Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

54, 3, pp. 945-961, Warsaw 2016
DOI: 10.15632/jtam-pl.54.3.945

GEOMETRICALLY NONLINEAR VIBRATIONS OF THIN VISCO-ELASTIC

PERIODIC PLATES ON A FOUNDATION WITH DAMPING:

NON-ASYMPTOTIC MODELLING

Jarosław Jędrysiak
Lodz University of Technology, Department of Structural Mechanics, Łódź, Poland

e-mail: jarek@p.lodz.pl

The objects under consideration are thin visco-elastic periodic plates with moderately lar-
ge deflections. Geometrically nonlinear vibrations of these plates are investigated. In order
to take into account the effect of microstructure size on behaviour of these plates a non-
-asymptotic modelling method is proposed. Using this method, called the tolerance model-
ling, model equations with constant coefficients involving terms dependent on the micro-
structure size can be derived. In this paper, only theoretical considerations of the problem
of nonlinear vibrations of thin visco-elastic periodic plates resting on a foundation with
damping are presented.

Keywords: thin visco-elastic periodic plates, nonlinear vibrations, effect of microstructure
size, analytical tolerancemodelling

1. Introduction

In this paper, thin visco-elastic plates with a periodic structure in planes parallel to the plate
midplane, interacting with a periodically heterogeneous foundation are considered. These plates
consist ofmany identical small elements, called periodicity cells (theyaredistinguishedbydotted
lines in Fig. 1). Plates of this kind can have deflections of the order of their thickness. Dynamic
problems of these plates are describedbynonlinear partial differential equationswith coefficients
being highly oscillating, periodic and non-continuous functions of x1,x2. Hence, these equations
are not a good tool to analyse various special problems of the plates under consideration. In
order to obtain governing equations with constant coefficients, various simplified approaches are
proposed, which introduce effective plate properties. Amongst them, it is necessary to mention
those based on the asymptotic homogenization, see Kohn and Vogelius (1984). Unfortunately,
the governing equations of these models usually neglect the effect of the microstructure size on
the plate behaviour.

Fig. 1. A fragment of a thin periodic plate on a foundation under consideration



946 J. Jędrysiak

Other variousmethods are also applied to describe differentmechanical problems of periodic
structures and composites.Multiscalemodels were applied to describe compositematerials rein-
forcedbymicro-particles byLurieet al. (2005); the two-scale asymptotic homogenizationmethod
was used to analyse honeycomb sandwich composite shells by Saha et al. (2007); a relationship
between the 3D and the homogenised Euler-Bernoulli beam limit was shown by Dallot et al.
(2009), where the homogenisation procedure was also justified using the asymptotic expansion
method. Results suggesting the relevance of the proposed algorithm towards the efficient multi-
scalemodelling of periodicmaterials such as woven composites were obtained byDeCarvalho et
al. (2011); a two-dimensional analytical solution of amultilayered rectangular platewith a small
periodic structure along one in-plane direction was obtained byHe et al. (2013), where the two-
-scale asymptotic expansionmethodwas employed to develop ahomogenizedmodel of each layer
in the plate and then the state-space approach was used. Heterogeneous plates were investiga-
ted by Schmitz and Horst (2014) using a finite element unit-cell method. The two-dimensional
stationary temperature distribution in a periodically stratified composite layer was analysed by
Matysiak and Perkowski (2014) within the framework of the homogenized model with microlo-
cal parameters. An asymptotic dispersive method for description of the problem of shear-wave
propagation in a laminated composite was proposed by Brito-Santana et al. (2015).

Mechanical problems of thin plates under moderately large deflections are described by the
known geometrically nonlinear equations presented by e.g. Timoshenko andWoinowsky-Krieger
(1959) andWoźniak (2001). Equations of vonKármán-typeplateswerederived fromequations of
the three-dimensionalnonlinear continuummechanicsbyMeenenandAltenbach (2001).Bending
problems of such plates can be analysed using variousmethods, e.g. proposed by Levy (1942) or
Timoshenko and Woinowsky-Krieger (1959). However, other new or modified methods are also
presented in a lot of papers. Some of them are mentioned below. An asymptotic approach for
thin rectangular plates with variable thickness clamped on all edges was used byHuang (2004).
Theoretical, numerical and experimental analysis of the stability and ultimate load of multi-
-cell thin-walled columns of rectangular cross-sections was shown byKrólak et al. (2009). Teter
(2011) analysed the dynamic critical load for buckling of columns, but global and local buckling
of sandwich beams and plates was examined by Jasion et al. (2012). The nonlinear bending
behaviour of moderately thick functionally graded plates on a two-parameter elastic foundation
was studied by Golmakani and Alamatian (2013), where the dynamic relaxation method and
the finite difference discretization techniquewere used to solve equations based on the first order
shear deformation theory and von Kármán theory.

Problems of nonlinear vibrations and/or visco-elastic damping of composite structures such
as beams, plates and shells, are considered by many researchers, applying various methods.
The influence of damping and/or stiffness on vibrations of nonlinear periodic plates was shown
by Reinhall and Miles (1989). Large amplitude flexural vibration characteristics of composite
plates using von Kármán’s assumptions and Galerkin’s method were obtained by Singha et al.
(2009). Geometrically nonlinear vibrations of free-edge circular plates with geometric imperfec-
tions described by von Kármán equations with using an expansion onto the eigenmode basis of
the perfect plate to discretise the equations of motion were analysed by Camier et al. (2009).
An approximate frequency equation of clamped visco-elastic rectangular plates with thickness
variations was derived by using the Rayleigh-Ritz technique byGupta et al. (2009). Magnucka-
Blandzi (2010) carried out a certain nonlinear analysis of dynamic stability of a circular plate.
Some oscillations of visco-elastic Timoshenko beams were investigated by Manevich and Koła-
kowski (2011). The variational method was used in nonlinear free vibration and post-buckling
analysis of functionally graded beams resting on a nonlinear elastic foundation by Fallah and
Aghdam (2011). Damping and forced vibrations of three-layered laminated composite beams
described in the framework of the higher-order zig-zag theories were investigated by Youzera et
al. (2012). Nonlinear free vibrations of orthotropic shells with variable thickness were analysed



Geometrically nonlinear vibrations of thin visco-elastic periodic plates... 947

by Awrejcewicz et al. (2013). Lei et al. (2013) used a transfer function method to obtain a
closed-form and uniform solution for damped visco-elastic vibrations of Euler-Bernoulli beams.
A linearized updated mode method was applied to solve nonlinear equations of geometrically
nonlinear free vibrations of laminated composite rectangular plates with curvilinear fibers by
Houmat (2013). Natural frequencies of free vibrations for functionally graded annular plates
resting on a Winkler’s foundation were predicted using the differential quadrature method and
theChebyshev collocation technique byYajuvindra andLal (2013). Yaghoobi andTorabi (2013)
presented large amplitude vibrations of functionally graded beams on a nonlinear elastic founda-
tion. Nonlinear bending vibrations of sandwich plates with a visco-elastic core were investigated
by Mahmoudkhani et al. (2014), where the 5th-order method of multiple scales was applied to
solve the equations of motion.

