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Stress concentration analysis of thick-walled laminate
composites with a loaded circular cut-out by using a
first-order shear deformation theory
W. Hufenbach, R. Gottwald, B. Grüber, M. Lepper, B. Zhou
To cite this version:
W. Hufenbach, R. Gottwald, B. Grüber, M. Lepper, B. Zhou. Stress concentration analysis
of thick-walled laminate composites with a loaded circular cut-out by using a first-order shear
deformation theory. Composites Science and Technology, Elsevier, 2009, 68 (10-11), pp.2238. .
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Accepted Manuscript
Stress concentration analysis of thick-walled laminate composites with a loaded
circular cut-out by using a first-order shear deformation theory
W. Hufenbach, R. Gottwald, B. Grüber, M. Lepper, B. Zhou
PII:
DOI:
Reference:
10.1016/j.compscitech.2008.04.005
CSTE 4030
To appear in:
Composites Science and Technology
Received Date:
Revised Date:
Accepted Date:
13 December 2007
2 April 2008
5 April 2008
S0266-3538(08)00128-0
Please cite this article as: Hufenbach, W., Gottwald, R., Grüber, B., Lepper, M., Zhou, B., Stress concentration
analysis of thick-walled laminate composites with a loaded circular cut-out by using a first-order shear deformation
theory, Composites Science and Technology (2008), doi: 10.1016/j.compscitech.2008.04.005
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers
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ACCEPTED MANUSCRIPT
Stress concentration analysis of thick-walled
laminate composites with a loaded circular
cut-out by using a first-order shear
deformation theory
W. Hufenbach, R. Gottwald, B. Grüber ∗, M. Lepper, B. Zhou
Institut für Leichtbau und Kunststofftechnik (ILK), TU Dresden, 01062 Dresden,
Germany
Abstract
Stress concentrations in the vicinity of cut-outs can often be regarded as design
drivers for the whole structure. Especially in thick-walled laminate composites not
only the extension-bending coupling effects but as well the influence of the sheardeformation has to be taken into account. For analyzing stress concentration effects
in the vicinity of circular cut-outs in thick-walled anisotropic composites, sophisticated semi-analytical solution methods based on a first-order shear-deformation
theory have been developed at the Institut für Leichtbau und Kunststofftechnik.
The comparison of the strains in the vicinity of cut-outs in textile-reinforced composites obtained by the new semi-analytical methods with experimentally measured
and numerically determined strain values show a good agreement.
Key words: Textile composites, Stress concentrations, Notch, Shear deformation
theory
Preprint submitted to Elsevier Science
11 April 2008
ACCEPTED MANUSCRIPT
1
Introduction
Nowadays multilayered composites are used more and more often even for loadcarrying and safety-relevant structures in all kind of applications for aviation
and space technology, for vehicles, for mechanical engineering as well as for the
sporting and leisure goods industry. While up to now the reinforcing structure
mostly is composed of uni- or bidirectional fibre-reinforced layers, currently
textile semi-finished products in form of multi-axial knitted, woven or braided
preforms are getting more and more into the focus of research and application.
In the past composites used in applications could mostly be considered as
rather thin, but now more and more often thick-walled laminates are used. In
the case of such thick composites besides the extension-bending coupling effects, as known from thin-walled laminates, the influence of the shear-deformations
has to be taken into consideration. So it is of great importance to provide
adapted calculation methods for critical areas like cut-outs to utilize the
large lightweight design potential of thick-walled multilayered composites. The
method presented in this study for the linear stress/strain analysis of thickwalled laminate composites with a circular cut-out can be regarded as one
module of such a design method and enables the user to compare the influence or different composite designs on the stress/strain distribution.
In the literature (for instance in [1–5]), analytical solutions for the analysis
of the stress concentration fields of fibre-reinforced single- and multilayered
composite plates with cut-outs or inclusions can be found for various cases of
geometry and loads. These solutions are mainly based on the classical laminate
∗ Corresponding author. Tel.: +49-351-463-38146; Fax: +49-351-463-38143
Email address: [email protected] (B. Grüber).
2
ACCEPTED MANUSCRIPT
theory and the fundamental works of Lekhnitskii [6]. Solutions for thickwalled multilayered composites based on a higher order laminate theory, for
instance a first-order shear deformation theory, are not known to the authors.
