Constitutive modeling of the viscoelastic mechanical response of foam rubber-like materials

Abstract

The viscoelastic mechanical behavior of foam rubber-like materials under large deformation is studied using thermodynamic methods. Because the strain energy function can be additively split into deformation energy and volume energy, the deformation of materials is decomposed into volumetric deformation and isochoric deformation. A nonlinear viscoelastic constitutive model for large deformation of foam rubber-like materials is established in this paper. The nonlinear viscoelastic mechanical responses of foam rubber-like materials under several common loading conditions are calculated by the new proposed constitutive model.

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1. Introduction

Due to their unique microstructure morphology, foam rubber-like materials have many excellent properties, such as high elasticity, shock resistance, wear resistance, impact resistance, insulating properties and physiological inertness. They have been widely used in areas such as aerospace, transportation, petrochemical industry and construction industry.^1–3 Because of the complexity of the mechanical properties and microstructures of the materials, the structural design and analysis of foam rubber-like materials is increasingly dependent on numerical methods; however, whether or not the numerical method is accurate mainly depends on the constitutive model used. Therefore, constitutive modeling of foam rubber-like materials has drawn great attention from researchers.^4,5

At present, the mechanical properties of rubber-like materials have been studied to a degree in the literature, and some theoretical models for describing the mechanical behavior of the materials were proposed.^6,7 These models can be categorized into two major types. The first type is the continuum elastic body model based on the phenomenological method. The representative works can be found in the literature by Ogden, Gent and Yeoh.^8–10 The second type is the dynamic model based on the statistical thermodynamic method. Typical models are proposed by Simo, Govindjee and Lion.^11–13 Almost all of these models are established by the assumption of incompressibility of solid rubber-like materials.^14–16 However, to the best knowledge of the authors, few viscoelastic models have been proposed to describe foam rubber-like materials.

The viscoelastic mechanical behavior of foam rubber-like materials was studied by theoretical analysis in this paper. Based on the additive splitting of the strain energy function into deformation energy and volume energy, the nonlinear viscoelastic constitutive model for large deformation of foam rubber-like materials was established using a thermodynamic method. The viscoelastic mechanical responses of foam rubber-like materials under several common loading conditions were calculated using the constitutive model.

2. Basic theory

In recent years, with the large-scale development and utilization of polymer materials, the mechanical properties of many new polymer materials belong neither to the category of elastic theory, nor viscous theory. It is well known that elastic theory is mainly suitable for solids, which have the ability to store energy, but not to dissipate it; viscous theory is mainly suitable for fluids, which have the ability to dissipate energy, but not to store it. The mechanical properties of foam rubber-like materials exceed the research scope of the above two theories. These materials can both store and dissipate energy. The basic characteristics of the material's mechanical response are manifested as instantaneous elastic, creep and relaxation. The stress response of these materials depends not only on the current state of stress, but also on the past history of stress. The stress state of the materials depends on the specific time or rate.¹⁷

The viscoelastic mechanical behavior of foam rubber-like materials under large deformation exhibits nonlinear characteristics. The deformation of the materials does not abide by the assumption of small deformation; therefore, the coordinate of each particle within the material in the reference configuration and in the current configuration must be considered. X is the coordinate of a typical particle in the reference configuration and refers to a given set of Cartesian coordinate system; x is the coordinate of a typical particle in the current configuration. The whole deformation history of a typical particle can be described as the following equation:¹⁸

x(τ) = x(X, τ) (0 < τ < t) (1)

where t is the current time, and τ is the time variable. The deformation gradient F is defined as

F = ∂x/∂X (2)

The deformation gradient F is used to describe the variable quantity of material shape. The large deformation of the materials can be described by the left Cauchy-Green tensor B or right Cauchy-Green tensor C

B = F·F^T C = F^T·F (3)

Green strain tensor E can be expressed as

E = 1/2(C – I) (4)

where I is the unit tensor, and E represents the deformation measure relative to the reference configuration. The three principal invariants of the left Cauchy-Green tensor B can be written as

