Introduction

Rubbers are categorised among nonlinear elastic or hyper-elastic materials. The molecular structure of hyper-elastic materials permits high flexibility in room temperature as well as high reversibility against deformation. The most important property of these materials is incompressibility, which is the reason for having Poisson’s ratio near 0.5. This causes the complexity of numerical calculations, especially in three-dimensional analysis. The unique properties of these materials make them highly applicable in everyday life and science. In addition to the wide application of these materials in the automotive industry, aerospace, and tires [1,2,3,4,5], railway industry [6,7,8], power devices and motors [9], and bearings [10], this kind of materials also are used in medical sciences. Investigations on heart muscles [11,12,13,14,15,16], tendon and ligaments under different loads [17,18,19,20], body skin under special circumstances like injections [21,22,23,24,25,26], brain tissues’ behaviour [27,28,29,30,31,32], cancer cure and the spread of cancer inside human body [33,34,35], rehabilitation of disabled or diabetic patients [36, 37] are some of these applications. Another highly applicable field for these materials is sports sciences. Other applications of the hyper-elastic material include their vast usage in sports’ instruments like clothing [38,39,40], basketball, baseball and golf balls [41, 42] and sports tracks [43] as well as injuries due to impact of the ball and other hyper-elastic materials to the human body in sports matches [44,45,46,47,48,49,50].

It is known that the mechanical properties of materials are defined by their response to the environment and circumstances loads. To investigate the mechanical properties, some experiments should be designed. The results of the experiments usually are in the form of diagrams and tables e.g. stress–strain diagrams. Unlike free energy models such as Helmholtz free energy function and Gibbs function in the constitutive models of hyper-elastic materials, the mechanical behaviour of rubber-like materials, some polymers, foams, and so on are expressed in terms of strain energy in isotropic, incompressible and constant temperature conditions. In elastic materials, stress tensor is obtained with the derivation of the strain energy density function with respect to the strain; however, for hyper-elastic material stress tensor, the derivation of the strain energy function should be performed with respect to deformation gradient tensor due to large deformation. In recent decades, several functions are introduced as the strain energy density function. In 1940, Mooney [51] introduced one of the most significant models of hyper-elastic materials. This model is defined based on the linear response of rubber under simple shear loads. In 1942, strain energy function for a single chaincase was found, using non-Gaussian assumption in the limiting extensibility of polymer chains [52]. In 1943, another model was introduced using molecular networks and static Gaussian law by Treloar [53]. Rivlin and Saunders [54] established standard experiments that led to discovering some new facts about the deformation of hyper-elastic materials. Later on, Isihara et.al [55] used the result data of Rivlin and Saunders experiment as reference data. Some novel models were introduced for strain energy function to investigate spherical membranes in 1966 [56], soft body tissues in 1967 [57], interactions between the balloon and its fluid with respect to height change in 1968 [58]. In 1981 [59], a novel model was introduced to express the mechanical properties of hyper-elastic based on network chains assuming their slide on their connections. Another model has presented under the assumption of real network chains general theory and restriction of movements of connection points between network chains and another adjacent in 1982 [60]. Considering Van der Waals forces and assuming gas behaviour for rubber network under influence of forces between quasi-particles, a novel model was introduced in 1986 [61]. The existence and behaviour of cracks in hyper-elastic materials were investigated in 1997 [62]. In 2000, stress was expressed in terms of guessed functions [63]. Later on, some other models were introduced which are a combination between previous models such as [64, 65], novel systematic experiments [66], and models sensitive to certain experiments [67].

In the present work, in Sect. 2, a discussion and classification on strain energy functions are performed. In Sect. 3 based on the introducing five standard loading physical constants, Cauchy stress tensor, hydrostatic pressure, and nominal stress tensor are obtained. In Sect. 4, the genetic algorithm (GA) method (one of the meta-heuristics that has been employed to find optimum solution of many combinational problems [68,69,70]) is discussed for optimal parameters and function estimation. In the final section, concluding remarks are presented.

