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## Über dieses Buch

This book presents studies on the inelastic behavior of materials and structures under monotonic and cyclic loads. It focuses on the description of new effects like purely thermal cycles or cases of non-trivial damages. The various models are based on different approaches and methods and scaling aspects are taken into account. In addition to purely phenomenological models, the book also presents mechanisms-based approaches. It includes contributions written by leading authors from a host of different countries.

## Inhaltsverzeichnis

### Creep Behavior Modeling of Polyoxymethylene (POM) Applying Rheological Models

Abstract
Polyoxymethylene (POM) is a semi-crystalline thermoplastic polymer with broad technical application. Microstructure after solidifying is strongly dependent on the thermodynamical conditions. As an outcome macroscopic observable time dependent behavior is complex and significantly non-linear. To describe creep behavior of POM a rheological model with five elements is utilized. Creep behavior of POM under monotonic loading and constant temperature conditions can be described in a satisfying manner according to experimental results. A three-dimensional generalization with a comparable backstress formulation will be given. Finally, influence of data scattering will be estimated applying statistical analysis.
Holm Altenbach, Anna Girchenko, Andreas Kutschke, Konstantin Naumenko

### Experiments and Numerical Simulations on Stress-State-Dependence of Ductile Damage Criteria

Abstract
The paper deals with a series of new experiments and corresponding numerical simulations to be able to study the effect of stress state on damage behavior of ductile metals. In this context, a thermodynamically consistent anisotropic continuum damage model is presented. It takes into account the effect of stress state on damage and failure conditions as well as on evolution equations of damage strains. Different branches of the respective criteria are considered corresponding to various damage and failure mechanisms depending on stress intensity, stress triaxiality and the Lode parameter. Since it is not possible to propose and to validate stress-state-dependent criteria only based on tests with uniaxially loaded specimens for a wide range of stress states, new experiments with two-dimensionally loaded specimens have been developed. Corresponding numerical simulations of these experiments show that they cover a wide range of stress triaxialities and Lode parameters in the tension, shear and compression domains. The new series of experiments allow validation of stress-state-dependent functions for the damage criteria and are used to identify parameters of the continuum model.
Michael Brünig, Steffen Gerke, Daniel Brenner

### Constitutive Modeling of Dissipative Phenomena in Austenitic Metastable Steels at Cryogenic Temperatures

Abstract
In the present paper the constitutive model of dissipative material at cryogenic temperature is presented. Three coupled dissipative phenomena: plastic flow, plastic strain induced phase transformation and evolution of damage are considered using a thermodynamically consistent framework. The theory relies on notion of local state, and involves one state potential for the writing of the state laws, and dissipation potential for the description of the irreversible part of the model. The kinetic laws for state variables are derived from the generalized normality rule applied to the plastic potential, while the consistency multiplier is obtained from the consistency condition applied to the yield function. The model is applied for simulation of two distinct dissipative phenomena taking place in FCC metals and alloys at low temperatures: plastic strain induced transformation from the parent austenitic phase to the secondary martensitic phase, and evolution of micro-damage.
Halina Egner, Błażej Skoczeń, Maciej Ryś

### Modeling of Continuous Damage Deactivation Accompanying Low Cycle Fatigue of Al-2024 Under Complex Loading

Abstract
Present paper deals with numerical modeling of the damage deactivation accompanying low cycle fatigue under complex loading. Based on kinetic theory of damage evolution by Lemaitre and Chaboche (Mécanique des Matériaux Solides, 1985) the continuous function of the crack closure parameter is proposed. Results of numerical simulation are verified with experimental tests for aluminum alloy Al-2024 by Abdul-Latif and Chadli (Int J Damage Mech 16:133–158, 2007). Detailed quantitative and qualitative analysis of solutions obtained for uniaxial and biaxial cyclic tests confirms the necessity and correctness of an application of proposed continuous damage deactivation effect.
Artur Ganczarski, Maciej Oleksy

