Nonlinear Viscoelastic Viscoplastic Material Model Including Stiffness Degradation for Hemp/Lignin Composites

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1 Nonlinear Viscoelastic Viscoastic Material Model Including Stiffness Degradation for Hep/Lignin Coposites Erik Marklund, Johannes Eitzenberger, Janis Varna To cite this version: Erik Marklund, Johannes Eitzenberger, Janis Varna. Nonlinear Viscoelastic Viscoastic Material Model Including Stiffness Degradation for Hep/Lignin Coposites. Coposites Science and Technology, Elsevier, 29, 68 (9), pp.256. <.6/j.copscitech.28.3.>. <hal-59497> HAL Id: hal Subitted on 22 May 2 HAL is a ulti-disciinary open access archive for the deposit and disseination of scientific research docuents, whether they are published or not. The docuents ay coe fro teaching and research institutions in France or abroad, or fro public or private research centers. L archive ouverte uridisciinaire HAL, est destinée au dépôt et à la diffusion de docuents scientifiques de niveau recherche, publiés ou non, éanant des établisseents d enseigneent et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Accepted Manuscript Nonlinear Viscoelastic Viscoastic Material Model Including Stiffness Degradation for Hep/Lignin Coposites Erik Marklund, Johannes Eitzenberger, Janis Varna PII: S (8)4- DOI:.6/j.copscitech Reference: CSTE 46 To appear in: Coposites Science and Technology Received Date: 2 June 27 Revised Date: 7 March 28 Accepted Date: 7 March 28 Please cite this article as: Marklund, E., Eitzenberger, J., Varna, J., Nonlinear Viscoelastic Viscoastic Material Model Including Stiffness Degradation for Hep/Lignin Coposites, Coposites Science and Technology (28), doi:.6/j.copscitech This is a PDF file of an unedited anuscript that has been accepted for publication. As a service to our custoers we are providing this early version of the anuscript. The anuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final for. Please note that during the production process errors ay be discovered which could affect the content, and all legal disclaiers that apy to the journal pertain.

3 Nonlinear Viscoelastic Viscoastic Material Model Including Stiffness Degradation for Hep/Lignin Coposites Erik Marklund (), Johannes Eitzenberger (2), Janis Varna* (3) (), (2), (3) Div of Polyer Engineering, Luleå University of Technology, SE-97 87, Luleå, Sweden () Swerea SICOMP AB, Box 4, SE-43 22, Mölndal, Sweden () Phone: , Fax: , E-ail address: (2) Phone: , Fax: , E-ail address: (3) Phone: , Fax: , E-ail address: Abstract In repeating tensile tests with increasing axiu strain for every loading cycle the hep/lignin coposites clearly showed a nonlinear behavior and hysteresis loops in loading and unloading. The exanation for this behavior is the inherent viscoelastic nature for this type of aterial, but also noticeable stiffness degradation with increasing strain level. Creep tests perfored at different stress levels revealed a nonlinear viscoelastic response and after recovery viscoastic strain was detected for high stress levels. It is deonstrated that Schapery s odel is suitable to odel nonlinear viscoelasticity whereas viscoastic strain ay be described by a nonlinear functional presented by Zapas and Crissan. In a creep test this functional leads to a power law with respect to tie and stress. In order to include stiffness reduction due to daage Schapery s odel has been odified by incorporating a axiu strain-state dependent function reflecting the elastic odulus reduction with increasing strain easured in tensile tests. A generalized increental odel of the constitutive equation for viscoelastic case has been used to validate the developed aterial odel in a linear stress controlled loading and unloading rap. The odel successfully

