Mesoscale modelling of crack nucleation from defects in steel

Defects such as pores and non-metallic inclusions have a significant influence on the long-life fatigue strength of high strength steels. The largest of these defects in a critical material volume is in the range of tens of micrometres which is on the same size scale as the grain size. At this scale materials are non-homogeneous since each grain in a polycrystalline material will have a different orientation. Finite element-based mesoscale modelling has been used to model the stress and strain in individual grains in the vicinity of a spherical defect. Microcrack nucleation and propagation models based on shear stress and plastic shear strain have been applied. Especially for low stress amplitudes near the endurance limit, critical grain orientation and defects are both essential for cracks to initiate and propagate.     

Grain orientation, deformation microstructure and flow stress

Dislocation structures in deformed metals have been analyzed quantitatively by transmission electron microscopy, high-resolution electron microscopy and Kikuchi line analysis. A general pattern for the microstructural evolution with increasing strain has been established and structural parameters have been defined and quantified. It has been found that two dislocation patterns co-exist in all grains, however, with very different characteristics dependent on grain orientation. This correlation with the grain orientation has been applied in modeling of the tensile flow stress and the flow stress anisotropy of fcc polycrystals. In conclusion some future research areas are briefly outlined.     

Numerical approach of cyclic behaviour of 316LN stainless steel based on a polycrystal modelling including strain gradients

A non-local polycrystal approach, taking into account strain gradients, is proposed to simulate the 316LN stainless steel fatigue life curve in the hardening stage. Material parameters identification is performed on tensile curves corresponding to several 316LN polycrystals presenting different grain sizes. Applied to an actual 3D aggregate of 316LN stainless steel of 1200 grains, this model leads to an accurate prediction of cyclic curves. Geometrical Necessary Dislocation densities related to the computed strain gradient are added to the micro-plasticity laws. Compared to standard models, this model predicts a decrease of the local stresses as well as a grain size effect.     

Local texture and fatigue crack initiation in a Ti-6Al-4V titanium alloy

ABSTRACT Fatigue crack initiation was studied in a bimodal TA6V titanium alloy. A ghost structure inherited from the forging process, the scale of which is roughly 100 times the apparent grain size, was found to govern the initiation process. In these macrograins, that we have labelled macrozones, most of the primary alpha grains (α_p) are found to display the same crystallographic orientation. Fatigue cracks are initiated on the basal plane or, if basal slip is difficult, on the prismatic plane. Thus in macrozones, where basal or prismatic slip is easy, numerous neighbouring tiny cracks appear over the whole macrozone, which have the size of the primary α_p grains. In these macrozones the contribution of crack coalescence to crack growth is consequently very significant. On the contrary, if basal and prismatic slips are both difficult in the macrozone, no crack can be found in the corresponding macrozone. The crack initiation process is thus highly heterogeneous at the scale of the macrozone. Furthermore, this microstructure is found to induce a large scatter in the fatigue life of notched samples.     

Theoretical calculation of creep and relaxation of polycrystals,and stress redistribution among constituent grains

Taking into account both transient and steady creep of slip systems in the grain, a theoretical method is developed to determine the overall creep and relaxation behaviour of polycrystals and, by which, the accompanying stress and strain distribution among the constituent grains can also be evaluated. This method extends the incremental self-consistent relation for grain interactions to the total form, and is further complemented with an iterative computational process. It is primarily intended for the calculation of creep under a constant stress, relaxation under a constant strain, and a combination of both. While maintaining almost the same degree of accuracy, this new method, as compared to the incremental one, is far more effective. Its theoretical predictions on the creep and relaxation of a 2618-T61 aluminium are shown to be in good accord with experiments. The heterogeneous nature of creep deformation and stress distribution among the constituent grains are also displayed for several selected grain orientations. Finally some implications and limitations of the model are assessed.     

