A multiple mechanism weakest link model for intergranular and transgranular brittle fracture is developed on the basis of experimental observations of a thermally aged low alloy steel. The model development is carried out in tandem with micro mechanical analysis of grain boundary cracking using crystal plasticity modeling of polycrystalline aggregates with the purpose to inform the weakest link model. The fracture modeling presented in this paper is carried out by using a non-local porous plastic Gurson model where the void volume fraction evolution is regularized over two separate length scales. The ductile crack growth preceding the final brittle fracture is well predicted using this type of modeling. When applied to the brittle fracture tests, the weakest link model predicts the fracture toughness distribution remarkably well, both in terms of the constraint and the size effect. Included in the study is also the analysis of a reference material.

A multiple mechanism weakest link model for intergranular and transgranularbrittle fracture is developed on the basis of experimental observations in a thermallyaged low alloy steel. The model development is carried out in tandemwith micro mechanical analysis of grain boundary cracking using crystal plasticitymodeling of polycrystalline aggregates with the purpose to inform theweakest link model. The fracture modeling presented in this paper is carriedout by using a non-local porous plastic Gurson model where the void volumefraction evolution is regularized over two separate length scales. The ductilecrack growth preceding the nal brittle fracture is well predicted using this typeof modeling. When applied to the brittle fracture tests, the weakest link modelpredicts the fracture toughness distribution remarkably well, both in terms ofthe constraint and the size eect. Included in the study is also the analysis of areference material.

The strong evidence for a size dependence of plastic deformation in polycrystalline metalsis the basis for the research presented in this thesis. The most important parameter for this, and arguably also the most well known, is the grain size. As the size of the grains in a microstructure is decreased the yield stress increases. This is known as the Hall–Petch relation and have been confirmed for a large number of materials and grain sizes. Other structural dimensions may also give rise to a similar strengthening effect, such as the thickness of films and surface coatings, the widths of ligaments and localization zones and the diameter of thin wires, to mention a few. The work presented in this thesis is shown to be able to model these effects.

Size dependent plastic deformation have here been modeled in a continuum mechanical setting by an extension of the standard theory of solid mechanics. Specifically, the work in this thesis is formulated in terms of the higher order strain gradient plasticity (SGP) theory presented by Gudmundson [Gudmundson, P., 2004. A unified treatment of strain gradient plasticity. Journal of the Mechanics and Physics of Solids, 52]. This allows size dependent plasticity phenomena to be modeled and a yield stress that is proportional to the inverse of the geometric dimension of the problem is predicted.

The ability to model interfaces have been of specific importance to the work presented here. The state at internal interface is shown, via a physically motivated constitutive description, to be of great importance to capture size effects. The surface energy at grain boundaries is shown to influence both the local and the macroscopic behavior. At the smallest scales an additional deformation mechanism have been introduced at the internal boundaries. This allowed the strengthening trend associated with decreasing grains size to be halted, in qualitative agreement with reported experiments on the behavior of ultrafine and nanocrystalline polycrystals. In the later part of the thesis the focus is aimed at modeling grains structures to bring some insight into the different regions of deformation mechanisms in relation to grainsize and interface strength. A deformation mechanism map for polycrystals is suggested based on the results from structures with both hexagons and log-normal size distributed Voronoi tessellations, and the implication of a statistical variation in grain size have been explored.

A finite element implementation of the theory have been developed that is a fully implicit backward-Euler algorithm with tangent operators consistent with the stress update scheme, which give excellent convergence properties and is numerically very stable. Higher order finite elements have been implemented for modeling of both bulk material and internal interfaces. A plane strain version have been used to model metal-matrix composites and explore the implication of some of the more exotic features of the theory.

A two-dimensional deformation field on an indented single crystal, where the only nonzero lattice rotation occurs in the plane of deformation and only three effective in-plane slip systems are activated, is investigated both experimentally and numerically. ElectronBackscatter Diffraction (EBSD) is utilized to probe the lattice rotation field on the sample. The lattice rotation field is utilized to calculate the two non-zero components of Nye'sdislocation density tensor, which serves as a link between plastic and elastic deformation states. The enhanced accuracy of EBSD enabled measurements of the net Burgers vector density, and a new quantity β, which monitors the activity of slip systems in the deformed zone. The β-field is compared to the slip system activity obtained by analytical solution and also by crystal plasticity simulations. A qualitative comparison of the three methods confirms that the β-field obtained experimentally agrees with the slip system activity obtained analytically and by numerical methods.

