A novel dual-hardening viscoelastic-plastic constitutive model for thermoplastic resins

doi:10.1007/s10483-026-3354-8

Abstract

Abstract:

This study examines the viscoelastic-plastic behavior of thermoplastic resin poly-ether-ether-ketone (PEEK) under high temperature and strain rate conditions, highlighting its potential in aerospace applications due to its impact resistance. A dual-hardening constitutive model that combines physical and phenomenological approaches is developed to simulate the mechanical behavior of PEEK. The model explicitly incorporates its marked tension-compression asymmetry in plasticity and relaxation, along with thermal softening at high strain rates, enabling accurate predictions over a wide range of temperatures and strain rates with minimal parameters. This study establishes a comprehensive workflow from experimentation to finite element (FE) simulation for thermoplastic resins. Uniaxial tensile and compression tests (23 °C–180 °C, 0.002 29 s^-1–0.193 61 s^-1) and split Hopkinson pressure bar (SHPB) tests (1 094.08 s^-1–5 957.88 s^-1) are performed to capture stress-strain responses across various conditions, with small-scale specimens enhancing fracture strain measurement accuracy, and quantify the Taylor-Quinney factor of the PEEK material during the adiabatic heating process. The findings demonstrate that the proposed constitutive model effectively predicts yield points across different strain rates and temperatures, with parameters easily obtainable through simple experimental methods, enhancing its practical applications.

Key words: constitutive model, viscoelastic-plastic, poly-ether-ether-ketone (PEEK), uniaxial tension, split Hopkinson pressure bar (SHPB)

2010 MSC Number:

O345

Feiyang ZHAO, Jinzhao HUANG, Shangyang YU, Jikai YU, Licheng GUO. A novel dual-hardening viscoelastic-plastic constitutive model for thermoplastic resins. Applied Mathematics and Mechanics (English Edition), 2026, 47(3): 599-622.

TrendMD

Figures/Tables 24

Table 1

Mechanical and thermal properties of PEEK-5600G"

Property	Test standard	Unit	Value
Density	ISO 1138	g/cm³	1.3
Melting point	DSC	°C	343
Distortion temperature	ISO 75-1/-2	°C	163
Thermal expansion	ASTM D696	1/°C	$4.7 × 10 − 5$
Poisson’s ratio	ISO 572-2		0.4

Table 1

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Table 2

Strain rates in uniaxial tensile test"

$u ˙ / (mm ⋅ min − 1)$	1	20	100
$ε ˙ / s − 1$	0.002 29	0.030 64	0.193 61

Table 2

Table 3

Fig. 5

Fig. 6

Table 4

Fig. 7

Table 5

Initial modulus and equilibrium modulus of materials at different temperatures"

Temperature/°C	Compressive relaxation				Tensile relaxation
Temperature/°C	$E 0$ /MPa	$E ¯ ∞$ /MPa	$η E$	$R fit 2$	$E 0$ /MPa	$E ¯ ∞$ /MPa	$η E$	$R fit 2$
23	3 389.21	2 991.66	$4 445.85 × 103$	0.67	3 299.54	3 053.63	$14 328.50 × 103$	0.94
100	2 897.90	2 519.62	$1 492.32 × 103$	0.31	2 856.40	2 587.80	$8 395.66 × 103$	0.97
140	2 634.26	2 359.99	$820.03 × 103$	0.39	2 402.26	2 124.45	$3 932.41 × 103$	0.90
180	1 102.80	598.93	$50.87 × 103$	0.36	988.80	383.32	$290.26 × 103$	0.96

Table 5

Table 6

Linear fitting results for temperature dependence of E0, E¯∞, and ηE"

Test	Relaxation parameter	$T < T g$		$T > T g$		$R fit 2$
Test	Relaxation parameter	Intercept	Slope	Intercept	Slope	$R fit 2$
C	E₀	$3.502 52 × 103$	$− 7.41$	$1.481 235 × 104$	$− 76.79$	0.965
C	$E ¯ ∞$	$3.262 57 × 103$	$− 7.68$	$1.760 157 × 104$	$− 95.65$	0.968
T	E₀	$3.598 61 × 103$	$− 6.44$	$1.577 411 × 104$	$− 81.50$	0.999
T	$E ¯ ∞$	$3.105 66 × 103$	$− 5.79$	$1.765 351 × 104$	$− 94.74$	0.990
C	$η E$	$4.816 00 × 106$	$− 28 117.95$	$4.816 000 × 106$	$− 28 117.95$	0.961
T	$η E$	$1.670 00 × 107$	$− 90 033.24$	$1.670 000 × 107$	$− 90 033.24$	0.994

