Issue 28
Pubblicazione animata
Year VIII Issue 28 April 2014
Rivista Internazionale Ufficiale del Gruppo Italiano Frattura Fondata nel 2007
Editor-in-chief:
Francesco Iacoviello Alfredo Navarro Robles Luca Susmel John Yates
ISSN 1971-8993
Associate Editors:
Editorial Advisory Board:
Harm Askes Alberto Carpinteri Andrea Carpinteri Donato Firrao M. Neil James Gary Marquis Robert O. Ritchie Ashok Saxena Darrell F. Socie Cetin Morris Sonsino Ramesh Talreja Shouwen Yu
Frattura ed integrità strutturale The International Journal of the ItalianGroupof Fracture
David Taylor
www.gruppofrattura.it
Frattura ed Integrità Strutturale, 28 (2014); International Journal of the ItalianGroup of Fracture
Table of Contents
P. Valentino, E. Sgambitterra, F. Furgiuele,M. Romano, I. Ehrlich, N. Gebbeken Mechanical characterization of basalt woven fabric composites: numerical and experimental investigation ... 1 M.Malnati Amethod for calculation of finite fatigue life under multiaxial loading in high-cycle domain…………... 12 A. Brotzu, F. Felli, C. Lupi, C. Vendittozzi, E. Fantini Fatigue behavior of lubricatedNi-Ti endodontic rotary instruments…..................................................... 19 B. Ye,M. Li, G. Qiu, J. Dong, F. Zeng, T. Yang, B. Bai Quantitative estimating size of deep defects in multi-layered structures from eddy current NDT signals using improved ant colony algorithm…………...................................................................................... 32 D. Gentile, G. Iannitti, N. Bonora Experimental measurement and model validation of COD in pipe under bending with off-centered circumferential crack…………............................................................................................................. 42
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Frattura ed Integrità Strutturale, 28 (2014); ISSN 1971-9883
Editor-in-Chief Francesco Iacoviello
(Università di Cassino e del LazioMeridionale, Italy)
AssociateEditors AlfredoNavarroRobles
(Escuela Superior de Ingenieros, Universidadde Sevilla, Spain)
Luca Susmel JohnYates
(University of Sheffield, UK) (University ofManchester, UK)
AdvisoryEditorial Board HarmAskes
(University of Sheffield, Italy) (Politecnicodi Torino, Italy) (Università di Parma, Italy) (Politecnico di Torino, Italy) (University of Plymouth, UK)
AlbertoCarpinteri AndreaCarpinteri DonatoFirrao M. Neil James GaryMarquis Ashok Saxena Darrell F. Socie ShouwenYu RameshTalreja DavidTaylor RobertO. Ritchie CetinMorris Sonsino Editorial Board StefanoBeretta NicolaBonora RobertoCitarella ClaudioDalleDonne Manuel de Freitas VittorioDi Cocco Giuseppe Ferro TommasoGhidini PaoloLeonetti CarmineMaletta LiviuMarsavina DanieleDini
(Helsinki University of Technology, Finland)
(University of California, USA)
(GalgotiasUniversity, GreaterNoida, UP, India; University ofArkansas, USA)
(University of Illinois at Urbana-Champaign, USA)
(TsinghuaUniversity, China) (Fraunhofer LBF,Germany) (TexasA&MUniversity, USA) (University ofDublin, Ireland)
(PolitecnicodiMilano, Italy)
(Università di Cassino e del LazioMeridionale, Italy)
(Università di Salerno, Italy) (EADS,Munich, Germany) (EDAMMIT, Portugal)
(Università di Cassino e del LazioMeridionale, Italy)
(Imperial College, UK)
(Politecnico di Torino, Italy)
(European SpaceAgency - ESA-ESRIN) (Università dellaCalabria, Italy) (Università dellaCalabria, Italy) (University of Timisoara, Romania) (University of Porto, Portugal)
Lucas FilipeMartins da Silva
HisaoMatsunaga
(KyushuUniversity, Japan) (University of Sheffield, UK) (Politecnico di Torino, Italy) (Università di Parma, Italy) (Università diMessina, Italy) (Università di Brescia, Italy) (Università di Bologna, Italy) (Università di Parma, Italy)
MahmoudMostafavi
MarcoPaggi Oleg Plekhov
(RussianAcademy of Sciences, Ural Section,MoscowRussianFederation)
AlessandroPirondi GiacomoRisitano RobertoRoberti
Marco Savoia
Andrea Spagnoli CharlesV.White
(KetteringUniversity,Michigan,USA)
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Frattura ed Integrità Strutturale, 28 (2014); International Journal of the ItalianGroup of Fracture
Journal description and aims Frattura ed Integrità Strutturale (Fracture and Structural Integrity) is the official Journal of the ItalianGroup of Fracture. It is an open-access Journal publishedon-line every threemonths (July,October, January, April). Frattura ed Integrità Strutturale encompasses the broad topic of structural integrity, which is based on the mechanics of fatigue and fracture, and is concerned with the reliability and effectiveness of structural components. The aim of the Journal is to promoteworks and researches on fracture phenomena, as well as the development of newmaterials and new standards for structural integrity assessment. The Journal is interdisciplinary and accepts contributions from engineers, metallurgists, materials scientists, physicists, chemists, andmathematicians. Contributions Frattura ed Integrità Strutturale is a medium for rapid dissemination of original analytical, numerical and experimental contributions on fracture mechanics and structural integrity. Research works which provide improved understanding of the fracture behaviour of conventional and innovative engineeringmaterial systems are welcome. Technical notes, letters and review papers may also be accepted depending on their quality. Special issues containing full-length papers presented during selected conferences or symposia are also solicitedby theEditorial Board. Manuscript submission Manuscripts have to be written using a standard word file without any specific format and submitted via e-mail to iacoviello@unicas.it. The paper may be written in English or Italian (with an English 1000 words abstract). A confirmation of receptionwill be sent within 48 hours. The review and the on-line publication process will be concluded within threemonths from the date of submission. Peer reviewprocess Frattura ed Integrità Strutturale adopts a single blind reviewing procedure. The Editor in Chief receives the manuscript and, considering the paper’smain topics, the paper is remitted to a panel of referees involved in those research areas. They can be either external ormembers of theEditorial Board. Eachpaper is reviewed by two referees. After evaluation, the referees produce reports about the paper, bywhich the paper canbe: a) accepted without modifications; the Editor in Chief forwards to the corresponding author the result of the reviewing process and the paper is directly submitted to the publishing procedure; b) accepted with minor modifications or corrections (a second review process of the modified paper is not mandatory); the Editor in Chief returns the manuscript to the corresponding author, together with the referees’ reports and all the suggestions, recommendations and comments therein. c) accepted with major modifications or corrections (a second review process of the modified paper is mandatory); the Editor in Chief returns the manuscript to the corresponding author, together with the referees’ reports and all the suggestions, recommendations and comments therein. d) rejected. The final decision concerning the papers publicationbelongs to theEditor inChief and to theAssociateEditors. The reviewing process is completedwithin threemonths. The paper is published in the first issue that is available after the endof the reviewing process.
