Models and analysis for adhesive contact, material damage, and cardiac waves

The thesis investigates mathematical models for contact processes with material damage, adhesion, and cardiac waves. Processes of material damage and adhesion are very important in industry and in everyday life. Electrical cardiac waves keep the heart functioning.;We consider two models which describe the evolution of mechanical damage in a vibrating string or a membrane which lies above an obstacle. Material damage is caused by the opening and growth of micro-cracks or micro-cavities as a result of internal stresses, leading to the reduction in the systems capacity to support loads. When the density of these micro-cracks or micro-cavities reaches a critical limit, the part or component breaks down. The evolution of material damage is modeled by an internal variable, the damage field, which measures the fractional decrease in the material strength.;We establish the existence of a unique weak solution to the problems using fixed point arguments or theory of evolution equations.;We next describe two dynamic models for the process of adhesive contact between a rod or a membrane and a rigid or a deformable obstacle. The adhesion process is modeled by the introduction of the bonding function, which measures the fraction of active bonds between the two bonded objects. The main new idea in this part of the dissertation is the choice of the adhesion rate exponent which allows for complete debonding in finite time, thus the bonded joint may fail (unlike models used in early publications). Mathematically, the rate is not a Lipschitz function which makes it necessary to study a harder problem carefully.;We establish the existence of a unique weak solution for the problems using fixed point arguments.;Finally, we use similar tools to prove the existence and uniqueness of a weak solution for a problem arising in biomechanics. We study a version of the bidomain model for the electrical properties of cardiac tissue. The model has an unusual mathematical structure and consists of a nonlinear parabolic equation coupled with a time dependent elliptic equation, and an ordinary differential equation.