Table 6

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Table 7

Material constants for PEEK-5600G"

JC property for tension
Parameter	$A T$	$C T$	$m T$	$B T$	n
Value	95.49	0.056 2	0.784 6	17.85	0.193 1
JC property for compression
Parameter	$A C$	$C C$	$m C$	$B 1 C$	$B 2 C$	$B 3 C$
Value	107.55	0.047 7	1.364 7	1 255.02	$− 8 630.37$	29 306.53
Strain rate and temperature property
Parameter	$ε ¯ 0$ /s^-1	$θ r / ∘ C$	$θ m / ∘ C$	$β int$	$C ν / (mJ ⋅ t − 1 ⋅ ° C -1)$
Value	0.002 29	23	343	0.48	$1.33 × 109$ ^[40]

Table 7

Fig. 12

Fig. 13

Fig. 14

Fig. 15

Table 8

Fig. 16

References 43

[1]	JOGUR, G., NAWAZ KHAN, A., DAS, A., MAHAJAN, P., and ALAGIRUSAMY, R. Impact properties of thermoplastic composites. Textile Progress, 50, 109–183 (2018)
[2]	ZHU, J., LI, L., and ZHU, N. Modification of Maxwell model for conductivity prediction of carbon nanotubes-filled polymer composites with tunneling effect. Applied Mathematics and Mechanics (English Edition), 46(1), 25–36 (2025) https://doi.org/10.1007/s10483-025-3210-9
[3]	ZHENG, B., WANG, H., HUANG, Z., ZHANG, Y., ZHOU, H., and LI, D. Experimental investigation and constitutive modeling of the deformation behavior of poly-ether-ether-ketone at elevated temperatures. Polymer Testing, 63, 349–359 (2017)
[4]	WANG, H., LIU, H., DING, Z., and LI, N. Experimental and constitutive modelling studies of semicrystalline thermoplastics under solid-state stamp forming conditions. Polymer, 228, 123939 (2021)
[5]	GARCIA-GONZALEZ, D., ZAERA, R., and ARIAS, A. A hyperelastic-thermoviscoplastic constitutive model for semi-crystalline polymers: application to peek under dynamic loading conditions. International Journal of Plasticity, 88, 27–52 (2017)
[6]	HAWARD, R. N. and THACKRAY, G. The use of a mathematical model to describe isothermal stress-strain curves in glassy thermoplastics. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 302, 453–472 (1968)
[7]	BOYCE, M. C., PARKS, D. M., and ARGON, A. S. Large inelastic deformation of glassy polymers, part I: rate dependent constitutive model. Mechanics of Materials, 7(1), 15–33 SEP (1988)
[8]	JAEKEL, D. J., MACDONALD, D. W., and KURTZ, S. M. Characterization of PEEK biomaterials using the small punch test. Journal of the Mechanical Behavior of Biomedical Materials, 4(7), 1275–1282 (2011)
[9]	EL-QOUBAA, Z. and OTHMAN, R. Characterization and modeling of the strain rate sensitivity of polyetheretherketone’s compressive yield stress. Materials & Design (1980–2015), 66, 336–345 (2015)
[10]	BARBA, D., ARIAS, A., and GARCIA-GONZALEZ, D. Temperature and strain rate dependences on hardening and softening behaviours in semi-crystalline polyers: application to PEEK. International Journal of Solids and Structures, 182, 205–217 (2020)
[11]	BORRUTO, A. A new material for hip prosthesis without considerable debris release. Medical Engineering & Physics, 32(8), 908–913 (2010)
[12]	CHANG, B., WANG, X., LONG, Z., LI, Z., GU, J., RUAN, S., and SHEN, C. Constitutive modeling for the accurate characterization of the tension behavior of PEEK under small strain. Polymer Testing, 69, 514–521 (2018)
[13]	GARCIA-GONZALEZ, D., GARZON-HERNANDEZ, S., and ARIAS, A. A new constitutive model for polymeric matrices: application to biomedical materials. Composites Part B: Engineering, 139, 117–129 (2018)
[14]	MULLIKEN, A. D. and BOYCE, M. C. Mechanics of the rate-dependent elastic-plastic deformation of glassy polymers from low to high strain rates. International Journal of Solids and Structures, 43(5), 1331–1356 (2006)
