L. BEK, R. ZEM^ÍK: MODEL OF PROGRESSIVE FAILURE FOR COMPOSITE MATERIALS ...
319–322
MODEL OF PROGRESSIVE FAILURE FOR COMPOSITE
MATERIALS USING THE 3D PUCK FAILURE CRITERION
MODEL POSTOPNEGA POPU[^ANJA KOMPOZITNEGA
MATERIALA Z UPORABO PUCKOVEGA TRIDIMENZIONALNEGA
KRITERIJA PORU[ITVE
Luká{ Bek, Robert Zem~ík
University of West Bohemia, European Centre of Excellence, Faculty of Applied Sciences, NTIS – New Technologies for the Information
Society, Univerzitní 22, 306 14 Pilsen, Czech Republic
lukasbek@kme.zcu.cz
Prejem rokopisa – received: 2014-09-16; sprejem za objavo – accepted for publication: 2015-05-07
doi:10.17222/mit.2014.233
A model for the progressive failure of composite materials that considers the materials’ non-linearity was developed and
implemented with the Abaqus FE software. An extended Puck failure criterion for the 3D stress state was used for the failure
prediction. Furthermore, a simplified approach for the simulation of the delamination was considered. For the progressive
failure simulation, the stiffness matrix degradation was used and the degradation parameters were a function of the fracture
angle. The model was tested on problems of a pin-loaded composite plate and of a composite tube subjected to compressive
loading perpendicular to the tube axis.
Keywords: progressive failure, composite, Puck criterion, finite-element analysis
Razvit je bil model postopnega popu{~anja kompozitnega materiala z upo{tevanjem nelinearnosti materiala, ki je bil uporabljen
v Abaqus FE programski opremi. Raz{irjeni Puckov kriterij poru{itve za tridimenzionalno napetostno stanje je bil uporabljen za
napoved poru{itve. Poleg tega je bil uporabljen tudi poenostavljen pribli`ek za simulacijo delaminacije. Za simulacijo napredo-
vanja popu{~anja je bila uporabljena degradacija togosti matrice. Degradacijski parametri pa so bili funkcija kota poru{itve.
Model je bil preizku{en na problemu obremenjevanja kompozitne plo{~e s konico in kompozitne cevi, izpostavljene tla~ni
obremenitvi pravokotno na os cevi.
Klju~ne besede: postopno popu{~anje, kompozit, Puckov kriterij, analiza kon~nih elementov
1 INTRODUCTION
Composite materials are frequently used in the aero-
space, automotive and marine industries, where extrem-
ely strong components and structures are necessary. Due
to the complex loading, finite-element (FE) analyses are
frequently used for the investigation of the stress state
and the failure of structures.1 Commercial FE software
systems are usually able to predict only the first failure,
which can occur at 20 % of the total strength of com-
posite structures. Some new releases of FE systems are
able to perform progressive failure analyses. However,
the analyses are often not sufficiently precise or have
problems with numerical stability. Therefore, new mo-
dels of progressive failure are developed and imple-
mented into the FE systems using a user-defined material
subroutine.2
The development, implementation and testing of the
progressive failure model for the 3D stress state based on
the Puck failure criterion and considering the material’s
non-linearity in the Abaqus FE software using the
UMAT material subroutine was the aim of this investi-
gation.
2 NON-LINEAR MATERIAL BEHAVIOUR
For the simulation of the non-linear material beha-
viour of composite materials, a non-linear function with
a constant asymptote was used for the calculation of the
shear modulus G12 and G13:3
G
G
G
n n
12 12
12
0
12
0
12
12
0
1
1
12
1( )


=
+
⋅⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
+
12
(1)
G
G
G
n n
13 13
12
0
12
0
13
12
0
1
1
12
1( )


=
+
⋅⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
+
12
(2)
where G120 is the initial shear modulus, 12 and 13 are
the shear strains, 120 is the asymptote value of the shear
stress and n12 is the shape parameter.
