CONSTITUTIVE MODELING OF POLYMERIC MATRIX UNDER MULTI-AXIAL STATIC AND DYNAMIC LOADING

Transcription

1 THE 9 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS CONSTITUTIVE MODELING OF POLYMERIC MATRIX UNDER MULTI-AXIAL STATIC AND DYNAMIC LOADING B. T. Werner and I. M. Daniel Rober R. McCormick School of Engineering and Applied Science Norhwesern Universiy, Evanson, IL, USA Keywords: Polymer marix, muli-axial loading, rae effecs, elasic-plasic behaviour, modelling.. General Inroducion Recen and ongoing research in fiber reinforced polymer composies has shown ha here is a significan rae effec on he resin dominaed siffness and srengh properies of hese maerials [, ]. Fabricaion and esing of hese composies is cosly due o he expense of he maerial iself, and is anisoropic behavior ha requires ime consuming muli-axial esing. In he case of carbon/epoxy composies, he fiber iself shows no rae dependence in is mechanical response. This suggess ha all of he rae effecs on he composie can be raced o he marix. Since he polymer marix is basically isoropic, a much less cosly evaluaion of a composie can be achieved by characerizing he bulk marix a various srain raes. The consiuive and srain rae behavior of epoxies under various loading condiions has been sudied by many researchers [3-3]. Some sudies focus on characerizaion of he resin a various raes [3, 4]. Many invesigaors have focused on viscoelasic analysis and he deerminaion of shif facors used in emperaure-ime superposiion modeling [5-8]. Some have aemped o link he monomer srucure o he bulk properies [9]. Goldberg e al. have published works on boh esing various resins under muliaxial condiions a various raes as well as modeling wih a sae variable approach [-3]. The objecive of his sudy was o characerize a marix resin under muli-axial loading a differen srain raes and develop a general hree-dimensional elaso-viscoplasic model ha incorporaes rae effecs including dilaaional and deviaoric saes of sress. Emphasis was placed on developmen of a relaively simple model ha would no require exensive esing for evaluaion of a given marix.. Maerial Characerizaion. Maerial and Specimen Preparaion The polymer marix invesigaed is a high siffness, high srengh epoxy (35-6) commonly used in composies. I has a highly crosslinked srucure ha provides siffness and srengh bu also reduces is duciliy. The B-saged resin is normally sored in a freezer. The parially cured and frozen resin was chipped ou, weighed, and placed in he mold. A small amoun of aceone was added o reduce he resin viscosiy and faciliae casing of he parially cured resin ino he mold. The emperaure was raised a a rae of C/min up o ºC, and held a ha level for one hour. During his sage he resin viscosiy was low and vacuum was drawn a 76 mm of Hg o remove any air ha may have been rapped and o boil off any aceone ha migh remain in he resin. Subsequenly, he vacuum was removed and air was slowly bled back ino he mold. For large volumes of resin, he amoun of solven is resriced o preven he aceone from igniing as i boils off during he exohermic sage of he cure. The maerial was cas ino closed molds o produce hin (3 mm) plaes for ensile coupons, hick blocks for prismaic compression coupons, and hin-wall cylinders for specimens o be esed under orsion and combinaions of orsion and axial ension or compression. In casing he cylinders, an

