#### On the mathematical models of the Timoshenko-type multi-layer flexible orthotropic shells

On the mathematical models of the Timoshenko-type multi-layer flexible orthotropic shells
V. A. Krysko 0 1 2 3
J. Awrejcewicz 0 1 2 3
M. V. Zhigalov 0 1 2 3
I. V. Papkova 0 1 2 3
T. V. Yakovleva 0 1 2 3
A. V. Krysko 0 1 2 3
0 T. V. Yakovleva
1 J. Awrejcewicz (
2 V. A. Krysko
3 A. V. Krysko Department of Applied Mathematics and Systems Analysis, Saratov State Technical University , Politehnicheskaya 77, Saratov, Russian Federation 410054
Mathematical models of multi-layer orthotropic shells were reconsidered based on the Timoshenko hypothesis. A new mathematical model with ε-regularisation was proposed, and the theorem regarding the existence of a generalised solution to the model was formulated and proved. The algorithms of numerical investigation of models studied with the aid of the variational-difference method were developed. The associated stability problem was also addressed. A comparison of the results yielded by the considered models was carried out and discussed for numerous factors and parameters.
Buckling; Vibrations; Shells; Structures; Finite differences
1 Introduction
The first papers devoted to the investigation of
vibrations of piecewise non-homogeneous (with respect to
thickness) multi-layer/composite plates and shells with
the account of the geometric nonlinearity appeared in
the sixties of the twentieth century. In these works,
numerical computations of three-layer structural
members with lightly stiffened internal layer (filler) have
been used to show that inclusion of transverse shear
and normal forces as well as deformations in the filler
is required during modelling. Usually, owing to the
complexity of nonlinear PDEs governing the
dynamics of multi-layer plates and shells made from either
orthotropic or anisotropic composite materials, the
solutions to PDEs have been constructed by taking into
account only small nonlinear vibrations and the
firstorder approximation.
The first pioneering results related to the
investigation of three-layer plates and shells have been reported
by [
1–8
], and other. In references [
1, 2
], three-layer
orthotropic rectangular plates subjected to normal loads
have been studied. Within the study, the governing
PDEs have been derived and the dependency of
nonlinear vibrations on the amplitude and frequency has
been obtained in the first approximation.
Yu [
5
] found the approximate solution to the
problem of nonlinear axially symmetric vibrations of a
three-layer closed circular cylindrical shell with a light
non-compressed filler and the membrane external
layers. Kholod [
7,8
] has utilised the same deformation
hypotheses regarding the filler while solving the
problem of nonlinear transverse vibrations of a three-layer
cylindrical simply supported panel symmetric with
respect to its height, having a stiff filler and carrying
loads by external layers.
The influence of deformations of the transverse
shear on nonlinear vibrations of composite plates and
shells has been studied in a vast number of
references such as [
6,9–12
]. In one of these works [
6
], it
was pointed out that the influence of the transverse
shear cannot be omitted while investigating nonlinear
vibrations and stability of freely supported multi-layer
plates, and it plays a crucial role in studying clamped
multi-layer plates.
Kogan and Yurchenko [
12
] studied the influence of
geometrical and mechanical parameters (relative plate
thickness h/b, slenderness l = a/b, stiffness of the
filler against the shear factor, bending stiffness of the
carrying load layers, parameter e responsible for
external damping of vibrations, the number of half-waves
m and n) as well as of the type of boundary
conditions on the character of free/forced nonlinear
vibrations and the shape of the amplitude–frequency bone
curves for a three-layer rectangular plate with a
nonsymmetric (with respect to thickness) structure with a
stiff, transversally isotropic filler transmitting the
transverse shear and two isotropic load-carrying layers.
In general, factors such as a multi-layer mechanical
structure, the presence of layers with different physical
and geometrical characteristics, interlayer contact
conditions, well-posed statements of the multi-layer strain
as well as proper implementation of boundary
conditions require more advanced modelling due to
phenomena that are more complex than those exhibited by
simple, reduced-order models.
On the other hand, mathematical models derived
in the form of coupled nonlinear PDEs require the
employment of more rigorous mathematical tools to
reduce infinite dimensional problems to finite
dimensional ones. In other words, it is necessary to reduce
PDEs to truncated systems of nonlinear ODEs, putting
emphasis on adequate modelling of the rich nonlinear
structural behaviour.
Particular features of numerous directions of the
theory development and the construction of
mathematical models of multi-layer plates and shells have been
described in references [
13,14
]. In general, one can
distinguish two main approaches to investigating
nonlinear vibrations of multi-layer plates and shells. Namely,
one can employ: (i) a hypothesis of a straight,
nondeformed normal (to all layers), and reduce the problem
to computing a quasi-homogeneous plate/shell with
reduced elastic parameters [
15–22
]; (ii) non-classical
2D theories based on integral hypotheses for the entire
packet of layers, taking into account deformations and
stresses of the transverse shear in all layers [
5,6,23–
28
].
Ugrimov [
29
] presented the layer-wise generalised
theory of multi-layer shells based on the expansion of
the displacement vector components of each layer into
power series of the transverse coordinate. The results
were validated by the analysis of the strain-stress states
of one- and three-layer structures.
Fernandes [
30
] proposed a so-called mixed
formulation to study elastic multi-layer plates for which the
continuity of transverse shear stress is satisfied in a
natural way. The introduced model was validated by
bilayer and sandwich plates.
Basar and Eckstein [
31
] developed a finite
multilayer shell element formulation with the account of
large inelastic strains and finite rotations. The derived
model was enforced into a multi-layer shell kinematics,
putting emphasis on the proper modelling of stresses
across the thickness. Also, a few examples validating
the introduced theory and computational algorithms
were reported.
Recently, Amabili and his co-workers have
constructed a series of geometric nonlinear theories of the
third order for the shells of general forms. For this
purpose, they have employed five, seven, or eight
parameters [
34–36
]. Since the terms of a higher order with
respect to the transverse coordinate are also taken into
account, the theory is feasible also for thick shells.
The developed theory allows for investigating
largeamplitude vibration excited by periodic load and the
obtained results coincide with those yielded by other
theories. Owing to complexity of the analysed
problems, the investigations have been carried out with
different numerical methods, i.e. the finite difference
method, the finite element method, and the Galerkin
methods.
Gutiérrez Rivera et al. [
37
] developed a new
12parameter finite element method of the shell in order
to analyse large deformations of composite shell
structures by using the third-order model.
