ProVerB

Creep test with concrete specimens

Photo: Max Kovalenko

In search of appropriate repositories for radioactive waste from nuclear power plants, disused rock salt mines are considered. A safe inclusion of the radioactive material requires the salt mines to be closed hermetically, which is realized by the use of salt concrete for which a limited crack growth for long timescales has to be guaranteed. Since conventional concrete structures are designed for a lifetime of 30 to 50 years, new approaches are necessary to model the respective material properties under given mechanical loading conditions where the constitutive behavior is influenced by aging as well as by creep and relaxation processes on long time-scales.

Micrograph of a salt concrete specimen

Photo: BGE TEC

The project “ProVerB” aims to develop material models for long time-scales using fractional calculus, i.e. derivatives of non-integer order with respect to time. Moreover, in contrast to existing approaches, the aging processes will be described by a time-dependent order of derivative. The resulting constitutive equations will be used in finite element simulations in order to calculate the deformation processes of concrete structures. A verification of the methodology is realized by comparison with experimental data from creep and relaxation tests with concrete samples.

The project “ProVerB” is conducted in cooperation with the Materials Testing Institute of the University of Stuttgart (MPA) and the engineering company GNS in Brunswick. It is supported by a subsidiary of the BGE, the federal company for radioactive waste disposal. Creep tests with salt concrete specimens, which are needed to validate material models, are carried out by the MPA. The GNS designs and implements efficient algorithms to integrate the models into a commercial engineering finite element software. The project is funded by the program “KMU-innovativ” of the German Federal Ministry of Education and Research (Grant No. 01IS17096).

Kooperation mit Prof. Nowak

Basierend auf den entwickelten Materialgesetzen soll abschließend der Einfluss der Unsicherheiten untersucht werden, die sich aus der Streuung der Messergebnisse ergeben. Anders als bei Elastomeren oder metallischen Werkstoffen ist die Bestimmung des Kriechverhaltens von Beton von einer deutlich höheren Varianz gekennzeichnet. Diese lässt sich durch verschiedene äußere Einflussfaktoren erklären, wird aber ganz wesentlich durch die Inhomogenität des Materials bestimmt, durch die kein Probekörper dem anderen gleicht. Damit können auch die Materialparameter nur mit einer entsprechenden Unsicherheit identifiziert werden, wodurch wiederum Simulationsrechnungen über lange Zeiträume zu „unscharfen“ Ergebnissen führen. Den sich daraus ergebenden Fragestellungen wird im Rahmen einer Kooperation mit Prof. Nowak (SimTech) nachgegangen. Über die Zusammenarbeit ist ein Artikel in der Zeitschrift "forschung leben" der Universität Stuttgart erschienen.