An equivalence is assumed between a microstructural length scale related to dislocation density and the constitutive length scale parameter in phenomenological strain gradient plasticity. An evolution law is formed on an incremental basis for the constitutive length scale parameter. Specific evolution equations are established through interpretations of the relation between changes in dislocation densities and increments in plastic strain and strain gradient. The length scale evolution has been implemented in a 2D-plane strain finite element method (FEM) code, which has been used to study a beam in pure bending. The main effect of the length scale evolution on the response of the beam is a decreased strain hardening, which in cases of small beam thicknesses even leads to a strain softening behavior. An intense plastic strain gradient may develop close to the neutral axis and can be interpreted as a pile-up of dislocations. The effects of the length scale evolution on the mechanical fields are compared with respect to the choice of length evolution equation.

A fully implicit backward-Euler implementation of a higher order strain gradient plasticity theory is presented. A tangent operator consistent with the numerical update procedure is given. The implemented theory is a dissipative bulk formulation with energetic contribution from internal interface to model the behavior of material interfaces at small length scales. The implementation is tested by solving some examples that specifically highlight the numerics and the effect of using the energetic interfaces as higher order boundary conditions. Specifically, it is demonstrated that the energetic interface formulation is able to mimic a wide range of plastic strain conditions at internal boundaries. It is also shown that delayed micro-hard conditions may arise under certain circumstances such that an interface at first offers little constraints on plastic flow, but with increasing plastic deformation will develop and become a barrier to dislocation motion.

Plane strain models of polycrystalline microstructures are investigated using strain gradient plasticity (SGP) and a grain boundary (GB) deformation mechanism. The microstructures are constructed using a non-linear constrained Voronoi tessellation so that they conform to a log-normal distribution in grain size. The SGP framework is used to model the grain size dependent strengthening and the GB deformation results in a cut-off of this trend below a certain critical grain size. Plastic strain field localization is discussed in relation to the non-local effects introduced by SGP and a material length scale. A modification of the Hall-Petch relation that accounts for, not only the mean grain size, but also the statistical size variation in a population of grains is proposed.

Correlation of sharp indentation problems is examined theoretically and numerically. The analysis focuses on elastic-plastic pressure-sensitive materials and especially the case when the local plastic zone is so large that elastic effects on the mean contact pressure will be small or negligible as is the case for engineering metals and alloys. The results from the theoretical analysis indicate that the effect from pressure-sensitivity and plastic strain-hardening are separable at correlation of hardness values. This is confirmed using finite element methods and closed-form formulas are presented representing a pressure-sensitive counterpart to the Tabor formula at von Mises plasticity. The situation for the relative contact area is more complicated as also discussed.

Small scale strain gradient plasticity is coupled with a model of grain boundaries that take into account the energetic state of a plastically strained boundary and the slip and separation between neighboring grains. A microstructure of hexagonal grains is investigated using a plane strain finite element model. The results show that three different microstructural deformation mechanisms can be identified. The standard plasticity case in which the material behaves as expected from coarse grained experiments, the nonlocal plasticity region where size of the microstructure compared to some intrinsic length scale enhances the yield stress and a third mechanism, active only in very fine grained microstructures, where the grains deform mainly in relative sliding and separation.

A laminate structure with varying lamina thicknesses is used as a qualitative model of grain size dependence on yield behaviour in metallic materials. Both strain gradient plasticity and slip between layers are considered. It is shown that an inverse Hall-Petch effect can be generated in this way. For very small thicknesses, corresponding to very small grain sizes, sliding is the dominant mechanism and the strength then decreases with decreasing thickness. For larger thicknesses, strain gradient plasticity is controlling the deformation and the strength is, instead, increasing with decreasing thickness. Numerical examples are presented that demonstrate these mechanisms.