3 FAILURE CRITERION
The failure criterion determines the occurrence of
failure and indicates the failure’s propagation. The Puck
Materiali in tehnologije / Materials and technology 50 (2016) 3, 319–322 319
UDK 621.785.7:620.168:620.17 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 50(3)319(2016)
criterion for the 3D stress state, described in 2,4, was
selected for this model because it provides the fracture
angle fr, later used for the stiffness degradation. Further-
more, the influence of the fibre parallel-stress extension
and the influence of the non-fracture plane extension
were used with this criterion.4
4 PROGRESSIVE FAILURE IN THE CASE OF
INTER-FIBRE FAILURE
The stiffness-matrix degradation method was used to
simulate the progressive failure. In order to simplify the
determination of the degradation parameters, the stiff-
ness matrix C, in UMAT, called DDSDDE, was trans-
formed from the material coordinate system (1, 2, 3) to
the crack coordinate system (x, y, z) described in Fig-
ure 1.
The transformation of the C matrix in the (1, 2, 3)
system to the C’ in the (x, y, z) system was carried out
using the Equation (3):
C T C T' ( )  fr = ⋅ ⋅
−1 (3)
where
T
c s sc
s c sc
c s
s c
sc sc
 =
−
−
−
1 0 0 0 0 0
0 0 0 2
0 0 0 2
0 0 0 0
0 0 0 0
0 0
2 2
2 2
0 2 2c s−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
(4)
is the transformation matrix for the stress vector and
T
−1 is the inverted transformation matrix
T
c s sc
s c sc
c s
s c
sc sc
 =
−
−
−
1 0 0 0 0 0
0 0 0
0 0 0
0 0 0 0
0 0 0 0
0 2 2 0
2 2
2 2
0 2 2c s−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
(5)
for the strain vector. In Equations (4) and (5), c repre-
sents cos fr and s represents sin fr.
The non-zero components of the C’ matrix Cij’ are
multiplied by (1 – dij) terms. The degradation parameters
dij ∈ 01, are constant values and differ for tensile and
compressive failure.
Afterwards, the C’ matrix is transformed back from
the (x, y, z) system to the (1, 2, 3) system using the trans-
formation matrices:
C d T C d T' ' ( ) ' ( , )ij fr ij= ⋅ ⋅
−
 
1 (6)
5 PROGRESSIVE FAILURE IN THE CASE OF
FIBRE FAILURE
The transformation of the C matrix is not necessary.
Therefore, the non-zero components of the C matrix Cij
are only multiplied by (1 – dij) terms, as in the case of
inter-fibre failure.
6 DELAMINATION
During the testing it was observed that delamination
must be considered because after the initial fibre or
inter-fibre failure, the crack often propagates in the form
of a delamination. Therefore, an approach for the simula-
tion of the delamination was also implemented.
A thin isotropic layer of brittle matrix was inserted
between each of the orthotropic layers in the FE model.
For the prediction of the matrix failure, the maximum
stress criterion, originally used for orthotropic materials,
was considered because it provides information about
which stress component suffered failure. The normal
stress components were compared to the compressive
and tensile strengths of the matrix, while the shear com-
ponents were compared to the shear strength of the
matrix.
In the case of the failure, the non-zero components of
the C matrix Cij are again multiplied by (1 – dij) terms as
in the case of inter-fibre failure.
320 Materiali in tehnologije / Materials and technology 50 (2016) 3, 319–322
L. BEK, R. ZEM^ÍK: MODEL OF PROGRESSIVE FAILURE FOR COMPOSITE MATERIALS ...
Figure 1: Description of the material coordinate system (1, 2, 3) and
the crack coordinate system (x, y, z)
Slika 1: Opis koordinatnega sistema materiala (1, 2, 3) in koordinat-
nega sistema razpoke (x, y, z)
Figure 2: Geometric properties of the pin-loaded plate
Slika 2: Geometrijske lastnosti s konico obremenjene plo{~e
7 CASE STUDY 1 – PIN JOINT
First, in order to test the model, the failure simula-
tions of pin-loaded carbon composite plates were
compared with the experiments. Two types of specimens
with different failure modes (shear-out and net-tension5)
were selected for the failure simulation. The geometric
properties of the specimens are described in Figure 2,
where the 0° layup orientation is parallel to the y axis
and the pin diameter D = 8 mm.
