Bachelor End Project: Characterization of the contitutive behavior of polymer foam R. van Eijden MT 05.27 Coach: Dr. ir. J.A.W. van Dommelen Eindhoven, April 21t 2005
Content Content Abtract Lit of ymbol Introduction i ii iii iv 1 Small deformation 1 1.1 Hooke law 1 1.2 Numerical 1 1.3 Reult 3 2 Large deformation 7 2.1 Elatic 7 2.2 Leonov model 7 2.3 2-D macrocopic contitutive behavior 9 2.4 Numerical 10 2.5 Reult 11 3 Concluion and recommendation 17 Bibliography 18 i
Abtract The predictive capabilitie of the commercial crah imulation code that are ued in the deign proce are limited by the quality of the available contitutive model for foam. The objective of thi project i to improve the current contitutive model for polymeric foam. Thi report conit of two chapter. In the firt chapter the macrocopic behavior of polymeric foam will be reearched by impoing only mall deformation. Uing Hooke law the Young modulu of the foam i calculated. Alo the influence of the volume percentage of void i dicued. In the econd chapter the behavior will be reearched uing large deformation. The model ued to decribe the contitutive behavior i the Leonov model. Firt a global repreentation of the model i given. Then the macrocopic contitutive behavior of polymeric foam in 2-D i given. Finally the implementation of the Leonov model in MARC i ued to imulate compreion tet on the polymeric foam and the reult are dicued. ii
Lit of ymbol Symbol Decription unit E Young modulu GPa Engineering tre Pa Engineering train [-] F Load N A 0 Original cro ectional m 2 area l Deformation elongation m l 0 Original length m Poion ratio [-] G Elatic hear modulu MPa Bulk modulu MPa G r Strain hardening modulu MPa Vicoity Pa σ Equivalent tre MPa p Hydrotatic preure MPa T Temperature K S Softening variable [-] A Deformed cro ectional m 2 area T True tre Pa S 0 Initial oftening variable [-] iii
Introduction Foam material are known for their good energy-aborbing and ergonomic propertie. That the reaon why they are ued by the automotive indutry. The foam material are ued in afety application; example are dahboard, bumper ytem and eat. Another advantage of the ue of foam material i their low denity. Thi reult in a reduction in weight and the ga conumption and pollution are reduced The quality of the available contitutive model for foam i limited. Therefore the automotive indutry i not able to optimize their product in a reliable manner uing commercial crah imulation. That i why there a need for the characterization of the contitutive behavior of polymeric foam. The energy-aborbing and ergonomic propertie are a reult of the macrocopic contitutive behavior of polymeric foam. Thi behavior i the reult of an interplay between the intrinic material behavior of the polymer bai-material and the complex microtructure that i preent in the foam. The outline of thi report i a follow: polymeric (in thi cae polycarbonate) foam material i canned uing a CT-canner. Then the can are mehed uing automatic 3D meh generation, a developed in the world of bone-biomechanic. A compreion tet will be imulated with the ue of a finite element code in thi cae MSC.MARC 2003 uing the 3D meh. Two different ort of tet are imulated: mall deformation and large deformation. The ued model are briefly explained and the reult of the imulation are preented. Finally remark will be placed and concluion will be drawn. iv
Chapter 1 Small deformation 1.1 Hooke law If the material i deformed with a compreion tet, the degree to which a material deform or train depend on the magnitude of an impoed tre. If the material i treed at a relatively low level, tre and train are proportional to each other through the relationhip: σ = Eε (1.1) Thi i known a Hooke law and the contant of proportionality E [GPa] i the modulu of elaticity, or Young modulu. The engineering tre σ and the engineering train ε in Hooke law are defined by the following relationhip: F σ = (1.2) A 0 and l ε = (1.3) l 0 In which F i the load applied perpendicular to the material cro ection, A 0 i the original cro ectional area before any load i applied, l 0 i the original length of the pecimen and l i the deformation elongation [1]. Thee relationhip will be ued to calculate the ratio between E-foam (Young modulu of the polymeric foam, in thi cae polycarbonate) and E-material (Young modulu of the bai material). 1.2 Numerical To predict the macrocopic repone of the polymeric foam the 3D meh baed on the CT-can ha to be incorporated in a finite element method. The finite element code ued i MSC.MARC 2003. The CT-can are baed on voxel. The term voxel i ued to characterize a volume element; it i a generalization of the notion of pixel that tand for a picture element. Two different 3D mehe of the ame piece of polycarbonate were made, one with eight-noded linear hexagonal element (136400 element and 153973 node) and the other one with four-noded linear tetrahedral element (790582 element and 172218 node). They will be called hexmeh and tetmeh repectively from now on. The file reulting from the 3D meh generation can be imported in MSC.Mentat if the meh 1