We develop a strain-gradient plasticity theory based on fractional derivatives of plastic strain and assess its ability to reproduce the scaling laws and size effects uncovered by the recent experiments of Mu et al. (2014, 2016, 2017) on copper thin layers undergoing plastically constrained simple shear. We show that the size-scaling discrepancy between conventional strain-gradient plasticity and the experimental data is resolved if the inhomogeneity of the plastic strain distribution is quantified by means of fractional derivatives of plastic strain. In particular, the theory predicts that the size scaling exponent is equal to the fractional order of the plastic-strain derivatives, which establishes a direct connection between the size scaling of the yield stress and fractionality.

A recently developed model based on fractional derivatives of plastic strain is compared with conventional strain-gradient plasticity (SGP) models. Specifically, the experimental data and observed model discrepancies in the study by Mu et al. (2016, "Dependence of Confined Plastic Flow of Polycrystalline Cu Thin Films on Microstructure," MRS Com. Res. Let. 20, pp. 1-6) are considered by solving the constrained simple shear problem. Solutions are presented both for a conventional SGP model and a model extension introducing an energetic interface. The interface allows us to relax the Dirichlet boundary condition usually assumed to prevail when solving this problem with the SGP model. We show that the particular form of a relaxed boundary condition does not change the underlying size scaling of the yield stress and consequently does not resolve the scaling issue. Furthermore, we show that the fractional strain-gradient plasticity model predicts a yield stress with a scaling exponent that is equal to the fractional order of differentiation.

An aluminum bicrystal with a symmetric tilt Σ43 (3 3 5)[1 1 0] coincident site lattice grain boundary was deformed plastically via wedge indentation under conditions that led to a plane strain deformation state. Plastic deformation is induced into both crystals and the initially straight grain boundary developed a significant curvature. The resulting lattice rotation field was measured via Electron Backscatter Diffraction (EBSD). The Nye dislocation density tensor and the associated Geometrically Necessary Dislocation (GND) densities introduced by the plastic deformation were calculated. The grain boundary served as an impediment to plastic deformation as quantified through a smaller lattice rotation magnitude and smaller GND density magnitudes in one of the crystals. There is evidence that the lattice rotations in one grain brought a slip system in that grain into alignment with a slip system in the other grain, upon which the impediment to dislocation transmission across the grain boundary was reduced. This allowed the two slip systems to rotate together in tandem at later stages of the deformation. Finite element crystal plasticity simulations using classical constitutive hardening relationship capture the general features observed in the experiments.

Experiments and numerical simulations of various angles of wedge indenters into face-centered cubic single crystal were performed under plane strain conditions. In the experiments, the included angles of indenters are chosen to be 60 degrees, 90 degrees and 120 degrees and they are indented into nickel single crystal into the < 00 (1) over bar > direction with its tip parallel to < 1 1 0 > direction, so that there are three effective in-plane slip systems on (1 1 0) plane. Indenters are applied 200 mu m in depth. The midsection of the specimens is exposed with a wire Electrical Discharge Machining (EDM) and the in-plane lattice rotations of the region around the indented area are calculated from the crystallographic orientation maps obtained from electron backscatter diffraction (EBSD) measurement. No matter which angles of indenters are applied, the rotation fields are very similar. There is a strong lattice rotation discontinuity on the line below the indenter tip. The magnitude of the lattice rotation ranges from -20 degrees to 20 degrees. Lower bounds on the Geometrically Necessary Dislocation (GND) densities are also calculated and plotted. The numerical simulations of the same experimental setup are performed. The simulation results of lattice rotation and slip rates are plotted and compared with the experimental result. There is high correlation between the experimental result and the numerical result.

A non-local continuum damage constitutive model has been developed. The evolution of the damage parameter is driven by a non-local plastic strain rate at low stress triaxiality and by a non-local rate of porosity at higher triaxiality. Nodular cast iron has been employed as a model material for which the model parameters have been calibrated utilizing a variety of tests. The non-local material model is able to explain the different outcome in two sets of tests in pure shear. 

