernel R(t, r) is determined as 
10X 
R(t, ~') - (t, ~'). 
E(t) ar 
(2.2.8) 
36 Chapter 2. Constitutive Models in Linear Viscoelasticity 
Substituting expressions (2.2.5) and (2.2.8) into Eq. (2.2.3), we obtain the constitutive 
equation of a linear viscoelastic medium 
E /0' l or(t) = E(t) e(t) - R(t, ~')e(~') d~" . (2.2.9) 
Equations (2.2.3) and (2.2.9) describe the viscoelastic response in aging vis- 
coelastic media, mechanical properties of which depend explicitly on time. For aging 
materials, the function X(t, ~') depends on two variables, t and ~-. Typical examples of 
aging media are polymers, concrete, and soils [see, e.g., experimental data presented 
in Arutyunyan et al. (1987) and Struik (1978)]. 
Aging elastic media provide the simplest example of aging viscoelastic materials. 
For an aging elastic solid, Young's modulus E(t) depends on time, whereas the 
relaxation function vanishes, 
Q(t, ~-) = 0. (2.2.10) 
Combining Eqs. (2.2.4) and (2.2.10), we find that 
X(t, 1") = E(~'). 
Substitution of this expression into Eq. (2.2.3) implies that 
fot~T or(t) = E( t )E ( t ) - (T)E(I")dI". (2.2.11) 
Differentiation of Eq. (2.2.11) with respect to time yields the differential constitutive 
equation with a time-dependent Young's modulus 
do- dE 
dt - E(t) d---t" (2.2.12) 
The mechanical response in nonaging viscoelastic media is time-independent, 
which means that the function X depends on the difference t - T only 
X(t, r) = Xo(t - r). (2.2.13) 
It follows from Eqs. (2.2.5), (2.2.6), and (2.2.13) that Young's modulus E of a 
nonaging viscoelastic medium is time-independent, 
E = X0(0), (2.2.14) 
and the relaxation function Q depends on the difference t - ~-, 
Q = EQo(t - ~). (2.2.15) 
Substituting expressions (2.2.14) and (2.2.15) into Eqs. (2.2.3) and (2.2.9), we obtain 
the constitutive equation of a nonaging viscoelastic material 
2.2. Integral Constitutive Models 37 
where 
[ /0 t 1 or(t) = E e(t) + Qo(t - r)e(r) dr 
=E[e( t ) - f tR ( t - r )e ( r )dr ] , (2.2.16) 
1 
E(r) - (2.2.21) 
Y(r, r) 
is the current Young's modulus, and 
C(t, z) = Y(t, r) - Y(r, r) 
is the creep measure, which satisfies the condition 
C(t, t) - O. 
Substitution of expressions (2.2.20) and (2.2.21) into Eq. (2.2.19) yields 
o-(t) 
e(t) - 
E(t) 
- ~ + C(t, r) or(r) dr. 
(2.2.22) 
(2.2.23) 
(2.2.24) 
dQo 
R(t) = - ~( t ) , (2.2.17) 
dt 
and the superimposed dot denotes differentiation. 
Another formulation of Boltzmann's superposition principle states that the strain 
e at the current instant t is a functional of the entire history of stresses. Assuming 
this functional to be linear and applying Riesz's theorem, we arrive at the constitutive 
equation similar to Eq. (2.2.1) 
J0 e(t) = Y(t, r) do'(r), (2.2.18) 
where Y(t, r) is a function integrable in r for any fixed t >-- 0. We suppose that 
the stress o" and the strain e are sufficiently smooth functions of time, integrate Eq. 
(2.2.18) by parts, and use Eq. (2.2.2). As a result, we obtain the constitutive equation 
of an aging, linear, viscoelastic medium 
foo t OY e(t) = Y(t , t )o(t ) - -~-~T(t, r)cr(r)dr. (2.2.19) 
The function Y (t, r) is presented in the form 
1 
Y(t, r) - + C(t, r), (2.2.20) 
E(r) 
where 
38 Chapter 2. Constitutive Models in Linear Viscoelasticity 
Introducing the creep kernel 
K(t, r) = -E ( t ) ~-~ + C(t, r) , 
we rewrite Eq. (2.2.24) as 
e(t) = -~ or(t) + K(t, r )~(r) d . 