Stored strain energy

As was mentioned before, in hyper-elastic materials, the stress tensor is defined as a derivative of stored energy function concerning the deformation gradient. This can be written in the form of the following equation:

$$ \begin{aligned} S_{ij} & = \frac{{\partial {\text{w}}}}{{\partial E_{ij} }} = 2\frac{{\partial {\text{w}}}}{{\partial C_{ij} }}, \\ E_{ij} & = \frac{1}{2}\left( {C_{ij} - \delta_{ij} } \right), \\ \end{aligned} $$
(1)

where, \({S}_{ij}\) is Piolla–Kirchhof second stress tensor components, \(w\) is stored strain energy function, \({E}_{ij}\) is Green–Lagrange strain tensor components, \({C}_{ij}\) is right Cauchy–Green deformation tensor components and \({\delta }_{ij}\) is Kronecker delta. Right Cauchy–Green deformation tensor is defined as a function of deformability gradient tensor, which is expressed in Eq. 2:

$$ \begin{gathered} C_{ij} = F_{ik} F_{kj} , \hfill \\ F_{ik} = \frac{{\partial {\text{x}}_{{\text{i}}} }}{{\partial {\text{X}}_{{\text{k}}} }}, \hfill \\ \end{gathered} $$
(2)

where \({F}_{ij}\) is deformation gradient components, \({X}_{i}\) is non-deformed body, \({x}_{i}={X}_{i}+{U}_{i}\) is deformed body and \({U}_{i}\) is displacement field. Eigenvalues of \({C}_{ij}\) exist, if and only if

$$ \det \left( {C_{ij} - {\uplambda }_{{\text{p}}}^{2} \delta_{ij} } \right) = 0, $$
(3)

from the characteristic equation

$$ \lambda_{p}^{6} - I_{2} \lambda_{p}^{4} + I_{2} \lambda_{p}^{2} - I_{3} = 0, $$
(4)

where

$$ \begin{gathered} I_{1} = tr\left( C \right) = \left( {\lambda_{1} } \right)^{2} + \left( {\lambda_{2} } \right)^{2} + \left( {\lambda_{3} } \right)^{2} , \hfill \\ I_{2} = \frac{1}{2}\left\{ {\left( {{\text{tr}}C} \right)^{2} - tr\left( {C^{2} } \right)} \right\} = \left( {\lambda_{1} \lambda_{2} } \right)^{2} + \left( {\lambda_{1} \lambda_{3} } \right)^{2} + \left( {\lambda_{2} \lambda_{3} } \right)^{2} , \hfill \\ I_{3} = \left( {\lambda_{1} \lambda_{2} \lambda_{3} } \right)^{2} = J^{2} , J = \det \left[ {F_{ik} } \right]. \hfill \\ \end{gathered} $$
(5)

The third invariant is not commonly used for these sorts of models, because it signifies the constant volume or Poisson’s ratio ~ 0.5. Where \({{\lambda }_{i}}^{2}\) are eigenvalues of right Cauchy–Green tensor and \({\lambda }_{i}\) are eigenvalues of deformation gradient tensor, \(J\) (Jacobian) is the ratio of deformed elastic volume to initial volume of the material. In these materials, the determinant of the deformation gradient is equal to unity. Therefore, stored energy function, which depends on deformations and is not depended on rigid body movements, is expressed in terms of first and second invariants as follows:

$$ w = w\left( {I_{1} \left( C \right),I_{2} \left( C \right)} \right). $$
(6)

These energy functions have to satisfy some essential conditions, which are

absence of energy, if there is no deformation

$$ w_{{{\text{un}}}} = 0. $$
(7)

The amount of energy and the resulting stress is infinite for huge deformations

$$ \begin{gathered} \mathop {\lim }\limits_{{{\uplambda }_{{\text{i}}} \to \infty }} {\text{w}} = + \infty \;\;\;\;\;\mathop {\lim }\limits_{{{\uplambda }_{{\text{i}}} \to 0}} {\text{w}} = + \infty , \hfill \\ \mathop {\lim }\limits_{{{\uplambda }_{{\text{i}}} \to \infty }} \frac{{\partial {\text{w}}}}{{\partial {\uplambda }_{{\text{i}}} }} = + \infty \;\;\;\mathop {\lim }\limits_{{{\uplambda }_{{\text{i}}} \to 0}} \frac{{\partial {\text{w}}}}{{\partial {\uplambda }_{{\text{i}}} }} = - \infty . \hfill \\ \end{gathered} $$
(8)

