On a relation between memory effects by Maxwell - Boltzmann and Kelvin - Voigt in linear viscoelastic theory
We study the smoothness properties of relaxation function such that a linear viscoelastic material system by Maxwell Boltzmann can be considered of Kelvin Voigt type; assuming that the relaxation function and its derivative decrease rapidly, and that the infinitesimal strain history is an analytical...
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1999
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irk-123456789-1771562021-02-12T01:25:59Z On a relation between memory effects by Maxwell - Boltzmann and Kelvin - Voigt in linear viscoelastic theory Matarazzo, G. We study the smoothness properties of relaxation function such that a linear viscoelastic material system by Maxwell Boltzmann can be considered of Kelvin Voigt type; assuming that the relaxation function and its derivative decrease rapidly, and that the infinitesimal strain history is an analytical function, the Cauchy stress tensor of the linear viscoelasticity is well approximated by a constitutive functional of rate type. Вивчаються властивостi гладкостi релаксацiйної функцiї для випадку, коли лiнiйно пружна за Максвеллом Больцманом матерiальна система може розглядатись як система типу Кельвiна Войгта. У припущеннi, що релаксацiйна функцiя та її похiдна швидко спадають, а iнфiнiтезiмальна функцiя деформацiї є аналiтичною, показано, що тензор напруження Кошi в лiнiйнiй теорiї пружностi добре апроксимується складовим (конститутивним) функцiоналом коефiцiєнтного типу. 1999 Article On a relation between memory effects by Maxwell - Boltzmann and Kelvin - Voigt in linear viscoelastic theory / G. Matarazzo // Нелінійні коливання. — 1999. — Т. 2, № 3. — С. 345-351. — Бібліогр.: 7 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177156 517.958 en Нелінійні коливання Інститут математики НАН України |
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We study the smoothness properties of relaxation function such that a linear viscoelastic material system by Maxwell Boltzmann can be considered of Kelvin Voigt type; assuming that the relaxation function and its derivative decrease rapidly, and that the infinitesimal strain history is an analytical function, the Cauchy stress tensor of the linear viscoelasticity is well approximated by a constitutive functional of rate type. |
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Article |
| author |
Matarazzo, G. |
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Matarazzo, G. On a relation between memory effects by Maxwell - Boltzmann and Kelvin - Voigt in linear viscoelastic theory Нелінійні коливання |
| author_facet |
Matarazzo, G. |
| author_sort |
Matarazzo, G. |
| title |
On a relation between memory effects by Maxwell - Boltzmann and Kelvin - Voigt in linear viscoelastic theory |
| title_short |
On a relation between memory effects by Maxwell - Boltzmann and Kelvin - Voigt in linear viscoelastic theory |
| title_full |
On a relation between memory effects by Maxwell - Boltzmann and Kelvin - Voigt in linear viscoelastic theory |
| title_fullStr |
On a relation between memory effects by Maxwell - Boltzmann and Kelvin - Voigt in linear viscoelastic theory |
| title_full_unstemmed |
On a relation between memory effects by Maxwell - Boltzmann and Kelvin - Voigt in linear viscoelastic theory |
| title_sort |
on a relation between memory effects by maxwell - boltzmann and kelvin - voigt in linear viscoelastic theory |
| publisher |
Інститут математики НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/177156 |
| citation_txt |
On a relation between memory effects by Maxwell - Boltzmann and Kelvin - Voigt in linear viscoelastic theory / G. Matarazzo // Нелінійні коливання. — 1999. — Т. 2, № 3. — С. 345-351. — Бібліогр.: 7 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT matarazzog onarelationbetweenmemoryeffectsbymaxwellboltzmannandkelvinvoigtinlinearviscoelastictheory |
| first_indexed |
2025-07-15T15:11:13Z |
| last_indexed |
2025-07-15T15:11:13Z |
| _version_ |
1837726196414021632 |
| fulltext |
т. 2 •№ 3 • 1999
УДК 517 . 958
ON A RELATION BETWEEN MEMORY EFFECTS BY MAXWELL
BOLTZMANN AND KELVIN VOIGT IN LINEAR VISCOELASTIC
THEORY∗
ПРО ЗВ’ЯЗОК МIЖ ЕФЕКТАМИ ПАМ’ЯТI ЗА МАКСВЕЛЛОМ
БОЛЬЦМАНОМ ТА КЕЛЬВIНОМ ВОЙГТОМ В ЛIНIЙНIЙ
ТЕОРIЇ ПРУЖНОСТI
G. Matarazzo
Univ. Salerno,
84084, Fisciano (Salerno), Italy
We study the smoothness properties of relaxation function such that a linear viscoelastic material system
by Maxwell Boltzmann can be considered of Kelvin Voigt type; assuming that the relaxation function
and its derivative decrease rapidly, and that the infinitesimal strain history is an analytical function, the
Cauchy stress tensor of the linear viscoelasticity is well approximated by a constitutive functional of rate
type.
