Analysis for constitutive models of high impact polystyrene in rubber⁃like state

doi:10.19491/j.issn.1001-9278.2022.04.014

Abstract

Abstract:

A series of rotary rheological experiments were conducted for high?impact polystyrene （PS?HI） under different temperatures to analyze the precision of conventional obtained from usually used viscous models. Then， a flow theory based on Phan?Thien Tanner （PTT） model was established and the corresponding algorithm was constructed to study the viscoelastic characteristics of rubber?like state PS?HI. Both theoretical and numerical results indicated that the Cross and Carreau?Yasuda models could describe the rheological characteristics of PS?HI better compared to other viscous models. The viscoelastic PTT model exhibited a significant improvement in precision， indicating that PS?HI presented some elasticity in the rubber?like state.

Key words: constitutive model, viscosity, viscoelastic, relaxation time spectrum

CLC Number:

TQ325.2

CHEN Tengfei, MO Qimian, ZHANG Shixun, CAO Wei. Analysis for constitutive models of high impact polystyrene in rubber⁃like state[J]. China Plastics, 2022, 36(4): 89-94.

Figures/Tables 6

Fig.1. Measured viscosities of PS?HI at 150~180 ℃

本构模型	本构模型关系式	默认参数	参数1	参数2	参数3	参数4	参数5	参数6
Cross⁃ WLF	$η = η 0 1 + λ γ ˙ m × H (T)$ $H T = c 1 T r - T a c 2 + T r - T a - c 1 T - T a c 2 + T - T a$	—	$η 0$ =2.22×10¹⁰	$m$ =0.721 924	$λ$ =10.292 414	$c 1$ =23.920 985	$c 2$ =100.939 89	—
Carreau⁃Yasuda⁃ WLF	$η = {η ∞ + (η 0 - η ∞) × [1 + (λ γ ˙) a] n - 1 a} × H (T)$	$η ∞$ =0	$η 0$ =2.22×10¹⁰	$n$ =0.238 951	$λ$ =7.643 827	$a$ =0.637 159	$c 1$ =23.920 985	$c 2$ =100.939 89
Bird⁃Carreau⁃ WLF	$η = [η ∞ + (η 0 - η ∞) × (1 + λ 2 γ ˙ 2) n - 1 2] × H (T)$	$η ∞$ =0	$η 0$ =2.22×10¹⁰	$n$ =0.316 841	$λ$ =16.358 112	$c 1$ =23.920 985	$c 2$ =100.939 89	—
Power⁃WLF	$η = K (λ γ ˙) n - 1 × H (T)$	K=1	$n$ =0.33	$λ$ =2.72×10^-7	—	$c 1$ =23.920 985	$c 2$ =100.939 89	—
Log⁃Log⁃WLF	$η = η 0 × E {a 0 + a 1 [l o g (γ ˙ / γ ˙ c)] + a 11 [l o g (γ ˙ / γ ˙ c) 2]} × H T$	—	$η 0$ =2.22×10¹⁰	$a 0$ =1.178 54	$a 1$ =-0.674 0	$a 11$ =0	$c 1$ =23.920 985	$c 2$ =100.939 89
Modified Cross⁃ WLF	$η = η 0 (1 + λ γ ˙) m × H (T)$	—	$η 0$ =2.22×10¹⁰	$m$ =0.709 469	$λ$ =12.672 873	$c 1$ =23.920 985	$c 2$ =100.939 89	—