Absence of stress, if there is no deformation; and its amount has to be minimum

$$ \begin{gathered} \frac{\partial w}{{\partial \lambda_{i} }}|_{{\lambda_{1} = \lambda_{2} = \lambda_{3} = 1}} = 0\;\;\;\;\;\;\;\;\frac{{\partial^{2} w}}{{\partial \lambda_{i}^{2} }}|_{{\lambda_{1} = \lambda_{2} = \lambda_{3} = 1}} > 0. \hfill \\ \det \left[ {H_{ij} } \right] > 0 , H_{ij} = \frac{{\partial^{2} w}}{{\partial \lambda_{i} \partial \lambda_{j} }}|_{{\lambda_{1} = \lambda_{2} = \lambda_{3} = 1}} , i,j \in \left\{ {1,2,3} \right\}, \hfill \\ \end{gathered} $$
(9)

where \({H}_{ij}\) are the components of the Hessian matrix of the stored energy function, without any deformations. Generally, there are two principles to construct stored energy function as a model of hyper-elastic.

The first principle uses the Rivlin series [68] with a known number of parameters or Ogden expansion [69, 70]. For incompressible materials, Rivlin series is described as

$$ w_{{{\text{Ri}}}} = \mathop \sum \limits_{p,q = 0}^{\infty } C_{pq} \left( {I_{1} - 3} \right)^{p} \left( {I_{2} - 3} \right)^{q} . $$
(10)

And Ogden expansion as

$$ w_{{{\text{Og}}}} = \mathop \sum \limits_{p = 1}^{\infty } \frac{{\mu_{p} }}{{\alpha_{p} }}\left( {\lambda_{1}^{{\alpha_{p} }} + \lambda_{2}^{{\alpha_{p} }} + \lambda_{3}^{{\alpha_{p} }} - 3} \right), $$
(11)

where \({C}_{pq}\), \({\mu }_{p}\) and \({\alpha }_{p}\) are material characteristics, under condition \({C}_{00 }=0\).

In the second principle, which is essential for modelling based on experimental data, \(\partial w/\partial {I}_{1}\) is independent of \({I}_{1}\) and \({I}_{2}\) for low values and dependent on \({I}_{1}\) for high values and \(\partial w/\partial {I}_{2}\) is independent of \({I}_{1}\) but a descending function of \({I}_{1}\).

Various stored strain energy models can be categorised in the following section.

Models based on first and second invariants

Polynomial function of first invariant

Although in these models, the effect of the second invariant is neglected, they have no valid predictions for some kinds of deformation. Moreover, they show poor results for shear and biaxial stress. These models are listed in Table1.

Table 1 List of polynomial function of first invariant
Full size table

Polynomial function of first and second invariants

These models use the following function:

$$ w_{{I_{1} - I_{2} - {\text{based}}}} = \mathop \sum \limits_{p = 1}^{\infty } \frac{{C_{p0} }}{p}\left( {I_{1} - 3} \right)^{p} + \mathop \sum \limits_{q = 1}^{\infty } \frac{{C_{0q} }}{q}\left( {I_{2} - 3} \right)^{q} . $$
(12)

Most of these models use the Mooney–Rivlin model’s expansion. Their prediction for biaxial deformation modes is poor. Generally speaking, models that have the term \({({I}_{2}-3)}^{q}\) for \(q\ge 1\) are not accurate for biaxial results. In Table 2 some of these models are presented.

Table 2 List of polynomial function of first and second invariants
Full size table

Function models with respect of first and second invariants

Generally, in this model, the following function is used:

$$w_{{I_{1} - I_{2} - {\text{based}}}} = f\left( {I_{1} } \right) + g\left( {I_{2} } \right).$$
(13)

In the function with respect to the second invariant, it is important to mention that \(\partial g/\partial {I}_{2}\) is a positive function with respect to the second invariant and has a descending value. This is why logarithmic functions are used to express these functions. These functions are the resultants of power expansion of \({I}_{2}-3\) with a bounded convergence radius. The first invariant function is expressed in several forms; power series, exponential function: this function can be the result of Taylor series expansion with unbounded convergence radius, mirror function; in this function, the reflection of function \(y\left(x\right)=1/{(a-x)}^{n}\) is used to express \(\partial w/\partial {I}_{1}\). Projection of this function is the reflection of \(z\left(x\right)=1/{x}^{n}\), functions \(y\) and \(z\) are symmetric with respect to \(x=a/2\) and also \(\mathrm{x}=\mathrm{a}\) is the asymptotic value of \(y\). This value can be introduced as limiting value for function with respect to the first invariant.

