Nonlinear dynamic finite element analysis of thick transversly flexible sandwich panel on elastic foundation with account of damage progression in time. Part 1. Three-dimensional formulation and two-dimensional plate theory
In order to construct a plate theory for a thick transversely compressible sandwich plate with composite laminated face sheets, the author makes simplifying assumptions regarding distribution o f transverse strain components in the thickness direction. It is assumed that the transverse strains...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2005
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| Цитувати: | Nonlinear dynamic finite element analysis of thick transversly flexible sandwich panel on elastic foundation with account of damage progression in time. Part 1. Three-dimensional formulation and two-dimensional plate theory / V.Y. Perel // Проблемы прочности. — 2005. — № 2. — С. 92-106. — Бібліогр.: 13 назв. — англ. |
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irk-123456789-476762013-07-25T12:00:16Z Nonlinear dynamic finite element analysis of thick transversly flexible sandwich panel on elastic foundation with account of damage progression in time. Part 1. Three-dimensional formulation and two-dimensional plate theory Perel, V.Y. Научно-технический раздел In order to construct a plate theory for a thick transversely compressible sandwich plate with composite laminated face sheets, the author makes simplifying assumptions regarding distribution o f transverse strain components in the thickness direction. It is assumed that the transverse strains, i.e., £xz, £yz, and £zz do not vary in the thickness direction within the face sheets and the core, but can be different functions of the in-plane coordinates in different sublaminates (the face sheets and the core). An algorithm, which takes account of damage progression in dynamic problems, is incorporated into the computational scheme based on the geometrically nonlinear formulation, and is applied to failure analysis o f a sandwich plate under ground impact. The proposed model o f the sandwich plate does not require many degrees o f freedom in the finite element computations and has a wide range of applicability: it can be applied to the sandwich plates with a wide range o f ratios of thickness to the in-plane dimensionswith both thin and thick face sheets (as compared to the thickness o f the core) and to the sandwich plates with both transversely rigid and transversely compressible face sheets and cores. Із метою розвитку теорії пластин для товстих багатошарових панелей, що стискувані у поперечному напряму, із зовнішніми шарами у вигляді композитних ламінатних листів запропоновано спрощену схему розподілу поперечних деформацій по товщині панелі. Припускається, що поперечні деформації є xz, £ yz та £ zz не змінюються по товщині панелі в інтервалі її зовнішніх листів і серцевини, але можуть описуватися різними функціональними залежностями від координати в площинах різних субламінатів (зовнішні листи і серцевина панелі). Алгоритм, що ураховує розвиток пошкодження для динамічних задач, використовується в розрахунковій схемі, що базується на геометрично нелінійному формулюванні, стосовно до аналізу руйнування багатошарової пластини від ударного стикання з грунтом. Модель багатошарової пластини характеризується малою кількістю степеней вільності у скінченноелементних розрахунках та широким використанням: для пластин із тонкими або товстими зовнішніми шарами (у порівнянні з товщиною серцевини), для випадків стисливості або нестисливості зовнішніх шарів та (або) серцевини в поперечному напряму. С целью развития теории пластин для толстых многослойных панелей, сжимаемых в поперечном направлении, с наружными слоями в виде композитных ламинатных листов предложена упрощенная схема распределения поперечных деформаций по толщине панели. Предполагается, что поперечные деформации Exz, Eyz и ezz не изменяются по толщине панели в пределах ее наружных листов и сердцевины, но могут описываться различными функциональными зависимостями от координат в плоскостях разных субламинатов (наружные листы и сердцевина панели). Алгоритм, учитывающий развитие повреждения для динамических задач, используется в расчетной схеме, базирующейся на геометрически нелинейной формулировке, применительно к анализу разрушения многослойной пластины от ударного соприкосновения с грунтом. Модель многослойной пластины характеризуется малым количеством степеней свободы в конечноэлементных расчетах и широким спектром применения: для пластин с тонкими или толстыми наружными слоями (по сравнению с толщиной сердцевины), для случаев сжимаемости или несжимаемости наружных слоев и (или) сердцевины в поперечном направлении. 2005 Article Nonlinear dynamic finite element analysis of thick transversly flexible sandwich panel on elastic foundation with account of damage progression in time. Part 1. Three-dimensional formulation and two-dimensional plate theory / V.Y. Perel // Проблемы прочности. — 2005. — № 2. — С. 92-106. — Бібліогр.: 13 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/47676 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Научно-технический раздел Научно-технический раздел |
| spellingShingle |
Научно-технический раздел Научно-технический раздел Perel, V.Y. Nonlinear dynamic finite element analysis of thick transversly flexible sandwich panel on elastic foundation with account of damage progression in time. Part 1. Three-dimensional formulation and two-dimensional plate theory Проблемы прочности |
