Mechanical Engineering Journal
Injection molding temperature dependence of elastic
coefficients obtained using three-point bending tests to
ascertain thermoplastic polymer coefficients
Tetsuo TAKAYAMA* and Yuki MOTOYAMA*
1. Introduction
Thermoplastic resins have many characteristics such as being lighter than metals and ceramics while providing excellent processability. Therefore, they are used in widely diverse applications from daily commodities to transportation equipment. One characteristic is that the resins have strength and an elastic modulus that are lower than those of other materials. This feature provides important benefits for applications requiring flexibility, but it presents shortcomings for applications requiring load-bearing capacity.
One method to improve load-bearing capacity is to compound thermoplastic resins with other materials. Among them, fiber-reinforced thermoplastic plastics (FRTP) compounded with glass fiber and carbon fiber have become well known as a material with improved load-bearing capacity. Particularly in recent years, molding technologies using the thermoplasticity of FRTP have been attracting attention. Structural analyses of the obtained composite material molded products are also being conducted (Kurkin, 2020). Elastic coefficients typified by Young's modulus and Poisson's ratio are necessary for structural analysis (Tsai, 2003). Young's modulus can be obtained from uniaxial tensile tests. Poisson's ratio can be found from non-contact displacement measurements using a laser and image analysis (Kumar, 2008, 2015). Other elastic modulus measurement methods use ultrasonic waves, but it is difficult to obtain accurate measurements because polymer materials are viscoelastic. For metal materials, contact displacement measurements using strain gauges are also performed. With this method, the strain is obtained directly. For that reason, one can derive Poisson's ratio with high accuracy, but the rigidity of the strain gauge cannot be neglected because thermoplastic resin has lower rigidity than
* Graduate School of Organic Materials Science, Yamagata University 4-3-16 Jonan Yonezawa, Yamagata, Japan 992-8510
E-mail: [email protected]
Abstract
Thermoplastic resins, which are lighter than metals and ceramics, have many features such as excellent processability. They are used in widely various fields for daily necessities and for transportation equipment. Even for the same material, the elastic properties of the thermoplastic resin can vary depending on the molecular weight, molding temperature and conditions. Therefore, the elastic properties required for structural analysis must be evaluated each time. After this study derived a formula for yield initiation stress of injection-molded thermoplastics, the formula was applied for a method of obtaining the longitudinal elastic modulus and Poisson's ratio from results of a three-point bending test. From actual examination using polypropylene (PP) and polystyrene (PS), the calculated longitudinal elastic modulus and Poisson's ratio showed good agreement with literature values. Results clarified that these values depend on the injection molding temperature. Furthermore, the bulk modulus and shear modulus were obtained from the longitudinal modulus and Poisson's ratio. Their injection molding temperature dependences were investigated. The shear modulus showed the same tendency as that of the longitudinal elastic modulus, but results indicated the dependence of bulk modulus as small in PP: bulk modulus of PS increased with increasing injection molding temperature.
Keywords : Thermoplastics, Injection molding, Elastic modulus, Poisson’s ratio, Three-point bending tests
metals or ceramics. In the cases of metals and ceramics, structural analysis using the elastic coefficients of literature values can be especially helpful, but for thermoplastic resins, even if the same material is used, characteristics will differ depending on the molecular weight, molding processing temperature and other conditions. Therefore, the elastic coefficients necessary for structural analysis must be evaluated for each test (Fujiyama, 1997).
Based on the background presented above, this study proposes a simple model for predicting the yield stress of injection-molded thermoplastics. We also propose a method of estimating Poisson's ratio and the longitudinal elastic modulus of thermoplastics based on three-point bending test results. We evaluated polypropylene and polystyrene injection-molded products, compared them with values referred from the literature, and discussed their usefulness. This method is not limited to injection-molded products, but is regarded as effective as a method for evaluating the elastic coefficients of press-molded products.
2. Theory
Figure 1 presents a relation between the residual stress generated inside the molded product and the shear angle assumed for these analyses.
Fig. 1 Relation between residual stress generated inside the molded product and the shear angle.
