Dart Impact Response of glassy Polymers MT E.D.Kleijne Bachelor Final Project 29th May 2005

Transcription

1 Dart Impact Response of glassy Polymers MT5.26 E.D.Kleijne Bachelor Final Project 29th May 25

2 Contents 1 Introduction 2 2 Experimental Introduction to Dart Impact Instrumented Puncture Test Experimental Results Brittle or Ductile Numerical Constitutive Approach Marc Mentat with implemented constitutive Leonov Model Back to Basic Improved Marc Mentat Simulations Conclusion 21 A List of symbols 23 B Molding conditions 24 C Thickness Measurements 25 D All Experimental Results 27 E Puncture Test 32 F Ductile - Brittle Test 35 G Typical Hydrostatic Pressure History Plot 38 1

3 Chapter 1 Introduction In today s modern fabricating industry more and more products are made out of plastics. In the process of development of new products a great deal of time is spend on testing. Therefore reliable simulations are a great addition. A fairly new method is employed to directly predict the development of yield stress distributions in injection molded product of glassy polymers [1]. The approach is based on the results of a study on temperature dependence of the evolution of yield stress during annealing of polycarbonate below T g. Computed yield stress distributions of injection molded plates prove to be in excellent agreement with the experimental values, including their dependence on mold temperature and thickness. The constitutive model derived from this approach is already validated for a 1-dimensional tensile bar tests. The next logical step is to check if this method is applicable for simulating 3-dimensional problems. An instrumental puncture test, as in a Dart Impact on a polycarbonate test-plate, is a 3D problem which gives the necessary information and is easy to simulate with a finite element program. Also, a Dart Impact Test gives information whether the specimen is tough or brittle, which is of great importance when designing products. Overall this approach provides two main goals in this report: First, the applicability of the simulation model with implemented constitutive Leonov model is tested and validated for this particular 3-dimensional problem. Secondly, an attempt is made to predict the failure behavior of polycarbonate test-plates. 2

4 Chapter 2 Experimental 2.1 Introduction to Dart Impact The mechanical properties of polycarbonate test samples can be measured in several ways. One of them is the Dart Impact Test. The International Organization for Standardization (ISO) specifies two methods for determination of impact properties on rigid plastic materials in the form of directly molded disks or squares of standard dimensions. In Part I of ISO 663 [2] the impact strength of suitable sized test specimens is determined by striking them with a weight dropped vertically from a known height (staircase method). Two methods of adjusting the energy at impact are permitted: altering the mass at constant height or altering the height at constant mass. In this technique, a uniform energy increment is employed during testing and the energy is decreased or increased after each specimen, depending upon the result (fail or not fail). When both brittle and ductile failures occur within a homogeneous group of test specimen a statistical method shall be employed on successive groups of at least ten specimens. With the lack of a staircase and the wish for comparing the force-deformation curve for simulation and experiment, Part II of ISO 663 [3] is used, and discussed in next paragraph. (a) (b) Figure 2.1: a) Staircase Method b) Instrumented Puncture Method 3

5 2.2 Instrumented Puncture Test Principle The test specimen is penetrated normal to the plane by a striker at a nominally uniform velocity as in figure (2.1b). The resulting force-deformation is electronically recorded. The force-deformation diagram, obtained in these tests, records the behavior under impact of the specimen form which several features of behavior of the material may be inferred. Furthermore, the area under the force-deformation curve represents the energy applied by the striker on the specimen, such as in figure G.1. This will give information concerning the failure behavior later on. Figure 2.2: A typical force-deformation curve with energy until failure E p and total energy E tot Specimens The specimens tested are (made of Lexan 141 Polycarbonate) molded square (7by7mm) plates. To investigate the deformation dependence on yield stress the plates are molded at a constant mold temperature of 3, 8, 13 Celsius. Besides the 1mm thickness also plates of 2mm thickness are used. To be able to couple numerical and experimental results, the 1mm plates are kept 6s in the mold and the 2mm 9s. Because of flashing at 9cmm/s at a thickness of 2mm, a smaller injection speed of 5ccm/s is chosen. All other parameters, shown in appendix B, were kept the same under all conditions. According to the ISO normalization thickness should not vary more than ±.1mm. Though three measurement points which are equidistant from one another on a circle with a radius of 1mm centered on the center of the specimen are prescribed, for practical reasons four measurement points are taken: the left, right, top and bottom of the prescribed circle. This is done with a thickness gauge with a precision of 1µm and the results are presented in appendix C. 4

