The book retains strong conceptual and mathematical modelling approach for two potentials combining theory of variational problem. The subject of this book concerns analytical and numerical studies of the energy landscape for an adhesive crack of a thin film adhered to a rigid foundation.The final goal is to find the optimal crack nucleation path of a thin film for Griffith and Dugdale potentials. The energy is obtained in closed analytical form and the numerical simulation is developed for plotting the energy landscape. Problem formulation using constraints and boundary conditions are stressed in the book. It serves as a learning resource for senior undergraduate, graduate, postgraduate students and researchers and also for practicing engineers. The students will learn to solve mathematical models of a variety of engineering problems.