In reference [
38
], the finite element method has
been successfully used to study laminated
composite skewed hyper shell roof, taking the account of an
oblique impact with friction boundary elements.
Tornabene et al. [
39
] studied a mechanical behaviour
of damaged laminated composite plates and shells by
employing the higher-order shear deformation theories
and using the accurate and stable generalised
differential quadrature method (GDQ).
Large vibrations of three- and multi-layer plates,
taking into account the initial imperfection, have been
studied in references [
12,24–26
]. In particular, in
references [
24,25
] the parabolic form of distribution of
deformation of the transverse shear along the thickness
of the whole layer has been investigated. Five input
differential equations have been reduced to one nonlinear
second-order equation with square and cubic
nonlinearity. Then, the equation has been solved using the
multiscale approach. It has been also proved that in the case
of either strong or weak nonlinearity, the behaviour of
the plate essentially depends on the initially introduced
imperfections [
25
].
More recently, Wang et al. [
32
] studied spherical
cloaking by using multi-layer ordinary dielectric
materials and optimised the total scattering cross section of
a spherical multi-layer shell with a metallic core.
Jiang et al. [
33
] implemented a reduced-order model
to evaluate the dynamics response of multi-layer plates
under impulsive loads. The introduced approach aimed
at either absorbing or transmitting the energy in
transverse directions is based on the reverberation matrix
method matched with the theory of generalised rays.
The dynamic response of multi-layer plates was
quantified using the generalised ray theory and the inverse
Fourier transformation.
The authors of the present paper have been
investigating the problems of statics and dynamics of
multilayer structural members for many years. In
particular, the multi-layer flexible Euler–Bernoulli and
Timoshenko beams were investigated in reference [
40
],
whereas multi-layer beams with a clearance have been
studied in references [
41,42
]. Also, the non-classical
mathematical models of coupled problems of
thermoelasticity were derived and studied for multi-layer shells
with initial imperfections in [43].
This paper extends the former investigations. It is
organised in the following way. Section 2 comprises the
description of fundamental hypotheses and relations
as well as the derivation of the governing equations.
Section 3 presents the modification of the Timoshenko
model and the ε-regularisation model. It includes
rigorous mathematical considerations including
formulation of the Theorem 1 and its proof. Numerical
investigations of stability of the multi-layer orthotropic shells
within the earlier introduced improved theories are
carried out in Sect. 4. Section 5 reports a comparison of
the stability curves of the multi-layer symmetric shells
based on the “load deflection” relations yielded by
the four studied different shell mathematical models.
Finally, Sect. 6 summarises the carried-out research.
2 Fundamental hypotheses
A multi-layer shallow shell being a part of the 3D space
R3 is considered. The rectangular coordinate system
of the space R3 is introduced as follows. An
arbitrary surface of the shell, called a reference surface
and defined as x3 = 0, is chosen; the axes 0x1,0x2
are directed along the main curvatures of this surface,
whereas the axis 0x3 is directed towards the reference
surface curvature axis. In the given system of
coordinates, the shell is defined in the following manner: =
{(x1, x2, x3)/ (x1, x2) ∈ [0, a] × [0, b] , δ0 − ≤ z ≤
δn+m − }, where the space [0, a] × [0, b] stands for
the rectangular shell planform; δn+m − δ0 = 2h0—
constant shell thickness; x3 = δ0 − —lower shell
surface; x3 = δn+m − —upper shell surface; —
shell thickness measured from the upper surface to the
reference surface x3 = 0.
To develop mathematical models (MM2) based on
the Timoshenko hypothesis, the following assumptions
regarding the shell construction, material properties,
layers and exploitation conditions are introduced.
1. The thickness of the i -th shell layer is denoted by
δi , i = 0, . . . , n + m the; m stands for the
number of layers up to the layer consisting the surface
x3 = 0; n is the number of the remaining layers.
The interval x3 ∈ (δ0 − , δn+m − ) is divided
into subintervals with respect to x3 within one layer
(δi − , δi+1 − ) .
2. It is assumed that the shell is fully shallow, i.e. the
coefficients of the first squared form of the
reference surface are taken as Ai ≈ 1. The main
curvatures and the twist curvature of x3 = 0 are constant
(the Gauss curvature is neglected). Furthermore, it
is assumed that the total thickness of layers is small
compared to the curvature radii, i.e. 1 + Ki j x3 ≈ 1.
3. It is assumed that normal stresses σ33 = 0 are small
compared to the stresses in the state equations.
4. It is assumed that the shell is under a stationary
temperature field. The temperature distribution is
yielded by a solution to the Fourier heat transfer
equation. If T0 stands for the input shell temperature
and T (x1, x2, x3) is the temperature at the time
instant t , then the function = T (x1, x2, x3) − T0
defines the grad as a vector of heat flow.
5. The considered piecewise homogeneous shell is
composed of an arbitrary number of layers of the
same thickness, but different stiffness. The layers
are arbitrarily located with respect to the reference
surface x3 = 0. If the layers are orthotropic, then
in each of them, there is one plane of elastic
symmetry which is parallel to the plane tangent to the
reference surface, and the two remaining planes are
perpendicular to the axes O x1, O x2.
6. The deformable shell state is considered based on
the assumption that deflections of the points of the
reference surface can be of the same order as the
shell thickness [
44
].
7. The components of the displacement vector of the
surfaces x3 = 0 are denoted by u1 = u1 (x1, x2),
u2 = u2 (x1, x2); u3 = u3 (x1, x2) stands for
the shell deflection; u1z = u1 (x1, x2, x3), u2z =
u2 (x1, x2, x3) are displacements of an arbitrary
shell point, and u3z = u3 (x1, x2, x3) is the shell
deflection at an arbitrary point.
8. The external shell surfaces are subjected to the load
vector of the form: q+ = q; q¯ = 0, i.e. the shell
behaviour caused only by the action of the
transverse force is studied.
9. Since only thin flexible shells are considered,
following monograph [
9
], the deformations in an
arbitrary shell layer have the following form
z ∂u1z 1
l11 = ∂ x1 + 2
z ∂u2z 1
l22 = ∂ x2 + 2
∂u3z 2
∂ x1
∂u3z 2
∂ x2
z
− K1u3,
z
− K2u3
z ∂u1z ∂u2z ∂u3z ∂u3z
l12 = ∂ x2 + ∂ x1 + ∂ x1 · ∂ x2
,
(1)
10. One can transit from 3D to 2D theory of shells by
taking into account the hypothesis of the second
approximation, i.e. the Timoshenko-type
hypothesis (Fig. 1).