4 describes the ain features for the investigated aterial and shows good agreeent with test data within the considered stress range. Keywords Polyer-atrix coposites; Creep; Non-linear behavior; Modeling; Viscoelasticity. Introduction Plant fibers as reinforceent in coposites have received uch attention over the recent years [-4]. Natural fibers, such as hep, flax, jute and sisal are renewable and biodegradable cellulosic aterials which offer relatively good specific echanical properties. Drawbacks are high oisture absorption and poor adhesion with non-polar polyer atrices. Natural fiber coposites with theroastic atrix are extensively used these days in the autootive industry where the fiber acts ainly as filler aterial in non-structural interior panels. Natural fiber coposites for structural purposes do exist, but then usually with oil-based synthetic theroset atrices. The future goal will be to develop environental friendly high perforance coposites which are recyclable and coe fro renewable resources. A coete biodegradable syste ay be obtained if the atrix aterial also coes fro a renewable resource. Exaes of such atrix aterials are lignophenolics, starch and polylactic acid (PLA). Soe of these systes are showing encouraging results. For exae, Oksan et al. [5] and Bodros et al. [6] have reported that flax fiber coposites with PLA atrix outperfor both flax/polypropylene and glass/polypropylene coposites in ters of stiffness and strength. Lignin is a coex non-crystalline aroatic acroolecule readily extracted in vast quantities fro the paper industry. Studies have suggested that it is possible to reace part of phenol by lignin in phenolic theroset atrices without loss of echanical properties [7]. 2

5 However, there sees to be little work done on echanical perforance of the hep/lignin coposite syste in particular. A very iportant feature of natural fiber coposites is that the echanical properties of both fibers and the polyer atrix are tie dependent. Therefore, natural fiber coposites experience coex tie dependent stress-strain behavior with loading rate effects and hysteresis loops. This behavior is due to viscoelastic effects of both constituents and ay also include icro daage evolution resulting in stiffness degradation and developent of irreversible viscoastic strains. Mechanical properties of natural fiber coposites have been studied previously by several authors. For exae, coposites ade fro wood fibers and theroset atrices have been studied [8,9]. Flax/polypropylene coposites were characterized in ters of viscoelastic behavior in []. Tensile loading unloading tests to different axia perfored on the hep/lignin coposites showed large increase in size of the hysteresis loops, peranent strains after unloading and stiffness reduction. Hence, the coposite has to be described as a nonlinear viscoelastic viscoastic aterial that also experience stiffness degradation. A general therodynaically consistent theory of nonlinear viscoelastic and nonlinear viscoastic aterials was developed by Schapery [] and it has been used in siulations by several authors [2,3]. In an earlier study [4] Lou and Schapery presented a constitutive equation for nonlinear viscoelasticity in uniaxial loading. The odel contains three stress dependent functions which characterize the nonlinearity. Actually these functions also depend on teperature and huidity [], but under fixed environental conditions they are functions of stress only [4]. A ethodology to deterine the nonlinearity paraeters for aterials which obey power law tie dependence was described. However, not all aterials obey the linear viscoelastic power law and a better fit to experiental data is often achieved if the viscoelastic creep coiance in for of Prony series is used [,5]. In the range of 3

6 linear viscoelasticity and in fixed environental conditions the stress dependent functions are equal to and the data reduction schee is significantly siified. The optial set of experients needed to deterine the stress dependent functions in the aterial odel and developent of reliable ethodology for data reduction is still a debatable issue. The developent of viscoastic strains will be described by a nonlinear functional presented by Zapas and Crissan [6]. The objective of the presented paper is to identify the aterial odel for hep/lignin coposites which accounts for the observed nonlinear viscoelasticity, viscoasticity and daage. The experiental part includes (i) deterination of the daage related stiffness degradation with strain in tensile loading unloading tests and incorporating the obtained axiu strain dependent degradation function in the aterial odel., (ii) analysis of the viscoelastic and viscoastic properties of this aterial in creep tests at several stress levels and easuring of the peranent strains after strain recovery. The developed odel will be validated in a test with a linear stress controlled loading and unloading rap using an increental for of the constitutive equation. This paper address the odeling of acroscale aterial properties and thus details regarding aterial icroarchitecture and orphology are not pertinent to the outcoe in this work. It will be shown that the aterial can be characterized using only a few speciens and that the stress dependent nonlinearity functions ay be expressed via sie polynoial functions. 2. Theory 2.. Constitutive odel 4

