2.1. Theory of Hyperelastic Models

This section shows the theoretical relationship between stress and strain for a proposed hyperelastic model based on the FOEC in the sense of Landau’s theory, Mooney–Rivlin and Ogden models.

Nonlinear FOECs are defined in the sense of Landau’s theory [43] to establish a strain energy function, considering the medium incompressible valid for the hyperelastic regime as defined Hamilton and Destrade [44,45],

where I1=trE,I2=trE2 and I3=trE3 are the classical invariant of deformation defined by Cemal et al. [46], E is the Green strain tensor, μ is the shear modulus and A and D are the Third and Fourth Order Elastic Constants of Landau respectively. The Second Piola–Kirchoff stress tensor is determined by a constitutive law as follows,

where E is the Green–Cauchy strain tensor defined in terms of displacement field as the difference between actual and initial position respectively, u=x−X. This strain tensor is defined, according to the large deformation theory, as,

Under the hypothesis of a tensile test setup, the initial conditions are described in Figure 1. where the displacements are defined in three directions as,

(Left): scheme of the uniaxial tensile test. (Right): zoom of a differential element of the sample.

In this case, the Green–Cauchy strain tensor defined in Equation (3) may be described in matrix form as,

To describe the Second Piola–Kirchoff stress tensor in a nonlinear regime, it is necessary to determine the invariant I3 in terms of strains.

The constitutive law for tensile test case in direction 1 is deduced by the expression,

The relationship between the Cauchy stress tensor and the Second Piola–Kirchoff stress tensor is defined as,

where F is the deformation gradient tensor and J=det(F).

The derivation of Cauchy stress tensor in the context of weakly nonlinear elasticity [47] yields the constitutive law defined in high order as follows,

In order to compare with the other two hyperelastic models, the aforementioned tensor is simplified (using μ and A) as follows:

where a is defined in Equation (4).

The Mooney–Rivlin model, originally derived by Mooney in 1940 [48] was formulated in terms of the Cauchy–Green deformation tensor invariants by Rivlin [49] as:

where c1 and c2 are the material parameters, I1 and I2 the first and second strain invariants respectively and Ψ the strain energy function.

In the case of an uniaxial tension (σ=σ1, σ2=σ3=0) the Cauchy stress as a function of the strain invariants is

where λ=λ1 (λ1 is the principal stretch in 1 direction) and the invariants from the Cauchy–Green tensor for an incompressible hyperelastic material subjected to a uniaxial tension are defined as [50],

For the Mooney–Rivlin model, the Cauchy stress obtained employing (12) and using two parameters (c1 and c2) is,

The strain energy function in the Ogden model, developed in 1972 [51], is described by,

where μr (infinitesimal shear modulus) and αr (stiffening parameter) are material constants, and λ1, λ2 and λ3 are the principal stretches. Taking into account that for an incompressible material, λ1=λ and λ2=λ3=1/λ [50], Equation (15) is simplified into,

The Cauchy stress tensor as a function of the principal stretches for an incompressible material is,

Finally, using Equation (17), the Cauchy stress using two parameters (μr and αr) is obtained as follows,

The shear modulus μ in the Ogden model results from the expression,