| description |
In order to construct a plate theory for a thick
transversely compressible sandwich plate with
composite laminated face sheets, the author
makes simplifying assumptions regarding distribution
o f transverse strain components in the
thickness direction. It is assumed that the transverse
strains, i.e., £xz, £yz, and £zz do not vary
in the thickness direction within the face sheets
and the core, but can be different functions of
the in-plane coordinates in different
sublaminates (the face sheets and the core). An
algorithm, which takes account of damage progression
in dynamic problems, is incorporated
into the computational scheme based on the
geometrically nonlinear formulation, and is applied
to failure analysis o f a sandwich plate under
ground impact. The proposed model o f the
sandwich plate does not require many degrees
o f freedom in the finite element computations
and has a wide range of applicability: it can be
applied to the sandwich plates with a wide
range o f ratios of thickness to the in-plane dimensionswith both thin and thick face sheets
(as compared to the thickness o f the core) and
to the sandwich plates with both transversely
rigid and transversely compressible face sheets
and cores. |
| format |
Article |
| author |
Perel, V.Y. |
| author_facet |
Perel, V.Y. |
| author_sort |
Perel, V.Y. |
| title |
Nonlinear dynamic finite element analysis of thick transversly flexible sandwich panel on elastic foundation with account of damage progression in time. Part 1. Three-dimensional formulation and two-dimensional plate theory |
| title_short |
Nonlinear dynamic finite element analysis of thick transversly flexible sandwich panel on elastic foundation with account of damage progression in time. Part 1. Three-dimensional formulation and two-dimensional plate theory |
| title_full |
Nonlinear dynamic finite element analysis of thick transversly flexible sandwich panel on elastic foundation with account of damage progression in time. Part 1. Three-dimensional formulation and two-dimensional plate theory |
| title_fullStr |
Nonlinear dynamic finite element analysis of thick transversly flexible sandwich panel on elastic foundation with account of damage progression in time. Part 1. Three-dimensional formulation and two-dimensional plate theory |
| title_full_unstemmed |
Nonlinear dynamic finite element analysis of thick transversly flexible sandwich panel on elastic foundation with account of damage progression in time. Part 1. Three-dimensional formulation and two-dimensional plate theory |
| title_sort |
nonlinear dynamic finite element analysis of thick transversly flexible sandwich panel on elastic foundation with account of damage progression in time. part 1. three-dimensional formulation and two-dimensional plate theory |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2005 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/47676 |
| citation_txt |
Nonlinear dynamic finite element analysis of thick transversly flexible sandwich panel on elastic foundation with account of damage progression in time. Part 1. Three-dimensional formulation and two-dimensional plate theory / V.Y. Perel // Проблемы прочности. — 2005. — № 2. — С. 92-106. — Бібліогр.: 13 назв. — англ. |
| series |
Проблемы прочности |
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| first_indexed |
2025-07-04T07:38:54Z |
| last_indexed |
2025-07-04T07:38:54Z |
| _version_ |
1836701172739604480 |
| fulltext |
UDC 539.4
Nonlinear Dynamic Finite Element Analysis of Thick Transversly
Flexible Sandwich Panel on Elastic Foundation with Account of
Damage Progression in Time. Part 1. Three-Dimensional Formulation
and Two-Dimensional Plate Theory
V. Y. Perel
Air Force Institute of Technology, Department of Aeronautics and Astronautics, Wright
Patterson Air Force Base, Ohio, USA
УДК 539.4
Нелинейный динамический конечноэлементный анализ гибкой в
поперечном направлении толстой многослойной панели на
упругом основании с учетом развития повреждения во времени.
Сообщение 1. Трехмерная формулировка задачи и двухмерная
теория пластин
В. Ю. Перель
Технологический институт ВВС США, Отделение аэронавтики и астронавтики,
Хобсон Бэй, Огайо, США
С целью развития теории пластин для толстых многослойных панелей, сжимаемых в
поперечном направлении, с наружными слоями в виде композитных ламинатных листов
предложена упрощенная схема распределения поперечных деформаций по толщине панели.
Предполагается, что поперечные деформации Exz, Eyz и ezz не изменяются по толщине
панели в пределах ее наружных листов и сердцевины, но могут описываться различными
функциональными зависимостями от координат в плоскостях разных субламинатов (на
ружные листы и сердцевина панели). Алгоритм, учитывающий развитие повреждения для
динамических задач, используется в расчетной схеме, базирующейся на геометрически
нелинейной формулировке, применительно к анализу разрушения многослойной пластины от
ударного соприкосновения с грунтом. Модель многослойной пластины характеризуется
малым количеством степеней свободы в конечноэлементных расчетах и широким спектром
применения: для пластин с тонкими или толстыми наружными слоями (по сравнению с
толщиной сердцевины), для случаев сжимаемости или несжимаемости наружных слоев и
(или) сердцевины в поперечном направлении.
Ключевые слова : сжимаемые в поперечном направлении многослойные
панели, упругое основание, нелинейная динамика, трехмерное напряженное
состояние.