6 Apparatus The apparatus consists of a mechanical part (the test device) for applying the test force and a instrument for data acquisition. The clamp and indentor are according to the ISO normalization [2, 3]. Only the clamping device for brittle plastics was available and therefor added with a newly fabricated one for ductile samples, shown as in figure 2.3. The impact test were performed on a servo-hydraulic MTS Elastomer Testing System 81. Impact speeds of more than 1ms 1 are not possible, therefor a impact speed range of 1mms 1 through 1ms 1 is chosen. Data Acquisition Figure 2.3: Clamping device for either tough or brittle plastics. The shortest impact time is in the order of 2ms. To be able to perform a numerical integration of the experimental force-deformation curve a data acquisition frequency of at least 5kHz is necessary. The software and data-acquisition of the MTS is able to record at a data acquisition frequency of approximately 6kHz and is therefore sufficient. 2.3 Experimental Results As a start the mechanical device is tuned properly in order to achieve the best possible constant speed. This is done by applying a square cyclic signal on the actuator. By comparing and evaluating the output signal with the command signal the proper parameters are found. In order to get some understanding of the force-deformation curve, a test is conducted with increasing depth of the indentor. From each specimen a polaroid image is taken and presented in appendix E. Before conducting further experiments the data acquisition is checked. 5

7 speed analysis speed analysis 1 1 normalised data [ ] normalised data [ ] time [s] (a) true speed command time [s] (b) command speed error force Figure 2.4: a) b) Figure 2.4a clearly shows the mechanical part having trouble achieving the command signal during a test experiment at 1m/s and a plate thickness of 2mm. However in the ISO normalization the velocity may not change during the penetration process more than 2% of it s value at the time of impact on the test specimen. Figure 2.4b shows that the velocity during impact (during the force-peak) changes no more than approximately 15%. Time and temperature are the two main parameters which have the most influence on instantaneous material properties of glassy polymers. Different time and temperatures scales of the thermal history of the specimens are caught in one parameter, the state parameter S. Therefor, all relevant information can be acquired if the right parameters are changed during the experiments. Changing the mold temperature when fabricating the test specimens changes the temperature history and therefor the state parameter S. Changing the impact velocity changes the time scale. Changing the experimental temperature is more complicated practically, but a rough indication of the experimental temperature dependency is possible. 6

8 Different Impact Speeds 6 Different Impact Speeds 25 5 reaction force [N] mm/s 1 cm/s 1 dm/s 1 m/s (a) reaction force [N] mm/s 1 1 cm/s 1 dm/s 1 m/s (b) 25 Different State Parameters mm 2 mm Different Plate Thickness reaction force [N] reaction force [N] T3 ~ S27.5 T8 ~ S29. T13 ~ S (c) (d) Figure 2.5: a) Plate Thickness 1mm and State Parameter 29 b) Plate Thickness 2mm and State Parameter 29 c) Impact Speed 1mm/s and Plate Thickness 1mm d) Impact Speed 1mm/s and State Parameter 29 In figure 2.5a,b a clear loading ratio dependency is shown. Figure 2.5c shows that the state parameter is of less influence on the force-deformation curve. Although these force-deformation curves (figure 2.5) are within expectations and show nice relations between force and impact speed, the applied energy until failure is far more interesting. Therefor a midpoint integration is employed on the curve, hence the integral of a force-deformation equals the working energy. When one only integrates till the point of maximum force (point of failure) the energy till failure is calculated. Figure 2.6 shows different relations of impact speed, state parameter and thickness in relation to the absorbed energy. Each specific experiment is repeated three time in order to get some more reliable results. The standard deviation of the average of three experiments are shown in table 2.1. Some standard deviations are quite large, but nevertheless the results show some interesting results concerning failure behavior. Figure 2.6a,b shows little influence of the impact speed (loading rate) on the absorbed energy and figure 2.6c,d shows very little influence of the state parameter on the absorbed energy. In appendix D all experimental data is shown. 7

