Abstract:
Abstract: Two different finite element models have been applied to the analysis of delami-nation growth in a multidirectional graphite/epoxy ENF specimen, which has previously been investigated experimentally. For a more accurate computation of energy release rates along the delamination front, and particularly the individual mode contributions, a layered 3D shell finite element has been used. A 2D finite element, based on Reissner-Mindlin plate theory, has been employed for simulation of the delamination growth. This element incorporates a process layer in which the delamination can grow. The virtual crack closure method is employed in both models to compute the energy release rates along the delamination front. It is postulated that these energy release rates control the development of the shape of the delamination front, as well as the final unstable growth of the delamination, as observed in the experiment. By a comparison between simulation and experiment it is found that in the present case of pure shear mode (combination of Modes II and III) the Griffith criterion predicts correctly the global delamination growth.
S. Weihe, S. Rinderknecht, R. Krüger,
and B. Kröplin.
Simulation of fracture and delamination propagation in fibre reinforced
materials.
Zeitschrift
für Angewandte Mathematik und Mechanik,
vol. 78, pp. S73-S76, 1998.
Abstract:
M. König, R. Krüger.
Delamination Growth under Cyclic Loading.
ECCM-8, European Conference on Composite Materials, Volume 4, ISBN
1 85573 410, Woodhead Publishing
Ltd., 1998, pp. 479-486.
Abstract: Cyclic loading has been applied to CFRP (Carbon Fiber Reinforced Plastic) laminates, containing artificial delaminations, as well as ply cuts at various interfaces. For the experimentally detected delamination contours local energy release rates have been computed. Plots of measured delamination progression per load cycle versus computed energy release rates have been included in a Paris Law diagram, as obtained experimentally using simple specimens for material characterization. Results obtained for delamination growth in a 0/0 interface suggest that growth prediction, based on Paris Law, is possible. Additional results for specimens where delamination growth occurs in a 0/45 interface indicate that Paris Law parameters depend on the orientation of the adjacent plies. It appears that - at least in the case of the shear mode - the slope of the Paris law is much lower than in the case of a 0/0 interface. This implies, that the possibility of a delamination growth approach should be considered in the design of CFRP structures.
M. König, R. Krüger, K. Kussmaul, M. v. Alberti, and M.
Gädke .
Characterizing Static and Fatigue Interlaminar Fracture Behaviour of
a First Generation Graphite/Epoxy Composite.
Composite Materials: Testing and Design - (13th Vol.), ASTM
STP 1242, S.J. Hooper, Ed., American Society for Testing and Materials,
1997, pp. 60-81.
Abstract: The characterization of interlaminar fracture -- with the goal to obtain a data base that can be used in design -- is demonstrated for a first generation graphite/epoxy composite widely used by European aircraft manufacturers. Critical energy release rates for mode I and mode II failure have been obtained from static tests using double cantilever beam, end notched flexure and transverse crack tension specimens. An interaction criterion for the mixed mode case is formulated, based on the results from mixed mode bending tests. Fatigue tests have been carried out to determine Paris Law parameters for pure mode I, pure mode II and mixed mode conditions as well as threshold energy release rates which could be used as design limit in a no-growth concept. In accordance with the static case an interaction criterion is formulated for the crack growth rate under mixed mode conditions. Delamination progression in more complex specimens has been measured and mixed mode energy release rates have been computed along the delamination fronts. Results lie well within the scatter band of Paris Law as obtained by the specimens employed for characterization. This confirms that the data obtained from the characterization of interlaminar fatigue growth can be applied for predictions in design.
R. Krüger and M. König.
Prediction of Delamination Growth Under Cyclic Loading.
Composite Materials: Fatigue and Fracture - (Sixth Volume)., ASTM
STP 1285, E.A. Armanios, Ed., American Society for Testing and Materials,
1997, pp. 162-178.
Abstract: The growth of delaminations in CFRE (Carbon Fibre Reinforced Epoxy) specimens during R=0.1 and R=-1 fatigue loading has been studied. Artificial circular and square delaminations as well as ply cuts have been introduced at various interfaces during manufacturing to simulate a pre-damaged structure and to cause delamination growth. Criteria based on fracture mechanics will be used to describe the delamination failure. Predicting delamination growth with this approach requires the distribution of the local energy release rate along the delamination front. For obtaining this energy release rate distribution the virtual crack closure method was found to be most favourable for three-dimensional finite element analysis as the separation of the total energy release rate into the contributing modes is inherent to the method and only one complete finite element analysis is necessary. Plots of measured delamination progression per load cycle (da/dN-values) versus computed energy release rates have been included in a Paris Law diagram as obtained experimentally using DCB specimens to characterize mode I and ENF and TCT specimens to characterize mode II failure, respectively. Computed mixed-mode results lie well within the scatter band of the experimentally determined Paris Law for mode I and mode II failure.
M. König and R. Krüger
Delamination Growth in CFRP-Laminates Under Fatigue Loading.
International Conference on Fatigue of Composites, S. Degallaix, C.
Bathias and R. Fougeres, eds., Societe Francaise de Metallurgie et de Materiaux
, pp. 299-306, 1997.
Abstract: Specimens from CFRP (Carbon Fibre Reinforced Plastic) laminates, containing artificial delaminations as well as ply cuts at various interfaces, were subjected to fatigue loading. For the experimentally detected delamination contours local energy release rates were computed, based on results obtained from three-dimensional finite element analysis. Plots of measured delamination progression per load cycle versus computed energy release rates have been included in a Paris Law diagram as obtained experimentally using simple specimens for material characterization. Results obtained for delamination growth in a 0/0 interface suggest that growth prediction based on Paris Law is an approach to be considered. Additional results for specimens where delamination growth occurs in a 0/45 interface indicate that Paris Law parameters depend on the orientations of the adjacent plies. A suitable growth law for this interface could be determined.
