ANALYTICAL ANALYSIS OF MULTI-DELAMINATED LAMINATES
UNDER EXTENSION USING A LAYERWISE STRESS MODEL
Navid SAEEDI
PhD student
Université Paris-Est, Laboratoire Navier (Ecole des Ponts ParisTech / IFSTTAR / CNRS)
École des Pont ParisTech, 6 et 8 avenue Blaise Pascal, 77455, Marne-la-Vallée,France
[email protected] *
Karam SAB
Professor
Université Paris-Est, Laboratoire Navier (Ecole des Ponts ParisTech / IFSTTAR / CNRS)
École des Pont ParisTech, 6 et 8 avenue Blaise Pascal, 77455, Marne-la-Vallée,France
[email protected]
Jean-François CARON
Professor
Université Paris-Est, Laboratoire Navier (Ecole des Ponts ParisTech / IFSTTAR / CNRS)
École des Pont ParisTech, 6 et 8 avenue Blaise Pascal, 77455, Marne-la-Vallée,France
[email protected]
Abstract
The objective of this work is to provide an accurate and efficient alternative to threedimensional finite element method (3D-FEM) for the analysis of delaminated laminates with
free-edges. In the present study, the application of a layerwise stress model, called the LS1
model, is extended to delaminated multilayered plates subjected to uniaxial extension. The
analytical LS1 solutions for a general multilayered plate with arbitrary delaminations in its
section are derived. In order to validate the proposed method, ( ) s composite laminates in
delaminated state are investigated and results of the LS1 model are compared to 3D-FE
solutions. The comparisons in terms of interlaminar stress and energy release rate estimations
between the LS1 and 3D-FE models reveal an excellent agreement in non-delaminated and
delaminated states. It is concluded that the proposed model, as an accurate and very efficient
model for the analysis of delaminations in multilayered plates under extension, can be used in
delamination problems using stress based or energy release rate based criteria.
Keywords: Composite, Delamination, Energy release rate, Interlaminar stresses, Laminate,
Layerwise model
1. Introduction
The free-edge problem is one of the most frequently problem in design and analysis of the
laminated composites. The difference in elastic properties of two adjacent layers in composite
materials can produce very high interlaminar stresses in multilayered structures near free
edges. These interlaminar stresses, theoretically singular, could lead to delamination of the
laminate which might cause the global failure of the structure. It has already been shown that
the classical lamination theory (CLT) is unable of predicting interlaminar stress singularities
near free edges or crack tips. Due to the complexity of the problem in the general case, there
is no exact solution yet to this problem except in some simple cases. As a consequence, many
various approximate analytical (e.g. [1-8]) and numerical (e.g. [9-13]) methods have been
used to overcome this lack of CLT in calculating interlaminar stresses in the vicinity of
singularities.
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In general, the analytical solutions can be classified as either equivalent single-layer theories
(ESL) or layerwise theories. In both cases, the 3D elasticity problem is reduced to a 2D
problem. The ESL methods consist in treating the heterogeneous multilayered laminate as a
homogeneous single-layer plate having equivalent effective elasticity. Since the number of
governing equations is independent of the total number of the layers, the ESL methods are
relatively simple and computationally efficient. These methods generally provide acceptable
results for global response of laminates but their results are not accurate enough near the
edges. On the other hand, in layerwise methods, the number of governing equations depends
on the number of the layers. Thus, these methods are generally more sophisticated and require
more computer costs; but they can provide very accurate results. Consequently, they are one
of the best alternatives to 3D models. The reader can refer to [14] for a complete review of
these approaches.
In this work, a layerwise stress model, called the Multiparticle Model of Multilayered
Materials (M4) in previous articles (e.g. [7-9]), is used for the analysis of delaminations in
multilayers under tensile loading. According to the Carrera’s nomenclature proposed in [14],
this model can be described as a LS1 approach which means a Layerwise Stress approach
with first order membrane stress approximations per layer in the thickness direction. Indeed,
the layers of the laminate are considered as Reissner-Mindlin plates linked together by
interfacial stresses in the model. Since the interfacial stresses are considered as generalized
stresses in the model, the continuity of these stresses at the interfaces is automatically
ensured. It should be noted that the LS1 model, inspired from Pagano’s model [4], is a pure
layerwise stress approach where there is no hypothesis on displacement fields; while the other
layerwise models are either displacement approaches or mixed displacement-stress
approaches.
The analytical solutions of the LS1 model for non-delaminated symmetric laminates under
uniaxial extension were obtained by Caron et al. [7-8] and validated by Carreira et al. [9] in
comparison with FEM. This investigation uses the LS1 model to analyse multilayered plates
in delaminated state. The analytical LS1 solution of a general angle-ply laminate with
multiple delaminations is derived analytically. As numerical examples, ( ) s composite
laminates are investigated and the results of the LS1 model are compared with those of a 3DFE model. Excellent agreement between results demonstrates the accuracy of the model
knowing that, contrary to 3D-FE calculations, the proposed model herein is very efficient in
computational calculations.
2. Formulation of the LS1 model
In this section, the formulation of the LS1 model (Layerwise Stress model with first-order
membrane stress approximations per layer) is briefly presented. The formulation of the model
is based on the Hellinger-Reissner functional. In the following formulation, x and y represent
the in-plane directions and z is the thickness coordinate. hi , hi and h i are the bottom, top
and mid-plane z-coordinate of layer i and e i  hi  hi denotes the thickness of layer. Greek
alphabet subscripts correspond to x, y.
2.1
Generalized stresses
The generalized internal stresses are chosen as follows (  ,   x, y):
 In-plane stress, moment and shear resultants of layer i, respectively:
Page 2 of 8
hi
   ( x, y, z ) dz
i
N
( x, y ) 
hi
i
M 
( x, y ) 
hi
 (z  h
i
)   ( x, y, z ) dz
hi
Qi ( x, y ) 
hi
   3 ( x, y, z ) dz
hi