An aging elastic medium is characterized by the condition 
C(t, r) = 0. 
This equality together with Eqs. (2.2.20) and (2.2.24) implies that 
1 
Y(t, r) .- 
E(r) 
and 
(2.2.25) 
(2.2.26) 
where 
E(t) = E, 
and the creep function depends on the difference t - r, 
1 
C = -~Co(t - r). (2.2.28) 
Substitution of expression (2.2.28) into Eqs. (2.2.24) and (2.2.25) implies the consti- 
tutive relation for a nonaging viscoelastic material 
1[ /0t e(t) = ff~ or(t) + K( t - r lor(r ld 
I l o t 1 1 tr(t) + (?0(t - r)o'(r)dr (2.2.29) E 
dCo (t). (2.2.30) K(t) = -~ 
Differentiating Eq. (2.2.27) with respect to time, we obtain the constitutive Eq. 
(2.2.12). 
For non-aging viscoelastic media, the function Y depends on the difference t - r 
only. According to Eqs. (2.2.21) and (2.2.22), this means that Young's modulus is 
constant, 
o-(t) ~ 'd ( 1 ) 
e(t) - E(t) - ~ ~ ~r(r) dr. (2.2.27) 
2.2. Integral Constitutive Models 39 
The constitutive Eqs. (2.2.9) and (2.2.26) describe homogeneous viscoelastic 
media. For a viscoelastic solid with an arbitrary nonhomogeneity, Young's modulus, 
and the creep and relaxation kernels depend explicitly on Lagrangian coordinates {~, 
1[ 
e(t, ~) - E(t, ~) or(t, ~) + /o t 1 K(t, r, ~)or(r, !~) dr , 
[ /0 t ] or(t, ~) = E(t, ~) e(t, ~) - R(t, r, ~)e('r, ~) dr . (2.2.31) 
For nonhomogeneously aging media, we assume that different portions were manu- 
factured at different instants that preceded the initial instant t = 0 [see Arutyunyan 
et al. (1987)]. To describe the manufacturing process, we introduce a piecewise con- 
tinuous and bounded function K(~), which equals the material age at a point ~ at 
the initial instant t = 0. Since the material response is characterized by the internal 
time t + K(~), the constitutive equations of a nonhomogeneously aging viscoelastic 
medium read 
1 [cr(t, ~) 
E(t + K(~)) 7o t ] + K(t + K(~), r + K(~))o-(r, ~)dr , e(t,~) = 
(2.2.32) or(t, ~) = E(t + K(~)) e(t, ~) - R(t + K(~), r + K(~))e(r, ~)dr . 
Three approaches may be distinguished: 
(i) K(~) is a prescribed function, which characterizes the age distribution in a 
medium. 
(ii) K(~) is a control function, which is chosen to ensure optimal properties of a 
structure. 
(iii) K({~) describes environmental dependent aging caused by temperature [see, e.g., 
Stouffer and Wineman (1971) and Struik (1978)], by humidity [see Aniskevich 
et al. (1992), Knauss and Kenner (1980), Makhmutov et al. (1983), Morgan et 
al. (1980), Panasyuk et al. (1987), Shen and Springer (1977)], and by radiation 
[see McHerron and Wilkes (1993) and Sharafutdinov (1984)]. 