R. Krüger, M. König and B. Kröplin
Delamination Growth in CFRP-Laminates: Development of a Predictive
Tool Based on Computational Fracture Mechanics.
AIAA-97-1130-CP. In The 38th
AIAA/ASME/ASCE/AHS/ASC
SSDM Conference, Kissimmee, FL, pp. 2064--2072, 1997.
Abstract: Specimens from CFRP (Carbon Fibre Reinforced Plastic) laminates, containing artificial delaminations as well as ply cuts at various interfaces, were subjected to tension-tension (R=0.1) and tension-compression (R=-1) fatigue loading and delamination growth was monitored. Local energy release rates along experimentally detected delamination contours were computed, based on results obtained from three dimensional finite element analysis. Plots of measured delamination progression per load cycle versus computed energy release rates have been included in a Paris Law diagram as obtained experimentally using simple specimens for material characterization. Results obtained for delamination growth in a 0/0 interface suggest that growth prediction based on Paris Law is an approach to be considered. However, other results for specimens where delamination growth occurs between plies of dissimilar orientations indicate that Paris Law parameters determined for a unidirectional interface are not valid in this case. A suitable growth law for a 0/45 interface could be determined. Growth in this case is considerably slower as compared to the unidirectional case. This is essential when considering a damage tolerance approach for design and operation of CFRP structures.
M. König, R. Krüger and H. Parisch
A Finite Element Approach for Predicting Delamination Growth
In Proceedings of NAFEMS World Congress '97 - Stuttgart, pp. 136-147,
NAFEMS
, 1997.
Abstract: For more brittle composites such as carbon fibre reinforced
epoxy laminates it is shown that criteria based on elastic fracture mechanics
are an appropriate tool for predicting delamination growth under quasi-static
and fatigue loading. The criteria are based on the distribution of the
local energy release rate, which is obtained using the virtual crack closure
method, implemented in a finite element analysis. For quasi-static loading
the concept has been verified by analyzing delamination fronts of the well
known unidirectionally reinforced double cantilever beam specimen. The
experimentally observed delaminations show contours for which the energy
release rate is constant along the front. For cyclic loading experiments
have been performed utilizing a specimen that contains a ply cut and an
adjacent deliberate delamination. Plots of measured delamination progression
per load cycle versus computed energy release rate have been included in
a Paris Law diagram as obtained experimentally using simple specimens for
material characterization. The results lie well within the scatter band
of the experimentally determined Paris Law.
M. König, R. Krüger, E. Kohler, M. Kurz and T. Ruckstuhl
Analytical and
Numerical Analysis of a Specimen Containing a Delamination Caused by a
Ply Cut.
ISD-Report No. 97/2, Institute for Statics und Dynamics of Aerospace
Structures, University of Stuttgart, 1997 (Revision B, 1999). - (1.1
MB)
Abstract: The growth of delaminations in CFRE (Carbon Fibre
Reinforced
Epoxy) specimens during R=0.1 fatigue loading has been studied. Two cuts
through the top and bottom layer across the entire width of the specimen
have been introduced during manufacturing to simulate a component with
a ply drop and to cause delamination growth. Criteria based on fracture
mechanics are used to describe the delamination failure. Predicting
delamination
growth with this approach requires the distribution of the local energy
release rate along the delamination front. For obtaining this energy release
rate distribution the virtual crack closure method was found to be most
favourable for three-dimensional finite element analysis, as the separation
of the total energy release rate into the contributing modes is inherent
to the method and only one complete finite element analysis is necessary.
Using the grid reflection method delamination growth was observed and the
size and shape of the affected area determined, yielding the delamination
contours. Plots of measured delamination progression per load cycle versus
computed energy release rates have been included in a Paris Law diagram,
as obtained experimentally using DCB specimens to characterize mode I and
ENF and TCT specimens to characterize mode II failure. Computed mixed-mode
results lie well within the scatter band of the experimentally determined
Paris Law for mode I and mode II failure. This is in agreement with results
obtained previously from another type of specimen. These results suggest
that growth prediction based on Paris Law is an approach to be considered
for further investigations.
R. Krüger, S. Rinderknecht and M. König
Two- and Three- Dimensional Finite Element
Analysis of Crack Fronts in a Multidirectional Composite ENF Specimen
ISD-Report No. 97/1, Institute for Statics und Dynamics of Aerospace
Structures, University of Stuttgart, 1997. - (444 K)
Abstract:Results are presented from a numerical investigation of the effect of the shape of the delamination front on energy release rate distribution in a laminated carbon fibre reinforced epoxy end-notched flexure test specimen. The deformation behaviour of the specimen has been computed by three-dimensional nonlinear finite element analyses using a recently developed layered 3D-shell element. The distributions of the local mixed mode energy release rates along the initial straight and one measured curved delamination fronts have been determined by the virtual crack closure method. The dependency of computed total energy release rate as well as the individual mode contributions from the shape of the delamination front is discussed. By means of 2D models for which a delamination process element has been used, a curved front was iteratively determined along which the total energy release rate is constant. The shape of this front is in good agreement with experimentally observed front shapes.
R. Krüger and M. König
Investigation of Delamination Growth Between
Plies of Dissimilar Orientations.
