Finite Element Analysis of V-Bending of Polypropylene Using Hydrostatic-Pressure-Dependent Plastic Constitutive Equation*

Transcription

1 Materials Transactions, Vol. 48, No. 1 (7). 6 to 664 #7 The Jaan Society for Technology of Plasticity Finite Element Analysis of V-Bending of Polyroylene Using Hydrostatic-Pressure-Deendent Plastic onstitutive Equation* Kunio Hayakawa 1, Yukio Sanomura, Mamoru Mizuno 3, Yukio Kasuga and Tamotsu Nakamura 1 1 Deartment of Mechanical Engineering, Shizuoka University, Hamamatsu , Jaan Deartment of Mechanical Engineering, Tamagawa University, Tokyo , Jaan 3 Deartment of Machine Intelligence and Systems Engineering, Akita Prefectural University, Yuri-Honjo 1-, Jaan In the resent aer, V-bending of olyroylene (PP) is analyzed by the finite element method using a lastic constitutive equation for hydrostatic-ressure-deendent olymers roosed by one of the resent authors. The yield surface is exressed by the first and second invariants of stress to describe the hydrostatic-ressure deendence. A lastic otential that is different from the yield surface is emloyed to describe the incomressibility of olymeric materials. Isotroic hardening is assumed. The roosed constitutive equations are imlemented in the finite element code MS.Marc with user subroutines. The calculated load-stroke curves aroriately describe the effect of introducing the hydrostatic-ressure deendence of PP. Moreover, the calculated results agree with the exerimental ones for various thicknesses of secimens. Finally, the calculated distributions of bending stress and bending strain in the secimen also show the effects of hydrostatic-ressure deendence. [doi:1.3/matertrans.p-mra7878] (Received March, 6; Acceted July 3, 7; Published Setember, 7) Keywords: V-bending, finite element method, constitutive equation, hydrostatic-ressure-deendent lasticity, olyroylene, load-stroke curve, user subroutines 1. Introduction Recently, a great many arts and comonents made of olymers have been used in, for examle, automobiles, cellular hones, electric aliances, and office automation aliances. For the structural analysis of these arts, the static finite element method using the J flow rule and von Misestye yield surface have mainly been erformed. 1 3) It has been reorted that olymer has a significant strain rate deendence and a hydrostatic-ressure deendence. 4) As a first aroximation, however, the strain-rate deendence can be negligible in the structural analyses if the distribution of the strain rate of the deformed art is regarded as being almost uniform. On the other hand, the consideration of the hydrostatic-ressure deendence of olymer is imortant to achieve more recise estimation of the stress distribution of the olymer art subject to deformation. One of the resent authors has develoed a timeindeendent lastic constitutive equation that roerly describes the hydrostatic-ressure deendence, from the abovementioned standoint. The effect and limitation of the roosed equation have been verified by calculating the uniaxial loading behavior of a olymer that has a significant hydrostatic-ressure deendence, such as olyroylene (PP). 7) In the resent study, the V-bending of PP is analyzed in order to verify the effect of using the hydrostatic-ressuredeendent lastic constitutive equation for the structural or rocessing analyses by the static finite element method. The hydrostatic-ressure-deendent constitutive equation used in *This Paer was Originally Published in Jaanese in J. Jn. Soc. Technol. Plasticity, (), the resent aer has been develoed in the framework of the isotroic hardening rule, the modified von Mises-tye yield surface and the non associate flow rule.,6) The equation is imlemented into