Volterra’s theory of elastic dislocations for a transversally isotropic homogeneous hollow cylinder
This work will consider Volterra’s theory of elastic dislocations in the case of a transversally isotropic homogeneous hyperelastic hollow cylinder. We obtain explicit equations for vector field of displacements, for tensor fields of strain and stress, and for forces upon the boundary.
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irk-123456789-1761652021-02-03T21:31:47Z Volterra’s theory of elastic dislocations for a transversally isotropic homogeneous hollow cylinder Laserra, E. Pecoraro, M. This work will consider Volterra’s theory of elastic dislocations in the case of a transversally isotropic homogeneous hyperelastic hollow cylinder. We obtain explicit equations for vector field of displacements, for tensor fields of strain and stress, and for forces upon the boundary. Розглянуто теорiю Вольтерри для пружних дислокацiй у випадку трансверсально iзотропних однорiдних надпружних порожнистих цилiндрiв. Одержано рiвняння для векторного поля перемiщень, тензорного поля напружень i сил на межi. 2003 Article Volterra’s theory of elastic dislocations for a transversally isotropic homogeneous hollow cylinder / E. Laserra, M. Pecoraro // Нелінійні коливання. — 2003. — Т. 6, № 1. — С. 56-73. — Бібліогр.: 8 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/176165 539.3 en Нелінійні коливання Інститут математики НАН України |
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This work will consider Volterra’s theory of elastic dislocations in the case of a transversally isotropic
homogeneous hyperelastic hollow cylinder. We obtain explicit equations for vector field of displacements,
for tensor fields of strain and stress, and for forces upon the boundary. |
| format |
Article |
| author |
Laserra, E. Pecoraro, M. |
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Laserra, E. Pecoraro, M. Volterra’s theory of elastic dislocations for a transversally isotropic homogeneous hollow cylinder Нелінійні коливання |
| author_facet |
Laserra, E. Pecoraro, M. |
| author_sort |
Laserra, E. |
| title |
Volterra’s theory of elastic dislocations for a transversally isotropic homogeneous hollow cylinder |
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Volterra’s theory of elastic dislocations for a transversally isotropic homogeneous hollow cylinder |
| title_full |
Volterra’s theory of elastic dislocations for a transversally isotropic homogeneous hollow cylinder |
| title_fullStr |
Volterra’s theory of elastic dislocations for a transversally isotropic homogeneous hollow cylinder |
| title_full_unstemmed |
Volterra’s theory of elastic dislocations for a transversally isotropic homogeneous hollow cylinder |
| title_sort |
volterra’s theory of elastic dislocations for a transversally isotropic homogeneous hollow cylinder |
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Інститут математики НАН України |
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2003 |
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http://dspace.nbuv.gov.ua/handle/123456789/176165 |
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Volterra’s theory of elastic dislocations for a transversally isotropic homogeneous hollow cylinder / E. Laserra, M. Pecoraro // Нелінійні коливання. — 2003. — Т. 6, № 1. — С. 56-73. — Бібліогр.: 8 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT laserrae volterrastheoryofelasticdislocationsforatransversallyisotropichomogeneoushollowcylinder AT pecorarom volterrastheoryofelasticdislocationsforatransversallyisotropichomogeneoushollowcylinder |
| first_indexed |
2025-07-15T13:50:13Z |
| last_indexed |
2025-07-15T13:50:13Z |
| _version_ |
1837721111677108224 |
| fulltext |
UDC 539.3
VOLTERRA’S THEORY OF ELASTIC DISLOCATIONS FOR
A TRANSVERSALLY ISOTROPIC HOMOGENEOUS HOLLOW CYLINDER
ТЕОРIЯ ВОЛЬТЕРРИ ДЛЯ ПРУЖНИХ ДИСЛОКАЦIЙ
ТРАНСВЕРСАЛЬНО IЗОТРОПНИХ ОДНОРIДНИХ
ПОРОЖНИСТИХ ЦИЛIНДРIВ
E. Laserra, M. Pecoraro
Universitá di Salerno, Via S.Allende, I-84081 Baronissi (SA)
e-mail: elaserra@unisa.it
pecoraro1@libero.it
This work will consider Volterra’s theory of elastic dislocations in the case of a transversally isotropic
homogeneous hyperelastic hollow cylinder. We obtain explicit equations for vector field of displacements,
for tensor fields of strain and stress, and for forces upon the boundary.
Розглянуто теорiю Вольтерри для пружних дислокацiй у випадку трансверсально iзотропних
однорiдних надпружних порожнистих цилiндрiв. Одержано рiвняння для векторного поля пере-
мiщень, тензорного поля напружень i сил на межi.
Introduction. In his note on distorsions 1, Volterra studies the equilibrium of multi-connected
elastic homogeneous bodies, particularly hollow cylinders, limiting his study only to isotropic
bodies (see e.g. [1 – 3]). Recently G. Caricato proposed an extension of that theory in the case
of a transversally isotropic homogeneous hiperelastic hollow cylinder (see e.g. [4, 5]).
In this work we will reconsider and expand the findings [4, 5]; we will obtain explicit formulas
for the equilibrium equations, for the boundary conditions, for the vector field of displacements,
and for tensor fields of strain and stress 2.
Thus we are presenting the following:
That the hypothesis in [4, 5] of the parallelism of the two vectors h and k, characteristic of
the displacement (3), plays no role in our research.
From the analysis of the equilibrium equations we show that the coefficient l3, present in
the displacement (3) and arbitrarily retained in the notes [4, 5], assumes instead the expression
(19), so the displacement (3) depends only on the parameters a1 and l4.The strain (35) and the
stress (23) are calculated from the following form (21) of the displacement; they only depend
on the parameters a1 and l4. From examining the boundary conditions we can then deduce
that the coefficient l3 vanishes and, as a consequence, the parameter a1 assumes the explicit
form (32).
1 Volterra calls the deformations which his theory refers to „distortions”. Love prefers to call them ”dislocations”
(see [1, p. 221], art. 156, note ).
2 We have utilized the Computer Algebra System Mathematica, which allows not only to verify the calculations
rapidly, but also automatically generates the LATEX sources of formulas.
c© E. Laserra, M. Pecoraro, 2003
56 ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
VOLTERRA’S THEORY OF ELASTIC DISLOCATIONS FOR A TRANSVERSALLY ISOTROPIC . . . 57
Fig. 1 Fig. 2
The final displacement (33), together with the strain (35) and stress (36) tensors, become
exclusively dependent on the arbitrary parameter l4. The only components of the strain and
stress tensors depending on the parameter l4 are ε13, ε23 and σ13, σ23, respectively.