ISD-Report No. 96/5, Institute for Statics und Dynamics of Aerospace
Structures, University of Stuttgart, 1996. - (2 MB)
Abstract: The growth of delaminations in CFRE (Carbon Fibre Reinforced Epoxy) specimens during R=0.1 fatigue loading has been studied. Artificial 10 x 10 mm square delaminations and an additional cut through two plies have been introduced during manufacturing to simulate a pre-damaged structure and to cause delamination growth. Criteria based on fracture mechanics are used to describe the delamination failure. Predicting delamination growth with this approach requires the distribution of the local energy release rate along the delamination front. For obtaining this energy release rate distribution the virtual crack closure method was found to be most favourable for three-dimensional finite element analysis as the separation of the total energy release rate into the contributing modes is inherent to the method and only one complete finite element analysis is necessary. Using C-scan delamination growth was observed, yielding the delamination contours. Plots of measured delamination progression per load cycle versus computed energy release rates have been included in a Paris Law diagram obtained experimentally for the unidirectional interface. The results indicate that Paris Law parameters determined for a unidirectional interface are not valid if delamination propagation occurs at interfaces between layers of dissimilar orientations. This is in agreement with results obtained previously from a delamination buckling specimen. Growth at the interface investigated is considerably slower as in the unidirectional case. This is essential when considering a damage tolerance approach for the design and operation of CFRE structures. A suitable growth law for the interface investigated was determined from the computed energy release rates and the corresponding measured delamination progression.
R. Krüger, C. Hänsel and M. König
Experimental-Numerical Investigation of
Delamination Buckling and Growth.
ISD-Report No. 96/3, Institute for Statics und Dynamics of Aerospace
Structures, University of Stuttgart, 1996. - (1.6 MB)
Abstract: The growth of delaminations in CFRE (Carbon Fibre
Reinforced
Epoxy) specimens during R=-1 fatigue loading has been studied. Artificial
circular delaminations have been introduced during manufacturing to simulate
a pre-damaged structure and to cause delamination growth. Criteria based
on fracture mechanics are used to describe the delamination failure. Predicting
delamination growth with this approach requires the distribution of the
local energy release rate along the delamination front. For obtaining this
energy release rate distribution the virtual crack closure method was found
to be most favourable for three-dimensional finite element analysis, as
the separation of the total energy release rate into the contributing modes
is inherent to the method and only one complete finite element analysis
is necessary. Plots of measured delamination progression per load cycle
versus computed energy release rates have been included in a Paris Law
diagram obtained experimentally for the unidirectional interface. The results
indicate that Paris Law parameters determined for a unidirectional interface
are not valid for delamination propagation that occurs at interfaces between
layers of dissimilar orientations. The corresponding Paris Law line seems
to be less steep as for the unidirectional case. This is essential when
considering a damage tolerance approach for the design and operation of
CFRE structures. However, the growth law for the interface considered should
be confirmed by material characterization tests to support the current
results.
B.D. Davidson, R. Krüger and M. König.
Effect of Stacking Sequence on Energy Release Rate Distributions in
Multidirectional DCB and ENF Specimens.
Engineering
Fracture Mechanics, Vol. 55, No. 4, pages 557--569, 1996.
Also appears as Syracuse University Technical Report No. MAME-95-101,
1995.
Abstract: Results are presented from a theoretical investigation of the effect of stacking sequence on energy release rate distributions in laminated composite double cantilever beam and end-notched flexure test specimens. Eight different stacking sequences are investigated; four of these will result in delamination growth at 30/30 interface and four will result in growth at a 30/-30 interface. Each set of four sequences are chosen to exhibit varying amounts of coupling between the primary bending curvature and either the transverse curvature or the twist curvature. For a set number of plies, sequences that minimize one type of coupling will have increased coupling of the other type. The sequences chosen for study span a range of possible choices for practical use. Three dimensional finite element analyses are used to obtain the total energy release rate and its distribution along an initially straight delamination front for the eight sequences under DCB and ENF loadings. As expected, for DCB loading, peak energy release rates occur near the center of the specimen's width, whereas for ENF loading the peak occurs at one or both edges. It is shown that larger bending-twisting coupling results in larger asymmetries in the energy release rate, whereas larger longitudinal-transverse bending coupling results in larger peak values. Practical application to DCB and ENF testing is discussed.
R. Krüger, S. Rinderknecht, C. Hänsel and M. König.
Computational Structural Analysis and Testing:
An Approach to Understand Delamination Growth
Fracture of Composites, E. A. Armanios
Ed., Key Engineering Materials,
Vols. 120-121, pp. 181-202, Trans Tech Publications, 1996.
Abstract: The so-called "computer aided test" (CAT) offers
possibilities
for experimental investigation of the behaviour of materials using complex
structural related specimens. For the determination of energy release rates
at delamination growth laminated Carbon Fibre Reinforced Epoxy (CFRE)
specimens,
containing an artificial delamination embedded near the surface, have been
tested under static compression and under tension-compression (R=-1) fatigue
loading. The out-of-plane (i.e. buckling) deformations of the delaminated
region have been measured via moiré technique. Using numerical
post-processing
techniques, the size and shape of the delaminated sublaminate as well as
the delamination front contour may be obtained from the information stored
in the digitized pictures. The postbuckling state of the specimens in the
compression phase of the test, which is assumed to drive the delamination
growth, is computed by two-dimensional (plate) and three-dimensional (layered
3D shell) finite element models in which the measured delamination front
contour is taken into account. Results for the computed displacements of
specimens with circular delaminations, which have been used as starter
delaminations in the tests, show satisfactory agreement with the measured
deformations. They are, however, rather sensitive to the load and the
delamination
size. The same holds to an even larger extend for the energy release rates
which have been computed along the circular delamination front contours
using the virtual crack closure method. The goal of the investigations
is to compute energy release rates along measured arbitrarily shaped
delamination
front contours and to correlate the results with the measured delamination
growth.
