The numerical treatment of incorporating constitutive models of solid materials in finite element programs is already known. This course, however, treats the interpretation of the entire space and time integration, where the results of spatial discretization using finite elements will yield, depending on the underlying problem, systems of algebraic, ordinary differential or even their combinations, i.e. differential-algebraic equations (DAE). The solution of these systems, particularly, of time-dependent problems (models of viscoelasticity, rate-independent plasticity, viscoplasticity, and thermo-mechanically coupled problems) is treated.
Course Outline (9x2 hours)
1. Recapitulation of 3D small strain elasticity
2. Viscoelasticity, rate-independent plasticity, viscoplasticity
3. Principle of virtual displacements and its discretization using finite elements
4. Temporal discretization using the Backward-Euler method to solve DAEs
5. Newton-Raphson method versus Multilevel-Newton method
6. Higher order time discretization using diagonally-implicit Runge-Kutta methods
8. Extension to finite deformations
Reference Texts (recommended):
- Ellsiepen, P. and Hartmann, S. (2001). Remarks on the interpretation of current non-linear finite-element analyses as differential-algebraic equations. International Journal for Numerical Methods in Engineering51:679–707.
- Hartmann, S. (2002). Computation in finite strain viscoelasticity: finite elements based on the interpretation as differential-algebraic equations. Computer Methods in Applied Mechanics and Engineering191(13-14):1439–1470.
- Hartmann, S. (2005). A remark on the application of the Newton-Raphson method in non-linear finiteelement analysis. Computational Mechanics36(2):100–116.
- Grafenhorst, M., Rang, J., Hartmann, S. (2017): Time-adaptive finite element simulations of dynamical problems for temperature-dependent materials, Journal of Mechanics of Materials and Structures12(1), 57 – 91.
A manuscript on Theory of Materials will be provided.
Who Will Benefit
Researchers in (thermo)mechanics and computational mechanics or those which are interested in the background of implicit finite element programs.