Introduction. Sandwich structures are used in a variety of load-bearing
applications. Sandwich plates have a well pronounced zigzag variation of the
in-plane displacements in the thickness direction, due to their high thickness-
to-length ratios and large difference of values of elastic moduli of the face sheets
and the core. Such characteristics of the sandwich plates make it necessary to use
a layerwise approach in their analysis, the idea of which is to introduce separate
© V. Y. PEREL, 2005
92 ISSN 0556-171X. Проблемы прочности, 2005, № 2
Nonlinear Dynamic Finite Element Analysis
simplifying assumptions regarding the through-thickness variation of
displacements, strains or stresses within each face sheet and the core. Many
researchers studied the sandwich plate with thick, light-weight, vertically stiff
cores and thin rigid face sheets, using discrete-layer (or layerwise) models. Most
of the layerwise models of such structures are based on the piecewise linear
through the thickness in-plane displacement approximations in addition to
constant (through the thickness) transverse displacements [1-9].
The modern cores are usually made of plastic foams and non-metallic
honeycombs, like Aramid and Nomex. These cores have properties similar to
those used traditionally (for example, metallic honeycombs), but due to their
transverse compressibility (i.e., ability of such cores to change height under
applied loads) the direct transverse strain e zz becomes important. Therefore, the
models of the sandwich plates with the cores made of plastic foams or
non-metallic honeycombs must not exclude the change of height of the core.
Frostig and Baruch [10] developed a theory of a sandwich beam with thin face
sheets in which account is taken of transverse compressibility of the core, and the
longitudinal displacement in the core varies nonlinearly in the thickness direction.
In this theory the longitudinal displacement in the face sheets varies linearly in
the thickness direction, and the transverse displacement of the face sheets does
not vary in the thickness direction, that is, the transverse direct strain e zz in the
face sheets is assumed to be equal to zero in the expression for the strain energy.
The transverse shear strain e xz in the face sheets is also considered to be
negligibly small in the expression for the strain energy that is used for variational
derivation of the differential equations for the unknown functions. The transverse
shear stress in the face sheets can be computed by integration of the pointwise
equilibrium equations a xxx + a xz z = 0.
Under certain circumstances , when the face sheets are thick, when the plate
is loaded by a concentrated or partially distributed load or when the plate is on an
elastic foundation, taking account of the direct transverse strain e zz in the face
sheets and the transverse shear strain e xz in the face sheets in the expression for
the strain energy allows one to obtain a higher accuracy of the stress computation.
Besides, in order to achieve a high accuracy of stress computation in the thick
face sheets, a model for such a plate must assume or lead to the nonlinear
through-the-thickness variation of the in-plane displacements not only in the core,
but also in the face sheets.
Construction of a computational scheme that satisfies these requirements can
be approached, for example, with the help of the layerwise laminated plate theory
of Reddy [11], which is a generalization of many other displacement-based
layerwise theories of laminated plates. In this theory the displacement field in the
kth layer is written as
m
u (k)(x , y , z , t) = 2 u j )(x , y , t) 0 j )( z ),
j =1
m
v (K\ x , y , z , t) = 2 v j )(x , y , t) 0 j )( z ),
j=1
ISSN 0556-171X. npodxeMbi npounocmu, 2005, N 2 93
V. Y. Perel
n
w (k}(x , y , z , t) = 2 w f \ x , y , t) V f )( z ),
j=i
where uf \ x ,y , t), v(k)(x ,y , t), and w(k)(x ,y , t) are the unknown functions and
^ )( z ) and V )(z ) are chosen to be Lagrange interpolation functions of the
thickness coordinate, to provide the required continuity of displacements and
discontinuity of the transverse strains across the interface between adjacent
thickness subdivisions.
This theory allows one to achieve a high accuracy of the transverse stress
computation in the composite laminates, but for this purpose it requires a large
number of thickness subdivisions of the laminate. This leads to a large number of
the unknown functions and degrees of freedom in a finite element model. In
effect, the finite element model, based on this generalized layerwise laminated
plate theory is equivalent to the three-dimensional finite element model. To
reduce the number of the unknown functions in the layerwise model of a
laminated plate, one can use the concept of a sublaminate (i.e., make the number
of thickness subdivisions less than the number of material layers) and deal with
the material properties, averaged through the thickness of a sublaminate. In a
model of the sandwich plate, it is natural to choose three sublaminates: the two
face sheets and the core. With such a small number of the sublaminates, the nature
of assumptions on the through-the-thickness variation of displacements can have
a large effect on the accuracy of the computed stresses. Besides, the actual
through-the-thickness variation of displacements can depend on the character of
applied loads and boundary conditions. Therefore, in a layerwise model of the
sandwich plate with only three sublaminates, it is desirable to have a flexibility in
the choice of the functions that represent through-the-thickness variation of
displacements. O f course, the Lagrange interpolation polynomials, which
represent the thickness variation of the displacements within a sublaminate in the
Reddy’s [11] layerwise theory, can be chosen to be of any desired degree, but
such an increase of the degree of the Lagrange interpolation polynomials leads to
the increase of the number of the unknown functions.