7 Lou and Schapery [4] presented a general nonlinear constitutive equation of viscoelasticity in the case of uniaxial loading. The sae constitutive equation with an additional ter for viscoastic strain accuulation ε ( t,σ ) was used previously by Marklund et al. [] where flax/polypropylene coposites were characterized using different fors of the viscoelastic creep coiance. No stiffness degradation was detected in [] and thus nonlinear viscoelasticity and viscoasticity were the echaniss responsible for the observed behavior. In the present study however, tensile tests revealed that the hep/lignin speciens do indeed experience stiffness degradation. The constitutive equation in this case has therefore been odified by incorporation of a axiu strain-state dependent function d ( ε ax ) which reflects the elastic odulus reduction with increasing strain. ( g σ ) t d ( ) 2 ε = d ε ( ) ( ) ax ε + g S ψ ψ dτ + ε t, σ dτ () In Eq. () integration is over reduced tie according to, t τ dt dt ψ = and ψ = a a σ σ (2) ε represents the initial strain which ay be nonlinear with respect to stress. S(ψ ) is the transient coponent of the linear viscoelastic creep coiance. g and g 2 are stress dependent aterial properties. a σ is the shift factor, which in fixed conditions is a function of stress only. For sufficiently sall stresses g g = a, and thus Eq. () turns into the = 2 σ = strain-stress relationship for linear viscoelastic viscoastic aterials. In the therodynaic analysis presented by Schapery [] it was shown that the viscoelastic creep coiance does not depend on the apied stress level and it ay therefore be deterined using loads in the linear region. Viscoelastic creep coiance in the for of Prony series was obtained, 5

8 S ( ψ ) = C exp τ ACCEPTED MANUSCRIPT ψ (3) C are constants and τ are called retardation ties. The retardation ties are chosen arbitrary, but the highest τ should at least cover the tie for the conducted creep test. A good approxiation to experiental data ay be achieved if the retardation ties are spread uniforly over the logarithic tie scale, typically with a factor of ten between the. In a creep test the stress is constant until soe tie instant t whereby the stress is reoved and the recovery period begins according to = σ [ H t) H ( t t )] σ where H (t) is ( the Heaviside step function. The expression () ay therefore be divided into creep strain and recovery strain in a creep test. Together with the creep coiance fro Eq. (3) we obtain the following for of creep strain and recovery strain respectively: ( ) t ε + ( ) c = d ε ax ε + gg 2σC exp ε t, σ (4) aστ ( ) t = + ( ) t t ε r d ε ax g 2σC exp exp ε t, σ (5) aστ τ 2.2. Viscoastic strain Assuing that the viscoastic strain ay be described via a nonlinear functional presented by Zapas and Crissan [6] we have, t M ε = C σ ( τ ) dτ (6) C, M and are constants and ust be deterined experientally. The following discussion is a short suary of the procedure needed for the paraeter identification which is described in detail in [,5]. First the tie dependence of viscoastic strains is deterined by perforing creep tests at a fixed stress level so that integration of Eq. (6) is trivial. After 6

9 strain recovery the reaining irreversible strain corresponding to the loading period is easured. Several creep tests with different lengths are perfored and the developed viscoastic strains are sued. The viscoastic strain after k steps of creep loading at the sae stress level σ will be, k ( t + t +... t ) ε = (7) M C σ 2 + k The developent of viscoastic strains at fixed stress should thus follow a power law in tie with coefficient B M = C σ and constant which are deterined as the best fit in logarithic axes. Furtherore, only one specien is necessary to obtain the tie dependence of viscoastic strains at a certain fixed stress level. The stress dependence of viscoastic strains has to be obtained perforing creep tests of the sae length at several stress levels. In result B is obtained as a function of stress and the best fit in logarithic axes yields the required constants C and M Increental for of the constitutive equation In structural analysis with nonunifor and coex stress state the aterial odel has to be ieented in FE codes which require an increental for of Eq. (). Substitution of Eq. (3) in () and integration gives, ( t ) d( ε ) ε ( σ ) + g ( σ ) g ( σ ) σ C g ( σ ) ε ( ψ ) + ε ( σ, t) ε = ax 2 (8) where ε ψ ψ ψ [ g ( σ ) σ ] τ d 2 = C dψ e dψ (9) 7

