RECENT ADVANCES IN THE ANALYSIS OF LAMINATED COMPOSITE
STRUCTURES
D.N. ALPHA & C.C. BETA
Institute of Laminated Structures,
Department of Space Engineering, Delta University, 54124 City, Country
E

mail:
ccb@
civil.auth.gr
D.C. GAMMA
Laboratory of Structures,
Department of Mechanical Engineering, Epsilon University, 55523 City, Country
E

mail:
dgamma@mecheng.eps.is
1
SUMMARY
Certain recent advances in the
analysis of laminated composite structures are presented in the
present paper. To study the failure phenomena, two different simulations are considered: The
first simulation assumes that in the structure there are
a priori
known critical surfaces whose
beh
avior is described by means nonmonotone, possibly multivalued laws including all the
nonlinearities, while the rest part of the structure behaves linear

elastically. These surfaces are
called interfaces. The second simulation assumes that the analysed stru
ctures obey to
nonmonotone, possibly multivalued stress

strain or reaction

displacement laws that describe
their structural behavior in a macroscopic way. As a matter of fact the type of the problem
defines the criteria for the suitable simulation. Nonmono
tone constitutive or boundary laws
are the result of nonconvex

nonsmooth energy potentials. They lead to a variational
expression for the principle of virtual work called hemivariational or "Panagiotopoulos"
inequalities and are equivalent to a substationa
rity problem of the potential or the
complementary energy of the structure.
2
INTRODUCTION
The study of the mechanical response of composite laminates has the last two decades
attracted the interest of many researchers. These effects can be described by
diagrams
obtained from experimental tests (Fig. 1b,c) [2

6].
(a) (b)
(c)
Figure 1: a) Stress

strain law for reinforced concrete in tension b) Reaction

displacement diagram in bending for brittle
polycarbonate composites reinforced with glass fibers [5] c) Stress

strain curve for composite with carbon after pull

out
experi
ments [6]
The description of the failure phenomena of composite laminates is possible by means of two
different simulations. The first simulation concentrates all the nonlinearities of the structure to
a priori
defined surfaces between the elements of th
e structure. Through these surfaces
(interfaces) significant tension and shear stresses are transferred leading to debonding
phenomena along the interfaces. The behavior of these interfaces is described by
nonmonotone, possible multivalued stress

strain or
reaction

displacement laws while the rest
parts of the structure remain in the elastic regime [7

10]. A nonmonotone law is the result of
nonconvex and nonsmooth energy superpotentials [11] and their study has been achieved by
means of the generalized grad
ient of Clarke

Rockafellar [12,13] that allowed to P.D.
Panagiotopoulos to generalize the variational inequalities and to formulate the theory of
hemivariational or “Panagiotopoulos” inequalities [14,15]. Hemivariational inequalities lead
to substationarit
y problems of the potential or the complementary energy of structure.
The numerical treatment of these problems in this way is possible applying the optimization
program NSOLIB based on the proximal bundle method [16]. The main advantage of this
method i
s that it can be applied for a large number of unknowns and for three

dimensional
problems combining short rate of convergence with absence of numerical instability.
3
FORMULATION OF THE PROBLEM
In
this section we will formulate the discretized problem f
or both the previously mentioned
simulations. A laminated composite structure
Ω
is first considered, which consists of n parts
Ω
n
that present linear elastic behavior and the corresponding interfaces
Γ
between each couple
of
Ω
n
in a global orthogonal Cartesian coordinate system O
x
1
x
2
x
3
of
3
. The formulation is
similar to this o
ne that is presented in [17]. Denoting with
Γ
i
and
Γ
i

1
(i = 1,n), the discretized
interfaces between the boundaries of two parts of the structure (Fig.2), with
P
n
the load vector
applied acting only
on the nodes of each substructure
Ω
n
and not on the nodes of the interfaces
and with
P
i
,
P
i
1
the developed unknown interfacial forces acting on the nodes of
Γ
i
and
Γ
i

1
respectively, the following total load vectors
are formed:
1
n
n
n
1
i
i
i
i
1
1
1

P
P
P
P
P
P
P
P
P
P
i = 2, n

1
(1)
Figure 2: Discretized structure with interfaces
We define as X the set of kinematically admissible displacements
u
for
Ω
n
and the relative
displacement at each interface as:
[u
j,i
] = u
j,i

u
j,i
(2)
where j = 1, m (m is the number of nodes at each interface), u
j,i
, u
j,i
are the 1, m node
displacements for the corresponding fronts of each interface and i = 1, n