the commercial finite element code MS.Marc emloying user subroutines. The effect of the hydrostatic-ressure deendence on the calculations is discussed. Furthermore, calculated results are comared with the corresonding exeriments of V-bending erformed by the resent authors.. Elastic-Plastic onstitutive Equation of Hydrostatic- Pressure-Deendent Polymer Let us summarize the elastic-lastic constitutive equation of the hydrostatic-ressure-deendent olymer used in the resent study.,6) The total strain tensor can be divided into elastic and lastic arts as " ij ¼ " e ij þ " ij : ð1þ The elastic deformation of the olymer is assumed to be isotroic linear elastic and is given as " e ij ¼ 1 þ E ij E kk ij ; ðþ where ij is Kronecker s delta, and E and are Young s modulus and Poisson s ratio, resectively. The hydrostatic-deendent yield surface is given as f ¼ð1 Þ eq þ I 1 ¼ I 1 ¼ kk ; eq ¼ ffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi = 3J ¼ ; ; ð3þ 3 s ijs ij where I 1 and J are the first invariant of stress tensor and the

2 66 K. Hayakawa, Y. Sanomura, M. Mizuno, Y. Kasuga and T. Nakamura second invariant of deviatoric stress tensor, resectively, and eq is the equivalent stress. Furthermore, reresents the isotroic strain hardening as a function of the equivalent lastic strain ". Moreover, the coefficient is introduced to exress the extent of the hydrostatic ressure deendence. If ¼, no hydrostatic ressure deendence is considered. Although the hydrostatic ressure deendence is significant, most olymers have been reorted to exibit incomressibility. Therefore, in order to take the incomressibility into account in the lastic constitutive equation, a lastic otential g is introduced as g ¼ ffiffiffiffiffiffiffi 3J : ð4þ According to the normality rule with resect to the otential g, a lastic constitutive equation is given as _" ij ¼ ; ij where _ is a ositive scalar function obtained using the consistency condition of the yield surface _f ¼. Equation () can be rewritten using eqs. (3) and (4) as _" ij ¼ 1 H ð1 Þ 3s kl ffiffiffiffiffiffiffi þ kl 3J H ¼ d Z t d " ; " ¼ 3 _" ij _" ij dt _ kl 3s ij ffiffiffiffiffiffiffi 3J : ð6þ For evolution equations of ð" Þ, Swift s law and an exonential function are emloyed so that the actual lastic behavior can be roerly exressed as ¼ Fðb þ " Þ n ; ¼ Y þðy YÞ½1 exð c" ÞŠ; where F, b, n, Y, Y and c are material constants. ð7aþ ð7bþ 3. Stress Rate-Strain Rate Relation of Hydrostatic- Pressure-Deendent Polymer The roosed equations are to be imlemented into the commercial finite element code MS.Marc emloying the user subroutine HYPELA for calculations of the V-bending of the olymer. The stress rate-strain rate relation is given as _ ij ¼ e ijkl _" kl ; ð8aþ e ijkl ¼ e ijkl 8 e ijq s q >< ¼ >: ; if f < or ð1 Þs mn mnkl e þ 3 eq mn mnkl e 4 eq H þð1 Þ e ijkl s ijs kl 1; if f ij ij _ ij < ; ð8bþ : ð8cþ For the lane-stress roblem, Eqs. (8a) (8c) can be rewritten, using the Voigt notation, as >< _ x _ y >: ¼ E 1 B A 1 S x x S x y ð1 ÞS x S xy B S y x S y y ð1 ÞS y S >< _" x xy A7 4 S H _" y _ >: ; xy S xy x S xy y S xy _ xy ðaþ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eq ¼ x x y þ y þ 3 xy; S x ¼ E 1 ðs x þ s y Þ; S y ¼ E 1 ðs x þ s y Þ; S xy ¼ E 1 þ xy; S eq ¼ E 3 1 : eq; S H ¼ 4 eq H þð1 ÞðS xs x þ S y s y þ S xy xy Þ x ¼ð1 ÞS x þ S eq ; y ¼ð1 ÞS y þ S eq ðbþ For the lane strain roblem, Equations (8a) (8c) can be rewritten as _ z ¼ ð _ x þ _ y Þ ¼ eq þ 1 E" ð eq þ E" Þ ; eq ¼fð þ 1Þx þð 1Þ x y þð þ 1Þy þ 3 xy g1= ð1aþ 8 1 >< _ x _ y >: ¼ E 6 B 1 4ð1 þ Þð1 _ xy 1 1 A 1 S H 138 S x x S x y ð1 ÞS x S xy B S y x S y y ð1 ÞS y S >< _" xy A7 _" y >: ; ð1bþ S xy x S xy y S xy _ xy