[15]	VARGHESE, A. G. and BATRA, R. C. Constitutive equations for thermomechanical deformations of glassy polymers. International Journal of Solids and Structures, 46(22-23), 4079–4094 (2009)
[16]	SAFARI, K. H., ZAMANI, J., GUEDES, R. M., and FERREIRA, F. J. The effect of heat developed during high strain rate deformation on the constitutive modeling of amorphous polymers. Mechanics of Time-Dependent Materials, 20(1), 45–64 (2016)
[17]	RICHETON, J., AHZI, S., DARIDON, L., and RÉMOND, Y. A formulation of the cooperative model for the yield stress of amorphous polymers for a wide range of strain rates and temperatures. Polymer, 46(16), 6035–6043 (2005)
[18]	RICHETON, J., AHZI, S., VECCHIO, K. S., JIANG, F. C., and ADHARAPURAPU, R. R. Influence of temperature and strain rate on the mechanical behavior of three amorphous polymers: characterization and modeling of the compressive yield stress. International Journal of Solids and Structures, 43(7-8), 2318–2335 (2006)
[19]	RICHETON, J., AHZI, S., VECCHIO, K. S., JIANG, F. C., and MAKRADI, A. Modeling and validation of the large deformation inelastic response of amorphous polymers over a wide range of temperatures and strain rates. International Journal of Solids and Structures, 44(24), 7938–7954 (2007)
[20]	HASHIGUCHI, K., UENO, M., and ANJIKI, T. Subloading-overstress model: unified constitutive equation for elasto-plastic and elasto-viscoplastic deformations under monotonic and cyclic loadings research with systematic review. Archives of Computational Methods in Engineering, 30(4), 2627–2649 (2023)
[21]	BOYCE, M. C., PARKS, D. M., and ARGON, A. S. Large inelastic deformation of glassy polymers, part II: numerical simulation of hydrostatic extrusion. Mechanics of Materials, 7(1), 35–47 (1988)
[22]	GARCIA-GONZALEZ, D., RUSINEK, A., JANKOWIAK, T., and ARIAS, A. Mechanical impact behavior of polyether-ether-ketone (PEEK). Composite Structures, 124, 88–99 (2015)
[23]	XU, Y. J., LU, H., GAO, T. L., and ZHANG, W. H. Predicting the low-velocity impact behavior of polycarbonate: influence of thermal history during injection molding. International Journal of Impact Engineering, 86, 265–273 (2015)
[24]	XU, Y. J., GAO, T. L., WANG, J., and ZHANG, W. H. Experimentation and modeling of the tension behavior of polycarbonate at high strain rates. Polymers, 8(3), 63 (2016)
[25]	CHEN, F., OU, H., GATEA, S., and LONG, H. Hot tensile fracture characteristics and constitutive modelling of polyether-ether-ketone (PEEK). Polymer Testing, 63, 168–179 (2017)
[26]	JINAGA, U. K., ZULUETA, K., BURGOA, A., COBIAN, L., FREITAS, U., LACKNER, M., MAJOR, Z., and NOELS, L. A consistent finite-strain thermomechanical quasi-nonlinear-viscoelastic viscoplastic constitutive model for thermoplastic polymers. International Journal of Solids and Structures, 321, 113517 (2025)
[27]	JONES, E. M. C. and IADICOLA, M. A. A good practices guide for digital image correlation. Standardization, Good Practices, and Uncertainty Quantification Committee, International Digital Image Correlation Society (2018)
[28]	YANG, C. C., TIAN, X. Y., LI, D. C., CAO, Y., ZHAO, F., and SHI, C. Q. Influence of thermal processing conditions in 3D printing on the crystallinity and mechanical properties of PEEK material. Journal of Materials Processing Technology, 248, 1–7 (2017)
[29]	HU, Y. X., ZHAO, W. L., WANG, L. Q., LIN, J. P., and DU, L.. Machine-learning-assisted design of highly tough thermosetting polymers. ACS Applied Materials & Interfaces, 14(49), 55004–55016 (2022)
[30]	WANG, X. Y., LI, Z. X., GUO, L. C., WANG, Z. X., and ZHAO, J. Z.. Analysis on the turning point of dynamic in-plane compressive strength for a plain weave composite. Defence Technology, 32, 485–495 (2024)