10 The integral in Eq. (9) ay be calculated in tie instant t k using the previously deterined value at tie t k where t k = tk + t. Equation (2) gives the relation between the tie increent and ψ according to, ψ = a t σ and ψ k = ψ + ψ () + k The recursive expression for Eq. (9) becoes, ε ψ ψ = () τ τ ( tk ) e ε ( t k ) + C e R( t k ) τ In () R ( t ) d = [ g ( σ ) σ ] dσ σ a (2) dt 2 k k σ σ k tk dσ The routine for siulation of ε ( t) is as follows; (i) choose the tie increent t ; (ii) for the tie instant t k find ψ, R ( ) and calculate all ε ( t ) using ( t ) t k k ε fro the previous step (they are zero in the -step); (iii) now use Eq. (8) to find ε ( t) and then repeat the steps. k 3. Experiental Lignin is a very brittle aterial at RT and therefore the coposites were anufactured containing a large fraction of asticizer. In addition to this flae retardant was also added. Weight fractions were hep fiber 3%, flae retardant 2%, asticizer 3% and finally the atrix aterial lignin 2%. Further detail regarding constituent orphology is irrelevant for the odel developent and thus oitted in the following discussion. The reduction in elastic odulus was easured using an Instron 44 tensile testing achine with a 5 kn load cell. Strain was easured by an Instron 263- series extensoeter with 5 gauge length. Cross-head speed was set to 5/in. All tests were perfored at RT and with relative huidity of 25 35%. The stiffness reduction was 8

11 easured repeatedly apying a load-unload rap with increasing axiu to introduce daage followed by low stress load-unload rap (after each cycle) to easure the elastic odulus. Due to viscoelastic effects the speciens after introducing daage had to recover for soe short period of tie before the elastic odulus was easured. The speciens were not reoved fro the grips during the recovery. The axiu strain in the first cycle was.2% with increent of approxiately.% in each following cycle until failure. In the nondaaging stiffness deterination raps axiu strain was.2%. A few exaes of stress-strain curves are shown in Figure. In the data reduction the elastic odulus was estiated in the stress region corresponding to.5 and.5% in strain for the undaaged specien. The saing rate was 2 points per second which gave about 25 points for establishing the odulus. The odulus was calculated for the loading part as well as for the unloading part of the curve and then taken as the average. The creep tests were perfored by hanging of dead-weights and easuring strain with standard extensoeters. One specien was first subjected to, 2 and 3 in of creep loading at 9 MPa with following recovery in order to estiate the viscoastic tie dependence at a fixed stress level. The viscoelastic properties were later deterined fro the sae specien by subjecting it to hour of creep loading at stress levels 9, 6 and 3 MPa respectively. Another specien was subjected to, 2 and 3 in of creep loading at 6 MPa for estiation of the viscoastic stress dependence. A sall degradation in stiffness was detected afterwards which indicates that viscoastic strains due to daage ight have developed. Unfortunately we were unable to easure these very sall viscoastic strains with sufficient accuracy. The liit for viscoastic strain developent was therefore set to 6 MPa. The sae specien was also tested in creep at MPa (, 2 and 3 in). A third 9