The prediction of the cyclic deformation behaviour in lath martensite-based high-strength steels requires constitutive models that reflect the local stress and strain fields as accurately as possible. At the same time, the constitutive models should act as efficiently as possible in order to achieve the required high number of cycles in a finite time. Only few research works have studied the sensitivity of the local cyclic deformation in lath martensite to the power law-based flow rule (Hutchinson or Chaboche-Cailletaud) and the active body-centred cubic (bcc) slip systems ({110}(111) and {112}(111)) in the crystal plasticity finite element method (CPFEM). This paper, therefore, aims to provide some guidance in the selection of suitable flow rule and slip systems. Based on full-field micromechanical modelling of a medium-carbon steel under symmetric strain-controlled cyclic loading, it can be shown that the two most commonly used flow rules according to Hutchinson and Chaboche-Cailletaud are equally capable of predicting the local stress and strain distributions within the hierarchical martensitic microstructure. However, using the Hutchinson flow rule increases the computational performance for the quasi-rate-independent problem considered here. The local distributions found differ strongly from those in the parent austenitic microstructure. If plastic deformation is assumed not only on the slip systems {110}(111), as often done, but also on the {112}(111) type, a redistribution of the bimodal distributed local stresses occurs at a significantly lower stress level. The unimodal distributed local strains are less affected by this. In addition, it is found that slightly different critical resolved shear stress (CRSS) values for both slip system types influence the local stress and strain distributions less severely than the additional plastic slip activation in the material.

To predict and better understand the creep-fatigue behaviour of austenitic cast iron D5S under tension and compression dwell at 800 degrees C, a physics-based crystal plasticity model that describes the complex rate-and temperature-dependent deformation of the material as a function of the dislocation density is implemented. In addition to the tension and compression dwell direction, the effect of three different dwell times (30, 180 and 600 s) on the creep-fatigue properties is investigated. The dislocation density-based crystal plasticity simulations are compared to experimental tests from a prior work. While relaxation tests and low-cycle fatigue (LCF) tests without dwell assist in systematically identifying the material parameters, creep-fatigue (CF) data is used to validate the predictions. The virtual testing is performed on a large-scale representation of the actual test specimen with a polycrystalline structure. To analyse the fatigue damage mechanism, small-scale predictions are also conducted using a micromechanical unit cell approach. Here, a single graphite nodule frequently found in the material is embedded into the austenitic matrix. In the present work, a close agreement is achieved between the predicted CF behaviour and the experimental results. Consistent with the experimental findings, the simulation results show that the addition of compression dwell leads to an uplift of the overall tensile stress level, which significantly reduces the fatigue life of the material. The unit cell studies demonstrate that during this uplift, a strong localisation of stresses and strains arises at the graphite/matrix interface, triggering the nucleation and growth of cavities and/or debonding.

To enhance the fundamental understanding for micromechanical lath martensite deformation, the microstructure as well as macro- and microscopic tensile properties of as -quenched 15-5 PH stainless steel are systematically analysed depending on the austenitisation temperature. Based on electron backscatter diffraction (EBSD) and backscattered electron (BSE) analysis, it is noted that the martensite morphology alters from a less defined to a more clearly defined parallel arrangement of the block and lath structure with increasing temperature. For an indepth quantification of the hierarchical boundary strengthening contributions in relation to local stress/strain heterogeneity, separate high-fidelity virtual microstructures are realised for the different scales (prior austenite grains, packets and blocks). This is consistent with the materials transformation process. The virtual microstructures are simulated employing the crystal plasticity finite element method (CPFEM) adapted for handling high dislocation density and encompassing all relevant strengthening mechanisms by boundaries, dislocations and solute atoms. While accurately capturing the measured size -dependent stress-strain behaviour, the simulations reveal in line with the experiments (Hall-Petch) that blocks are the most effective dislocation motion barrier, causing increased strain hardening and stress/strain heterogeneity. Furthermore, since strain localisation is predicted strongest in the distinct block structure, the experimentally observed early plastic material yielding is thought to be favoured here.

Many experimental studies have shown a plastic strengthening effect for structural length scales approaching microstructural dimensions. Both increases in initial yield strength and strain hardening have been observed. Over the last 30 years different strain gradient plasticity (SGP) theories have been developed in order to capture these length scale dependences. However, up to now no generally accepted theory has emerged. In the present presentation, focus is directed into a physically based SGP model for initiation of plastic deformation.

