MULTI-SCALE MODELLING OF TIME-DEPENDENT RESPONSE OF COMPOSITE STRUCTURES MADE OF ORTHOTROPIC VISCOELASTIC MATERIALS

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1 3 th Internatonal Conference on Steel, Space and Composte Structures 3 January - February 8, erth, Australa MULTI-SCALE MODELLING OF TIME-DEENDENT RESONSE OF COMOSITE STRUCTURES MADE OF ORTHOTROIC VISCOELASTIC MATERIALS Sardar Malek *, Thomas Gereke, Nma Zobery, and Reza Vazr *School of Cvl and Envronmental Engneerng Unversty of Technology Sydney, Ultmo, NSW, Australa e-mal: sardar.malek@uts.edu.au Keywords: Compostes, Mult-scale Modellng, Fnte Element Method, Vscoelastcty Abstract. Modern structural compostes such as advanced fbre-renforced and strand-based compostes are ncreasngly beng used n the new generaton of aerospace, automotve, marne and cvl engneerng structures. To analyse the tme-dependent response of these orthotropc compostes, an effcent and easyto-mplement approach n the context of 3-D mult-scale modellng, s presented. The modellng approach s based on computatonal homogenzaton technque and dfferental form of vscoelastcty proposed recently for modellng the response of sotropc and transversely sotropc materals. The fnte element analyses wll be conducted at dfferent length scales n a general purpose fnte element code, ABAQUS. Although the modellng approach s general, t s specfcally appled to predct the tme-dependent behavour of a specal structural wood composte product, parallel strand lumber (SL) beam, subjected to a three-pont bendng load. The effect of mcrostructural parameters (e.g. wood strand sze and orentaton dstrbuton, as well as resn area coverage, volume fracton and relaxaton modulus) on the creep response of SL wll be demonstrated. INTRODUCTION Composte materals wth varous mcrostructures and propertes are ncreasngly beng developed to meet the needs of desgn engneers n varous ndustres. Havng predctve tools whch are capable of relatng the materal propertes at dfferent scales to structural response of compostes under complex loadng condtons would enable us to accelerate the development of novel composte materals for structural applcatons. Analytcal mcromechancs equatons are currently avalable and used wth great success for predctng the effectve elastc propertes of the sold undrectonal crcular fbre compostes as well as rectangular strand or veneer-based wood compostes. However, ther applcaton n predctng the effectve vscoelastc propertes of composte or cases where one phase could become softer (e.g. due to temperature ncrease) than the other phase has not been nvestgated fully. Woven compostes and strand-based compostes where the fbre (strand) has a rectangular cross-secton are examples of such compostes. Most of these composte materals show drectonally-dependent (orthotropc) propertes whch make ther structural analyss complex. To predct the structural TU Dresden, Insttute of Textle Machnery and Hgh erformance Materal Technology, Dresden, Germany SNC-Lavaln, Toronto, ON, Canada Department of Cvl Engneerng, The Unversty of Brtsh Columba, Vancouver, BC, Canada

2 response of compostes exhbtng tme-dependent characterstcs specal treatment of consttutve models s requred. Several mult-scale models have been developed to take nto account the herarchcal nature of compostes and solvng problems arsng from the complex mcrostructure of new compostes. The concept of mult-scale modellng, ts orgns, recent developments and current challenges n ths feld have been dscussed n several research papers [], []. Currently, mult-scale modellng s a well-establshed approach n lterature for analysng composte materals. However, a mult-scale approach stll needs to be effcent enough to be employed for analysng the behavour of large composte structures n practce. Effcency becomes a partcularly mportant ssue n those applcatons where tme-dependent (vscoelastc) response of the structure s of nterest such as process modellng or predctng the long-term behavour (creep) of structural elements made of new compostes. In addton to the effcency requrement for a mult-scale modellng tool, accuracy s also mportant for new compostes wth drectonally dependent (orthotropc) propertes that are beng developed for aerospace and constructon applcaton. To address the above requrements, an effcent mult-scale modellng strategy whch relates mcrostructural parameters (e.g. strand sze and orentaton) to the tme-dependent response (e.g. creep) of composte structures s presented below. MODELLING FRAMEWORK The mult-scale approach presented n ths paper s based on other well-establshed approaches and models and nvolves analyses at two major resoluton levels; namely, mcro-level and macrolevel as depcted n Fgure for modellng the response of parallel strand lumber (SL) to a constant load. It should be noted that throughout ths paper, the macro-scale s the scale at whch we are nterested n determnng the structural response. Scales below the macro-scale such as the mesoand mcro-scale may be dentfed for dfferent compostes. Lower scales, n general, are referred to as mcro-scale unless stated otherwse. Idealzed materal mcro-structure Mcro-mechancal Smulatons Macro-mechancal Smulatons Fgure : Schematc of the mult-scale modellng approach for smulatng the structural behavour of SL