The vector field displacement becomes dependent on the ratio N/A of only two of the five
elastic constants, which characterize the transversally isotropic case.
Finally we have found under what conditions Volterra’s formulas (2) for the isotropic case
can be attained again.
It remains to be calculated, in a following paper, a generic auxiliary displacement u′, to
obtain the complete explicit form of Volterra’s dislocations in the case under examination (see
[5], §2.1, note 5).
1. Volterra’s dislocations in the case of an isotropic homogeneous hollow cylinder. Briefly
let’s refer to Volterra’s dislocations theory, limiting it to the case of cylinder C, circular, hollow
(therefore doubly connected), homogeneous, hyperelastic and isotropic, which is found, at a
certain assigned temperature τ , in a natural state C∗τ , and assumed as a reference configuration 3.
So we introduce into an ordinary space a Cartesian rectangular reference
Ox1x2x3 of respective versors {c1, c2, c3}. We choose the axis Ox3 coinciding with the sym-
metry axis of the cylinder and the coordinate plane Ox1x2 placed over the base α∗1; d =
= (x3)α∗2 > 0 is the height of the base α∗2 (Fig. 1). Finally Σ∗ is the lateral surface of C∗, made
from the two cylindrical coaxial surfaces Σ∗1, internal surface of radius r, and Σ∗2, external
surface of radius R (Fig. 2). P ∗ is the generic point of C∗τ , θ = arctan
x2
x1
is the anomaly of P ∗
and ρ =
√
x1
2 + x2
2 is the distance of P ∗ from the axis of the cylinder. Since the cylinder C is
doubly connected, many-valued displacements u are possible (see e.g. [1, p. 221], art. 156).
Volterra used Weingarten’s note [6] as a starting point, where it is shown that an elastic
body occupying a dominion, not simply connected, can find itself in a state of tension also in
the absence of external forces. Volterra developed a general theory, with some improvements
from Cesaro (see e.g. [2] and [1, p. 221], art. 156). Volterra began with the observation that
Weingarten’s considerations could not be validated in the case of simply-connected bodies in
the range of regular deformations. With this in mind he constructed his well-known Volterra’s
formulas, which obtain the displacements of the points of an elastic body, once assigned the
3 The theory of dislocations initially has a very general character, but subsequently is substantially focused to
obtain explicit results in the study of equilibrium of hollow homogeneous and isotropic cylinders.
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
58 E. LASERRA, M. PECORARO
linearized tensor of deformation. Then he examined the field of displacements whose Cartesian
components are 4
u1 =
1
2π
( l + qx3 − rx2) θ + ( ax1 + bx2 + cx3 + e) log ρ2,
u2 =
1
2π
(m+ rx1 − px3) θ + ( a′x1 + b′x2 + c′x3 + e′) log ρ2, (1)
u3 =
1
2π
(n+ px2 − qx1) θ + ( a′′x1 + b′′x2 + c′′x3 + e′′) log ρ2,
where the two triplets (l,m, n) and (p, q, r) are the respective Cartesian components of the two
assigned constant vectors h ≡ (l,m, n) and k ≡ (p, q, r). He determined the twelve constants
a, b, c, e; a′, b′, c′, e′; a′′, b′′, c′′, e′′ so that the three functions (1) would verify the equilibrium
equations (see e.g. [2, p. 428]); so he obtained the formulas
u1 =
1
2π
{
(l + qx3 − rx2) θ +
1
2
(
−m+ px3 +
rµ
λ+ 2µ
x1
)
log ρ2
}
,
u2 =
1
2π
{
(m+ rx1 − px3) θ +
1
2
(
l + qx3 +
rµ
λ+ 2µ
x2
)
log ρ2
}
, (2)
u3 =
1
2π
{
(n+ px2 − qx1) θ − 1
2
(px1 + qx2) log ρ2
}
,
where λ and µ are the two Lamé constants (see e.g. [3]).
He observed that the displacement (2) generates a distribution of forces not identically
vanishing on the surface of the cylinder. So he calculated a supplementary field of displacements
u′(P ∗) single-valued, which would satisfy the indefinite equations of elastic equilibrium in the
absence of forces of mass and would generate the same distribution of surface forces on the
boundary of the cylinder. The field of displacements,
u′′(P ∗) = u(P ∗)− u′(P ∗),
satisfies the indefinite equations of equilibrium equally, but does not generate any distribution
of forces on the boundary of the cylinder and is many-valued like u(P ∗) 5.
The many-valued field of displacements, u′′(P ∗), can be physically interpreted in terms of
the following operations (see e.g. [1, p. 224]):
1. By making a transversal cut on an axial semi-plane, we make the hollow homogeneous
cylinder Cτ simply-connected and it assumes a natural state C∗τ . We’ll denote the two faces of
the cut by γ∗1 and γ∗2 .
4 Conforming to [3, 5] the p, q, r have opossite signs as compared with Volterra’s original work (see e.g. [2,
p. 427]).
5 So we obtain the real Volterra’s dislocation, which consists of two parts: the many-valued main displacement
u(P ∗) and the single-valued supplementary displacement −u′(P ∗).
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
VOLTERRA’S THEORY OF ELASTIC DISLOCATIONS FOR A TRANSVERSALLY ISOTROPIC . . . 59
2. We’ll impose a translatory displacement h ≡ (l,m, n) and a rotatory displacement k ≡
≡ (p, q, r) to one of the two faces, e.g. γ∗1 , with respect to the other. The two characteristic
vectors h and k together will be parallel to the semi-plane π. In this way making the face
γ∗1 penetrate into γ∗2 or distance itself from γ∗2 according to the vector k will levogyrous or
dextrogyrous accordingly.
3. If k is levogyrous we’ll remove a thin slice of matter, a thickness proportional to the
distance from the axis of the cylinder. If instead k is dextrogyrous we’ll add a thin slice of matter,
the same material as the cylinder, between the two faces of the cut, a thickness still proportional
to the distance from the axis of the cylinder. In this way we create a state of deformation in the
cylinder and therefore of stress.
4. Finally we’ll remake the cylinder doubly connected by soldering the two faces of the
cut. In this way the cylinder assumes a helicoidal configuration absent of superficial forces and
results in a state of regular internal stress.
Collectively, Volterra called the described operations a dislocation whose characteristics are
l,m, n, p, q, r (see e.g. [2]).