The precise analysis of the stress and strain fields in the vicinity of cut-outs is
often of great importance for the design of thick-walled composites in multimaterial design, since stress concentrations can often be regarded as design
drivers for the whole structure. At the Institut für Leichtbau und Kunststofftechnik (ILK) sophisticated analytical solutions for basic problems concerning thin-walled notched single- and multilayered composites have already
been developed during recent years [7–11]. Based on the experience gathered
during these developments, it was reasoned that for a realistic stress concentration analysis of thick-walled generally structured multilayered composites
(MLC), it is important to provide methods which enable a layer-by-layer precalculation of the entire stress and distortion field. Hereby it is necessary to
pursue superordinate approaches in the expanded stress-deformation analysis,
which not only take into consideration the extension-bending coupling effects
occurring in non-symmetrical composites but which also account for the influence of the shear deformation and thus have less restrictions with regard to
the lamina materials, the composite lay-up and the thickness.
2
Analytical calculation methods for multilayered composites using a first-order shear deformation theory
2.1 Generalized plate equation for thick-walled laminates
3
ACCEPTED MANUSCRIPT
Starting point for the derivation of a generalized plate equation for thickwalled laminates is the deformation law for the single lamina. In this study,
Hooke’s law for orthotropic materials
σij = Cijkl εkl
(1)
is used to describe the stress-strain-relationship, with the stress and strain
tensors σij and εkl and the stiffness tensor Cijkl (i, j, k, l = 1 . . . 3). The use
of this deformation law implicates, that the single lamina shall be deemed to
be made up of a homogeneous orthotropic material by using homogenization
techniques, while in reality the single lamina is inhomogeneous as it is composed of fibres and matrixmaterial. Since in this study neither temperature
nor moisture effects are taken into consideration, the terms describing these
effects have been omitted in (1).
For the incorporation of the shear deformation effects, the following kinematic
assumption, which are well known from the Mindlin-Reissner plate theory,
are made 1
u(x, y, z) = u0 (x, y) + zψx (x, y),
v(x, y, z) = v0 (x, y) + zψy (x, y),
w(x, y, z) = w0 (x, y).
(2)
The displacement functions u0 , v0 and w0 describe the displacements of the
midplane of the plate and ψx and ψy the rotations of cross-sections, that
were normal to the midplane before deformation. This formulation of the
displacement functions was first given by Reisner [12,13] and Mindlin [14]
1
x, y, z: global coordinates with x-,y-axis as in-plane axis and z-axis as out of plane
axis
4
ACCEPTED MANUSCRIPT
in their plate theories. Some examples for theories of even higher order –
including not only the shear deformation effects but also, e. g., warping of
crossections – can be found in [15].
With the kinematic assumption (2) the strain-displacement relationships for
small displacements can be written in the form
⎡
⎡
⎤
⎤
⎡
⎤
⎡
⎤
⎡
⎤
∂ψx
∂u0
0
⎢
⎥
⎢ εx ⎥
⎢
⎥
⎢ εx ⎥
⎢ κx ⎥
∂x
∂x
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
∂ψy
⎢
⎥
∂v
0
0
⎢ε ⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
ε
κ
⎢ y⎥
⎢
⎥
⎢
⎥
⎢
⎥
y
y
∂y
⎢
⎥
∂y
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎥
∂ψ
∂ψ
∂u
∂v
y
⎢γ ⎥ = ⎢
⎢
⎥
⎢
⎥
⎢
⎥
0
x
0
0
⎢
⎥+z⎢
=:
+
z
+
γ
κ
+
⎢ xy ⎥
⎥
⎢
⎥
⎢
xy ⎥ .
xy
⎢ ∂y
⎥
∂x
∂y
∂x
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢ ∂w
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
0
0
⎢
⎥
γ
0
⎢γxz ⎥
⎢
⎥
⎢
⎥
⎢
⎥
+
ψ
0
x⎥
xz ⎥
⎢ ∂x
⎢
⎥
⎢
⎥
⎢
⎢
⎥
⎢
⎥
⎣
⎦
⎣
⎦
⎣
⎦
⎣
⎦
⎣ ∂w
⎦
0
0 +ψ
γyz
γ
0
0
y
yz
∂y
(3)
Based on the strain-displacement-relations (3) and Hooke’s deformation law
(1), finally the structural law for thick-walled multilayered laminates according
to the first-order shear deformation theory is derived
⎡
⎤
⎢ Nx ⎥
⎢
⎥
⎢
⎥
⎢N ⎥
⎢ y ⎥
⎢
⎥
⎢
⎥
⎢N ⎥
⎢ xy ⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢ Mx ⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢ My ⎥
⎢
⎥
⎣
⎦
⎡
=
⎢A11 A12 A16 B11 B12
⎢
⎢
⎢
A22 A26
B22
⎢
⎢
⎢
⎢
A66 symm.
