# Theory of Continuum Mechanics

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The goal of continuum mechanics is to model and understand complex material behaviour at the macroscopic level. As the name suggests, all state variables such as mass and temperature distribution are considered as continuous quantities.
Using universal laws of nature, such as the conservation of energy, general equations can be derived that allow the prediction of physical behaviour.
For this purpose, it is necessary to supplement the equations with specific material laws. The scope of application is not limited to the description of solids, but also applies to fluids and gases.
In addition to gaining theoretical knowledge, continuum mechanics serves as the basis for various calculation and simulation methods. The fields of application range from structural mechanics and biomechanics to fluid mechanics.

Based on the axiom of Hamilton's principle, the theory of continuum mechanics is further explored at IKM.
An example of this is the exploration of "Space-Time-Methods". On the basis of Hamilton's principle, a comprehensive description for space and time can be derived for the physical processes in a material. For the mathematical equations thus obtained, solution methods based, e.g., on the "Virtual Elements Method" in combination with error estimators are investigated.
Another application is the derivation of evolution equations for materials with microstructure evolution. Advantages include a small number of material parameters, the thermodynamic consistency of the resulting evolution equations, and the ability to accurately consider constraints of the thermodynamic state variables. For the simulation of complex material behaviour, so-called multiscale approaches are used. Based on continuum mechanics motivated integration rules, approaches to reduce the otherwise very high computational effort are investigated.
In the context of the classical Hamilton's principle, which focuses on purely mechanical problems, there is the possibility to extend it by including electric and magnetic field quantities. Here, the electric and magnetic field quantities are linked via the Maxwell equations. These equations can partly, analogous to the equation of motion, be directly derived from an extended Hamilton's principle.