### Computational Multiscale Modeling of Nickel-Based Superalloys Containing Gamma-Gamma’ Precipitates

Abstract
A hierarchical crystal plasticity constitutive model, comprising three different scales for polycrystalline microstructures of Ni-based superalloys, is developed. Three scales, dominant in models of polycrystalline Ni-based superalloys, are: (i) the sub-grain scale of $$\gamma$$$$\gamma '$$ microstructure, characterized by $$\gamma '$$ precipitate size and their spacing; (ii) grain-scale characterized by the size of single crystals; and (iii) the scale of polycrystalline representative volume elements. A homogenized activation energy-based crystal plasticity (AE-CP) FEM model is developed for the grain-scale, accounting for characteristic parameters of the sub-grain scale $$\gamma$$$$\gamma '$$ morphology. A significant advantage of this AE-CP model is that its high efficiency enables it to be effectively incorporated in polycrystalline crystal plasticity FE simulations, while retaining the accuracy of detailed sub-grain level representative volume element (SG-RVE) models. The SG-RVE models are created for variable morphology, e.g. volume fraction, precipitate shape and channel-widths. The sub-grain crystal plasticity model incorporates a dislocation density-based crystal plasticity model augmented with mechanisms of anti-phase boundary (APB) shearing of precipitates. The sub-grain model is homogenized for developing parametric functions of morphological variables in evolution laws of the AE-CP model. Micro-twinning initiation and evolution models are incorporated in the single crystal AE-CP finite element models for manifesting tension-compression asymmetry. In the next ascending scale, a polycrystalline microstructure of Ni-based superalloys is simulated using an augmented AE-CP FE model with micro-twinning. Statistically equivalent virtual polycrystals of the alloy CMSX-4 are created for simulations with the homogenized model. The results of simulations at each scale are compared with experimental data with good agreement.
Somnath Ghosh, Shahriyar Keshavarz, George Weber

### Homogenized Elastic-Viscoplastic Behavior of Thick Perforated Plates with Pore Pressure

Abstract
The homogenized elastic-viscoplastic behavior of thick perforated plates with pore pressure is investigated for macro-material modeling. To this end, homogenized stress-strain relations of a periodic unit cell of pore-pressurized thick perforated plates under uniaxial and multiaxial loadings are analyzed using a finite element method with periodic boundary conditions. It is assumed in the analysis that the base metal of the perforated plates exhibits elastic-viscoplasticity based on Hooke’s law and Norton’s power law and has the material parameters of Mod. 9Cr-1Mo steel at 550 $$^\circ$$C. The resulting homogenized stress-strain relations are simulated using a macro-material model in which the pore-viscoplastic macro-strain rate is represented as an anisotropic power function of Terzaghi’s effective stress. It is demonstrated that this macro-material model suitably represents the macro-anisotropy, macro-volumetric compressibility, and pore pressure effect revealed in the viscoplastic range in the finite element homogenization analysis.
Kazutaka Ikenoya, Nobutada Ohno, Naoto Kasahara

### Experimental and Numerical Investigations of the Effects Associated to Complex Loading Combinations

Abstract
The paper presents experimental results concerning evaluation of an influence of cyclic torsion on stress variations during monotonic deformation carried out on the X10CrMoVNb9-1 steel. All strain controlled tests were performed at room temperature using thin-walled tubular specimens. The experimental programme contained selected combinations of monotonic and cyclic loadings, i.e. the torsion-reverse-torsion cycles were superimposed on the monotonic tension. It is shown that such cycles associated with monotonic tension caused essential variations of tensile stress. A significant decrease of the axial stress was visible. A single specimen method for yield surface determination was used to evaluate variations of yield point at different combinations of tension and torsion. The yield surface concept was also used to check permanency of the stress reduction during tension assisted by cyclic torsion. The effects observed during monotonic and cyclic loading combinations were theoretically described using the Maciejewski-Mróz model. It enabled to predict kinematic and isotropic softening or hardening of the material in question. The results exhibited that the model can be used successfully to simulate material behaviour during various combinations of monotonic and cyclic loadings.
Zbigniew L. Kowalewski, Tadeusz Szymczak, Jan Maciejewski

### Fracture of Cortical Bone Tissue

Abstract
In this chapter, mechanical behaviours of a unique type of composite material—cortical bone tissue—are considered for different length scales. Both experimental and computational approaches are discussed in this study to evaluate the effects of mechanical anisotropy and structural heterogeneity on the fracture process of cortical bone. First, variability and anisotropic mechanical behaviour of cortical bone tissue are characterised and analysed experimentally for different loading conditions and orientations. Then, results from the experimental studies are used to develop finite-element models across different length-scales to elucidate mechanical and structural mechanisms underpinning the anisotropic and non-linear fracture processes of cortical bone.