2.2.2 Connections Between Creep and Relaxation Measures 
Let us derive an integral equation which expresses creep and relaxation measures of 
an aging viscoelastic medium in terms of each other. For this purpose, we substitute 
expression (2.2.19) into Eq. (2.2.3), take into account Eqs. (2.2.5) and (2.2.21), and 
obtain 
,~(t) = E(t) [,~(t) _ fo' or" ] [E(t) -g-ss (t' s)cr(s) cls 
fO t oqX [ or(s) - Ts (t's~ Y(-;f fo 
s OY ] 
-~r (s, r)o'(r) dr ds 
40 Chapter 2. Constitutive Models in Linear Viscoelasticity 
= or ( t ) - fOt [E(t)aY(t,s) + 
! Os 
log ] 
E(s) --~s (t' s) or(s) ds 
fot OX fo ~ OY + --~s(t, s) ds -~T(s, r)or(r) dr. 
This equality implies that 
OY 
E(t)--~s (t,s) + fs t OY l_~ OX (t ' s) = ~( t , T)-~s (Z, s) dr. (2.2.33) E(s) Os 
Substitution of expressions (2.2.4) and (2.2.20) into Eq. (2.2.33) yields 
E(t)~ +C(t,s) + 1 0 
E(s) Os [E(s) + Q(t, s)] 
fs t 0 = ~ [E(T) + Q(t, ~')]~ + C(~',s) d~'. (2.2.34) 
Integrating Eq. (2.2.34) from T to t, we obtain 
E(t) [(E-~t)+ C(t, t)) - ( E~T) + C(t, T)) fT t 1 0 + ~ ~ [E(s) + Q(t, s)] ds 
fT t fs t 0 = ds ~ [E(T) + Q(t, ~')]~ + C(~',s) d~'. (2.2.35) 
We change the order of integration in the right-hand side of Eq. (2.2.35) and find that 
ds ~[E(~') + Q(t, ~')]~ss ~ + C(T,s) dr 
fr t 0 = ~-~T [E(T) + Q(t, r)]dT Os -E~ + c(r,s) ds. 
We calculate the integral with the use of Eq. (2.2.23) and obtain 
fr t 0
~ [E(T) fT ~ 0 + Q(t, ~')] aT 
1 ] 
-k~+C(~,s) ds 
f r t 0 = ~-~T[E(r) + Q(t, ~')] (' 1 +C(r , r ) - E(T) + c(T,r)/] d~ 
fr t 0 = ~-~r [E(~') + Q(t, r)] 1 1 1 E(~') - E(T) - C(T, T) d~'. 
2.2. Integral Constitutive Models 41 
Substitution of this expression into Eq. (2.2.35) yields 
1 - E(t) 
1 
E(T) E 1 ] + C(t, T) = - ~-~r[E(~') + Q(t, 1")] E(T) + C(~', T) dr. 
(2.2.36) 
Integration of the right side of Eq. (2.2.36) by parts with the use of Eqs. (2.2.7) and 
(2.2.23) implies that 
f 
t 0 
~-~r [E( T ) 
[1 ] 
+ Q(t, ~')] E(T) + C(T, T) dT 
I 1 ] = [E(t) + Q(t, t)] E(T) + C(t, T) - [E(T) [1 ] + Q(t, T)] E(T) + C(T, T) 
~ t 0C - [E(r) + Q(t, r)]-~r (r, T ldr 
=E(t) IE~T ) +C( t ,T ) ] -1 
Q(t,T) 
E(T) 
fT t °3C - [E(I") + Q(t, ~')]-~r (~', T) d~'. 
Substitution of this expression into Eq. (2.2.36) results in 
Q(t,s) 
E(s) f 
t OC 
+ [E(I-) + Q(t, ~-)]-~r (1-, s) d~- = 0. (2.2.37) 
Equation (2.2.37) is a linear Volterra equation for the relaxation measure Q(t, s) pro- 
vided that the creep measure C(t, s) and Young's modulus E(t) are given. Introducing 
the notation 
M(t, s) = 1 + E(s)C(t, s), (2.2.38) 
we rewrite Eq. (2.2.37) in the form 
f tOM ft~T Q(t, s) + --~r (l-, s)Q(t, r) dr = - E(r) (r, s) dr. (2.2.39) 
Equation (2.2.39) can be solved using the standard numerical methods for linear 
Volterra equations [see, e.g., Brunner and van der Houwen (1986) and Linz (1985)]. 