It is assumed that tangential displacements u1z, uz ,
2
u3z are distributed along the shell thickness in a way
described by the following linear form
u1z = u1 + x3γ1, u2z = u2 + x3γ2, u3z = u3, (2)
where γ1 = γ1 (x1, x2), γ2 = γ2 (x1, x2) denote angles
of the normal rotation with respect to the surface x3 =
0 generated by deformations in the planes x20x3 and
x20x3, respectively. Then, equation (1) with the account
of equation (2) yields
z ∂u1 1
l11 = ∂ x1 + 2
z ∂u2 1
l22 = ∂ x2 + 2
l1z2 = ∂∂ux21 + ∂∂ux12 + ∂∂ux13 · ∂∂ux23 + x3
l1z3 = γ1 + ∂∂ux13 ; l2z3 = γ2 + ∂∂ux23 .
Now, if one denotes by ε11, ε22, ε12 the tangential
deformations of the middle surface as follows
ε11 = ∂∂ux11 + 21 ∂∂ux13 2 − K1u3,
∂γ2 ∂γ1
∂ x2 + ∂ x1
;
∂u2 1
ε22 = ∂ x2 + 2
the shear deformations, then the expressions for
deformation of each layer can be presented in the form of
the linear series development with respect to x3, i.e.
one obtains
z
l12 = ε12 + x3H12,
z
z
l11 = ε11 + x3H11, l22 = ε22 + x3H22,
z z
l13 = ε13, l23 = ε23, x3 ∈ (δi − ,δi+1 − ).
The first three equations of (6), taking into account (8),
The Duhamel–Neuman principle [
45
] employed to
each orthotropic (i-th) layer results in the following
formulas
i i
⎪⎪⎪⎪⎧ l1i1 = E11iiσ1i1 − νE11i2σ2i2 − Eν12ii3σ3i3 + α2i2 ,
ν1i3σ3i3 + α1i1 ,
⎪⎪⎪⎨ l2i2 = −Eν2i2i1σ1i1 + E1i2iσ2i2 − E2
⎪⎪⎪⎪⎪ l3ii3 = −Eνi3i31σ1i1 − νEσ313iii23σ,2il22i3+=E13iGσσ2ii2333i3, + α3i3 ,
σ1i2 ,l1i3 = G13
⎪⎪⎩ l12 = G12
and hence
i
σ22 =
E2i ν1i2 + ν1i3ν3i2 l1i1 +
E2i 1 − ν3i1ν1i3 l2i2
E2i ν3i2 + ν3i1ν1i2 l3i3 − β2i2 ,
E1i 1 − ν2i3ν3i2 l1i1 +
The equilibrium equations and the boundary
conditions are obtained using the principle of virtual
displacements [
47
]. This implies δV = δ A, where V
stands for the potential energy of the shell
deformation, and A is the work done by the external forces.
One gets
δV =
To obtain 2D equations of the shell equilibrium, the
series (4) and the rule for computing of the triple
integrals are used. For each layer, the interval with respect
to x3 ∈ (δ0 − , δntm − ) is divided into intervals
with respect to x3, i.e.
where: E1i, E2i, E3i are the elasticity moduli; ν1i2, ν1i3,
ν3i2, ν3i1, ν2i3—Poisson’s coefficients; Gi12, Gi13, Gi23—
shear moduli of the i -th layer. For the chosen form of
orthotropy of the material layers, the following
relations hold [
5
]:
Now, one can take into account the static hypothesis
σ33 = 0, find a formula for deformations ε3i3, and
subi
stitute it into the first two relations of (6) using (8), (11).
Then, in each shell layer, stress tensor components take
the following form:
and taking into account (4), one can recast the stress
formula to the following form suitable for further
considerations:
k ai+1
k ai+1
Then, the notation n + m − 1 = k, δi − = ai , δi+1 −
Δ = ai+1 can be used and, based on the analogy to the
one-layer shell, one denotes by T11, T22, T12 the internal
stresses; by Q1, Q2—transverse forces; byM11, M22—
bending moments, and byM12—torsional moment:
ϕij · x3dx3,
ϕij · νijl · x3dx3,
ϕij · x32dx3,
ϕij · νijl · x32dx3,
D66 =
Pj j =
Pjl =
K Pj j =
K Pjl =
A44 =
A55 =
and, owing to (10), ϕ1i · V1i2 = ϕ2i · V2i1. It means that
C12 = C21, K12 = K21, D12 = D21. Furthermore,
formulas (18) imply that if the packet of layers is
symmetrical, then the coefficients K j j , K jl , K Pj j , K Pjl
are equal to zero. Then, relations (17), solvable with
respect to deformations, are used:
,
εi j from (3a) is substituted into to the first integral on the
right-hand side of (16), whereas the remaining integrals
take into account expressions from (19). Furthermore,
Mi j is taken from (20) and Hi j and εi3—from (3c).
As a result, the variational principle takes the
following form
M11
∂δγ1
∂ x1
+ ∇k2u3 + 21 L (u3, u3) δ FdS
−
+
+
+
+
+
+
−
(S)
0
0
0
0
0
0
a
a
b
b
b
0
a
a
0
b
0
M12δγ2|ax=0 dx2 +
Q1δu3|ax=0 dx2
= ∂∂x2 (M P22)
3 Modification of the Timoshenko MM2 and the
ε-regularisation MM3
The Timoshenko mathematical model MM1 consists of
differential equations of the 10th order and 5 boundary
conditions. In other mathematical models (for instance,
the Kirchhoff–Love model of the first approximation,
the Pelekh–Sheremetev–Reddy–Levinson model of the
third approximation), a biharmonic operator is present
in the equation considering deflection. The operator
improves stability and convergence of the numerical
algorithm in geometrically nonlinear problems as
compared to the mathematical Timoshenko model of the
second-order approximation. However, construction of
the computational schemes for these models does not
belong to easy tasks and is more difficult than in the
case of MM1. This observation is the motivation to
employ the ε-regularisation [
48
] to establish one more
variant of the improved mathematical model, called
further MM3, which can be treated as a bridge between
the Kirchhoff-Love (MM1) and Timoshenko (MM2)
models.
In one of its variants, MM3 is a system of
differential equations, in which the first equation includes
the term ε 2u3 = ε
∂∂4xu143 + 2 ∂∂2xu223 ∂∂2xu123 + ∂∂4xu243 . The
term u3 − 1−ρu2 · ∂∂un3 ∂ = 0 is added, where: 2—
biharmonic operator, —Laplace operator, ρ—radius
of curvature of the contour ∂ , ν =const > 0 [
49
].