R. Krüger, M. König and M. Gädke.
Prediction of Delamination Growth Under Cyclic Loading Using Fracture
Mechanics.
Progress in Durability Analysis of
Composite Systems, Proceedings of the international conference DURACOSYS
95, Brussels.
ISBN 90-5410-809-6, A.A. Balkema Publishers, pp. 45-52, 1996.
Abstract: Results are presented which show that criteria based on elastic fracture mechanics are an appropriate tool for predicting delamination growth in carbon fibre reinforced epoxy laminates caused by fatigue loading. To check the validity of the approach local energy release rates along experimentally determined delamination fronts have been determined using the virtual crack closure method, implemented in a finite element analysis. Plots of computed energy release rate versus measured delamination progression per load cycle have been included in a Paris Law diagram as obtained experimentally using simple specimens for material characterization. The results lie well within the scatter band of the experimentally determined Paris Law.
R. Krüger and M. König
Two- and Three-Dimensional Finite Element
Analysis of a Specimen Containing a Rectangular Artificial Delamination
and Additional Ply Cut
ISD-Report No. 95/11, Institute for Statics und Dynamics of Aerospace
Structures, University of Stuttgart, 1995. - (3 MB)
Abstract: The growth of delaminations in CFRE (Carbon Fibre Reinforced Epoxy) specimens during R=0.1 fatigue loading has been studied. Artificial 10 x 10 mm square delaminations and an additional cut through one ply have been introduced during manufacturing to simulate a pre-damaged structure and to cause delamination growth. Criteria based on fracture mechanics are used to describe the delamination failure. Predicting delamination growth with this approach requires the distribution of the local energy release rate along the delamination front. For obtaining this energy release rate distribution the virtual crack closure method was found to be most favourable for three-dimensional finite element analysis, as the separation of the total energy release rate into the contributing modes is inherent to the method and only one complete finite element analysis is necessary. Using ultrasonic C-scan, delamination growth was observed and the size and shape of the affected area determined, yielding the delamination contours. Plots of measured delamination progression per load cycle versus computed energy release rates have been included in a Paris Law diagram as obtained experimentally using DCB specimens to characterize mode I and ENF and TCT specimens to characterize mode II failure, respectively. Computed mixed-mode results lie well within the scatter band of the experimentally determined Paris Law for mode I and mode II failure. These results suggest that growth prediction based on Paris Law is an approach to be considered for further investigations.
B.D. Davidson, R. Krüger and M. König.
Three Dimensional Analysis and Resulting Design Recommendations for
Unidirectional and Multidirectional End- Notched Flexure Tests.
Journal of Composite
Materials, Vol. 29, No. 16, pages 2108--2133, 1995.
Also appears as Syracuse University Technical Report No. MAME-94-401,
1994.
Abstract: Results are presented from a theoretical investigation
of the effects of stacking sequence on the energy release rate in laminated
composite end-notched flexure test specimens. Deflections and energy release
rates of unidirectional and multidirectional ENF specimens are obtained
by classical laminated plate theory, shear deformable plate theory, and
three dimensional finite element analyses. It is shown that the distribution
of energy release rate varies across the front of an initially straight
delamination. The percentage of mode II and mode III energy release rates
for the specimen, as well as the local peak values of the mode II, mode
III and total energy release rates that occur at the specimen's free edge
are shown to correlate with a nondimensional ratio comprised of the specimen's
flexural rigidities. The results of the study are used as a basis for a
proposed "ENF test design procedure" that may be used for the determination
of appropritate specimen stacking sequences and test geometries for studying
delamination growth at interfaces between plies at various orientations.
The test design procedure minimizes the contributions to the energy release
rate from residual thermal stresses, geometric nonlinearities, local mode
II concentrations at the specimen's free edges and local mode III effects.
B.D. Davidson, R. Krüger and M. König.
Three Dimensional Analysis of Center Delaminated Unidirectional and
Multidirectional Single Leg Bending Specimens.
Composites
Science and Technology, Vol. 54, No. 4, pages 385--394, 1995.
Also appears as: Technical Report No. MAME-94-901, Department of
Mechanical,
Aerospace and Manufacturing Engineering, Syracuse University, 1994.
Abstract: Results are presented from a theoretical investigation
of the effects of stacking sequence on the energy release rate in laminated
composite single leg bending test specimens. Deflections and energy release
rates of unidirectional and multidirectional SLB specimens are obtained
by classical laminated plate theory based methods and by three dimensional
finite element analyses. It is shown that the distribution of energy release
rate varies across the front of an initially straight delamination and,
regardless of stacking sequence, all three energy release rate components
will occur. The percentage of mode III energy release rate is generally
small. Its relative magnitude, as well as the differences between the maximum
values of all the energy release rate components, are shown to correlate
with a nonbdimensional ratio comprised of the specimen's flexural rigidities.
The classical plate theory based methods are shown to accurately predict
both total energy release rate and the average mode ratio, i.e., with respect
to the specimen's full width, for all stacking sequences evaluated.
R. Krüger, M. König and M. Gädke.
Predicting Delamination Growth under Cyclic Loading: An Approach Using
Computational Structural Analysis and Testing.
In Proccedings of the The Tenth International Conference on Composite
Materials,Vol. I,
A. Poursartip and K. Street, editors, Woodhead
Publishing Ltd., ISBN 1-85573-222-1, pages 269-276, 1995.
