Material data are usually not available as idealized, fixed values. In contrast, due to microstructure fluctuations or limitations of the measurements, material parameters fluctuate stochastically around some expectation value. The fluctuations of the material parameters result, in turn, in fluctuations of important engineering information such as the stresses. For an improved materials usage, the inclusion of the stochastic behavior constitutes an important aspect of research in the field of continuum mechanics which is referred to as stochastic material modeling. Several approaches are already available which, unfortunately, consume high computational resources rendering the possible applications rather limited. A prominent example of such models is given by Monte Carlo simulations: for each realization of the stochastic input data, a simulation of the material and structure behavior is performed. The set of individual simulations is used to compute important stochastic information, e.g., the expectation value and the standard deviation of the stresses. The results converge to the mathematically accurate results. However, the asymptotic convergence rate is small such that several hundreds of realizations, each evaluated by means of a (nonlinear) finite element simulation, need to be investigated. This prevents Monte Carlo simulations, although being the most accurate approach, to be used for industrially relevant problems.
To improve the modeling of materials with stochastic properties, a novel approach has been developed in close cooperation with Prof. Jan Nagel from TU Dortmund. The fundamental idea of this method is to separate the stochastic information from the specific load and geometry of the component. Consequently, the method is referred to as TimeSeparated Stochastic Mechanics (TSM). Due to the separation, the stochastic properties can be analyzed in advance of any concrete boundary value problem. Effectively, this results in an extended set of material parameters which do not only consider the usual expectation values but furthermore they account for the stochastic behavior. These additional material parameters can be computationally determined from experimental data through Monte Carlo simulations: they need to be performed only once per each material and are furthermore only to be executed for lowdimensional quantities as, e.g., the elasticity tensor.
Timeseparated stochastic mechanics

TSM for dynamic processesModeling and simulation of materials with stochastic properties is typically computationally expensive especially for nonlinear materials or dynamic simulations. The Timeseparated stochastic mechanics (TSM) can be extended for the dynamic analysis of stochastic viscoelasic materials by incorporation of the transient terms. In transient timedomain simulations a good approximation of the expectation and variance of the reaction force and the stresses for the dynamic response can be observed. A numerical extra cost of 10% compared with one deterministic finite element simulation is reported. However, the Monte Carlo simulation needs a minimum number of 400 finite element computations to arrive at results, that can be considered converged. Therefore, the TSM provides a fast yet accurate procedure for the dynamic simulation of viscoelastic structures witch stochastic properties.Leitung: P. Junker, J. NagelTeam:Jahr: 2022

TSM for damage processesIn industrial applications the reliability of components with high cycle load is of interest. In addition to the TSM a damage model needs to be used. The combination of TSM with a damage model enables to predict the evolution of the reaction forces, which are influenced by the stochastic nature of the materials and the damage processes. The advantages of the TimeSeparated Stochastic Mechanics remain. The stochastic properties can be computed in advance of any concrete finite element simulation. This results in a low computational effort in comparison to Monte Carlo Simulations and enables to simulate industrially relevant problems with moderate computational resources.Leitung: P. Junker, J. NagelTeam:Jahr: 2021

TSM for viscoelastic structuresThe local TSM model for viscoelastic materials can also be employed for finite structures. To this end, it is evaluated for each integration point within a finite element routine. It is also possible to find analytic formulas for the expectation value and the standard deviation of each component of the reaction force. The numerical extra costs are less than 5% needed for a deterministic finite element simulation. Considering a minimum number of 400 finite element computations for a Monte Carlo simulation reveals that TSM provides a fast yet accurate procedure for the modeling of viscoelastic components with stochastic properties.Leitung: P. Junker, J. NagelJahr: 2019

TSM for viscoelastic materialsThe TSM has successfully been derived for viscoelastic materials. Here, two internal variables need to be computed which approximate the usual viscous part of the total strain. These internal variables are timedependent and thus also vary for varying loads; however, they are deterministic. The expectation and standard deviation for both the stochastic viscous part of the strains and the stresses is computed by means of the two internal variables and deterministic coefficients which depend on the stochastic behavior of the material. Once they are computed, they remain constant even if the external load is changing. This results in a computational effort which is doubled as compared to classical (deterministic) material models for viscoelasticity. However, for a Monte Carlo approach, at least 400 realizations need to be performed. This renders the TSM approach to be faster than Monte Carlo simulations by a factor of approximately 200.Leitung: P. Junker, J. NagelJahr: 2018
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