FE Analysis of Shearing of Granular Bodies in a Direct Shear Box

A plane strain analysis of a deformation and stress field in cohesionless granular bodies during shearing in a direct shear tester was performed with a finite element method on the basis of a hypoplastic constitutive law enhanced by polar quantities: rotations, curvatures, couple stresses, and a mean grain diameter used as characteristic length. The constitutive law takes into account the effect of pressure, void ratio, direction of deformation rate, mean grain diameter, and grain roughness on the material behavior. The FE calculations were carried out with a different initial void ratio, vertical load, mean grain diameter, and specimen length. Attention was focused on the size effect caused by the size of microstructure related to the specimen dimensions and the effect of side boundaries on the shear zone formation. The FE results show that the thickness of the shear zone increases with increasing initial void ratio, pressure level, mean grain diameter, and specimen length. Due to the effect of boundary conditions, the thickness changes along a horizontal midsection (it is widest in the mid-region).     

Statistics and probabilistic modeling of simulated intergranular cracks

Using Monte Carlo simulation, the statistical properties of intergranular crack trajectories in polycrystalline materials are estimated. The polycrystalline microstructures are two dimensional and are modeled by a Poisson–Voronoi tessellation for the grain geometry and a uniform orientation distribution function for the crystallographic orientation. A heuristic is introduced for determining the path of crack propagation when the crack tip arrives at a grain boundary triple junction. This heuristic applies a combination of two criteria for determining the direction of crack propagation, the maximum circumferential stress criterion, and a criterion in which the crack is assumed to propagate in the direction with the least material resistance. The resistance of grain boundaries is assumed to be related to the crystallographic misorientation at the grain boundary. The trajectories of microcracks can be treated as a random process, and simulation results indicate that the crack process exhibits linear variance growth, the rate of which is related to the importance attached to the circumferential stress and the material resistance in determining the direction of propagation. The rate of variance growth is shown to vary with the average grain diameter, so that microcracks in polycrystals with small grain size will exhibit less spatial uncertainty. The statistics and distributions of the increments of the crack process are also given. Through a small change made to the normalization applied to non-dimensionalize the statistics, the results are extended to polycrystals that have spatially varying grain size. Finally, a probabilistic model is proposed that is able to produce synthetic crack trajectories that replicate the important statistical properties of the simulated cracks. Such a model may prove useful in studies of the transition from micro to macrocracking.     

CYCLIC DEFORMATION OF A1-4%Cu ALLOY POLYCRYSTALS CONTAINING θ" PRECIPITATES

The cyclic deformation of polycrystalline A1-4%Cu alloy containing θ precipitates was investigated by strain controlled tests, and compared to monocrystalline cyclic behavior. The cyclic deformation of the polycrystals was found to depend on the grain size, and in coarse-grained material (grain diameter = 1 mm) a definite plateau was observed in the cyclic stress-strain (CSS) curve. In order to explore the relationship of cyclic behavior between single crystals and polycrystals, three available orientation factors were examined. This relationship can be interpreted by selection from the different types of orientation factors (Taylor factor, Sachs factor and Maximum Schmid factor), which operate to different degrees depending on the magnitude of the strain amplitude and the grain size.     

Modeling the effect of microstructural features on the nucleation of creep cavities

A crystal plasticity based finite element model has been applied to study the deformation of metals at the microstructural length scale, in order to determine the effect of various microstructural features on the nucleation of creep cavities. The deformation model captures the non-uniform distributions of the equivalent plastic strain and the hydrostatic stress within the different grains of the microstructure when subjected to cyclic loading conditions. The influence of various microstructural features such as grain boundaries, triple junctions, and second-phase particles, on the strain and stress fields is examined through the simulations. The results indicate that the various microstructural parameters, such as grain orientation, presence of the precipitates and their shape, and alignment of the boundaries with respect to the loading direction influence the strain and stress distributions, and therefore, the conditions that favor the nucleation and growth of creep cavities.     