2. Volterra’s dislocations in the case of a transversally isotropic homogeneous hollow cyli-
nder. Now let’s consider a transversally isotropic 6 hyperelastic homogeneous hollow cylinder
and let’s suppose it is found in a natural state C∗τ 7 at temperature τ .
In analogy to Volterra’s procedure, conforming to [5], let’s consider a displacement of the
following type:
u(P ∗) =
1
2π
(h + k ∧OP ∗) θ +
+
[
(a ·OP ∗ + a4) c1 + (b ·OP ∗ + b4) c2 + (l ·OP ∗ + l4) c3
]
log ρ2, (3)
where we can assign the two vectors h and k, characteristic of the dislocation, while we have to
determine yet the vectors a, b, l and the constants a4, b4, l4.
If we project (3) onto the axis we obtain
u1 =
1
2π
(h1 + k2x3 − k3x2) θ + (a1x1 + a2x2 + a3x3 + a4) log ρ2,
u2 =
1
2π
(h2 + k3x1 − k1x3) θ + (b1x1 + b2x2 + b3x3 + b4) log ρ2, (4)
u3 =
1
2π
(h3 + k1x2 − k2x1) θ + (l1x1 + l2x2 + l3x3 + l4) log ρ2.
The displacement gradient ∇u ≡
∥∥∥∥∂uh∂xk
∥∥∥∥ relative to the displacement (3), (4) is
6 Since it conserves its mechanical characteristics along any direction perpendicular to the axis of symmetry.
7 Therefore absent of external mass and/or superficial forces.
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
60 E. LASERRA, M. PECORARO
∂u1
∂x1
=
1
2πρ2
(4πa4x1 − h1x2 + 4πa1x
2
1 + 4πa2x1x2 + 4πa3x1x3 + k3x2
2 − k2x2x3) +
+ a1 log ρ2,
∂u1
∂x2
=
1
2πρ2
(h1x1 + 4πa4x2 − (k3 − 4πa1)x1x2 + 4πa2x
2
2 + k2x1x3 + 4πa3x2x3)−
− k3
2π
arctan θ + a2 log ρ2,
∂u1
∂x3
=
k2
2π
arctan θ + a3 log ρ2,
∂u2
∂x1
=
1
2πρ2
(4πb4x1 − h2x2 + 4πb1x2
1 − (k3 − 4πb2)x1x2 + 4πb3x1x3 + k1x2x3) +
+
k3
2π
arctan θ + b1 log ρ2,
∂u2
∂x2
=
1
2πρ2
(h2x1 + 4πb4x2 + k3x1
2 + 4πb1x1x2 + 4πb2x2
2 − k1x1x3 + 4πb3x2x3) +
+ b2 log ρ2, (5)
∂u2
∂x3
= − k1
2π
arctan θ + b3 log ρ2,
∂u3
∂x1
=
1
2πρ2
(4πl4x1 − h3x2 + k2x1x2 − k1x
2
2 + 4πl1x2
1 + 4πl2x1x2 + 4πl3x1x3)−
− k2
2π
arctan θ + l1 log ρ2,
∂u3
∂x2
=
1
2πρ2
(h3x1 + 4πl4x2 − k2x
2
1 + (k1 + 4πl1)x1x2 + 4πl2x2
2 + 4πl3x3) +
+
k1
2π
arctan θ + l2 log ρ2,
∂u3
∂x3
= l3 log ρ2
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
VOLTERRA’S THEORY OF ELASTIC DISLOCATIONS FOR A TRANSVERSALLY ISOTROPIC . . . 61
and the components of the strain tensor
ε =
1
2
(∇uT +∇u) ⇔ εhk =
1
2
(
∂ uh
∂ xk
+
∂ uk
∂ xh
)
are 8
ε11 =
1
2πρ2
(4πa4x1 + 4πa1x1
2 − h1x2 + 4πa2x1x2 + k3x2
2 + 4πa3x1x3 − k2x2x3) + a1 log ρ2,
ε22 =
1
2πρ2
(h2x1 + k3x1
2 + 4πb4x2 + 4πb1x1x2 + 4πb2x2
2 − k1x1x3 + 4πb3x2x3) + b2 log ρ2,
ε33 = l3 log ρ2,
(6)
ε12 =
1
4πρ2
((h1 + 4πb4)x1 + 4πb1x1
2 − (h2 − 4πa4)x2 − 2(k3 − 2πa1 − 2πb2)x1x2 +
+ 4πa2x2
2 + (k2 + 4πb3)x1x3 + (k1 + 4πa3)x2x3) +
1
2
(a2 + b1) log ρ2,
ε13 =
1
4πρ2
(4πl4x1 + 4πl1x1
2 − h3x2 + (k2 + 4πl2)x1x2 − k1x2
2 + 4πl3x1x3) +
+
1
2
(a3 + l1) log ρ2,
ε23 =
1
4πρ2
(h3x1 − k2x1
2 + 4πl4x2 + (k1 + 4πl1)x1x2 + 4πl2x2
2 + 4πl3x2x3) +
+
1
2
(b3 + l2) log ρ2.
In analogy to Volterra’s procedure, to calculate the unknown constants a1, a2, a3, a4, b1, b2, b3, b4,
l1, l2, l3, l4, we have to impose the verification of the indefinite equations of equilibrium and the
boundary conditions on the field of displacements. 9
2.1. Constitutive equations. If a homogeneous body, linearly elastic and transversally isotro-
pic, experiences an isotermic displacement at an assigned temperature τ , and departs from its
natural state C∗τ , then its isotermic strain-energy-function Wτ can be written in the form
Wτ (ε) =
1
2
A(ε211 + ε222) +
1
2
Cε233 + (A− 2N)ε11ε22 + F (ε11 + ε22)ε33 + 2L(ε223 + ε213) + 2Nε212
8 In engineering practice the characteristics of strain, ehk = εhk if h = k, ehk = 2εhk if h 6= k, are usually used
(see e.g. [1, p. 39], art.10).
9 It’s evident the strain (6) proves to be congruent, in that De Saint-Venant’s conditions of congruence are
automatically verified, independently of the value of the unknown constants.
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
62 E. LASERRA, M. PECORARO
(see e.g. [1, p. 160], (16) or [7], Capter V, §2), where the coefficients A, C, F , L, N are the elastic
constants10 of the cylinder Cτ and are, by hypotesis, not vanishing and different from each other.