In addition to the problem (23), (24), one can define
the problem of the form
L1 (u3, F, γ1, γ2) + ε 2u3 = q,
L2 (u3, F, γ1, γ2) = q1,
L3 (u3, F, γ1, γ2) = q2,
L4 (u3, F, γ1, γ2) = qF ,
with the boundary conditions
(26)
At first, the “helping problem” was considered, where
the biharmonic operator was replaced with an
arbitrary positive definite operator T with the natural-type
boundary conditions, the energetic space of which is
o
embedded into W22 ( ) ∩ W21 ( ). The fact that the
operator T can be chosen arbitrarily allows for
“construction” of the properties of a total algebraic system
aimed at increasing the computational efficiency of the
employed algorithm.
Introduction of the “helping problem” yielded a
strong convergence of some series of the approximate
solutions Wεn to the exact solution W 0, which is
important while constructing the practical realisation of the
computational algorithm. The similar approach can be
employed to the searched functions γ1, γ2, F if one
substitutes the terms L2 (γ1), L3 (γ2), L4 (F ) in the system
(23, 24) by the following ones: ε1T1γ1, ε2T2γ2, ε3T3 F ,
respectively, where for arbitrary i = 1, 2, 3, εi > 0, Ti
serve as positive operators with natural boundary
conditions, the energetic space of which is compact and
o
embedded into W22 ( ) ∩ W21 ( ). Obviously,
choosing the operators T , Ti properly, one can achieve any
required convergence of the sequence of
approximations of the solutions tending to the exact value without
adding any additional constraints on the input data of
the considered problems.
The numerical implementation of the MM can be
illustrated and discussed on the example of nonlinear
differential equations governing dynamical behaviour
of a flexible shallow multi-layer shell, each layer of
which is made from a non-homogeneous orthotropic
material within the Timoshenko kinematic model.
To get a reliable approximate solution, the Bubnov–
Galerkin method (BGM) and the finite difference
method (FDM) can be employed.
The input problem is formulated in the following
way. The aim is to find a solution to the system (27)
in the space understood as a bounded part of the
Euclidean space E2(x1, x2) (a point in E2) with the
boundary ∂ satisfying the condition of the Sobolev
embedding theorem [
11
]:
L1 (u3) = −L (u3, F ) − ∇K2 F
∂ ∂u3
− ∂ x1 A44 γ1 + ∂ x1
0 < α1 ≤ A44(55) ≤ β1; 0 < α1 ≤ Aii j j ( Aiiii ) ≤ β1;
0 < α2 ≤ aii j j (aiiii ) ≤ β2; 0 < α2 ≤ a16 ≤ β2;
0 < α¯ 1 ≤ A1212 ≤ β1; α1C1 < Aiiii − Aii j j < β1;
α2 < 2 aiiii − aii j j < β2;
0 < C1 < 1; i = 1, 2; j = 2, 1;
α¯ 1 < C2 < 1.
0 < C1 − α1
(29)
A vector u = (u3, γ1, γ2, F ) ∈ H1 is called the
generalised solution to the mentioned problem, and the
following integral relations are satisfied:
(L1 (u3, γ1, γ2, F ) , v1) = (g, v1) ;
(L2 (u3, γ1, γ2, F ) , v2) = 0;
(L3 (u3, γ1, γ2, F ) , v3) = 0;
(L4 (u3, γ1, γ2, F ) , v2) = 0;
∀v = (v1, v2, v3, v4) ∈ H1. (30)
Besides the considered problem (25), (26), the
following “helping problem” is introduced:
⎧ L1 (u3, γ1, γ2, F ) + ε 2u3 = g,
⎪⎪⎨ L2 (u3, γ1, γ2, F ) = 0,
L3 (u3, γ1, γ2, F ) = 0,
⎪⎪⎩ L4 (u3, γ1, γ2, F ) = 0,
including the boundary conditions
∂ F
u3 = γ1 = γ2 = F = ∂n ∂ = 0,
(31)
u3 − 1 −ρ u2 ∂∂un3 ∂ = 0, (32)
where: 2—biharmonic operator, —Laplace
operator, ρ—radius of the contour ∂ curvature, ν=const >
0 [
43
].
By the general solution, one should understand a
vector uε = u3ε, γxε, γyε, Fε ∈ H2 satisfying the
following integral identities:
(R (uε) , v) ≡ (L1 (u3ε) , v1) + (L2 (γxε) , v2)
+ L3 γyε , v3
+ (L3 (Fε) , v4) + ε
u3ε v1 + 2 (1 − u2)
×
(2) α1 − β3 ε1 > 0; α1 − β4ε2 > 0; α¯ − ε2β42 H > 0;
2
where {χi } , {χ2i } , {χ1i } are basis systems in W22 ( ) ∩
o o o
W21 ( ) , W22 ( ) , W21 ( ), respectively.
Moreover:
o
u3ε → u30ε strong in W21 ( ) and weak in
n
o
W22 ( ) ∩ W21 ( ) ,
o
γxnε → γx0ε weak in W21 ( ) ,
o
γyε → γy0ε weak in W21 ( ) ,
n
o
Fεn → Fε0 weak in W22 ( ) .
The theorem conditions imply that the operator A
exhibits the property of coercivity. Then, owing to the
Bauer’s theorem [
48
], the system can be solved to define
the coefficients of the series (36).
The coercivity of the operator A and orthogonality
of the operator B allow one to write the following a
priori estimate for a set of the approximate solutions
(36):
√εu3nε Wo22( ) ≤ C 0;
γynε Wo21( ) ≤ C 0;
γxnε Wo21( ) ≤ C 0;
F
εn Wo22( ) ≤ C 0;
C 0 > 0.
(37)
The above inequalities and continuity of the
functional (34) as well as weak discontinuity of the
functional (33) (see [
50
]) imply Theorem 1. One has:
o
u3ε → u03 strong in W21 ( ) and weak in
o
γxε → γx0 weak in W21 ( ) ,
o
γyε → γy0 weak in W21 ( ) ,
o
Fε → F 0 weak in W22 ( ) .
Indeed, the following estimations hold
The inequalities (38) are yielded by (37) through
the limiting transition regarding n, which allows one to
choose a certain subsequence {uε} = u3ε, γxε, γyε, Fε .
The subsequence is weakly convergent in H2 and,
together with the Sobolev embedding theorem,
guarantees the possibility of a limit transition with respect
to ε → 0 in the integral identity (37) and yields the
integral identity (33) after closing the set {χ1} ∈ H2
within the norm of the space H1.