3 3 MICRO-SCALE ANALYSIS Sardar Malek, Thomas Gereke, Nma Zobery, and Reza Vazr As a frst step n mult-scale modellng of vscoelastc compostes, the effectve vscoelastc propertes of compostes through a mcro-scale analyss s requred. The vscoelastc propertes of the materal unt cell can be estmated analytcally usng the correspondence prncple employed by most researchers, e.g. see [3] [5]. Usng the correspondence prncple, the lnear vscoelastc heterogeneous problem n the real tme doman s frst transformed to a vrtual lnear elastc problem n Laplace space. The latter s then solved usng lnear mcromechancal schemes. Fnally, the effectve vscoelastc propertes are obtaned usng numercal nverson to tme doman. As an alternatve to the analytcal approach, the effectve vscoelastc behavour of a unt cell of materal can be obtaned numercally usng fnte element method. Assumng a perodc mcrostructure for the materals, a unt cell of the materal can be dentfed. Ths unt cell s subjected to sx elementary loads whch are held constant durng tme. Components of the effectve stffness tensor are determned durng tme usng the volume average stress and stran tensors at each tme step. Therefore, the stffness tensor components are functons of tme and each component can be expressed as a set of contnuous mathematcal functons, known as rony seres. 4 MACRO-SCALE ANALYSIS A dfferental form (DF) of vscoelastcty has recently been presented by Zobery and others [6], [7] as an effcent approach for modellng the response of polymer composte materals. The formulatons n [6] were developed for transversely sotropc materals and extended by Malek [8] for modellng vscoelastc behavour of orthotropc compostes. The 3-D formulaton for orthotropc vscoelastc materals presented n [8] can be easly mplemented n vrtually any exstng code. To llustrate ths, such formulaton s brefly revewed here. A user materal subroutne (UMAT) has been developed and mplemented n the ABAQUS software. The mplementaton of the formulaton has been verfed and presented here through a benchmark example. 4. Dfferental Approach to Modellng Generally Orthotropc Materals In a general 3-D state of stress, the consttutve equaton relatng the stress σ j and the stran ε j for a lnear elastc materal can be wrtten as follows: C j jkl kl () where C jkl are components of the materal stffness tensor. Usng matrx notaton, the stffness tensor can also be wrtten n a 6 6 matrx form, denoted by the symbol C wth the double underscore ndcatng matrx quanttes and sngle underscore denotng vectors. For an sotropc materal, the stffness matrx can be expressed n terms of two ndependent constants (modul), G and K, representng the shear and bulk behavour of the materal. Therefore, the stress-stran relatonshp can also be wrtten as follows: 3 ( 33) 33 ( ) ( 33 ) 33 G K 3 3 () In other words, the total stress vector for an sotropc materal can be decomposed nto two vectors: G K (3) where the vectors σ G and σ K, epresent the shear and bulk components of the total stress, respectvely. In order to descrbe the behavour of an elastc, orthotropc materal, nne constants are needed and the stress-stran relatonshp can be expressed n terms of these nne constants (stffness 3