3 Finite Element Analysis of V-Bending of Polyroylene Using Hydrostatic-Pressure-Deendent Plastic onstitutive Equation 661 S H ¼ 4 eq H þ 3 ð1 Þ E ; 1 þ S eq ¼ E 3 ð1 þ Þð1 Þ eq; S x ¼ E 1 þ s x; S y ¼ E 1 þ s y; S xy ¼ E 1 þ xy; x ¼ð1 ÞS x þ S eq ; y ¼ð1 ÞS y þ S eq : ð1cþ Note that the elastic-lastic stiffness tensor (8b) and tangent stiffness matrices (a) and (1a) are asymmetric, because the non associate flow rule, in which the yield surface f is different from the lastic otential g, is emloyed. 4. Exeriments and Finite Element Analyses of V- bending of PP 4.1 Exerimental conditions and aaratus In the resent exeriments, secimens were lates of PP (GrandPolyro J16W) that had been reared by milling from extruded round bars. The width and length were b ¼ 1 mm and l ¼ mm, resectively. The secimens had three different thicknesses of t ¼ 4, and 6 mm, in order to examine the effect of the thickness on the bending behavior. Figure 1 shows the exerimental aaratus and the geometries of the unch and die used. The ti radius of the unch was R ¼ mm, the bottom radius of the die was R D ¼ 1 mm, the interval between the die shoulder was L ¼ 7 mm, and the radius of the die shoulder was r D ¼ 1 mm. The unch seed was set to v ¼ : mm/min. This slow seed was selected to avoid the effect of strain rate during the bending deformation of the secimens. Plenty of mechanical oil was sulied to the secimens and tools to avoid the effect of friction. 4. Analytical conditions Figure shows a discretized model of the finite element analysis. Only the right side of whole secimen was modeled because of symmetry. The elements used were 4-node quadrilateral isoarametric elements. The numbers of elements and nodes were 446 and. The unch and die were modeled as rigid bodies. Friction between the secimen and tools was neglected by adoting oulomb friction coefficient ¼. The material constants in the constitutive equation were determined so that the equation can roerly exress the exerimental results of the uniaxial tension and comression, and were given as E ¼ 1:7 GPa; ¼ :36 F ¼ 67: MPa; b ¼ 6:3 1 4 ; n ¼ :17 Y ¼ 14: MPa; Y ¼ 3:7 MPa; c ¼ 6: ¼ :17 ; ð11þ so that the results of uniaxial tensile and comressive tests were exressed roerly. Figure 3 shows the exerimental and calculated results of the uniaxial tensile and comressive tests of PP. The ordinate and abscissa show the absolute values of uniaxial stress and strain, resectively. It is found that the roosed constitutive Die Punch Secimen R D R P (a) 7 7 (b) Fig. 1 V-bending aaratus and geometry of unch and die (a) V-bending aaratus (b) geometries of unch and die. equation can accurately redict the difference between tensile and comressive stresses. The stresses estimated using Swift s law (7a) are higher than those estimated using exonential function (7b) in the range of large strain. 4.3 alculated results and discussion on V-bending Figure 4 shows the influence of Swift s law (7a) and

4 66 K. Hayakawa, Y. Sanomura, M. Mizuno, Y. Kasuga and T. Nakamura Punch Secimen Load FB / N β =.17, lane stress β =.17, lane strain β =, lane stress β =, lane strain 1 Die 1 1 Fig. Load-stroke curves for V-bending under lane stress and lane strain with hydrostatic-ressure deendence ¼ :17 and. 3 Fig. Discretization of model analyzed. Absolute true stress σ / MPa comression tension hydrostatic-ressure deendence β =.17 Exerimental alculated [ Eq. (7a)] alculated [ Eq. (7b)] Absolute true strain ε.1 Fig. 3 Identification of material constants under uniaxial tension and comression. Fig. 6 Distribution of lastic bending strain for V-bending at stroke S ¼ 1: mm under lane stress. Load F B / N Exonential law [Eq. (7b)] hydrostatic-ressure deendence β =.17 lane stress t = mm t = 4 mm 1 1 Fig. 4 Load-stroke curves for