In the present paper, a computational scheme for analysis of the sandwich
plate is constructed in which the simplifying assumptions that lead to a plate-type
theory are made with respect to the variation of the transverse strains in the
thickness direction of the face sheets and the core of the sandwich plate. The
displacements are then obtained by integration of these assumed transverse
strains, and the constants of integration are chosen to satisfy the conditions of
continuity of the displacements across the borders between the face sheets and the
core. In such a method, the required continuity of displacements in the thickness
direction is satisfied regardless of the assumed type of through-the-thickness
distribution of the transverse strains, and the transverse flexibility of the plate can
be taken into account. This leads to a larger number of choices of simplifying
assumptions about the variation of strains (and, therefore, displacements) in the
thickness direction, and, therefore, allows a better adjustment of the computational
scheme to the conditions under which the sandwich plate is analyzed by a
layerwise method with only three sublaminates (being the face sheets and the
94 ISSN 0556-171X. npo6n.eMH npounocmu, 2005, N9 2
Nonlinear Dynamic Finite Element Analysis
core). The transverse stresses are computed by integration of the pointwise
equilibrium equations that leads to satisfaction of conditions of continuity of the
transverse stresses across the boundaries between the face sheets and the core and
satisfaction of stress boundary conditions on the upper and lower surfaces of the
plate.
In the present paper, the model is considered on the basis of the simplest of
such assumptions that do not ignore, in the expression for the strain energy, the
transverse shear and normal strains in the face sheets. It is assumed here that the
transverse strains do not vary in the thickness direction within the face sheets and
the core, but can be different functions of the in-plane coordinate in the face
sheets and the core. In the post-process stage of analysis, these first
approximations of the transverse strains can be improved by substituting the
transverse stresses, obtained by integration of the pointwise equations of motion
(Appendix) into the strain-stress relations. These improved values of the
transverse strains vary in the thickness direction and are sufficiently accurate as
compared to those of the known exact solutions, based on the linear three
dimensional theory [12]. In the theory, discussed in this paper, the transverse
displacement, obtained by integration of the assumed transverse normal strain,
varies linearly in the thickness direction within a sublaminate (therefore,
transverse compressibility of the plate is taken into account), and the in-plane
displacement, obtained by integration of the assumed transverse shear strains,
varies quadratically within the thickness of a sublaminate. The developed theory
does not require many degrees of freedom in finite element models, despite its
ability to capture the transverse flexibility of the plate and non-linear through-
the-thickness variation of the in-plane displacements.
Three-Dimensional Formulation. The sandwich plate is divided into three
conventional layers (sublaminates): the two face sheets and the core. Within each
sublaminate, the simplifying assumptions of the plate theory are made separately.
In the following text, the superscript k denotes the number of a sublaminate:
k = 1 denotes the lower face sheet, k = 2 denotes the core, and k = 3 denotes the
upper face sheet (Fig. 1).
L
h
**-2 «-
upper face sheet, k=3
Z*~ 2
core, k=2
X*
t 4 ^
2 2
lower face sheet, k=l
h
Fig. 1. The coordinate system and notations for the sandwich plate. (Axis z is in the thickness
direction, h is a thickness of the whole plate, and t is the core thickness.)
ISSN 0556-171X. npoôneMU npoHHocmu, 2005, № 2 95
V. Y. Perel
In the subsequent text, both index and non-index notations for the
displacements will be used interchangeably without a preliminary notice, the
correspondence between them being established as follows: u1 = u, u2 = v ,
u 3 = w.
As energy-conjugate measures of strain and stress, the Green-Lagrange
strain tensor and the second Piola-Kirchhoff stress tensor are used. The analysis
is limited to the important case of small strains, moderate displacements (of the
order of thickness of the plate) and moderate rotations (10-15 degrees). This
means that of all the higher order terms in the Green-Lagrange strain-
displacement relations
1
£ij = 2 (u i j + uj i + us,ius,j ) (1)
only u3 ,au3/g (a , y3 = 1, 2 ) are not negligible compared to ua i (a = 1, 2 ; i = 1, 2 ,
3) [11, 13]. Therefore, the strain-displacement relations become
1 2
«XX = u,x + 2 (w x ) , (2)
1 2
« ^ + 2 (w ,^) ’ (3)
1
;x.y = 2 ( u, y + V ,x + w,xw, y )> (4)
= 2 ( u , 2 + W ,X )’ (5)xz
i(k) = ^ ( v
yz 2 ( V, (6)
« zz w ,z . (7)
Now one needs to find the simplified equations of motion and boundary
conditions, such that their accuracy corresponds to the accuracy of the adopted
von Karman strain-displacement relations. These equations of motion are used
for computation of the transverse stresses in the post-processing stage of the finite
element analysis. The equations of motion and boundary conditions, consistent
with the von Karman [13] strain-displacement relations (2)-(7), are received
using the virtual work principle (see Appendix). The equations of motion are
written for each of the three conventional layers: the upper and lower face sheets
and the core. The boundary conditions are applied to the upper and lower surfaces
of the plate and to the interfaces between the face sheets and the core, with the
result that
96 ISSN 0556-171X. npoôëeMbi npounocmu, 2005, N9 2
Nonlinear Dynamic Finite Element Analysis
O 0, o Ĵ] = 0, o (i) -
at the lower surface,
o (3) = 0u xz o (3) = 0 ̂yZ V,5 O ZZ} = qu
(8)
(9)
at the upper surface, and
° Xz(z 2 ) = ^ X2)( z 2 ). ff(J!z)( z 2 ) = ° (y z (z 2 ). ^ ZZ)( z 2 ) (z2)( z 2 X (10)
X̂2)(z3 ) = ^XP(z 3 ) o(y z (z 3 ) = °% )(z 3). ^ g )(z3 ) (i )(z3). (11)
at the interfaces.