4 matrx components) as follows: C C C C C C C C C (4) C C 33 C C C C66 C C3 C3 where the stress tensor has been decomposed nto nne tensors each correspondng to a stffness matrx component or a materal property, p:... C C C3 p p 9 (5) For smplcty, the above equaton may also be wrtten n vector form as: C C C C44 C55 C C C3 C3 (6) For a vscoelastc orthotropc materal, smlar to the sotropc or transversely sotropc case n [6], a dfferental equaton s employed to relate the assocated stresses and strans as follows: ; C, C,..., C N u (7) 3 In dervng the above equaton, for each component of the stffness matrx (e.g. C ) a generalzed Maxwell model has been employed. For each element of the generalzed Maxwell model a governng dfferental equaton can be wrtten as follows: ; C, C,..., C (8) 3 In the above equaton, the sprng stffness and tme constant of Maxwell element assocated wth component of the stffness matrx are denoted by and (τ ), respectvely. The unrelaxed values of each component of stffness matrx, denoted by superscrpt u, s obtaned by summng over all Maxwell elements. Therefore, the unrelaxed value of component s obtaned as: N u ; C, C,..., C 3 where N s the number of Maxwell elements. The overall governng dfferental equaton s obtaned by summng over all nne stress vectors, (9) 4

5 therefore the overall governng dfferental equaton can be wrtten as: 4. Implementaton N u C () Rewrtng Equaton (8) at tme steps n and n + for element of the generalzed Maxwell model, ncludng the free stran ε pf (e.g. stran generated due to cure shrnkage or thermal expanson) leads to: n n n n ( f ) n () n n n n ( ) f n As t s descrbed Zobery et al. (6), usng the fnte dfference method wth a central dfference scheme and rearrangng, the stresses n each Maxwell element can be calculated as: t n n n ( n n ) ( n n ) f f t t () n n The total stress vector at tme ncrement (n + ) can be calculated by summng over all Maxwell elements ( =,, 3,, N) and materal propertes, as follows: n N n ; C, C,..., C 3 (3) Detals of the fnte element formulaton and soluton convergence condtons could be found n prevous publcatons of the authors (e.g. [6], [7]). 4.3 Verfcaton example To verfy the mplementaton of the DF formulatons, numercal results are compared wth avalable analytcal solutons for a smple example here. The example s taken from [9], n whch an orthotropc rectangular rod s subjected to an appled unaxal stress n the -drecton prescrbed as: t t, t (4) The vscoelastc propertes of the orthotropc materal are gven n Table. roperty C g j (Ga) C r j (Ga) τ k (s) C C C C C C C C C Table : Vscoelastc propertes of the orthotropc materal taken from [9]. 5

6 The varatons of non-zero stran components are plotted n Fgure. In ths fgure, the analytcal solutons have been obtaned usng Laplace transform and shown wth sold lnes. The numercal predctons, depcted wth symbols, show very good agreement wth analytcal exact results n all three drectons. Ths valdates the capabltes of dfferental form of vscoelastcty n modellng orthotropc vscoelastc materals. () t -3 t (sec) () t 3 Fgure : An orthotropc rod subjected to an appled unaxal stress n the longtudnal drecton, and free of stress n other drectons [9]. Comparson of numercal predctons (usng FE) and analytcal soluton for strans n the longtudnal and two transverse drectons. 5 ALICATION TO CREE MODELLING OF SL There are many studes throughout the lterature showng the tme-dependent behavour of both phenol formaldehyde (F) resn and wood, especally at dfferent mosture contents. One of the man applcatons of mult-scale modellng framework s predctng the response of strand-based wood composte structures. Strand-based compostes are structural materals manufactured through combnng orthotropc wood strands wth small amount of adhesve (~5% by volume). The adhesve (e.g. a thermoset resn) glues the strands together. There are advantages assocated wth strand-based wood compostes (e.g SL) over sold wood []. The structural members made from these compostes are often consdered to be stronger, more relable and more dmensonally stable than sold sawn lumber. The use of such structural composte lumber products as constructon materals demands certan requrements of ther mechancal propertes be known such as stffness and strength as well as some servceablty requrement such as the allowable amount of creep due to mosture, heat or long duraton of loads (e.g. snow loads). A numercal stochastc mult-scale modellng approach has been presented n [] for predctng the elastc behavour of SL beams. Havng developed a dfferental approach for orthotropc vscoelastc materals, the tme-dependent behavour of SL beams subjected to a constant load can be smulated. In order to predct the creep of a SL beam, a unt cell representng the mcrostructure of SL whch s consstng of resn and wood s employed for mcro-mechancal smulatons. It should be mentoned that both consttutng phases (wood and resn) n SL exhbt vscoelastc behavour. However, for smplcty reasons, only the resn phase (F) s consdered to be vscoelastc and ts behavour s descrbed wth the vscoelastc parameters used n []. The three-dmensonal orthotropc materal constants for wood are average values for dfferent pne speces taken from [3] representng the actual SL products. In order to predct the creep response of the beams, the components of the unt cell effectve relaxaton matrx, C j, are requred. These can be estmated ether analytcally usng the correspondence prncple or numercally as descrbed n [8]. Here, the numercal procedure presented n [8] was employed to derve components of the unt cell relaxaton matrx. These components are requred to characterze the vscoelastc behavour of the materal on the mcroscale. Varaton of these components for fully bonded strands durng tme s shown n Fgure 3. 6