Usually, those proposed modelling approaches for microstructured media lead to governing
equations neglecting the effect of the microstructure size which can play a crucial role in dyna-
mical problems of such media, e.g. for periodic plates under consideration, see Jędrysiak (2003,
2009). In order to take into account this effect, some special methods are adopted sometimes to
analyse particular problems. For example, Zhou et al. (2014) investigated the problem of free
flexural vibration of periodic stiffened thin plates using Bloch’s theorem and the center finite
difference method.

However, in order to obtain equations of themodel,whichdescribe the aforementioned effect,
new non-asymptoticmodels of thin periodic plates based on the nonlinear theorywere proposed
by Domagalski and Jędrysiak (2012, 2015). These models are called the tolerance models and
are obtained in the framework of a certain general modelling approach called the tolerance
averaging technique, seeWoźniak et al. (2008, 2010). The derived equations, in contrary to the
exact ones, have constant coefficients. Some of them explicitly depend on the characteristic size
of the periodicity cell.

The tolerance method is general and is useful to model various problems described by diffe-
rential equations with highly oscillating non-continuous functional coefficients. It can be applied
in analysis of various thermo-mechanical problems of microheterogeneous solids and structures.
Some applications of this method for different periodic structures were presented in a series
of papers, e.g. for dynamics of plane periodic structures by Wierzbicki and Woźniak (2000);
for dynamics with near-boundary phenomena in stratified layers by Wierzbicki et al. (2001);
for vibrations of periodic wavy-type plates by Michalak (2001); for thin plates reinforced by a
system of periodic stiffeners by Nagórko and Woźniak (2002); for stability problems of perio-
dic thin plates by Jędrysiak (2000); for stability analysis of periodic shells by Tomczyk (2007);
for vibrations of periodic plates by Jędrysiak (2003, 2009); for dynamics problems of medium
thickness plates on a periodic foundation by Jędrysiak and Paś (2014); for vibrations of thin
functionally graded plates with plate thickness small comparing to the microstructure size by
Kaźmierczak and Jędrysiak (2011) and for stability of such plates by Jędrysiak and Michalak
(2011); for vibrations of thin functionally graded plates with themicrostructure size of an order
of the plate thickness by Jędrysiak (2013); for stability of thin functionally graded annular plates
on an elastic heterogeneous subsoil by Perliński et al. (2014). Moreover, the tolerancemethod is
also used to analyse dampedvibrations of periodic plate strips byMarczak and Jędrysiak (2014)
and nonlinear vibrations of periodic beams resting on a visco-elastic foundation by Domagalski
and Jędrysiak (2014). An extended list of papers can be found in the books by Woźniak et al.
(2008, 2010).

The main aim of this theoretical contribution is to formulate and discuss the nonlinear
tolerance and asymptotic models of dynamic problems for thin visco-elastic periodic plates
with moderately large deflections resting on a foundation with damping, on various levels of
accuracy.Thesenewtolerancemodelsare anextensionandgeneralization of the tolerancemodels



948 J. Jędrysiak

presented and applied byDomagalski and Jędrysiak (2012, 2014, 2015), Marczak and Jędrysiak
(2014).

2. Fundamental equations

Let us denote by 0x1x2x3 the orthogonal Cartesian co-ordinate system in the physical space
and by t the time co-ordinate. Let the subscripts α,β,. . . (i,j, . . .) run over 1, 2 (over 1, 2, 3)
and the indicesA,B,. . . (a,b, . . .) run over 1, . . . ,N (1, . . . ,n). The summation convention holds
for all aforementioned indices. Denote also x ≡ (x1,x2) and z ≡ x3. Let us assume that the
undeformed plate occupies the region Ω≡{(x,z) :−d(x)/2<z<d(x)/2,x∈Π}, whereΠ is
themidplane with length dimensionsL1,L2 along the x1- and x2-axis, respectively, and d(x) is
plate thickness.

It is assumed that plates under consideration have a periodic structure along the x1- and
x2-axis directions with periods l1, l2, respectively, in planes parallel to the plate midplane.
The periodicity basic cell on 0x1x2 plane is denoted by ∆ ≡ [−l1/2, l1/2]× [−l2/2, l2/2]. It
is assumed that the cell size is described by a parameter l ≡ [(l1)

2 +(l2)
2]1/2, satisfying the

condition max(d)≪ l≪min(L1,L2). Thus, l will be called the microstructure parameter. Let
us denote partial derivatives with respect to a space co-ordinate by (·),α ≡ ∂/∂xα.

Moreover, thicknessd(x) canbeaperiodic function inx, elasticmoduliaijkl = aijkl(x,z) and
mass density ρ= ρ(x,z) can be also periodic functions in x. In general, these plate properties
are not assumed to be even functions in z. Let aαβγδ, aαβ33, a3333 be non-zero components
of the elastic moduli tensor. Denote cαβγδ ≡ aαβγδ −aαβ33aγδ33(a3333)

−1. Proper visco-elastic
moduli are denoted by c̃αβγδ.

It is also assumed that the periodic plates interact with a periodic visco-elastic foundation
which rests on a rigid undeformable base, see Vlasov and Leontiev (1960). A fragment of such a
plate is presented in Fig. 1. The heterogeneous foundation is assumed to be periodic in planes
parallel to the plate midplane, i.e. along the x1- and x2-axis directions with periods l1 and l2,
respectively; however, it has constant properties along the z-axis direction. Hence, the founda-
tion properties, i.e. mass density per unit area µ̂ = µ̂(x), Winkler’s coefficient k = k(x) and
the damping parameter c = c(x) can be periodic functions in x = (x1,x2). These foundation
parameters can be defined following the book byVlasov andLeontiev (1960). It is also assumed
that the plate cannot be torn off from the foundation.