2 aluminum ube was used as he mold wih a Teflon rod insered as he core. The geomery and dimensions of he specimens used are shown in Figure. Specimens for uniaxial ensile esing were hin dogbone coupons wih a gage secion of.7 mm x 5.8 mm machined from 3 mm hick shees. Uniaxial compressive ess were conduced on hick prismaic coupons.7 mm long wih a cross secion of 7.6 mm x 8.89 mm. I was imporan o mainain an aspec raio of around.5 because lower aspec raios would lead o end effecs producing a muli-axial sress sae ha would resul in a higher apparen modulus and criical sress. A higher aspec raio would cause buckling of he specimen and a decrease in he criical sress. Aspec raios from.5 o.7 were esed and i was found ha low aspec raios yielded siffness values approaching he plane srain siffness value (C ) and he higher aspec raios yielded he Young s modulus E of he maerial. An aspec raio of.5 was chosen for he deerminaion of Young s modulus E (Figure ). Tess under pure shear and combinaions of shear and normal ensile or compressive sress were conduced on hin-wall cylindrical specimens. Since he resin used has a shear srengh higher han ha of available bonding epoxies, he cylindrical specimens were designed and machined wih enlarged ends and filles as shown in Fig.. The cylinders were machined using a lahe o produce a.7 mm wall hickness in he gage secion and a fille radius of.4 mm. The cylindrical specimens were mouned on a seel pos used o apply compression and orsion.. Tesing Experimens were conduced a various srain raes ranging from -5 o 5 s -. Lower rae experimens a less han s - were conduced in a servo-hydraulic esing machine while he higher rae ess were conduced in a Spli Hopkinson Pressure Bar sysem. Tension and pure shear ess a lower raes, -4 o s -, did no show much rae dependence in sress-srain behavior. Similarly, he brile failure under ensile loading did no show much variaion (Figure 3). The rae effec was much more pronounced under uniaxial compression. Dynamic esing a high srain raes was conduced in a Spli Hopkinson Pressure Bar (SHPB) sysem Typically, in such ess, eiher polymeric bars such as polycarbonae or meallic bars such as seel or aluminum are used. However, he resin esed was siffer and sronger han mos polymers and i would damage he ends of he bars during esing. For meallic bars, no only is here a very large impedance mismach, he resin reaches ulimae failure a srains above %. The wave speed in meallic bars is very high and he lengh of he bars needed o produce a long enough sress pulse o achieve he required ulimae srain was impracical. A compromise was reached wih composie (G-) rods made of sacked layers of woven glass/epoxy. The impedance of he bars was no oo differen from ha of he resin, a fac which allowed sress equilibraion wihin he specimen o occur much sooner han in he seel or aluminum Hopkinson bar sysems (Figure 4). The wave speed in he composie bars was much lower han ha in he meallic bars, so ha a.83 m inciden bar was sufficien o capure mos of he resin behavior. Because he glass fibers in he G- bars are oriened along he axis of he bar, he sress wave did no show much dispersion or aenuaion, and hus, here was no need for a more complex viscoelasic analysis of he signals. The only concern was he lack of axisymmery in he rods, bu no significan difference was observed in he propagaing pulse moniored by srain gages along he principal planes of he bar cross secion..3 Sress-Srain Behavior Sress-srain curves were obained over a wide range of srain raes. The compressive sress-srain curves shown in Figure 5 have he same overall form and share many key feaures. They exhibi a linear elasic region up o he yield poin, followed by a plasic region up o a criical poin of maerial insabiliy, he criical sress. Nex, follows a sofening plaeau reaching a local minimum, followed by srain hardening. All curves share he same iniial modulus. The rae dependence appears only in he nonlinear regime. The characerisic feaures of he curves expand radially from he origin wih increasing srain rae as highlighed in Fig. 5.

3 These characerisic properies vary linearly wih he logarihm of srain rae as shown in Figure 6. The resuling rae dependence can be expressed as P P mlog () where P is he rae dependen propery, P is he propery a he reference srain rae ( -4 s - in his case) and m is he slope of he linear logarihmic relaion (.96 in his case). Using his relaion, he sress-srain curves were ransformed ino one maser curve a he reference srain rae as shown in Fig Consiuive Modeling of Polymeric Marix 3. Poenial Funcion Mos models for polymers are expressed in sress space. However, because of he sofening and subsequen hardening behavior of he sress-srain curves, he srain dependence of he sress is no unique. For his reason an approach formulaed in srain space was used. I is also valuable o break up he consiuive response ino wo regions, one precriical sress and he oher pos-criical sress region. Here he emphasis was placed on he maerial behavior in he pre-criical sress region. In a general nonlinear elasic-plasic sress-srain response shown in Figure 8, he oal sress incremen can be decomposed ino an elasic par and a plasic par as e p di di di () The elasic sress incremen is linearly relaed o he oal srain incremen by he siffness C ij as e di Cijd j (3) The plasic sress incremen is relaed o a poenial (or loading) funcion hrough he associaed flow rule and normaliy rule as follows: p f di d (4) i where i =,,, 6; f is a plasic poenial funcion; and dλ is a scalar funcion of proporionaliy. The consiuive model sough is obained from he plasic poenial and scalar funcions. A plasic poenial funcion was proposed based on a modified Drucker-Prager [4] yield crierion. The proposed model describes he muli-axial sress-srain behavior and accouns for coupling of deviaoric and dilaaional deformaion and he sign of he normal sress (ension or compression). The poenial funcion in srain space is expressed as: a a f b v 3 (5) where a, a v, and b are plasic anisoropy numbers, and is he effecive srain ha conribues o plasic work. All srains are elasic-plasic srains defined as: where i i i y i (6) y is he oal srain and i is he yield For his maerial, under mos loading srain. condiions he yield srain is effecively zero as he maerial shows nonlinear behavior from he sar. However, under compression, he normal srain is linear up o a cerain poin. This region does no conribue o he plasic work performed, as shown in Figure 8. For he maser compressive sresssrain curve, his normal yield srain is.6%. 3. Consiuive Relaions The incremenal plasic srain energy per uni volume is given in erms of he effecive plasic sress as p p dwp idi d (7) p Where d is he incremen in effecive plasic sress. Replacing he plasic sress incremen wih he gradien of he poenial funcion by he flow rule of Eq. (4) we obain, p f d i d d (8) i Therefore, he scalar funcion of proporionaliy is given by p d d (9) Assuming a power law relaionship beween he effecive plasic sress and effecive srain as p n A () we obain he following expression for he scalar funcion of proporionaliy: n d na d ()