In the laminate coordinate system (x, y , z), whose axes are aligned with the
sides of the plate, the stress-strain relations for an orthotropic material have the
form [11]
(12)
O xx C 11 C 12 C 13 0 0 C 16 «xx
yyo C 12 C 22 C 23 0 0 C 26 «« yy
o zz C 13 C 23 C 33 0 0 C 36 « zz
o yz 0 0 C 44 C 45 0 2« yz
xzo 0 0 C 45 C 55 0 2« xz
xyo C 16 C 26 C 36 0 0 C 66. 2« xy
or
{o}= [C ]{«}, (13)
where C j are the elastic coefficients, referred to the laminate coordinate system.
In addition, the displacements must be continuous at the interfaces between
the faces and the core:
,(i) _ (2) (2) = « (3)z—z — Ui z=z2 1 z=z2 ’ ui Z=Z3 Ui (14)
Two-Dimensional Plate Theory.
Simplifying Assumptions o f the Plate Theory. To construct a two-dimensional
plate theory, simplifying assumptions regarding a distribution of the transverse
strain components in the thickness direction are made. It is assumed that within
the face sheets and the core the transverse strains do not depend on the
z-coordinate, but they can be different functions of coordinates x , y , and time t
in different face sheets and the core:
« {xz] = « Xz)( x , y ,t X
« = «yZ)(x , y , o>
« {zz] = « {zz)( x, y , t X
( z = 1, 2 , 3),
(15)
ISSN 0556-171X. npoôëeMbi npounocmu, 2005, N 2 97
xz
V. Y. Perel
where the superscript k denotes the number of a sublaminate: k = 1 denotes the
lower face sheet, k = 2 denotes the core, and k = 3 denotes the upper face sheet.
An accuracy of the theory, based on these assumptions, is studied in [12].
The assumed transverse strains of equations (15), together with displacements
of the middle surface of the plate
u 0(x ,y , t) = u(2)| z=0 ,
v 0(x ,y , t) = v (21 z=0 , (16)
w0 ( x ,y , t) = w(2)| z=0
are the unknown functions of the problem that will be computed by the finite
element method. Therefore, all displacements, strains and stresses must be
expressed in terms of these functions.
Displacements in Terms o f the Unknown Functions. In order to obtain
expressions for the displacements in terms of the unknown functions e %\ e k ,
e %\ uo, v o, and w0, the strain-displacement relations (5)-(7) are integrated
with the following result:
w (1)(x ,y ,z , t) = w0(x ,y , t) + e {z z ( x ,y , t )z2 + e zz)(x ,y ,t)(z - z 2 )
(17)
( z 1 < z < z 2 ),
w (2)(x ,y , z , t) = Wo(x,y , t) + e ̂ ( x ,y , t)z , (18)
w (3)(x ,y ,z ,t) = w0(x ,y ,t) + e 2 )(x ,y ,t )z 3 + e z3)(x ,y ,t)(z - z 3)
(19)
( z 3 < z < z 4 ),
u(1) = u 0 + (2e x2} - w0,x )z2 - 2 e (2 )x z 2 + (2exz - w0,x - e (z22)x z 2 )(z - z 2) -
- ^ e {̂ ,x (z - z 2 )2 ( z 1 < z < z 2 X (20)
u(2) = u 0 + (2e x22} - w0,x )z - ^ e I22)x z 2 ( z 2 < z < z 3 X (21)
(3) _l^)„(2) \ 1 „(2) 2 ,^ > = u 0 + (2e x / - w 0,x ) z 3 - ^ e zz ,x z 3 +
+ (2e (x3z} - w0,x - e z2)x z 3 )( z - z 3 ) - 2 e S x ( z - z 3)2 ( z 3 < z < z 4 ), (22)
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V(1) = v 0 + {2e (yZ - w0 ,y )Z2 - 2 £{zz,y z 2 + (2£ § - w0 ,y -
-£ (z2)y Z 2 )(z - z 2 ) — 2 £ S y (z - z 2 ) 2 ( Z 1 ^ z ^ z 2 X (23)
v (2) = vo + (2£(yz) - Wo,y)z - 1 £ (z2)yz 2 ( z 2 ^ z ^ z 3 X (24)
v (3) = v 0 + (2£ (yz] - w0 ,y ) z 3 - 1 £ (zl)yz 3 + (2£ (y ! - w0 ,y - £ z22)y )(z - z 3 ) -
- 1 £(z3)y(z - z 3) 2 (z 3 ^ z ^ z 4). (25)
It can be verified easily that these expressions for the displacements are
continuous across the boundaries between the faces and the core, that is, at
z = z 2 and z = z 3 .
