Úvod FME 23004541


Title of the subject: Nonlinear mechanics and continuum mechanics

Semester_of_study (WT – winter/ST – summer): WT

Learning outcomes: Students will acquire knowledge of non-linear problems and methods of their solutions in continuum mechanics. Get an overview of fundamental concepts of physical and geometric non-linearities, the principles used in non-linear mechanics in simplification, which leads to a linear formulation. To gain basic knowledge of paying respects to nonlinear mechanics and foundations for understanding how the computer programs work in this area.

Brief content of course: Lecture Topics: 1. Fundamentals of tensor algebra and analysis in Riemann space, duality, covariant, contravariant and mixed tensors, space metrics, covariant derivative. 2. Lie derivative, symmetric and antisymmetric tensors, rotation. 3. Continuum deformation in Lagrangian and Eulerian description, deformation gradient. 4. Polar decomposition of deformation gradient, types of deformation tensors. 5. Isochoric and volumetric deformation, rate of change of deformation tensor, spin, operations pull-back push-forward. 6. Strain tensors - Cauchy, Kirchhoff, I. and II. Piola-Kirchhoff, Mandel, corotation. 7. Energy conjugated stresses and strains. 8. Derivative with respect to time, velocity and acceleration. 9. Equations of equilibrium, conservation of mass, momentum, angular momentum. 10. Kinetic energy, law of conservation of energy. 11. Principle of virtual work, constitutive equations, objectivity of tensors, elastic material, isotropic elastic and hyperelastic material. 12. Linearization and comparison of different material models, fundamentals of plasticity. 13. Computer implementation of theoretical relationships. Topics of practices: 1. Tensors on differential varieties. 2. Operations with tensors. 3. Displacements and deformation gradient. 4. Numerical computation of polar decomposition. 5. Change of volume and shape during deformation. 6. Meaning of individual stress tensors. 7. Energy during deformation. 8. Examples for material time derivative. 9. Equilibrium equations. 10. Conservation of energy formulations in various reference frames. 11. Virtual work, formulation of constitutive equations. 12. Linearization, plasticity. 13. Principle of work of FEM programs.

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