1 is the number of
interfaces. The behavior of the interfaces is governed by
a three

dimensional nonmonotone
reaction

displacement 1aw that describes the concentrated nonlinearities of the structure
incorporating failure effects and is described by means of a relation (inclusion) between the
vectors of the interfacial forces and t
he relative displacements of the opposite nodes of the
fronts of the interface:
P
j
j
( [u
j,i
] )
(3)
The function
3
j
:
is continuous and locally Lipschitz.
Following the same procedure as in [17] we conclude to the f
ollowing problem:
Find
u
X such that:
X
)
(
])
[u
]
[u
],
([u
)
(
...
)
(
...
)
(
)
(
...
)
(
...
)
(
n
1
=
i
*
T
i
1

n
1
=
i
i
j,
*
i
j,
i
j,
k
1
j
j
n
*
n
T
n
1
o
T
o
i
*
i
T
i
1
o
T
o
1
*
1
T
1
1
o
T
o
n
*
n
n
T
n
i
*
i
i
T
i
1
*
1
1
T
1
i
i
n
n
i
i
1
1
*
u
u
u
P
u
u
G
F
e
u
u
G
F
e
u
u
G
F
e
u
u
K
u
u
u
K
u
u
u
K
u
(4)
where
*
'
n
*
n
*
n
*
'
i
*
i
*
i
*
i
*
1
*
1
*
1
u
u
u
u
u
u
u
u
u
u
i = 2, n

1 (5)
etc….
4
NUMERICAL APPLICATIONS
The structure depicted in Fig. 3 is first analyzed. A three

dime
nsional nonconvex

multivalued
superpotential law (depicted in plane projection in Fig. 4a,b) governs the behavior of the
interface and the respective superpotential is of the form:
)
,
(
j
T
N
u
u
S
(6)
where
S
are the develope
d forces of the interface and
u
N
,
u
T
are the relative normal and
tangential displacements of the fronts of the interface respectively with:
u
u
u
u
u
N
x
T
x
x
3
1
2
,
2
2
(7)
The problem was solved for simultaneous application of normal and ta
ngential loading to the
structure and for five different load cases by applying the optimization program NSOLIB. The
nonlinear structural response of the composite structure under investigation is depicted in
Fig.4. The exact response of the laminated stru
cture according to the laboratory testing
performed at the ISSTUT is very similar to the response of the numerical models under
consideration. Similar tests have been carried out having as scope to investigate the structural
response in the space where com
pletely different boundary conditions hold and gravity loads
are absent.
(a) (b)
(c)
Figure 3: a) The analysed laminat
ed composite structure and the corresponding loading
b) Composite laminated structures in axial loading and the respective analysis model
c) Composite laminated structures in bending and the respective analysis model
(a) (b) (c)
Figure 4: a, b) Nonmonotone reaction

displacement law for the interface of the structure of Fig. 3a
c) Stress

strain constitutive laws
for the materials of the composite laminated structures of Fig. 3b,c.
5 REFERENCES
[1]
Scanlon A.,
Time

dependent Deflections of Reinforced Concrete Slabs.
Ph. D. Thesis,
University of Alberta at Edmonton, Alberta (1971).
[2]
Schlimmer M., Zur Berechnung der Fe
stigkeit von Werkstoffverbunden mit polymer
Zwischenschicht. In:
Verbundwerkstoffe und Werkstoffverbunde
(eds.
Leonhardt G. and
Ondracek G.)
DGM Informationsgesellschaft, Oberursel, (1993).
[3]
Panagiotopoulos, P.D., Inequality Problems in Mechanics and Applic
ations. Birkhaeuser,
Boston (1985).
[4]
Cheon S.S., Lee D.G. and Jeong K.S., Composite side

door impact beams for passenger
cars.
Composite Structures
,
38
, (1997), 229

239.
[5]
Hampe A., Messung der Scherfestigkeit in der Grenzflae
che zwischen Polymer and Glas
dur
ch Einzelfaserexperimente. In:
Haftung bei Verbundwerkstoffen und
Werkstoffverbunden
(eds. Brockman W.), DGM Informationsgesellschaft, Oberursel,
(1989).
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