⎢
⎢
⎢
⎢
D11 D12
⎢
⎢
⎢
⎢
⎢ symm.
D22
⎢
⎣
Mxy
⎡
Qy
B16 ⎥
⎡
=
⎤
2
⎢−k2 A55
⎢
⎣
symm.
−k1 k2 A45 ⎥
−k12 A44
⎥
⎦
ε0x
⎤
⎢
⎥
⎢
⎥
⎢
⎥
⎢ ε0 ⎥
⎢ y⎥
⎢
⎥
⎢
⎥
⎢γ 0 ⎥
⎢ xy ⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢ κx ⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢ κy ⎥
⎢
⎥
⎣
⎦
⎥
⎥
B26 ⎥
⎥
⎥
⎥
B66 ⎥
⎥
⎥
⎥
D16 ⎥
⎥
⎥
⎥
⎥
D26 ⎥
⎥
⎦
D66
⎤
⎢Qx ⎥
⎢ ⎥
⎣ ⎦
⎡
⎤
,
(4)
κxy
⎤
⎡
0
⎢−γxz ⎥
⎢
⎣
0
−γyz
⎥
⎦
,
with the shear correction factors 2 k1 , k2 , the stress resultants and moment
2
For details on the shear correction factors see e. g. [16]
5
ACCEPTED MANUSCRIPT
resultants
⎡
⎤
⎢ Nx ⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢ Ny ⎥
⎢
⎥
⎢
⎥
⎣
⎦
Nxy
⎡
⎤
⎢ σx ⎥
⎥
h/2 ⎢
⎢ ⎥
⎢ ⎥
⎢ σy ⎥ dz
=
⎢ ⎥
⎥
−h/2 ⎢
⎣ ⎦
τxy
⎡
⎡
⎤
⎢
⎢
⎢
⎢
,⎢
⎢
⎢
⎣
Mx ⎥
Mxy
Qy
⎡
=
,
τxy
(5)
⎤
h/2
⎢ τxz ⎥
⎢
⎥
⎢
⎥ dz
⎣
⎦
−h/2
⎤
⎢ σx ⎥
⎥
h/2 ⎢
⎢ ⎥
⎢ ⎥
⎢ σy ⎥ z dz
=
⎢ ⎥
⎥
−h/2 ⎢
⎣ ⎦
⎥
⎥
⎥
My ⎥
⎥
⎥
⎦
⎤
⎢Qx ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦
⎡
,
τyz
the extensional, extension-bending coupling and bending stiffnesses, as known
from the classical laminate theory
⎡
⎡
⎤
⎢ Aij ⎥
⎢
⎥
⎢
⎥
⎢B ⎥
⎢ ij ⎥
⎢
⎥
⎣
⎦
h
2 =
Dij
− h2
Ci3
Cij −
C3j
C33
⎤
⎢1⎥
⎢ ⎥
⎢ ⎥
⎢ z ⎥ dz
⎢ ⎥
⎢ ⎥
⎣ ⎦
(i, j = 1, 2, 6) ,
(6)
z2
and the additional shear stiffnesses due to the incorporation of the shear
deformation effects
h
2
Aij =
Cij dz
(i, j = 4, 5) .
(7)
− h2
In multilayered textile- or fibre-reinforced composites, a separation of the
plate-bending and membrane problems often is not possible due to coupling of
force resultants and curvatures as well as coupling of moment resultants and
strains occurring in non-symmetrical composites. Therefore, the MindlinReisner plate theory is expanded in such a way, that a unified approach for
the solution of both, the plate bending and the membrane problem is found.
