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- Fatigue damage (2)
- Accelerated temporal integration (1)
- Adaptive time stepping (1)
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- Continuum damage (1)
- Cycle jump (1)
- Cycle-by-cycle (1)
- Cycle-by-cycle integration (1)

#### Organisationseinheit der BAM

Quasi-brittle materials exhibit strain softening. Their modeling requires regularized constitutive formulations to avoid instabilities on the material level. A commonly used model is the implicit gradient-enhanced damage model. For complex geometries, it still Shows structural instabilities when integrated with classical backward Euler schemes. An alternative is the implicit–explicit (IMPL-EX) Integration scheme. It consists of the extrapolation of internal variables followed by an implicit calculation of the solution fields. The solution procedure for the nonlinear gradient-enhanced damage model is thus transformed into a sequence of problems that are algorithmically linear in every time step. Therefore, they require one single Newton–Raphson iteration per time step to converge. This provides both additional robustness and computational acceleration. The introduced extrapolation error is controlled by adaptive time-stepping schemes. This paper introduced and assessed two novel classes of error control schemes that provide further Performance improvements. In a three-dimensional compression test for a mesoscale model of concrete, the presented scheme was about 40 times faster than an adaptive backward Euler time integration.

Accurate models for the long term behavior of concrete structures are important to ensure a durable and reliable design.
A variety of interacting phenomena, such as the loss of prestress, the degradation due to chemical reactions or creep and shrinkage, influence the fatigue resistance. Therefore, a reliable numerical model to predict the performance of concrete over its lifetime is required.
The presented fatigue model is an extension of a static damage model to allow easy coupling in a multiphysics context. The evolution equation of the damage driving variable is enhanced to allow damage growth below the static limit. The model is defined in the time domain and does not include the number of cycles as a parameter.
Thus, it can capture both static and cyclic failure.
Additionally, this allows calibrating the majority of the model parameters static experiments. The model is integrated by resolving each loading cycle, requiring about ten time steps per cycle. The high computational costs are handled via a time scale separation.
The short time scale describes one cycle with marginal changes in the internal variables. These changes are integrated along the large time scale of material deterioration. Various high-order time integration schemes are compared.
Wöhler curves relate loading amplitudes to the number of cycles that the material endures. They are used to validate the model against experimental data.

Damage caused by stress concentrations in the complex mesoscopic geometry of concrete leads to continuous stress redistribution over the material’s life time. The presented fatigue damage model
captures this by resolving each load cycle in a cycle-by-cycle time integration. The model extends a static damage model to failure caused by the (time dependent) strain amplitudes and, thus, allows calibrating the majority of the material’s parameters in static experiments.

Lifetime aspects including fatigue failure of concrete structures were traditionally only of minor importance. Because of the growing interest in maxing out the capacities of concrete, its fatigue failure under compression has become an issue. A variety of interacting phenomena such as e.g. loss of prestress, degradation due to chemical reactions or creep and shrinkage influence the fatigue resistance. Failure due to cyclic loads is generally not instantaneous, but characterized by a steady damage accumulation. Therefore, a reliable numerical model to predict the performance of concrete over its lifetime is required, which accurately captures order effects and full three-dimensional stress states.
Many constitutive models for concrete are currently available, which are applicable for specific loading regimes, different time scales and different resolution scales.
However, a key limitation of those models is that they generally do not address issues related to fatigue on a structural level. Very few models can be found in the literature that reproduce deterioration of concrete under repeated loading-unloading cycles. This is due to the computational effort necessary to explicitly resolve every cycle which exceeds the currently available computational resources. The limitation can only be overcome by the application of multiscale methods in time.
The objective of the paper is the development of numerical methods for the simulation of concrete under fatigue loading using temporal multiscale methods.
First, a continuum damage model for concrete is developed with a focus on fatigue under compressive stresses [1]. This includes the possibility to model stress redistributions and capture size effects. In contrast to cycle based approaches, where damage is accumulated based on the number of full stress cycles, a strain based approach is developed that can capture cyclic degradation under variable loading cycles including different amplitudes and loading frequencies. The model is designed to represent failure under static loading as a particular case of fatigue failure after a single loading cycle. As a consequence, most of the material parameters can be deduced from static tests. Only a limit set of additional constitutive parameters is required to accurately describe the evolution under fatigue loading. Another advantage of the proposed model is the possibility to directly incorporate other multi-physics effects such as creep and shrinkage or thermal loading on the constitutive level.
Second, a multiscale approach in time is presented to enable structural computations of fatigue failure with a reduced computational effort. The damage rate within the short time scale corresponding to a single cycle is computed based on a Fourier based approach [2]. This evolution equation is then solved on the long time scale using different implicit and explicit time integration schemes. Their performance and some limitations for specific loading regimes is discussed.
Finally, the developed methods will be validated and compared to experimental data.
[1] Vitaliy Kindrachuk, Marc Thiele, Jörg F. Unger. Constitutive modeling of creep-fatigue interaction for normal strength concrete under compression, International Journal of Fatigue, 78:81-94, 2015
[2] Vitaliy Kindrachuk, Jörg F. Unger. A Fourier transformation-based temporal integration scheme for viscoplastic solids subjected to fatigue deterioration, International Journal of Fatigue, 100:215-228, 2017

Damage caused by stress concentrations in the complex mesoscopic geometry of concrete leads to continuous stress redistribution over the material's life time. The presented fatigue damage model captures this by resolving each load cycle in a cycle-by-cycle time integration. The model extends a static damage model to failure caused by the (time dependent) strain amplitudes and, thus, allows calibrating the majority of the material's parameters in static experiments.