The plastic behavior is governed by a dissipative part that primarily controls the hardening at moderate plastic strains and an energetic part that is of importance for the initiation of plastic flow. It is shown that a model based on the self-energies of dislocations can be translated into an internal free energy in terms of plastic strain gradients. Similarly, the dissipative part of the model is based on the Taylor model, which also gives a direct connection to dislocation theory.

In this way, a physical connection is made between the SGP framework and dislocation mechanics. It is shown that the same length scale emerges for both the energetic and the dissipative part of the model. Apart from a non-dimensional factor of the order of unity, the length scale can be defined as the Burgers vector divided by the strain for initiation of plastic deformation.

When the structural length scale approaches this microstructural length scale, strengthening effects result. The three-dimensional SGP model is specialized to the simple load cases of tensile tension with a passivation layer that prohibits plastic deformation on the surfaces as well as pure bending with free and fixed boundary conditions for plastic strain. Simulations of initial yield stress for varying thicknesses are compared to experimental observations reported in the literature. It is shown that the model in a good way can capture the length scale dependences. Also upper bound solutions are presented for a spherical void in an infinite volume as well as torsion of a cylindrical rod. The model is as well applied to derive a prediction for the Hall-Petch effect.

A dislocation mechanics based isotropic strain gradient plasticity model is developed. The model is derived from self-energies of dislocations and the Taylor model for plastic dissipation. It is shown that the same microstructural length scale emerges for both the energetic and the dissipative parts of the model. Apart from a non-dimensional factor of the order of unity, the length scale is defined as the Burgers vector divided by the strain for initiation of plastic deformation. When the structural length scale approaches this microstructural length scale, strengthening effects result. The present model predicts an increased initial yield stress that is controlled by the energetic contribution. For larger plastic strains, the hardening is governed by the dissipative part of the model. The theory is specialized to the simple load cases of tension with a passivation layer that prohibits plastic deformation on the surfaces as well as pure bending with free and fixed boundary conditions for plastic strain. Simulations of initial yield stress for varying thicknesses are compared to experimental observations reported in the literature. It is shown that the model in a good way can capture the length scale dependencies. Also upper bound solutions are presented for a spherical void in an infinite volume as well as torsion of a cylindrical rod. The model is as well applied to derive a prediction for the Hall-Petch effect.

Many experimental studies have shown a plastic strengthening effect for structural length scales approaching microstructural dimensions. Both increases in initial yield strength and strain hardening have been observed. Over the last 30 years different strain gradient plasticity (SGP) theories have been developed in order to capture these length scale dependences. However, up to now no generally accepted theory has emerged. In the present paper, focus is directed into a physically based SGP model for initiation of plastic deformation. The plastic behavior is governed by a dissipative part that primarily controls the hardening at moderate plastic strains and an energetic part that is of importance for the initiation of plastic flow. It is shown that a model based on the self-energies of dislocations can be translated into an internal free energy in terms of plastic strain gradients. In this way a physical connection is made between the SGP framework and dislocation mechanics. A microstructural length scale can then be defined as the Burgers vector divided by the strain for initiation of plastic deformation. When structural length scales approach this microstructural length scale, strengthening effects result. It the Taylor model is used for the dissipative part, the same microstructural length scale appears. The so developed three-dimensional SGP model is specialized to the simple load cases of tensile tension with a passivation layer that prohibits plastic deformation on surfaces as well as pure bending with free and fixed boundary conditions for plastic strain. Simulations for varying thicknesses are compared to experimental observations reported in the literature. It is shown that the model in a good way can capture the length scale dependences. Suggestions for improvement of the dislocation theory based model for the internal free energy are discussed.

A continuum modeling approach that considers two separate failure mechanisms of steels subjected to hydrogen embrittlement is proposed based on experimental observations. The brittle failure is modeled using a cohesive zone approach, where both the cohesive strength and the fracture energy are degraded when exposed to hydrogen. The ductile failure is modeled using the Gurson model that includes a strain driven nucleation of void. Here, the nucleation model also incorporates hydrogen degradation where an increase in hydrogen is assumed to increase the volume of nucleated voids. This modeling approach is divided into two parts where the first step is to utilize a conceptual degradation of both failure modes and calibrate modeling parameters, and the second part incorporates a coupled diffusion-mechanical approach.