Here the formulations of the equilibrium of force and moment resultants at the
differential plate element is supplemented by the membrane force resultants
6
ACCEPTED MANUSCRIPT
and a so called generalized plate equation is derived. This system of coupled
partial differential equations (PDES) can be written down by means of a differential operator matrix in compact and clear form as a matrix equation
analogously to generalized plate equation for thin-walled laminate composites
found in [8]
⎡
⎡
⎤
⎢ u0 ⎥
⎢
⎥
⎢
⎥
⎢
⎥
0⎥
⎢ v0 ⎥
⎥
⎢
⎥
⎥ T⎢
⎥
⎥
⎢
0 ⎥ Δ ⎢ w0 ⎥
⎥
⎥
⎢
⎥
⎢
⎥
⎦
⎢
⎥
K
⎢ ψx ⎥
⎢
⎥
⎣
⎦
⎤
⎢A
⎢
⎢
Δ⎢
⎢B
⎢
⎣
B
D
0 0
= −P
(8)
ψy
with
⎡
Δ=
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
∂
∂x
0
∂
∂y
0 0 0
0
0
∂ ∂
∂y ∂x
0 0 0
0
0 0 0 0 0 0
0 0 0
∂
∂x
0 0 0 0
0
∂
∂y
∂ ∂
∂y ∂x
0 ⎥
⎥
⎥
0 ⎥
⎥
∂
− ∂x
∂
− ∂y
−1
0
0
−1
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
(9)
A, B, D: matrices of extensional, extension-bending coupling and bending stiffnesses, q. v. (6),
K: matrix of shear correction factors and shear stiffness, q. v. (7),
P : vector of external loads.
2.2 Boundary conditions and solution method
Due to the structure of the PDES (8) it is not possible to transform it into
a single partial differential equation in one function, where all derivations
7
ACCEPTED MANUSCRIPT
of the function are of the same order. So the solution method of complex
valued displacement functions, as used for the solution of the generalized plate
equation in context of the classical laminate theory (for example in [10]), is
not applicable here.
In this study, an infinite plate with a finite circular cut-out is selected as a
mathematical equivalent model for the notched plate and the following restrictions as for the loading conditions are introduced. The notched plate should
be loaded at the infinite outer boundary by constant edge forces per unit
length Nx , Ny , Nxy (Fig. 1(a)) and constant edge moments per unit length
Mx , My , Mxy (Fig. 1(b)) in such a way, that the static equilibrium conditions
are fulfilled automatically when looking on the overall plate. At the edge of
the notch, the plate can be loaded by a constant normal edge force per unit
length Ni and a constant edge bending moment per unit length Mi (Fig. 1(c)).
(a) force resultants
(b) moment resultants
(c) force and moment resultant
Fig. 1. Boundary conditions on cut-out and outer boundary
In order to take into consideration the effect of the loads on the outer boundary
on the stress concentration problem of an infinite plate, the actual state of
stress is decomposed using the superposition principle into three states of
stress (Fig. 2) as follows
I:
a finite, unnotched plate with loads at the outer edge,
II: an infinite notched plate with loads at the edge of the notch, adapted
in such a way that, with superposition of I and II, an overall unloaded
8
ACCEPTED MANUSCRIPT
Fig. 2. Decomposition of the coupled membrane-plate problem by means of superposition, here shown for in-plane loading.
notch edge results,
III: an infinite notched plate with the corresponding loads at the notch edge.
Looking at this decomposition from a different point of view, the solution for
the stress concentration problem of a notched plate with a loaded cut-out
is derived by superpositioning of the solutions for the problem of a notched
plate with an unloaded hole and a loaded outer boundary (subproblem I and
II) and the problem of an notched plate with a loaded cut-out and an unloaded
outer boundary (subproblem III). Of course, if dealing with the problem of an
unloaded cut-out, subproblem III can be omitted.
Subproblem I, the finite plate loaded on the outer boundary by constant edge
forces and moments, can be solved elementarily and the solutions provide the
dynamic boundary conditions for subproblem II, i. e. the force and moment
resultants that have to be applied to the boundary of the notch in subproblem II so, that by superposition of I and II an overall unloaded notch edge
results. Since the method of complex-valued displacement functions is not applicable here, the Ritz-method is called upon for the solution of each of the
subproblems II and III, respectively. By superposing the solutions of all three
9
ACCEPTED MANUSCRIPT
subproblems finally the solution for the stress concentration problem of an
infinite plate is derived. To apply the Ritz-method, all equations are transformed into a polar coordinate system with its origin in the centre of the notch
first. Afterwards the PDES is rewritten into a so-called weak formulation.