Interlaminar shear and normal stresses at interface i, i + 1:
 i, i 1 ( x, y )    z ( x, y, hi )    z ( x, y, hi 1 )
 i, i 1 ( x, y )   zz ( x, y, hi )   zz ( x, y, hi 1 )
2.2
Generalized displacements and generalized strains
Since the LS1 model is a layerwise stress approach, there is no hypothesis on the form of the
displacement fields and the displacements stem from the model. By introducing the assumed
stress fields into the Hellinger-Reissner functional and integrating with respect to z over the
thickness of each layer, expressions of generalized displacements are deduced as follows:
U i ( x, y ) 
i ( x, y ) 
U zi ( x, y ) 
Generalized strains, associated
generalized displacements:
1
i
( x, y )  U i ,  U i , ;
 
2


1
ei
hi
 U  ( x, y, z ) dz
hi
hi
12


2
e i hi
1
ei
(z  h i )
ei
U  ( x, y, z ) dz
hi
 U z ( x, y, z ) dz
hi
with the generalized stresses, are deduced from the


1 i
i
i
i
  ,   i , ; d 
 ( x, y )     U z ,
2
 ei
e i 1 i 1 

Di,i 1 ( x, y )  U i 1  U i    i 
; D zi,i 1 ( x, y )  U zi 1  U zi
2

2


2.3
i
( x, y ) 

Equilibrium equations
The derivation of the Hellinger-Reissner functional with respect to generalized displacements
leads to 5 equilibrium equations per layer as:
Page 3 of 8
N i
  i , i 1   i 1, i  0
  , 

e i i , i 1
 i
  i 1, i  Qi  0

M  ,  
2

i
i
,
Q
  i 1   i 1, i  0


,

The derivation of the Hellinger-Reissner functional with respect to generalized stresses yields
the constitutive equations of the model (8 constitutive relations per layer and 3 constitutive
relations per interface).


3. Analysis of multi-delaminated multilayered plate
A general (1 , 2 ,...,  n ) multilayered rectangular plate with a length of 2l , a width of 2b and
a thickness of 2h respectively in the x, y and z directions is considered (Fig. 1a). It is
assumed that the middle plane of the plate is located at z  0 . The behavior of all layers is
considered orthotropic. A general multi-delamination state in the section of the plate is
considered in which there can be several interfacial cracks with different lengths in the y
direction as shown in Fig. 1b.
(b)
(a)
Figure 1 : Laminate geometry, imposed displacements and coordinate system (a);
Laminate section with several cracks at different interfaces - subdivision in the section (b)
The plate is subjected to uniform displacements   at the edges x  l while the other edges
are free. It is assumed that the plate is so long in the x direction ( l  b  h ); thus the strain
components are independent of the x-coordinate far from the ends. The generalized
displacements can be written as follows:
U xi ( x, y )  u ix ( y ) 