7 (a) (b) (c) Fgure 3: Varaton of the effectve relaxaton matrx components C j versus tme obtaned from numercal analyss of SL unt cell. Symbols denote numercal results and lnes are the assocated ftted curves. In Fgure 3, numercal results are shown wth symbols whle the ftted curves are plotted wth lnes. The dfferental approach for modellng the macroscopc behavour of vscoelastc structures requres the components of relaxaton matrx of the orthotropc materal be defned by rony seres expansons followng the formulaton descrbed n [8]. The unrelaxed and relaxed values and the rony seres parameters assocated wth each component are obtaned from a curve fttng procedure on the numercal results. These parameters are gven n Table and Table 3 for reference. Component C j u (Ga) C j r (Ga) C C C C C C C C C Table : Unrelaxed and relaxed values of components of the relaxaton matrx for a SL unt cell. 7

8 w w w 33 w 44 w 55 w 66 w w 3 w 3 τ (mn) e e E E E E E E E E E E+ Table 3: rony seres parameters obtaned from curve fttng of the numercal results. Havng characterzed the vscoelastc behavour of the materal unt cell numercally (mcromechancal step), the creep behavour of the beam can now be determned from FE analyss of a beam consstng of several strands dstrbuted wthn t (macro-mechancal step). SL beams under threepont bendng are analysed for ths purpose. Each beam conssts of 4 unt cells (strands wth resn) arranged n a regular structure. The beam s assumed to have a square cross-secton of 39 mm 39 mm and a span of 38 mm between the supports as descrbed n [4]. A constant dstrbuted load wth the total magntude of.5 kn s appled at the centre of the beam on the top surface. The average dsplacement at the m centre nodes at the bottom surface of the beam s defned as follows: m (5) m and recorded durng tme whle the appled loads s mantaned. The average dsplacement, δ, s used to construct deflecton tme curves. To evaluate the effect of strand orentaton and resn area coverage of strands (.e. vods), three cases are consdered. In the frst case, all strands are assumed to be fully bonded and algned wth the beam length. In the second case, random strand orentaton s consdered wthn each beam and s ntroduced by defnng the materal orentaton wthn each unt cell makng up the beam accordngly. For SL beam consstng of random strands, strands are randomly dstrbuted accordng to the orentaton dstrbuton descrbed n [] whch s also gven n fgure below: Fgure 4: Strand orentaton n the SL beam consstng of random strands accordng to []. 8