doen t contain too many element (ee figure 1.1). When the number of element ued i too large ome problem arie when importing the CT file in MSC.Mentat. Therefore it i neceary to rewrite the CT mehfile o that Marc can read them in a an input file [2], in thi way MSC.Mentat doen t have to viualize the 3D meh. To rewrite the CT meh file into an input file for MSC.MARC an m-file i made in MATLAB (ee tetmeh.m). Firt the information from the 3D meh generation file i reordered o it can be ued in an inputfile. Then the actual inputfile can be written that contain all the material propertie, geometry information, boundary condition, nodal tie etc. For the tetmeh it i alo neceary to check if the meh contain any element that are inide out. The m-file doe thi and if there are any element inide out they will be flipped. Another advantage of writing input file in thi way i the poibility to change the ize of the 3D meh with the m-file. Figure 1.1: Hexmeh in MSC.Mentat with ize 0.57*0.55*0.5 [mm]. Figure 1.2: Choen boundary condition. 2
The imulation in MARC are baed on a compreion tet. A compreion tet ha a particular et of boundary condition o it important that the imulation ha the ame boundary condition. The boundary condition are choen a follow: On the left ide of the block (ee figure 1.2) one node i fixed in pace at the left lower angular point, the right lower angular point i fixed in the y-direction for antirotation and the whole left ide i fixed in the x-direction. A diplacement in the negative x-direction i acting on the right ide of the block. The other node on the right ide are linked to the node where the diplacement act on (o they will follow the diplacement of the node they are tied to), imulating a compreion tet. For the imulation of compreion tet in other direction other boundary condition have to be choen. 1.3 Reult The compreion tet will be imulated in the x, y and z-direction and with tetmehe of different ize, thi i done with the m-file (ee figure 1.3). The mehe have a different volume percentage of void. (a) 27 vol. % void; 57634 element and 13815 node (b) 23 vol. % void; 169400 element and 38815 node (c) 21 vol. % void; 373698 element and 83584 node (d) 19 vol. % void; 701661 element and 154223 node Figure1.3: The different ize of tetmehe ued with (a) being the mallet and (d) being the larget meh. 3
The imulation are done with tetrahedral element of type 134 (full integration). The material propertie ued are a Young modulu of 1.0 GPa and a poion ratio of 0.3. A diplacement of 1.0e-5 m i impoed in negative x, y or z-direction. The olver ued i of the type iterative pare. To calculate E-foam the reaction force F ha to be known. If F i known the modulu of elaticity can be calculated uing equation (1.2), (1.3) and (1.1) becaue A 0, l 0 and l are known. All the node on the ide of the block where the diplacement act on are tied to one node with the o-called nodal tie. Only one variable i tied at a time (x, y or z-direction). In thi way the total reaction force F that applie perpendicular to the block cro ection will be obtained from one node, otherwie all the reaction force of the different node have to be added individually. Figure 1.4: Ratio between E-foam and E-material againt the height of the tetmeh in different direction. In figure 1.4 the ratio between the Young modulu (E-foam) that i a reult of the imulation and the Young modulu (E-material) that i ued a input in MARC i plotted againt the height of the tetmeh. The graph how that an increaing height of the meh reult in a higher E-foam. It i expected that at a certain height the value will not change anymore. That certain height in t reached yet in the can that were ued in thi imulation. In figure 1.4 it i een that with larger height the volume percentage of void decreae. That not alway the cae becaue if the meh wa reduced to a region that didn t contain any void E-foam and E-material would be the ame. In figure 1.5 the influence of the vol. % of void i een. The graph how that an increaing vol. % void reult in a lower value of E-foam. If the foam ha a low vol. % of void E-foam will go to the ame value of E-material. For lower percentage the graph i linear, then E-foam can be predicted. It i alo een that E-foam ha different value in different direction. So the difference in vol. % of void in t the only reaon for the E-foam value to differ. The boundary condition ued in the different direction differ from each other. The area of the meh that i fixed by the boundary condition differ for the imulation in different direction. For example in figure 1.3(d) it i een 4