Denote displacements, strains and stresses by ui, eij and sij, respectively; virtual displace-
ments and virtual strains by ui and eij; loadings (along the z-axis) by p.

Now, the fundamental relations of the nonlinear thin plates theory, see Levy (1942) and
Woźniak et al. (2001), are reminded.

• Kinematic assumptions of thin plates

uα(x,z,t) =u
0
α(x, t)−z∂αw(x, t) u3(x,z,t) =w(x, t) (2.1)

withw(x, t) as the deflection of themidplane, u0α(x, t) as the in-plane displacement. Simi-
larly, these are for virtual displacements

uα(x,z) =u
0
α(x)−z∂αw(x) u3(x,z) =w(x) (2.2)

• Strain-displacement relations

eαβ =u(α,β)+
1

2
u3,αu3,β (2.3)



Geometrically nonlinear vibrations of thin visco-elastic periodic plates... 949

• Stress-strain relations (it is assumed that the plane of elastic symmetry is parallel to the
plane z=0)

sαβ = cαβγδeγδ + c̃αβγδėγδ (2.4)

with

cαβγδ = aαβγδ −aαβ33a33γδ/a3333 cα3γ3 = aα3γ3−aα333a33γ3/a3333

c̃αβγδ = ãαβγδ − ãαβ33ã33γδ/ã3333 c̃α3γ3 = ãα3γ3− ãα333ã33γ3/ã3333
(2.5)

• The virtual work equation

∫

Π

d/2∫

−d/2

ρüiui dz da+

∫

Π

d/2∫

−d/2

sαβeαβ dz da=

∫

Π

pu3
(
x,
d

2

)
da

−

∫

Π

(ku3+ µ̂ü3+ cu̇3)u3
(
x,−
d

2

)
da

(2.6)

is satisfied for arbitrary virtual displacements (2.2), assuming these displacements neglect
the plate boundary; moreover: da = dx1dx2; the virtual displacements are sufficiently
regular, independent functions.

The plate properties are periodic functions in x, i.e. stiffness tensors: bαβγδ, dαβγδ, hαβγδ,

visco-elastic tensors: b̃αβγδ, d̃αβγδ, h̃αβγδ, and inertia properties: µ, j, i are defined as

bαβγδ(x)=

d/2∫

−d/2

cαβγδ(x,z) dz dαβγδ(x)=

d/2∫

−d/2

cαβγδ(x,z)z
2 dz

hαβγδ(x)=

d/2∫

−d/2

cαβγδ(x,z)z dz b̃αβγδ(x) =

d/2∫

−d/2

c̃αβγδ(x,z) dz

d̃αβγδ(x)=

d/2∫

−d/2

c̃αβγδ(x,z)z
2 dz h̃αβγδ(x)=

d/2∫

−d/2

c̃αβγδ(x,z)z dz

µ(x)=

d/2∫

−d/2

ρ(x,z) dz j(x) =

d/2∫

−d/2

ρ(x,z)z2 dz i(x)=

d/2∫

−d/2

ρ(x,z)z dz

(2.7)

Using assumptions (2.1)-(2.4) of the nonlinear two-dimensional thin plate theory, applying
the divergence theorem and the duBois-Reymond lemma to equation (2.6), after somemanipu-
lations the governing equations of thin visco-elastic plates resting on foundations with damping
can be written in the form:
— constitutive equations

mαβ =−hαβγδuγ,δ +dαβγδw,γδ −
1

2
hαβγδw,γw,δ− h̃αβγδu̇γ,δ + d̃αβγδẇ,γδ

+
1

2
h̃αβγδ(ẇ,γw,δ +w,γẇ,δ)

nαβ = bαβγδuγ,δ −hαβγδw,γδ +
1

2
bαβγδw,γw,δ + b̃αβγδu̇γ,δ − h̃αβγδẇ,γδ

+
1

2
b̃αβγδ(ẇ,γw,δ +w,γẇ,δ)

(2.8)



950 J. Jędrysiak

—equilibrium equations

mαβ,αβ − (nαβw,α),β +µẅ− jẅ,αα+kw+ µ̂ẅ+ cẇ+ iüα,α = p

−nαβ,β+µüα− iẅ,α =0
(2.9)

or after substituting equations (2.8) into (2.9) as

{
−hαβγδ

(
u0γ,δ +

1

2
w,γw,δ

)
+dαβγδw,γδ − h̃αβγδ

[
u̇0γ,δ −

1

2
(ẇ,γw,δ +w,γẇ,δ)

]

+ d̃αβγδẇ,γδ
}

,αβ
−
{[
bαβγδ

(
u0γ,δ +

1

2
w,γw,δ

)
−hαβγδw,γδ

+ b̃αβγδ
[
u̇0γ,δ +

1

2
(ẇ,γw,δ+w,γẇ,δ)

]
− h̃αβγδẇ,γδ

]
w,α
}

,β

+µẅ− jẅ,αα+kw+ µ̂ẅ+ cẇ+ iü
0
α,α = p

−
{
bαβγδ

(
u0γ,δ +

1

2
w,γw,δ

)
−hαβγδw,γδ + b̃αβγδ

[
u̇0γ,δ +

1

2
(ẇ,γw,δ +w,γẇ,δ)

]

− h̃αβγδẇ,γδ
}

,β
+µü0α− iẅ,α =0

(2.10)

It can be observed that coefficients of equations (2.8) and (2.9) (or (2.10)) can be discontinu-
ous and highly oscillating, periodic functions in x, cf. (2.7). Hence, solutions to these equations
are very difficult to obtain.

The main aim of this paper is to propose a replacement of original equations with approxi-
mate models, which describe (or not) the information about the microstructure of considered
plates by using systems of equations with constant coefficients.

3. Outline of the tolerance modelling

3.1. Introductory concepts

In the tolerance modelling, certain introductory concepts are used. FollowingWoźniak et al.
(2008, 2010) some of them are reminded below.

A cell atx∈Π∆ is denoted by∆(x)=x+∆,Π∆ = {x∈Π : ∆(x)⊂Π}. The fundamental
concept of the modelling technique is the averaging operator, defined by

〈φ〉(x) =
1

l1l2

∫

∆(x)

f(y1,y2) dy1 dy2 x∈Π∆ y∈∆(x) (3.1)

for an integrable functionϕ. If the functionϕ is periodic inx, its averaged value calculated from
(3.1) is constant.