4 The incremenal effecive srain, as defined in Eq. (4), is f d d j () j Using Eqs. (-4), and (9-) we obain he consiuive elaso/viscoplasic model for he oal sress and srain incremens as n f f di Cij na d j (3) i j In order o accoun for he rae effec, compressive normal sresses and srains mus be ransformed o he reference srain rae using Eq. (). 3. Model Parameers The consiuive law is expressed in erms of hree parameers associaed wih he poenial funcion, a, a v, and b, and wo power law parameers, A and n, which relae he effecive plasic sress wih he effecive srain. However, he hree parameers in he poenial funcion are no independen. They are used o ransform any given sae of srain ino an effecive srain. For his reason, he poenial funcion can be normalized by one of he parameers, or any loading condiion can be chosen o represen a srain equivalen o he effecive srain. Since he pure shear and ension ess produced relaively low srains before failure, he uniaxial compression es was chosen o yield he effecive srain. In his es, denoes he axial srain, he shear srains being zero, and 3 (4) Subsiuing hese ino he poenial funcion, we obain he following relaion f a a b (5) v Since, under uniaxial compression, he axial srain is chosen as he effecive srain a av b (6) This provides a direc relaion for he sign dependen dilaaional consan b in erms of a, and a v a a v b (7) A direc expression for parameer a can be found by comparing he pure shear and uniaxial compression ess a he criical poin. A he criical poin he slope of he sress-srain curve is zero d d 6 (8) d d 6 Using he incremenal consiuive law in Eq. (3) and Eq. (8), we obain he following relaions n f C na (9) n f 66 C na () 6 By definiion, from Eq. (5) for a uniaxial compression es, f () and for a pure shear es f 6a () 6 Thus, by combining Eqs. (9-), an expression for parameer a can be found C66 G a.3 (3) 6C 3E The final poenial funcion parameer, a v, can be found from he ension ess. I is sensiive o he value of a v, so ha i is no difficul o fi i o he remaining daa. The parameer a v was found o have a value of 5.8.The power law parameers A and n were deermined by using he criical poin under uniaxial compression. Equaion (9) can be rewrien in erms of he produc na a he criical poin c n n na C c E c (4) Then, subsiuing in he oal srain value from Eq. (6) we obain y na E n c (5) The criical sress is hen found by inegraing he incremenal consiuive law, Eq. (3), from zero o he oal criical srain, c c n y c E na d (6)

5 Subsiuing in Eq. (5) and inegraing, we obain an expression for n y E c n (7) E c c Finally, rearranging Eq. (5) provides an expression for A y n E c A (8) n All five model parameers can be obained using he basic elasic consans of he maerial, he characerisic properies of he sress-srain curve ( y c, and criical c) and by fiing he sress-srain curve under uniaxial ension. Using hese parameers and he experimenal daa from uniaxial ension and compression, pure shear, and combined loading ess, as well as he ransformed sress-srain curves a various srain raes, all of he effecive srain-effecive plasic sress curves collapse ino a maser curve as shown in Figure 9. The heavy solid black curve is he power law approximaion for his relaionship obained as discussed above. 3.3 Model Validaion The model was formulaed using uniaxial ension and compression and pure shear ess a differen srain raes. Figures and, show ha he predicions of he proposed consiuive model are in excellen agreemen wih experimenal resuls. All of he ess and model predicions repored hus far have been uniaxial sress or srain ess, bu he model is valid for combined loadings as well. Two minor changes mus be made o he model derivaion described above. Firs, o accoun for he rae effec, he srain raes used in Eq. () mus be convered o effecive srain raes. This exends he he model o include srain rae effecs under muliaxial saes of sress and srain. P P m log (9) This change does no affec any of he resuls presened earlier, applied only for uniaxial compression ess. Under hose loading condiions, he effecive srain is equivalen o he axial srain. The oher modificaion is ha he yield srain used before is a principal yield srain and i mus be changed o he axial yield srain under he given loading condiion. For combined loading ess, he same hin wall cylindrical specimen and procedure were used as for he pure shear ess. The servo-hydraulic esing machine was conrolled o apply a consan displacemen rae in compression and roaion o provide he desired loading condiion. For hese ess, he compressive srain rae was specified o be 75% of he shear srain rae. This would provide a loading condiion ha would exhibi nonlineariy in boh compression and shear. Tess were conduced a wo differen srain raes o confirm ha he model could accoun for rae effecs under combined loading. Figure shows excellen agreemen beween predicions of he proposed consiuive model and experimenal resuls under combined compression and shear a wo srain raes. Acknowledgemen The sudy described here was sponsored by he Office of Naval Research wih Dr. Yapa D. S. Rajapakse as Program Manager. References [] I. M. Daniel, J.-M. Cho, B. T.Werner and J. S. Fenner "Characerizaion and consiuive modeling of composie maerials under saic and dynamic loading," AIAA Journal, Vol. 49, No. 8, pp ,. [] I. M. Daniel, B. T. Werner and J. S. Fenner, "Srainrae-dependen failure crieria for composies," Composies Science and Technology, Vol. 7, No. 3, pp ,. [3] Y. M. Liang and K. M. Liechi "On he large deformaion and localizaion behavior of an epoxy resin under muliaxial sress saes," Inernaional Journal of Solids and Srucures, Vol. 33, Vol., pp , 996. [4] C. P. Buckley, e al. "Deformaion of hermoseing resins a impac raes of srain. Par I: Experimenal sudy," Journal of he Mechanics and Physics of Solids, Vol. 49, No. 7, pp ,. [5] R. A. Schapery "On characerizaion of nonlinear viscoelasic maerials," Polymer Engineering and Science, Vol. 9, No. 4, p.95, 969. [6] K. Padmanabhan "Time-emperaure failure analysis of epoxies and unidirecional glass/epoxy omposies in compression," Composies: Par A, Vol. 7A, pp , 996.