In the case of the chosen mathematical model of an infinite plate, the shell
stress resultants and moments resultants have to disappear for r → ∞. Due to
this, the displacements u0 , v0 , w0 and the shear deformations ψx and ψy have
to disappear as well for r → ∞. For the Ritz method, the ansatz functions
have to be chosen in such a way, that they fulfill the kinematic boundary
conditions, i. e. they have to tend to zero for r → ∞. Since on the edge of the
cut-out only dynamic boundary conditions, i. e. force and moment resultants,
are applied, there are no restrictions for the ansatz functions on the notch
edge. The ansatz functions are chosen in the form of a series in such a way,
that in ϕ-direction a 2π-periodical Fourier series is used while in r-direction a
power series is applied
u(r, ϕ) =
M1
c
Um0
m=1
M1
N m
a
r
+
a
r
c
(Umn
a
r
c
s
(Vmn
cos(nϕ) + Vmn
sin(nϕ)) ,
m
cos(nϕ) +
n=1 m=1
m
M2
a
c
Vm0
+
v(r, ϕ) =
r
m=1
M2 m
N n=1 m=1
M3
c
Wm0
w(r, ϕ) =
m=1
sin(nϕ)) ,
m
a
r
M3 m
N a
n=1 m=1
M4
Ψcm0
ψr (r, ϕ) =
m=1
s
Umn
r
+
c
s
(Wmn
cos(nϕ) + Wmn
sin(nϕ)) ,
m
a
r
+
10
(10)
ACCEPTED MANUSCRIPT
M4 m
N a
r
n=1 m=1
M5
a m
c
Φm0
ψϕ (r, ϕ) =
r
m=1
m
M5
N n=1 m=1
a
r
(Ψcmn
cos(nϕ) +
Ψsmn
cos(nϕ) +
Φsmn
sin(nϕ)) ,
+
(Φcmn
sin(nϕ)) .
where a is the diameter of the circular notch. By application of the Ritz
method the looked-after coefficients for the displacement functions can be
computed. Hereby the summation limits are normally chosen in the following
form
M1 = M2 = M3 , M4 = M5 = M1 + 1, N = free.
(11)
The values for M1 and N should be chosen with regard to the given problem.
In this study M1 = 16 and N = 20 was sufficient for all problems. The (semi)analytical calculation method developed here has finally been implemented
into a calculation tool and can now be used on standard office PCs.
3
Experimental and numerical verification
For the verification of the developed semi-analytical solution method for thickwalled multilayered composites, extensive experimental and numerical finite
element (FE) investigations have been carried out. Within the scope of the
Collaborative Research Centre SFB 639, these studies have been conducted
on multilayered composites built from bidirectionally reinforced knitted layers
made from hybrid yarn containing glass fibres and polypropylene matrix fibres
(GF/PP). The material parameters (12) of the single lamina were determined
11
ACCEPTED MANUSCRIPT
experimentally and by the means of X-FEM [17,18]
E11 = 15.4 GPa, E22 = 15.4 GPa, E33 = 4.6 GPa,
G12 = 1.6 GPa, G13 = 0.9 GPa, G23 = 0.9 GPa,
ν12 = 0.134,
ν13 = 0.417,
(12)
ν23 = 0.417.
3.1 Comparison to experimental results
In Fig. 3 the experimental set-up for the stress concentration analysis is shown.
In this test a textile reinforced GF/PP plate with an un-loaded cut-out is
loaded in a unidirectional tension test. For the determination of the displacement and strain fields the grey-scale correlation method is used as a contactless
optical 3D-field measurement method. From the experimental investigations
the εx and εy strains along the 0◦ , 90◦ and the ±45◦ radians (Fig. 3(b)) are
determined and are compared to semi-analytically calculated results.
(a) experimental setup
(b) directions
Fig. 3. Experimental set-up and directions of result evaluation
In Fig. 4, some of the results for the calculated and experimentally determined
strain decay behaviours along the 0◦ , 90◦ and the ±45◦ radians are shown.
From the figures, an overall good correlation of the semi-analytical and experimental results can be observed. Never the less, the correlation seems to
12
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be not so good at the edge of the notch. This is mainly caused by the used
3D gray-scale measurement equipment used. Due to the limited resolution of
the cameras used, the precipitous gradient of the strains near the edge of the
cut-out can not be detected with adequate accuracy.