I₁ = tr(B) I₂ = 1/2[I₁² − tr(B²)] I₃ = det(B). (5)

The relationship between the principal invariants and the principal stretches of B can be expressed as

Local entropy inequality of the materials can be expressed as^19,20

ρTJ̇ − ρr + Q_i,i − Q_i − (T_i/T) ≥ 0 (7)

Local energy equilibrium equation of the materials can be expressed as^19,20

ργ − ρ[Ẇ + ṪJ + TJ̇] + S_ij[small epsi, Greek, dot above]_ij − Q_i,j = 0 (8)

where ρ is mass density, r is the heating function per unit mass, W is the Helmholtz potential per unit mass, T is the absolute temperature, J is the entropy per unit mass, and Q_i is the Cartesian components of the measured heat flow vector per unit area. Under the assumption that the two equations are in isothermal condition, simultaneous eqn (7) and (8) leads to

−ρẆ + S_iju_ij ≥ 0 (9)

where S_ij is the component of Cauchy stress tensor in the current configuration, and u_ij is the velocity gradient of material deformation, written as follows:

The Helmholtz potential W can be expressed as the functional of deformation history by time-dependent effects of the material mechanical properties and can be written as follows²¹

Assuming that eqn (11) is a continuous functional of strain history, the strain tensor E and its first-order time derivative is continuous. Since the Helmholtz potential W is the time derivative form in eqn (9), the Frechet differential of the Helmholtz free energy functional is given by

where is linear, based on δE(t − s). From eqn (11), the time derivative of the Helmholtz free energy functional can be expressed as

Inserting eqn (12) into eqn (13) gives

Inserting the time derivative Ẇ of the Helmholtz free energy functional into inequality (9) then yields

The kinematic relation is given by the time differential of eqn (4) and using eqn (10):

Ė(t) = u_ijF·F^T (16)

Inserting eqn (16) into inequality (15) then yields

In order to satisfy inequality (17), the coefficient of u_ij is equal to zero for a given material deformation history, and using eqn (11) and (14), leads to

The hydrostatic pressure p is added to eqn (18). The nonlinear viscoelastic constitutive equation for large deformation of foam rubber-like materials is finally obtained as

Because the strain energy function can be additively split into deformation energy and volume energy,^22,23 the deformation of materials is decomposed into volumetric deformation and isochoric deformation in this paper. Volumetric deformation mainly depends on the microstructure morphology of the foam; isochoric deformation is mainly determined by the mechanical properties of the rubber.^23,24 The derivative form of the Helmholtz free energy functional for the isochoric deformation part is given by the assumption of incompressibility of solid rubber-like materials.^5,22 In this case, I₃ = 1.

where θ₁ is the relaxation factor of the materials at low strain rate, and θ₂ is the relaxation factor of the materials at high strain rate. Based on the deformation mechanism analysis of the materials under load, it can be deduced that the squeezing properties of foam rubber-like materials mainly depend on the level of material porosity. The derivative form of the Helmholtz free energy functional for volumetric deformation is given by the influence law of the material porosity on the stress–strain relationship.where n_f is the material porosity, E_S is Young's modulus of the rubber matrix material, d is a constant that reflects the pore structure of foam rubber-like materials, and J(t) is the determinant of the deformation gradient tensor F. The nonlinear viscoelastic constitutive model independent of the coordinate system of foam rubber-like materials under large deformation can be obtained by inserting eqn (20) and (21) into eqn (19)

3. Numerical example

To illustrate the application of the new model developed in the preceding section, a parametric study of sample cases is conducted here for three-dimensional rubber foams, whose modeling motivated the present study. Under triaxial loading conditions, the elongation ratio of rubber foams in the loading direction can be denoted as λ₁, λ₂, λ₃. The material is assumed to be homogeneous and isotropic. The deformation of sample cases under triaxial loading can be defined as the following form

The deformation gradient F(t) is given by the differential of eqn (23) and using eqn (2):

Green strain tensor E is given by inserting eqn (24) into eqn (4):

By substituting eqn (24) and (25) into eqn (19), the principal stress components of sample cases under triaxial loading is finally obtained as

The stress expressions in several common experiments can be derived from eqn (26).