Publisher Gruppo ItalianoFrattura (IGF) http://www.gruppofrattura.it ISSN 1971-8993 Reg. Trib. di Cassinon. 729/07, 30/07/2007
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Frattura ed Integrità Strutturale, 28 (2014); ISSN 1971-9883
Somenews about our Journal…
Dear Friend, this is the 28 th issue of the IGF Journal. Born in 2007, after only seven years and thanks to the contribution and to the support of many friends, the Journal has obtained some incredible results. The Journal is now very well indexed (Scopus, EI Compendex, DOAJ, Index Copernicus, EBSCO host, ProQuest, Google Scholar…) and is available in somany on line library that it is quite impossible to give a short list of them, also considering only the most prestigious ones. Considering the first obtained evaluations, we can remember the evolution of the IndexCopernicus evaluation: After 28 issues, the journal is still characterizedbe the same peculiaritieswe defined from the first issue: - a fast but careful reviewing process: up to now, all the submitted papers has been always published (or rejected) within threemonths from the first submission; - an “obsession” for the open access approach: our Journal is completely free of charge both for readers and for writers. - indexing as a service: it is not sufficient that the papers are published on line… the Journal must assist the spreadof the publishedpapers improving the Journal indexation. Considering these developing lines, in the last months we had a little “revolution” in Frattura ed Integrità Strutturale. First of all, new components joined the Advisory Editorial Board : Ashok Saxena and Shouwen Yu. All the components of this Board are part of the elite of the Fracture and Structural Integrity world, and we are really proudof it!! Concerning the Journal Review Board , now this has been changed in an Editorial Board : this is not only a mere name modification, but also the components’ responsibilities have been changed. Furthermore, also the composition of this Board has been renewed: thanking the Colleagues that worked for the old JRB , we wish to thank all the friends of the new EB for all the efforts theywill produce to improve to journal quality. - 2011: 5.09 - 2012: 6.96 - 2013 (new!!): 7.67
Francesco Iacoviello Direttore F&IS
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P.Valentino et alii, Frattura ed Integrità Strutturale, 28 (2014) 1-11; DOI: 10.3221/IGF-ESIS.28.01
Mechanical characterizationof basaltwoven fabric composites: numerical and experimental investigation
PiergiorgioValentino, Emanuele Sgambitterra, FrancoFurgiuele University of Calabria, Department of Mechanical, Energy and Management Engineering, Ponte P. Bucci, 44C, 87036 Rende (CS), Italy MarcoRomano, IngoEhrlich University of Applied Sciences Regensburg, Department of Mechanical Engineering, Laboratory for Composite Technology, Galgenbergstrasse 30, 93053Regensburg, Germany NorbertGebbeken University of the Bundeswehr Munich, Institute for Engineering Mechanics and Structural Mechanics, Werner-Heisenberg Weg 39, 85577Neubiberg, Germany A BSTRACT . Basalt fabric composite, with different twill wave reinforcements, i.e. twill 2/2 and twill 1/3, have been studied in this work by means of experimental tests and numerical finite element (FE) simulations. As fabric reinforcements show repeating undulations of warp and fill yarn, simplemixtures law cannot be applied. As a consequence, the mesoscopic scale, lying between the microscopic and the macroscopic one, has to be taken into account to mechanically characterize a fabric reinforced composite. The aim of this work is to evaluate the stiffness of a fabric reinforced composite in warp and fill direction. In particular a numerical FE model, assuming elliptical sections and sinusoidal shape of the yarns, has been implemented and experimental tests have been carried out in order to validate the proposedmodel. Finally, the strength and the failuremodes of the composite material, for each analysed structure and textile orientation, have been experimentally investigated. S OMMARIO . Diverse tipologie di tessuti compositi con fibre di basalto, i.e. tessitura 2/2 e tessitura 1/3 con fibre di basalto sono stati studiati in questo lavoromediante prove sperimentali e simulazioni agli elementi finiti (FE). Poiché i tessuti rinforzanti sono caratterizzati da ondulazioni ripetute di trama e ordito la semplice legge delle misture non può essere applicata. Per questomotivo la scalamesoscopica, via di mezzo tra quellamicroscopica emacroscopica, è utilizzata per la caratterizzazionemeccanica di questa tipologia di compositi. Scopo del presente lavoro è quello di determinare la rigidezza sia nella direzione della trama che in quella dell’ordito dei compositi rinforzati con tessuti. In particolare è stato implementato un modello numerico agli elementi finiti, (FE) realizzato ipotizzando le sezioni dei fasci di forma ellittica e con andamento sinusoidale e i risultati ottenuti sono stati validati mediante confronto sperimentale. Infine, si è investigato sulla resistenza e il modo di cedimentodel materiale composito, per ogni struttura analizzata ed orientazione del tessuto. K EYWORDS . Basalt fibre, Basalt fibre reinforced plastic, Fabric reinforcement, Woven fabric, Tensile testing, RepresentativeVolumeElement (RVE).