Journal articles

Hinze, M., Schmidt, A., & Leine, R. I. (2021). Finite element formulation of fractional constitutive laws using the reformulated infinite state representation. Fractal Fractional, 5(132), Article 132. https://doi.org/10.3390/fractalfract5030132
Abstract
In this paper, we introduce a formulation of fractional constitutive equations for finite element analysis using the reformulated infinite state representation of fractional derivatives. Thereby, the fractional constitutive law is approximated by a high-dimensional set of ordinary differential and algebraic equations describing the relation of internal and external system states. The method is deduced for a three-dimensional linear viscoelastic continuum, for which the hydrostatic and deviatoric stress-strain relations are represented by a fractional Zener model. One- and two-dimensional finite elements are considered as benchmark problems with known closed form solutions in order to evaluate the performance of the scheme.
BibTeX
@article{Hinze&Schmidt&Leine2021, abstract = {In this paper, we introduce a formulation of fractional constitutive equations for finite element analysis using the reformulated infinite state representation of fractional derivatives. Thereby, the fractional constitutive law is approximated by a high-dimensional set of ordinary differential and algebraic equations describing the relation of internal and external system states. The method is deduced for a three-dimensional linear viscoelastic continuum, for which the hydrostatic and deviatoric stress-strain relations are represented by a fractional Zener model. One- and two-dimensional finite elements are considered as benchmark problems with known closed form solutions in order to evaluate the performance of the scheme.}, author = {Hinze, M. and Schmidt, A. and Leine, R. I.}, journal = {Fractal Fractional}, number = 132, pages = {1--22}, title = {Finite element formulation of fractional constitutive laws using the reformulated infinite state representation}, url = {https://doi.org/10.3390/fractalfract5030132}, volume = 5, year = 2021 }
Link
https://doi.org/10.3390/fractalfract5030132
Documents
- fractalfract-05-00132-v2.pdf
Hinze, M., Schmidt, A., & Leine, R. I. (2020). The direct method of Lyapunov for nonlinear dynamical systems with fractional damping. Nonlinear Dynamics, 102, 2017--2037. https://doi.org/10.1007/s11071-020-05962-3
Abstract
In this paper, we introduce a generalization of Lyapunov’s direct method for dynamical systems with fractional damping. Hereto, we embed such systems within the fundamental theory of functional differential equations with infinite delay and use the associated stability concept and known theorems regarding Lyapunov functionals including a generalized invariance principle. The formulation of Lyapunov functionals in the case of fractional damping is derived from a mechanical interpretation of the fractional derivative in infinite state representation. The method is applied on a single degree-of-freedom oscillator first, and the developed Lyapunov functionals are subsequently generalized for the finite-dimensional case. This opens the way to a stability analysis of nonlinear (controlled) systems with fractional damping. An important result of the paper is the solution of a tracking control problem with fractional and nonlinear damping. For this problem, the classical concepts of convergence and incremental stability are generalized to systems with fractional-order derivatives of state variables. The application of the related method is illustrated on a fractionally damped two degree-of-freedom oscillator with regularized Coulomb friction and non-collocated control.
BibTeX
@article{Hinze&Schmidt&Leine2020NODY, abstract = {In this paper, we introduce a generalization of Lyapunov’s direct method for dynamical systems with fractional damping. Hereto, we embed such systems within the fundamental theory of functional differential equations with infinite delay and use the associated stability concept and known theorems regarding Lyapunov functionals including a generalized invariance principle. The formulation of Lyapunov functionals in the case of fractional damping is derived from a mechanical interpretation of the fractional derivative in infinite state representation. The method is applied on a single degree-of-freedom oscillator first, and the developed Lyapunov functionals are subsequently generalized for the finite-dimensional case. This opens the way to a stability analysis of nonlinear (controlled) systems with fractional damping. An important result of the paper is the solution of a tracking control problem with fractional and nonlinear damping. For this problem, the classical concepts of convergence and incremental stability are generalized to systems with fractional-order derivatives of state variables. The application of the related method is illustrated on a fractionally damped two degree-of-freedom oscillator with regularized Coulomb friction and non-collocated control.}, author = {Hinze, M. and Schmidt, A. and Leine, R. I.}, doi = {https://doi.org/10.1007/s11071-020-05962-3}, journal = {Nonlinear Dynamics}, pages = {2017--2037}, title = {The direct method of {Lyapunov} for nonlinear dynamical systems with fractional damping}, volume = 102, year = 2020 }
Documents
- Hinze2020_Article_TheDirectMethodOfLyapunovForNo.pdf
Hinze, M., Schmidt, A., & Leine, R. I. (2020). Lyapunov stability of a fractionally damped oscillator with linear (anti-)damping. International Journal of Nonlinear Science and Numerical Simulation, 21(5), Article 5. https://doi.org/10.1515/ijnsns-2018-0381
Abstract
In this paper, we develop a Lyapunov stability framework for fractionally damped mechanical systems. In particular, we study the asymptotic stability of a linear single degree-of-freedom oscillator with viscous and fractional damping. We prove that the total mechanical energy, including the stored energy in the fractional element, is a Lyapunov functional with which one can prove stability of the equilibrium. Furthermore, we develop a strict Lyapunov functional for asymptotic stability, thereby opening the way to a nonlinear stability analysis beyond an eigenvalue analysis. A key result of the paper is a Lyapunov stability condition for systems having negative viscous damping but a sufficient amount of positive fractional damping. This result forms the stepping stone to the study of Hopf bifurcations in fractionally damped mechanical systems. The theory is demonstrated on a stick-slip oscillator with Stribeck friction law leading to an effective negative viscous damping.
BibTeX
@article{Hinze&Schmidt&Leine2020IJNS, abstract = {In this paper, we develop a Lyapunov stability framework for fractionally damped mechanical systems. In particular, we study the asymptotic stability of a linear single degree-of-freedom oscillator with viscous and fractional damping. We prove that the total mechanical energy, including the stored energy in the fractional element, is a Lyapunov functional with which one can prove stability of the equilibrium. Furthermore, we develop a strict Lyapunov functional for asymptotic stability, thereby opening the way to a nonlinear stability analysis beyond an eigenvalue analysis. A key result of the paper is a Lyapunov stability condition for systems having negative viscous damping but a sufficient amount of positive fractional damping. This result forms the stepping stone to the study of Hopf bifurcations in fractionally damped mechanical systems. The theory is demonstrated on a stick-slip oscillator with Stribeck friction law leading to an effective negative viscous damping.}, author = {Hinze, M. and Schmidt, A. and Leine, R. I.}, doi = {https://doi.org/10.1515/ijnsns-2018-0381}, journal = {International Journal of Nonlinear Science and Numerical Simulation}, number = 5, pages = {425--442}, title = {Lyapunov stability of a fractionally damped oscillator with linear (anti-)damping}, volume = 21, year = 2020 }
Documents
- 10.1515_ijnsns-2018-0381.pdf
Hinze, M., Schmidt, A., & Leine, R. I. (2019). Numerical solution of fractional-order ordinary differential equations using the reformulated infinite state representation. Fractional Calculus and Applied Analysis, 22(5), Article 5. https://doi.org/10.1515/fca-2019-0070
Abstract
In this paper, we propose a novel approach for the numerical solution of fractional-order ordinary differential equations. The method is based on the infinite state representation of the Caputo fractional differential operator, in which the entire history of the state of the system is considered for correct initialization. The infinite state representation contains an improper integral with respect to frequency, expressing the history dependence of the fractional derivative. The integral generally has a weakly singular kernel, which may lead to problems in numerical computations. A reformulation of the integral generates a kernel that decays to zero at both ends of the integration interval leading to better convergence properties of the related numerical scheme. We compare our method to other schemes by considering several benchmark problems.
BibTeX
@article{Hinze&Schmidt&Leine2019, abstract = {In this paper, we propose a novel approach for the numerical solution of fractional-order ordinary differential equations. The method is based on the infinite state representation of the Caputo fractional differential operator, in which the entire history of the state of the system is considered for correct initialization. The infinite state representation contains an improper integral with respect to frequency, expressing the history dependence of the fractional derivative. The integral generally has a weakly singular kernel, which may lead to problems in numerical computations. A reformulation of the integral generates a kernel that decays to zero at both ends of the integration interval leading to better convergence properties of the related numerical scheme. We compare our method to other schemes by considering several benchmark problems.}, author = {Hinze, M. and Schmidt, A. and Leine, R. I.}, doi = {https://doi.org/10.1515/fca-2019-0070}, journal = {Fractional Calculus and Applied Analysis}, number = 5, pages = {1321-1350}, title = {Numerical solution of fractional-order ordinary differential equations using the reformulated infinite state representation}, volume = 22, year = 2019 }
Documents
- 10-FCAA-576-HCL.pdf