Let δ be an arbitrary positive number and X be a linear normed space. The tolerance
relation ≈ for a certain positive constant δ, called the tolerance parameter, is defined by

(∀(x1,x2)∈X
2) [x1 ≈x2 ⇔‖x1−x2‖X ¬ δ] (3.2)

Let ∂kϕ denote the k-th gradient of the function ϕ= ϕ(x), x∈Π, k = 0,1, . . . ,α, α­ 0,
and ∂0ϕ≡ϕ. Let φ̃(k)(x, ·) be a function defined inΠ×Rm, and δ be the tolerance parameter.
Introduce alsoΠx ≡Π∩

⋃
z∈∆(x)

∆(z), x∈Π.



Geometrically nonlinear vibrations of thin visco-elastic periodic plates... 951

The function ϕ ∈ Hα(Π) is called the tolerance-periodic function (with respect to cell ∆
and tolerance parameter δ), ϕ∈ TPαδ (Π,∆), if for k=0,1, . . . ,α, the following conditions are
satisfied

(i) (∀x∈Π)(∃φ̃(k)(x, ·)∈H0(∆))
[∥∥∥∂kφ

∥∥∥
Πx
(·)− φ̃(k)(x, ·)

∥∥∥
H0(Πx)

¬ δ
]

(ii)

∫

∆(·)

φ̃(k)(·,z) dz∈C0(Π)
(3.3)

The function ϕ̃(k)(x, ·) is a periodic approximation of ∂kϕ in∆(x), x∈Π, k=0,1, . . . ,α.

The function F ∈Hα(Π) is a slowly-varying function, F ∈SVαδ (Π,∆), if

(i) F ∈TPαδ (Π,∆)

(ii) (∀x∈Π)
[
F̃(k)(x, ·)

∣∣∣
∆(x)
= ∂kF(x), k=0, . . . ,α

] (3.4)

The function φ∈Hα(Π) is a highly oscillating function, φ∈HOαδ (Π,∆), if

(i) φ∈TPαδ (Π,∆)

(ii) (∀x∈Π)
[
φ̃(k)(x, ·)

∣∣∣
∆(x)
= ∂kφ̃(x), k=0, . . . ,α

]

(iii) ∀F ∈SVαδ (Π,∆) ∃ϕ=φF ∈TP
α
δ (Π,∆)

ϕ̃(k)(x, ·)
∣∣∣
∆(x)
=F(x)∂kφ̃(x)

∣∣∣
∆(x)
, k=1, . . . ,α

(3.5)

For α=0, let us denote ϕ̃≡ ϕ̃(0).

Let us introduce two highly oscillating functions defined on Π, f(·), g(·), f ∈ HO1δ(Π,∆),
g∈HO2δ(Π,∆).

Let the function f(·) be continuous and have a piecewise continuous and bounded gra-
dient ∂1f. The function f(·) is a fluctuation shape function of the 1st kind, FS1δ(Π,∆), if it
depends on l as a parameter and the conditions hold

(i) ∂kf ∈O(lα−k) for k=0,α, α=1, ∂0f ≡ f

(ii) 〈f〉(x)≈ 0 ∀x∈Π∆
(3.6)

where l is the microstructure parameter. Condition (3.6)(ii) can be replaced by 〈µf〉(x)≈ 0 for
every x∈Π∆, where µ> 0 is a certain tolerance-periodic function.

However, let g(·) be a continuous function together with the gradient ∂1g and with the
piecewise continuous andboundedgradient∂2g. The function g(·) is a fluctuation shape function
of the 2-nd kind, FS2δ(Π,∆), if it depends on l as a parameter and the conditions hold

(i) ∂kg∈O(lα−k) for k=0,1, . . . ,α, α=2, ∂0g≡ g

(ii) 〈g〉(x)≈ 0 ∀x∈Π∆
(3.7)

where l is the microstructure parameter. Condition (3.7)(ii) can be replaced by 〈µg〉(x)≈ 0 for
every x∈Π∆, where µ> 0 is a certain tolerance-periodic function.



952 J. Jędrysiak

3.2. Fundamental assumptions of the tolerance modelling

The tolerance modelling is based on two fundamental modelling assumptions which are
formulated in general form in the books by Woźniak et al. (2008, 2010). Here, they are shown
below in the form for thin periodic plates.

The micro-macro decomposition is the first assumption in which it is assumed that the
deflection and the in-plane displacements can be decomposed as

w(x, t) =W(x, t)+gA(x)VA(x, t) A=1, . . . ,N

u0α(x, t)=Uα(x, t)+f
a(x)Taα(x, t) a=1, . . . ,m

(3.8)

and the functions W(·, t),VA(·, t) ∈ SV 2δ (Π,∆), Uα(·, t),T
a
α(·, t) ∈ SV

1
δ (Π,∆) are the basic

unknowns; gA(·) ∈ FS2δ(Π,∆), f
a(·) ∈ FS1δ(Π,∆) are the known fluctuation shape functions.

The functions W(·, t) andUα(·, t) are called the macrodeflection and the in-plane macrodispla-
cements, respectively; VA(·, t) andTaα(·, t) are called the fluctuation amplitudes of the deflection
and the in-plane displacements, respectively. The fluctuation shape function can be obtained as
solutions to eigenvalue problems posed on the periodicity cell, cf. Jędrysiak (2009). However,
in most cases, they are assumed in an approximate form as: trigonometric functions (gA) or
saw-type functions (fa), see Jędrysiak (2003, 2013).

Moreover, similar assumptions to (3.8) are introduced for virtual displacements w(·), u0α(·)

w(x)=W(x)+gA(x)V
A
(x) A=1, . . . ,N

u0α(x)=Uα(x)+f
a(x)T

a
α(x) a=1, . . . ,m

(3.9)

with slowly-varying functionsW(·),V
A
(·)∈SV 2δ (Π,∆),Uα(·),T

a
α(·)∈SV

1
δ (Π,∆).

