Applicability and Time Variation Law of Constitutive Model of Scrap Tire Rubber Pads Considering Aging Effect

doi:10.12068/j.issn.1005-3026.2025.20239062

Abstract

Abstract:

At present， many hyperelastic constitutive models are widely used in the finite element analysis of rubber isolation bearings. The selection of proper constitutive models and corresponding parameter determination are crucial in this process. This study aims to investigate the applicability of various hyperelastic constitutive models for different brands of tire rubber and the variation law of rubber parameters after aging. Three different brands of waste tires were selected for uniaxial tensile tests， while a batch of tires were selected for accelerated aging tests using hot air at 77， 154， 231， and 308 h. Employing the determined model parameters， the scrap tire rubber pads （STP） were modeled and compared with vertical experiments after aging using ABAQUS software. The results show that the stress-strain curves of the rubber in all three tire brands exhibit typical reverse “S” shapes， and the Yeoh model is determined to be optimal after comparing multiple models. The Yeoh model parameters approximately show monotonic changes with increasing aging time. The errors between the STP vertical stiffness test and the finite element simulation values before and after aging are within an acceptable range， which verifies the accuracy of the time-varying law of the aging parameters. This research provides theoretical support for the whole life design of STP and other waste tire products.

Key words: Yeoh model, scrap tire rubber pads, aging, finite element analysis

CLC Number:

TU 352.12

Guang-tai ZHANG, Jun-fu ZHANG, Dong-liang LU. Applicability and Time Variation Law of Constitutive Model of Scrap Tire Rubber Pads Considering Aging Effect[J]. Journal of Northeastern University(Natural Science), 2025, 46(5): 126-133.

Figures/Tables 16

Fig.1 Comparison of dumbbell specimens before and after stretching

Table 1 Constitutive function type of rubber

本构函数	应变能函数形式	工程应力与伸长率的关系
Neo-Hookean	$W = C 10 I 1 - 3$	$σ = 2 C 10 λ - λ - 2$
Mooney-Rivlin	$W = C 10 I 1 - 3 + C 01 I 2 - 3 + 1 D 1 J - 1 2$	$σ = 2 C 10 λ - λ - 2 + 2 C 01 （ 1 - λ - 3 ）$
Yeoh	$W = C 10 I 1 - 3 + C 20 I 1 - 3 2 + C 30 I 1 - 3 3$	$σ = ∑ i = 1 N 2 i C i 0 λ 2 + 2 λ - 1 - 3 i - 1 （ λ - λ - 2 ）$
Ogden	$W = ∑ i = 1 N μ i α i λ 1 α i + λ 2 α i + λ 3 α i - 3$	$σ = ∑ i = 1 N μ i λ α i - 1 + λ - 12 α i - 1$

Table 1 Constitutive function type of rubber

本构函数	应变能函数形式	工程应力与伸长率的关系
Neo-Hookean	$W = C 10 I 1 - 3$	$σ = 2 C 10 λ - λ - 2$
Mooney-Rivlin	$W = C 10 I 1 - 3 + C 01 I 2 - 3 + 1 D 1 J - 1 2$	$σ = 2 C 10 λ - λ - 2 + 2 C 01 （ 1 - λ - 3 ）$
Yeoh	$W = C 10 I 1 - 3 + C 20 I 1 - 3 2 + C 30 I 1 - 3 3$	$σ = ∑ i = 1 N 2 i C i 0 λ 2 + 2 λ - 1 - 3 i - 1 （ λ - λ - 2 ）$
Ogden	$W = ∑ i = 1 N μ i α i λ 1 α i + λ 2 α i + λ 3 α i - 3$	$σ = ∑ i = 1 N μ i λ α i - 1 + λ - 12 α i - 1$

Fig.2 Stress-strain relationships for three brands of rubber

Table 2 Hyperelastic constitutive parameters and fitting errors of Bridgestone tires

C₁₀=1.469

C₁₀=2.775，C₀₁=-3.42

C₁₀=0.623 9，C₂₀=0.096 81

C₃₀=-0.001 968

μ 1 = 0.692 5, α 1 = 3.385

R²

Table 2 Hyperelastic constitutive parameters and fitting errors of Bridgestone tires

模型

Neo-Hookean

Mooney-Rivlin

Yeoh

Ogden

参数

C₁₀=1.469

C₁₀=2.775，C₀₁=-3.42

C₁₀=0.623 9，C₂₀=0.096 81

C₃₀=-0.001 968

μ 1 = 0.692 5, α 1 = 3.385

R²

0.842 4

0.978 1

0.998 3

0.997 5

Table 3 Hyperelastic constitutive parameters and fitting errors of Dunlop tires

C₁₀=1.357

C₁₀=2.547，C₀₁=-2.706

C₁₀=0.828 3，C₂₀=0.050 06

C₃₀=0.004 193

μ 1 = 0.676 5, α 1 = 3.535

R²

Table 3 Hyperelastic constitutive parameters and fitting errors of Dunlop tires

模型

Neo-Hookean

Mooney-Rivlin

Yeoh

Ogden

参数

C₁₀=1.357

C₁₀=2.547，C₀₁=-2.706

C₁₀=0.828 3，C₂₀=0.050 06

C₃₀=0.004 193

μ 1 = 0.676 5, α 1 = 3.535

R²

0.860 5

0.965 3

0.997 3

0.993 4

Table 4 Hyperelastic constitutive parameters and fitting errors of Michelin tires

C₁₀=1.26

C₁₀=2.13，C₀₁=-2.529

C₁₀=0.607 5，C₂₀=0.059 65

C₃₀=-0.001 166

μ 1 = 0.748 3, α 1 = 3.057

R²

Table 4 Hyperelastic constitutive parameters and fitting errors of Michelin tires

模型

Neo-Hookean

Mooney-Rivlin

Yeoh

Ogden

参数

C₁₀=1.26

C₁₀=2.13，C₀₁=-2.529

C₁₀=0.607 5，C₂₀=0.059 65

C₃₀=-0.001 166

μ 1 = 0.748 3, α 1 = 3.057

R²

0.880 6

0.987 9

0.999 7

0.993 4

Fig.3 Comparison of different hyperelastic models and test data

Fig.4 Stress-strain relationship of Dunlop tires aging 0~100 years

Table 5 Hyperelastic constitutive parameters of aged Dunlop tires

时间/a	0	25	50	75	100
C₁₀	0.757 5	0.687 8	0.657 4	0.557 9	0.501 4
C₂₀	0.115 40	0.109 30	0.078 34	0.069 45	0.065 04
C₃₀	-0.001 764	-0.001 933	-0.001 637	-0.001 504	-0.001 097
R²	0.997 1	0.999 4	0.999 8	0.999 8	0.999 4

Fig.5 Modelling diagram of type 2 dumbbell specimen

Fig.6 Corresponding stress-strain relationship with decreasing C10

Fig.7 Corresponding stress-strain relationship with decreasing C20

Fig.8 Corresponding stress-strain relationship with decreasing C30

Fig.9 Comparison of Yeoh constitutive parameters fitting with time

Fig.10 Scrap tire rubber pads modelling diagram

Fig.11 Test and simulation values of vertical stiffness at 6 MPa

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