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Research Papers

A Visco-Hyperelastic-Damage Constitutive Model for the Analysis of the Biomechanical Response of the Periodontal Ligament

[+] Author and Article Information
Arturo N. Natali, Emanuele L. Carniel, Piero G. Pavan

Centre of Mechanics of Biological Materials,  University of Padova, Via F. Marzolo 9, Padova 1-35131, Italy

Franz G. Sander, Christina Dorow, Martin Geiger

Poliklinik fuer Kieferorthopaedie,  University of Ulm, Ulm 89073, Germany

J Biomech Eng 130(3), 031004 (Apr 22, 2008) (8 pages) doi:10.1115/1.2900415 History: Received October 24, 2006; Revised December 20, 2007; Published April 22, 2008

The periodontal ligament (PDL), as other soft biological tissues, shows a strongly non-linear and time-dependent mechanical response and can undergo large strains under physiological loads. Therefore, the characterization of the mechanical behavior of soft tissues entails the definition of constitutive models capable of accounting for geometric and material non-linearity. The microstructural arrangement determines specific anisotropic properties. A hyperelastic anisotropic formulation is adopted as the basis for the development of constitutive models for the PDL and properly arranged for investigating the viscous and damage phenomena as well to interpret significant aspects pertaining to ordinary and degenerative conditions. Visco-hyperelastic models are used to analyze the time-dependent mechanical response, while elasto-damage models account for the stiffness and strength decrease that can develop under significant loading or degenerative conditions. Experimental testing points out that damage response is affected by the strain rate associated with loading, showing a decrease in the damage limits as the strain rate increases. These phenomena can be investigated by means of a model capable of accounting for damage phenomena in relation to viscous effects. The visco-hyperelastic-damage model developed is defined on the basis of a Helmholtz free energy function depending on the strain-damage history. In particular, a specific damage criterion is formulated in order to evaluate the influence of the strain rate on damage. The model can be implemented in a general purpose finite element code. The accuracy of the formulation is evaluated by using results of experimental tests performed on animal model, accounting for different strain rates and for strain states capable of inducing damage phenomena. The comparison shows a good agreement between numerical results and experimental data.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

Scheme of the viscoelastic Zener model

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Figure 2

Transversal section of a tooth unit (a) and specimen adopted for tensile loading tests (b). The sample is composed of alveolar bone, dentine, and the PDL in between.

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Figure 3

Initial bulk modulus (a) and shear modulus (b) versus strain rate. Experimental data (open circles) are interpolated by exponential functions (continuous lines).

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Figure 4

Viscohyperelastic analysis of the compression behavior of the PDL. A comparison between experimental (open circles) and model (continuous lines) results is reported in terms of first Piola–Kirchhoff stress (nominal stress) versus stretch.

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Figure 5

Initial axial stiffness of fibers versus strain rate. Experimental data (open circles) are interpolated by exponential functions (continuous lines).

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Figure 6

Comparison of the tensile mechanical response of the PDL interpreted by using a viscohyperelastic model and a visco-hyperelastic-damage model: EXP represent experimental data, VH numerical result obtained with the viscohyperelastic model, and VHD numerical results obtained with visco-hyperelastic-damage model. The strain rates are 0.42% (Curves 1), 12.5% (Curves 2), and 83.4% (Curves 3). The curves VH-1 and VHD-1 are almost coincident. The first Piola–Kirchhoff stress (nominal stress) is reported versus stretch.

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Figure 7

Visco-hyperelastic-damage analysis of the tensile behavior of the PDL in terms of first Piola–Kirchhoff stress (nominal stress) versus stretch. A comparison between experimental (open circles) and model (continuous lines) results is reported up to complete failure of the tissue for a strain rate of 8.34%.

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Figure 8

Mean square error versus number of viscous branches adopted in the constitutive model. The error is related to the fitting of data taken from loading/unloading tensile tests at different strain rates: 0.42%/s, 0.83%/s, 4.16%/s, 8.33%/s, and 83.4%/s.

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Figure 9

Comparison of numerical results and experimental data for a tensile loading. The stress-stretch curves refer to experimental data (open circles), results obtained with a viscohyperelastic constitutive model with four viscous branches (VH 4-proc), a viscohyperelastic model with six viscous branches (VH 6-proc), and a visco-hyperelastic-damage model with four viscous branches (VHD 4-proc).

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