Isotropic damage models are widely used for the finite element simulation of softening materials, e.g. in mesoscale simulations of concrete. Regularization techniques must be employed to obtain a physically meaningful fracture energy upon mesh refinement.
In regularized local damage models the strains localize in single elements allowing them to represent weak or strong discontinuities. In implicit integration schemes, these models can exhibit convergence Problems caused by an ill-conditioned tangent stiffness. This corresponds to the loss of ellipticity of the local rate equilibrium equations.
Oliver et al. developed the implicit/explicit (IMPL-EX) integration scheme which overcomes These problems in local damage models. The internal damage driving variable is extrapolated based on previous implicitly determined values. This provides two main benefits: First, it always results in a symmetric positive semi-definite algorithmic stiffness matrix which precludes ill-posedness. Second, the system becomes incrementally linear and converges in one Newton-Raphson iteration. Even though the IMPL-EX algorithm, like explicit algorithms in general, requires smaller time steps than implicit schemes to obtain the same accuracy, it leads to a computational speedup.
The gradient enhanced damage model by Peerlings is a nonlocal damage model that provides the regularization by limiting the curvature of the damage-driving strains. These models do not lose their ellipticity. However, structural instabilities often require tiny time steps and many iterations to obtain convergence. Here, the second aspect of the IMPL-EX scheme reduces the computational costs. This is shown in simulations of the complex geometry of concrete mesostructures, where only the gradient enhanced matrix material and linear elastic aggregates are considered.
With regard to future mesoscale simulations, the remaining component of the mesoscopic structure, the interfacial transition zone and its degradation, has to be included. This adds a local damage model to the nonlocal problem. Thus, an IMPL-EX implementation has to be provided for both models to benefit from the increase of robustness and performance.

A key limitation of the most constitutive models that reproduce a Degradation of quasi-brittle materials is that they generally do not address issues related to fatigue. One reason is the huge computational costs to resolve each load cycle on the structural level. The goal of this paper is the development of a temporal Integration scheme, which significantly increases the computational efficiency of the finite element method in comparison to conventional temporal integrations.
The essential constituent of the fatigue model is an implicit gradient-enhanced formulation of the damage rate. The evolution of the field variables is computed as amultiscale Fourier series in time.On a microchronological scale attributed to single cycles, the initial boundary value problem is approximated by linear BVPs with respect to the Fourier coefficients. Using the adaptive cycle jump concept, the obtained damage rates are transferred to a coarsermacrochronological scale associated with the duration of material deterioration. The performance of the developedmethod is hence improved due to an efficient numerical treatment of the microchronological problem in combination with the cycle jump technique on the macrochronological scale. Validation examples demonstrate the convergence of the obtained solutions to the reference simulations while significantly reducing the computational costs.

Concrete is a complex material. Its properties evolve over time, especially at early age, and are dependent on environmental conditions, i.e. temperature and moisture conditions, as well as the composition of the material.
This leads to a variety of macroscopic phenomena such as hydration/solidification/hardening, creep and shrinkage, thermal strains, damage and inelastic deformations. Most of these phenomena are characterized by specific set of model assumptions and often an additive decomposition of strains into elastic, plastic, shrinkage and creep components is performed. Each of these phenomena are investigated separately and a number of respective independent models have been designed. The interactions are then accounted for by adding appropriate correction factors or additional models for the particular interaction. This paper discusses the importance of reconsider even in the experimental phase the model assumptions required to generalize the experimental data into models used in design codes. It is especially underlined that the complex macroscopic behaviour of concrete is strongly influenced by its multiscale and multiphyscis nature and two examples (shrinkage and fatigue) of interacting phenomena are discussed.

The problem of polydisperse sphere packings is applied to concrete mesoscale geometries in finite sized specimens. Realistic sphere diameter distributions are derived from concrete grading curves. An event-driven molecular dynamics simulation using growing particles is introduced. Compared to the widely used random sequential addition algorithm, it reaches denser aggregate packings and saves computation time at high volume fractions.
A minimal distance between particles strongly influences the maximum aggregate content. It is essential to obtain undistorted elements when meshing the geometry for finite element simulations. The algorithm maximizes this value and produces meshable concrete mesostructures with more than 70% aggregate content.