9 3 1mm 7 2mm 25 6 FaalEnergie [J] FaalEnergie [J] T3 T8 T velocity decades mm/s (a) 1mm 1 T3 T8 T velocity decades mm/s (b) 2mm 25 6 FaalEnergie [J] mm/s 1 cm/s 1 dm/s 1 m/s Mal Temperature (c) FaalEnergie [J] mm/s 1 1 cm/s 1 dm/s 1 m/s Mal Temperature (d) Figure 2.6: 1mm T 3 1mm/s mm T 3 1mm/s mm T 3 1cm/s mm T 3 1cm/s.723 1mm T 3 1dm/s mm T 3 1dm/s mm T 3 1m/s mm T 3 1m/s mm T 8 1mm/s mm T 8 1mm/s mm T 8 1cm/s mm T 8 1cm/s mm T 8 1dm/s mm T 8 1dm/s mm T 8 1m/s mm T 8 1m/s mm T 13 1mm/s mm T 13 1mm/s mm T 13 1cm/s mm T 13 1cm/s mm T 13 1dm/s mm T 13 1dm/s.758 1mm T 13 1m/s mm T 13 1m/s.3667 Table 2.1: Standard Deviation of absorbed Energy 8

10 Temperature Depedency 1mm 1 Temperature Depedency 2mm 4 8 reactionforce [N] K 213K 193K 173K (a) State Parameter Depedency 1mm reactionforce [N] K 193K 173K (b) State Parameter Depedency 2mm 5 8 reactionforce [N] K T13 173K T (c) reactionforce [N] K T13 213K T (d) Figure 2.7: In figure 2.7 it is obvious that the temperature has a mayor influence on the force-deformation curve. Also some strange oscillations in some curves are visible which could mean that the data acquisition was not properly functioning. 2.4 Brittle or Ductile In the introduction two main goals are described. Precise simulations are necessary but without a correct failure criterium it is impossible to predict when a product is going to fail brittle. Failure of plastic materials is initiated by localization of the strain. When the localization occurs at a higher rate than the strain hardening brittle behavior occurs and when localization occurs at a lower rate than the strain hardening ductile behavior occurs. Many plastics materials undergo ductile/brittle transitions that result from low temperatures, high deformation rates, aging, surface embrittlement or combinations thereof. Aging and different deformation rates are already shown in figure 2.6. Both seems to have very little influence on the absorbed energy. Surface embrittlement is in this context of intrinsic behavior not really interesting. Therefore low temperatures remains to be investigated. In figure 2.7 the acquired data from the temperature experiments are shown, also see the test setting in figure 2.8. As mentioned earlier, the data is possibly not reliable enough for good analysis. Therefore a visible examination for determining brittle behavior is conducted 9

11 according to the ISO normalization. Because it is expected that the state parameter S is of influence also plates molded on different temperatures are examined. Clamping Device ThermoCouple Hemispherical Indentor (a) (b) Figure 2.8: a) Climate Box b) Inside the climate Box (a) (b) Figure 2.9: a) Ductile at 233K b) Brittle at 173K 1

12 (a) (b) Figure 2.1: a) Ductile at T mold 3 C b) Brittle at T mold 13 C In figure 2.9 a clear different result after impact is shown between ductile failure (a) and brittle failure (b). Also in figure 2.1 a clear different result after impact is shown between ductile failure (a) and brittle failure (b). Only in this experiment the temperature is kept the same at 6 C, and there the difference between ductile and brittle is the result of a different mold temperature. When testing a specimen thickness of 2mm also around 6 C the same result occurs, although less obvious. These and all other pictures are shown in appendix F. 11

13 Chapter 3 Numerical 3.1 Constitutive Approach The numerical part consists of the standard Finite Element Method, by means of the program MARC from MSC.Software Corp., in cooperation with a constitutive model that was developed at the Technical University of Eindhoven. The constitutive model used is a finite strain elastoviscoplastic model which is capable of describing post-yield behavior. This model was initially developed by Tervoort et al. [4] to describe the mechanical behavior of amorphous polymers. The constitutive model is a 3D model for non-linear elasto-viscoplastic behavior using an Eyring formulation for the viscosity. Later refinements were made with respect to strain hardening and strain softening. The refinements with respect to the strain-hardening were developed by Tervoort et al. [5]. The refinements with respect to strain-softening are based upon an approach proposed by Hasan et al. [6] which was implemented by Govaert et al. [7], and later modified [8] to give a better description of the softening behavior and to account for physical ageing. The model consists of the driving stress, σ s, which represents the contribution of the secondary interactions, and the hardening stress, σ r, representing the contribution of the hardening network. Together they form the total Cauchy stress tensor, σ: σ = σ s + σ r (3.1) The driving stress, σ s is divided into a deviatoric part, σ s d, and a hydrostatic part, σ s h. Superscript d indicates the deviatoric part, whereas superscript h indicates the hydrostatic part: σ s d = G B d e and σ s h = K(J 1)I (3.2), in which G is the elastic shear modulus, B e the isochoric elastic left Cauchy Green deformation tensor, K the bulk modulus, J the volume change ratio and I the unity tensor. A neo-hookean relation is used for the hardening stress σ r : σ r = G r Bd (3.3), where G r is the strain-hardening modulus and B the isochoric left Cauchy Green deformation tensor. The evolution of J and B are given by: J = Jtr(D) and o B e = (D d D p ) B e + B e (D d D p ) (3.4) 12