V-bending under lane stress calculated using Eqs. (7a) and (7b) with hydrostatic-ressure deendence ¼ :17. exonential function (7b) on load F B -stroke S curves under the lane-stress state. Solid and dashed lines are the results obtained using eqs. (7a) and (7b), resectively. Both equations give similar F B before reaching the maximal value. After the maximal value, F B obtained with eq. (7a) is more stable and less scattered than that obtained using eq. (7b). This is because the stress increment in eq. (7b) becomes almost zero as the strain increases. By considering the result of Fig. 4, Equation (7a) was emloyed in the calculations hereafter in the resent study. 3 Figure shows the F B -S curves in the case of t ¼ 6 mm. The results without the effect of hydrostatic ressure ( ¼ ) are also shown for reference. Regardless of, results obtained under the lane-strain state yield higher estimates of load than under the lane stress. This is because the deformation in the direction of width is constrained in the case of the lane-strain state. For the effect of hydrostatic ressure, F B when ¼ :17 is larger than that when ¼. From the results, it can be concluded that the conventional von Mises-tye yield surface essentially yields a value of F B smaller than the actual value. The lane-stress state is used hereafter, because this state yields the better aroximation in the range of the ratio of width to thickness of the secimen used in the resent study, b=t ¼ :{3:7. 8) Figure 6 shows the distribution of the bending lastic strain when S ¼ 1: mm, with the effect of the hydrostaticressure deendence taken into account. Tensile bending lastic strain is observed on the tensile side of the bent secimen, whereas no comressive bending lastic strain is observed on the comressive side. This is because of the consideration of the effect of hydrostatic-ressure deendence. As observed in Fig. 3, the tensile yield stress is less than that of comression because of the hydrostatic-ressure deendence. Therefore, yielding on the tensile side takes lace in advance of that on the comressive side.

5 Finite Element Analysis of V-Bending of Polyroylene Using Hydrostatic-Pressure-Deendent Plastic onstitutive Equation 663 Load FB / N Exerimental alculated Plane stress β =.17 t = mm Load FB / N Without friction, µ =. With friction, µ =.3 Plane stress t = 6mm β =.17 1 t = 4 mm Fig. 7 alculated and exerimental results of load-stroke curves for V- bending of various thicknesses. Fig. 8 Effect of friction coefficient on load-stroke curve for V-bending. Bending strain ε B β =.17 S = mm S = 1 mm S = mm S = mm S = 1 mm S = mm Bending strain ε B β =. S = mm S = 1 mm S = mm S = mm S = 1 mm S = mm (a) (b) Fig. Effect of hydrostatic-ressure deendence on bending strain distribution at each stroke (a) With hydrostatic-ressure deendence ¼ :17 (b) Without hydrostatic-ressure deendence ¼. Bending stress σ B / MPa - -1 β =.17 S = mm S = mm S = 1 mm S = 1 mm S = mm S = mm Bending stress σ B / MPa - -1 β =. S = mm S = mm S = 1 mm S = 1 mm S = mm S = mm (a) (b) Fig. 1 Effect of hydrostatic-ressure deendence on bending strain distribution at each stroke (a) With hydrostatic-ressure deendence ¼ :17 (b) Without hydrostatic-ressure deendence ¼. Figure 7 shows the exerimental and calculated F B -S relations for t ¼ 4, and 6 mm. The solid lines and symbols show the calculated and exerimental results, resectively. Good accordance between the exeriments and calculations during the loading rocess is observed. On the other hand, the behavior is not exressed roerly during the unloading rocess. This is because the simle isotroic hardening rule is emloyed in the resent constitutive equation. To achieve a more recise descrition, constitutive equations that can redict the comlicated hardening and time-deendent behaviors will be necessary.,1) Figure 8 shows the effect of friction on the F B -S relation. The solid and dashed lines are results calculated without friction ( ¼ :) and with friction ( ¼ :3), resectively. It is observed that F B increases because of the existence of friction. However, the effect of friction is less significant in V-bending than in other rocesses such as usetting. Therefore, the assumtion of no friction seems reasonable in the resent study. 