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P. Valentino et alii, Frattura ed Integrità Strutturale, 28 (2014) 1-11; DOI: 10.3221/IGF-ESIS.28.01
I NTRODUCTION oven fabric (WF) composite materials are of particular interest in the scientific community due to their mechanical performances compared to the unidirectional laminates. In particular, textile composites offer better dimensional stability over a large range of temperatures, better impact resistance and tolerance; subtle conformability anddeepdrawmoldability, compared to the common unidirectional laminated composites. The variety ofmanufacturingmethods havemade the textile composites cost-competitive, therefore they are used inmany application fields and they are being considered for intra and interlaminar strength and damage resistance. Among the various textile forms, woven fabrics are the most widely used in composites, they provide more balanced properties in the fabric plane and the interlacing of yarns provides higher out of plane strength, which can take up the secondary loads due to loadpath eccentricities, local buckling, etc. Simplified theoretical approaches to predict themechanical behaviour of such a kind of compositematerial are quite little, in factmost of themodels are limited to the unidirectional reinforced layers and they use the homogenization technique. Several studies, on 2-D and 3-D geometries, have been carried out tomodel and analyse themechanical properties of the reinforced composite materials. In particular, Barbero et al [1], developed an accurate model of a plain wave fabric in order to evaluate its mechanical properties assuming a sinusoidal shape of the tow fibres. Stress and strain averaging procedure has been studied by Yiwie Jiang et al. [2], for local/global analysis of plain-weave fabric composites, where, within a representative volume cell, using uniform stress and uniform strain assumptions, the constitutive equations are averaged along the thickness direction. Chou and Ito [3] analysed the strength and failure behaviour of plain weave composites. In particular, the geometrical characteristics of yarn shape, laminate stacking configuration, fibre volume fraction, and yarn packing fraction were investigated using three-dimensional geometrical models. Based on the geometrical characteristics, iso-strain approach was developed to predict elastic properties, stress distributions, and strengths under tensile loading. N. K. Naik et al. [4] developed a two-dimensional closed-form analytical method for the thermo-elastic analysis of two dimensional orthogonal plain weave fabric laminas, considering the volume fraction of fibres and the possible gap between the two adjacent strands. Finally, analytical models of the plain weave laminated composites to evaluate the elastic properties of woven fabric composites are reported in [5]. However, most of these works are related to plainwave fabrics, while not many efforts to study the twill wave type have been carried out. The aim of this work is to evaluate the stiffness of a fabric reinforced composite inwarp and fill direction. In particular a numerical FE model, assuming elliptical sections and sinusoidal shape of the yarns, has been implemented and experimental tests have been carriedout inorder to validate the proposedmodel. Results in terms of stress-strain response and stiffness, for the different analysed structures are reported and discussed. Finally, the strength and the failure modes of the composite material, for each analysed structure and textile orientation, have been experimentally investigated. W oven fabric composites made in continuous basalt fibres and polymeric matrix has been investigated. Test panels have been produced by placing hand all the constituent layers, as shown inFig. 1, and a curing treatment at room temperature in vacuum bag has been carriedout, inorder to get the final product. All test panels have been produced using the same epoxy matrix system [6] and two different orientations of the fibres have been considered, as reported in Tab. 2. According to the German standardDINEN 2747 [7], each panel has been processedwith the aim to get a thickness of approx. 2mm. Two different fabric types have been used in this investigation, i.e. twill 2/2 and twill 1/3, details are reported in [5]. The corresponding layups and technical data, according to the provided data sheet [12] are listed inTab. 2. Tensile test specimens, with plane dimensions of 250 x 25mm [7], have been obtained bywater jet cutting from the panel produced in autoclave as shown in Fig. 2. In order to avoid the failure of the specimens near to the gripping zone, due to the local stress concentration, and with the aim to prevent eccentricity of load and limit bending phenomena during the setup, tabs have been bonded to both the ends of each sample, Fig. 3. According to [7], they were made of glass fiber reinforcedplasticswith a±45°-layup andwith dimensions of 25 x 45 x 1.5mm. W M ATERIALSANDTESTPROCEDURES
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P.Valentino et alii, Frattura ed Integrità Strutturale, 28 (2014) 1-11; DOI: 10.3221/IGF-ESIS.28.01
The volume fraction of fibre has been evaluated by calculating the mass variation of a sample after heat treatment. In particular, a certain number of specimens, with plane dimensions of 20 x 10 mm, has been collected from the created panels, and each of them has been weighted by using a precision balance M ETTLER T OLEDO 204-S. A heat treatment, needed to completely burn the polymericmatrixwithout effecting the reinforcement [8-11], has been performed by using a Muffle furnace C ARBOLITE EML 11/6 for 15 minutes at 620 °C. After cooling, all the treated samples have been weighted again, therefore, the density of thematrix and fibres and, consequently, the fibre volume content, φ f , according to the standardDINEN 2559 [13], has been calculated as below:
m
f
f V V V V V m m m f f
f
(1)
c
c
m
f
c
f
f
m
where the subscripts f , m and c indicate the fibre, matrix and composite properties, respectively, V mass and is the density.
m is the
is the volume,
Figure 1 : Scheme of the layup used in the processing: 1 sealant tape, 2 vacuum connector, 3 release film, 4 peel ply, 5 composite layup, 6 peel ply, 7 perforated foil, 8 bleeder, 9 vacuum bag.