Abstract: For more brittle composites such as carbon fibre reinforced epoxy laminates it is shown that criteria based on elastic fracture mechanics are an appropriate tool for predicting delamination growth under fatigue loading. The distribution of the local energy release rate along experimentally determined delamination fronts is obtained using the virtual crack closure method, implemented in a finite element analysis. Plots of measured delamination progression per load cycle versus computed energy release rate have been included in a Paris Law diagram as obtained experimentally using simple specimens for material characterization. The results lie well within the scatter band of the experimentally determined Paris Law.
M. König, R. Krüger and S. Rinderknecht.
Numerical Simulation of Delamination Buckling and Growth.
In Proccedings of the The Tenth International Conference on Composite
Materials, Vol. I,
A. Poursartip and K. Street, editors, Woodhead
Publishing Ltd., ISBN 1-85573-222-1, pages 561-568, 1995.
Abstract: Buckling induced growth of delaminations has been tested for tension-compression (R=-1) fatigue loading. Via Moiré-technique the out-of-plane (i.e. buckling) deformation of the delaminated region has been measured. Using numerical post-processing techniques the size and the shape of the delamination can be determined from this measurement. The post-buckling state of the specimens has then been analyzed by two-dimensional (plate) and three-dimensional (layered 3D shell) finite element models. First comparisons between experiment and analysis indicate that a prediction of the growth rate of delaminations might be possible via simulation of the post-buckling behaviour and computation of the energy release rates along the delamination front.
M. König and R. Krüger
Computation of Energy Release Rates: A Tool for Predicting Delamination
Growth in Carbon Fibre Reinforced Epoxy Laminates.
Computational Plasticity. - Fundamentals and Applications - Proceedings
of 4th International Conference, Barcelona, E. Onate, D.R.J Owen and E.
Hinton, editors, ISBN 0-906674-85-9. Pineridge Press, pages 1167--1178,
1995.
Abstract: For more brittle composites such as carbon fibre reinforced
epoxy laminates it is shown that criteria based on elastic fracture mechanics
are an appropriate tool for predicting delamination growth under quasi-static
and fatigue loading. The criteria are based on the distribution of the
local energy release rate, which is obtained using the virtual crack closure
method, implemented in a finite element analysis. For quasi-static loading
the concept has been verified by analyzing delamination fronts of
unidirectionally
reinforced double cantilever beam and end notched flexure specimens. The
experimentally observed delaminations show contours for which the energy
release rate is constant along the front. For cyclic loading experiments
have been performed utilizing a specimen that contains a ply cut and an
adjacent deliberate delamination, a model-configuration for a laminate
after foreign object impact. Plots of measured delamination progression
per load cycle versus computed energy release rate have been included in
a Paris Law diagram as obtained experimentally using simple specimens for
material characterization. The results lie well within the scatter band
of the experimentally determined Paris Law.
M. König und R. Krüger
Bruchmechaniche Modellierung von Faserverbund-Laminaten.
In Space
Course 1995, Lehrgang über Raumtransportsysteme,
D. Dinkler und E. Messerschmid (Herausgeber), ISBN 3-930683-03-2.
Institut für Statik und Dynamik der Luft- und Raumfahrtkonstruktionen,
Universität Stuttgart, Seiten 397-414, 1995.
Abstract: Das Konzept der Bruchmechanik und deren Bedeutung für
die Auslegung und den Betrieb von Luft- und Raumfahrtstrukturen wird
dargestellt.
An einem für zukünftige heiße tragende Strukturen relevanten
Beispiel, der Delamination von Faserverbund-Laminaten, wird versucht, einen
Einblick in den Stand der Technik der bruchmechanischen Modellierung zu
geben.
M. König und R. Krüger.
Bruchmechanik des Delaminationswachstums in zyklisch belasteten
CFK-Laminaten
In 27. Vortragsveranstaltung des DVM-Arbeitskreises Bruchvorgänge,
Köln. Deutscher Verband für Materialforschung und -prüfung
e.V., Seiten 29-38, 1995.
Abtract: Es werden Ergebnisse aus einem Vorhaben vorgestellt,
das zum Ziel hat, die Möglichkeit der Vorhersage des
Delaminationswachstums
mittels elastischer Bruchmechanik zu überprüfen. Dazu bedient
man sich einer experimentell-numerischen Vorgehensweise. Es wird eine komplexe
Flachprobe vorgestellt, die aus kohlenstoffaserverstärktem Epoxidharz
(Prepreg T300/914C) hergestellt wurde. Durch Einbetten einer doppelten
Trennfolie während der Fertigung wurde eine künstliche, quadratische
Delamination eingebracht und ferner die darunterliegende Schicht durchtrennt,
um so eine vorgeschädigte Struktur simulieren zu können bei der
Delaminationswachstum auftritt. Die Probe wurde mit Zugschwellbelastung
(R= 0,1) bei Oberspannungen von 30%, 40% und 50% der Zugfestigkeit des
Laminats belastet. Die Delaminationsfrontverläufe wurden in
regelmäßigen
Abständen mittels Ultraschallprüfung (C-Scan) ermittelt. Zur
Auswertung der Meßergebnisse wurde die auf der Methode der Finiten
Elemente basierende Modifizierte Rißschließungsmethode eingesetzt,
mit deren Hilfe die lokalen Energiefreisetzungsraten entlang der
Delaminationsfront
bestimmt werden können. Die Meßergebnisse für den lokalen
Delaminationsfortschritt pro Lastzyklus (da/dN-Werte) in Abhängigkeit
von der berechneten Energiefreisetzungsrate wurden in ein Paris-Gesetz-Diagramm
aufgenommen, das aus einfachen Tests ermittelt worden war. Die Ergebnisse
lassen den Schluß zu, daß es möglich sein sollte das Wachstum
eingebetteter Delaminationen vorherzusagen, indem man die
Energiefreisetzungsraten
entlang der Delaminationfront berechnet und die erhaltenen Ergebnisse mit
den Werten aus dem Paris-Gesetz vergleicht, das mit Hilfe einfacher Proben
ermittelt wurde.