The failure simulation for the first type of specimens
with the shear-out failure mode, a composite layup
[0°|45°|–45°|90°] s, ratios E/D = 1 and W/D = 3, and a
thickness t = 2.32 mm, is illustrated in Figure 3. The
black colour indicates the elements with a degraded
stiffness matrix and represents the failure of the material.
All the layers representing the isotropic matrix were also
degraded. The error for the ultimate load F was 6.8 %
(compared to the average value from the experiments).
The error for the ultimate load F investigated using
the failure simulation of the second type of specimens
with a net-tension failure mode, a composite layup
[90°|45°|–45°|0°] s, ratios E/D = 4 and W/D = 2, and a
thickness t = 2.32 mm was 10.9 % (compared to the ave-
rage value from the experiments as well).
L. BEK, R. ZEM^ÍK: MODEL OF PROGRESSIVE FAILURE FOR COMPOSITE MATERIALS ...
Materiali in tehnologije / Materials and technology 50 (2016) 3, 319–322 321
Figure 4: Load-displacement diagrams of the experiment and the
numerical simulation: a) first failure investigated using experiment, b)
first failure investigated using numerical simulation and c) loss of
numerical stability
Slika 4: Diagram obremenitev-raztezek eksperimenta in numeri~ne
simulacije: a) prva preiskovana poru{itev pri eksperimentu, b) prva
preiskovana poru{itev pri numeri~ni simulaciji in c) izguba numeri~ne
stabilnosti
Figure 3: Numerical simulation of the final shape of failure in the
case of specimens with a shear-out failure mode; different layers
displayed
Slika 3: Numeri~na simulacija kon~ne oblike poru{itve v primeru
vzorca s poru{itvijo z izstri`enjem; prikazane so razli~ne plasti
Figure 5: Comparison of the position and shape of the first failure
investigated using the experiment and the numerical simulation
Slika 5: Primerjava polo`aja in oblike prve poru{itve pri preizkusu in
pri numeri~ni simulaciji
8 CASE STUDY 2 – COMPOSITE TUBE
In addition, the testing was carried out on a thin-
walled composite tube subjected to compressive loading
perpendicular to the tube’s axis. The tube consisted of
carbon fibres with a composite layup [45°|–45°], a
wall-thickness of 1 mm and an outer diameter of 42 mm.
The length of the tested tube was 200 mm.
A stiffness comparison of the experiment and the
numerical simulation is illustrated in Figure 4. A com-
parison of the position and the shape of the first failure
investigated using the experiment (Figure 4a) and the
numerical simulation (Figure 4b) is illustrated in Figure
5. Unfortunately, the numerical model was not able to
simulate the whole specimen failure due to a loss of
numerical stability. In Figure 6, the failure just before
the loss of numerical stability in both layers is illu-
strated. The error of the simulation at this point (Figure
4c) is 13.8 %.
9 CONCLUSION
Our model of progressive failure using the extended
Puck failure criterion for the 3D stress state and con-
sidering the simplified approach for the simulation of
delamination and the material’s non-linearity showed
very good agreement between the numerical simulation
and the experiments. The error for all the simulations
was below 14 %. In future work, the problem of nume-
rical stability will be further investigated.
Acknowledgements
This work was supported by the European Regional
Development Fund (ERDF), project "NTIS – New Tech-
nologies for Information Society", European Centre of
Excellence, CZ.1.05/1.1.00/02.0090, by the research
project GACR_P101/11/0288 and by the grant project
SGS-2013-036.
10 REFERENCES
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L. BEK, R. ZEM^ÍK: MODEL OF PROGRESSIVE FAILURE FOR COMPOSITE MATERIALS ...
322 Materiali in tehnologije / Materials and technology 50 (2016) 3, 319–322
Figure 6: Numerical simulation of the shape and the position of
failure just before the loss of numerical stability
Slika 6: Numeri~na simulacija oblike in polo`aja po{kodbe tik pred
izgubo numeri~ne stabilnosti