12 specien was tested in creep at MPa and experienced creep rupture after less than 8 in of loading. The fracture surface revealed a dry region and any voids. 4. Results and discussion 4.. Stiffness degradation The initial odulus E was deterined fro the first loading cycle corresponding to.2% in strain. The initial Young s odulus was in the range GPa. Tensile strength was 4 6 MPa. Figure 2 shows the reduced odulus noralized with respect to the initial odulus. No stiffness degradation could be seen for strain values lower than.3% and consequently this strain level was therefore set as liit for stiffness reduction. At a strain level of.9% the stiffness reduction was roughly 7-8%. The regression line in Figure 2 deterines the axiu strain-state dependent function required in Eq. (), never loaded above.3 % d( ε ax ) = (3) otherwise.6ε ax +.33 Stiffness degradation is the anifestation of daage evolution. In Figure 3 in situ icrographs of poorly ipregnated regions in one of the tested hep/lignin speciens are shown. The load is in horizontal direction. The icrographs give an understanding of the condition of the edge surface. There are any sall randoly oriented surface cracks (a) with an opening alost independent on the load. Their contribution to the stiffness reduction is ost likely negligible. The large crack (b) opens ore with increasing load which eans that it is ore a volue crack than a surface crack. This indicates that the large crack is one of the contributors to the stiffness reduction Viscoastic strain

13 The viscoastic strains in tension were easured as the reaining recovery strains after 6 ties the loading period in creep. The viscoastic strains accuulated in all steps were sued and otted against tie in log-log scale according to the description in section 2.2. The slope of the trendline was straight which eans that the developent of viscoastic strains at fixed stress indeed follows a power law with high accuracy. However, the power law which is a consequence of assuption Eq. (6) states that the exponent is constant for all stress levels and certainly that was not true in the present case. At MPa the exponent is higher than at 9 MPa. In light of this result the decision was to use an average value of in the forthcoing calculations. Experiental results and odel predictions for viscoastic tie dependence at fixed stress using an averaged value is shown in Figure 4. The corresponding values of B are also shown. The stress dependence for developent of viscoastic strains was obtained by assuing a strain value of.% at 6 MPa in order to have three values of B (reeber that we were unable to easure the very sall viscoastic strains at this stress level and at MPa the specien experienced creep rupture). Consequently, values M = 2. 7 and C =.65 3 (for tie in s, stress in MPa and strain in %) were deterined Nonlinear viscoelastic odel with Prony series We suggest that each specien in the viscoelastic analysis is analyzed separately which is a preferable strategy since the data reduction procedure otherwise easily becoes both tedious and ipractical and ay contain soe artificial trends when averages are used. After recovery in the 9 MPa creep test a sall irreversible strain was detected. Thus viscoastic strains are developing during the test and the creep strain is a su of viscoelastic and viscoastic strains. In order to obtain a pure nonlinear viscoelastic response the viscoastic strains that develop ust therefore be subtracted fro experiental data.

14 Following the work in [5] the developent of viscoastic strains in the current creep test is expressed by, / ( ) / ( ) ( ) = k t ( ) + k t ( ) k ε t ε ε ε (4) tk tk t k is the length of the creep period in question and ε is the total viscoastic strain k developent in k creep tests. =. 49 was deterined in the previous section. The specien had experienced creep loading three ties at 9 MPa in the viscoastic characterization procedure earlier with a total duration of one hour so ε =.43% was known fro the previous creep tests. After the 9 MPa creep test in the viscoelastic characterization procedure ε =.55% was easured and Eq. (4) could be used calculating the pure nonlinear viscoelastic response for this stress level. No viscoastic strains developed during the 6- and 3 MPa tests. Tensile tests before and after the viscoastic characterization procedure showed that the stiffness of the specien was reduced by approxiately 8% and easureents after 9-, 6- and 3 MPa in the viscoelastic characterization creep tests showed no further stiffness degradation. Thus d ( ε ax ) =. 8 was used in the following calculations. The first step in the deterination of the paraeters in the nonlinear viscoelastic odel is to deterine the paraeters in the linear viscoelastic range, i.e. to deterine τ and C. The retardation ties τ were chosen to be uniforly spread over the logarithic tie scale and with the highest value covering the tie for the creep tests in question. The coefficients were deterined fro experiental recovery data for the 3 MPa creep test with the assuption that we are within the region of linear response (all stress dependent nonlinearity functions are thus equal to ) and the result is shown in Table. This procedure, and also the forthcoing calculations of the stress dependent nonlinearity functions were perfored using the ethod of least squares written in MATLAB code. C 2