H. Eggers, R. Krüger and M. König
Test and Calibration of the Special Beam Lattice Model.
ISD-Report No. 94/8, Institute for Statics und Dynamics of Aerospace
Structures, University of Stuttgart, 1994.
Abstract: It is commonly accepted that criteria based on fracture
mechanics are a meaningful tool to describe the failure of a delaminated
structure. Therefore the knowledge of the distribution of the local energy
release rate along the delamination front becomes essential for predicting
delamination growth. Due to the tremendous effort for the numerical modeling
of a damaged laminate by finite elements (especially three dimensional
finite elements) numerical solutions are rare. In order to overcome this
disadvantage the beam lattice model (BLM) was developed. In this model
the sublaminates are approximated by two beam lattices coupled outside
of the damaged zone. The adaption of the BLM to a given specimen configuration
is rather simple, because the number and locations of corner and side nodes
can be chosen arbitrarily. The stiffness matrices are evaluated via the
solution for the buckled beam, such that rather coarse grids can be used.
Therefore a complete specimen with an embedded delamination can be analyzed
with reasonable numerical effort. For statistical investigations, however,
an even smaller fast running program is needed, which produces some parameters
describing the potential endangering of components by local delaminations.
In order to meet this demand research focussed on the development of a
special beam lattice model (SBLM), generated with the components of the
BLM for a minimal lattice structure, which means that in the delaminated
zone the sublaminates are modeled by two beam crosses stacked one over
another and coupled at the delamination front. The legs of the beam crosses
are extended to the undamaged area in order to approximate the stiffness
behaviour of the remaining structure. For the deformed laminate the energy
release rates are evaluated at the delamination front in length and cross
direction. The implemented subroutines for evaluating the energy release
rates are easy to handle and cover different cases like inner and outer
delaminations with arbitrary size and stacking sequence of the laminate.
The actual loading of the undamaged laminate is taken into account as well
as the local unloading due to the local buckling of the sublaminate. It
is demonstrated that using the SBLM different damage configurations can
be evaluated fast, such that tendencies of damage progress and damage tolerance
of the composite structure can be estimated. The lengths of the extended
beams and an enlargement factor for their tensile stiffnesses are the free
parameters of the model, which were calibrated using the results from a
full 3D-analysis of a compression specimen containing an embedded circular
delamination.
J. Albinger, C. Hänsel, M. König, R. Krüger, H. Parisch
und S. Rinderknecht.
Ein kombiniertes experimentelles und numerisches Verfahren zur Bestimmung
der Energiefreisetzungsraten beim Delaminationsfortschritt in CFK.
In:
Numerische und experimentelle Methoden in der Statik und Dynamik,
D. Dinkler und U. Hänle (Herausgeber), ISBN 3-930683-00-8.
Institut für Statik und Dynamik der Luft- und Raumfahrtkonstruktionen,
Universität Stuttgart, Seiten 265-289, 1994.
Abtract: Das computerunterstützte Experiment (computer aided test, CAT) bietet die Mögichkeit, das Verhalten von Materialien in komplexen, bauteilähnlichen Proben experimentell zu untersuchen. Zur Bestimmung der Energiefreisetzungsraten bei Delaminationswachstum wurden CFK-Flachproben mit künstlicher, oberflächennaher Delamination quasi-statisch unter Druck und zyklisch mit Zug-Druck-Wechselbelastung (R=-1) belastet. Die Verformungen des delaminierten Bereichs wurden mit Hilfe der Moiré-Technik gemessen. Mittels numerischer Weiterverarbeitung der digitalisierten Bilder kann aus den gemessenen Verformungen die Größe und die Form des lokal ausbeulenden Sublaminats und somit der Delaminationsfrontverlauf bestimmt werden. Dies wiederum ermöglicht die Analyse der Vorgänge beim Delaminationsfortschritt mit Hilfe der Finite-Elemente (FE)-Methode. Mittels zweidimensionaler Modelle (aus Plattenelementen) und dreidimensionaler Modelle aus 3D-Schalenelementen wurde das Nachbeulverhalten der Probe während der Druckphase des Versuchs - in der das Delaminationswachstum erwartet wird - untersucht. Erste Ergebnisse für die Beulverformung einer Probe mit kreisförmiger Ausgangsdelamination zeigen im Vergleich zum Versuch zufriedenstellende Ergebnisse. Lokale Energiefreisetzungsraten entlang der kreisförmigen Delaminationfront wurden mittels der modifizierten Rißschließungsmethode berechnet.
R. Krüger.
Three Dimensional Finite Element Analysis
of Multidirectional Composite DCB, SLB and ENF Specimens.
ISD-Report No. 94/2, Institute for Statics und Dynamics of Aerospace
Structures, University of Stuttgart, 1994. - (1.4 MB)
Abstract: Results are presented from a numerical investigation
of the effect of stacking sequence on energy release rate distribution
in laminated carbon fibre reinforced epoxy double cantilever beam, single
leg bending and end-notched flexure test specimens. The deformation behaviour
of the specimens has been computed by three-dimensional nonlinear finite
element analyses using a recently developed layered 3D-shell element. The
distributions of the local mixed mode energy release rates along straight
and measured curved delamination fronts have been determined by the virtual
crack closure method. The dependency of computed total energy release rate
and the individual mode contributions on element types and local mesh sizes
is discussed. It is shown that for all three specimen types the distribution
depends on the ply angles at the interface of crack propagation. In addition,
it also depends on the bending-twisting coupling and the
longitudinal-transverse
bending coupling in the individual arms of the specimens caused by different
stacking sequences. Results are used to verify existing and proposed new
design recommendations for all three types of specimens.