2.2.3 A Mode l of Adapt ive L inks 
Our objective now is to demonstrate that the response in an aging viscoelastic medium 
may be described by a network containing only elastic elements (without dashpots) 
provided the springs replace each other according to a given law. For this purpose, 
42 Chapter 2. Constitutive Models in Linear Viscoelasticity 
we transform Eq. (2.2.3) as follows: 
fo t O~x fO t O~x or(t) = X(t, t)e(t) - -~r (t, r)e(t)dr + -~r (t, r)[e(t) - e(r)] dr 
foo t OX = X(t, 0)e(t) 4- -~-T (t, r)[e(t) - e('r)] dr 
~o t OX = X(t, 0)e(t) 4- -~- (t, r)eO(t, r )dr , (2.2.40) 
where e¢(t, r) = e(t) - e(r) is the relative strain for transition from the actual 
configuration at instant r to the actual configuration at instant t. 
For definiteness, an interpretation of Eq. (2.2.40) is provided for polymeric 
materials. However, the proposed concept may be applied to an arbitrary viscoelastic 
medium. 
Let us consider a system of parallel elastic springs (which model links between 
chain molecules). At the initial instant t = 0, the system consists of X.(0, 0) links 
in the natural (stress4ree) state. Rigidity of any spring equals c. Within the interval 
[ r , r + dr], 
aX, 
~( t , r)]t=~ dr 
8r 
new links merge with the system. These links are connected in parallel to the initial 
links, and they are stress-free at the instant of their appearence. The latter means 
that the natural configuration of links arising at instant r coincides with the actual 
configuration of the system at that instant. The strain at instant t in links arising at 
instant r equals e <>(t, r). 
Due to the breakage process, some links annihilate. The number of initial links 
existing at instant t equals X,(t, 0), whereas the amount 
8X, 
~(t , r) dr 
determines the number of links arising within the interval [r, r + dr] and existing at 
instant t. 
To calculate the response in a network of parallel links, stresses in all the links 
should be added 
o'(t) = o'o(t) + do'(t, "r). (2.2.41) 
Here o0(t) is the stress at instant t in the initial links, and do'(t, r) is the stress at 
instant t in links joining the system at instant r. It follows from Hooke's law that 
~ro(t) = cX,(t, 0)e(t), 
c tgX, t tgX, dtr(t, "r) = -~z ( , r)e~(t, 7")d'r = c-~r(t, "r)[e(t) - e(r)] d'r. 
2.2. Integral Constitutive Models 43 
Substitution of these expressions into Eq. (2.2.41) yields 
{ fO t°gx* } or(t) = c X,(t, O)e(t) + --~r (t, ~')[e(t)- e(T)] aT 
fO 
t °~X 
= X(t, O)e(t) + -~r (t, ~')[e(t) - e(r)] dr, 
where 
(2.2.42) 
X(t, T) = cX.(t, ~'). (2.2.43) 
Since expressions (2.2.40) and (2.2.42) coincide, the behavior of this system of 
adaptive links coincides with the behavior of an aging linear viscoelastic medium, 
which means that a system of adaptive links may model the mechanical response in 
a linear viscoelastic material. 
The reason for this assertion lies deeper than a simple coincidence of equations. 
A polymeric material may be treated as a network of long molecules mutually linked 
by chemical and physical crosslinks and entanglements. The chains move relatively to 
each other (micro-Brownian motion). When the relative displacement of two portions 
connected by a link reaches some ultimate value, the link breaks, and chains acquire 
"free edges" that are ready to create new links. These links emerge when appropriate 
free edges are located sufficiently close to each other owing to random wandering. 
After their onset, new links oppose the displacements of chains relative to their 
positions at the instant when the links arise. 