(a) 0◦ - and 90◦ -radian
(b) ±45◦ -radian
Fig. 4. Experimentally determined and analytically calculated strain decay behaviour
3.2 Comparison of analytical and numerical results
To reduce the number of expensive and time consuming experimental investigations, an extensive number of comparative finite-element calculations are
performed. Therefore different multilayered GF/PP composites (properties of
the single layer according to (12)) with different lay-ups and different external
loadings are investigated. Representative results are presented here from the
large number of performed verifying calculations. Fig. 6 shows a comparison of
the numerically and analytically obtained strains on the laminate top surface
for a [±4510 ]s GF/PP laminate with a circular cut-out (diameter ∅ = 30 mm)
loaded with a bending moment resultant Mx = 1000 Nm/m according to
Fig. 5.
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Fig. 5. FE model for a notched GF/PP plate with with a practically far-field
plate-bending moment
In Fig. 6 the strains directly on the edge of the notch (r = 15 mm) and the
strains along a concentric circle with radius r = 16 mm are given. From the
subfigures a very good correlation of the analytical and numerical results can
be observed.
(a) notch boundary (r = 15 mm)
(b) concentric circle (r = 16 mm)
Fig. 6. Strains on laminate top surface
Overall, it turns out that in case of all examined combinations of composite
and load, the numerically determined results and the results calculated by
use of the developed analytical methods show a high degree of agreement.
Thus, the developed fundamentals for calculations of stress concentrations at
notches in thick-walled multilayered composites represent a quick, dependable
and easily manageable alternative to a time-consuming FE analysis.
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4
Parameter studies on thick-walled multilayered composite plates
with cut-outs
4.1 Comparison with classical laminate theory
In the first parameter study presented here the effect of a first-order shear
deformation theory (SDT1) in comparison to the classical laminate theory
(CLT) is studied. As exemplary results, the strain distribution on the laminate top surface at the edge of the notch for a [±4510 ]s GF/PP multilayered
composite plate with a circular cut-out (diameter ∅ = 30 mm) is shown. The
plate was loaded under unidirectional tension (membrane load, Fig. 7(a)) and
under plate-bending load (Fig. 7(b))
(b) bending load Mx = 100 Nm/m
(a) membrane load Nx = 10 N/m
Fig. 7. Strains on laminate top surface
In the case of pure membrane load and the symmetric multilayered composite,
no differences between the results using the CLT and the SDT1 can be observed, as expected (see Fig. 7(a)). Naturally, the effects of the shear deformation (Fig. 7(b)) are only relevant in cases where shear force initiated bending
effects occur, either caused by the material (non-symmetrical laminates) or
caused by the loads(bending load). In these cases a great difference between
the results calculated using the CLT or SDT1, respectively, can be observed.
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This phenomena is further studied in the next parameter study concentrating
on the influence of the laminate thickness.
4.2 Influence of the laminate thickness
In a second parameter study, the influence of the laminate thickness on the
strain distribution on the surface of the composite is examined. The platebending moment Mx was chosen according to
Mx =
h
h0
2
· 100
Nm
m
(13)
with h as laminate thickness and h0 = 1 mm as reference thickness. The investigated parameter combinations are shown in Table 1, the material parameter
are chosen according to (12). With this choice of Mx in the case of classical
laminate theory the strain fields on the laminate top and bottom respectively
are independent of the thicknesses and therefore should be comparable. Some
of the results of this parameter study are presented in Fig. 8.
Table 1
Laminates and loading parameters for studying the influence of the laminate thickness
thickness h
plate bending moment
[mm]
Mx [Nm/m]
[±452 ]s
1
100
[±453 ]s
1.5
225
[±454 ]s
2
400
[±458 ]s
4
1 600
[±4516 ]s
8
6 400
[±4532 ]s
16
25 600
[±4564 ]s
32
102 400
[±45128 ]s
64
409 600
[±45256 ]s
128
1 638 400
lay-up
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(a) radial strain
(b) radial/tangential shear strain
Fig. 8. Influence of laminate thickness
As expected, the strain fields on the laminate top and bottom surface are independent of the laminate thickness in case of classical laminate theory and
the chosen bending load. In the case of the SDT1, the situation is different.