6 [7] A. E. Mayr, W. D. Cook and G. H. Edward "Yielding behaviour in model epoxy hermoses - I. Effec of srain rae and composiion," Polymer, Vol. 39, No. 6, pp , 998. [8] S. Behzadi and F. R. Jones "Yielding behavior of model epoxy marices for fiber reinforced composies: effec of srain rae and emperaure," Journal of Macromolecular Science: Physics, Vol. 44, No. 6, pp , 5. [9] S. A. Y. Yamini "The mechanical properies of epoxy resins: Par Mechanisms of plasic deformaion," Journal of maerials Science, Vol. 5, pp. 84-8, 98. [] R. K. Goldberg, G. D. Robers and A. Gila "Implemenaion of an associaive flow rule including hydrosaic sress effecs ino he high srain rae deformaion analysis of polymer marix composies," Journal of Aerospace Engineering, Vol. 8, Vol., pp. 8-7, 5. [] R. K. Goldberg e al. "Approximaion of nonlinear unloading effecs in he srain rae dependen deformaion analysis of polymer marix maerials uilizing a sae variable approach," Journal of Aerospace Engineering, Vol., No. 3, pp. 9-3, 8. [] J. D. Liell e al. "Measuremen of epoxy resin ension, compression, and shear sress-srain curves over a wide range of srain raes using small es specimens," Journal of Aerospace Engineering, Vol., No. 3, pp. 6-73, 8. [3] C. P. Buckley e al. "Deformaion of hermoseing resins a impac raes of srain. Par : consiuive model wih rejuvenaion," Journal of he Mechanics and Physics of Solids, Vol. 5, No., pp , 4. [4] D. C. Drucker and W. Prager "Soil mechanics and plasic analysis or limi design," Quarerly of Applied Mahemaics, Vol., pp , 95. Fig.. Effec of aspec raio on compressive response Fig. 3 Tension and shear sress-srain curves a wo srain raes Fig.. Specimen geomeries and dimensions (a) Tensile, (b) Compressive, (c) Shear and combined sress specimens

7 Sress, * (MPa) 5 Fig. 4. Sresses a ends of resin specimen for differen Hopkinson bar maerials. 5 ( ) *( P = P m log ε ) K + ε ε = 4 s ( ) *( ) -4 s m =.96 K - - s K mlog - 5 s - m.96 3 s s Srain, * (%) Fig. 7. Maser compressive sress-srain curve a reference srain rae Fig. 5. Compressive sress-srain curves a various srain raes showing radial alignmen of characerisic feaures Fig. 8. Definiion of plasic sress and plasic work Fig. 6. Variaion of normalized criical and yield sresses wih srain rae Fig. 9. Collapsed curves of ension, shear, compression and combined loadings a various srain raes wih power law model

8 Compressive Sress, (MPa) Shear Sress, (MPa) Tensile Sress, (MPa) Sress Componen,, (MPa) s - s - Model Experimen 4 3 Model Experimen 8 6 g s - 4 g 4 s Tensile Srain, (%) Shear Srain, g (%) Model Experimen Fig.. Comparison beween model predicions and experimenal resuls for (a) uniaxial ension and (b) shear ess Srain Componen,, g (%) Fig.. Combined loading experimens and model predicions a differen srain raes Table Model parameers Parameer Value a.3 a v 5.4 b -.8 A 3 n Model Experimen - s - s s - 4 s Compressive Srain, (%) Fig.. Comparison beween model predicions and experimenal resuls for uniaxial compression ess a differen srain raes