Basalt fibres
Epoxy resin
Mechanical Property
Value
Mechanical Property
Value
2.75 g/cm 3
1.15 g/cm 3 2.65GPa 0.98GPa
Density,
Density,
Y OUNG ’smodulus, E 1 Y OUNG ’smodulus, E 2 Shearmodulus, G 12 Shearmodulus, G 23 P OISSON ’s ratio, ν 12 P OISSON ’s ratio, ν 23
Y OUNG ’smodulus, E
89GPa 89GPa
Shearmodulus, G P OISSON ’s ratio, ν
21.7GPa 21.7GPa
0.35
0.26
0.26 Table 1 :Mechanical properties of basalt fibres [12] and epoxy resin [6].
Fibre volume content f ρ in% 51.85
Fabric type
Warp yarns per cm
Fill yarns per cm
Yarn type Specificweight in g/m²
Layup
Layers Thickness inmm
Thread of direct roving 11.5 µm 110 tex
Twill 2/2
334
16
9
[0°/90°] 6
6
~1,8
6
~1,8
47.88
Twill 2/2
334
16
9
[90°/0°] 6
Twill 1/3 Direct roving 11.5 µm 110 tex Twill 1/3
362 362
18 18
8 8
[0°/90°] 6 [90°/0°] 6
6 6
~1,9 ~1,8
49.17 54.75
Table 2 : Types of fabrics, specificweights and layups.
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P. Valentino et alii, Frattura ed Integrità Strutturale, 28 (2014) 1-11; DOI: 10.3221/IGF-ESIS.28.01
Figure 2 : Typical composite plate after cutting usingwater jet inorder toproduce tensile test samples.
Figure 3 : Side viewof the tensile test specimenswith tabs according toGerman standardDINEN 2747 [7].
Experimental tensile tests, in order to evaluate the mechanical properties of the fabric composites, in terms of stiffness and strength, have been carried out according to [7]. Each test has been performed using a universal servo-hydraulic testing machine (Instron 8500), at room temperature T=298 K and the strain has been measured by a resistance extensometer with a gauge lengthof 20mm.
F INITE -E LEMENT -A NALYSES
A
representative volume element (RVE) of the fabric composite has been geometrically defined and numerically modelled. In order to evaluate the geometric dimensions of the RVE, top view of the dry fabric and side view of a cross section of the test panels have been investigated using an optical microscope. Information about length and width have beenobtained and they have beenproperlymatchedwith the geometric dimensions provided in the data sheet. Fig. 4 shows a depiction of the cross-section of a single lamina for both of the analysed structures. The height of theRVE has been simply obtained by dividing the thickness of the test panel by the number of constituent laminas. The geometry of the RVE has been carefully modelled, in different steps, by using a commercial CAD software as follows: 1. Modeling of warp and fill yarn (corresponds to the dry fabric); 2. Separatelymodeling of the surroundingmatrix; 3. Assembling of the fabric andmatrix inorder to get the final geometricmodel of theRVE. Fig. 5 shows a 3-dimensional depiction, inwireframe, of the final obtainedRVEs for bothof the fabric reinforcements. According to [5], inorder tobetter simulate the real shape of the fabric composite, the trend of thewarp and fill yarnhave beenmodelled using a sinusoidal shapewhich canbe expressed as [5, 14] 2 cos x y A c (2) where A is the amplitude, and c is the pitch of the tow path curve, Fig. 6(a). An elliptical shape has been assumed to model the cross-sections of thewarp and fill yarn [15, 16].
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P.Valentino et alii, Frattura ed Integrità Strutturale, 28 (2014) 1-11; DOI: 10.3221/IGF-ESIS.28.01
(a)
(b) Figure 4 : Schematic depictionof sample cross-sections for the two types of dry fabric reinforcements. (a) Twill wave 2/2; (b) Twill weave 1/3.
(a)
(b)
Figure 5 : 3Dmodels of theRVE. (a) Twill weave 2/2; (b) Twill weave 1/3.
(a) (b) Figure 6 : Simplifying assumptions for analytical description of the geometry: (a) Sinusoidal spline about centreline of an undulated yarn; (b) Elliptical cross-sectionof a yarn. Starting from the created geometry, the FEMmodel has been generated by using a commercial finite element software. Themesh distribution, with a proper size, has been generated bymeans of an automatic option of the FEM software and a depictionof themeshedRVE for the twill weave 2/2 and twill weave 1/3, respectively, is shown inFig. 7. As a first approach an idealized contact has been obtained by imposing the coincidence of nodes in the matrix-fibres interface, therefore failure mechanisms and friction effects were neglected. Furthermore, this assumption results in no relative displacement between the three different regions, i.e. warp yarns, fill yarns andpurematrix. In order to properly assign the mechanical properties of each component of the RVE, i.e. matrix and fibres tow, two different reference systems have been considered, respectively. Linear elastic properties have been imposed to the yarns, but, in order to simulate the orthotropic behaviour of thematerial two additional reference systems have been introduced:
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P. Valentino et alii, Frattura ed Integrità Strutturale, 28 (2014) 1-11; DOI: 10.3221/IGF-ESIS.28.01
the one to assign the mechanical properties of the warp yarns and the other one to assign the fill yarns properties, respectively. Isotropic linear elastic properties have been assumed for the matrix and data listed in Tab. 1 have been assigned to it. This assumption requires no further specificationof themeshorientation.