15 Next step will be to deterine the stress dependent nonlinearity functions fro recovery and creep data and the ethodology will be the sae for all stress levels: a σ is altered in the recovery strain expression (5) in tie interval t < t < 2t and for every value of a σ we obtain the corresponding value of g 2 directly via ethod of least squares. This procedure is continued until we find the value of a σ which gives a nice visual fit to experiental data. Equation (4) is then used to find ε and g g 2 fro experiental creep data for the sae stress level. When all the nonlinearity paraeters have been deterined for all stress levels they ay be approxiated with sie analytical functions. Figure 5a) shows the initial strain ε as a function of stress and b) the nonlinearity values for the creep tests of 3-, 6- and 9 MPa and their approxiations as function of stress. In order to check whether the approxiations lead to acceptable results the odel is copared to experiental creep and recovery data as shown in Figure 6. Clearly, the accuracy of the odel based on Prony series and polynoial approxiation of the stress dependent functions is good for all stress levels in the considered stress range. The developed odel ay be used to predict the nonlinear viscoelastic behavior for this type of coposite (with siilar properties) for any stress fro -9 MPa. However, we ust ephasize that all the viscoelastic tie dependences have been obtained fro creep tests on one single specien and so far the odel is apied to that particular specien. It is therefore crucial for the characterization that the specien is representative for the analyzed aterial. The specien was singled out on the basis that its elastic properties were interediate in this group of speciens. The accuracy of the developed odel is best understood coparing the siulated creep curves with other speciens not used in the data reduction and it has been checked that the variation of properties between different speciens is larger than the sall deviation shown in Figure 6a). 3

16 4.4. Model validation in linear loading and unloading rap The developed viscoelastic viscoastic aterial odel including stiffness degradation was used to siulate the coposite behavior in a linear loading and unloading schee: constant load rate of. MPa/s up to 9 MPa, unloading with. MPa/s and finally loading with.2 MPa/s up to 3 MPa according to the solid line in Figure 7. The dashed line in Figure 7 shows the shift in tie which is necessary when the viscoastic strains are calculated. According to Eq. (6) the integration is over a continuous function fro tie t =. However, in this case we have no viscoastic strains developing for stresses lower than 6 MPa (between < t < t and t 3 < t < t4 ). The stress as a function of tie (with tie coordinate t ) for the different parts of the loading rap which needs to be integrated is also shown in Figure 7. The increental for of the constitutive equation was prograed using MATLAB code following the routine described in section 2.3. A tie step of. sec was used. Saller steps were not considered since the difference in result fro using a tie step of sec was less than %. The elastic (initial) strain, the viscoelastic- and viscoastic strains are all calculated separately and then added. Finally the total strain due to stiffness degradation is calculated with the requireent that d ( ε ax ) = before ( t) d ( ε ax ) > following Eq. (3). Since ( ) strain state it will increase until 2 ε ax ε passes.3% the first tie and then d is a function of the highest previously known t and then have a constant value of ( ( t 2 )) d ε until next tie it reaches a higher strain value (around t = 225 sec and σ = MPa). Figure 8 shows the result of the siulation. The largest part of the tie dependent strain is clearly nonlinear viscoelastic. The viscoasticity and stiffness reduction do not contribute uch to the total strain except in the end of the loading rap where we have high stresses and actually are very close to rupture for these coposites. 4