Heat expansion and cooling shrinkage that occur during processing cause residual strain, which eventually generates this residual stress. Because this residual strain is essentially isotropic, it does not persist as actual strain. Plastic deformation of the material results from shear deformation. Therefore, the resistance force that is necessary for yield initiation, which is the initiation of plastic deformation, can be regarded as shear deformation initiation stress. As described in this section, this resistance force is regarded as frictional force generated on the shear plane. According to the adhesion theory, frictional force N is expressed by the following equation (1) (Ernst and Merchant, 1940; Holm, 1938; Tomlinson, 1927).
𝑁 = 𝐴
𝑟𝜏
𝑦(1)
In that equation, Ar represents the true contact area; τy denotes the yield initiation shear stress. Because the friction inside the solid is assumed, Ar is set to 1. The above-described residual strain is inherent in the molded article of thermoplastic polymer material. From residual stress resulting from this strain, the stress component perpendicular to the shear plane is extracted and set as σt. It is expressed as equation (2).𝑁 = 𝜏
𝑦= 𝜇𝑊 (3)
In that equation, µ is the friction coefficient, W represents the force generated in the direction perpendicular to the friction surface, where σt is applicable. According to this theory, µ is set to 1. Substituting equation (2) into equation (3) and considering it together with equation (1), τy can be expressed as equation (4).𝜏
𝑦= 𝜇𝑊 = 𝜎
𝑡= 𝛼∆𝑇𝐸 cos 𝜃 (4)
Here, when the Mises yield condition for uniaxial tension is applied, the yield initiation tensile stress σy is expressed by equation (5) as shown below.𝜎
𝑦=
√
3𝜏
𝑦=
√
3𝛼∆𝑇𝐸 cos 𝜃
(5)
From this equation, results show that the yield initiation tensile stress of thermoplastic polymer materials depends on the processing temperature because of ΔT.The yield initiation tensile stress described in Eq. (5) above is the yield initiation stress under uniaxial tensile loading accompanied by Poisson deformation. Generally, the evaluation of mechanical properties is performed using a uniaxial tensile test. The maximum value obtained at the initial stage of loading is defined as the yield stress. The yield initiation tensile stress described in this section is the stress at which the material yields and shear deformation begins. It differs from the yield stress obtained in the uniaxial tensile test. To evaluate the yield initiation stress quantitatively, conducting a three-point bending test and obtaining the yield initiation flexural stress is simple and effective. Figure2 shows deformation model caused by 3-point bending load. In a three-point bending load, deformation concentrates in the loaded area. Therefore, Poisson contraction is suppressed, especially when tested flatwise. The deformation is similar to pure extensional deformation. Assuming pure elongation deformation, equation (5) can be transformed into equation (6) as presented below.
Fig.2 Deformation model caused by 3-point bending load.
𝜎
𝑓𝑦=
√
3(1 + υ)𝜏
𝑦=
√
3(1 + υ)𝛼∆𝑇𝐸 cos 𝜃 (6)
However, Timoshenko reported that flexural modulus Ef, obtained by a three-point bending load is obtainable as the sum of the term derived from the bending moment and the term derived from shear stress shown in the following equation (7) (Timoshenko, 1963).𝐸
𝑓= { 1
𝐸
11+ 32(
ℎ
𝐿)
21
𝐺
13}
−1(7)
Therein, Ε11 is the apparent elastic modulus obtained during pure extensional deformation, G13 is the out-of-plane shear elastic modulus, h represents the specimen thickness, and L stands for the distance between spans. In addition, the longitudinal elastic moduli E and Ε11 and G13 have the relation of the following equations (8) and (9) when Poisson's ratio is υ.
(1 − 2𝜐)(1 + 𝜐)
(1 − 2𝜐)(1 + 𝜐)
𝜐
= 𝐸
𝐺
13(9)
The relation between the flexural modulus and the longitudinal elastic modulus is expressed by the following equation (10) in which Ε11 and G13 are eliminated using equations (8) and (9) in equation (7).𝐸
𝑓= {(1 + 𝜐)(1 − 2𝜐)
(1 − 𝜐)
+ 32(
𝐿)
ℎ
2
(1 + 𝜐)(1 − 2𝜐)
𝜐
}
−1
𝐸 (10)
Substituting equation (10) into equation (9), the flexural yield stress is expressed as equation (11) below.
