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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In this work, we use the rule of mixtures to develop an equivalent material model in which the total strain energy density is split into the isotropic part related to the matrix component and the anisotropic energy contribution related to the fiber effects. For the isotropic energy part, we select the amended non-Gaussian strain energy density model, while the energy fiber effects are added by considering the equivalent anisotropic volumetric fraction contribution, as well as the isotropized representation form of the eight-chain energy model that accounts for the material anisotropic effects. Furthermore, our proposed material model uses a phenomenological non-monotonous softening function that predicts stress softening effects and has an energy term, derived from the pseudo-elasticity theory, that accounts for residual strain deformations. The model’s theoretical predictions are compared with experimental data collected from human vaginal tissues, mice skin, poly(glycolide-co-caprolactone) (PGC25 3-0) and polypropylene suture materials and tracheal and brain human tissues. In all cases examined here, our equivalent material model closely follows stress-softening and residual strain effects exhibited by experimental data.

It is well-known that most of the constitutive relations available in the literature can not predict most biological material behaviors well, such as the multiaxial behavior of muscles, the softening of biological tissues, damage and healing, among others, because of the molecular and cellular contributions to the behavior at the tissue and organ levels, soft tissue anisotropy, transverse isotropy by tendons and ligaments, cylindrical orthotropy by arteries and complex symmetries by planar tissues [

Based on these energy models on which the energy is split into two parts, it is clear that an equivalent energy model that considers the matrix (isotropic part) and the fiber (anisotropic part) effects could be used to predict the material behavior that is exhibited by biological tissues and biocompatible materials [

The paper has been organized as follow. In Section 2, we introduce a brief review of the required equations that describe finite deformations of hyperelastic materials. In Section 3, we introduced an equivalent strain energy density representation form that combines, by using the rule of mixtures, the isotropic and the anisotropic energy material parts. In Section 4, we have derived the corresponding stress-stretch constitutive equations that are based on the amended non-Gaussian strain energy density model, a non-monotonous stress-softening function and a residual strain effects material model that is derived from the pseudo-elasticity theory concepts. Furthermore, we have included the Dorfmann and Ogden material model with slight modifications to capture Mullins and permanent set effects. Comparison of the model’s prediction with experimental data is done in Section 5. Finally, some conclusions related to theoretical predictions and experimental data are addressed in Section 6.

Since biocompatible materials tend to exhibit large deformations, in this section, we introduce some basic definitions related to finite deformations that are needed to characterize the material behavior. Let Ω be a fixed reference configuration of a body, and use the notation χ : Ω → ℝ^{3} to denote the body deformation, which transforms a material point, _{c}

which describes an isochoric deformation, since

in which the relation ^{2} = _{3} has been used [_{k}

where tr is the trace operation. Furthermore, the Cauchy–Green deformation tensor B ≡ ^{T}

where e_{jk}_{j}_{k}_{i}_{i}_{k}

In the undeformed state B = 1, the identity tensor and

The main motivation on deriving an equivalent strain energy density model not only comes from the ideas previously developed by the aforementioned research works in which the material energy density were split into two parts, but also from the experimental findings obtained in samples of vulcanized natural rubber during uniaxial deformation tests in which the usage of synchrotron X-rays allowed for the determination of the isotropic and anisotropic energy contributions to the material response behavior [_{T}

where _{iso}(_{1}) is the strain energy density related to the isotropic material behavior, _{aniso} (_{4}_{i}_{5}_{i}_{4i}_{5}_{i}

_{i}_{1}_{i}_{1} + _{2}_{i}_{2} + _{3}_{i}_{3} in the initial configuration, _{ji}_{iso}(_{1}) and _{aniso}(_{4}_{i}_{5}_{i}

where λ_{r}

_{chain} is the chain deformation, which can be computed from:

^{−1} (λ_{r}

_{aniso}(I_{4}_{i}_{5}_{i}_{T}_{iso}(_{1}) as:

We next follow a procedure similar to the one developed in [

has been

since the fiber direction cosines have been assumed to have the following possible orientations, (1_{1} and _{2} are energy density fitting parameters. Other forms for the strain energy density are possible, but we prefer to use expression

Before we use _{1} = 0 MPa_{2} = 0.0001 MPa, ^{3},

Encouraged by the accuracy of the predicted results obtained from the equivalent strain energy density given by

in which T is the Cauchy stress, B is the left Green–Cauchy deformation tensor,

To characterize the stress-softening effect, as well as residual strains, we use the material model introduced in [

where:

Here, _{a}_{max} _{a}, a

Thus, the Cauchy stress-stretch equivalent material model components for the virgin material are obtained from

Eliminating the pressure,

The corresponding constitutive equation for a non-monotonous stress-softened material model is provided by the following equation:

where

thus, the relative chain stretch, λ_{r}

Before we assess the accuracy achieved by our proposed energy material model

Here, we modify the pseudo-elastic material model proposed by Dorfmann and Ogden in [

which describe Mullins and residual strain effects in which:

wherein _{max} = _{T}_{max}), _{2} = _{1}, _{1}, _{2} and

We next use the aforementioned material models to predict uniaxial extension experimental data of human vaginal tissue, mice skin, two suture materials, tracheal and brain human tissue samples.