(a)
(b)
Figure 7 : FEmodels of theRVEs (a) Twill weave 2/2; (b) Twill weave 1/3.
In order to correctly assign the mechanical properties to the yarns, it has to consider that each tow is characterized by a big percentage of fibres immersed in epoxy matrix. Furthermore, the calculation of the material properties should be based on the experimentally determined values of the fibre volume content φ f , previously described. However, the latters have been calculated based on a bigger volume of material than the RVE, therefore, in order to match the data and to correctly assign the material properties to the yarns, the fibre volume content of the yarns in the RVE, φ f,y , has been determined bymodifying the one experimentally calculated, φ f , as below:
1 V V X RVE y
f
(3)
, f y
y
y
where V RVE is the RVE volume, V y is the tow volume of the RVE, x is the relative volume of yarns in the RVE. Details of the new data are listed inTab. 3.
φ f experimentally determined
φ f,y basedon exp. values
φ f,y standardized to 50% in theRVE
Relative volume x in theRVE
Fabric type and test direction Twill weave 2/2
Region in theRVE
Warp and fill yarns Polymericmatrix Warp and fill yarns Polymericmatrix Warp and fill yarns Polymericmatrix Warp and fill yarns Polymericmatrix
76.00% 24.00% 76.00% 24.00% 79.00% 21.00% 79.00% 21.00%
51.85%
68.22%
65.79%
Warp
Twill weave 2/2
47.88%
63.00%
65.79%
Fill
Twill weave 1/3
49.17%
62.24%
63.29%
Warp
Twill weave 1/3
69.30%
63.29%
54.75%
Fill
Table 3 : Calculated fibre volume content, φ f,y
, to used inFE-analyses.
The new calculated values of the fibre volume content φ f,y the RVEs and on the fibre volume content experimentally calculated, φ f
are based on the relative volumes ratio of the different phase of . Therefore, the geometric dimensions of the
yarns, the thickness of the RVEs [15, 16, 18] as well as the accuracy of the experimental determination of φ f calculation. The yarn is considered as a unidirectional reinforced phase with a transversally isotropic behaviour, therefore, nine elastic constants (five are independent) are required to assign the transversally isotropic properties to the reinforced regions of theRVE. The evaluation is described in the following. Inparticular, in the predominant direction the stiffness canbe calculated bymeans of themixture law, as below: 1 1 , , f y f f y m E E E (4) , affect this
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P.Valentino et alii, Frattura ed Integrità Strutturale, 28 (2014) 1-11; DOI: 10.3221/IGF-ESIS.28.01
As the basalt fibre and the polymeric matrix can be considered as homogeneous isotropic materials, the respective shear moduli can be calculated by G=E/2(1+ ν) , where the corresponding values are listed in Tab. 1. Mixture law, according to Chamis [19, 22], canbe applied inorder to calculate threemore independent linear-elastic properties of the yarns, namely:
E
m
E E
(5)
1
2
E E
1 m
1
, f y
f
G
m
12 G G
(6)
13
G G
1 m
1
, f y
f
12 13 (7) The remaining parameters, needed to fully define thematerial properties of the RVE, have been calculated exploiting the Maxwell-Betti's law, E i ν ji =E j ν ij , as below: 1 , f y f , f y m
E E
2
(8)
21 31
12
1
Finally, the last independent parameters, i.e. G 23 , G 32 , 23
and 32
have been calculated
G
m
23 G G
(9)
32
G
m
1
1
, f y
G
23
f
,
E G
2
1
(10)
23 32
2
23
Finally, appropriate boundary conditions have been applied to theRVE inorder to: 1. ensure a purely longitudinal deformation; 2. allow contractionof the cross-sections due toPoisson’s effects; 3. avoid twisting andbending effects.
R ESULTSAND D ISCUSSION
Experimentally and numerically determined stiffness esults, in terms of stress-strain response, for both of the fabric reinforcements, i.e. twill 2/2 and twill 1/3, are reported inFig. 8(a) andFig. 8(b), respectively. Figures clearly exhibit an high repeatability of the response in all the cases. This can be attributed to a constant material quality over the whole test panel. However, whereas the 2/2 fabric type shows a clear difference, in terms of stiffness and strength, along the warp and fill direction, the 1/3 fabric type does not show substantial variations. Furthermore, the textile exhibits higher stiffness and strength along the warp direction than the fill one for both of the fabric types. Using the weighted averagemethod, as a first approximation [21, 22], the stiffness of the specimens have been calculated as secant modulus E s.0.5% at a strain value of ε =0.5% and tangent modulus E t.0.1% (as a Secant Modulus at a strain of ε =0.1%) according toGerman standardDINEN 2747 [7]. For each textile specimen and layup the average values and standard deviations have been calculated, respectively [23-25]. Fig. 9 and Fig. 10 show the stiffness related to each textile semi-finished product as secant modulus E s.0.5% and tangent modulus E t.0.1% with the respective standard deviations. In particular, whereas Fig. 9 shows the secantmoduli and tangent R
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P. Valentino et alii, Frattura ed Integrità Strutturale, 28 (2014) 1-11; DOI: 10.3221/IGF-ESIS.28.01
moduli based on the experimentally determined values of the fibre volume content, Fig. 10 shows the values standardized to a fibre volume content φ f =50% .