𝜎
𝑓𝑦=
√
3𝛼∆𝑇 𝐸
𝑓cos 𝜃 {(1 + 𝜐)
2
(1 − 2𝜐)
(1 − 𝜐)
+ 32(
𝐿)
ℎ
2
(1 + 𝜐)
2(1 − 2𝜐)
𝜐
} (11)
When equation (6) is transformed, the longitudinal elastic modulus can be expressed as equation (12) below.
𝐸 =
√
𝜎
𝑓𝑦3(1 + 𝜐)𝛼∆𝑇 cos 𝜃
(12)
Poisson's ratio can be estimated based on Eq. (11) from the flexural yield stress and flexural modulus obtained from three-point bending tests. Furthermore, it is possible to obtain the longitudinal elastic modulus from Poisson's ratio and Eq. (12). Two elastic coefficients can be found by conducting a three-point bending test based on this theory. According to material mechanics, if two elastic coefficients are known, then all elastic coefficients are calculable. Therefore, all elastic coefficients are obtainable from three-point bending test results.
3. Experimental Procedure
Pellets of one type of polystyrene (PS; Toyo Styrene Co. Ltd.) were used as the amorphous polymer. Polypropylene of two types with differing degrees of rigidity (PP; Japan Polypropylene Corp.) was used as the crystalline polymer: PP with low rigidity is called Homo-PP; PP with high rigidity is called HM-PP. Table 1 presents the trade names, grades, melt flow rates (MFRs), and densities of the respective materials.
Table 1 Material properties.
*1 MFRs were obtained at 230°C, 2.160 kgf.
After these pellets were filled into a micro injection molding machine (Shinko Cellbic Co., Ltd., C. Mobile0813), injection molding was applied to obtain beam-shaped molded products. Figure3 shows geometry of beam specimen. The molded product has 2 mm thickness, 5 mm width, and 50 mm length. Table 2 presents the injection molding conditions. For this study, the mold temperature, Tmold, was fixed. The shape of the molded product is not strictly a rectangular parallelepiped because it has a draft. However, the error between the moment of inertia of area calculated with the exact shape and that calculated as a rectangular parallelepiped is less than 0.1%, and it was judged that it can be regarded as a rectangular parallelepiped approximately. Since no sink marks or warpages that changed the cross-sectional shape were found in the molded product obtained under the conditions specified in this paper, it was judged that it could be used for the 3-point bending test. Then the injection molding temperature, Tinj, was set at three levels within the range in which good products were obtained.
Table 2 Injection molding conditions.
Fig.3 Geometry of beam specimen.
Fig.4 Determinations of flexural properties obtained from results of three-point bending tests.
bending test conducted in this study, it was judged that the influence of the flow behavior near the gate was extremely small. From the obtained load–deflection curve, flexural strength σf and flexural modulus Ef were calculated using the following equations (13) and (14).
𝜎
𝑓= 3𝑃
𝑓𝐿
2𝑏ℎ
2(13)
𝐸
𝑓=
𝜎
𝜀
𝑓1− 𝜎
𝑓2𝑓1
− 𝜀
𝑓2(14)
Here, b in Eq. (13) represents the specimen width; also, Pf denotes the maximum bending load. In Equation (14), σf1 stands for the bending stress when εf1 reaches 0.0025. Also, σf2 represents the bending stress when εf2 reaches 0.0005. The εf1 and εf2 bending strains are calculated using the following equation (15).𝜀
𝑓=
6𝛿
𝐿
𝑓2ℎ
(15)
In that equation, δf represents the deflection. E and υ were obtained from the obtained σf and Ef based on the theory described in Section 2. The test was conducted five times for each sample. Then the average value was taken as the characteristic value.In addition, tensile tests were conducted using a small universal testing machine (IMADA Co. Ltd., FSA-1KE-1000N-L) in a test environment of 23 ± 1 ° C at a 2 mm/min loading rate. Nominal stress σ – true strain εt curves were obtained from the obtained load–displacement curves using the following equations.