14 o, in which B e is the objective Jaumann derivative of B e, D the rate of deformation tensor and subscript p indicates the plastic part of the tensor in question. Now the plastic part of the rate of deformation tensor D p can be related to the deviatoric part of the driving stress tensor σ d s through a three dimensional non-newtonian flow rule with a stress dependent Eyring viscosity: D p = σ s d 2η( τ, p, T, S) (3.5) In this relation the viscosity η is dependent on the equivalent shear stress τ, the hydrostatic pressure p, the temperature T and the state parameter S: η( τ, p, T, S) = A rej τ exp ( U RT + µp τ/τ + S) τ sinh( τ/τ ) (3.6), where U is the activation energy, R the gas constant, A rej a pre-exponential factor defining the nul-viscosity and µ quantifies the pressure dependence of η. The equivalent shear stress, τ, the hydrostatic pressure, p, and the characteristic stress τ can be written as: τ = 1 2 tr(σ s d σ sd ), p = 1 3 tr(σ) and τ = RT V (3.7), V the activation volume and T the absolute temperature. This all together gives a final model completely governed by a stress, pressure and state dependent viscosity η, which is defined as; η( τ, p, S) = A rej τ τ/τ sinh( τ/τ ) } {{ } (I) ( ) µp exp exp (S(t, γ p )) τ }{{}}{{} (II) (III) (3.8) Here the part marked (I) represents the stress dependency of the viscosity governed by the parameter τ. Part (II) yields the pressure dependency governed by the parameter µ. Part (III) represents the dependency of the viscosity on the state of the material expressed by the state parameter S. The parameter S increases with ageing and decreases with strain softening and thus with plastic strain. In the present approach these dependencies are incorporated into the constitutive model in a simple but effective approach [9], which effectively separates both contributions: S(t, T, γ p ) = S a (t, T) R γ ( γ p ) (3.9) The softening function R γ ( γ p ) is expressed as: R γ ( γ p ) = ( 1 + (r exp( γ p )) r 1 ) r 2 1 r 1 (3.1) (1 + r r 1 ) r2 1 r 1 13

15 , where γ p is the equivalent plastic strain and r, r 1 and r 2 are effectively fitparameters which are independent of the molecular weight of the material used. The state parameter S a, which defines the thermo-dynamic state of the material, is described by: S a (t, T ) = c + c 1 log(t eff (t, τ, T ) + t a ) (3.11), where c and c 1 are constants determined at a constant strain rate of 1 2 s 1, t a is the initial age and t eff the effective time. The dependency of the parameter S on time, as given by Equation 3.11, has only to be taken into account when the experimental time-scale is of the same, or higher order than the apparent age of the material. In the case of short-term experiments, such as uniaxial tensile and compressive constant strain rate experiments, the influence of ageing is negligible since the experimental time-scales are much lower than the apparent age of the material. The same holds for the indentation experiments performed in this study. Only the current value of S a is needed for numerical evaluation. The following definition of the uniaxial yield stress can be used to obtain this value: σ y ( ɛ ) = 3τ (ln (2 ) ) 3 ɛ η,r /τ + S a µ 3 G r (λ 2 y λ 1 y 3 + µ ) (3.12) or vice versa: 3 + µ S a = σ y ( ɛ ) ln (2 ) 3 ɛ η,r /τ 1 G r (λ 2 y λ 1 ) y 3τ 3 τ (3.13) where σ y ( ɛ ) is the yield stress measured at a strain rate ɛ, G r is the strain hardening modulus and λ y the draw ratio at the yield point. 14