This scenario for the interaction of polymeric molecules coincides with the pre- 
ceding scenario for a system of elastic springs, provided crosslinks and entanglements 
are treated as appropriate springs. The function X(t, T) is an average (deterministic) 
characteristic of random motion of chains at the microlevel. The quantity X(t, T) is 
proportional to the number of links arising before instant ~" and existing at instant t. 
The derivative 
OX 
~( t , "r) 
determines the rate of creation (at instant ~-) of new links which have not been broken 
before instant t. 
To determine potential energy of a network of parallel elastic springs, we add 
together the mechanical energies of individual links. The potential energy of the 
initial links existing at instant t equals 
C 
-~X.(t, 0)EZ(t). 
The potential energy (at instant t) of links joining the system at instant ~" is calculated 
as 
c OX,(t ' ,r)[e~(t ' T)]2 dT. 
2 0~- 
44 Chapter 2. Constitutive Models in Linear Viscoelasticity 
Summing up these expressions, we obtain the potential energy of the entire network 
(strain energy density of an aging viscoelastic medium) 
W(t) = -~ X,(t, O)e2(t) + ---~r (t, r)[e<>(t, r)] 2 dr 
{ f0 t } 1 X(t,o)eZ(t) + ~(t , r)[e(t) -- e(r)] 2 dr . (2.2.44) 2 
Dafermos (1970) employed an expression similar to Eq. (2.2.44) as a Lyapunov 
functional for an aging, linear viscoelastic medium. 
A rheological model of elastic links between polymeric chains was suggested 
by Green and Tobolsky (1946). In that work, one-dimensional constitutive equations 
were proposed for elongation and shear of non-aging polymers with the exponential 
relaxation kernel. 
Yamamoto (1956) generalized the Green-Tobolsky concept and developed a sta- 
tistical theory that permits relaxation kernels to be determined under some assump- 
tions regarding breakage of polymeric chains. As a result, integro-differential equa- 
tions were derived for a chain-distribution function, a chain-reformation function, 
and a chain-breakage function. For a comprehensive exposition of statistical models 
in viscoelasticity, see, e.g., Lodge (1989). Two shortcomings of the Yamamoto ap- 
proach may be mentioned: (i) it is too cumbersome for engineering applications, since 
it requires integro-differential equations for chain-distribution functions to be solved, 
and (ii) no experimental confirmation exists for relations between chain-distribution 
and chain-reformation functions. 
2.2.4 Spectral Presentation of the Function X(t, r) 
Our purpose now is to derive an integral equation for the function X.(t, r) and to 
solve it. We suppose that adaptive links
are divided into two types: the links of type I 
are not involved in the process of replacement, whereas the links of type II take part 
in this process. Denote by X E [0, 1 ] concentration of links of type I, and by 1 - X 
concentration of links of type II. 
Let g(t - r, r) be the relative number of links which have arisen at instant r and 
have lost before instant t. We can write 
where 
X.(t, 0) = X.(0, 0){X + (1 - X)[1 - g(t, 0)1}, 
OX, 
Or 
~(t , "r) = ~(r)[1 -- g(t -- r, r)], 
OX, 
• (r) = --~--(t, r){t=~ 
(2.2.45) 
(2.2.46) 
2.2. Integral Constitutive Models 45 
is the rate of creation for new links. The total number of links at instant t is calculated 
as 
f0 t t~X, X, ( t , t) = X , ( t , O) + -~T (t, ~') d~'. (2.2.47) 
Substitution of expressions (2.2.45) into Eq. (2.2.47) with the use of Eqs. (2.2.5) and 
(2.2.43) implies that 