The space can be divided into non-intersecting
parts 1, . . . , p, such that = ∪iP=1 i . By h the
maximum diameter of the partitioned spaces is denoted.
o o
The spaces of the finite measure W22h ( ), W21h ( ),
2 o o
W2h ( ) of the spaces W22 ( ), W21 ( ), W22 ( ),
respectively, are constructed in the following way. The
associated axes should be subspaces of the type of
the finite elements, i.e. their basic functions should
possess “small” carriers composed of a few partial
o o o o
spaces. The notation H1h = W21h × W21h × W21h × W22h ;
o o o o
H2h = W22h ∩ W21h × W21h × W21h × W22h is employed.
It is assumed that the sequence of the spaces H1h , H2h ,
obtained for different h, is full in H1 and H2, i.e. an
arbitrary element from H1 and H2 can be (for
sufficiently small h) approximated with arbitrary accuracy
in the norm of the space H1, H2 by elements from
H1h , H2h . The variational-difference problem (31),
(32) can be formulated as follows. First, the vector
uεh = u3hε, γxhε, γyhε, Fεh ∈ H2h satisfying the
following integral identity:
+ ε
h h
u3ε · v1 + 2 (1 − ν)
The difference scheme is constructed by imposing a
mesh on the general solution (33) in a way similar to
the already discussed. To keep the properties of
orthogonality of the operator B uεh , the nonlinear operators
L (u3, F ) and L (u3, u3) are approximated as follows
L (·, ·) = (·)y¯ y (·)W x¯x + (·)x¯x (·)y¯ y
1
− 2 (·)x y (·)x y +(·)x¯ y (·)x¯ y +(·)x y¯ (·)x y¯ + Fx¯ y¯ u3x¯ y¯ ;
L (·, ·) = (·)y¯ y (·)x¯x
1
− 4
(u3)2xy + (·)2x¯ y + (u3)2x y¯ + (·)2x¯ y¯ .
Stability of the obtained system of equations is
implied by the coercivity of the operator A uεh and
orthogonality of the operator B uεh , i.e. the
following estimations hold under applicability conditions of
Theorem 1:
√εu3hε W22( ) ≤ C 0; γxε W21( ) ≤ C 0;
h
γyhε Wo21( ) ≤ C 0;
Fεh W22( ) ≤ C 0.
Owing to (42), the set of points uεh is weakly
compact in H2, i.e. its arbitrary subset allows for extraction
of a subset weakly convergent in H2 for h → 0.
(40)
(41)
(42)
Theorem 2 Any of the mentioned limits stands for a
general solution to the problems (31), (32). In order to
prove this, one can transit to a limit for h → 0 in the
identity (39).
It is clear that conditions of the Theorem 1 can be
always satisfied when the plate Kx = K y = 0 is
considered. Furthermore, the analogous results hold
also for other boundary conditions, for instance, when
the type of the boundary conditions changes along a
contour [
51
] (for instance, this is guaranteed for an
appropriate transition of boundary conditions for the
Timoshenko-type model).
Remark Modifications of the Timoshenko
mathematical model of the second approximation (MM2) are
possible. The MM2 is created by means of
employment of the kinematic hypothesis regarding
tangential displacements, which yields f (x3) ≡ 1 as a
result of using the shell theory. However, one can
impose an additional static hypothesis onto the
tangential stresses σ13, σ23 and take into account their
variations along the thickness of each packet of the
layers in the form of a squared parabola, i.e. when
f (x3) = (2h0−1 ) (x3 + ) (2h0 − − x3). Such a
mathematical model will be further called MM4.
4 Numerical investigation of stability of a multi-layer orthotropic shallow shell in the frame of various improved theories
The static stability problem is investigated similarly as
described in [
52
], where the problem of stability was
considered in terms of “load-deflection” coordinates
measured at the shell centre q [u3 (0.5; 0.5)].
The so far introduced mathematical models have
been solved using the variational-difference method.
The advantages of this over other possible approaches
are briefly addressed below.
1. Minimal preliminary work (for instance, in contrast
to FEM) is required to prepare models for computer
simulations.
2. Universality in terms of using various boundary
conditions.
3. Possibility of development of a unique/one
program devoted to investigate stability of multi-layer
shells taking into account different MM.
4. Coincidence of approximations of derivatives of
an arbitrary order in the given points of the space
boundaries with the approximation in the internal
points under imposing a mesh on the integral form
of the equations of the shell equilibrium.
The following notation for the Sobolev spaces [
53
]
is employed:
W2m ( ) = u/Dαu ∈ L2 ( ), ∀α |α| ≤ m ,
1◦. The variational-difference scheme for the
Timoshenko second-order approximation model (MM2).
The vector u = (u3, γ1, γ2, F) ∈ H1 serves as a
generalised solution to the problems (23), (24) where
the identity:
+ (L4 (F) , ξ ) = (q, v)
R (u, v) ≡ (L1 (u3) , ϕ) + (L2 (γ1) , ψ) + (L3 (γ2) , ζ )
(43)
is satisfied for an arbitrary vector v = (ϕ, ψ, ζ, ξ ) ∈
H1.
The functional R (u, v) can be recast to its
counterpart form:
1
2
(S)
−
−
(S)
∂2 F ∂u3 ∂ϕ ∂2 F ∂u3 ∂ϕ
∂ x1∂ x2 ∂ x1 ∂y2 + ∂ x1∂ x2 ∂ x2 ∂ x1
∂2 F ∂u33 ∂ϕ
+ ∂ x12 ∂ x2 ∂ x2
dS
K1 ∂∂2xF22 + K2 ∂∂2xF12
ϕdS
+
+
+
+
+
+
+
+
+
−
−
−
+
(S)
(S)
(S)
(S)
(S)
(S)
(S)
(S)
(S)
(S)
(S)
(S)
(S)
= qψdS + q1ψdS
+ q2ζdS + qFξdS,
u32¯ u31 ξ2¯ 1 +
u32¯ u31¯ ξ2¯ 1¯
N −1 f (x1i ) − f (x1i−1) v (x1i ) + f · v|xx11==0N ,
= −h h
h
N −1
i=1
and the difference relations regarding differentiation of
the mesh functions are as follows
Then, the shell equations and natural boundary
conditions formulated in the difference form can be written
as
The difference analogue of the integration by parts [
53,
54
] is employed:
h
N −1
i=0
+ b31 (γ2)12¯ (F1)i j j=N2
+
N1−1
i=0
× (F1)i j j=0 +
i=1 2h2 a16 F2011¯
N1−1 1
i=0 2h2
Here, the terms of approximation of nonlinear operators
corresponding to the limiting values of the variable x1
and x2 were not shown owing to their complexity and
due to fact that the aim of this work is not to study
all boundary conditions. Furthermore, the construction
of MM with the boundary conditions (23a) and (23b)
does not need nonlinear operators to be defined on the
boundary.