In the tolerance averaging approximation, the termsO(δ) are assumed to be negligibly small
in the course of modelling, i.e. they can be omitted in the following formulas

(i) 〈ϕ〉(x) = 〈ϕ̃(x)+O(δ)

(ii) 〈ϕ〉F〉(x) = 〈ϕ〉(x)F(x)+O(δ)

(iii) 〈ϕ〉(gF),γ(x)= 〈ϕ〉g,γ(x)F(x)+O(δ)

x∈Π; γ=1,α; α=1,2; 0<δ≪ 1;

ϕ∈TPαδ (Π,∆); F ∈SV
α
δ (Π,∆); g∈FS

α
δ (Π,∆)

(3.10)

3.3. The modelling procedure

The above concepts and fundamental assumptions are used in themodelling procedure.This
procedure can be divided into four steps.

In thefirst step,micro-macro decompositions (3.8) and (3.9) are substituted intovirtualwork
equation (2.6) of such a plate resting on a foundation. Then, in the second step, the averaging
operation is used to average the resulting equation over the periodicity cell, see Jędrysiak (2003).



Geometrically nonlinear vibrations of thin visco-elastic periodic plates... 953

In the next step, we arrive at the tolerance averaged virtual work equation after using formu-
las (3.10) of the tolerance averaging approximation (Jędrysiak, 2003). Applying the following
denotations of some averaged parameters, being averaged constitutive relations

Mαβ ≡−
〈 d/2∫

−d/2

sαβz dz
〉

MA ≡−
〈
gA,αβ

d/2∫

−d/2

sαβz dz
〉

Nαβ =
〈 d/2∫

−d/2

sαβ dz
〉

Naα ≡
〈 d/2∫

−d/2

sαβf
a
,β dz
〉

QAα ≡
〈 d/2∫

−d/2

sαβg
A
,β dz
〉

RAB ≡
〈 d/2∫

−d/2

sαβg
A
,αg
B
,β dz
〉

(3.11)

this tolerance averaged virtual work equation can be written as

∫

Π

(〈µ〉Ẅ + 〈µgB〉V̈B)δW da+

∫

Π

(〈µgA〉Ẅ + 〈µgAgB〉V̈B)δVA da

+

∫

Π

(〈µ〉Üα+ 〈µf
b〉T̈bα−〈i〉Ẅ,α−〈ig

B
,α〉V̈

B)δUα da

+

∫

Π

(〈µfa〉Üα+ 〈µf
afb〉T̈bα−〈if

a〉Ẅ,α−〈if
agB,α〉V̈

B)δTaα da

−

∫

Π

(−〈i〉Üα,α−〈if
b〉T̈bα,α+ 〈j〉Ẅ,αα+ 〈jg

B
,α〉V̈

B
,α)δW da

+

∫

Π

(−〈igA,α〉Üα−〈if
bgA,α〉T̈

b
α+ 〈jg

A
,α〉Ẅ,α+ 〈jg

A
,αg
B
,α〉V̈

B)δVA da

−

∫

Π

Nαβ,βδUα da+

∫

Π

NaαδT
a
α da

+

∫

Π

[Mαβ,αβ − (NαβW,α+Q
A
βV
A),β]δW da

+

∫

Π

(MA+QAαW,α+R
ABVB)δVA da

=

∫

Π

pδW da−

∫

Π

(〈k〉W + 〈kgB〉VB)δW da

−

∫

Π

(〈kgA〉W + 〈kgAgB〉VB)δVA da

−

∫

Π

(〈µ̂〉Ẅ + 〈µ̂gB〉V̈B)δW da−

∫

Π

(〈µ̂gA〉Ẅ + 〈µ̂gAgB〉V̈B)δVA da

−

∫

Π

(〈c〉Ẇ + 〈cgB〉V̇B)δW da−

∫

Π

(〈cgA〉Ẇ + 〈cgAgB〉V̇B)δVA da

(3.12)

Then, using the divergence theorem and the du Bois-Reymond lemma to equation (3.12),
after somemanipulations, governing equations of the proposed approximate tolerancemodel can
be obtained.