Strains in Terms o f the Unknown Functions. Expressions for the in-plane
strains £ Xx , £ Xy) , and £ (y* in terms of the unknown functions are obtained by
substituting expressions (17)-(25) for displacements in terms of the unknown
functions into the strain-displacement relations (2), (3) and (4). The transverse
strains £ (k\ £k , and £ ) are the unknown functions themselves.
Extended H am ilton’s Principle, Written fo r This Specific Problem. In order
to derive either differential equations for the unknown functions with boundary
conditions, or the finite element formulation, one can use the extended Hamilton’s
principle
h h
(5/ (T - n )dt + / d' Wncdt = 0, (26)
where T is a kinetic energy of the system, n is a total potential energy of the
system, and d'W nc is a virtual work of external non-conservative forces.
Therefore, the extended Hamilton’s principle for the sandwich plate on an elastic
foundation can be written as follows:
ô f [(kinetic energy of plate) — (strain energy of plate) -
h
— (strain energy of elastic foundation) —
— (potential energy of plate in gravity field)]Jt +
t2
+ f (virtual work of damping forces)dt +
ti
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t
V. Y. Perel
2
+ f (virtual work of surface forces)dt = 0 . (27)
In order to derive the differential equations for the unknown functions with the
boundary conditions, or to obtain a finite element formulation, all terms in the
Hamilton’s principle (27) need to be written in terms of the unknown functions
,(1) *(1) *(1) *(1) c (2) c (2) c (2) c (3) c(3) and £ (3)u0 , v0, wo, ■ ' xz * yz> yz> ̂zz ’ ̂xz ’ yz > ̂zz ’ ̂xz ’ ^ yz '■
Kinetic Energy o f the Sandwich Plate. Considering that the mass density of
the face sheets is constant, kinetic energy of the lower face sheet, core and the
upper face sheet can be written as follows:
u(k ) '
r
u(k ) '
v (k ) • ■v (k )
* (k ) v,(k )
dV ( k = 1,2,3), (28)
t
where dots over the letters denote partial derivatives with respect to time. The
displacements in equation (28) are expressed in terms of the unknown functions
by formulas (17)-(25). The kinetic energy of the sandwich plate is the sum of
kinetic energies of the face sheets and the core:
T = T (1) + T (2) + T (3). (29)
Strain Energy o f the Sandwich Plate. The face sheets of a sandwich plate
are made from composite laminates, which are built up of fiber-reinforced plies.
The orientation of the fibers can vary from ply to ply, and, therefore, values of the
stiffness coefficients C f in the Hooke’s law (referred to the laminate coordinate
system) can vary from ply to ply in the face sheets. Let us introduce the following
notation for a stiffness coefficient in the Hooke’s law for a ply of the lower face
sheet, in the laminate coordinate system:
a C f , (30)
where the right superscript (1) denotes that a stiffness coefficient is associated
with the first sublaminate (i.e., the lower face sheet), the left superscript a is a
number of a ply in a lower face sheet, subscripts i and denote a position of the
stiffness coefficient in the stiffness matrix. The stiffness matrix with components
a C f will be denoted as [C0(1)]. So, the strain energy of a lower face sheet’s
ply with a number a is
U r = 2 f f f {£ (1Y [Cf№ "'I d V ,
2 (V«)
(31)
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where V^ is volume of a ply with number a, of the lower face sheet, and the
column-matrix of strains {£(1)} is defined as follows:
(32)
Unlike the material coefficients ac ij \ the strains do not have a subscript a,
which denotes a number of a ply of the lower face sheet, because assumptions
sheet, not for each individual ply of the lower face sheet. Therefore, each strain in
the lower face sheet, as a function of z-coordinate, is represented with a single
expression through the thickness of the lower face sheet.
Similarly, one can write an expression for the strain energy of the upper face
orthotropic medium. But the failure in the core can be distributed non-uniformly
in the thickness direction. As a result of this, in the presence of failure, the
coefficients C j of the stress-strain relation of the core can vary in the thickness
direction. To take account of this, the core is nominally divided into layers
parallel to the x — y-plane, such that within each layer the coefficients of the
stress-strain relation can be considered approximately constant in the thickness
direction. Thus, the core is treated as a laminated plate, the same way as the face
sheets. The strain energy of the sandwich plate is the sum of the strain energies of
the core and the face sheets
Potential Energy o f the Sandwich Plate in the Gravity Field. The potential
energy of the sandwich plate in the gravity field, n g , is equal to the sum of
about through-the-thickness variation of strains are made for the whole lower face
The strain energy U (1) of the whole lower face sheet is a sum of strain
energies of the plies of the lower face sheet:
n
U (1) i 1)- (33)
a=1
sheet, U (3). The core of the sandwich plate is considered to be a homogeneous
U = U (1) + U (2) + U (3). (34)
where
B L zk+1
n gk) = p (k) g f f f w (k) dzdxdy. (36)
0 0 zk
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V. Y. Perel
Strain Energy o f Elastic Foundation. The strain energy of the elastic
foundation, modeled as a Winkler foundation, is defined by the expression
U
, B L
= ~ f f s( x )( w (1) z=z ) dxdy, (37)
0 0
where s(x ) is a modulus of the foundation.