The following important steps of constructing a
computational scheme should be emphasised. It should
be mentioned that the total identity (46) does not allow
to get mesh approximation of the derivatives ∂∂3xγ131 and
∂∂3xγ232 while approaching the boundary layer for the
boundary condition of clamping in the case of a
nonsymmetric packet of layers.
In order to solve the problem, the additional
conditions regarding smoothness of the functions γ1 (x1, x2)
and γ2 (x1, x2) should be imposed. Namely, the values
of (γ1)−1, j and (γ2)i,−1 in the contour-out points are
yielded by difference approximation of two differential
equations of the equilibrium with respect to γ1 and γ2
after their additional differentiation.
+ F1
+ F2
Using the notations employed in programming of
MM2, the following non-dimensional formulas are
obtained
where
The similar-like approach aimed at constructing the
difference scheme guarantees approximation of the
problem with an error of O h2 in the internal space and
of O (h) on its boundary.
In the obtained nonlinear algebraic equations, the
denominator is removed by multiplying each equation
by h1α1 · h2α2 , where α1 + α2 = α stands for the largest
power of the step of the space partition appeared in
the nominators of all equations. Then, the values of the
searched functions are placed on the left-hand side of
equations regarding the central node, i.e. the system is
transformed to its counterpart form that to be solved
directly by the Jacoby method:
ui j F 1h22 + F 2h21 = h12h22(L˜ (u3, F )
+ K1l2 (F ) + K2l1 (F ) + l¯1 (u3)
and l¯i stands for the difference operator without
values of the corresponding functions in the node (i, j );
ψ˜ i (·)—difference analogue of the right-hand sides of
the equations (24), L˜ (·)—difference analogues of
nonlinear operators.
The differential equations are given in
nondimensional forms. The relation between dimensional
and non-dimensional (with bars) quantities is as
follows:
M¯ 12 = M12ab/ (2h0)4 ϕ10 ,
M¯ 112 = M112a2/ (2h0)4 ϕ10 ,
M¯ 113 = M113a2/ (2h0)4 ϕ10 ,
M¯ 222 = M222b2/ (2h0)4 ϕ10 ,
M¯ 223 = M223b2/ (2h0)4 ϕ10 ,
M¯ 122 = M122ab/ (2h0)4 ϕ10 ,
M¯ 123 = M123ab/ (2h0)4 ϕ10 .
Note that bars over non-dimensional quantities were
already omitted in equations (24). Let us consider thin
shallow shells of the thickness from the interval:
(51)
(52)
(53)
1
1000
,
where R denotes the radius of the shell curvature, and
the Reissner’s criterion (< ρ = 600) is satisfied [
8
].
If one considers square shells, then λ = a/b = 1
and (51) allow one to employ parameters λ1 = λ2 for
fixed values of the non-dimensional curvatures. Indeed,
since 1000 · K¯ 1 < λ21 < 50 · K¯ 1, then for K¯ 1 = K¯ 2 =
9 : λ1 = λ2 ∈ (21,2; 94,8);
K¯ 1 = K¯ 2 = 15 :
K¯ 1 = K¯ 2 = 18 :
K¯ 1 = K¯ 2 = 24 :
K¯ 1 = K¯ 2 = 36 :
λ1 = λ2 ∈ (27.4; 122.5) ,
λ1 = λ2 ∈ (30.0; 134.2) ,
λ1 = λ2 ∈ (34.6; 154.9) ,
λ1 = λ2 ∈ (42.4; 189.7) .
Condition (52) implies that R1 ≤ a1 , and hence
K¯ 1 ≤ λ1.
However, if one takes fixed K¯ 1 = K¯ 2 = 9 : λ1 > 21.2
in (53), then the inequality (53) is also satisfied: λ1 > 9.
Consequently, satisfaction of conditions (52) governing
the interval of the geometric changes of the parameter
λ1 = λ2 guarantees satisfaction of the condition (51).
Recall that λ1, λ2 stand for the geometric parameter
characterising a ratio of the shell length and width to
the shell thickness. Obviously, for each fixed value of
the shell curvature, the change in λ1, λ2 from minimum
to maximum implies the shell thickness decrease, and
hence there is no need to take angles of rotation and
twisting of the normal into account. In this case,
deformations of transverse shear follow: ε13 ≈ 0, ε23 ≈ 0,
i.e. γ1 ≈ − ∂∂ux13 , γ2 ≈ − ∂x2
∂u3 . It means that transverse
tangential stresses σ13 ≈ σ23 ≈ 0, and consequently,
also transversal forces Q1, Q2 ≈ 0, i.e. beginning with
the defined λ1 = λ2, one transits into the space of
application of the classical Kirchhoff-Love model.
Employing the variational-difference method and
introducing the parameters λ¯1 = A¯1313 · λ21 and λ¯2 =
A¯2323 · λ22, he matrix of the system of algebraic
equations is defined, where also both shell geometry and
materials of layers are included by means of the ratio
Gi3/ϕ10 (i = 1, 2).
Now, if λ1, λ2 are increased to large values (within
the possible interval), then the matrix of algebraic
equations corresponding to MM1 becomes badly
conditioned and convergence of the iterational process
becomes too slow to employ MM1 for such kind of
problems. It is clear that for large values of λ1, λ2,one
can decrease λ¯1, λ¯2 by choosing properly the material
of layers i.e. every time MM1 is employed, there is a
need to follow the values of λ¯1, λ¯2. This drawback is
cancelled in the case of using MM2, which works well
in the case of both large and small values of λ¯1, λ¯2. This
observation exhibits an advantage and universality of
MM2 in comparison with the Timoshenko-type model
for f (x3) ≡ 1.
To avoid problems with finding a solution to the
system of nonlinear algebraic equations of large and
badly conditioned matrix (for large values of λ1, λ2),
the iteration was employed instead of the direct
GaussSeidel method, the main advantage of which is the
selfimprovement property.
The materials of layers considered in the present
study are presented in Table 1 (in non-dimensional
form). To get their dimensional counterparts, the values
need to be multiplied by 10−6 (kG/cm2).