We first start by considering uniaxial extension experimental data collected from samples of vaginal tissue subjected to loading and unloading cyclic tests along the longitudinal and transverse axes of the biological tissue samples [^{3}, _{1} = 1.8, _{1} = 1.0001, _{2} = 0.1 and _{1}, _{2},

We next use experimental data collected from cyclic loading and unloading of 18-month male and female mice skin [^{3}, _{1} = 0.0525, _{1} = 1.00001, _{2} = 0.55 and ^{3}, _{1} = 0.05, _{1} = 1.001, _{2} = 0.04 and

To further assess the accuracy of our proposed equivalent energy material model, we now use cyclic loading-unloading uniaxial stress-stretch data from poly(glycolide-co-caprolactone) (PGC25 3-0) and polypropylene suture material samples collected from an Instron tensile machine model 3365 with a maximum cell load capacity of 1.6 kN [

We next model the material behavior of human tracheal specimens by considering uniaxial test experimental data collected by Teng _{max} < 1.215.

As a final example, let us consider the experimental data collected from human brain tissue that exhibits Mullins and residual strain effects, which are qualitatively similar to that observed in filled elastomers. Experimental data plotted in

In this paper, we have used the rule of mixtures to develop material models that are based on the equivalent representation form of the strain energy density of hyperelastic materials. This equivalent representation form of the strain energy density follows the idea of finding the isotropized energy form of polymeric materials reinforced with carbon nanotubes. Here, we adopted that isotropized energy form and used the non-Gaussian amended strain energy density form

This work was funded by Tecnológico de Monterrey-Campus Monterrey, through the Research Chair in Nanomaterials for Medical Devices and the Research Chair in Intelligent Machines. Additional support was provided from the European Commission project, IREBID(FP7-PEOPLE-2009-IRSES-247476) and from Consejo Nacional de Ciencia y Tecnología (Conacyt), México. This work is dedicated to Francisco J. Cantú Ortiz with esteem and appreciation.

The authors declare no conflicts of interest.

Comparison between the experimental strain energy density and the theoretical predictions obtained from

Cauchy stress-stretch data collected from human vaginal tissue compared with theoretical predictions. (

Cauchy stress-stretch data for mice skin compared with the theoretical predictions. The material parameter values used to obtain the theoretical predictions are summarized in

Engineering stress-stretch data for PGC25 3-0 sutures compared with theoretical predictions. The material parameter values used to obtain theoretical predictions from ^{3}, _{1} = 1.65, _{1} = 1.1, _{2} = 0.6 and

Engineering stress-stretch data for polypropylene sutures compared with theoretical predictions. The material parameter values used to obtain theoretical predictions from ^{3}, _{1} = 0.85, _{1} = 1.0001, _{2} = 0.35 and

Cauchy stress-stretch data for specimens of the mucosa and submucosa human tracheal membrane. The material parameter values used to obtain the theoretical predictions are summarized in

Engineering stress-stretch data for samples of brain tissue harvested from the frontal lobe in the sagittal direction compared with the theoretical predictions. The material parameter values used to obtain the theoretical predictions from ^{3}, _{1} = 0.095, _{1} = 1.001, _{2} = 0.8 and

Engineering stress-stretch data for samples of brain tissue harvested from the occipital lobe in the frontal direction compared with the theoretical predictions. The material parameter values used to obtain the theoretical predictions from ^{3}, _{1} = 0.006, _{1} = 2.5, _{2} = 1.8 and

Material constants used to fit experimental data.

Material samples | _{1} (MPa) |
_{2} (MPa) |
|||||
---|---|---|---|---|---|---|---|

Vaginal tissue (longitudinal axis) | 0.085 | 3.25 | −6.5 | 70 | 1.3 | 0.7 | 0.2 |

Vaginal tissue (transverse axis) | 0.085 | 3.25 | −6.5 | 3.93 | 1.3 | 0.7 | 0.2 |

Male mouse skin | 0.95 | 1.082 | 0 | 30 | 2.8 | 0.98 | 9.0 |

Female mouse skin | 0.77 | 1.18 | 0 | 20 | 2.55 | 1.2 | 9.0 |

PGC25 suture material | 100 | 2.35 | 0 | 1300 | 0.95 | 0.008 | 10 |

Polypropylene suture material | 300 | 50.5 | −7500 | −2100 | 0.6 | 0.0024 | 1.35 |

Material constants used to fit the human tracheal experimental data.

Material samples | _{1} (MPa) |
_{2} (MPa) |
|||||
---|---|---|---|---|---|---|---|

Circumferential mucosa and submucosa membrane (CSM) | 5 | 1.029 | 0 | 2, 500 | 1.3 | 2.1 | 0.8 |

Axial mucosa and submucosa membrane (ASM) | 5 | 1.029 | 0 | 15 | 1.3 | 2.1 | 0.8 |

Circumferential adventitial membrane (CAM) | 10 | 1.045 | 0 | 35, 000 | 1 | 3.4 | 0.6 |

Axial adventitial membrane (AAM) | 10 | 1.045 | 0 | 1.5 | 1 | 3.4 | 0.6 |

Material constants used to fit the brain tissue experimental data.

Material samples | _{1} (kPa) |
_{2} (kPa) |
|||||
---|---|---|---|---|---|---|---|

Frontal lobe (sagittal direction: tension) | 2.6 | 1.065 | −150 | −3000 | 2.7 | 4.6 | 1 |

Frontal lobe (sagittal direction: compression) | 2.6 | 1.065 | −150 | −3000 | 2.5 | 0.6 | 1 |

Occipital lobe (frontal direction: tension) | 2.65 | 2.5 | −350 | 650 | 2.7 | 3.8 | 0.93 |

Occipital lobe (frontal direction: compression) | 2.65 | 2.5 | −350 | 650 | 2.7 | 3.8 | 0.93 |