Given the tensor field of stress σ ≡ ||σ(τ)
hk || at a temperature τ , the constitutive equations
take the form11
σ
(τ)
hh = −∂Wτ
∂εhh
, h = 1, 2, 3, σ
(τ)
hk = −1
2
∂Wτ
∂(εhk)
, (h, k) = (2, 3), (1, 3), (1, 2) (7)
(see e.g. [1, 5]).
Writing (7) in an explicit form, we obtain the stress-strain relations:
σ
(τ)
11 = −Aε11 − (A− 2N)ε22 − Fε33, σ
(τ)
12 = −2Nε12,
σ
(τ)
22 = − (A− 2N)ε11 −Aε22 − Fε33, σ
(τ)
13 = −2Lε13, (8)
σ
(τ)
33 = − F (ε11 + ε22)− Cε33, σ
(τ)
23 = −2Lε23.
Taking into account the relationship between the displacement gradient and the strain tensor,
the preceding relations can also be written:
σ
(τ)
11 = −A∂ u1
∂ x1
− (A− 2N)
∂ u2
∂ x2
− F ∂ u3
∂ x3
, σ
(τ)
12 = −N
(
∂ u1
∂ x2
+
∂ u2
∂ x1
)
,
σ
(τ)
22 = −A∂ u2
∂ x2
− (A− 2N)
∂ u1
∂ x1
− F ∂ u3
∂ x3
, σ
(τ)
13 = −L
(
∂ u1
∂ x3
+
∂ u3
∂ x1
)
, (9)
σ
(τ)
33 = − F
(
∂ u1
∂ x1
+
∂ u2
∂ x2
)
− C∂ u3
∂ x3
, σ
(τ)
23 = −L
(
∂ u2
∂ x3
+
∂ u3
∂ x2
)
(see e.g. [5]).
Through the stress-strain relations (8) or the preceding equations (10), we obtain the followi-
ng explicit expression for the components of the stress tensor σ (relative to the field of di-
10 Since the cylinder is homogeneous, the coefficients A, C, F , L, N are constant.
11 The signs of σhk are chosen conforming to [4, 7], so a pressure is a positive stress and a tension is a negative
stress, as it is usual in theoretical mechanics (see e.g. [8]). Many authors define the stress tensor with the opposite
sign from the definition adopted here (see e.g. [1]), as it is almost universal in engineering practice.
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
VOLTERRA’S THEORY OF ELASTIC DISLOCATIONS FOR A TRANSVERSALLY ISOTROPIC . . . 63
splacements (4))
σ
(τ)
11 =
1
2πρ2
(((A− 2N)h2 + 4π Aa4)x1 + (−Ah1 + 4π b4 (A− 2N))x2 + ((A− 2N)k3 +
+ 4π Aa1)x1
2 + 4π(Aa2 + (A − 2N) b1)x1 x2 + (Ak3 − π (N − A) b2)x2
2 +
+ ((2N −A) k1 + 4π Aa3)x1 x3 + (−Ak2 + 4π(A− 2N ) b3)x2 x3)−
− (Aa1 + (2N −A)b2 + Fl3) log ρ2,
σ
(τ)
22 = − 1
2πρ2
(Ah2 + 4 a4 (A− 2N) π)x1 + 4a1 (A− 2N)πx1
2 + (h1 (−A+ 2N) +
+ 4Ab4 π)x2 + 4 (Ab1 + a2 (A− 2N))π x1 x2 − 2 (k3N − 2Ab2 π)x2
2 +
+ (−Ak1 + 4 a3 (A− 2N) π) x1 x3 + (k2 (−A+ 2N) + 4Ab3 π) x2 x3)− k3
2π
A−
− (Ab2 + F l3 + a1 (A− 2N)) log ρ2,
σ
(τ)
33 = − F k3
2π
− F
2πρ2
((h2 + 4 a4 π)x1 + 4 a1 π x1
2 + (−h1 + 4 b4 π) x2 + 4 (a2 + b1) π x1 x2 +
+ 4 b2 π x2
2 (−k1 + 4 a3 π) x1 x3 + (−k2 + 4 b3 π) x2 x3)− (F (a1 + b2) + C l3) log ρ2,
(10)
σ
(τ)
12 = − N
2πρ2
((h1 + 4 b4 π)x1 + 4 b1 π x1
2 + (−h2 + 4 a4 π)x2 + (−2 k3 + 4 a1 π +
+ 4 b2 π)x1 x2 + 4 a2 π x2
2 + (k2 + 4 b3 π)x1 x3 + (k1 + 4 a3 π)x2 x3)−
− (a2 + b1)N log ρ2,
σ
(τ)
13 = − L
2πρ2
(
4 l4 π x1 + 4 l1 π x1
2 − h3 x2 + (k2 + 4 l2 π) x1 x2 − k1 x2
2 + 4 l3 π x1 x3
)
−
− L (a3 + l1) log ρ2,
σ
(τ)
23 = − L
2πρ2
(
h3 x1 − k2 x1
2 + 4 l4 π x2 + (k1 + 4 l1 π)x1 x2 + 4 l2 π x2
2 + 4 l3 π x2 x3
)
−
− L (b3 + l2) log ρ2.
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64 E. LASERRA, M. PECORARO
Remark. Since the cylinder is transversally isotropic, the stress-strain relation (8) are invari-
ant with respect to the exchange of the axes x1 and x2. We can obtain from (8), (10) the stress-
strain relations for a homogeneous and isotropic cylinder by making them invariant with respect
to the exchange of any pair of axes or to the directional change of any one of the axes (see e.g.
[1, p. 102]). The five elastic constants therefore, must verify the following three conditions of
isotropy:
C = A, L = N, F = A− 2N (11)
reduce the independent elastic constants to only A and N .
2.2. Indefinite equation. Because C is initially found in a natural state, Cauchy’s static equati-
ons, in absence of the force of mass, must be verified:
divσ = 0 ∀P ∗ ∈ C∗τ . (12)
If (12) is projected onto the axes, and expressions (9) of the stress-tensor are taken into account,
we obtain
A
∂2u1
(∂ x1)2
+N
∂2u1
(∂ x2)2
+ L
∂2u1
(∂ x3)2
+ (A−N)
∂2u2
∂ x1∂ x2
+ (F + L)
∂2u3
∂ x1∂ x3
= 0,
N
∂2u2
(∂ x1)2
+A
∂2u2
(∂ x2)2
+ L
∂2u2
(∂ x3)2
+ (A−N)
∂2u1
∂ x1∂ x2
+ (F + L)
∂2u3
∂ x2∂ x3
= 0, (13)
L
∂2u3
(∂ x1)2
+ L
∂2u3
(∂ x2)2
+ C
∂2u3
(∂ x3)2
+ (F + L)
∂2u1
∂ x1∂ x3
+ (F + L)
∂2u2
∂ x2∂ x3
= 0
(see e.g. [5]).