954 J. Jędrysiak

4. Governing equations

4.1. Tolerance model equations

Let us introduce denotations

Bαβγδ ≡〈bαβγδ〉 B
a
αβγ ≡〈bαβγδf

a
,δ〉

Babαγ ≡〈bαβγδf
a
,βf
b
,δ〉 Dαβγδ ≡〈dαβγδ〉

DAαβ ≡〈dαβγδg
A
,γδ〉 D

AB ≡〈dαβγδg
A
,αβg

B
,γδ〉

FABCα ≡ l
−3〈bαβγδg

A
,βg
B
,γg
C
,δ〉 F

A
αβγ ≡ l

−1〈bαβγδg
A
,δ〉

FABαβ ≡ l
−2〈bαβγδg

A
,γg
B
,δ〉 F

aB
αγ ≡ l

−1〈bαβγδf
a
,βg
B
,δ〉

FaBCα ≡ l
−2〈bαβγδf

a
,βg
B
,γg
C
,δ〉 F

ABCD ≡ l−4〈bαβγδg
A
,αg
B
,βg
C
,γg
D
,δ〉

GAαβγ ≡ l
−1〈hαβγδg

A
,δ〉 G

AB
αβ ≡ l

−2〈hαβγδg
A
,γg
B
,δ〉

GABγ ≡ l
−1〈hαβγδg

A
,αβg

B
,δ〉 G

ABC ≡ l−2〈hαβγδg
A
,αβg

B
,γg
C
,δ〉

Hαβγδ ≡〈hαβγδ〉 H
A
αβ ≡〈hαβγδg

A
,γδ〉

Haαγδ ≡〈hαβγδf
a
,β〉 H

aB
α ≡〈hαβγδf

a
,βg
B
,γδ〉

B̃αβγδ ≡〈b̃αβγδ〉 B̃
a
αβγ ≡〈b̃αβγδf

a
,δ〉

B̃abαγ ≡〈b̃αβγδf
a
,βf
b
,δ〉 D̃αβγδ ≡〈d̃αβγδ〉

D̃Aαβ ≡〈d̃αβγδg
A
,γδ〉 D̃

AB ≡〈d̃αβγδg
A
,αβg

B
,γδ〉

F̃ABCα ≡ l
−3〈b̃αβγδg

A
,βg
B
,γg
C
,δ〉 F̃

A
αβγ ≡ l

−1〈b̃αβγδg
A
,δ〉

F̃ABαβ ≡ l
−2〈b̃αβγδg

A
,γg
B
,δ〉 F̃

aB
αγ ≡ l

−1〈b̃αβγδf
a
,βg
B
,δ〉

F̃aBCα ≡ l
−2〈b̃αβγδf

A
,βg
B
,γg
C
,δ〉 F̃

ABCD ≡ l−4〈b̃αβγδg
A
,αg
B
,βg
C
,γg
D
,δ〉

G̃Aαβγ ≡ l
−1〈h̃αβγδg

A
,δ〉 G̃

AB
αβ ≡ l

−2〈h̃αβγδg
A
,γg
B
,δ〉

G̃ABγ ≡ l
−1〈h̃αβγδg

A
,αβg

B
,δ〉 G̃

ABC ≡ l−2〈h̃αβγδg
A
,αβg

B
,γg
C
,δ〉

H̃αβγδ ≡〈h̃αβγδ〉 H̃
A
αβ ≡〈h̃αβγδg

A
,γδ〉

H̃aαγδ ≡〈h̃αβγδf
a
,β〉 H̃

aB
α ≡〈h̃αβγδf

a
,βg
B
,γδ〉

m≡〈µ〉 mA ≡ l−2〈µgA〉

mAB ≡ l−4〈µgAgB〉 ϑ≡〈j〉

ϑAα ≡ l
−1〈jgA,α〉 ϑ

AB
αβ ≡ l

−2〈jgA,αg
B
,β〉

θ≡〈i〉 θa ≡ l−1〈ifa〉

θAα ≡ l
−1〈igA,α〉 θ

aA
α ≡ l

−2〈ifagA,α〉

C ≡〈c〉 CA ≡ l−2〈cgA〉

CAB ≡ l−4〈cgAgB〉 K ≡〈k〉

KA ≡ l−2〈kgA〉 KAB ≡ l−4〈kgAgB〉

m̂≡〈µ̂〉 m̂A ≡ l−2〈µ̂gA〉

m̂AB ≡ l−4〈µ̂gAgB〉 ma ≡ l−1〈µfa〉

mab ≡ l−4〈µfafb〉 P ≡〈p〉 PA ≡ l−2〈pgA〉

(4.1)

Using the tolerancemodelling procedure, a systemof equations for the in-planemacrodispla-
cementsUα, fluctuation amplitudes of the in-planemacrodisplacements T

a
α, macrodeflectionW ,

fluctuation amplitudes of the deflection VA can be derived:



Geometrically nonlinear vibrations of thin visco-elastic periodic plates... 955

—constitutive equations

Mαβ =−Hαβγδ
(
Uγ,δ +

1

2
W,γW,δ

)
−HaαβγT

a
γ +DαβγδW,γδ +D

A
αβV

A

− lGAαβγ[W,γV
A+(Ẇ,γV

A+W,γV̇
A)]−

1

2
l2GABαβ [V

AVB +(V̇AVB +VAV̇B)]

− H̃αβγδ
[
U̇γ,δ −

1

2
(Ẇ,γW,δ+W,γẆ,δ)

]
− H̃aαβγṪ

a
γ + D̃αβγδẆ,γδ + D̃

A
αβV̇

A

MA =−HAαβ

(
Uγ,δ +

1

2
W,γW,δ

)
−HaAα T

a
α +D

A
αβW,γδ +D

ABVB

− lGABγ W,γV
B −
1

2
l2GABCVBVC − H̃Aαβ

[
U̇γ,δ +

1

2
(Ẇ,γW,δ+W,γẆ,δ)

]

− H̃aAα Ṫ
a
α + D̃

A
αβẆ,γδ + D̃

ABV̇B − lG̃ABγ (Ẇ,γV
B +W,γV̇

B)

−
1

2
l2G̃ABC(V̇BVC +VBV̇C)

Nαβ =Bαβγδ
(
Uγ,δ +

1

2
W,γW,δ

)
+BaαβγT

a
γ −HαβγδW,γδ−H

A
αβV

A+ lFAαβγW,γV
A

+
1

2
l2FABαβ V

AVB + B̃αβγδ
[
U̇γ,δ +

1

2
(Ẇ,γW,δ+W,γẆ,δ)

]
+ B̃aαβγṪ

a
γ − H̃αβγδẆ,γδ

− H̃AαβV̇
A+ lF̃Aαβγ(Ẇ,γV

A+W,γV̇
A)+

1

2
l2F̃ABαβ (V̇

AVB +VAV̇B)

Naα =B
a
αγδ

(
Uγ,δ +

1

2
W,γW,δ

)
+BabαγT

b
γ −H

a
αγδW,γδ −H

aB
α V

B + lFaBαγ W,γV
B

+
1

2
l2FaBCα V

BVC + B̃aαγδ

[
U̇γ,δ +

1

2
(Ẇ,γW,δ+W,γẆ,δ)

]
+ B̃abαγṪ

b
γ − H̃

a
αγδẆ,γδ

− H̃aBα V̇
B + lF̃aBαγ (Ẇ,γV

B +W,γV̇
B)+

1

2
l2F̃aBCα (V̇

BVC +VBV̇C)

QAα = lF
A
αγδ

(
Uγ,δ +

1

2
W,γW,δ

)
+ lFaAαγ T

a
γ − lG

A
αγδW,γδ − lG

AB
α V

B + l2FABαγ W,γV
B

+
1

2
l3FABCα V

BVC + lF̃AαγδU̇γ,δ + lF̃
aA
αγ

[
Ṫaγ +

1

2
(Ẇ,γW,δ +W,γẆ,δ)

]

− lG̃AαγδẆ,γδ − lG̃
AB
α V̇

B + l2F̃ABαγ (Ẇ,γV
B +W,γV̇

B)+
1

2
l3F̃ABCα (V̇

BVC +VBV̇C)

RAB = l2FABγδ

(
Uγ,δ +

1

2
W,γW,δ

)
+ l2FaABγ T

a
γ − l

2GABγδ W,γδ − l
2GABCVC + l3FABCγ W,γV

C

+
1

2
l4FABCDVCVD+ l2F̃ABγδ

[
U̇γ,δ +

1

2
(Ẇ,γW,δ+W,γẆ,δ)

]
+ l2F̃aABγ Ṫ

a
γ − l

2G̃

(4.2)