[31]	MIAO, Y. G., LI, Y. L., LIU, H. Y., DENG, Q., SHEN, L., MAI, Y. W., GUO, Y. Z., SUO, T., HU, H. T., XIE, F. Q., ZHAO, L., MAO, Y. J., and QI, W. Determination of dynamic elastic modulus of polymeric materials using vertical split Hopkinson pressure bar. International Journal of Mechanical Sciences, 108-109, 188–196 (2016)
[32]	BRINSON, H. F. and BRINSON, L. C. Polymer Engineering Science and Viscoelasticity: An Introduction, 2nd ed., Springer, New York (2015)
[33]	RAVICHANDRAN, G. and SUBHASH, G. Critical-appraisal of limiting strain rates for compression testing of ceramics in a split Hopkinson pressure bar. Journal of the American Ceramic Society, 77(1), 263–267 (1994)
[34]	FAN, J. T., WEERHEIJM, J., and SLUYS, L. J. Dynamic compressive mechanical response of a soft polymer material. Materials & Design, 79, 73–85 (2015)
[35]	FAN, J. T., WEERHEIJM, J., and SLUYS, L. J. Deformation to fracture evolution of a flexible polymer under split Hopkinson pressure bar loading. Polymer Testing, 70, 192–196 (2018)
[36]	BOUVARD, J. L., FRANCIS, D. K., TSCHOPP, M. A., MARIN, E. B., BAMMANN, D. J., and HORSTEMEYER, M. F. An internal state variable material model for predicting the time, thermomechanical, and stress state dependence of amorphous glassy polymers under large deformation. International Journal of Plasticity, 42, 168–193 (2013)
[37]	GARCIA-GONZALEZ, D., ZAERA, R., and ARIAS, A. A hyperelastic-thermoviscoplastic constitutive model for semi-crystalline polymers: application to PEEK under dynamic loading conditions. International Journal of Plasticity, 88, 27–52 (2017)
[38]	SHAO, G. J., ZHU, S. X., WANG, Y., and ZHAO, Q. L. An internal state variable thermodynamic model for determining the Taylor-Quinney coefficient of glassy polymers. International Journal of Mechanical Sciences, 126, 261–269 (2017)
[39]	BENAARBIA, A., CHATZIGEORGIOU, G., KIEFER, B., and MERAGHNI, F. A fully coupled thermo-viscoelastic-viscoplastic-damage framework to study the cyclic variability of the Taylor-Quinney coefficient for semi-crystalline polymers. International Journal of Mechanical Sciences, 163, 105128 (2019)
[40]	HAO, P., SPRONK, S. W. F., VAN PAEPEGEM, W., and GILABERT, F. A. Hydraulic-based testing and material modelling to investigate uniaxial compression of thermoset and thermoplastic polymers in quasistatic-to-dynamic regime. Materials & Design, 224, 111367 (2022)
[41]	BÉGUELIN, P., BARBEZAT, M., and KAUSCH, H. H. Mechanical characterization of polymers and composites with a servohydraulic high-speed tensile tester. Journal de Physique III, 1, 1867–1880 (1991)
[42]	SWALLOWE, G. M., FERNANDEZ, J. O., and HAMDAN, S. Crystallinity increases in semi crystalline polymers during high rate testing. Journal de Physique IV, 7(C3), 453–458 (1997)
[43]	RAE, P. J., BROWN, E. N., and ORLER, E. B. The mechanical properties of poly(ether-ether-ketone) (PEEK) with emphasis on the large compressive strain response. Polymer, 48, 598–615 (2007)

Strain rate/s^-1	Temperature/°C
Strain rate/s^-1	23	100	140	180
	Yield strength for tension/MPa
0.002 29	94.93	71.78	54.37	39.57
0.030 64	107.34	83.12	67.35	41.65
0.193 61	120.42	94.61	80.40	43.43
	Yield strength for compression/MPa
0.002 29	107.55	93.19	82.63	67.19
0.030 64	112.38	97.20	86.02	68.56
0.193 61	123.12	102.98	90.75	68.84

Strain rate/s^-1	1 094.08	2 859.31	4 018.49	5 641.70	5 957.88
Yield strength/MPa	202.90	208.77	213.61	217.28	219.41

Strain rate/s^-1	Simulation/MPa	Experiment/MPa	Deviation/%
1 094.08	197.96	202.90	2.43
2 859.31	205.69	208.77	1.47
4 018.49	208.45	213.61	2.41
5 641.70	211.60	217.28	2.61
5 957.88	212.18	219.41	3.29