𝜎 = 𝑃
𝑏ℎ
(16)
𝜀
𝑡= 𝑙𝑛 (1 + 𝛿
𝐿
𝑐
)
(17)
Therein, P, t, and w respectively represent the load, specimen thickness, and the width of parallel part. Also, δ and Lc respectively denote the displacement and chuck distance. Lc was fixed at 22 mm. From the obtained σ – εt curve, tensile modulus Et were calculated using the following equation (18).𝐸𝑡= 𝜎𝜀1− 𝜎2
𝑡1− 𝜀𝑡2 (18) In Equation (14), σt1 stands for the nominal stress when εt1 reaches 0.0025. Also, σt2 represents the nominal stress when
εt2 reaches 0.0005. The test was conducted five times for each sample. Then the average value was taken as the characteristic value.
4. Results and Discussions
(a) Homo-PP (b) HM-PP
(c) PS
Fig. 5 Flexural stress–flexural strain curves obtained from three-point bending test.
Figure 6 presents the injection molding temperature dependence of flexural strength. Flexural strength of all materials tended to decrease as the injection molding temperature increased. Figure 7 presents the injection molding temperature dependence of the flexural modulus. The flexural modulus of each material tended to decrease concomitantly with increasing injection molding temperature.
Fig. 7 Injection molding temperature dependences of flexural modulus, Ef. Table3 Mechanical properties obtained from three-point bending tests.
(Values within parentheses denote standard deviation.) *1 These values were calculated by Eq. (12).
*2 These values were estimated by Eq. (11). *3 These values were calculated by Eq. (5).
Fig. 8 Injection molding temperature dependences of Poisson’s ratio, υ.
We estimated υ based on results obtained in Section 2. Figure 8 and Table 3 portrays the injection molding temperature dependence of υ. The figure also presents values for comparison obtained from the literature (Baltenneck, 1997; Sjögren, 1997).
Results show that υ exhibits a tendency to increase as the injection molding temperature increased. The value of PS was close to the value reported by Baltenneck. The υ of PP obtained in this study showed good agreement with the value reported by Sjögren. Comparison of the results obtained for PP and PS confirmed that υ estimated using this method is valid.
For this study, E was calculated from the estimated υ and Ef. Figure 10 shows the injection molding temperature dependence of E.
Results demonstrate that E has the same injection molding temperature dependence as that shown by Ef. Moreover, it tends to decrease concomitantly with increasing temperature. This tendency was more prominent than that found for Ef.
Furthermore, the bulk modulus K and shear modulus G were calculated from the estimated υ and E using the following equations (16) and (17).
𝐾 =
3(1 − 2𝜐) (16)
𝐸
𝐺 =
2(1 + 𝜐) (17)
𝐸
Fig. 10 Injection molding temperature dependences of bulk modulus, K.
Table 4 Comparison of tensile modulus and elastic modulus obtained from three-point bending tests.
(Values within parentheses denote standard deviation.) *1 These values were calculated by Eq. (12).
Fig. 11 Injection molding temperature dependences of shear modulus, G.
Figure 11 shows the injection molding temperature dependence of G. Actually, G showed the same injection molding temperature dependence as E. It tended to decrease concomitantly with increasing temperature.
modulus of PP. This is partly because the load region of the elastic modulus evaluated in the tensile test and the load region assumed in theory are different, and the elastic modulus at the start of yield and the elastic modulus at the initial stage of loading are different in PP with a large non-linearity. In order to treat the load region up to the start of yield as an elastic region, a model that can reproduce the non-linear behavior in detail is required, and this problem will be an issue for the future.
The discussion presented above demonstrates that the elastic modulus of injection-molded thermoplastic resin depends on the injection molding temperature and on the orientation of molecular chains. Results also demonstrate that the bulk modulus tends to be different from other moduli.
5. Conclusion
For this study, we derived a formula for the yield initiation stress of injection-molded thermoplastics. Subsequently, we applied this formula to demonstrate a method for obtaining the longitudinal elastic modulus and Poisson's ratio from three-point bending test results. Actual examinations of PP and PS revealed that the obtained Poisson's ratio showed good agreement with values referred from the literature. Results clarified that these values are dependent on the injection molding temperature.
The bulk modulus and shear modulus were ascertained from the longitudinal modulus and Poisson's ratio. Their injection molding temperature dependence was evaluated. The shear modulus showed the same tendency as that of the longitudinal modulus, but results indicated the dependence of bulk modulus as small in PP: bulk modulus of PS increased with increasing injection molding temperature.
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