Вивчаються властивостi гладкостi релаксацiйної функцiї для випадку, коли лiнiйно пружна за
Максвеллом Больцманом матерiальна система може розглядатись як система типу Кельвi-
на Войгта. У припущеннi, що релаксацiйна функцiя та її похiдна швидко спадають, а iн-
фiнiтезiмальна функцiя деформацiї є аналiтичною, показано, що тензор напруження Кошi в
лiнiйнiй теорiї пружностi добре апроксимується складовим (конститутивним) функцiоналом
коефiцiєнтного типу.
1. As is posed in evidence in the study of the quasistatic problem in linear viscoelasticity
theory [1,2] the crucial point for a good and exaustive formulation of viscoelastic materials
theories [3,4] consists in the determination of general and physically admissible conditions [5]
so that materials with more fading or negligible memory effects can be classified by a good
approximation as particular viscoelastic materials; these conditions must be in accordance with
the structural properties [6, 7] of viscoelastic materials and with the pattern that describes them.
This problem is resolved partially in [5, 6], were have been formulated conditions, with the
above properties, so that materials of linear elastic type can be considered as particular linear
viscoelastic materials.
Purpose of the present paper is to prove that materials of linear rate type can be considered
as particular viscoelastic materials; if stronger smoothness hypotheses of relaxation and
Boltzmann functions are verified and if the infinitesimal strain history is an analytic function,
it is possible to approximate the constitutive functional of linear viscoelasticity theory by a
particular constitutive equation of Kelvi Voigt type.
It is interesting to observe that the coefficient of the memory term of this constitutive
* This research has been performed under the auspices of the G.N.F.M. of the C.N.R. and has been partially
supported by Italian Ministry of University and Technology Research M.U.R.S.T.
c© G. Matarazzo, 1999 345
relation is equal to the value of the dynamic viscosity tensor, when the frequency w approaches
to zero, where this tensor has an eliminable discontinuity in virtue of the assumed hypothe-
ses [5].
Finally we conclude proving an existence and uniqueness theorem for the quasistatic
problem of material systems expounded by the above functional; the solution is determined
as limit of a solution of the quasistatic problem for a strictly viscoelastic material system when
w → 0 [6].
2. Let β be a linear viscoelastic and homogeneous material system described by the
following constitutive functional:
T (x, t) = G0(x)E(x, t) +
+∞∫
0
G′(x, s)Et(x, s)ds =
= G∞(x)E(x, t) +
+∞∫
0
[G(x, s)−G∞(x)] Ėt(x, s)ds,
(1)
T (x, t) = T T (x, t), (x, t) ∈ Ω× [0,+∞) = Q,
where T (x, s) is the Cauchy stress tensor, G(x, s) and G′(x, s) are respectively the relaxation
and Boltzmann fourth-order Cartesian tensors, G0(x) and G∞(x) denote respectively the
instantaneous and equilibrium elastic moduli, that are so defined:
G0(x) = lim
s→0
G(x, s) = G(x, s)−
s∫
0
G′(x, τ)dτ,
G∞(x) = lim
s→+∞
G(x, s) = G0(x) +
+∞∫
0
G′(x, τ)dτ ;
E(x, t) =
1
2
[
∇u+ (∇u)T
]
, where u(x, t) denotes the displacement vector, is the second-order
infinitesimal strain tensor, whileEt(x, s) = E(x, t− s), s ∈ [0,+∞) with respect to every fixed
t ∈ [0,+∞), denotes the history of the infinitesimal strain tensor at instant t; finally Ω is an
open and bounded domain of R3 with sufficiently regular boundary ∂Ω.