x
l
;
U iy ( x, y )  u iy ( y )
 ix ( x, y )   xi ( y )
;
;
U zi ( x, y )  u iz ( y )
 iy ( x, y )   yi ( y )
As shown schematically in Fig. 1b, the solving method consists in dividing vertically the
laminate section following the y direction at every crack tip. In this way, the laminate is
divided into some sublaminates (some zones). Then the solution of the problem on every zone
is found. By imposing continuity conditions between adjacent sublaminates and free-edge
conditions at y  b , the global solution of the problem is derived.
Introducing the displacement fields into the strain-displacement relations yields the
generalized strain components for each zone. By using the constitutive and equilibrium
equations, and imposing the delamination conditions (zero interlaminar stress values at
delaminated interfaces), a system of 5n second-order differential equations is extracted for
each zone as follows:
X  M . X
d
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where X is an unknown vector of dimension 5n and M d is a 5n×5n matrix which depends
on the mechanical material properties, orientation and thickness of the layers and also the
position of the cracks. The obtained system of equations is solved by applying the eigenvector
expansion method. Knowing that there may be complex and repeated eigenvalues, the
analytical solution of the system of equations will be generally in the form of exponential,
trigonometric and polynomial functions as follows:
2n


X   ei y Pi ( y ) Sin i y   Qi ( y )Cos  i y 
i 1
where the components of the vectors Pi ( y ) and Qi ( y ) are polynomial functions with constant
coefficients. For q zones, there will be q×5n second-order differential equations and thus
q×10n unknown constants of integration. In order to determine these constants, q×10n limit
conditions are required. There are 10n free edge conditions at the edges y  b . Moreover,
there are 10 stress and displacement continuity conditions per layer between every two
adjacent zones. Therefore, totally 10n  (q  1)  10n  q  10n conditions are obtained. These
conditions yield a system of q×10n linear algebraic equations with q×10n unknown constants
which can be easily solved.
4. Validation by 3D finite element calculations
In order to validate the proposed model, some numerical examples are treated and the results
of the LS1 model are compared to those of a 3D finite element model performed in the
commercial software ABAQUS. For this reason, (10) s and (20) s rectangular composite
laminates with a width of 2b  20mm and a total thickness of et  4e  0.76mm are
considered. It is assumed that the laminate is very long so that there is no variation in the x
direction far from the ends. The laminates are made up of four G947/M18 carbon-epoxy plies
whose mechanical properties are as follows:
E L  97.6 GPa ; ET  E N  8.0 GPa ; G LT  G LN  3.1 GPa ; G TN  2.7 GPa
 LT   LN  0.37 ;  TN  0.5
e  0.19 mm
The laminates are subjected to a uniaxial longitudinal strain  xx  0.001 in the x direction. As
shown in Fig. 3, the delamination consists of four interfacial cracks of length a located
symmetrically at the interfaces  /   . It should be noted that due to the mirror symmetry of
the laminate with respect to the xy-plane, only the half thickness of the laminate is modelled.
Figure 2 : Laminate section with four interfacial cracks at the interfaces θ/-θ
In the 3D-FE modeling, invariance conditions are exploited in order to reduce the size of the
problem and ensure the longitudinal invariance. By making use of the invariance in the x
direction, only one element in this direction is considered with the following invariance
conditions:
u x ( x1 , y, z )  u x ( x0 , y, z )  ( x1  x0 ) ;  xx u y ( x1 , y, z )  u y ( x0 , y, z ) ; u z ( x1 , y, z )  u z ( x0 , y, z )
Page 5 of 8
In order to obtain an appropriate accuracy, the mesh must be greatly refined near the crack tip
(see Fig. 4). The size of the smallest elements near the crack tip is almost 1 μm.
Figure 3 : Typical 3D Finite element model of the laminate in delaminated state
Regarding the LS1 modeling, in order to increase the accuracy of the model near singularities,
each physical layer is subdivided through the thickness into some mathematical layers in the
model. This refinement strategy through the thickness of the laminate, called layerwise mesh
in this study, allows us to increase the accuracy of the model. Instead of a regular layerwise
mesh in which the thicknesses of all mathematical layers are the same, the authors propose an