17 The odel and its strain response to the linear loading and unloading rap was also validated by coparing it to experiental data for two speciens, see Figure 9. The accuracy of the odel is rearkably good and it captures ost features fro the experiental curves except for a sall discrepancy in the unloading part. The odel also sees to predict the behavior of the coposites rather well even for the very high stresses prior to rupture which is surprising since we are well outside the region for which it was designed. Figure on the other hand illustrates the proble that ight occur if we apy the odel to a specien tested in creep at MPa. The specien experienced secondary creep and ruptured after only 8 inutes. Clearly the creep echaniss are different at this high stress level. 5. Conclusions The analysis of tie dependent properties for the hep/lignin coposites has been liited to tension case. The behavior in copression can certainly be different. Creep tests perfored on the coposites showed that they ay be described as a nonlinear viscoelastic aterial for stresses higher than 3 MPa (for this particular tie of creep loading). For stresses higher than 6 MPa the aterial ay be described as nonlinear viscoelastic and viscoastic. The aterial also showed icro daage evolution which resulted in reduction of elastic odulus for strain levels higher than.3%. The largest part of the tie dependent strain is viscoelastic and this aterial behavior has been odeled using the theory of nonlinear viscoelasticity developed by Schapery. The stress dependent nonlinearity functions in Schapery s expression was successfully described by sie polynoial functions. Prony series was used to describe viscoelastic creep coiance. The viscoastic strain was described by a nonlinear functional originally presented by Zapas and Crissan. This functional contains three constants which ust be deterined fro experients. It was found that one of these constants was in fact not a 5

18 constant, but rather a function of stress. Therefore, this ter was approxiated using an average value in the considered stress range. A axiu strain-state dependent function reflecting the elastic odulus reduction with increasing strain has also been incorporated in the aterial odel. The developed odel has been validated in a linear loading and unloading rap using an increental for of the constitutive equation. The accuracy of the odel is rearkably good within the stress range for which it is designed and captures ost features of the copared experiental curves. For very high stresses when we are well outside the odel stress range (and actually close to rupture) the odel description gets inaccurate. It was shown that the tie dependent properties of the aterial ay be characterized using only a few speciens. It was therefore crucial for the characterization procedure that the speciens were representative for the analyzed aterial. 6. References [] Bledzki A.K, Gassan J., Coposites Reinforced with Cellulose Based Fibres. Progress in Polyer Science 999; 24(2): [2] Mueller D.H., Krobjilowski A., New Discovery in the Properties of Coposites Reinforced with Natural Fibers. Journal of Industrial Textiles 23; 33(2):-29. [3] Sain M., Suhara P., Law S., Bouilloux A., Interface Modification and Mechanical Properties of Natural Fiber-Polyolefin Coposite Products. Journal of Reinforced Plastics and Coposites 25; 24(2):2-3. [4] Eichhorn S.J., Baillie C.A., Zafeiropoulos N., Mwaikabo L.Y., Ansell M.P., Dufresne A., Entwistle K.M., Herrera-Franco P.J., Escailla G.C., Groo L.H., Hughes M., Hill C., Rials T.G. and Wild P.M., Review: Current international research into cellulosic fibres and coposites. Journal of Materials Science 2; 36(9):