(a) (b) Figure 8 : Stress-strain diagrams as results of the experimental tensile tests: (a) fabric type twill 2/2 inwarp and fill direction; (b) Fabric type twill 1/3 inwarp and fill direction.
Figure 9 : Secant modulus and tangent modulus of the respective textile semi-finished products based on the experimentally determined values of the fibre volume content of φ f .
Figure 10 : Secant modulus and tangent modulus of the respective textile semi-finished products standardized to a fibre volume content of φ f =50% .
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P.Valentino et alii, Frattura ed Integrità Strutturale, 28 (2014) 1-11; DOI: 10.3221/IGF-ESIS.28.01
The stiffness values, numerically calculated, show good agreement with the corresponding experimental ones in both the considered cases, i.e. experimental and standardized, φ f =50%, fibre volume content. In particular, the stiffness determined by the FE-analyses lies in the range of the secant modulus E s.0.5% and the tangent modulus E t.0.1% . As an exception the numerical results for the stiffness of the twill weave 1/3 in fill direction yield slightly lower values than the experimental ones. Experimentally determined strengths and corresponding failure modes Tensile strength values σ z.B , based on the experimental results, have been calculated, according to [22] and [12], and shown in Fig. 12, respectively. In particular the experimental values related to the twill weave 2/2 reinforcement are higher than the ones calculated by the indicated values in the data sheets, while a good agreement has been observed for the twill weave 1/3 reinforcement. With the aim to provide a proper interpretation of the experimental tensile tests a careful analysis on the failure modes and crack pathpropagationhas been carriedout. In particular, in Fig. 11 is shown a depiction of a typical fracturemode of a specimenwith twill weave 2/2 reinforcement. Woven fabric with 0°-90° and 90°-0° orientation exhibited analogous failuremodes. The crack started in the gauge length andproceeded perpendicularly to the applied load.
Figure 11 : Fractured specimen with twill weave 2/2 fabric reinforcement in the warp direction. The area enclosed in the square box show details of the damagemechanism. InFig. 13 is shown a depictionof a typical fracturemode of a specimenwith twill weave 1/3 reinforcement. Similar to the previous case study, the orientation of the woven fabric did not affect the failuremode and the crack started in the gauge length. However, the crack path is inclinedof approx. 45° to the applied load axis.
Figure 12: Calculated tensile strengths of the respective textile semi-finished products standardized to a fibre volume content of φ f =50%.
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P. Valentino et alii, Frattura ed Integrità Strutturale, 28 (2014) 1-11; DOI: 10.3221/IGF-ESIS.28.01
Figure 13 : Fractured specimen with twill weave 1/3 fabric reinforcement in the fill direction. The areas enclosed in the square boxes show details of the damagemechanism leading to sample failure.
C ONCLUSIONS
B
asalt fabric composite, with different twill wave reinforcements, i.e. twill 2/2 and twill 1/3, have been studied in this work by means of experimental tests and numerical finite element (FE) simulations. In particular, the mechanical response and the stiffness of a fabric reinforced composite in warp and fill direction has been analysed. The numerical FEmodel has been properly implemented assuming elliptical sections of the tows and sinusoidal shape of the yarns and particular attention has been applied to generate the RVEs geometry. The obtained results have been compared with the experimental data in order to validate the proposed model and a good agreement has been observed. Therefore the FE-method can be considered an adequate way to predict the stiffness of woven fabric composite with different geometries in themesoscopic scale or evendifferent kindof fibre reinforcement. Finally, the strength and the failure modes of the composite material, for each analysed structure and textile orientation, have been experimentally investigated. A CKNOWLEDGEMENTS NCOTELOGY LTD. is acknowledged for providing the fabrics of basalt fibres. Further thanks go to Mr M. Eisenried (Laboratory for Composite Technology (LFT - Labor für Faserverbundtechnik) at theDepartment ofMechanical Engineering at the University of Applied Sciences Regensburg) for proofreading and for generating the schematic illustrations in the CAD System. REFERENCES [1] Barbero, J., Trovillion, J., Mayugo, J.A., Sikkil, K.K., Finite element modelling of plain weave fabrics from photomicrographmeasurements, Composite Structures, 73 (2006) 41–52. [2] Jiang, Y., Tabiei, A., Simitses, G. J., A novel micromechanics-based approach to the derivation of constitutive equations for local/global analysis of a plain-weave fabric composite, Composites Science Technology, 60 (2000) 1825-1833. [3] Ito, M., Chou, T.-W., An analytical and experimental study of strength and failure behaviour of plain weave composites, Journal of CompositeMaterials, 32 (1) (1998) 2-30. [4] Naik, N. K., Ganesh, V. K., An analytical method for plainweave fabric composites, Composites, 26(4) (1995) 281 289. I
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P.Valentino et alii, Frattura ed Integrità Strutturale, 28 (2014) 1-11; DOI: 10.3221/IGF-ESIS.28.01