Proceedings

Hinze, M., Schmidt, A., & Leine, R. I. (2019). Numerical simulation of fractionally damped mechanical systems using infinite state representation. Proceedings of the 2019 International Conference on Fractional Calculus Theory and Applications (ICFCTA 2019).
- BibTeX
- Documents
BibTeX
@inproceedings{Hinze&Schmidt&Leine2019ICFCTA, address = {Bourges, France}, author = {Hinze, M. and Schmidt, A. and Leine, R. I.}, booktitle = {Proceedings of the 2019 International Conference on Fractional Calculus Theory and Applications (ICFCTA 2019)}, title = {Numerical simulation of fractionally damped mechanical systems using infinite state representation}, year = 2019 }
Documents
- Hinze_x_Schmidt_x_Leine-Numerical_Simulation_of_Fractionally_Damped_Mechanical_Systems_Using_Infinite_State_Representation.pdf
Hinze, M., Schmidt, A., & Leine, R. I. (2018). Mechanical representation and stability of dynamical systems containing fractional springpot elements. Proceedings of the ASME 2018 International Design Engineering Technical Conferences (IDETC2018), DETC2018-85146, Article DETC2018-85146. https://doi.org/10.1115/DETC2018-85146
BibTeX
@inproceedings{Hinze&Schmidt&Leine2018ASME, address = {Quebec City, Canada}, author = {Hinze, M. and Schmidt, A. and Leine, R. I.}, booktitle = {Proceedings of the ASME 2018 International Design Engineering Technical Conferences (IDETC2018)}, doi = {10.1115/DETC2018-85146}, month = {08}, number = {DETC2018-85146}, title = {Mechanical representation and stability of dynamical systems containing fractional springpot elements}, url = {https://asmedigitalcollection.asme.org/IDETC-CIE/proceedings/IDETC-CIE2018/51838/Quebec City, Quebec, Canada/276205}, year = 2018 }
Link
https://asmedigitalcollection.asme.org/IDETC-CIE/proceedings/IDETC-CIE2018/51838/Quebec City, Quebec, Canada/276205
Documents
- Hinze_x_Schmidt_x_Leine-Mechanical_Representation_And_Stability_Of_Dynamical_Systems_Containing_Fractional_Springpot_Elements.pdf

Project partner

Gesellschaft für numerische Simulation

Materials Testing Institute University of Stuttgart

BGE, the federal company for radioactive waste disposal

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André Schmidt

Remco I. Leine

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