Referring to (13) or the expressions of stress (10), the equations of equilibrium (12) can be
written in the explicit forms:
(divσ)1 =
1
2πρ4
((A−N)(4πa4 + h2)x1
2 + 2(A−N)(4πb4 − h1)x1x2 −
− (A−N)(4πa4 + h2)x2
2 − 2(2πa1(A+N) + 2π(A−N)b2 −Nk3 +
+ 2π(F + L)l3)x1
3 − 4π((3N −A)a2 − (A−N)b1)x1
2x2 +
+ (A−N)(4πa3 − k1)x1
2x3 + 2(2π(N − 3A)a1 + 2π(A−N)b2 +Nk3 −
− 2π(F + L)l3)x1x2
2 + 2(A−N)(4πb3 − k2)x1x2x3 − 4π((A+N)a2 +
+ (A−N)b1)x2
3 − (A−N)(4πa3 − k1)x2
2x3) = 0, (14)
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VOLTERRA’S THEORY OF ELASTIC DISLOCATIONS FOR A TRANSVERSALLY ISOTROPIC . . . 65
(divσ)2 =
1
2πρ4
((A−N)(h1 − 4πb4)x1
2 + 2(A−N)(h2 + 4πa4)x1x2 + (A−N)(4πb4 −
− h1)x2
2 − 4π((A−N)a2 + (A+N)b1)x1
3 + 2(2π(A−N)a1 + 2π(−3A+
+N)b2 +Nk3 − 2π(F + L)l3)x1
2x2 + (A−N)(k2 − 4πb3)x1
2x3 +
+ 4π((A−N)a2 + (A− 3N)b1)x1x2
2 + 2(A−N)(4πa3 − k1)x1x2x3 −
− 2(2π(A−N)a1 + 2π(A+N)b2 −Nk3 + 2π(F + L)l3)x2
3 −
− (A−N)(4πb3 − k2)x2
2x3) = 0, (15)
(divσ)3 =− 1
2πρ2
((4π(F + L)a3 + (L− F )k1 + 8πLl1)x1 + (4π(F + L)b3 +
+ (L− F )k2 + 8πLl2)x2) = 0. (16)
If we make the coefficients of the various monomials (x1)j(x2)k(x3)l equal to zero, we arrive at
a system of linear equations, not all independent, for the unknowns ah, bh, lh, h = 1, 2, 3, 4. By
annulling the coefficients of the monomials x2
1x3 (or x2
2x3 ), x2
1 (or x2
2), x1x2x3, x1x2, in (14) (or
in (15)) we find respectively (in the aforementioned hypothesis A 6= N)
a3 =
k1
4π
, a4 = −h2
4π
, b3 =
k2
4π
, b4 =
h1
4π
. (17)
If we annul the coefficients of the monomials x2
1x2, x3
2 in (14) (or the coefficients of the monomi-
als x1x
2
2, x3
1 in (15)), we obtain the homogeneous system
(3N −A)a2 + (N −A)b1 = 0,
(N +A)a2 + (N −A)b1 = 0
where the only solution is a2 = 0, b1 = 0 (in the same hypothesis A 6= N). If we annul the
coefficient of x1 in (16), we obtain the equation
4π(F + L)a3 − (F − L)k1 + 8πLl1 = 0.
And if we take into account the calculated value of a3, we find l1 = − k1
4π
. By annulling the
coefficient of x2 in (16), we obtain the equation
4π(F + L)b3 − (F − L)k2 + 8πLl2 = 0.
And by taking into account the calculated value of b3, we find l2 = − k2
4π
.
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66 E. LASERRA, M. PECORARO
Finally, by annulling the coefficients of the monomials x3
1, x1x
2
2 in (14) and the coefficients
of the monomials x3
2, x2
1x2 in (15), we obtain the following subsystem of four linear equations
in the three unknowns a1, b2, l3:
2π[(A+N)a1 + (A−N)b2 + (F + L)l3] = Nk3,
2π(3A−N)a1 − 2π(A−N)b2 + 2π(F + L)l3 = Nk3,
(18)
2π(A−N)a1 + 2π(A+N)b2 + 2π(F + L)l3 = Nk3,
2π(N −A)a1 + 2π(3A−N)b2 + 2π(F + L)l3 = Nk3
which has the∞ solutions12
b2 = a1, l3 =
Nk3 − 4πAa1
2π(F + L)
. (19)
In summary we can say that the displacement (3) satisfies the indefinite equations of elastic
equilibrium, and therefore can be considered as an elastic displacement in an equilibrium
problem if the following conditions are verified (and A 6= N):
a2 = 0, a3 =
k1
4π
, a4 = −h2
4π
,
b1 = 0, b2 = a1, b3 =
k2
4π
, b4 =
h1
4π
, (20)
l1 = −a3 = − k1
4π
, l2 = −b3 = − k2
4π
, l3 =
Nk3 − 4πAa1
2π(F + L)
,
while the constants a1 and l4 are still undetermined.
So the Cartesian components of the displacement (3), which satisfy the indefinite equations
(12), take the form
u1 =
1
2π
(h1 − k3x2 + k2x3) θ +
(
−h2
4π
+
k1
4π
x3 + a1x1
)
log ρ2,
u2 =
1
2π
(h2 + k3x1 − k1x3) θ +
(
h1
4π
+
k2
4π
x3 + a1x2
)
log ρ2, (21)
u3 =
1
2π
(h3 − k2x1 + k1x2) θ −
(
k1
4π
x1 +
k2
4π
x2 − l3x3 − l4
)
log ρ2
where l3 has the expression (19).
12 It is sufficient to consider any two of (18) to obtain (19)1. Therefore (18) reduce to the unique equation
4πAa1 −Nk3 + 2π(F + L)l3 = 0.