Some of these models are summarised in Table 3.

Table 3 List of models based on function of first and second invariants
Full size table

Model based on principal tensions of deformation gradient tensor

In these models, the focus is on eigenvalues of deformation gradient tensor. Mainly, Ogden expansion is used in these models which are listed in Table 4.

Table 4 List of models based on principal tensions of deformation gradient tensor
Full size table

Models based on combinations of principal tensions of deformation gradient tensor and first and second invariants

In previous models, the focus was on constructing the functions, based on invariants or principal tensions only. It is possible to construct a function to express the behaviour of these materials, based on both of them. The list of these models is presented in Table 5.

Table 5 List of models based on combination of principal tensions and invariants
Full size table

Models based on chemical structures and quasi-rubber material network

This type of modelling focuses on the microscopic response of polymer chains in the material network. The process and instructions for making these materials play the main role in these models. The mechanical behaviour of quasi-rubbers is influenced by entropy changes that depend on molecular chain networks. In Table 6 a list of these models is presented.

Table 6 List of models based on chemical structures
Full size table

Physical parameter calculations of models

As it was shown before, strain energy functions have some parameters depending on natural parameters of materials. Finding these parameters causes the model to be able to express the mechanical behaviour of the rubber [102, 103]. In high strains hyper-elasticity, two stress tensors are introduced: true stress tensor (Cauchy) and nominal stress tensor (Piolla–Kirchhof first stress). In these materials, Cauchy stress tensor depends on the strain and an arbitrary parameter which is calculated from equilibrium equations. This parameter is called hydrostatic pressure, the unknown pressure in reaction to incompressibility constant, such that

$${\sigma }_{i}=-p+{\lambda }_{i}\left(\frac{\partial w}{\partial {I}_{1}}\frac{\partial {I}_{1}}{\partial {\lambda }_{i}}+\frac{\partial w}{\partial {I}_{2}}\frac{\partial {I}_{2}}{\partial {\lambda }_{i}}\right)$$
(14)

And nominal stress is given as follows:

$${P}_{i}=-\frac{1}{{\lambda }_{i}}p+\frac{\partial w}{\partial {I}_{1}}\frac{\partial {I}_{1}}{\partial {\lambda }_{i}}+\frac{\partial w}{\partial {I}_{2}}\frac{\partial {I}_{2}}{\partial {\lambda }_{i}}. $$
(15)

To find these parameters, some experiments are required through which the strain is measured as a function of input forces. By curve fitting, the stored energy function on experimental results, the unknown parameters are taken into account. Different types of usual loadings, in which the strain energy function is used to express hyper-elastics’ behaviour are uniaxial [104,105,106,107], equi-biaxial, pure shear, and biaxial loading [108]. The deformation gradient for each type of loading is as follows:

For equi-axial loading

$${\varvec{F}}=\left[\begin{array}{ccc}\uplambda & 0& 0\\ 0& {\uplambda }^{-\frac{1}{2}}& 0\\ 0& 0& {\uplambda }^{-\frac{1}{2}}\end{array}\right]\to \left\{\begin{array}{c}{P}_{1}=2\left[\frac{\partial w}{\partial {I}_{1}}+\frac{1}{\lambda }\frac{\partial w}{\partial {I}_{2}}\right]\left[\lambda -\frac{1}{{\lambda }^{2}}\right]\\ {\sigma }_{1}=2\left[\frac{\partial w}{\partial {I}_{1}}+\frac{1}{\lambda }\frac{\partial w}{\partial {I}_{2}}\right]\left[{\lambda }^{2}-\frac{1}{\lambda }\right]\end{array}\right.$$
(16)

for equi-biaxial loading

$${\varvec{F}}=\left[\begin{array}{ccc}\uplambda & 0& 0\\ 0&\uplambda & 0\\ 0& 0& {\uplambda }^{-2}\end{array}\right]\to \left\{\begin{array}{c}{P}_{1}={P}_{2}=2\left[\frac{\partial w}{\partial {I}_{1}}+{\lambda }^{2}\frac{\partial w}{\partial {I}_{2}}\right]\left[\lambda -\frac{1}{{\lambda }^{5}}\right]\\ {\sigma }_{1}={\sigma }_{2}=2\left[\frac{\partial w}{\partial {I}_{1}}+{\lambda }^{2}\frac{\partial w}{\partial {I}_{2}}\right]\left[{\lambda }^{2}-\frac{1}{{\lambda }^{4}}\right]\end{array}\right.$$
(17)