R. Krüger, M. König, J. Albinger and C. Hänsel.
Combined Experimental- Numerical Approach for the Determination of
Mixed- Mode Energy Release Rates at Delamination Growth.
AIAA-94-1460-CP. In the 35th
AIAA/ASME/ASCE/AHS/ASC
SSDM Conference, Hilton Head, SC, pages 1212--1222, 1994.
Abstract: The growth of delaminations embedded near the surface
of structural related, laminated CFRE-specimens during tension-compression
fatigue loading has been studied by measuring the out-of-plane (i.e. buckling)
deformation of the delaminated region via moiré technique. From
the measured deformations the delamination front contours have been determined.
The postbuckling state of the specimens in the compression phase of the
test, which is assumed to drive the delamination growth, is computed by
two-dimensional (plate) and three-dimensional (layered 3D shell) finite
element models in which the measured delamination front contour is taken
into account. First results for the computed displacements of a specimen
with circular delamination, which has been used as starter delamination
in the tests, show satisfactory agreement with the measured deformation.
Energy release rates have been computed along the circular delamination
front contour using the Modified Crack Closure Method. The goal of future
investigations is to compute energy release rates along measured arbitrarily
shaped delamination front contours and to correlate the results with the
measured delamination growth.
M. König, J. Albinger and C. Hänsel.
Delamination Buckling: Numerical Simulation of Experiments
In ICCM-9, Madrid, Vol. VI, page 535--542, 1993
Abstract: The growth of delaminations near the surface of laminated CFRP-specimens during tension-compression fatigue loading has been studied by measuring the out-of-plane (i.e. buckling) deformation of the delaminated region via moiré technique. From the measured deformations the delamination front contours have been determined. The postbuckling state of the specimen in the compression phase of the test, which is assumed to drive the delamination growth, can be computed by a two-dimensional (plate) finite element model in which the measured delamination front contour is taken into account. First results for the computed displacements of a specimen with artificial circular delamination, as used as starter-delamination in the tests, show satisfactory agreement with the measured deformation. The goal of the investigations is, to compute energy release rates along measured delamination front contours and to correlate the results with the measured delamination growth.
R. Krüger, M. König and T. Schneider.
Computation of Local Energy Release Rates Along Straight and Curved
Delamination Fronts of Unidirectionally Laminated DCB- and ENF - Specimens,
AIAA-93-1457-CP. In the 34th
AIAA/ASME/ASCE/AHS/ASC
SSDM Conference, La Jolla, California , pages 1332--1342, 1993.
Abstract: Several methods which utilize finite element (FE) analysis
for the calculation of energy release rates have been investigated.
Two-dimensional
models of double cantilever beam (DCB) and end notched flexure (ENF) specimens
were used for numerical tests and the investigation included methods such
as virtual crack extension and modified crack closure method. Results obtained
from FE-analysis were compared with experimental results using standard
methods for the evaluation of measured data to calculate the energy release
rate. From the numerical methods investigated the virtual crack entension
method and the modified crack closure method have been chosen for further
evaluation using three-dimensional analysis which provides the possibility
of calculating the distribution of the local energy release rate along
a crack front. For the ENF-specimen the energy release rate G_II was almost
uniform along an assumed straight crack front, for the DCB-specimen however
the energy release rate G_I was found to be constant only near the center
of the specimen. Towards the edges G_I decreases significantly. These results
confirm experimetal observations of straight crack fronts for ENF- and
curved crack fronts for DCB-specimens. Rather detailed investigations of
DCB-specimens with assumed curved crack fronts resulted in an almost constant
distribution of the local energy release rate across the width of the specimen.
M. König und R. Krüger.
Örtliche Energiefreisetzungsraten: Kriterium für den
Delaminationsfortschritt
in CFK - Laminaten ?
In 25. Vortragsveranstaltung des DVM-Arbeitskreises Bruchvorgänge,
Karlsruhe, pages 505--514. Deutscher Verband fur Materialforschung und
-prüfung e.V., Februar 1993.
Abtract:Die am häufigsten verwendeten Proben zur Bestimmung
der Rißzähigkeit bei der Delamination von
endlosfaserverstärkten
CFK (Carbon-Faser-verstärkten Kunststoff)-Laminaten sind die DCB (Double
Cantilever Beam)-Probe für Rißöffnungsmodus I und die ENF
(End Notched Flexure)-Probe für Rißöffnungsmodus II. Bei
quasi-statischem Delaminationsfortschritt beobachtet man für die DCB-Probe
gekrümmte und für die ENF-Probe gerade Delaminationsfronten.
Durch Berechnung der lokalen Energiefreisetzungsraten entlang gemessener
Delaminationsfronten mittels dreidimensionaler Finite-Elemente-Modelle
soll untersucht werden, ob die örtliche Energiefreisetzungsrate ein
Kriterium für den Delaminationsfortschritt, d.h. für die Entwicklung
der Form und Größe von Delaminationen, darstellt. Für eine
angenommene gerade Riß wurde für die ENF-Probe ein nahezu konstanter
Verlauf von G_II über die Probenbreite ermittelt, während sich
bei der DCB-Probe nur im Bereich der Probenmitte ein konstanter G_I-Verlauf
ergab. Zum Probenrand hin fällt G_I deutlich ab. Detaillierte
Untersuchungen
der DCB-Probe mit gekrümmter Rißfront zeigen einen nahezu konstanten
Verlauf der Energiefreisetzungsrate G_I entlang der Rißfront. Somit
kann die Arbeitshypothese, daß die lokale Energiefreisetzungsrate
als Kriterium für die Entwicklung einer Delaminationsfront betrachtet
werden kann, weiterverfolgt werden.