Reliability of the derived algorithms was verified
by comparing the results obtained for testing examples
with the exact solutions for the following mode
problems:
(i) Boundary conditions (23a)—MM2: u3 = sin π x1
× sin π x2, γ1 = sin 2π x1 × sin π x2, γ2 =
sin π x1 ×sin 2π x2, F = sin2 π x1 ×sin2 π x2, q¯ =
π, MM3, MM4: W = sin2 π x1 × sin2 π x2, γ1 =
sin 2π x1 ×sin π x1, γ2 = sin π x1 ×sin 2π x2, F =
sin2 π x1 × sin2 π x2, q¯ = π ;
(ii) Boundary conditions (23b)—MM2: W = sin π x1
× sin π x2, γ1 = cos π x1 × sin π x2, γ2 =
sin π x1 × cos π x2, F = sin2 π x1 × sin2 π x1, q¯ =
π, MM3, MM4: W = sin2 π x1 × sin2 π x2, γ1 =
Table 2 Results of testing the MM2 model: mesh resolution
N × N and accuracy between successive iterations ε
The Timoshenko model—MM2
Table 3 Comparison of the results obtained using the exact
Brucker’s solution and the solutions yielded by Timoshenko
MM2 and MM4 models
cos π x1 × sin π x2, γ2 = sin π x1 × cos π x2, F =
sin2 π x1 × sin2 π x2, q¯ = π.
In Table 2, the results of investigation of mesh
convergence are reported for various numbers of nodes N
(1/4 of the planform) and accuracy between iterations
ε for the three-layer shells of the same thickness with
curvatures K¯ 1 = K¯ 2 = 24, λ1 = λ2 = 55.56. The
exact value of the load is q¯ = 3.1416 for the boundary
condition (23a).
Convergence of the numerical algorithm aimed at
analysing the packet of layers of non-symmetric
structure, where mesh approximations were used for
differential equations with respect to γ1, γ2 in the
approximation of the out-of-contour nodes for the boundary
condition (23a), was verified on the model problems for
shells of the same structure as in the case of the
symmetric packet. However, the coordinate surface was moved
from the packet centre to an arbitrary point x3 located
along the shell thickness ( = h/2). Values of loads
for both symmetric and non-symmetric packets of
layers coincide, which validates the use of the numerical
algorithm employed to the non-symmetric multi-layer
shell.
3. Reliability of the numerical algorithm for MM2
with f (x3) = 1 and for MM4 with f (x3) = 1 − xh3 2
was checked by estimating deflection of the shell
centre u3(0; 0) = W00 and comparing it with the exact
Brucker’s solution [
55
] for the counterpart
geometrically linear variant of the PDEs.
The following three-layer plate was considered: the
middle layer h2 = 0, 6 · h1, where h1 stands for height
of the external layers (h1 = 0.0096, λ1 = λ2 = 40,
E1 = 10 · E2, ν12 = ν21 = 0.3), the boundary
condi∂ F
tion: u3 = M11 = γ2 = F = ∂n x=0 = 0, and the
load q = q0 · sin π x1 · sin π x2.
If the load and shell deflection at the Brucker’s centre
are denoted by qo, W¯ 3, respectively, and the load and
deflection regarding the MM2 by q¯oK , W¯ K , then the
following relations hold: W¯ 3 = W¯ K · E1/ 104q0 , q0 =
q¯0K · E2/ 1 − ν12 · 12 · λ1 · λ22 .
2 2
Therefore, one finally gets W¯ 3 =
2
W¯ K ·E1 1 − ν12 ·
12 · λ12 · λ22 / 104q¯0K · E2 . The obtained data used for
comparison/validation are reported in Table 3.
A comparison of the stability curves for MM2 and
MM4 (K1 = K2 = 15, λ1 = λ2 = 33.3) with N ×
N = 11 × 11, 13 × 13, 15 × 15 is shown in Fig. 2.
One can conclude that in spite of computations of
the “load-deflection” curve using different improved
models, the qualitative character of the numerically
obtained curves is similar. Based on extensive and
numerous numerical experiments, convergence of all
numerical algorithms investigating the improved MMs
Fig. 2 Dependencies q(u3) for the Kirchhoff–Love (MM1) and
the Timoshenko (MM2) models obtained for different numbers
of mesh partitions (N ). -MM4-N13, -MM2-N13,
-MM411, -MM2-N11. (Color figure online)
was found. The number of partition points N of spatial
coordinates was changed with respect to the values of
the curvature parameters, the geometric parameters λ1,
λ2, the number of layers and their localisation regarding
the reference surface x3 = 0 in order to keep the same
accuracy while comparing the numerically estimated
values of critical loads.
The carried-out qualitative investigations aimed at
finding the critical loads in relation to the initial data
were carried out for the following numbers of mesh
nodes: N × N = 11 × 11, 13 × 13, 15 × 15. One can
conclude that real shells should be computed choosing
an optimal mesh for particular input data, keeping the
same required interval of the computational accuracy.
5 Comparison of stability curves of the multi-layer symmetric shells in the “load-deflection” coordinates
Since the shells composed of interlacing layers of
aluminium and an orthotropic material were studied,
this section presents the investigation of one-layer
shells made only of either aluminium or an orthotropic
material. Figures 3, 4 report the stability curves for
shells made of aluminium and the orthotropic
material, respectively, for fixed parameters K1 = K2 = 9,
Fig. 3 Dependency q(u3) for one-layer shell made from an
orthotropic material for K1 = K2 = 9, λ1 = λ2 = 21.7
for the Timoshenko model (MM2), ε-regularisation (MM3), and
the modified Timoshenko model (MM4) ( -MM2, -MM3,
MM4). (Color figure online)
λ1 = λ2 = 21.7. For small loads, when the nonlinear
terms do not have a strong influence on the solution,
the curves are similar to each other. It holds, in
particular, for the Timoshenko models with f (x3) = 1
(MM2) and with f (x3) = 1 − xh3 2 (MM4). For
the orthotropic material, the results yielded by the
εregularisation model (MM3) strongly differ from the
results obtained by MM2 and MM4 models in “the
linear part” of the q(u3) curves. However, for large
deflections, the difference between the results starts to
decrease. In the case of one-layer shells made of
aluminium (Fig. 4), the largest differences between results
for MM2, MM4 and MM3 models are exhibited in the
nonlinear part of the q(u3) dependencies.