16 3.2 Marc Mentat with implemented constitutive Leonov Model Carapelluci and Yee [1] describe a comparison between a finite-element simulation and experimental results. Although a fairly simple linear stress-strain behavior in the finite-element modelling calculations is used (no softening behavior for example) the simulations describes the experimental data surprisingly well. Confident that the more up to date modelling of stress-strain behavior (discussed in previous chapter) is more accurate, promising simulations are expected. The constitutive model discussed in the previous paragraph has been implemented in as an user defined subroutine. First a fairly simple model of the indentation test is tried with the assumption that including an extra clamping rigid with included loadcase or a more fine mesh will not make much difference. Inc: 75 Time: 1.35e+1 rand 1.35e+1 center 1.215e+1 indentor 1.8e+1 randvast 9.45e+ 8.1e+ 6.75e+ 5.4e+ 4.5e+ 2.7e+ 1.35e+ Y 1.3e-12 Y Z X Z X indentor 1 Displacement X 1 (a) (b) Figure 3.1: a) Simulated model with boundary conditions b) Simulation result In Figure 3.1b a clear example of the principle of the Leonov model can be seen. Shown is value of the state parameter S. At the tip of the indentor the material is plastically deformed (mechanically rejenuvenated) and the state parameter S approaches zero (dark < light). 15

17 exp vs sim exp vs sim reaction force [N] reaction force [N] exp sim (a) 5 exp sim (b) Figure 3.2: a) Increasing the Strain Hardening b) Denying Hydrostatic Pressure and adding increased Strain Hardening Figure 3.2 shows the history plot of a similar simulation as shown in Figure 3.1. When comparing the simulation with the experimental data it is obvious that the force-deformation curve of the simulation is to low. Different (more fine) meshes or including a extra clamping loadcase does not make significant difference. Therefore further simulation will be conducted with the same mesh as shown in figure 3.1. In figure 3.2a the Strain Hardening is as a test increased. In figure 3.2b, first the contribution of the hydrostatic pressure is denied and furthermore is the increased Strain Hardening included. Done this, the simulation describes the experimental data very well. Hence there is no theoretical foundation for altering the parameters and the convergency problems (distortion at the end of the simulations curves) only one conclusion can be drawn: back to basic and redefine the constitutive model fit parameters. 3.3 Back to Basic The constitutive model parameter have to be analyzed again. Here we use the experimental data from Edwin Klompen [8]. In figure 3.3a the experimental data is shown. For this analysis we only use the curve with a strain rate of 1 2 [s 1 ] because this most comparable with the strain rate of the 3D Dart Impact analysis. The constitutive model consist of three different parts. The linear elastic part, the softening behavior part and the linear strain hardening part. The shear modulus G, is proportional to the angle of the first linear elastic part and is earlier fit on 634 [MPa]. First in figure 3.3b the linear fit to the strain hardening is made. This hardening fit is the value given for G r in equation 3.3. Then in figure 3.3c the softening is shown by extracting the hardening behavior of the experimental data. Then the yield point is determined. From that point we only have softening behavior. In figure 3.3d the softening is plot on a semilog scale. On this curve a second degree polynom is fit. These are the parameters r r1 and r2 in equation

18 1 Experimental Data 1 Experimental Data, Hardening fit 8 8 True Stress [ ] 6 4 Stress [ ] Strain rate 1e 2 Strain rate 1e 3 Strain rate 1e True Strain [ ] (a) 2 Strain Rate 1e 2 Hardening fit at λ 2 λ 1 [ ] (b) 1 Softening as Experimental minus Hardenig fit 1 fit function 8 σ y True Stress [ ] Strain Rate 1e 2 Hardening fit at 42.5 Softening True Strain [ ] (c) Stress [ ] Plastic Strain [ ] (d) Figure 3.3: a) Experimental Data from Klompen b) Hardening Fit c) Softening d) Fit function 17

19 True Stress [MPa] Fit Parameters 2 Experimental Data Simulation Data Shear 38 MPa Simulation Data Shear 634 MPa True Strain [ ] (a) yield stress [MPa] T g 155 C 15 C 155 C 15 C 4mm 1mm model predictions mold temperature [ C] (b) S a [ ] Figure 3.4: a) Fit Parameters b) Determining S a Parameters Old Value New Value [ ] r [-] r [-] r [-] G r [MPa] G [MPa] A rej [s] Table 3.1: Fit Parameters With these new parameters some simple one element compression simulations are an easy way to validate and possibly slightly adjust the parameters. When using these parameters in simple one-element compression simulation, figure 3.4a, the dashed line is the experimental data, the dotted line is a simulation with a shear modulus of 38 [MPa] and the solid line is a simulation with a shear modulus of 634 [MPa]. The simulation look like they do not describe the experimental data very well. There are two reasons for explaining the differences. One, though the experimental data seems to have a shear modulus of 38 [MPa], other experiments have shown that a shear modulus of 634 [MPa] is more appropriate. This different shear modulus also requires a different A rej -rejuvenated. Two, because of a higher strain hardening, a new state parameter is necessary, figure 3.4b. These new parameters have as a results that the one-element compression simulation is not in line with the experimental data from Edwin Klompen, hence the differences. Also, the fit parameter r 2 is altered from to 3.2 because the results improved radically. Concluded are the old and new parameters shown in table