irregular progressive mesh in which the thicknesses of mathematical layers are reduced by
approaching physical interfaces. In this study, we consider three mathematical layers per
physical layer and three models are compared: 3D-FE model, LS1 model (with regular
layerwise mesh) and refined LS1 model (with proposed irregular layerwise mesh).
For investigated laminates, the interlaminar shear stress  xy is always dominant at the
interfaces  /   . Therefore, the comparisons are made on the distribution of this interlaminar
shear stress near singularities. Fig. 5 shows the distributions of the interlaminar shear stress in
non-delaminated state (free edge singularity) where y indicates the distance from the
singularity point (i.e. free edge), e is the carbon-epoxy ply thickness and k xz   xz ( E x  xx )
denotes the normalized interlaminar stress ( E x is the longitudinal modulus of the laminate) .
Fig. 6 plots the same distribution in delaminated state (crack tip singularity) for interlaminar
cracks of length a  e . In this case, y is the distance from the crack tip (i.e. singularity
point). As expected, in both cases far from the singularity point ( y  0.02e ), the three models
give the same results. By nearing the stress singularity point ( y  0.02e ), it is realized that the
3D finite element model is more accurate than the LS1 model (with regular layerwise mesh)
while the refined LS1 model (with the proposed irregular layerwise mesh) is more accurate
than the 3D-FE model. Indeed, via the suggested refined LS1 model, the interlaminar stress
singularity is much better captured compared with the LS1 model and even the 3D-FE model.
2
Fig. 7 plots the normalized incremental energy release rate A(a)  G inc (e E x  xx
) [15] versus
the normalized crack length a / e for the considered laminates. As shown, for the small crack
lengths ( a  e ), the estimations of the LS1 model with regular layerwise mesh is not
accurate enough. The smaller the crack length is, the more significant the error becomes for
this model. As a result, contrary to the 3D theory, the curve of the incremental energy release
rate corresponding to the LS1 model doesn’t pass through the origin. To obtain accurate
results via this model, the regular layerwise mesh should be extremely refined. This
refinement greatly reduces the efficiency of the model. In the refined LS1 model with the
irregular layerwise mesh strategy proposed in this study, this drawback is easily overcomed.
Page 6 of 8
As shown in Fig. 7, the refined LS1 model is as accurate as the 3D-FE model even for too
small crack lengths; knowing that the proposed model is enormously more efficient than the
3D-FEM.
Figure 4 : Distribution of the interlaminar shear stress σxz at the interface θ/-θ – Free edge singularity
Figure 5 : Distribution of the interlaminar shear stress σxz at the interface θ/-θ – Crack tip singularity
Figure 6 : The normalized incremental energy release rate A(a) versus the normalized crack length a / e
5. Conclusion
In the present work, the LS1 model was extended to the analysis of multilayered plates
subjected to uniaxial extension in multi-delaminated state. The proposed method allows us to
model general multilayered long plates under uniaxial extension with any multi-delamination
configuration in plate section plane. As application examples, ( ) s composite laminates
were investigated and the results of the 3D-FE model (performed in ABAQUS), LS1 model
(with regular layerwise mesh) and refined LS1 model (with irregular progressive layerwise
mesh proposed in this study) were compared. The interlaminar stress comparisons between
the three models in non-delaminated state (free edge singularity) and delaminated state (crack
Page 7 of 8
tip singularity) demonstrate the accuracy and the efficiency of the refined LS1 model so that
the capture of stress singularities by the proposed refined LS1 model was even better than the
3D-FEM. Regarding the energy release rate, it is found that the refined LS1 model (with the
proposed irregular layerwise mesh) estimates the energy release rate as accurately as the 3DFEM while using a regular layerwise mesh leads to significant errors. It is important to keep
in mind that in this study, the total number of degrees of freedom (unknowns) in the refined
LS1 model is less than 1/200 of the total number degrees of freedom in the 3D-FE model.
This illustrates that the proposed model is an accurate and very efficient alternative to the 3DFEM for delamination analyses under tensile loading.
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