We assume that the following hypotheses are verified ∀x ∈ Ω:
sG′(x, s) ∈ L1(0,+∞),
G(x, ·)−G∞(x) = −
∞∫
s
G′(τ)dτ ∈ H1,1(0,+∞) ∩H1,2(0,+∞),
lim
s→+∞
s2 [G(x, s)−G∞(x)] = 0;
G(x, s) = −G(x,−s)
G′(x, s) = G′(x,−s)
∀s ∈ [0,+∞).
(2)
346 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 3
If and only ifG0(x) = G∞(x), then, for all t > 0
lim
a→+∞
+∞∫
−at
a
[
G
(
x, t+
y
a
)
−G∞(x)
] [sin y − y cos y
y2
]
dy = 0,
where y = a(s− t) and a > 0.
1. It is assumed that G′(x, ·) is continuous ∀x ∈ Ω while G′′(x, ·) is piecewise
continuous; furthermore G′(x, ·) verifies Dini condition in every point of discontinuity and
in a neighbourhood of such pointsG′′(x, ·) is bounded.
2. The fourth-order symmetric tensors G0(x) and G∞(x) are positive definite and
continuous in Ω; furthermore G(x, ·) and G′(x, ·) are continuous in Ω with respect to every
fixed s.
We remark that conditions (3)1,2,3 are all verified if we suppose:
∃ α ≥ 3: lim
s→∞
sα+1G′(s) = 0. (3)
By the assumed hypotheses we can state the following:
Theorem 1. If hypoteses (2) hold and if (3) holds by a suitable value of α and ifE(x, t− s) ∈
∈ H1,1(0,+∞)∩H1,2(0,+∞) ∀x ∈ Ω is an analytic function, then the body β is of Kelvin Voigt
type, i.e.:
T (x, t) = G∞(x)E(x, t) +K(x)Ė(x, t) ∀(x, t) ∈ Ω× [0, dpα), dpα < +∞, (4)
whereK(x) =
+∞∫
0
[G(x, s)−G∞(x)]ds is such that:
β1A : A > −A :
+∞∫
0
sG′(x, s)dsA = A : K(x)A > β2A : A > 0
∀x ∈ Ω, ∀A ∈ Sym(V ) \ {0},
(5)
and β1, β2 are positive constants.
Proof. By Maclaurin formula we have:
Et(x, s) = E(x, t) + (−s)Ė(x, t) + σ(x, s2) ∀x ∈ Ω, (6)
where lim
s→0
σ(x, s)
s2
= 0.
Using (6) we can rewrite (1)1, by a suitable value of t, in this manner:
T (x, t) = G0(x)E(x, t) +
t∫
0
G′(x, s)
[
E(x, t)− sĖ(x, t)
]
ds+
+
+∞∫
t
G′(x, s)
[
E(x, t)− sĖ(x, t) + σ(x, s2)
]
ds; (7)
ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 3 347
remarking that, in the assumed hypotheses, it is possible to replace the limit of the integral with
the integral of the limit, and that, within a linear theory, we have that:
lim
t→+∞
G′(x, s)
[
E(x, t)− sĖ(x, t) + σ(x, s2)
]
= 0 ∀x ∈ Ω;
if we pass to the limit of the last integral of (7) with t→ +∞, finally by a suitable value of α in
(3) and by the analiticity hypothesis of Et(x, s) we have that:
T (x, t) = G∞(x)E(x, t)−
+∞∫
0
sG′(x, s)dsĖ(x, t) = G∞(x)E(x, t)+
+
+∞∫
0
[G(x, s)−G∞(x)] dsĖ(x, t) ∀ (x, t) ∈ Ω× [0, dpα), dpα < +∞, (8)
that implies (4) setting
K(x) =
+∞∫
0
[G(x, s)−G∞(x)] ds;
finally (5) is a consequence of theorem IV of [5], because we get ∀x ∈ Ω
β1A : A > A : lim
w→0
Ĝc(x, w)A = −A :
+∞∫
0
sG′(x, s)dsA =
= A :
+∞∫
0
[G(x, s)−G∞(x)] dsA > β2A : A,
where Ĝc(x, w) =
+∞∫
0
[G(x, s)−G∞(x)] coswsds is the dynamic viscosity tensor.