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VOLTERRA’S THEORY OF ELASTIC DISLOCATIONS FOR A TRANSVERSALLY ISOTROPIC . . . 67
2.2.1. Deformation, stress and forces on the boundary. If (21) are introduced into (6), we
find the following expression for strain tensor, relative to the vector field of displacements (21)
ε11 =
1
2πρ2
(−h2x1 − h1x2 + 4πa1x1
2 + k3x2
2 + k1x1x3 − k2x2x3) + a1 log ρ2,
ε22 =
1
2πρ2
(h2x1 + h1x2 + 4πa1x2
2 + k3x1
2 − k1x1x3 + k2x2x3) + a1 log ρ2,
ε33 = l3 log ρ2,
(22)
ε12 =
1
2πρ2
(h1x1 − h2x2 − (k3 − 4πa1)x1x2 + k2x1x3 + k1x2x3,
ε13 =
1
4πρ2
(4πl4x1 − h3x2 + 4πl3x1x3)− k1
4π
,
ε23 =
1
4πρ2
(4πl4x2 + h3x1 + 4πl3x2x3)− k2
4π
.
If (22) are substituted into (10), which express the components of the stress tensor by those of
the tensor of deformation, we obtain the stress which satisfies Cauchy’s equations of equilibri-
um (12):
σ
(τ)
11 =
N
πρ2
(
h2x1 + h1x2 + k3x1
2 + 4πa1x2
2 − k1x1x3 + k2x2x3
)
−
− A
2π
(k3 + 4πa1) + [2a1(N −A)− Fl3] log ρ2,
σ
(τ)
22 =
N
πρ2
(
−h2x1 − h1x2 + k3x2
2 + 4πa1x1
2 + k1x1x3 − k2x2x3
)
−
− A
2π
(k3 + 4πa1) + [2a1(N −A)− Fl3] log ρ2,
σ
(τ)
33 = − F
2π
(k3 + 4πa1)− (2Fa1 − Cl3) log ρ2,
(23)
σ
(τ)
12 =
N
πρ2
[−h1x1 + h2x2 + (k3 − 4πa1)x1x2 − k2x1x3 − k1x2x3] ,
σ
(τ)
13 =
L
2πρ2
(h3x2 − 4πl4x1 − 4πl3x1x3) +
Lk1
2π
,
σ
(τ)
23 =
L
2πρ2
(−h3x1 − 4πl4x2 − 4πl3x2x3) +
Lk2
2π
.
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68 E. LASERRA, M. PECORARO
2.2.2. Boundary conditions. Finally, the boundary conditions must be verified,
σ · n− f = 0 ⇐⇒ σh1n1 + σh2n2 + σh3n3 = fh, h = 1, 2, 3, ∀Q∗ ∈ ∂C∗τ , (24)
where f(Q∗) are the vectorial surface forces and n ≡ (n1, n2, n3) is the unitary vector normal
to ∂C∗τ , with an internal orientation.
To calculate the forces which we must apply on the boundary of the cylinder to determine
the principal displacement (21), we divide the boundary ∂C∗τ into the parts Σ∗1, Σ∗2, α∗1 and
α∗2. By taking into account (23) and projecting (24) onto the axes, we obtain four groups of
equations.
First we consider the boundary conditions on the two lateral surfaces:
1) on Σ∗1 ( n = c1 cos θ + c2 sin θ, x1 = r cos θ, x2 = r sin θ, 0 ≤ x3 ≤ d )
[
σ
(τ)
11 cos θ+σ(τ)
12 sin θ
]
Σ∗1
=
1
2πr
[2Nh2 + ((2N −A)k3 − 4πAa1)x1 − 2Nk1x3 +
+ 4π(N −A)a1x1 log ρ2 − 2πF l3x1 log ρ2]Σ∗1 = (f1)Σ∗1 ,
[
σ
(τ)
12 cos θ+σ(τ)
22 sin θ
]
Σ∗1
=
1
2πr
[−2Nh1 + ((2N −A)k3 − 4πAa1)x2 − 2Nk2x3 + (25)
+ 4π(N −A)a1x2 log ρ2 − 2πF l3x2 log ρ2]Σ∗1 = (f2)Σ∗1 ,
[
σ
(τ)
13 cos θ+σ(τ)
23 sin θ
]
Σ∗1
=
L
2πr
[−4πl4 + k1x1 + k2x2 − 4πl3x3] = (f3)Σ∗1 ;
2) on Σ∗2 ( n = −c1 cos θ − c2 sin θ, x1 = R cos θ, x2 = R sin θ, 0 ≤ x3 ≤ d )
[
−σ(τ)
11 cos θ − σ(τ)
12 sin θ
]
Σ∗2
=
1
2πR
[−2Nh2 − ((2N −A)k3 − 4πa1)x1 + 2Nk1x3 −
− 4π(N −A)a1x1 log ρ2 + 2πF l3x1 log ρ2]Σ∗2 = (f1)Σ∗2 ,
[
−σ(τ)
12 cos θ − σ(τ)
22 sin θ
]
Σ∗2
=
1
2πR
[2Nh1 − ((2N −A)k3 − 4πa1)x2 + 2Nk2x3 − (26)
− 4π(N −A)a1x2 log ρ2 + 2πF l3x2 log ρ2]Σ∗2 = (f2)Σ∗2 ,
[
−σ(τ)
13 cos θ − σ(τ)
23 sin θ
]
Σ∗2
=
L
2πR
[4πl4 − k1x1 − k2x2 + 4πl3x3] = (f3)Σ∗2 .
