Dynamic frictional contact of a viscoelastic beam
In this paper, we study the dynamic frictional contact of a viscoelastic beam with a deformable obstacle. The beam is assumed to be situated horizontally and to move, in both horizontal and tangential directions, by the effect of applied forces. The left end of the beam is clamped and the right one is free. Its horizontal displacement is constrained because of the presence of a deformable obstacle, the so-called foundation, which is modelled by a normal compliance contact condition. The effect of the friction is included in the vertical motion of the free end, by using Tresca's law or Coulomb's law. In both cases, the variational formulation leads to a nonlinear variational equation for the horizontal displacement coupled with a nonlinear variational inequality for the vertical displacement. We recall an existence and uniqueness result. Then, by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives, a numerical scheme is proposed. Error estimates on the approximative solutions are derived. Numerical results demonstrate the application of the proposed algorithm.
Numerical analysis and simulations of a quasistatic frictional contact problem with damage in viscoelasticity
We consider a quasistatic problem which models the bilateral contact between a viscoelastic body and a foundation, taking into account the damage and the friction. The damage which results from tension or compression is then involved in the constitutive law and it is modelled using a nonlinear parabolic inclusion. The variational problem is formulated as a coupled system of evolutionary equations for which we state the existence of a unique solution. Then, we introduce a fully discrete scheme using the finite element method to approximate the spatial variable and the Euler scheme to discretize the time derivatives. Error estimates are derived and, under suitable regularity hypotheses, the convergence of the numerical scheme obtained. Finally, a numerical algorithm and results are presented for some two-dimensional examples.
Semiautomated quantification of the fibrous tissue response to complex three-dimensional filamentous scaffolds using digital image analysis.
Fibrosis represents a relevant response to the implantation of biomaterials, which occurs not only at the tissue-material interface (fibrotic encapsulation) but also within the void fraction of complex three-dimensional (3D) biomaterial constructions (fibrotic ingrowth). Usual evaluation of the biocompatibility mostly depicts fibrosis at the interface of the biomaterial using semiquantitative scores. Here, the relations between encapsulation and infiltrating fibrotic growth are poorly represented. Virtual pathology and digital image analysis provide new strategies to assess fibrosis in a more differentiated way. In this study, we adopted a method previously used to quantify fibrosis in visceral organs to the quantification of fibrosis to 3D biomaterials. In a proof-of-concept study, we transferred the "Collagen Proportionate Area" (CPA) analysis from hepatology to the field of biomaterials. As one task of an experimental animal study, we used CPA analysis to quantify the fibrotic ingrowth into a filamentous scaffold after subcutaneous implantation. We were able to demonstrate that the application of the CPA analysis is well suited as an additional fibrosis evaluation strategy for new biomaterial constructions. The CPA method can contribute to a better understanding of the fibrotic interactions between 3D scaffolds and the host tissue responses.