本构模型	本构模型关系式	默认参数	参数1	参数2	参数3	参数4	参数5	参数6
Cross⁃ WLF	$η = η 0 1 + λ γ ˙ m × H (T)$ $H T = c 1 T r - T a c 2 + T r - T a - c 1 T - T a c 2 + T - T a$	—	$η 0$ =2.22×10¹⁰	$m$ =0.721 924	$λ$ =10.292 414	$c 1$ =23.920 985	$c 2$ =100.939 89	—
Carreau⁃Yasuda⁃ WLF	$η = {η ∞ + (η 0 - η ∞) × [1 + (λ γ ˙) a] n - 1 a} × H (T)$	$η ∞$ =0	$η 0$ =2.22×10¹⁰	$n$ =0.238 951	$λ$ =7.643 827	$a$ =0.637 159	$c 1$ =23.920 985	$c 2$ =100.939 89
Bird⁃Carreau⁃ WLF	$η = [η ∞ + (η 0 - η ∞) × (1 + λ 2 γ ˙ 2) n - 1 2] × H (T)$	$η ∞$ =0	$η 0$ =2.22×10¹⁰	$n$ =0.316 841	$λ$ =16.358 112	$c 1$ =23.920 985	$c 2$ =100.939 89	—
Power⁃WLF	$η = K (λ γ ˙) n - 1 × H (T)$	K=1	$n$ =0.33	$λ$ =2.72×10^-7	—	$c 1$ =23.920 985	$c 2$ =100.939 89	—
Log⁃Log⁃WLF	$η = η 0 × E {a 0 + a 1 [l o g (γ ˙ / γ ˙ c)] + a 11 [l o g (γ ˙ / γ ˙ c) 2]} × H T$	—	$η 0$ =2.22×10¹⁰	$a 0$ =1.178 54	$a 1$ =-0.674 0	$a 11$ =0	$c 1$ =23.920 985	$c 2$ =100.939 89
Modified Cross⁃ WLF	$η = η 0 (1 + λ γ ˙) m × H (T)$	—	$η 0$ =2.22×10¹⁰	$m$ =0.709 469	$λ$ =12.672 873	$c 1$ =23.920 985	$c 2$ =100.939 89	—

Tab.1. Fitted parameters of the six viscous models

温度/℃	估计量	Cross	Carreau⁃Yasuda	Bird⁃Carreau	Power	Log⁃Log	Modified Cross	PTT
155	平均相对误差	0.233 63	0.205 89	0.204 27	0.320 58	0.201 804	0.200 58	0.053 45
	最大相对误差	0.460 16	0.347 38	0.465 29	0.432 36	0.497 638	0.393 77	0.219 95
	方差	0.025 27	0.012 51	0.023 70	0.009 87	0.031 457	0.014 42	0.002 36
165	平均相对误差	0.058 50	0.027 35	0.122 82	0.152 17	0.136 356	0.080 43	0.031 06
	最大相对误差	0.327 73	0.177 04	0.361 85	0.526 02	0.415 751	0.279 95	0.052 61
	方差	0.006 88	0.001 65	0.009 53	0.024 62	0.012 198	0.004 34	2.284 2×10^-4
175	平均相对误差	0.201 41	0.215 31	0.259 22	0.536 74	0.266 152	0.234 90	0.033 71
	最大相对误差	0.289 12	0.340 16	0.348 44	1.732 12	0.417 907	0.332 32	0.081 19
	方差	0.009 34	0.012 89	0.008 55	0.155 25	0.009 253	0.010 51	5.016 8×10^-4

Tab. 2. Fitted relative errors at three temperature

Fig.2. Comparison between the measured and calculated viscosities at different temperature

温度/℃	$α$	$λ 1$	$η 1$	$λ 2$	$η 2$	$λ 3$	$η 3$
155	1	20.7	365 238	1.13	9 912	0.16	5 134
165	1	21.5	175 515	0.90	8 941	0.11	3 156
175	1	20.3	99 678	0.97	7 951	0.14	3 132

温度/℃	$α$	$λ 1$	$η 1$	$λ 2$	$η 2$	$λ 3$	$η 3$
155	1	20.7	365 238	1.13	9 912	0.16	5 134
165	1	21.5	175 515	0.90	8 941	0.11	3 156
175	1	20.3	99 678	0.97	7 951	0.14	3 132

Tab.3. Parameters of PTT model at three temperature

Fig.3. Effects of nonlinear parameter of PTT on viscosities at 165 ℃

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