— equilibrium equations

Mαβ,αβ − (NαβW,α+Q
A
βV
A),β +(m+ m̂)Ẅ + l

2(mA+ m̂A)V̈A−ϑẄ,αα− lϑ
A
αV̈
A
,α

+KW + l2KAVA+CẆ + l2CAV̇A+θÜα,α+ lθ
aT̈aα,α =P

MA+QAαW,α+R
ABVB + l2(mA+ m̂A)Ẅ + lϑAαẄ,α+ l

2(l2mAB + l2m̂AB +ϑABαβ )V̈
B

+ l2KAW + l4KABVB + l2CAẆ + l4CABV̇B − lθAαÜα− l
2θaAα T̈

a
α = l

2PA

−Nαβ,β +mÜα+ lm
aT̈aα −θẄ,α− lθ

A
α V̈
A =0

Naα + lm
aÜα+ l

2mabT̈bα− lθ
aẄ,α− l

2θaAα V̈
A =0

(4.3)

Equations (4.2) and (4.3) together with micro-macro decompositions (3.8) constitute the
nonlinear tolerance model of thin visco-elastic periodic plates resting on a foundation with
damping if the plate properties are not even functions of z. This model describes the effect
of the microstructure size on the overall plate behaviour by terms with the microstructure



956 J. Jędrysiak

parameter l. For the considered plates, boundary conditions have to be formulated only for the
macrodeflection W and the in-plane macrodisplacements Uα. Moreover, the basic unknowns of
equations (4.2) and (4.3) have to satisfy the following conditions:W(·, t),VA(·, t)∈SV 2δ (Π,∆),
Uα(·, t),T

a
α(·, t)∈SV

1
δ (Π,∆), i.e. they are slowly-varying functions in x.

In the next considerations, it is assumed that the plate properties are even functions of z,
i.e. plates under consideration have the symmetry plane z = 0. Hence, some coefficients (4.1)
are equal to zero

Hαβγδ =H
A
αβ =H

a
αγδ =H

aB
α =G

A
αβγ =G

AB
αβ =G

AB
γ =G

ABC =0

H̃αβγδ = H̃
A
αβ = H̃

a
αγδ = H̃

aB
α = G̃

A
αβγ = G̃

AB
αβ = G̃

AB
γ = G̃

ABC =0

θ= θa = θAα = θ
aA
α =0

(4.4)

Equations (4.2) and (4.3) take the following form:
— constitutive equations

Mαβ =DαβγδW,γδ +D
A
αβV

A+ D̃αβγδẆ,γδ+ D̃
A
αβV̇

A

MA =DAαβW,γδ+D
ABVB + D̃AαβẆ,γδ + D̃

ABV̇B

Nαβ =Bαβγδ
(
Uγ,δ +

1

2
W,γW,δ

)
+BaαβγT

a
γ + lF

A
αβγW,γV

A+
1

2
l2FABαβ V

AVB

+ B̃αβγδ
[
U̇γ,δ +

1

2
(Ẇ,γW,δ +W,γẆ,δ)

]
+ B̃aαβγṪ

a
γ

+ lF̃Aαβγ(Ẇ,γV
A+W,γV̇

A)+
1

2
l2F̃ABαβ (V̇

AVB +VAV̇B)

Naα =B
a
αγδ

(
Uγ,δ +

1

2
W,γW,δ

)
+BabαγT

b
γ + lF

aB
αγ W,γV

B +
1

2
l2FaBCα V

BVC

+ B̃aαγδ

[
U̇γ,δ +

1

2
(Ẇ,γW,δ +W,γẆ,δ)

]
+ B̃abαγṪ

b
γ

+ lF̃aBαγ (Ẇ,γV
B +W,γV̇

B)+
1

2
l2F̃aBCα (V̇

BVC +VBV̇C)

QAα = lF
A
αγδ

(
Uγ,δ +

1

2
W,γW,δ

)
+ lFaAαγ T

a
γ + l

2FABαγ W,γV
B +
1

2
l3FABCα V

BVC

+ lF̃AαγδU̇γ,δ + lF̃
aA
αγ

[
Ṫaγ +

1

2
(Ẇ,γW,δ+W,γẆ,δ)

]
+ l2F̃ABαγ (Ẇ,γV

B +W,γV̇
B)

+
1

2
l3F̃ABCα (V̇

BVC +VBV̇C)

RAB = l2FABγδ

(
Uγ,δ +

1

2
W,γW,δ

)
+ l2FaABγ T

a
γ + l

3FABCγ W,γV
C +
1

2
l4FABCDVCVD

+ l2F̃aABγ Ṫ
a
γ + l

2F̃ABγδ

[
U̇γ,δ +

1

2
(Ẇ,γW,δ+W,γẆ,δ)

]

+ l3F̃ABCγ (Ẇ,γV
C +W,γV̇

C)+
1

2
l4F̃ABCD(V̇CVD+VCV̇D)

(4.5)

— equilibrium equations

Mαβ,αβ − (NαβW,α+Q
A
βV
A),β +(m+ m̂)Ẅ + l

2(mA+ m̂A)V̈A−ϑẄ,αα− lϑ
A
αV̈
A
,α

+KW + l2KAVA+CẆ + l2CAV̇A =P

MA+QAαW,α+R
ABVB + l2(mA+ m̂A)Ẅ + lϑAαẄ,α+ l

2(l2mAB + l2m̂AB +ϑABαβ )V̈
B

+ l2KAW + l4KABVB + l2CAẆ + l4CABV̇B = l2PA

−Nαβ,β +mÜα+ lm
aT̈aα =0

Naα + lm
aÜα+ l

2mabT̈bα =0

(4.6)



Geometrically nonlinear vibrations of thin visco-elastic periodic plates... 957

Similarly to equations (4.2) and (4.3), equations (4.5) and (4.6) together with micro-macro
decompositions (3.8) constitute thenonlinear tolerancemodel of thinvisco-elastic periodicplates
resting on a foundation with damping, but only for plates with the symmetry plane z = 0. It
can be observed that all above equations (4.2), (4.3) and (4.5), (4.6) have constant coefficients.

