With applications to the periodontal ligament: a nonlinear large strain viscoelastic law

The soft tissue, called periodontal ligament (PDL), that connects the alveolar bone and the tooth root accounts to a large extent for the mobility, stress-distribution and, due to its viscoelastic properties, damping of the bone-tooth complex. The accurate prediction of these quantities is essential to new solutions in dental health care making the PDL one of the most important materials in dental biomechanics. Experimental data clearly shows that both the elastic and viscous responses of the PDL are highly nonlinear and different in tension and compression, furthermore its elasticity is stiffening at large strains whereas its viscosity is thinning, i.e. pseudo-plastic, at low strain-rates, both realized within the physiological range of the PDL. The mechanical models for the PDL found in literature, except for one, are simplistic because they use only linear elasticity and few incorporate a viscous contribution. In this thesis a new nonlinear large strain viscoelastic three-dimensional law is developed and applied to the PDL. To begin, a one-dimensional viscoelastic law is proposed based on the standard linear model with the linear springs and dash-pots replaced by nonlinear power law analogs possessing adjustable exponents. The elastic exponent controls the elastic nonlinearity rendering the elastic law either hardening or softening. Likewise, the viscous exponent controls the viscous nonlinearity, rendering the viscous law either thickening or thinning. A thorough investigation of this new nonlinear viscoelastic standard model reveals two key features: first, a finite stress relaxation time as a response of a step strain for a certain choice of viscous and elastic exponents; second, the controllable width of the phase-shift spectrum through the choice of combinations of the exponents. Since viscoelastic biological tissues show a large phase-shift spectrum this feature is an advantage of this original nonlinear law over linear ones. A family of (hyper)elastic closed-form invertible material laws is proposed and it is shown that the three-dimensional extension of the one-dimensional power law falls into that family. Now, the three-dimensional extension of the nonlinear model can be formulated using the appropriate potentials. After implementing the law into a finite element software, representative experiments, namely elastic traction and shear tests at low strain-rates, stress relaxation tests as responses to step strains and sinusoidal strain excitations, are performed in order to obtain reliable data needed for the identification of the law's parameters. The results of these experiments agree with existing data in the literature. After calibrating the law to the PDL's parameters using finite element simulations corresponding to the previous experiments the experimental and the numerical results agree well for the elastic tests and for the phase-shift spectra of the sinusoidal viscoelastic tests indicating nonlinear stiffening elastic and nonlinear pseudo-plastic viscous behavior. They agree less well for the amplitude as a function of frequency. Overall the proposed new nonlinear viscoelastic law based on the power law is capable of simulating the PDL better than a linear viscoelastic law.