We can divide the surface forces (25), (26) applied on Σ∗1 and on Σ∗2 into the following vector
fields, all equivalent to zero:
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VOLTERRA’S THEORY OF ELASTIC DISLOCATIONS FOR A TRANSVERSALLY ISOTROPIC . . . 69
i)
{
P ∗, f (1)′(P ∗) dΣ∗1
}
Σ∗1
, consisting of couples of zero arms and so equivalent to zero, in
fact
[
f
(1)
1
′
(x1, x2, x3)
]
Σ∗1
=
1
2πr
[
((2N −A)k3 − 4πAa1)x1 − 2Nk1x3 + 4π(N −A)a1x1 log ρ2−
−2πF l3x1 log ρ2
]
Σ∗1
= −
[
f
(1)
1
′
(−x1,−x2, x3)
]
Σ∗1
,
[
f
(1)
2
′
(x1, x2, x3)
]
Σ∗1
=
1
2πr
[
((2N −A)k3 − 4πAa1)x2 − 2Nk2x3 + 4π(N −A)a1x2 log ρ2−
−2πF l3x2 log ρ2
]
Σ∗1
= −
[
f
(1)
2
′
(−x1,−x2, x3)
]
Σ∗1
,
[
f
(1)
3
′
(x1, x2, x3)
]
Σ∗1
=
L
2πr
[−4πl4 + k1x1 + k2x2 − 4πl3x3] = −
[
f
(1)
3
′
(−x1,−x2, x3)
]
Σ∗1
;
ii)
{
P ∗, f (2)′(P ∗)dΣ∗2
}
Σ∗2
, which, analogously, consists of couples of zero arms;
iii) the pair of two constant vector fields, parallel and opposite,
{
f (1)′′dΣ∗1
}
Σ∗1
≡
{
f (1)′′2πrdx3
}
Σ∗1
and
{
f (2)′′dΣ∗2
}
Σ∗2
≡
{
f (2)′′2πRdx3
}
Σ∗2
(f (1)
1 )′′Σ∗1dΣ∗1 = 2Nh2dx3, (f (1)
2 )′′Σ∗1dΣ∗1 = −2Nh1dx3, (f (1)
3 )′′Σ∗1dΣ∗1 = −4πLl4dx3,
(27)
(f (2)
1 )′′Σ∗2dΣ∗2 = −2Nh2dx3, (f (2)
2 )′′Σ∗2dΣ∗2 = 2Nh1dx3, (f (2)
3 )′′Σ∗2dΣ∗2 = 4πLl4dx3,
that together have, by symmetry, their center coincident with the center of the cylinder, and are
equivalent to their resultant applied to the center. Since their two resultants r′′(1) and r′′(2) are13
r
(1)
1
′′
= 2Nh2d , r
(1)
2
′′
= −2Nh1d , r
(1)
3
′′
= −4πLl4d,
(28)
r
(2)
1
′′
= −2Nh2d , r
(2)
2
′′
= 2Nh1d , r
(2)
3
′′
= 4πLl4d,
it is obvious that, being r′′(1) + r′′(2) = 0, also the vector field {(P ∗, f (1)′′)Σ∗1 , (P
∗, f (2)′′)Σ∗2} is
equivalent to a couple of zero arm.
Finally we consider the boundary conditions on the two bases:
13 r(1)′′ =
R
Σ∗1
f (1)′′dΣ∗1 = f (1)′′2πrd, r(2)′′ =
R
Σ∗2
f (2)′′dΣ∗2 = f (2)′′2πRd .
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70 E. LASERRA, M. PECORARO
3) on α∗1 ( n = c3 ≡ (0, 0, 1) , x1 = ρ cos θ, x2 = ρ sin θ, x3 = 0),
[σ(τ)
13 ]α∗1 =
L
2π
[
k1 +
1
ρ2
(h3x2 − 4πl4x1)
]
α∗1
= (f1)α∗1 ,
[σ(τ)
23 ]α∗1 =
L
2π
[
k2 −
1
ρ2
(h3x1 + 4πl4x2)
]
α∗1
= (f2)α∗1 , (29)
[σ(τ)
33 ]α∗1 =
[
F
2π
(k3 + 4πa1) + (2Fa1 + Cl3) log ρ2
]
α∗1
= (f3)α∗1 ;
4) on α∗2 ( n = −c3 ≡ (0, 0,−1), x1 = ρ cos θ, x2 = ρ sin θ, x3 = d)
[−σ(τ)
13 ]α∗2 =− L
2π
[
k1 +
1
ρ2
(h3x2 − 4π(l3d+ l4)x1)
]
α∗2
= (f1)α∗2 ,
[−σ(τ)
23 ]α∗2 =− L
2π
[
k2 −
1
ρ2
(h3x1 + 4π(l3d+ l4)x2)
]
α∗2
= (f2)α∗2 , (30)
[−σ(τ)
33 ]α∗2 =−
[
F
2π
(k3 + 4πa1) + (2Fa1 + Cl3) log ρ2
]
α∗2
= (f3)α∗2 .
Also the surface forces exerting on the two bases must be equivalent to zero. So, taking into
consideration (20), (30), we must put
l3 = 0, (31)
and consequently we obtain from (19)
a1 =
N
A
k3
4π
. (32)
Condition (31) together with (32) must be verified so that the surface forces corresponding to the
dislocation are equivalent to zero, therefore l3 and a1 are not arbitrary.
2.3. Vector field of displacements. From (31), (32) we can now write the definitive expression
for the vector field of displacements (21),
u1 =
1
2π
{
(h1 − k3x2 + k2x3) θ +
1
2
(
−h2 + k1x3 +
N
A
k3x1
)
log ρ2
}
,
u2 =
1
2π
{
(h2 + k3x1 − k1x3) θ +
1
2
(
h1 + k2x3 +
N
A
k3x2
)
log ρ2
}
, (33)
u3 =
1
2π
{
(h3 − k2x1 + k1x2) θ − 1
2
(k1x1 + k2x2 − 4πl4) log ρ2
}
.
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VOLTERRA’S THEORY OF ELASTIC DISLOCATIONS FOR A TRANSVERSALLY ISOTROPIC . . . 71
These formulas demonstrate that the vector field of displacements depends exclusively
on the ratio N/A of only two of the five elastic constants which characterize a transversally
isotropic elastic body.
Remark. If we impose the isotropy conditions (11) on the five elastic constants, and put
A = λ + 2µ, N = µ and l4 = 0, we once again obtain, from (33), Volterra’s formulas (2)
relative to the isotropic case.