Hydrogen delayed fracture, also known as hydrogen-induced cracking, is a type of brittle fracture that occurs due to the slow diffusion and accumulation of hydrogen atoms, leading todecreased ductility and eventual cracking under constant load. This paper presents an in-situobservation, using neutron imaging, of delayed crack propagation caused by hydrogen embrittlement in a high strength martensitic steel specimen. The experiments involved mechanicalloading of a single-edge-notch bend specimen while submerged in an electrolyte solution (H2O+ 3.5% NaCl) under cathodic polarization to facilitate hydrogen ingress. Neutron transmission images were obtained in-situ and used to monitor intermittent crack propagation wasrecorded over a period of 12 hours. The stress state at each crack configuration was extracted from a three-dimensional elastic-plastic finite element simulation, which was tailoredto match the quantitative information acquired from the neutron radiographs of the fractureprocess. To gain insight into the evolution of hydrogen concentration with crack propagation,a modeling scheme for stress-assisted hydrogen diffusion was employed. These simulationsprovided qualitative information on the relation between intermittent crack propagation andthe subsequent supply of hydrogen to the crack tip. Finally, a failure locus was constructedbased on the calculated hydrogen concentration levels and the experimentally determinedcrack growth resistance.

Discontinuous precipitation at high temperatures in (Ti,Zr)C was modeled using the phase field method. The modelling was based on previous treatments of diffusion induced grain boundary migration and spinodal decomposition. A fair agreement with experimental observations was obtained. However, a more refined treatment of the effect of strain energy is left for future work.

Background: The mechanical characterization of the cyclic elastoplastic response of structural materials at elevated temperatures is crucial for understanding and predicting the fatigue life of components in nuclear reactors. Objective: In this study, a comprehensive mechanical characterization of 304L stainless steel has been performed including metallography, tensile tests, fatigue tests, fatigue crack growth tests and cyclic stress-strain tests. Methods: Isothermal tests were conducted at both room temperature and 300 °C for both the rolling direction and the transverse direction of the hot rolled steel. Mechanical properties were extracted from the uniaxial experiments by fitting relevant material models to the data. The cyclic plasticity behavior has been modelled with a radial return-mapping algorithm that utilizes the Voce nonlinear isotropic hardening model in combination with the Armstrong-Frederick nonlinear kinematic hardening model. The plasticity models are available in commercial FE software and accurately capture the stabilized hysteresis loops, including a substantial Bauschinger effect. Results: The material exhibits near isotropic properties, but its mechanical performance is generally reduced at high temperatures. Specifically, in the rolling direction, the Young’s modulus is reduced by 16 % at 300 °C, the yield strength at 0.2 % plastic strain is lower by 23 %, and the ultimate tensile strength is lower by 30 % compared to room temperature. Fatigue life is also decreased, leading to an accelerated fatigue crack growth rate compared to room temperature. A von Mises radial return mapping algorithm proves to be effective in accurately modelling the cyclic plasticity of the material. The algorithm has also been used to establish a clear correlation between energy dissipation per cycle and cycles to failure, leading to the proposal of an energy-based fatigue life prediction model. Conclusions: The material exhibits reduced mechanical performance at elevated temperatures, with decreased monotonic strength, compared to room temperature. Fatigue life is also compromised, resulting in accelerated fatigue crack growth. The material’s hardening behavior differs at room temperature and elevated temperature, with lower peak stress values observed at higher temperatures. The radial return mapping algorithm can be used to determine the dissipated energy per cycle which together with fatigue testing has been used to propose a low cycle fatigue life prediction model at both temperatures.

Metastable austenitic stainless steel (304L) samples with a rectangular cross-section were plastically deformed in torsion during which they experienced multiaxial stresses that led to a complex martensitic phase distribution owing to the transformation induced plasticity effect. A three-dimensional character-ization of the phase distributions in these cm-sized samples was carried out by wavelength-selective neutron tomography. It was found that quantitatively correct results are obtained as long as the samples do not exhibit any considerable preferential grain orientation. Optical microscopy, electron backscatter diffraction, and finite element modeling were used to verify and explain the results obtained by neutron tomography. Altogether, neutron tomography was shown to extend the range of microstructure charac-terization methods towards the meso-and macroscale.