S. Fan, R. Krüger und M. König.
Anwendung numerischer Methoden der Bruchmechanik auf die Delamination
von CFK-Laminaten.
ISD-Report No. 92/11, Institut für Statik und Dynamik der Luft-
und Raumfahrtkonstruktioen, Universität Stuttgart, 1992.
Abstract: Verschiedene numerische Verfahren zur Berechnung der
Energiefreisetzungsraten G_I, G_II und G_III, die auf der Methode der Finiten
Elemente basieren, sind eingehend an zweidimensionalen Modellen von DCB-
und ENF-Proben untersucht worden. Die Ergebnisse linearer und nichtlinearer
Finite-Elemente-Analysen konnten mit jenen Energiefreisetzungsraten verglichen
werden, die mit Hilfe standardisierter Auswerteformeln aus den Versuchsdaten
ermittelt wurden. Von den untersuchten Verfahren wurden aufgrund guter
Ergebnisse nur die virtuelle Rißerweiterungsmethode (virtual crack
extension method) und die modifizierte Rißschließungsmethode
(modified crack closure method) für nichtlineare Analysen mit
dreidimensionalen
Modellen weiterverwendet. Mit solchen Analysen kann der Verlauf der
örtlichen
Energiefreisetzungsraten entlang der Delaminationsfront ermittelt werden.
Für eine angenommene gerade Rißfront wurde für die ENF-Probe
ein nahezu konstanter Verlauf von G_II über die Probenbreite ermittelt,
während sich bei der DCB-Probe nur im Bereich der Probenmitte ein
konstanter G_I-Verlauf ergab. Zum Probenrand hin fällt G_I deutlich
ab. Die Ergebnisse bestätigen die im Experiment beobachteten geraden
Rißfronten der ENF- und die zum Rand hin gekrümmten Rißfronten
der DCB-Probe. Detaillierte Untersuchungen der DCB-Probe mit gekrümmten
Rißfronten zeigen einen nahezu konstanten Verlauf der
Energiefreisetzungsrate
G_I entlang der Rißfront. Somit kann davon ausgegangen werden, daß
die lokale Energiefreisetzungsrate als Kriterium für die Entwicklung
einer Delaminationsfront Gültigkeit besitzt und daß die numerischen
Methoden zu deren Bestimmung qualifiziert sind.
R. Krüger und T. Schneider.
Berechnung der Energiefreisetzungsraten bei der Delamination
kohlenstoffaserverstärkter
Kunststoffe mit der Methode der Finiten Elemente.
In DGLR-Jahrbuch II, Deutscher Luft-
und Raumfahrtkongress, Bremen , pages 953--962, 1992. GW ISSN 0070-4083.
Abstract: Bei faserverstärkten Kunststoffen ist die Delamination, d.h. die flächige Ablösung zweier benachbarter Laminatschichten, eine der Hauptschädigungsformen. Unter Betriebsbelastung wird sich eine durch Fertigungsfehler oder Schlagbeanspruchung (impact) verursachte Delamination zunächst allmählich und dann mit zunehmender Geschwindigkeit vergrößern. Die dadurch verursachte Steifigkeits- und Festigkeitsabnahme des Bauteils kann bereits unterhalb der Belastungsgrenze, für die das Bauteil ausgelegt wurde, zum Versagen der Struktur führen. Konstruktion und Auslegung komplexer Bauteile aus faserverstärkten Kunststoffen mit Hilfe empirisch ermittelter Sicherheitsfaktoren können zwar ein Delaminationsversagen der Struktur verhindern, werden die Strukturen jedoch nicht optimal ausnutzen sondern überdimensionieren und so zu einem unerwünscht hohen Gewicht führen. Für eine optimale Ausnutzung der günstigen Eigenschaften faserverstärkter Kunststoffe, sowie für die Ermittlung von Inspektionsintervallen ist es notwendig, die Ausbreitung von Delaminationen vorhersagen zu können.
Für die folgenden Untersuchungen wird vorausgesetzt, daß
die linear-elastische Bruchmechanik für CFK anwendbar ist, wobei jedoch
Nichtlinearitäten aufgrund großer Verschiebungen (geometrische
Nichtlinearitäten) berücksichtigt sind. Ferner wird angenommen,
daß eine kritische Energiefreisetzungsrate G_c existiert und diese
eine Materialeigenschaft ist, die von der Geometrie und vom Lagenaufbau
unabhängig ist. Ein Delaminationsfortschritt kann folglich dann einsetzen,
wenn die durch die äußeren Lasten und die gespeicherte
Dehnungsenergie
zur Verfügung stehende Energiefreisetzungsrate G den kritischen Wert
G_c erreicht. Um nun die Ausbreitung von Delaminationen in einem CFK-Bauteil
vorhersagen zu könen, sind - neben der Kenntnis der kritischen
Energiefreisetzungsraten
des Werkstoffs für die drei Rißöffnungsarten und für
ihre Überlagerung - numerische Verfahren erforderlich, mit denen die
verfügbaren Energiefreisetzungsraten G_I, G_II und G_III für
ein komplexes Bauteil unter der gegebenen Belastung berechnet werden
können.
Hierzu stehen verschiedene Verfahren zur Verfügung, die auf der Methode
der Finiten Elemente beruhen.