2.4. Deformation, stress and surface forces corresponding to the established displacement.
If we substitute the values l3 = 0 and a1 =
N
A
k3
4π
in (5), (22), (23), and (25) – (30), we obtain
the following explicit definitive expressions (corresponding to the displacement (33)):
1) for the gradient of displacement ∇u,
∂u1
∂x1
=
1
2πρ2
(−h2 x1 − h1 x2 +
N
A
k3 x1
2 + k3 x2
2 + k1 x1 x3 − k2 x2 x3) +
N
A
k3
4π
log ρ2,
∂u1
∂x2
=
1
2πρ2
(
h1 x1 − h2 x2 +
(
N
A
− 1
)
k3 x1 x2 + k2 x1 x3 + k1 x2 x3
)
− k3
2π
arctan θ,
∂u1
∂x3
=
k2
2π
arctan θ +
k1
4π
log ρ2,
∂u2
∂x1
=
1
2πρ2
(
h1 x1 − h2 x2 +
(
N
A
− 1
)
k3 x1 x2 + k2 x1 x3 + k1 x2 x3
)
+
k3
2π
arctan θ,
∂u2
∂x2
=
1
2πρ2
(h2 x1 + h1 x2 +
N
A
k3 x2
2 + k3 x1
2 − k1 x1 x3 + k2 x2 x3) +
N
A
k3
4π
log ρ2, (34)
∂u2
∂x3
= − k1
2π
arctan θ +
k2
4π
log ρ2,
∂u3
∂x1
=
1
2πρ2
(4π l4 x1 − h3 x2 − k1 x1
2 − k1 x2
2)− k2
2π
arctan θ +
k1
4π
log ρ2,
∂u3
∂x2
=
1
2πρ2
(4π l4 x2 + h3 x1 − k2 x1
2 − k2 x2
2) +
k1
2π
arctan θ − k2
4π
log ρ2,
∂u3
∂x3
= 0;
2) for the tensor field of strain,
ε11 =
1
2πρ2
(
−h2x1 − h1x2 +
N
A
k3x1
2 + k3x2
2 + k1x1x3 − k2x2x3
)
+
N
A
k3
4π
log ρ2,
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
72 E. LASERRA, M. PECORARO
ε22 =
1
2πρ2
(
h2x1 + h1x2 +
N
A
k3x2
2 + k3x1
2 − k1x1x3 + k2x2x3
)
+
N
A
k3
4π
log ρ2,
ε33 = 0,
(35)
ε12 =
1
2πρ2
(
h1x1 − h2x2 +
(
N
A
− 1
)
k3x1x2 + k2x1x3 + k1x2x3
)
,
ε13 =
1
4πρ2
(4πl4x1 − h3x2)− k1
4π
,
ε23 =
1
4πρ2
(4πl4x2 + h3x1)− k2
4π
;
3) for the tensor field of stress,
σ
(τ)
11 =
N
πρ2
(
h2x1 + h1x2 +
N
A
k3x2
2 + k3x1
2 − k1x1x3 + k2x2x3
)
−
− (N +A)
k3
2π
+ (N −A)
N
A
k3
2π
log ρ2,
σ
(τ)
22 = − N
πρ2
(
h2x1 − h1x2 +
N
A
k3x1
2 + k3x2
2 + k1x1x3 − k2x2x3
)
−
− (N +A)
k3
2π
+ (N −A)
N
A
k3
2π
log ρ2,
σ
(τ)
33 = − F
(
N
A
+ 1
)
k3
2π
− F N
A
k3
2π
log ρ2,
(36)
σ
(τ)
12 =
N
πρ2
(
−h1x1 + h2x2 −
(
N
A
− 1
)
k3x1x2 − k2x1x3 − k1x2x3
)
,
σ
(τ)
13 = − L
2πρ2
(4πl4x1 − h3x2) +
Lk1
2π
,
σ
(τ)
23 = − L
2πρ2
(4πl4x2 + h3x1) +
Lk2
2π
;
4) for the surface forces, respectively, on
Σ∗1 =
(f1)Σ∗1 =
1
2πr
[
2Nh2 +
(
N
A
− 1
)
(A+N log ρ2)k3x1 − 2Nk1x3
]
Σ∗1
,
(f2)Σ∗1 =
1
2πr
[
−2Nh1 +
(
N
A
− 1
)
(A+N log ρ2)k3x2 − 2Nk2x3
]
Σ∗1
,
(f3)Σ∗1 =
L
2πr
[−4πl4 + k1x1 + k2x2]Σ∗1 ,
(37)
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
VOLTERRA’S THEORY OF ELASTIC DISLOCATIONS FOR A TRANSVERSALLY ISOTROPIC . . . 73
Σ∗2 =
(f1)Σ∗2 =
1
2πR
[
−2Nh2 −
(
N
A
− 1
)
(A+N log ρ2)k3x1 + 2Nk1x3
]
Σ∗2
,
(f2)Σ∗2 =
1
2πR
[
2Nh1 −
(
N
A
− 1
)
(A+N log ρ2)k3x2 + 2Nk2x3
]
Σ∗2
,
(f3)Σ∗2 =
L
2πR
[4πl4 − k1x1 − k2x2]Σ∗2 ,
(38)
α∗1 =
(f1)α∗1 =
L
2π
[
k1 +
1
ρ2
(h3x2 − 4πl4x1)
]
α∗1
,
(f2)α∗1 =
L
2π
[
k2 −
1
ρ2
(h3x1 + 4πl4x2)
]
α∗1
,
(f3)α∗1 = −Fk3
2π
[
1 +
(
1 + log ρ2
) N
A
]
α∗1
,
(39)
α∗2 =
(f1)α∗2 =
L
2π
[
−k1 −
1
ρ2
(h3x2 − 4πl4x1)
]
α∗2
,
(f2)α∗2 =
L
2π
[
−k2 +
1
ρ2
(h3x1 + 4πl4x2)
]
α∗2
,
(f3)α∗2 =
Fk3
2π
[
1 +
(
1 + log ρ2
) N
A
]
α∗2
.
(40)
Conclusion. If a hollow elastic cylinder, homogeneous and transversally isotropic C, is initi-
ally found in a natural state C∗τ and experiences a many-valued isotermic displacement C∗τ → Cτ
as in (21) (and consequently a regular deformation), then, in the equilibrium configuration Cτ ,
stress (36) and congruent deformation (35) are present in every internal point; and surface
forces (37) – (40) equivalent to zero are exerted on the boundary.
Remark. While displacement (33) and tensor field of strain (35)) depend only on the ratio
N/A, the tensor field of stress (36) depends on four of the five elastic constants.
Acknowledgements. The authors wish to express their heartfelt gratitude to professor
G. Caricato for his very helpful suggestions.
1. Love A. E. H. The mathematical theory of elasticity. — Fourth edition. — Cambridge Univ. Press, 1952.
2. Volterra V. Sur l’equilibre des corps elastiques multiplement connexes. — Paris: Gauthier-Villars, Imprimeur-
libraire, 1907.
3. Grioli G. Le distorsioni elastiche e l’opera di Vito Volterra // Atti Convegni Lincei. — 1992. — № 92. —
P. 271 – 289.
4. Caricato G. On the Volterra’s distortions theory // AIMETA’ 99, 14th Ital. Conf. Theor. and Appl. Mechanics,
Villa Olmo (Como), Italy.
5. Caricato G. On the Volterra’s distortions theory // Meccanica. — 2000. — 35. — P. 411 – 420.
6. Weingarten M. Sur les surfaces de discontinuitè dans la théorie des corps solides // Rend. Accad. Lincei. —
1901. — 10. — P. 57 – 60.
7. Signorini A. Lezioni di Fisica Matematica. — Roma: Libreria Eredi Virgilio Veschi, 1953.
8. Symon K. R.Mechanics. — New York: Addison-Wesley, 1979.
Received 11.03.2002
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
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