By the above theorem, the theorem I VI of [5] and the definitions I, II of [6] we are able to
complete the examination of viscoelastic materials, that have stronger or weaker or negligible
memory, formulating the following definitions:
Definition 1. A continuum material system expressed by the constitutive functional (1) is said
strictly viscoelastic if and only if, in hypotheses (2), the following conditions are verified:
i)G(x, ·)−G0(x) 6∈ L1(0,+∞) ∀x ∈ Ω,G(x, s) = GT (x, s) ∀(x, s) ∈ Ω× [0,+∞);
ii) there exist two constants µ1 > µ2 > 0, such that:
µ1A : A > A : [G0(x)−G∞(x)]A > µ2A : A ∀A ∈ Sym(V ) \ {0} and ∀x ∈ Ω,
where Sym(V ) is the second-order Cartesian symmetric tensor space of R3;
iii) the dynamic viscosity tensor Ĝc(x, w) =
+∞∫
0
[G(x, s)−G∞(x)] coswsds is positive
definite and bounded, i.e. there exist two constants β1 > β2 > 0, independent of w, such that:
β1A : A > A : Ĝs(x, w)A > β2A : A ∀A ∈ Sym(V ) \ {0},
∀ w ∈ (−∞,+∞) and ∀ x ∈ Ω;
348 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 3
particularly, ∀x ∈ Ω, we have:
β1A : A > A : lim
w→0
Ĝc(x, w)A = −A :
+∞∫
0
sG′(x, s)dsA =
= A :
+∞∫
0
[G(x, s)−G∞(x)] dsA > β2A : A
and lim
w→±∞
Ĝc(x, w) = 0;
iv) ∀x ∈ Ω, ∀A ∈ Sym(V ) \ {0}, ∃ν1, ν2 > 0 such that:
A :
[
G0(x) + Ĝ
′
c(x, w)
]
A = A :
[
G∞(x) + wĜs(x, w)
]
A ≥ ν1A : A ∀ w ∈ R,
w2A : Ĝc(x, w)A = −wĜ
′
s(x, w) ≥ ν2A : A ∀ w 6= 0,
where
Ĝ
′
c(x, w) =
+∞∫
0
G′(x, s) coswsds, Ĝs(x, w) =
+∞∫
0
[G(x, s)−G∞(x)] sinwsds,
Ĝ
′
s(x, w) =
+∞∫
0
G′(x, s) sinwsds
and ν1, ν2 don’t depend on w, particularly, ∀x ∈ Ω, we have:
A : lim
w→0
[
G0(x) + Ĝ
′
c(x, w)
]
A = A : lim
w→0
[
G∞(x) + wĜs(x, w)
]
A =
= A : G∞(x)A ≥ ν1A : A,
A : lim
w→±∞
[
G0(x) + Ĝ
′
c(x, w)
]
A = A : lim
w→±∞
[
G∞(x) + wĜs(x, w)
]
A =
= A : G0(x)A > ν1A : A.
Definition 2. If and only if G(x, ·) − G0(x) 6∈ L1(0,+∞) ∀x ∈ Ω, a continuum material
system described by the constitutive functional (1) is a linear viscoelastic body of Kelvin Voigt
type, if by hypotheses of theorem 1 the following conditions are verified:
i) T (x, t) = G∞(x)E(x, t)+K(x)Ė(x, t) ∀(x, t) ∈ Ω× [0, dpα), dpα < +∞, whereK(x) =
=
+∞∫
0
[G(x, s)−G∞(x)] ds is such that:
β1A : A > A : lim
w→0
Ĝc(x, w)A = −A :
+∞∫
0
sG′(x, s)dsA =
= A :
+∞∫
0
[G(x, s)−G∞(x)] dsA > β2A : A ∀ x ∈ Ω, ∀A ∈ Sym(V ) \ {0}
and β1, β2 are positive constants;
ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 3 349
ii)G(x, s) = GT (x, s) ∈ Ω× [0,+∞).
Definition 3. If hypotheses (2) hold, and if and only if G(x, ·)−G∞(x),G(x, ·)−G0(x) ∈
∈ L1(0,+∞) ∀x ∈ Ω, body β is of linear elastic type, i.e.
T (x, t)−G0(x)E(x, t) = G∞(x)E(x, t) ∀ t ∈ [0, Tc) where Tc < +∞;
if and only if
T (x) = G0(x)E(x) = G∞(x)E(x)
then body β is linear elastic.