Figure 1.5: Ratio between E-foam and E-material againt the volume percentage of void of the tetmeh in different direction. that the right ide of the meh contain void. So on that ide le area of the meh ha a pre-decribed boundary condition than if that ide of the meh wouldn t contain any void. Thee boundary condition have an influence on the value of E-foam, o the value of E-foam in the different direction can differ. Boundary condition reult in more tre in the meh o it i expected that the boundary condition reult in a larger E-foam. The two different 3D mehe contain different element o the foam i mehed differently. In figure 1.6 it i een that a void i mehed differently and a a reult the two different mehe have a different volume percentage of void. It i alo een that the foam i mehed more realitic in the tetmeh. In table 1.1 the volume percentage of void for the two mehe are given. Figure 1.8 how that a lower vol. % of void reult in a lower value of E-foam. Figure 1.6: Void mehing for hexmeh and tetmeh repectively. 5
Table 1.1: Volume percentage of void for the tetmeh and the hex meh Size of the meh Tetmeh Hexmeh Fig 1.3(a) 27 % 26 % Fig 1.3(b) 23 % 23,3 % Fig 1.3(c) 21 % 21,5 % Fig 1.3(d) 19 % 19,3 % Figure 1.7: Ratio between E-foam and E-material againt the height of the different mehe. If the difference of E-foam i only the reult of the volume percentage of foam, the value of E-foam would be the ame for the different mehe if the vol. % of void i the ame. In figure 1.8 it i een that thi i not the cae. The reulting E-foam for imulation uing a tetmeh are higher than uing a hexmeh. So the ue of different element ha influence on the value of E-foam. Figure 1.8: Ratio between E-foam and E-material againt the vol. % of void for different mehe. 6
Chapter 2 Large deformation 2.1 Elatic deformation In a compreion tet a pecimen will be expoed to large deformation. Beide elatic deformation there will be platic deformation too. Becaue elatic- platic imulation are more complex it i wie to look what happen if the meh i expoed to large deformation uing the theory of linear elaticity. The imulation will be done with a reduced hexmeh. The hexmeh i choen becaue it ha fewer element then the tetmeh, o the imulating time will be horter. The imulation are done with hexagonal element of type 7 (full integration). The value of E and the poion ratio of polycarbonate are 2.0 GPa and 0.3 repectively. They will be ued a the material parameter in MARC. During the imulation ome problem arie. Becaue the deformation are large ome element deform too much and they will be puhed inide out. If the tetmeh i ued the ame problem occur at approximately the ame train. A olution for the imulation problem i to remeh the meh when the element deformation i too large or the ue of quadratic element intead of linear. 2.2 Leonov model For the imulation of elatic-platic behavior of polycarbonate the Leonov model i ued. The Leonov model i a nonlinear vicoelatic model. In figure 2.1(b) a onedimenional repreentation of the contitutive model i preented. Figure 2.1: True tre-train curve with (b) correponding graphical repreentation [3]. The contitutive model conit of two parallel part: the part decribing the yield and train oftening and the other decribing the train hardening part. Phyically thi can be interpreted a the part of weak econdary force and the part of network entanglement. The firt i decribed with a generalized Maxwell pring-dahpot element including an Eyring vicoity, the econd a a neo-hookean pring. 7
The Cauchy tre tenor σ i additively decompoed in an effective/driving tre tenor σ and a hardening tre tenor σ (ee figure 2.1), according to σ = σ + σ r (2.1) The effective/driving tre tenor can be plit in two part: the deviatoric and the hydrotatic part. So σ i given by the relation: r σ = σ + σ (2.2) h d and (2.1) become σ = σ + σ + σ (2.3) h d r h d Where σ i the hydrotatic tre and σ i the deviatoric tre. Auming only d mall volumetric deformation, σ i related to the deviatoric part of the elatic d iochoric left Cauchy Green deformation tenor B ~ e through the generalized Hookean relation σ = G B ~ d e (2.4) d h where G repreent the hear modulu. The hydrotatic tre σ i coupled to the volume change by h σ = κ( J - 1)I (2.5) with the bulk modulu, J the volume change ratio and I the unity tenor. The Leonov model i completed by expreing the platic deformation rate tenor D p in d the effective tre. D p i related to the effective deviatoric tre tenor σ by a non- Newtonian flow rule with an Eyring vicoity D p = 2η d σ = G ( σ, p, T, S ) 2η B ~ (2.6) d e Where σ i the equivalent Von Mie tre, p the hydrotatic preure, T the temperature and S the oftening variable. For additional information about the Leonov model concerning kinematic, balance law and tre calculation i referred to van Breemen [2]. If the Leonov model i compared with experimental data (figure 2.2) it can be een that the model can be ued for a good decription of the oftening and hardening of the material. But it ha it hortcoming in the elatic region. 8