Based on the compressibility of rubber foams, the principal stretches of the material under uniaxial compression and tension can be expressed as λ₁ = λ, λ₂ = λ₃ = λ^k, where k is a material constant. Considering the boundary conditions of sample cases, the stress expressions under uniaxial compression and tension is given by eqn (27):

The principal stretches of the material under pure shear deformation can be expressed as λ₁ = λ^k, λ₂ = 1, λ₃ = λ^−k. Considering the boundary conditions of sample cases, the stress expressions under pure shear is given by eqn (28):

The principal stretches of the material under simple shear deformation can be expressed as λ₁ = λ^k, λ₂ = λ^−k, λ₃ = 1. Considering the boundary conditions of sample cases, the stress expressions under simple shear is given by eqn (29):

The principal stretches of the material under equi-biaxial tension deformation can be expressed as λ₁ = λ, λ₂ = λ, λ₃ = λ^k. Considering the boundary conditions of sample cases, the stress expressions under equi-biaxial tension is given by eqn (30):

The viscoelastic mechanical responses of foam rubber-like materials in several common loading conditions can be calculated by the above stress expressions.

3.1. Viscoelastic mechanical responses with uniaxial loading

By substituting eqn (22) into eqn (27), the stress of foam rubber-like materials under uniaxial compression deformation can be expressed aswhere
and where a, b, c are material constants related to strain rate, and d is a constant that reflects the pore structure of the materials. Left Cauchy-Green tensor B is given by

The three principal invariants of B is given by

Inserting eqn (33) into eqn (31) then yields

The nonlinear viscoelastic mechanical behavior of the materials under uniaxial loading is described by eqn (34). From eqn (34), using integration by parts leads to

where λ = 1 + ε₁₁. Introducing the unit step function H(t) to represent constant strain as follows,

The constant strain is given by

ε₁₁ = ε₀H(t) (37)

Inserting eqn (36) and (37) into eqn (35) then yields

The stress relaxation test of sample cases at low strain rate was carried out restricted to special purpose applications. The specimens are porous silicon rubber thin sections; the size of specimens is 60 mm × 12 mm × 0.65 mm. The experiments were conducted on a stress relaxation testing machine. The density of the specimens is ρ = 0.45 × 10⁻³ kg m⁻³; the porosity is n_f = 60%; the SEM micrograph of the materials is shown in Fig. 1. The experimental data for eqn (38) is fitted; the parameter values are given in Table 1, and the fitting results are shown in Fig. 2 and 3.

Fig. 1 SEM micrograph of a foam rubber.

Table 1 The model parameters determined by fitting the experimental data

a	b	c	d	k	θ1
95.722	7.443	−0.0079	3.275	0.658	6.887

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Fig. 2 Comparison between stress relaxation fitting curve and experimental results (t = 100 s).

Fig. 3 Comparison between stress relaxation fitting curve and experimental results (t = 1000 s).

The fitting curves from Fig. 2 to Fig. 3 show that the quality of the fit is essentially comparable between the two cases. In order to examine the validity of the constitutive model, the mechanical properties of three porous silicon rubber thin sections with different densities (0.56 g cm⁻³, 0.78 g cm⁻³ and 0.90 g cm⁻³) under uniaxial compression are theoretically forecasted by substitution of the parameter values in Table 1 into eqn (35). The comparison of the theoretical prediction curve and the experimental curve is shown in Fig. 4, and the error analysis is shown in Fig. 5.

Fig. 4 Comparison between theoretical prediction curve and experimental results of materials of various densities.

Fig. 5 Comparison of model prediction errors for each kind of density.

As shown in Fig. 4, the theoretical prediction curve by the compressible viscoelastic model is in good agreement with the available experimental data under uniaxial compression. In addition, it can be seen from the error analysis chart that the residual stress error is approximately in the range of −0.02 MPa < Δ < 0.02 MPa and is approximately negligible compared with the corresponding engineering stress amplitude.