20 3.4 Improved Marc Mentat Simulations With the new parameters the simulation finally almost resembles the experiment. Figure 3.5a shows the simulation data in comparison to the experimental data. At the first the simulation was again to low, but when the hydrostatic pressure contribution, µ in equation 3.8, was again turned off the simulation is almost perfectly on the experiment, hence the description of the hydrostatic isn t optimized yet. In appendix G the hydrostatic pressure increases far to high during indentation. Also interesting is the improved convergence, the simulation results in a straight line in sted of the distortions in for example figure 3.1. This is a result of the improvement of the script by Lambert Breemen [11]. Experiment vs Simulation reaction force [N] experiment sim with µ sim without µ (a) reaction force [N] sim 1 mm/s 5 sim 1 cm/s sim 1 dm/s sim 1 m/s (b) Figure 3.5: a) Simulation vs Experiment b) Simulation with different strain ratios In figure 3.5b also the dependency of the different strain ratios is clearly shown. It seems to have the same effect on the force-deformation curve as in the experiments, figure 2.5a. In figure 3.6a already a slight difference between simulation and experiment is noticeable and in figure 3.6b this difference is emphasized. 1 dm/s 1 m/s reaction force [N] reaction force [N] experiment simulation (a) experiment simulation (b) Figure 3.6: a) Simulation vs Experiment at 1 dm/s b) Simulation vs Experiment at 1 m/s 19

21 25 1 mm/s 6 5 Experiment Simulation 1 mm and 2 mm reaction force [N] S a = 4 S a = 42 S a = (a) reaction force [N] (b) Figure 3.7: a) Simulation with different S a values b) Simulation with different plate thickness In figure 3.7a the result of simulations with different values of S a are shown. In the experiments was already shown, see figure 2.5c, that different values of S a have little to no effect on the forcedeformation curve. This is also a result of the simulation. In figure 3.7b shows that geometric differences in for example a different thickness of the specimen has no effect on the accuracy of the simulation. In figure 3.7 is a more detailed example of mechanical rejuvenation at the tip of the indentor shown during the simulation of the 2mm thickness. Inc: 35 Time: 6.3e e e e e e e e e e e e+ Y Z X indentor User Defined Variable 2 1 Figure 3.8: Mechanical Rejuvenation at the tip of the indentor 2

22 Chapter 4 Conclusion In the introduction, two main goals are described. Testing and validating the applicability of the simulation model with implemented constitutive Leonov model for the particularly problem of a Dart Impact test. And another goal was to make an attempt for describing the failure behavior of the polycarbonate test-plates. Both goals shall be discussed separately and finally concluded in a final conclusion. Simulations At first few problems were expected involving simulating the Dart Impact Test with Marc Mentant in combination with the implemented constitutive Leonov Model. The first simulations seemed promising, as in figure 3.1, and the bending away from the experimental data was allocated to either denying friction contribution or convergence problems. Both proved not to be the case. Although the simulations are quite precise at the end, still the bend in the force-deformation curve of every simulation versus the straight line of the experimental data is still a mystery. Also describing of the strain rate dependency in the constitutive model is probably not accurate enough. Although the strain rate dependency is very clear to see, figure 3.5d, at higher strain rates of 1 ms 1 the simulations is are not that precise as in the simulations of 1 mms 1. The geometric dependency is very promising, the simulation is almost as precise for a plate thickness of 2 mm as for a plate thickness of 1 mm. Overall can be said that the constitutive model has evolved to a point that it can certainly be usable in complex three dimensional problems, such as a Dart Impact Test. This said with a marginal comment that the description of the strain rate dependency and the mystery of the bend is not optimized yet. Failure Behavior As mentioned earlier, precise simulations are much less worth without a correct failure criterium. It is very interesting to investigate which parameters are of influence on this failure behavior. In the experiment three parameters are tested. The influence of the strain rate and thermal history, or better the state parameter are shown in figures 2.6. One can conclude from this figures that the state parameter has little to no influence, it doesn t change the absorbed energy. Also the different strain rates seems to have very little influence on the absorbed energy. Here a interesting comment can be made. In contradiction to the state parameter does the different strain ratios change the force-deformation curves. What either mathematically or experimental can be proved or shown is that a higher strain rate results in a higher reaction force. With the assumption that absorbed energy is the same, the specimen has no choice then to fail at less deformation, in order to keep the area under the force-deformation curve (absorbed energy) the same. Is the failure behavior of polycarbonate only dependent of the strain rate? No, as all other plastic materials the temperature of the material is of great importance. During testing in a climate 21