[5] Niranjan, N. K., Woven fabric composites, Technomic publication, Indian institute of Technology, Bombay, India (1994). [6] Epoxidharz, L., Härter, L., TechnischeDaten. Technical Data Sheet byR&G, (2011). [7] DINEN 2747 – Luft- undRaumfahrt –GlasfaserverstärkteKunststoffe –Zugversuch. Normenstelle Luftfahrt (NL) imDINDeutsches Institut fürNormung e.V., BeuthVerlag, Berlin, (1998). [8] Jungbauer, B., Romano, M., Ehrlich, I., Reproduzierbare Herstellung und definierte Vorschädigung von Probekörpern aus basaltfaserverstärktemKunststoff zurDämpfungsmessung. Bachelor¬thesis, University of Applied Sciences Regensburg, Laboratory of CompositeTechnology, Regensburg, (2012). [9] Lauterborn, E., Dokumentation Ultraschalluntersuchung Eingangsprüfung, Internal Report WIWeB Erding, Erding, (2011). [10]Schmid, V., Jungbauer, B., Romano, M., Ehrlich, I., Gebbeken, N., Diminution of mass of different types of fibre reinforcements due to thermal load. In: Proceedings of theAppliedResearchConference, Regensburg, (2012). [11]Schmid, V., Jungbauer, B., Romano, M., Ehrlich, I., Gebbeken, N., The influence of different types of fabrics on the fibre volume content and porosity in basalt fibre reinforced plastics. In: Proceeding of the Applied Research Conference, Regensburg, (2012). [12]QualityCertificates for Fabrics andRovings. Incotelogy Ltd., Bonn, (2012). [13]DIN EN 2559 – Luft- und Raumfahrt – Kohlenstoffaser-Prepregs – Bestimmung des Harz- und Fasermasseanteils und der flächenbezogenen Fasermasse. Normenstelle Luftfahrt (NL) imDINDeutsches Institut für Normung e.V., BeuthVerlag, Berlin, (1997). [14]Ottawa, P., Romano, M., Wagner, M., Ehrlich, I., Gebbeken, N., The influence of ondulation in fabric reinforced composites on dynamic properties in a mesoscopic scale in composites reinforced with fabrics on the damping behaviour, In: Proceedings of the 11. LS-DYNAForum, Ulm, (2012). [15]Barbero, E. J., Luciano, R., Micromechanics Formulas for the Relaxation Tensor of linear Viscoelastic Composites withTransversely Isotropic Fibers, In: International Journal of Solid Structures, 32 (1995) 1859-1872. [16]Ozgen, B., Gong, H., Yarn geometry inwoven fabrics, TextileResearch Journal, 81 (2010) 738-745. [17]Luciano, R., Sacco, E., Variational Methods for the Homogenization of Periodic Heterogeneous Media, European Journal ofMechanics –A/Solids, 17(4) (1998) 599-617. [18]Lopresto, V., Leone, C., De Iorio, I., Mechanical characterisation of basalt fibre reinforced plastic, Composites: Part B, 42(4) (2011) 717-723. [19]Chamis, C. C., Simplified Composite Micromechanics Equations for Hygral, Thermal and Mechanical Properties, SAMPEQuarterly, (1984) 14-23. [20]Moser, K., Faser-Kunststoff-Verbund–Entwurfs- undBerechnungsgrundlagen, VDI-Verlag, Düsseldorf, (1992). [21]Schürmann, H., Konstruieren mit Faser-Kunststoff-Verbunden, Springer-Verlag, Berlin/Heidelberg/New York, (2005). [22]Stellbrink, K.,Micromechanics of Composites, Hanser-Verlag,München/Wien, (1996). [23]Fahrmeir, L., Künstler, R., Pigeot, I., Tutz, G., Statistik – Der Weg zur Datenanalyse. 5. Auflage, Springer Verlag, Berlin/Heidelberg, (2005). [24]DIN V 65352 – Luft- und Raumfahrt – Verfahren zur statistischen Auswertung der Prüfergebnisse bei der Qualifikations- und Abnahmeprüfung von Faserverbundwerkstoffen. Normenstelle Luftfahrt (NL) im DIN Deutsches Institut fürNormung e.V., Beuth-Verlag, Berlin, (1987). [25]Papula, L., Mathematische Formelsammlung für Naturwissenschaftler und Ingenieure, Auflage, Vieweg-Teubner, Wiesbaden, (2009) 10.
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M.Malnati, Frattura ed Integrità Strutturale, 28 (2014) 12-18; DOI: 10.3221/IGF-ESIS.28.02
A method for calculation of finite fatigue life under multiaxial loading inhigh-cycledomain
M.Malnati RUAGAerospace Services GmbH - RUAGAviation, Claude-Dornier-Str., 82231Wessling, Germany mario.malnati@ruag.com , mario.malnati@yahoo.fr
A BSTRACT . Amethod for fatigue life assessment in high-cycle domain under multiaxial loading is presented in this paper. This approach allows fatigue assessment under any kind of load history, without limitations. The methodology lies on the construction - at amacroscopic level - of an “indicator” in the form of a set of cycles, representing plasticity that can arise atmesoscopic level throughout fatigue process. During the advancement of the loading history new cycles are created and a continuous evaluationof the damage ismade. K EYWORDS . High-cycle multiaxial fatigue; Metal fatigue; Multiaxial rainflow counting; Crack initiation; Non proportional loading; Continuous damage. I NTRODUCTION umerous methods for multiaxial fatigue analysis exist in literature (see e.g. [1]) and they are widely applied in different industrial contexts. Nevertheless many of them are limited to specific loading conditions - typically proportional loading: see [2, 3] for a review - or intended only for an evaluation of the unlimited fatigue life (see [1, 2, 4]). Themotivation of themethod presented in this paper comes from the need felt during years by the author in the fatigue assessment of industrial structures, to conceive a simple and easily applicable tool capable to estimate finite life crack initiation under any kind of loading, in the domain of High-Cycle Fatigue (HCF). Every type of variable amplitude multiaxial stress history canbe treatedwithout presenting incorrect filtering of significant cycling events. G ENERALPRESENTATIONOFTHEMETHOD t is a widely recognized and well accepted phenomenon [5] that the fatigue damage in theHCF domain is related to the amount of plasticity created at least in some grains under amacroscopically elastic stress history. The ref. [5] gives for instance a synthetic understanding: “… the common principal mechanism responsible of the crack initiation is the plastic strains and the damage developed in the grains due to irreversible dislocations motion. The essential difference between HCF and LCF regimes is that the scale of the plastic localization in amaterial volume is mesoscopic and respectively macroscopic ”. Some criteria for the calculation of the fatigue life are actually based on the evaluation of the plasticity amount. They can be either direct - as described for instance in [6] and [7], where a computational crystal plasticitymodel is used at a refined microstructural scale - or indirect, like for example the work of Jabbado and Maitournam [8] where a macro-meso relationship is used. N I