E(t) = E(0){X + (1 - X)[1 - g(t, 0)]} + c ~('r)[1 - g(t - % "r)] d-r. (2.2.48) 
Setting 
E(t) X , ( t , t) c~( t ) 1 OX, 
E , ( t ) - - dO, U) - - - - (t, t), 
E(0) X,(0, 0)' E(0) X,(0, 0) &- 
we rewrite Eq. (2.2.48) as 
E, ( t ) = X + (1 - X)[1 - g(t, 0)] + ~,('r)[1 - g(t - %-r)] d-r. (2.2.49) 
For a given dimensionless Young's modulus E,( t ) , Eq. (2.2.49) imposes restrictions 
on the functions ~,(t ) and g(t - ~', "r). In the general case, these functions cannot be 
found uniquely from Eq. (2.2.49). However, for non-aging media this equation allows 
us to derive explicit expressions for ~,(~') and g(t - "r, ~'). Indeed, for a non-aging 
material, 
E, ( t ) = 1, dO, U) = alp,, g(t - ~', ~') = go(t - ~'). (2.2.50) 
Substitution of expressions (2.2.50) into Eq. (2.2.49) yields 
/o t /o (1 - X)go(t) = ~, [ 1 - go(t - "r)] dr = ~, [ 1 - g0(r)] dr. (2.2.51) 
Differentiation of Eq. (2.2.51) with respect to time implies that 
dgo ~, 
- ~ (1 - go), g0(0) = 0. (2.2.52) 
dt 1 - X 
The unique solution of Eq. (2.2.52) is 
(°,) = - - t . (2.2.53) go(t) 1 exp 1 - X 
To find the function X(t , q-), we substitute expressions (2.2.50) and (2.2.53) into the 
equality 
t OX 
X(t , T) = X(t , t) - --~s (t, s) ds 
[ x * f t ~ x * l a s = c (t, t) - --z--(t, s) ds (2.2.54) 
46 Chapter 2. Constitutive Models in Linear Viscoelasticity 
and obtain with the use of Eqs. (2.2.45) and (2.2.53) 
X(t, ~') = E(0) 1 - ~ , [1 - go(t - s)] ds 
= E(0) {1 - (1 - X) [1 - exp ( - 1 _ X 
Equation (2.2.55) expresses the relaxation function of a non-aging viscoelastic 
medium in terms of the rate of reformation q~, and the breakage function go(t). 
It follows from Eqs. (2.2.4) and (2.2.55) that the only relaxation measure for a 
non-aging viscoelastic material coincides with the relaxation measure of the standard 
viscoelastic solid [see Eq. (2.1.7)], 
~*t )1 . (2.2.56) Q0(t) = - (1 - X) [1 - exp ( - 1 _ X 
The constitutive relations (2.2.42) and (2.2.55) describe a network with only one 
kind of links. Observations demonstrate that several different kinds of links may be 
distinguished, such as "elastically active long chains" "elastically active slide and 
entanglement chains," and "elastically active short chains" [see He and Song (1993)]. 
Drozdov (1992, 1993) proposed a version of the model of adaptive links with M 
different kinds of links. Any kind of links is characterized by its strain energy density 
and relaxation measure. Links of different kinds arise and break independently of 
one another. 
Denote by ~m concentration of the mth kind of links (the ratio of the number of 
links of the mth kind to the total number of links), by ~m(~') and gm(t - r, T) the rates 
of creation and breakage for these links, and by Xm concentration of nonreplacing 
links (m = 1 . . . . . M). The parameters T~m are assumed to be time-independent. 
The balance law for mth kind of links states that the total number of links of the 
mth kind at instant t 
rlmX,(t, t) 
equals the sum of the number of initial links existing at instant t 
'l~mX,(0 , 0){Xm "q- ( l -- Xm)[ 1 - gm(t, 0)]} 
and the number of links arising within the interval (0, t] and existing at instant t. The 
latter quantity is calculated as follows. Within the interval D', ~" + d~'], 
T/mX, (0 , 0)(I)m, (T) d~- 
new links of the mth kind appear. At instant t, their number reduces to 
~mX,(O, 0)(I)m,(T)[ 1 - gm(t - ~', "r)] dr. (2.2.57) 
2.2. Integral Constitut ive Models 47 
Summing up these amounts for various intervals, we obtain 
f0 t rlmX,(O, 0) @m,('r)[1 - gm(t -- T, ~')] d~'. 