Figure 2.2: True tre-train curve of a compreion tet; experiment ( ), Leonov model with claic oftening function [5] (- -) and Leonov model with inverted oftening function [6]( ) [3]. 2.3 2-D macrocopic contitutive behavior Before imulating in 3D, the macrocopic contitutive behavior of polymeric foam in 2-D i given a tudied by Smit [4]. The darker area in figure 2.3 repreent yielded zone. The correponding tre-train curve i depicted in figure 2.4. Point f and h correpond with the tage of deformation figure 2.3 i in. Figure 2.3: Shear zone in a uniaxially tretched polycarbonate with 30 vol.% void [4]. The change in train oftening behavior originate from the irregularity of the microtructure. Thi i reflected in the yield proce, which occur a a equence of iolated hear event ditributed over the whole microtructure. A hear proce involve the development of hear band. The material inide a hear band will firt train often (untable deformation zone) and after that it will train harden (table deformation zone). Since the overall mechanical repone i the averaged behavior over both table and untable deformation zone, the temporary untable behavior of the yield zone i evened out in the global macrocopic repone. In figure 2.3 i een that the yielding of one ligament invoke ligament in the neighborhood to yield alo. In thi way the platic deformation i pread out over the whole microtructure. 9
Figure 2.4 : True tre- linear train curve of a tenile tet. The material ued i polycarbonate [4]. 2.4 Numerical The Leonov model wa already implemented in MSC.MARC 2003 by van Breemen [2]. Thi wa done uing the uer-ubroutine HYPELA2. Firt a compreion tet wa imulated uing a one-element block, the diplacement wa impoed at a contant velocity. The imulation reult in the following characteritic graph: Figure 2.5: Compreion tet imulation of one hexagonal element howing the equivalent von mie tre and the oftening variable againt the total equivalent train. In figure 2.5 it i alo een that when the material tart yielding, the initial oftening variable S 0 with a value of 30 (characteritic for polycarbonate foam) tart to decreae. So thi parameter can be ued to repreent the platic deformation in the material. Thi parameter can be viualized in the reult of MARC. Then it can be een were platic deformation tart and how it develop in the material. 10
The implemented Leonov model will be ued to imulate a compreion tet for a homogeneou block, a block with different ize of void and a block with two void. Finally the implementation i ued to imulate a compreion tet uing a reduced hex and tetmeh. 2.5 Reult When a material i expoed to large deformation, the cro-ectional area will change over time. Therefore the engineering tre can t be ued anymore and the true tre T i ued. True tre T i defined a the load F divided by the intantaneou cro-ectional area A over which deformation i occurring [1], or F σ T = (2.7) A Variou imulation were done with block with different void and different number of void. All the imulation were done with block of dimenion 1*1*1 [m] and a diplacement of 0.5 m i impoed in negative x-direction. The diplacement i ubdivided in 200 increment and i impoed at contant velocity of 0.005 m/ o the train rate i not contant. The olver ued i of the type multifrontal pare. In figure 2.6 and 2.7 it can be een how the platic deformation develop uing the oftening variable. The darker area repreent area with the mot platic deformation. The platic deformation tart above and under the void (figure 2.6) and develop in the direction of the corner (figure 2.7). While the material left and right of the void i till in it elatic region. In figure 2.8 the compreive true tre for different ize of void are hown. The train oftening for the block with 1 vol. % void i maller than for the homogenou one, the ame count for the Young modulu (a void mean le material). The reduction in train oftening can be explained in the ame way a wa done for the 2- D tenile tet imulation: the overall mechanical repone i the averaged behavior over both the table (elatic and hardening) and untable deformation (oftening) zone. Becaue the volume percentage of the void i low train oftening i till preent. Larger void reult in a lower Young modulu and the material will yield at a lower tre. Larger void alo reult in earlier imulation problem o not much can be aid on it influence on the train oftening. A wa tated before a olution for the imulation problem i to remeh the meh when the element deformation i too large or the ue of quadratic element intead of linear. 11
Figure 2.6: Value of the oftening variable S in a block with one void; early tage with trainε = 0. 0875 Figure 2.7: Value of the oftening variable S; Further tage with trainε = 0. 125 12
Figure 2.8: The compreive true tre veru the engineering train uing block with different vol. % of void. Figure 2.9 repreent the platic deformation developing for a block with two void. The platic deformation develop in the ame way a for the block with one void. The only difference i the region between the two void. The deformation develop from one hole to another. The reult for the Leonov implementation are a predicted. The implementation i now ued to decribe the behavior of polymer foam. For the imulation of a compreion tet on the canned polymeric foam a reduced hexmeh i ued (27 vol. % void, 4371 element and 5682 node). All the imulation were done with block of dimenion 0.21*0.19*0.14 [mm] and a diplacement of 0.1 mm i impoed in negative x-direction. The diplacement i ubdivided in 500 increment of the ame ize and i impoed at contant velocity of 0.001 mm/. The olver ued i of the type multifrontal pare. In figure 2.10 the development of platic deformation in the reduced hexmeh i preented. The platic deformation i preent in ome part of the hexmeh. But the bigget part of the meh i till deformed in it elatic region. Thi i a reult of the imulation with MARC, becaue the deformation of ome element were too large in an early tage of the imulation. 13