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M.Malnati, Frattura ed Integrità Strutturale, 28 (2014) 12-18; DOI: 10.3221/IGF-ESIS.28.02
Similarly, themethodology proposed in the present paper has its foundation on a plasticity approach, since usage is made of “yield surfaces” created in the stress space during the advancement of the stress history. But on the other side, the description remains here merely in the stress space: no explicit use of the mesoscopic plastic strain is made, but the equivalent fatigue stress is calculated directly on each created stress cycle. The yield surfaces used in the present approach just represent a way to describe at the macroscopic level the underlying phenomenon related to the plasticity arising at a mesoscopic level. From amore general point of view, the present work follows the same scope of the initial work of Jabbado [3] to extend the endurance criterion of Dang Van [9] to a finite life assessment using a continuous damage evaluation. Nevertheless, differently from [3] a notion of stress cycle is used here, even though the cycles are not counted and extracted from the global sequence - as it is done in a rainflowmethod (see e.g. [1]) - but they are simply created and updated in a continuous way while the stress history advances. This continuous damage approach has the same viewpoint of the cited work of Jabbado [3] or for example of the methodology proposed by Stefanov [10]. On the other hand, the principle to adapt a multi-surface concept to fatigue assessment by creating the stress cycles during the stress history advancement has some similaritieswith thework ofHerbland [11]. he basic ingredient of the present approach is the geometrical creation of closed surfaces in the stress space while the stress point is moving on its path, in a way identical to the classical plasticity theory (see e.g. [12, 13]). Each created yield surface is afterward associated to an amount of fatigue damage, evaluated using the basic fatigue material properties. This usage of the yield surfaces is analogous towhat is done for the stress cycles after their extraction by a classical cycle-counting, like the rainflowmethod. For this reason, the terms “yield surface” and “stress cycle”will be used hereafter as synonyms. The rules describedhere below resume the stress cycles creation. The equation defining a yield surface in the stress space has the form f P ( – X C ) = y (1) where f P is a scalar function that in accordance with the hypothesis used by Jabbado [3] and by many classical multiaxial criteria (see [1, 4]) takes theVonMises equivalent stress: T C ONSTRUCTIONOFTHESTRESSCYCLES
1 2
2
2
2
( ) =
I
f P
(2)
[
]
II
II
III
III
I
where I
, II , III are the principal stresses of . Let us note explicitly that f P
is function of the only deviatoric part s
of the stress tensor : f P ( ) = f P ( s )
(3) (4)
s = dev = - p H I
= tr( ) / 3
where p H
Hence each yield surface is a VonMises hypersphere fully described by its centre X C
(back-stress tensor) and its size
y . In the same way of a classical plasticity criterion, when the current stress state moves along the stress history a yield surface is hardened if the stress lies on the surface itself and is moving outwards. These conditions will be integrated in the Eqs. (5.a), (5.b) and (12.a), (12.b) written below. As a consequence the stress point will be always inside - or exactly on – a yield surface, but never outside. A multi-surface model is used, inspired to the one proposed by Mroz and described e.g. in [12]. In such a model, more than one yield surface can exist at the same time. However a significant dissimilaritywith the concept ofMroz is that intersections between surfaces are allowed in the present approach. As a consequence the existing surfaces are not necessarily all nested inside each other: this issue in the frame of multi-surface plasticity models is discussed for instance in ref. [14]. The following hardening rules are used. When the deviatoric part s of the stress tensor has an increment d s , among the surfaces that are hardened at a given instant the one havingmaximum y is hardenedby the following superposed isotropic / kinematic hardening law:
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M.Malnati, Frattura ed Integrità Strutturale, 28 (2014) 12-18; DOI: 10.3221/IGF-ESIS.28.02
1 2 1 2
H( P
d X C
=
)
(5.a)
2 3
d y
H( P
=
)
(5.b)
where:
H(x) = 1 if x≥ 0, H(x) = 0 if x< 0
(6) (7)
P
( s; X C (8) and n is the local outward vector normal to the surface in the deviatoric stress space, having unit length in the following sense: n : n = 1 (9) The symbol : in Eqs. (5.a), (5.b) and (9) designates the following scalar product in the stress space, between two tensors s A , s B : SinceVonMises stress is used, it is possible - as done in [3] - to give to n the following explicit form: n = 3 2 ( s – X C ) / f P ( s – X C ) (11) If at a given instant the same maximum y belongs to two or more yield surfaces contemporarily then the one to be hardened according to the Eqs. (5.a) and (5.b) is chosen conservatively as the one having the maximum value of the scalar product d s : n . Additionally, if two or more surfaces have not only the same maximum y but also the same maximum d s : n , then the choice for the one to be hardened according to the Eqs. (5.a) and (5.b) becomes physically arbitrary: we simply choose the onewhich hadbeenpreviously created firstly. The other surfaces that are hardened at a given instant - because the stress point lies on the surface and is moving towards the outside - simplymove rigidlywith a pure kinematic hardening ( y remain constant) defined by: d X C =H( P )
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