Virtual Work o f Surface Forces. It is assumed that the upper and lower
surfaces of the plate are loaded by distributed forces in the transverse direction
(along z-axis). Let qu (x ,y , t) and qt (x ,y , t) be forces per unit area in the
transverse direction, acting on the plate’s upper and lower surfaces respectively.
Then the virtual work ô' W of these forces is
B L
0 0
B L
ô ' Ws = f f qu(x ,y , t)ôw(3) z=z4 dxdy + f f q i(x ,y , t)ôw(1) z=z1 dxdy ■ (38)
0 0
Virtual Work o f Damping Forces. The damping force per unit volume will
be denoted as O. It will be considered, as it is generally accepted, that the
damping force is proportional to the velocity. Then, for the kth sublaminate, one
can write the following expression for the column-matrix of components of the
damping force per unit volume:
e : s u
o .= _ „ * > p (-> l . (k ) (3
e ( k)
where ( k and p ^ are, respectively, a damping parameter and mass density of
the kth sublaminate. The virtual work of the damping force in the kth sublaminate
can be written as follows:
B L zk +i ôu (k) T 0 (xk y
ô'W<k> = f f f j ô v (k ) ■ ■o (k )y dzdxdy
0 0 zk ôw (k ) o (k ). z v
(40)
The virtual work of the damping forces in the whole sandwich plate is the sum of
the virtual works in the face sheets and the core:
d' Wd = d' w f + 6' wd2) + d' wd3). (41)
The extended Hamilton’s principle, written in terms of the unknown functions,
can be used for deriving either differential equations and boundary conditions for
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the unknown functions, or it can be used for a finite element formulation. In the
following section, a finite element formulation for the sandwich plate in cylindrical
bending will be developed.
Appendix.
Pointwise Equilibrium Equations Variationally Consistent with the von
Karman Strain-Displacement Relations. In the equations of this Appendix, the
upper superscripts (k ), which denote the numbers of the sublaminates, will not be
used, because these equations have a very general character and their validity is
not limited to the layerwise plate theory, presented in this paper.
In order to derive pointwise equations of motion, consistent with the von
Karman strain-displacement relations, let us substitute variations of the von
Karman strain-displacement relations, written with the us of index notations,
eafi= 2 (ua,/J + ufi,a + u3,au3,/J ) (« , £ = 1 2X (A.1)
£ i3 = Ui,3 + U3 ,i ) ( i =1>2,3), (A.2)
into the virtual work principle
/ / / 0 ij^ i jd V = / / / (Fi - Pui ) dut dV + / / ttdut d S ,
(V) (V) (S) (A.3)
where Fi are components of the body force per unit volume, p is density, and
ft are components of the surface traction. The variations of these strains of
equations (A.1) and (A.2) have the form
^e ay3 = 2 (6ua,(5 + 6u/3,a + u3,a6u3,fi + u3,fidu3,a ) (a = 1, 2 £ = 1, 2), (A.4)
<̂e i3 = 1 (&ui,3 + ^u3,i ) ( i = 1, 2, 3). (A.5)
Expression о j i j can be presented in the form
0 i j i j = 0 a}^ a } 2o a 3 ^ a 3 0 33 ̂ 33
(A.6 )
(a = 1,2; y3 = 1,2; i = 1, 2, 3; j = 1, 2,3).
Substitution of Eqs. (A.4) and (A.5) into Eq. (A.6 ) produces the result
0 ij & ij = 0 ij <4- j + 0 a} u 3,adu 3,}
(A.7)
(a = 1,2; } = 1,2; i = 1, 2, 3; j = 1, 2,3).