Figure 2.9: Value of the oftening variable S. With trainε = 0. 125 Figure 2.10: Value of the oftening variable S for a reduced hexmeh. With trainε = 0. 0457 14
In figure 2.11 the reult of the imulation i plotted. A a reference the reult of a imulation of a homogeneou block of the ame ize a the reduced hexmeh i plotted in the ame figure. It can be een that the deformation i till in it elatic region becaue there are not enough hear band preent to initiate the yielding in the global macrocopic repone. Figure 2.11: The compreive true tre veru the engineering train for a homogeneou block and a reduced hexmeh (ame ize). Figure 2.12: The compreive true tre veru the engineering train for different mehe uing the Leonov implementation. Finally a imulation uing a reduced tetmeh wa done ( 27.8 vol. % void, 23780 element and 5928 node). In thi imulation there are even le hear band preent when imulation problem occur than with the imulation uing a reduced hexmeh. Thi i een in figure 2.12. In contrat to the reult of the imulation with mall 15
deformation, the Leonov model predict the ame value of E-foam (lope of the graph) in pite of the difference of vol. % of void between the two mehe and the different element ued. 16
Concluion and recommendation The ize of the can of the polycarbonate foam in t large enough to predict the Young modulu E of the polymeric foam. Thi i een in fig. 1.4. The value of the ratio between E-foam and E-material till increae when the height of the meh get larger. It i expected that with larger mehe E-foam will go to a contant. The value of E-foam alo differ in the different direction, thi can be the reult of the different boundary condition ued. The influence of the boundary condition wan t reearched in thi project. In figure 1.5 it i een that lower vol. % of void reult in a lower value of E- foam. For lower percentage the graph i linear, then E-foam can be predicted. The two 3D mehe have a different vol. % of void. Thi i a reult of the different mehing of the foam (ee fig.1.6). The difference in volume percentage of the void (ee table 1.1 and figure 1.7) reult in different value of E-foam. The tetmeh reproduce the void in a more realitic way. So when uing a canned piece of polymer foam for imulation it i preferred that the meh conit of tetrahedral element. The difference in E-foam in t only the reult of the vol. % of void in the meh (ee figure 1.8). The different element alo ued have an influence on the value of E-foam. The Leonov model i ued to give a decription of the macrocopic contitutive behavior of polymer foam. It wa implemented in MSC.MARC 2003 by van Breemen [2]. The reult from the imulation on block with different ize and number howed a reduce in train oftening (ee figure 2.8). In contrat to the reult of the imulation with mall deformation, the Leonov model predict the ame value of E-foam for the two different 3D mehe in pite of the difference of vol. % of void and the different element ued. When imulating a compreion tet on the polymer foam mehe ome problem occur. Some element are under uch a large deformation that they are puhed inide out. Thi happen with both mehe. So the deformation that act on the mehe i too low to reult in ignificant platic deformation (ee fig. 2.10). Ue of the tetmeh in imulation reult in imulation problem atε = 0. 047 and for the hexmeh at ε = 0. 037 (ee fig. 2.12). A olution for thi i to remeh the meh when the element deformation i too large or the ue of quadratic element intead of linear. The olution will reult in an increae of imulation time. The imulation uing the Leonov implementation were done by impoing a diplacement at contant velocity. It i recommended that the loading rate ued i a contant logarithmic train rate becaue the Leonov model i time dependant (ee figure 2.1(b)). 17
Bibliography [1] William D. Calliter, Material Science and Engineering an Introduction, fifth edition, 2000, p. 114-118, 131. [2] MARC uer manual A and C. [3] L.C.A. van Breemen, Implementation and validation of a 3D model decribing glay polymer behavior, 2004. [4] R.J.M. Smit, Toughne of Heterogeneou Polymeric Sytem. A Modeling Approach, 1998, p. 46-48. [5] L.E. Govaert, P.H.M. Timmerman, and W.A.M. Brekelman. The influence of intrinic train oftening on train localization in polycarbonate. J. eng. Mat. Techn., 2000. [6] E.T.J. Klompen, T.A.P. Engel, L.E. Govaert, and H.E.H. Meijer. Elatovicoplatic modeling of large train deformation of glay polymer: incorporation of ageing kinetic. J. Rheol., 2004. 18