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Material spin and finite-strain hypo-elasticity for two-dimensional orthotropic media

Zubko I. Y.

Engineering Transactions

2018

Vol. 66, iss. 4

443--459

A constitutive material spin tensor in the case of purely elastic finite-strain deformation is introduced for a two-dimensional orthotropic media using the minimizing principle applied to obtain the reloaded configuration of the material volume. This material spin explains the rotation of the orthonormal vector frame which coincides with the material symmetry axes in the initial configuration of the material volume and uniquely corresponds to a set of these axes in the current configuration although it does not coincide with the latter. The given definition is followed by the exact expression which includes the deformation gradient tensor, unit vectors of the initial material anisotropy axes and their axial parameters. This definition allows obtaining a new variant of decomposing any elastic finite-strain motion onto rigid and deformational parts and introducing the material corotational rate. The latter is used for the formulation of the anisotropic rate-type elastic law in the current configuration based on the strain measure which does not belong to the Seth-Hill family. For isotropic as well as for tetragonal media, the introduced material rotation tensor coincides with the rotation tensor from the polar decomposition of a deformation gradient.

Visco-hypoelastic model of photo-polymerization process for small changes of temperature

Gambin W.

Archives of Mechanics

2010

Vol. 62, nr 5

379-403

The aim of the paper is to propose a model for estimation of the shrinkage stress in photo-cured dental restorations. Up to now, the elastic and viscoelastic models of photo-curing process use an incremental approach with a large number of time steps, with a fixed Young’s modulus and viscosity within each of the time increments. The elastic approach with a stepped increasing Young’s modulus gives the stress values too high. On the other hand, the incremental viscoelastic approach requires long-lasting computations. In the present paper, a consistent model of the photo-curing process for the case of small temperature changes is proposed. The proposition bases on the Maxwell model, in which the Young’s modulus and the viscosity are continuous functions of time. The assumptions of the model follow from the dental practice, as well as from a physical nature of the process and from the rules of continuum mechanics. A performed incremental analysis of the process enables to formulate an integral model of the process, with an explicit rule for the shrinkage stress for 1D and 3D cases. The model has been tested for the material data of dental composite Clearfil F2. Results of the calculations coincide with the values of stresses measured in thin layers of Clearfil F2.