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V. Y. Perel
If one substitutes expression (A.7) into the left-hand side of equation (A.3), one
receives
/ / / a ij5£ ijdV = / / [aajnj 5ua + (a 3 j nj + a a0u3,an0 ) 5u3 ]dS -
(V) (5)
- / / / aj j 5ua + [a3 j ,j + (a a0u3,a ) ,0 ]5 u3} dV
(V) (A.8)
(a = 1,2; 0 = 1,2; i = 1, 2, 3; j = 1, 2,3),
where «1 , n 2 , and n 3 are components of the outward unit normal vector to the
surface. The substitution of expression (A.8) into the virtual work principle (A.3)
yields
0 = / / / a j &£ijdV - / / / (Fi - Pp i )5uidV - / / ti5uidS =
(V) (V) (S)
= / / [(°a jnj - ta )5 uadS + (a 3 j nj + a a0u3,an0 - t3 )5u3 ]dS -
(S)
- / / / {(a aj,j Fa Pu'a )5ua [a3 j,j (a a0u3,a ) ,0 F3 Pu3 ]5u3 }dV
(V)
(a = 1,2; 0 = 1,2; j = 1,2,3). (A.9)
If one equates to zero the coefficients of variations of displacements, one obtains
the equations of motion
a aj,j Fa = Pu'a; a 3 j,j (a a0 u3,a ) ,0 F3 = pu3 (a 10 )
(a = 1,2; 0 = 1,2; j = 1,2,3) '
and natural boundary conditions
a ajn j = ta ; a 3 j nj a a0 u3,an 0 = 13 at S a . )
(a = 1,2; 0 = 1,2; j = 1, 2, 3),
where S a is a part of the surface on which displacement constraints are not
imposed. Equations of motion (A.10) in expanded form are
a , + a xy, y + a xz, z + Fx = Pu, (A-12)
a yx,x a yy,y a yz,z F y pV, (A.13)
d
a zx,x a zy,y a zz,z dx (a xxW,x a yxw, y )
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Nonlinear Dynamic Finite Element Analysis
(A.14)
The boundary conditions (A.11) in expanded form are
(A.15)
(A.16)
+ xy (w ,xn y + w,ynx ) = t (A.17)
In the postprocessing stage of the finite element analysis, the computation of
the transverse stresses is done with the use of the pointwise equations of motion
(A.12), (A.13) and (A.14), variationally consistent with the von Karman strain-
displacement relations (2)-(7).
Р е з ю м е
Із метою розвитку теорії пластин для товстих багатошарових панелей, що
стискувані у поперечному напряму, із зовнішніми шарами у вигляді компо
зитних ламінатних листів запропоновано спрощену схему розподілу попе
речних деформацій по товщині панелі. Припускається, що поперечні дефор
мації є xz, £ yz та £ zz не змінюються по товщині панелі в інтервалі її
зовнішніх листів і серцевини, але можуть описуватися різними функці
ональними залежностями від координати в площинах різних субламінатів
(зовнішні листи і серцевина панелі). Алгоритм, що ураховує розвиток по
шкодження для динамічних задач, використовується в розрахунковій схемі,
що базується на геометрично нелінійному формулюванні, стосовно до ана
лізу руйнування багатошарової пластини від ударного стикання з грунтом.
Модель багатошарової пластини характеризується малою кількістю степе
ней вільності у скінченноелементних розрахунках та широким викорис
танням: для пластин із тонкими або товстими зовнішніми шарами (у порів
нянні з товщиною серцевини), для випадків стисливості або нестисливості
зовнішніх шарів та (або) серцевини в поперечному напряму.
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No. 7, 435-440 (1948).
2. E. I. Grigolyuk, “Equations of three-layered shells with soft filler,” Izv. Tekh.
Nauk, No. 1, 77-84 (1957).
3. Y. Y. Yu, “A new theory of elastic sandwich plates - one-dimensional case,”
J. Appl. Mech., 26, 415-421 (1959).
4. F. J. Plantema, Sandwich Construction, John Wiley, New York (1966).
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5. H. G. Allen, Analysis and Design o f Structural Sandwich Panels, Pergamon
Press, Oxford, UK (1969).
6. G. R. Monforton and I. M. Ibrahim, “Analysis of sandwich plates with
unbalanced cross-ply faces,” Int. J. Mech. Sci, 17, 227-238 (1975).
7. H. H. Kanematsu, Y. Hirano and H. Iyama, “Bending and vibration of
CFRP-faced rectangular sandwich plates,” Compos. Struct., 10, No. 2, 145
163 (1988).
8. A. K. Mukhopadhyay and R. L. Sierakowski, “On sandwich beams with
laminate facings and honeycomb cores subjected to hydrothermal loads. Pt I.
Analysis,” J. Compos. Mater., 24, 382-400 (1990).
9. K. H. Lee, P. B. Xavier and C. H. Chew, “Static response of unsymmetrical
sandwich beams using an improved zig-zag model,” Compos. Eng., 3, No. 3,
235-248 (1993).
10. Y. Frostig and M. Baruch, “Localized load effects in high-order bending of
sandwich panels with flexible core,” J. Eng. Mech., 122, No. 11, 1069-1076
(1996).
11. J. N. Reddy, Mechanics o f Laminated Composite Plates and Shells: Theory
and Analysis, CRC Press, Boca Raton, Florida (2003).
12. V. Y. Perel, Three-Dimensional Dynamic Stress Analysis o f Sandwich
Panels, Ph.D. Dissertation, AFIT/DS/ENY/00-02, Air Force Institute of
Technology (2000), https://research.maxwell.af.mil/papers/ay2001/afit/afit-
ds-eny-00-02.pdf.
13. T. von Karman, “Festigkeitsprobleme in Maschinenbau,” Encycl. Math.
Wissenschaften (1910), Vol. IV/4C, SS. 311-385.
Received 17. 11. 2004
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