for pure shear loading

$$ F = \left[ {\begin{array}{*{20}c} \lambda & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & {\lambda^{ - 1} } \\ \end{array} } \right] \to \left\{ {\begin{array}{*{20}c} {P_{1} = 2\left[ {\frac{\partial w}{{\partial I_{1} }} + \frac{\partial w}{{\partial I_{2} }}} \right]\left[ {\lambda - \frac{1}{{\lambda^{3} }}} \right]} \\ {\sigma_{1} = 2\left[ {\frac{\partial w}{{\partial I_{1} }} + \frac{\partial w}{{\partial I_{2} }}} \right]\left[ {\lambda^{2} - \frac{1}{{\lambda^{2} }}} \right]} \\ \end{array} } \right., $$
(18)

for biaxial loading

$$ \begin{gathered} F = \left[ {\begin{array}{*{20}c} {\lambda_{1} } & 0 & 0 \\ 0 & {\lambda_{2} } & 0 \\ 0 & 0 & {\lambda_{1}^{ - 1} \lambda_{2}^{ - 1} } \\ \end{array} } \right] \hfill \\ \to \left\{ {\begin{array}{*{20}c} {P_{1} = 2\left[ {\frac{\partial w}{{\partial I_{1} }} + \lambda_{2}^{2} \frac{\partial w}{{\partial I_{2} }}} \right]\left[ {\lambda_{1} - \frac{1}{{\lambda_{1}^{3} \lambda_{2}^{2} }}} \right]} \\ {P_{2} = 2\left[ {\frac{\partial w}{{\partial I_{1} }} + \lambda_{2}^{2} \frac{\partial w}{{\partial I_{2} }}} \right]\left[ {\lambda_{2} - \frac{1}{{\lambda_{2}^{3} \lambda_{1}^{2} }}} \right]} \\ \end{array} } \right., \hfill \\ \end{gathered} $$
(19)

for simple shear

$$ \begin{gathered} F = \left[ {\begin{array}{*{20}c} 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] \to P_{12} = 2\gamma \left[ {\frac{\partial w}{{\partial I_{1} }} + \frac{\partial w}{{\partial I_{2} }}} \right], \hfill \\ \lambda_{1} = \sqrt {1 + \frac{{\gamma^{2} }}{2} + \gamma \sqrt {1 + \frac{{\gamma^{2} }}{4}} } , \hfill \\ \lambda_{2} = \sqrt {1 + \frac{{\gamma^{2} }}{2} - \gamma \sqrt {1 + \frac{{\gamma^{2} }}{4}} } , \hfill \\ \end{gathered} $$
(20)

in which \({\lambda }_{1}\) and \({\lambda }_{2}\) are tension and compression eigenvalues, respectively.

Parameters estimation

In this part, the problem of finding the best model among the Tables 1, 2, 3, 4, 5, 6, to express the mechanical behaviour of a hyper-elastic material in the presence of the associated standard loading experimental data is defined in the form of an optimisation problem. Having experimental data as accuracy points, each model can be fitted to the data by finding the model’s unknown parameters. On the other hand, for each model, there is no unique solution for unknown parameters and it depends on the selected estimation method. Each method achieves a set of the unknown parameter which possessing its model accuracy. Therefore, to compare the suitability of each model first of all the best and the most accurate possible form each model should be found. To this end, the well-known optimisation method namely GA is used.

The GA method is a search based method inspiring from the genetic evolution of species. In this algorithm, a population of nominated solutions to an optimisation problem is measured toward better solutions, each candidate solution has a set of properties, which can be mutated and solution are expressed in binary as strings of 0 s and 1 s [109]. The function of this algorithm is based iteration and most of its parts are selected as random processes. The cost function for the optimisation problem is the root mean square of error (RMS) of model and accuracy points as reference data. In the case of having \(n\) accuracy point,s the cost function is defined as follows:

$$\mathrm{RMS}=\sqrt{\frac{1}{\mathrm{n}}\sum_{\mathrm{i}=1}^{\mathrm{n}}{\left({\stackrel{\sim }{\mathrm{y}}}_{\mathrm{i}}-{\mathrm{y}}_{\mathrm{i}}\right)}^{2}}$$
(21)

where, \({y}_{i}\) is the observed value for the ith observation and \({\stackrel{\sim }{y}}_{i}\) is the estimated value.