19 [5] Oksan K., Skrifvars M., Selin J.-F., Natural fibres as reinforceent in polylactic acid (PLA) coposites. Coposites Science and Technology 23; 63(9): [6] Bodros E., Pillin I., Montrelay N., Baley C., Could biopolyers reinforced by randoly scattered flax fibre be used in structural apications?. Coposites Science and Technology 27; 67(3-4): [7] Paiva J.M.F., Frollini E., Sugarcane Bagasse Reinforced Phenolic and Lignophenolic Coposites. Journal of Apied Polyer Science 22; 83(4): [8] Zadorecki P., Flodin P., Properties of Cellulose-polyester Coposites. Polyer Coposites 986; 7(3):7-75. [9] Hua L., Zadorecki P., Flodin P., Cellulose Fiber Polyester Coposites with Reduced Water Sensitivity ()-cheical Treatent and Mechanical Properties. Polyer Coposites 987; 8(3): [] Marklund E., Varna J., Wallströ L., Nonlinear Viscoelasticity and Viscoasticity of Flax/Polypropylene Coposites. Journal of Engineering Materials and Technology 26; 28(4): [] Schapery R.A., Nonlinear Viscoelastic and Viscoastic Constitutive Equations Based on Therodynaics. Mechanics of Tie-Dependent Materials 997; : [2] Guedes R.M., Marques A.T., Cardon A., Analytical and Experiental Evaluation of Nonlinear Viscoelastic-Viscoastic Coposite Lainates under Creep, Creep-recovery, Relaxation and Rap Loading. Mechanics of Tie-Dependent Materials 998; 2(2):3-28. [3] Nordin L-O., Varna J., Nonlinear Viscoelastic Behavior of Paper Fiber Coposites. Coposites Science and Technology 25; 65(): [4] Lou Y.C., Schapery R.A., Viscoelastic Characterization of a Nonlinear Fiber-reinforced Plastic. Journal of Coposite Materials 97; 5:

20 [5] Nordin L-O., Varna J., Nonlinear Viscoastic and Nonlinear Viscoelastic Material Model for Paper Fiber Coposites in Copression. Coposites Part A: Apied Science and Manufacturing 26; 37(2): [6] Zapas L.J., Crissan J.M., Creep and Recovery Behavior of Ultra-high Molecular Weight Polyethylene in the Region of Sall Uniaxial Deforations. Polyer 984; 25(): Table Captions Table. Coefficients in Prony series. τ (s) C (%/MPa) Figure Captions Stress (MPa) Strain (%).2%.4%.8% Figure. Tensile stress-strain curves. 8

21 .2.98 E x /E E x /E = -.6ε Strain (%) Figure 2. Stiffness reduction with increasing strain for three speciens fro tensile tests and d. ε ax regression line that deterines the function ( ) a) b) Figure 3. An in situ hep/lignin specien. 9

22 .6 =.57 ; B =.26 Viscoastic strain (%) =.49 ; B =.229 =.49 ; B =.785 =.4 ; B = Tie (s) Figure 4. The developent of viscoastic strains at fixed stress levels: Experiental data at 9 MPa (), MPa () creep tests and odel predictions using the averaged value () for both stress levels copared to true odel predictions (solid lines) Initial strain (%) ε =.238σ σ a) Stress (MPa) 2

23 .6 g *g 2 =.8274σ g 2 =.6963σ /a σ = σ b) Stress (MPa) Figure 5. a) Initial strain ε as a function of stress b) stress dependence of the nonlinearity functions. Strain (%) Tie (s) 9 MPa 6 MPa 3 MPa a) b) Strain (%) Figure 6. a) Viscoelastic strain response fro creep tests, odel prediction (solid line) vs. experiental data (dots) b) strain recovery fro creep tests. t-t (s) 2

24 4 2 Stress (MPa) σ =.t' + 6 t 2 t t 4 t 3 σ = -.t' + 2 σ =.2t' - 6 t' Tie (s) Figure 7. Linear loading and unloading raps and the required tie shift for viscoastic strain calculation..2 Strain (%) Nonlinear elastic Nonlinear viscoelastic Nonlinear viscoelastic & viscoastic with stiffness degradation Tie (s) Figure 8. The contribution of nonlinear elastic strain and nonlinear viscoelastic strain copared to total strain (viscoelastic strain, viscoastic strain and stiffness reduction) in the linear loading and unloading rap. 22

25 .2 Strain (%) Model prediction Specien H323-2 Specien H Tie (s) Figure 9. Strain response to the linear loading and unloading rap, odel prediction (solid line) and experiental values for two speciens (dots)..2 Strain (%) Model prediction Experient Tie (s) Figure. Creep strain at MPa and odel prediction. 23