In this case a great influence of the thickness on the strain distribution is
observed. Hereby the thickness not only quantitatively influences the curves,
especially the extreme values, but also can change the qualitative form of the
curves (see Fig. 8). This change in the curve characteristic can be seen exceptionally well for the shear-stress curves in Fig. 8(b). Here the characteristic
clearly changes from the curves for the laminate thickness of 1 mm over the
curve for the thickness 2 mm to the final curve characteristic for a thickness of
8 mm. As a further result of the investigation, it is also found that even for a
laminate thickness of 1 mm the influence of the shear deformation can clearly
be observed in comparison to the CLT (q. v. Fig. 8(b)). Simular results were
observed for other lay-ups and different materials.
4.3 Effects of FE mesh refinements
One of the greatest advantages of the developed semi-analytical calculation
method compared with the FE method is, that no discretization – as for exam17
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ple the creation of a FE mesh – is needed. The developed methods are mesh
independent. In the scope of this study, this advantage is proven by a vast
number of comparative FE calculations with different mesh refinements. The
FE calculation were carried out using the programm system I-Deas Master Series 11 with linear and quadratic 4-node and 8-node shell elements,
respectively. Some selected results of this study are presented in Fig. 9 for a
[±4510 ]s GF/PP multilayered composite plate with a circular cut-out (diameter ∅ = 30 mm) under a plate-bending load Mx = 1000 Nm/m.
Fig. 9. Shear distortion on laminate top surface for different FE mesh refinements
(linear 4-node shell elements)
It is obvious that there is a great influence of the number of elements at the
edge of the notch, i. e. the mesh density (q. v. Fig. 9). Generally the parameter
studies on the influence of the FE mesh density lead to the conclusion, that in
all cases there is a great influence on the stress and strain fields in the vicinity
of the notch. In the case presented here, more than 360 elements on the notch
edge are needed to at least qualitatively follow the curve of the analytically
calculated strains, and even 1440 elements are not enough to quantitatively
reach the analytical results. To overcome this disadvantage of the FE analysis, it is planned to combine the developed semi-analytical mesh-independent
calculation methods (for the local stress concentration analysis) with the FE
method (for the global structural analysis) by the means of analytical super
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elements.
5
Conclusion
During recent years the application of thick-walled textile- and fibre-reinforced
multilayered composites has expanded considerably. For the application of
these materials, local stress concentrations, as they occur in the vicinity of
notches, can often be regarded as design drivers for the whole structure. In this
study, the mechanically equivalent model of an infinite plate with a circular
cut-out, which is subject to combined membrane and plate loads, is dealt
with by means of a expanded plate theory based on the Mindlin-Reisner
plate theory and semi-analytical methods for a layer-by-layer stress/strain
analysis of thick-walled multilayered composites. The presented methods not
only calculate stresses, strains and displacements directly at the edge of the
notch, but also their distribution throughout the entire area of the plate.
For the verification of the developed calculation method experimental and a
vast number of numerical FE analyses were performed with symmetrical and
non-symmetrical composite structures. The comparison of the experimental
and numerical results, respectively, with the results obtained by means of the
developed solutions show a good agreement and thus indorse the developed
theory in an impressive way.
The performed parameter studies demonstrate that very complex mechanisms
are acting in thick-walled multilayered anisotropic composite plates in the
areas of notches. In the case of pure membrane loads and symmetric laminates,
the shear deformation effects don’t have to be taken into account. In this
case, the more easy-to-handle classical laminate theory should be used for the
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analysis of stress concentration effects in multilayered composites, for example
according to [8,5].
From the studies carried out it must be concluded, that for dimensioning
thick-walled multilayered composites in line with material and component
characteristics, a precise layer-by-layer analysis of the distribution of stresses
and distortions is absolutely indispensable and especially in the case of the
occurrence of bending effects the influence of shear deformation has also to be
taken into account. Starting from this layer-by-layer approach, the development of new failure criteria for notched multilayered laminates is currently one
of the focal points of ongoing research work at the ILK. These failure criteria
take into consideration so called physically based failure criteria (e. g. according to Puck [19] or Cuntze [20]) and additionally consider the so-called
microsupport effect according to Neuber [21].
Acknowledgements
We would like to express our gratitude towards the Deutsche Forschungsgemeinschaft (DFG), who funds the subproject B2 within the scope of the Collaborative Research Centre SFB 639 ”Textile-Reinforced Composite Components in Function-Integrating Multi-Material Design for Complex Lightweight
Applications”.
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