Construction steel is characterized by linear elasticity until reaching the yield stress. In the inelastic case the material description has to be extended to properly describe the material behavior. Depending on the loading condition other phenomena like yield plateau, plasticity, viscoplasticity, creep, hardening, softening, and damage have to be taken into account.
The kinetic energy of steel structures under significant dynamic excitation like earthquakes is partially dissipated by the inelastic material behavior. Regions of plastic flow occur in which the material experiences cyclic loading and hardening, but is also impaired by microcracks. With regard to the number of cycles this is often referred to as ultra low cycle fatigue.
The mathmatical description of the inelastic behavior of steel is realizes with a phenomenological model based on the work by Chaboche & Rousselier. Assuming small strains the material is described by a set of coupled nonlinear evolution equations for the inner variables. The proposed model includes viscoplasticity to account for the rate depency as well as kinematic and isotropic hardening. A strain memory surface is introduces to account for the influence of the strain amplitude history on the hardening of the material.
Strain Memory Surface
The evolution of the above mentioned strain memory surface is shown in the 2D strain space for the given stress states A and B. Starting in A the stress state exceeds the yield level for the first time. The resulting inelastic strain leads to the formation of the strain memory surface with the support vector β and radius q. With a new direction of the inelastic strain evolution for stress state B the radius increases and the center of the surface moves in the direction of the inelastic strain. The hardening evolution is coupled to the development of the strain memory surface radius.
The material equations describe the evolution for the undamaged material matrix with effective stresses and strains according to the concept of energy equivalence. Therefore the elastic behavior is influenced by existing damage. To account for the fact that damage only develops for significant inelastic strain level a threshold value governs the damage evolution. The yield condition is modified and takes into account the first and second stress invariant. With increasing damage the yield cylinder contracts and forms caps along the cylinder axis. Therefore damaged material may behave inelastically for hydrostatic stress states.
In structures the inelastic strain and the evolution of microcracks focuses on highly stressed regions. Regarding the finite element analysis the material softening leads to a concentration of inelastic strains within a small number of elements. The results become mesh dependent and the dissipations energy may vanish for further mesh refinement. This shortcoming can be overcome with the introduction of a nonlocal damage variable. The distribution of the local damage is governed by an implicit gradient formulation with an internal length which controls the spatial expansion of the damaged region.
Example: Damage evolution in a CT-specimen
The figures show the distribution of material damage within a compact-tension specimen under a uniform force at the left edge. Highly damaged regions are shown in red color.
Current Research Projects
S. Heinrich: Materialverhalten von Stahl unter dynamischer Belastung
Completed Research Projects
J.Velde: 3D Nonlocal Damage Modeling for Structures Subject to Earthquakes, Dissertation, TU Braunschweig, 2010
T. Zümendorf: Ein gradientenabhängiges Modell für Schädigung bei viskoplastischem Materialverhalten, Dissertation, TU Braunschweig, 2006