23 box, a clear transition between ductile failure behavior to brittle failure behavior was detected. Also around a temperature of 6 C specimens with a lower state parameter failed ductile while specimens with a higher state parameter failed brittle. At this point, the conclusion that the state parameter has no influence on the failure behavior proved to be incorrect. Overall there s too little information to exactly described the failure behavior of polycarbonate. But it is clear that absorbed energy and ductile or brittle behavior are dictating the point of failure of a polycarbonate. 22

24 Appendix A List of symbols Symbol Description S State Parameter [-] σ s Driving Stress [MP a] σ r Hardening Stress [MP a] σ Cauchy Stress Tensor [M pa] a d Deviatoric part of a a h Hydrostatic part of a G Elastic shear modulus [M P a] B e Isochoric elastic left Cauchy Green deformation tensor [-] K Bulk modulus [M P a] J Volume change ration [ ] I Unity tensor [ ] G r Strain hardening tensor [ ] o B e Objective Jaumann derivative of B e [ ] D Rate of deformation tensor [ ] a p Plastic part of a η Viscosity [P a s] τ Equivalent shear stress [M pa] p Hydrostatic pressure [P a] T Absolute temperature [K] U Activation energy [kj/mol/k] A rej pre-exponential factor defining nul-viscosity [s] µ Pressure dependency [ ] τ Characteristic stress [MP a] R Gas constant [J/mol/K] k Boltzmann s constant [J/K] V Activation volume [m 3 /mol] γ p Equivalent plastic strain [ ] r, r 1, r 2 Effective fit parameters [ ] R γ Softening function t a Initial age [s] t eff Effective time [s] σ y Yield stress [MP a] ε Strain rate [ ] λ y Draw ratio at yield point [ ] 23

25 Appendix B Molding conditions Table B.1: 1mm thickness Code Value [ ] Description V [ccm] Doseervolume Q35 9. [ccm/s] Injectiestroom p35 22 [bar] Injectiedruk V [ccm] Omschakelpunt t311.1 [s] Reactietijd p311 5 [bar] Referentiepunt 1 (Nadruk) t [s] Tijd (Nadruk) Table B.2: 2mm thickness Code Value [ ] Description V [ccm] Doseervolume Q35 5. [ccm/s] Injectiestroom p35 22 [bar] Injectiedruk V [ccm] Omschakelpunt t311.1 [s] Reactietijd p311. [bar] Referentiepunt 1 (Nadruk) t312. [s] Tijd (Nadruk) 24

26 Appendix C Thickness Measurements specimen left [mm] right [mm] top [mm] bottom [mm] Average [mm] 1t31ms t31ms t31ms t81ms t81ms t81ms t131ms t131ms t131ms t31ms t31ms t31ms t81ms t81ms t81ms t131ms t131ms t131ms t31dms t31dms t31dms t81dms t81dms t81dms t131dms t131dms t131dms t31dms t31dms t31dms t81dms t81dms t81dms t131dms t131dms t131dms

27 specimen left [mm] right [mm] top [mm] bottom [mm] Average [mm] 1t31cms t31cms t31cms t81cms t81cms t81cms t131cms t131cms t131cms t31cms t31cms t31cms t81cms t81cms t81cms t131cms t131cms t131cms t31mms t31mms t31mms t81mms t81mms t81mms t131mms t131mms t131mms t31mms t31mms t31mms t81mms t81mms t81mms t131mms t131mms t131mms

28 Appendix D All Experimental Results 27

29 1t31mms 1t31cms Ef1 = [J] Ef2 = [J] Ef3 = 2.92 [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = 1.41 [J] Ef1 = 2.31 [J] Ef2 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = [J] (a) 1t31dms (b) 1t31ms Ef1 = [J] Ef2 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = [J] Ef1 = [J] Ef2 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = [J] (c) 1t81mms (d) 1t81cms 25 Ef1 = [J] Ef2 = [J] 25 Ef1 = [J] Ef2 = [J] Ef3 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = 1.54 [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = [J] (e) (f) 28