3. The quasistatic problem for a strictly viscoelastic body expressed by definition 1 is
formulated by the following Dirichlet problem:
∇ ·
G∞(x)∇u(x, t) +
+∞∫
0
[G(x, s)−G∞(x)]∇u̇t(x, s) ds
+ b(x, t) =
= ∇ ·
G∞(x)∇u(x, t)+
+∞∫
0
G′(x, s)∇ut(x, s) ds
+ b(x, t)=0, (x, t) ∈ Q, (9)
u(x, t)
∣∣
∂Ω
= 0,
where
u(x, t) = u(x, t)− u∞(x), lim
t→+∞
u(x, t) = u∞(x), b(x, t) = b(x, t)− b∞(x)
and
lim
t→+∞
b(x, t) = b∞(x).
Relating to this problem we have proved [6] the following
Theorem 2. If and only if body β is strictly viscoelastic according to definition 1, if b(x, t) ∈
∈ L1(R;H1,2(Ω)) ∩ L2(R;H1,2(Ω)), b(x, ·) ∈ S∞(R) and has compact support in R, there exists
one and only one solution with compact support u(x, t) ∈ H1,1(R;H1,2(Ω))∩H1,2(R;H1,2(Ω)),
u(x, ·) ∈ S∞(R), such that:
∫
Ω′
G∞(x)∇u(x, t) +
+∞∫
0
[G(x, s)−G∞]∇u̇t(x, s) ds
: ∇H(x,x′, t)dx′ =
=
∫
Ω′
G0(x)∇u(x, t) +
+∞∫
0
G′(x, s)∇ut(x, s) ds
∇H(x,x′, t) dx′ =
=
∫
Ω′
b(x, t)H(x,x′, t)dx′
∀H(x,x′, t) ∈ L∞(−∞,+∞;H1,2(Ω)×H1,2(Ω′)) : H(x,x′, t)
∣∣
∂Ω
= 0,
(10)
350 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 3
whereH(x,x′, t) is strongly measurable, if x′ 6= x, and S∞(R) denote the class of infinitely many
time differentiable functions u(x, t) with respect to t for which there exists a set of constants Cpq,
dependent on same function u(x, t) and on numbers p and q, such that:∫
Ω
∣∣∣tp∂(q)
t u(x, t)
∣∣∣2 dx < C2
pq,
∫
Ω
∣∣∣tp∂(q)
t ∇u(x, t)
∣∣∣2 dx < C2
pq.
In [6] we verify that (10) holds if we inversely transform by Fourier the solution of the
Fourier transformed problem of (9); this solution is null outside at a compact interval of the
time origin, because of a condition of compatibility with the meaning itself of the quasistatic
problem.
This consideration and definition 2 imply that a solution of the quasistatic problem for a
viscoelastic body descrebed by definition 2 must be determined as limit of this Fourier inverse
transformed solution when w → 0.
Consequently we can state relating to the Dirichlet problem:
∇ ·
G∞(x)∇u(x, t) +
+∞∫
0
[G(x, s)−G∞(x)] ds∇u̇(x, t)
+ b(x, t) =
= ∇ ·
G∞(x)∇u(x, t)−
+∞∫
0
sG′(x, s)ds∇u̇(x, t)
+b(x, t)=0,
(x, t) ∈ Ω× [0, dpα), dpα <∞,
(11)
u(x, t)
∣∣
∂Ω
= 0,
the following
Theorem 3. If body β is a linear viscoelastic material system according to definition 2, if
b(x, t) ∈ L2(Iα;H1,2(Ω)), where Iα = [0, dpα), is analytic and has compact support, only null
solution solves the problem (11).
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2. Fichera G. Sul principio della memoria evanescente // Rend. Mat. Univ. Padova. 1982. 68. P. 245
259.
3. Coleman B.D., Noll W. Foundation of linear viscoelasticity // Rev. Modern Phisics. 1961. 33. P.239
249.
4. Green A.E., Rivlin R.S. The mechanics of nonlinear material with memory // Arch. Ration. Mech. and Anal.
1957 1958. 1. P. 1 21.
5. Matarazzo G. Symmetry of relaxation function in viscoelasticity // Proc. Int. Sci. Conf. "Asymptotic and
Qualitative Methods of Nonlinear Mechanics"(Kiev, August 1997).
6. Matarazzo G. Time unreversal and existence and uniqueness problems in linear viscoelasticity // Ukr. Math.
J. (To appear).
7. Rivlin R.S. A note on the Onsager Casimir relations // J. Appl. Math. and Phys. 1973. 24. P. 897
900.
Received 22.03.99
ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 3 351
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