The variables of the optimisation problem are the unknown parameter of each model. Having the cost function, variables, and probable constraints, the GA procedure is ready to use. Considering the flowchart depicted in Fig. 1, each step starts with defining an initial population. Then fitness of the population members (cost function for each individual) is calculated. Some genetic operators such as mutation and crossover are applied to the population to create the next generation from the previous population with more elite members. These steps are repeated to reach the stop criterion. The stop criterion is satisfied when no remarkable improvement occurs in the fitness of the population’s elite members. Now let’s consider the case that the constitutive mathematical model of a hyper-elastic material is required.

Fig. 1
figure 1

The GA procedure for finding the most accurate possible form of each model

Full size image

This model might be used in FEM software for stress analysis or used in analytical analysis. For example, experimental data of vulcanised rubber are extracted from three tests including; uniaxial tension, pure shear, and equi-biaxial tension, and depicted by three curves in Fig. 2, Treloar [104]. Practically, working with these data raises two problems. First, there is no mathematical function for expressing stress–strain relation and second, the data do not cover the whole range of deformation, e.g. equi-biaxial and pure shear test data exist for a range of stretch from 1 to 4.5 while for the uniaxial test from 1 to 7. That is why using the constitutive models of Tables 1, 2, 3, 4, 5, 6 is inevitable in applications. Using Tables 1, 2, 3, 4, 5, 6 and the proposed GA, the best possible model’s unknown parameters for predicting the uniaxial loading of Treloar [104] data are summarised in Table 7. Also, the best possible function for each model is depicted in Fig. 3. Both Table 7 and Fig. 3 shows that Swanson model is the most accurate model for predicting the uniaxial behaviour of this set of reference data. The Fig. 3 shows the Neo–Hookean and Gent–Thomas have the worst accuracy for uniaxial loading. Another study of Treloar [104] is related to the pure shear loading data. When the pure shear loading data are assumed as the reference data, it is concluded from Table 8, and Fig. 4 that the model Carrol has the most accuracy while Neo–Hookean has the worst.

Fig. 2
figure 2

Experimental data of vulcanised rubber containing 8% sulphur [104]

Full size image
Table 7 Evaluated parameters of models with uniaxial tension test data
Full size table
Fig. 3
figure 3

The best accurate function of each model for uniaxial loading

Full size image
Table 8 Evaluated parameters of models with pure shear test data
Full size table
Fig. 4
figure 4

The best accurate function of each model for pure shear loading

Full size image

Table 9 and Fig. 5 summarise the result obtaining form finding the best possible model fitted on the equi-biaxial loading data from Treloar [104]. It is seen that Ogden is the most suitable model for prediction of mechanical behaviour of the specimen under the equi-biaxial loading, while Isihara and Neo–Hookean have the worst accuracy. From Eqs. 16 to 20, it is clear that each constitutive model has different parameters and different mathematical functions for different loadings, e.g. for three types of loading, three functions for the 3-chain model are obtained as follows:

Table 9 Evaluated parameters of models with equi-biaxial tension test data
Full size table
Fig. 5
figure 5

Comparison between models and equi-biaxial experiment data

Full size image
$${\sigma }_{\mathrm{UT}}=\frac{\mu }{3}\left(\frac{3\lambda N-{\lambda }^{3}}{N-{\lambda }^{2}}-{\lambda }^{-2}\frac{3N-{\lambda }^{-1}}{N-{\lambda }^{-1}}\right),$$
$$\mu =0.2596, N=76.31,$$
(22)
$${\sigma }_{\mathrm{ET}}=\frac{\mu }{3}\left(\frac{2\lambda N-{\lambda }^{3}}{N-{\lambda }^{2}}-{\lambda }^{-5}\frac{3N-{\lambda }^{-4}}{N-{\lambda }^{-4}}\right),$$
$$\mu =0.3126, N=154.4,$$
(23)
$${\sigma }_{\mathrm{PS}}=\frac{\mu }{3}\left(\frac{3\lambda N-{\lambda }^{3}}{N-{\lambda }^{2}}-{\lambda }^{-3}\frac{3N-{\lambda }^{-2}}{N-{\lambda }^{-2}}\right),$$
$$\mu =0.4269, N=165.41,$$
(24)