### A Nonlocal Model of Plasticity and Damage with Different Internal Lengths

Abstract
A nonlocal thermodynamically consistent model of plasticity and damage is presented using an integral approach. The theory is developed in the framework of the generalized standard material and the constitutive model is identified by the specification of a nonlocal first law of thermodynamics and of a local second one. The constitutive model is then addressed by defining a suitable expression of the free energy which yields a nonlocal plastic model in the stress space and a nonlocal damage model in the strain space. A variational formulation depending on local and nonlocal state variables is thus provided.
Francesco Marotti de Sciarra

### Hysteresis Loop Analysis in Cyclically Strained Materials

Abstract
The generalized statistical theory of the hysteresis loop is adopted to describe the stress-strain relations, preferably in cyclic straining. The effective stress and the distribution of the internal critical stresses in cyclic straining are evaluated in two materials cycled at room and at elevated temperatures using the analysis of the hysteresis loop shape. The evolution of the shape of the probability density function of the internal critical stresses yields deeper insight into the mechanisms of cyclic plastic straining. It indicates the important role of cyclic plastic strain localization in room temperature fatigue softening. The approximation of the probability density function by Weibull distribution leads to the assessment of the effective and internal stresses and allows the simulation of the relations between the stress and strain in case of different cyclic histories.
Jaroslav Polák, Roman Petráš

### Creep, Plasticity and Fatigue of Single Crystal Superalloys: Physics-Based Life Prediction for Turbine Components in Severe Operating Environments

Abstract
The chapter reports the process and computer methodology for a physics-based prediction of overall deformation and local failure modes in cooled turbine airfoils, blade outer air seals, and other turbomachinery parts operating in severe high temperature and high stress environments. The computational analysis incorporated coupled aero-thermal CFD with non-linear deformation finite element calculations with a crystallographic slip-based constitutive model. The methodology utilized a fully-coupled elastic-viscoplastic model that was based on crystal viscoplasticity, and a semi-empirical lifing model introduced the use of dissipated energy to estimate the remaining part life in terms of cycles to failure. The viscoplastic model used an incremental large strain formulation additively that decomposed the inelastic strain rate into components along the octahedral and cubic slip planes of single crystal nickel-based superalloys. This crystallographic-based viscoplastic constitutive model based on Orowan’s law was developed to represent sigmoidal creep behavior. Inelastic shear rate along each slip system was expressed as a sum of a time dependent creep component and a rate independent plastic component. A new robust and computationally efficient rate-independent crystal plasticity formulation was developed and combined with the creep flow model. The transient variation of each of the inelastic components included a back stress for kinematic hardening and latent hardening parameters to account for the stress evolution with inelastic strain as well as the evolution for dislocation densities. The model was evaluated at real engine characteristic mission times and flight points for part life prediction. The method was effective for use with three-dimensional finite element models of realistic turbine airfoils using commercial finite element applications. The computationally predicted part life was calibrated and verified against test data for deformation and crack growth.
Alexander Staroselsky, Thomas J. Martin

### Ratchetting of Snake Skin: Experiments and Viscoelastic-Plastic Constitutive Model

Abstract
In this paper, the cyclic deformation of snake skin is experimentally observed by the in vitro tests under uniaxial cyclic loading conditions and at room temperature for the first time. The effect of loading level on the cyclic deformation and the anisotropic deformation of snake skin are investigated, respectively. The results show that ratchetting (i.e., a cyclic accumulation of strain) occurs during the cyclic tension-tension tests of snake skin, and depends greatly on the loading orientations and levels. Based on the experimental results of uniaxial ratchetting, a simplified version of the finite viscoelastic-plastic model (Zhu et al. in J Biomech 47:996–1003, 2014) for soft biological tissues is obtained. The comparison shows that the simulated results are in qualitative agreement with the experimental ones.
Yilin Zhu, Guozheng Kang
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