9 In order to nvestgate the effect of partal resn coverage of strands, vods are ncorporated n the resn phase (% by volume n the resn phase to represent 8% strand coverage by resn) accordng to [] and consdered as the thrd case. A convergence study showed that a mesh sze of 6 provdes converged values for the node dsplacement on the bottom face of the beam n both cases. The results for the three SL beams consdered are presented n Fgure 5. Fgure 5: Creep curve for SL beams consstng of algned strands and random strands. Note that the average deflecton has been normalzed by the ntal elastc deflecton of the beam consstng of algned strands ( o strands) n Fgure 5. As t s expected, the SL beam wth fully bonded and algned strands experences lower deflectons durng tme due to more contrbuton of elastc stff strands. Fgure 5 demonstrates the capablty of the mult-scale framework n predctng vscoelastc behavour of strand-based composte structures consderng several varables ncludng consttuents propertes, mcrostructural features and geometrcal parameters. 6 CONCLUSIONS An effcent modellng strategy s presented for the analyss of vscoelastc composte structures. The presented approach enables engneers to smulate the vscoelastc behavour of orthotropc compostes that consst of at least one vscoelastc phase. Therefore, the tme-dependent response of large complex structures to varous loads at the macro-scale (e.g. meter level) can be predcted usng the nput parameters obtaned from the output results of the mcromechancal analyses at the mcro-scale (e.g. mm level). The modellng approach s specfcally appled to predct the tme-dependent behavour of a specal structural wood product, parallel strand lumber beam, subjected to a three-pont bendng load. As t s demonstrated, mcrostructural parameters (e.g. wood strand sze and orentaton dstrbuton, as well as resn volume fracton and relaxaton modulus) and ther effect on the creep response of SL can be consdered usng the developed approach. Results suggest that characterzng the correct vscoelastc behavour of consttuents (e.g. wood and resn) would enable future researchers to optmze the mcrostructure of structural compostes for specfc load scenaros. 7 ACKNOWLEDGEMENTS The authors would lke to thank the Natural Scences and Engneerng Research Councl of Canada (NSERC) for ther fnancal support of ths research. The authors are also grateful to the nsghts provded by rofessors Anoush oursartp and Fernand Ellyn from UBC, rofessor Carole Nadot-Marn from Insttut prme (France) durng varous research dscussons held at UBC. 9

10 REFERENCES []. Kanouté, D.. Boso, J. L. Chaboche, and B. A. Schrefler, Multscale Methods for Compostes: A Revew, Arch Comput Methods Eng, vol. 6, pp. 3 75, 9. [] M. G. D. Geers, V. G. Kouznetsova, and W. A. M. Brekelmans, Mult-scale computatonal homogenzaton: Trends and challenges, J. Comput. Appl. Math., vol. 34, no. 7, pp. 75 8,. [3] R. Chrstensen, Theory of vscoelastcty: an ntroducton. New York: Academc ress Inc., 98. [4] N. Zobery, Vscoelastc consttutve models for evaluaton of resdual stresses n thermoset compostes durng cure. 6. [5] A. Matzenmller and S. Gerlach, Mcromechancal modelng of vscoelastc compostes wth complant fber-matrx bondng, Comput. Mater. Sc., vol. 9, no. 3, pp. 83 3, 4. [6] N. Zobery, S. Malek, R. Vazr, and A. oursartp, A dfferental approach to fnte element modellng of sotropc and transversely sotropc vscoelastc materals, Mech. Mater., vol. 97, pp. 76 9, 6. [7] N. Zobery, R. Vazr, and A. oursartp, Dfferental mplementaton of the vscoelastc response of a curng thermoset matrx for compostes processng, J. Eng. Mater. Technol. Asme, vol. 8, no., pp. 9 95, Jan. 6. [8] S. Malekmohammad, Effcent mult-scale modellng of vscoelastc compostes wth dfferent mcrostructures, The Unversty of Brtsh Columba, 4. [9] H. oon and M. F. Ahmad, A materal pont tme ntegraton procedure for ansotropc, thermo rheologcally smple, vscoelastc solds, Comput. Mech., vol., no. 3, pp. 36 4, Apr [] K. J. Frdley, Wood and wood-based materals: current status and future of a structural materal, J. Mater. Cv. Eng., vol. 4, no., pp. 9 96,. [] T. Gereke et al., Multscale Stochastc Modelng of the Elastc ropertes of Strand-Based Wood Compostes, J. Eng. Mech., pp ,. [] S. Malekmohammad, B. Tressou, C. Nadot-Martn, F. Ellyn, and R. Vazr, Analytcal mcromechancs equatons for elastc and vscoelastc propertes of strand-based compostes, J. Compos. Mater., vol. 48, no. 5, pp , 4. [3] Forest roducts Laboratory - USDA, Wood Handbook: Wood as an Engneerng Materal, vol. General Te.. [4] S. Arwade,. Clouston, and R. Wnans, Measurement and Computatonal Modelng of the Mechancal ropertes of arallel Strand Lumber, J. Eng. Mech., vol. 35, no. 9, pp , 4.