3.2. Viscoelastic mechanical responses with two-dimensional loading

By substituting eqn (22) into eqn (30), the stress of foam rubber-like materials under equi-biaxial tensile deformation can be expressed aswhere

From eqn (39), using the integration by parts leads to

where λ = 1 + ε.

In the same way, by substituting eqn (22) into eqn (28), the stress of foam rubber-like materials under pure shear deformation can be expressed as

M(t) = a(I₁ − 3) + b(I₂ − 3) + c(I₁ − 3)(I₂ − 3) = a(λ^2k + λ^−2k − 2) + b(λ^−2k + λ^2k − 2) + c(λ^2k + λ^−2k − 2)² (41)

The experimental data are fitted by eqn (40) and (41), and the material parameters are obtained as shown in Tables 2 and 3. All the experimental data used here are derived from Treloar's classic experiments and the experiments of James et al.^25–27 These experimental data have been widely used to validate the quantitative model of rubber-like materials under different loading conditions.^28–33 The experimental data from James et al. are used to fit the parameters of the model (Fig. 6 and 7)the well-known data from Treloar are used to examine the validity of the constitutive model. For solid rubber-like materials, take n_f = 0. By substitution of the parameter values into the constitutive equations, the prediction results are obtained by the present model, shown in Fig. 8 and 9 with other classical models. It can be seen from the figures that the present model has better prediction ability in comparison with other classical models.

Table 2 The model parameters determined by fitting the experimental data (equi-biaxial extension)

a	b	c	d	k	θ1
20.526	7.141	−1.467	7.073	0.120	9.239

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Table 3 The model parameters determined by fitting the experimental data (pure shear)

a	b	c	d	k	θ1
41.627	6.554	−1.558	3.457	0.770	7.106

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Fig. 6 Comparison between stress–stretch fitting curve and experimental results (equi-biaxial extension).

Fig. 7 Comparison between stress–stretch fitting curve and experimental results (pure shear).

Fig. 8 Comparison between theoretical prediction curve and experimental results (equi-biaxial extension).

Fig. 9 Comparison between theoretical prediction curve and experimental results (pure shear).

Through the analysis and research above, it can be seen that the new proposed viscoelastic model is suitable to describe the nonlinear viscoelastic mechanical response of foam rubber-like materials under large deformation.

4. Conclusions

Based on the viscoelastic constitutive theory and mathematical derivation methods, the nonlinear viscoelastic constitutive model for large deformation of foam rubber-like materials is finally established as:

The nonlinear viscoelastic mechanical responses of foam rubber-like materials under several common loading conditions are calculated by the above constitutive model. The theoretical values predicted by the current model are in good agreement with results based on the available experimental data.

5. Appendices

x	The coordinate of a typical particle in the current configuration
X	The coordinate of a typical particle in the reference configuration
τ	Time variable
t	Current time
F	Deformation gradient
B	Left Cauchy-Green tensor
C	Right Cauchy-Green tensor
E	Green strain tensor
I1 I2 I3	Three principal invariants of B
λ1 λ2 λ3	Three principal stretches of B
J(t)	The determinant of the deformation gradient tensor F
ρ	Mass density
r	Heating function per unit mass
W	Helmholtz potential per unit mass
T	Absolute temperature
Qi	Cartesian components of the measured heat flow vector per unit area
Sij	Deviator stress components of Cauchy stress tensor in the current configuration
uij	Velocity gradient of material deformation
p	Hydrostatic pressure
σij	Cauchy stress components
δij	Kronecker delta
θ	Relaxation factor
nf	Material porosity
ES	Young's modulus of rubber matrix material
a b c	Material constants related to strain rate
d	A constant which reflects pore structure of the materials
k	Material constant
H(t)	Unit step function

Acknowledgements

The authors would like to thank J. S. Lei and J. L. Tao for fruitful discussions on developing the constitutive model framework employed in this paper. The work was funded by the grant from the National Natural Science Foundation of China.

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