4.2. Asymptotic model equations

The asymptotic model equations can be obtained, from the formal point of view, using
the asymptotic modelling procedure, see Woźniak et al. (2010). Below, this is done by simply
neglecting terms of the order ofO(ln), n=1,2, . . ., in equations (4.2), (4.3) and (4.5), (4.6).
Hence, from equations (4.2) and (4.3), the equations of the nonlinear asymptoticmodel take

the form:
— constitutive equations

Mαβ =−Hαβγδ
(
Uγ,δ +

1

2
W,γW,δ

)
−HaαβγT

a
γ +DαβγδW,γδ +D

A
αβV

A

− H̃αβγδ
[
U̇γ,δ −

1

2
(Ẇ,γW,δ+W,γẆ,δ)

]
− H̃aαβγṪ

a
γ + D̃αβγδẆ,γδ + D̃

A
αβV̇

A

MA =−HAαβ

(
Uγ,δ +

1

2
W,γW,δ

)
−HaAα T

a
α +D

A
αβW,γδ +D

ABVB

− H̃Aαβ

[
U̇γ,δ +

1

2
(Ẇ,γW,δ+W,γẆ,δ)

]
− H̃aAα Ṫ

a
α + D̃

A
αβẆ,γδ + D̃

ABV̇B

Nαβ =Bαβγδ
(
Uγ,δ +

1

2
W,γW,δ

)
+BaαβγT

a
γ −HαβγδW,γδ−H

A
αβV

A

+ B̃αβγδ
[
U̇γ,δ +

1

2
(Ẇ,γW,δ +W,γẆ,δ)

]
+ ãBaαβγṪ

a
γ − H̃αβγδẆ,γδ − H̃

A
αβV̇

A

Naα =B
a
αγδ

(
Uγ,δ +

1

2
W,γW,δ

)
+BabαγT

b
γ −H

a
αγδW,γδ −H

aB
α V

B

+ B̃aαγδ

[
U̇γ,δ +

1

2
(Ẇ,γW,δ +W,γẆ,δ)

]
+ B̃abαγṪ

b
γ − H̃

a
αγδẆ,γδ− H̃

aB
α V̇

B

QAα =0 R
AB =0

(4.7)

— equilibrium equations

Mαβ,αβ − (NαβW,α+Q
A
βV
A),β +(m+ m̂)Ẅ −ϑẄ,αα+KW +CẆ +θÜα,α =P

MA =0 −Nαβ,β +mÜα−θẄ,α =0 N
a
α =0

(4.8)

where all coefficients are constant.
It can be observed that equations (4.7) and (4.8) with micro-macro decompositions (3.8)

constitute the nonlinear asymptotic model of thin visco-elastic periodic plates resting on a
foundation with damping for plates without the symmetry plane z=0.
On the other side, from equations (4.5) and (4.6), similar equations of the nonlinear asymp-

totic model can be derived in the form:
— constitutive equations

Mαβ =DαβγδW,γδ +D
A
αβV

A+ D̃αβγδẆ,γδ+ D̃
A
αβV̇

A

MA =DAαβW,γδ+D
ABVB + D̃AαβẆ,γδ + D̃

ABV̇B

Nαβ =Bαβγδ
(
Uγ,δ +

1

2
W,γW,δ

)
+BaαβγT

a
γ

+ B̃αβγδ
[
U̇γ,δ +

1

2
(Ẇ,γW,δ +W,γẆ,δ)

]
+ B̃aαβγṪ

a
γ

Naα =B
a
αγδ

(
Uγ,δ +

1

2
W,γW,δ

)
+BabαγT

b
γ + B̃

a
αγδ

[
U̇γ,δ +

1

2
(Ẇ,γW,δ+W,γẆ,δ)

]
+ B̃abαγṪ

b
γ

QAα =0 R
AB =0

(4.9)



958 J. Jędrysiak

—equilibrium equations

Mαβ,αβ − (NαβW,α),β +(m+ m̂)Ẅ −ϑẄ,αα+KW +CẆ =P

MA =0 −Nαβ,β +mÜα =0 N
a
α =0

(4.10)

with all coefficients constant.
It is necessary to observe that equations (4.9) and (4.10) and micro-macro decompositions

(3.8) constitute the nonlinear asymptotic model of thin visco-elastic periodic plates resting on
a foundation with damping for plates with the symmetry plane z=0.

5. Final remarks

Anewnonlinear non-asymptoticmodel for dynamicproblemsof thinvisco-elastic periodicplates
resting on a foundation with damping is proposed in this note. This model is based on the
assumptions of von Kármán nonlinear thin plate theory. In order to derive themodel governing
equations, the tolerance modelling is applied.
Summarizing, it can be concluded that:
• The proposed approach replaces governing equations of plates having highly oscillating,
periodic, non-continuous functional coefficients by the model equations with constant co-
efficients, which can be solved using suitable well-known methods. Thus, the nonlinear
tolerance model can be a useful tool in investigations of various dynamic phenomena of
the considered plate structures.

• In contrast to the original formulations, the new proposed nonlinear tolerance model in-
troduces some averaged, effective properties of the plate structure.

• Dynamic behaviour of the plates under consideration is described in this model by some
new unknowns as averaged deflections (macrodeflections) and averaged in-plane displa-
cements (in-plane macrodisplacements) and amplitudes of their disturbances due to in-
homogeneity of the structure. These new kinematic unknowns have to be slowly-varying
functions in x, which constitutes conditions of physical reliability of the solutions.

• The very important feature of the proposed nonlinear tolerancemodel is that its governing
equations involve terms with the microstructure parameter. Hence, this tolerance model
makes it possible to investigate the effect of themicrostructure size on the overall dynamic
behaviour of thin visco-elastic periodic plates resting on a foundation with damping in
the framework of vonKármán nonlinear thin plate theory. Using this model, some pheno-
mena in dynamic problems caused by the internal periodic structure of the plates under
consideration can be investigated.

• It can be observed that the transition from the governing equations of geometrically non-
linear tolerance models to the equations of the linear tolerance models may take place
on two levels – micro, when the effect of nonlinear terms with fluctuation amplitudes is
omitted, andmicro-macro, when all nonlinear terms are neglected.

• It should be noted that the proposed model is a kind of generalization in relation to
the known tolerance models shown by Domagalski and Jędrysiak (2012, 2015), where
some bending nonlinear problems of thin periodic plates were described. Domagalski and
Jędrysiak (2014) analysed nonlinear vibrations of slender periodic beams resting on a
foundation with damping, wherreas Marczak and Jędrysiak (2014) investigated damped
vibrations of plate strips with periodically distributed concentrated masses.

Various applications of the proposed tolerance and asymptotic models to dynamics of thin
visco-elastic periodic plates withmoderately large deflections resting on a foundationwith dam-
ping will be analysed in the forthcoming papers.



Geometrically nonlinear vibrations of thin visco-elastic periodic plates... 959

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Manuscript received September 23, 2015; accepted for print December 22, 2015