As a result, we arrive at the integral equations 
/0 t E, ( t ) = Xm + (1 - Xm)[1 - gm(t, 0)] + ~m,('r)[1 - gm(t -- 7, "r)] d'r, (2.2.58) 
which should be satisfied for m = 1 . . . . . M. It follows from Eqs. (2.2.45) and (2.2.57) 
that 
olX, M 
O---~(t, 1") = X,(0, 0) Z ~mCI)m*('r')[1 -- gm(t -- "r, 1")]. (2.2.59) 
m=l 
Substitution of expression (2.2.59) into Eq. (2.2.54) implies that 
{ s t / X(t , T) = E(O) E , ( t ) - "l~m (I)m,(S)[1 - gm(t - s, s)] ds . 
m=l 
For a non-aging medium (2.2.50), Eq. (2.2.58) is solved explicitly 
~m,t ) 
gm 0(t) = 1 - exp - 1 - Xm " 
Substitution of expressions (2.2.50) and (2.2.61) into Eq. (2.2.60) yields 
{ £ X( t ,T )=E(O) 1 - " r lm(1- -Xm ) 
m=l 
(I)m, 
1-exp -1 -Xm 
It follows from Eqs. (2.2.4) and (2.2.62) that 
(2.2.60) 
(2.2.61) 
Qo(t) = - Z ]'Lm 1 - exp -~m ' 
m=l 
(2.2.63) 
where 
1 - Xm 
].Lm = rim(1 -- Xm), Tm - ~ . (2.2.64) 
dPm, 
Equation (2.2.63) implies that the relaxation measure of an arbitrary non-aging vis- 
coelastic material equals a sum of exponential functions with positive coefficients. 
Differentiating Eq. (2.2.63) with respect to time and using Eq. (2.2.17), we find the 
relaxation kernel 
M m,m exp - 
1 m- 
48 Chapter 2. Constitutive Models in Linear Viscoelasticity 
Assuming that M ---, oo and 
]£m ~ ]£(Tm)(Tm+m - Tm), 
we arrive at the presentation of the relaxation kernel for a non-aging viscoelastic 
medium 
R(t)= j0 "°°/z(T) exp ( t ) T - -~ dT (2.2.65) 
with a nonnegative relaxation spectrum/x(T). The nonnegativity condition for the 
relaxation spectrum was discussed in details by Beris and Edwards (1993) and Pipkin 
(1972). 
2.2.5 Three-Dimensional Loading 
It follows from Eqs. (2.2.9) and (2.2.26) that in order to construct a constitutive 
equation in linear viscoelasticity it suffices to replace Young's modulus in Hooke's 
law by an appropriate Volterra operator. In Eq. (2.2.9), Young's modulus E is replaced 
by the relaxation operator E(I - R), and in Eq. (2.2.26), the elastic compliance E -1 
is replaced by the creep operator E-I( I + K). Here I is the unit operator, and for an 
arbitrary smooth function f(t), 
fOO t K f = K(t, r ) f (r) dr, ~0 t Rf = R(t, r)f(r) dr, (2.2.66) 
where K(t, r) and R(t, r) are the creep and relaxation kemels. 
It is natural to suppose that the same procedure (replacement of elastic moduli 
by Volterra operators) may be carried out for three-dimensional loading as well. 
However, even for an isotropic elastic medium, several different versions of consti- 
tutive equations exist, and any version is determined by at least two elastic moduli. 
When we replace these moduli (or only one of them) by integral operators, we obtain 
different versions of constitutive equations in viscoelasticity, and we are faced with 
the problem of choosing appropriate constitutive relations. 
We confine ourselves to two different versions of constitutive equations for an 
isotropic elastic medium. According to the first, we write 
E (5+ v j ) ~= 1 
6" - 1 + v 1 - 2------~ ' ~7 [(1 + v )8 - vor?], (2.2.67) 
where 6" is the stress tensor, 5