As it is depicted in Figs. 3, 4, 5, three functions of Eqs. 22 to 24 accurately describe the mechanical behaviours of the 8% Sulphur vulcanised rubber of Fig. 2. In spite of the accurate description of each type of loading test data by of these three functions, no one can describe the whole mechanical behaviour completely. To obtain a unique function which, not only be based on 3-chain model but also can relatively describe whole data of Fig. 2 let’s consider function \({\sigma }_{o}\) similar to one of the functions in Eqs. 22 to 24 e.g. Equation 22. This function’s parameters again are determined using GA; however, this time the cost function is defined such that the data of the three types of loading is considered as the reference or accuracy points. Defining the cost function as follows:

$$\begin{array}{ll} {\mathrm{CF}}& = {{W}_{1}} \sqrt{\frac{1}{n_1}\sum_{i=1}^{n1} \left({\tilde{y}_{{\rm UTi}}}- y_i\right)}^{2} +W_2 \sqrt{\frac{1}{n_2} \sum_{i=1}^{n_2}\left({\tilde{y}_{{\rm ETi}}}-y_i\right)}^{2}\\ &\quad +\,W_3 \sqrt{\frac{1}{n_3} \sum_{i=1}^{n_3} \left({\tilde{y}_{{\rm PSi}}}-y_i \right)}^{2} \end{array}$$
(25)

where \(\mathrm{CF}\) is overall cost function, \({W}_{1}\)–\({W}_{3}\) are weight of each single cost function that show the relative importance of each single cost function among the overall cost function, \({n}_{1}\)–\({n}_{3}\) are number of reference data in each type of loading, and \({\stackrel{\sim }{\mathrm{ y}}}_{\mathrm{UTi}}\), \({\stackrel{\sim }{y}}_{\mathrm{ETi}}\) and \({\stackrel{\sim }{y}}_{\mathrm{PSi}}\) are reference or accuracy points of each type of loading test data. After optimise the function \({\sigma }_{\mathrm{UT}}\) with respect to the cost function 25 the parameters \(\mu \) and \(N\) are obtained as follows:

$$\mu =0.3329, N=87.91.$$
(26)

The new optimised \({\sigma }_{\mathrm{o}}\) function is depicted in Fig. 6.

Fig. 6
figure 6

A unique function to be relatively close to the three types of loading test data

Full size image

It is seen that the function is slightly distant from each reference data such that it is placed between them. This function obtained using \({W}_{1}=1\) , \({W}_{2}=1\) and \({W}_{3}=2\). Selecting another set of \({W}_{i}\)’s, the function approaches to the associated reference data with larger\({ W}_{i}\). For \(\mathrm{M}\) type of loading the cost function can be generally defined as follows:

$$ {\text{CF}} = \mathop \sum \limits_{j = 1}^{M} \left[ {W_{j} \sqrt {\frac{1}{{n_{j} }}\mathop \sum \limits_{i = 1}^{{n_{j} }} \left( {\tilde{y}_{ji} - y_{i} } \right)^{2} } } \right], $$
(27)

in which \({\stackrel{\sim }{y}}_{ji}\) is \(\mathrm{i}\) th accuracy point of loading test data \(j\) th, and \({n}_{j}\) is number of data of test data \(j\) th.

Conclusion

In this research, 50 constitutive models of hyper-elastic materials are collected and classified into the six categories including; polynomial function of the first invariant, the polynomial function of first and second invariants, functions of first and second invariants, functions of principal tensions of deformation gradient tensor, functions of combinations of principal tensions of deformation gradient tensor and first and second invariants, and models based on chemical structures and quasi-rubber material network. The mathematical function of each model is not unique and depends on the type of loading, physical parameters, Cauchy stress tensor, hydrosstatic pressure, and nominal stress. Five standard loadings were introduced including; uniaxial, equi-biaxial, pure shear, biaxial, and simple shear loading. The physical parameters were taken into account for each type of loading. It was presented how to determine the most possible accurate unknown parameters of each model using GA, when experimental test data exist. Also, it was shown that using this algorithm and using a combinational cost function, all types of loading test data can be explained by an approximate unique function.

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.