Furthermore, Figs. 5, 6 show the functions q(u3)
for the three-layer shell with layers of equal
thickness h = 0.046. Figure 5 presents the layers sequence
aluminium–orthotropic material–aluminium, whereas
the sequence orthotropic material–aluminium–
orthotropic material is presented in Fig. 6. The
Timoshenko model with a parabola (MM4) is very
sensitive to the order of layers. In the first case (Fig. 5), the
MM4 model yields the results close to the results of
the ε-regularisation model (MM3). In the second case
(Fig. 6), the obtained results are similar to those given
by the MM2 model. It should be emphasised that the
results obtained using the MM2 model qualitatively
differ from the results obtained by other models.
Fig. 4 Dependency q(u3) for one-layer shells made from
aluminium for K1 = K2 = 9, λ1 = λ2 = 21.7 for the Timoshenko
model (MM2), ε-regularisation (MM3), and the modified
Timoshenko model (MM4) ( -MM2, -MM3, -MM4). (Color figure
online)
Fig. 6 Dependency q(u3) for the three-layer shells: the upper
and the lower layers made from the orthotropic material, and
the middle layer made from aluminium for the Timoshenko
model (MM2), ε-regularisation (MM3), and modified
Timoshenko models (MM4) ( -MM2, -MM3, -MM4). (Color
figure online)
Fig. 5 Dependency q(u3) for the three-layer shells: the upper
and the lower layers made from aluminium, and the middle
layer made from the orthotropic material for the Timoshenko
model (MM2), ε-regularisation (MM3), and modified
Timoshenko models (MM4) ( -MM2, -MM3, -MM4). (Color
figure online)
The comparison of the obtained numerical results
for thinner shells of the thickness h = 0.018 and for
different shell models is presented in Fig. 7 (the order of
layers: aluminium–orthotropic material–aluminium).
Comparing the results shown in Figs. 5 and 7, one
can conclude that the change of the layers thickness
(and thus the change of parameters λ1, λ2) yields
divergence of the results given by the Timoshenko model
Fig. 7 Dependency q(u3) for three-layer shells made from
aluminium–orthotropic material–aluminium for K1 = K2 = 9,
h = 0.018 for the Timoshenko model (MM2), ε-regularisation
(MM3), and the modified Timoshenko model (MM4) (
MM2, -MM3, -MM4). (Color figure online)
with f (x3) = 1 (MM2) and either the ε-regularisation
model (MM3) or the modified Timoshenko model with
f (x3) = 1 − xh3 2 (MM4). However, the results
yielded by the MM3 and MM4 models are similar to
each other.
Fig. 8 Distribution of
stresses σ11 (a), σ11 (b) and
σ13 (c) along the thickness
for the Timoshenko model
(MM2), ε-regularisation
model (MM3), and the
modified Timoshenko
model (MM4) ( -MM2,
-MM3, -MM4). (Color
figure online)
Numerical investigation of stability of multi-layer
shells was also carried out based on simultaneous
observations of stress-strain states, i.e. the stresses σ11,
σ12, σ13 for the shell deflection in the neighbourhood
of the critical loads. Figure 8 present the results of
the following problems: three-layer shell composed
of aluminium–orthotropic material–aluminium for the
fixed curvatures K1 = K2 = 9 and the same layer
thickness h = 0.046. The results shown in Fig. 8a allow
one to conclude that normal stresses σ11 are similar to
each other. The stresses σ12 (Fig. 8b) exhibit visible
differences on the shell edge and the shell centre. The
stresses σ13 (Fig. 8c) are qualitatively different in the
case of the Timoshenko model with f (x3) = 1 (MM2)
and the two remaining ones, i.e. the ε-regularisation
model (MM3) and the modified Timoshenko model
with f (x3) = 1 − xh3 2 (MM4).
All the carried-out investigations validated the
statement that fundamental stresses of all tested models
are uniquely defined with an accuracy allowed by the
employed model for all physical-geometrical
parameters with and without temperature fields. However, the
shear stresses are non-uniquely defined for all
investigated models. It means that the satisfaction of static
conditions of the layers should be supplemented by
using the discrete models while studying the
stressstrain states of multi-layer shells.
6 Concluding remarks
The carried-out investigations yielded the following
conclusions:
1. The mathematical models of geometrically
nonlinear multi-layer rectangular shallow shells were
derived. The governing equations were obtained
in the mixed form, i.e. with respect to the stress
function and deflection function. All presented
models are based on the Timoshenko hypothesis.
In addition, a new mathematical model with
εregularisation was constructed, and the theorem on
existence of its general solution was formulated and
proved.
2. The conservative difference schemes were
developed for the boundary value problems, based on
the fundamental variation equations of the shallow
shells. The method based on the difference
approximation of differential operators of the odd order
yielded by the theory of multi-layer shells with
nonsymmetric packet of layers was developed.
3. Investigations devoted to feasibility of the
constructed mathematical models were carried out.
Reliability of the obtained results was analysed
and validated by means of comparing the obtained
results with the analytical and/or numerical results
obtained by other authors.
4. The carried-out numerical experiments showed
convergence of numerical algorithms of all
developed mathematical models. It was demonstrated
that the stability curves in the “load-deflection”
coordinates depend on the layers position, layers
thickness and the layers material.
5. The one-layer shells made from two different
materials were investigated within all constructed
models. It was detected that the mathematical models of
the Timoshenko type with f (x3) = 1 (MM2) and
f (x3) = 1 − xh3 2 (MM4) yield similar results,
in particular in the case of small deflections. The
ε-regularisation gives different results, particularly
in the case of an orthotropic material.
6. Investigation of the three-layer shell composed
of aluminium–orthotropic material–aluminium as
well as of the orthotropic material–aluminium–
orthotropic material showed that for the
Timoshenko model with parabola (MM4), the position
of the stress-deflection curve strongly depends on
the order of layers in the packet.
7. The 2.5 times decrease in the layer thickness yields
convergence of the results obtained by the
Timoshenko model with parabola (MM4) and by the
ε-regularisation model (MM3).
8. The comparison of the stress-deformation states
shows that normal stresses yielded by all models
are very similar to each other. Also, the stresses
σ12 are qualitatively similar for all models, but the
ε-regularisation model exhibits a difference on both
the shell edges and the shell centre. The largest
difference was obtained for the stress σ13, where for
MM2 (MM4) and MM3, the detected difference is
both qualitative and quantitative.
Acknowledgements This work was supported by the Russian
Science Foundation RSF 16-11-10138.
provided you give appropriate credit to the original author(s) and
the source, provide a link to the Creative Commons license, and
indicate if changes were made.
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