4.4 Distribution of bending stress and strain Figures and 1 show the effects of hydrostatic-ressure

6 664 K. Hayakawa, Y. Sanomura, M. Mizuno, Y. Kasuga and T. Nakamura deendence on the distributions of bending stress and strain, resectively in the secimen. These calculations were carried out under the lane-stress state and with the thickness of t ¼ 6 mm. The abscissa of these figures shows the vertical osition from the center of the thickness and is denoted by y. In Fig., the distribution of the bending strain is almost linear, namely, elastic, when S is small. The neutral axis corresonds to the center of the thickness (y ¼ ). As S increases, the distributions become nonlinear, or lastic. Furthermore, the neutral axis moves to the comression side of the bent secimen. This is because the width of the secimen on the comression side becomes larger than that on the tensile side due to the incomressibility of PP. The movement of the axis becomes greater when the effect of hydrostatic-ressure deendence is taken into consideration. The effects of are observed significantly in case of the bending stress shown in Fig. 1. The comressive stress in ¼ :17 is much larger than that in ¼. It is confirmed that the PP characteristic of the comressive strength being higher than the tensile one can be exressed roerly by considering the effect of hydrostatic-ressure deendence. Note that the roosed constitutive equations and calculated results can also be verified by measuring the geometry of the cross section of the deformed secimen.. onclusion Finite element analyses of the V-bending behavior of hydrostatic-ressure-deendent PP were carried out using the hydrostatic-ressure-deendent lastic constitutive equation roosed by one of the resent authors. As a result, the loadstroke curves and the distributions of bending stress and bending strain were revealed to be considerably influenced by the hydrostatic-ressure deendence. It is found that more recise analysis of arts or comonents made of olymer can be realized by using the constitutive equations that can roerly describe the effect of hydrostatic ressure. Acknowledgement A art of the resent aer is the result of the activity of the Grou of Develoment of onstitutive Equation and Imlementation to General-Purose Finite Element ode. The authors would like to acknowledge the Simulation-Integrated-System ommittee of the Jaan Society of Technology of Plasticity for the financial suort of the grou. REFERENES 1) Y. Sugimoto, S. Otonari, H. Itou and H. Tomizawa: SAE Technical Paers, (18), 818. ) T. Takahara, M. Mikami, J. hen and Y. Sugimoto: SAE Technical Paers, (1), ) M. Todo, K. Arakawa and K. Takahashi: Key Engng. Mat () ) I. M. Ward and J. Sweeney: An Introduction to the Mechanical Proerties of Solid Polymers, nd Ed., (John Wiley & Sons, 4). ) Y. Sanomura: Mat. Sci. Res. Int. (3) ) Y. Sanomura: J. Soc. Mat. Sci. Jaan (1) ) Y. Sanomura and K. Hayakwa: J. Soc. Mat. Sci. Jn 3 (4) (in Jaanese). 8) K. Kazama and Y. Nagai: J. Jn. Soc. Technol. Plasticity 4 (4) 4 44 (in Jaanese). ) M. Mizuno and Y. Sanomura: Proc. ANTE 4, (4), ) Y. Sanomura and M. Mizuno: J. Jn. Soc. Strength Fract. Mat. 38 (4) 7 13 (in Jaanese).