30 1t81dms 1t81ms Ef1 = [J] Ef2 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd =.8654 [J] Ef1 = [J] Ef2 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = 1.57 [J] (g) 1t131mms (h) 1t31cms 25 Ef1 = [J] Ef2 = [J] 25 Ef1 = 2.31 [J] Ef2 = [J] Ef3 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = [J] (i) (j) Ef1 = [J] Ef2 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = [J] 1t131dms Ef1 = [J] Ef2 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = [J] 1t131ms (k) (l) 29

31 Ef1 = [J] Ef2 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = [J] 2t31mms (m) Ef1 = [J] Ef2 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd =.7227 [J] 2t31cms (n) 2t31dms 2t31ms Ef1 = [J] Ef2 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = [J] Ef1 = [J] Ef2 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = [J] (o) 2t81mms (q) Ef1 = [J] Ef2 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = [J] Ef1 = [J] Ef2 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = [J] (p) 2t81cms (r) 3

32 Ef1 = [J] Ef2 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = [J] 2t81dms Ef1 = [J] Ef2 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = [J] (s) 2t131mms Ef1 = [J] Ef2 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd =.758 [J] (u) 2t131dms (w) Ef1 = [J] Ef2 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = [J] 2t81ms Ef1 = [J] Ef2 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = 3.16 [J] (t) 2t131cms Ef1 = [J] Ef2 = [J] Ef3 = [J] Ff1 = [N] Ff2 = [N] Ff3 = [N] Egm = [J] Estd = [J] (v) 2t131ms (x) Figure D.1: 31

33 Appendix E Puncture Test reactionforce [N] mm I H G F E D C B (a) J K L (b) (c) (d) 32

34 (e) (f) (g) (h) (i) (j) 33

35 (k) (l) Figure E.1: a) Force-Deformation Curve b-l) Polaroid image of the specimen after impact 34

36 Appendix F Ductile - Brittle Test (a) (b) (c) (d) 35

37 (e) (f) (g) (h) (i) (j) 36

38 (k) (l) (m) (n) Figure F.1: a-n) Image after impact test 37

39 Appendix G Typical Hydrostatic Pressure History Plot 14 A typical hydrostatic pressure history plot 12 Hydrostatic Pressure [MPa] Displacement [mm] Figure G.1: A typical hydrostatic pressure history plot 38

40 References [1] L.E.Govaert, T.A.P.Engels, E.T.J.Klompen, G.W.M.Peters, and H.E.H.Meijer. Processing induced properties of glassy polymers: Development of the yield stress in polycarbonate. Dutch Polymer Institute (DPI), Section Materials Technology (MaTe), 1, 23. [2] International Organisation of Standardisation. Determination of multiaxial impact behaviour of rigid plastics - part 1: Falling dart method , [3] International Organisation of Standardisation. Determination of multiaxial impact behaviour of rigid plastics - part 2: Instrumented puncture test , [4] T.A.Tervoort, R.J.M Smit, W.A.M Brekelmans, and L.E.Govaert. A constitutive equation for the elasto-viscoplatic deformation of glassy polymers. Mechanics of Time-Dependent Materials, (1(3)): , [5] T.A.Tervoort and L.E.Govaert. Strain-hardening behavior of polycarbonate in the glassy state. Journal of Rheology, (44(6)): , 2. [6] O.A.Hasan, M.C.Boyce, X.S.Li, and S.Berko. An investigation of the yield and post-yield behavior and corresponding structure of poly(methylmethacrylate). Journal of Polymer Science, Part B: Ploymer Physics, (31(2)): , [7] L.E.Govaert, P.H.M.Timmermans, and W.A.M.Brekelmans. The influence of intrinsic strain softening on strain localization in polycarbonate: Modelling and experimental validation. Journal of Engineering Materials and Technology, (122): , 2. [8] E.T.J.Klompen. Mechanical Properties of solid polymers: Constitutive modelling of long and short term behavior, volume 1. Technische Universiteit Eindhoven, [9] T.A.P.Engels, L.E.Govaert, and H.E.H.Meijer. How strong is your product. Journal of Engineering Materials and Technology, (1):1 1, 23. [1] L.M.Carapellucci and A.F.Yee. Some problems associated with the puncture testing of plastics. Polymer engineering and science, 27, [11] L.C.A.v.Breemen. Implementation and validation of a 3d model describing glassy polymer behavior. MT4.24, 1,