Microstructure control for developing Fe–Pd ferromagnetic shape memory alloys

The effect of microstructure on the ferromagnetic shape memory effect in Fe–Pd alloys was examined focusing on the orientation distribution, the grain boundary character and the grain size. Fe–29.7–31.0 at.%Pd polycrystals composed of columnar or equiaxed grains were prepared and their magnetostriction was measured under a magnetic field up to 10 kOe. The magnetostriction of the columnar crystals was much larger than that of the equiaxed ones due to the development of 〈100〉 fibre texture and 〈100〉 tilt grain boundary. In particular, a large magnetostriction of 4.5×10⁻⁴ was obtained even in the polycrystals. The grain boundaries of the columnar crystals also influenced the variant selection of martensites, resulting in strong dependence of the magnetostriction on the direction of magnetic field. Larger grain size did not always result in larger magnetostriction in Fe–Pd alloys. An increase in the grain size of the alloys was sometimes accompanied by the development of several sets of twin variants within a grain, resulting in a decrease in the magnetostriction.     

Micromechanical modeling of intrinsic and specimen size effects in microforming

Size effect is a crucial phenomenon in the microforming processes of metallic alloys involving only limited amount of grains. At this scale intrinsic size effect arises due to the size of the grains and the specimen/statistical size effect occurs due to the number of grains where the properties of individual grains become decisive on the mechanical behavior of the material. This paper deals with the micromechanical modeling of the size dependent plastic response of polycrystalline metallic materials at micron scale through a strain gradient crystal plasticity framework. The model is implemented into a Finite Element software as a coupled implicit user element subroutine where the plastic slip and displacement fields are taken as global variables. Uniaxial tensile tests are conducted for microstructures having different number of grains with random orientations in plane strain setting. The influence of the grain size and number on both local and macroscopic behavior of the material is investigated. The attention is focussed on the effect of the grain boundary conditions, deformation rate and the grain size on the mechanical behavior of micron sized specimens. The model is intrinsically capable of capturing both experimentally observed phenomena thanks to the incorporated internal length scale and the crystallographic orientation definition of each grain.     

Influences of grain size and precracking load on the critical stress intensity factor of mild steel

The effects of grain size and precracking load on the critical stress intensity factor are studied. A plane stress model of elastic-plastic stress distribution which includes the strain hardening effects is used. The effects of residual stresses and strain hardening due to fatigue load are calculated by choosing plastic zone size as fracture criterion. Experimental results are obtained to demonstrate the reliability of theoretical calculations.     

Development and validation of a 3D computational tool to describe concrete behaviour at mesoscale. Application to the alkali-silica reaction

A finite element analysis of the large deformation of three-dimensional polycrystals is presented using pixel-based finite elements as well as finite elements conforming with grain boundaries. The macroscopic response is obtained through volume-averaging laws. A constitutive framework for elasto-viscoplastic response of single crystals is utilized along with a fully-implicit Lagrangian finite element algorithm for modeling microstructure evolution. The effect of grain size is included by considering a physically motivated measure of lattice incompatibility which provides an updated shearing resistance within grains. A domain decomposition approach is adopted for parallel computation to allow efficient large scale simulations. Conforming grids are adopted to simulate flexible and complex shapes of grains. The computed mechanical properties of polycrystals are shown to be consistent with experimental results for different grain sizes.     

A crystal plasticity study of cyclic constitutive behaviour, crack-tip deformation and crack-growth path for a polycrystalline nickel-based superalloy

Crystal plasticity has been applied to model the cyclic constitutive behaviour of a polycrystalline nickel-based superalloy at elevated temperature using finite element analyses. A representative volume element, consisting of randomly oriented grains, was considered for the finite element analyses under periodic boundary constraints. Strain-controlled cyclic test data at 650 °C were used to determine the model parameters from a fitting process, where three loading rates were considered. Model simulations are in good agreement with the experimental results for stress–strain loops, cyclic hardening behaviour and stress relaxation behaviour. Stress and strain distributions within the representative volume element are of heterogeneous nature due to the orientation mismatch between neighbouring grains. Stress concentrations tend to occur within “hard” grains while strain concentrations tend to locate within “soft” grains, depending on the orientation of grains with respect to the loading direction. The model was further applied to study the near-tip deformation of a transgranular crack in a compact tension specimen using a submodelling technique. Grain microstructure is shown to have an influence on the von Mises stress distribution near the crack tip, and the gain texture heterogeneity disturbs the well-known butterfly shape obtained from the viscoplasticity analysis at continuum level. The stress–strain response near the crack tip, as well as the accumulated shear deformation along slip system, is influenced by the orientation of the grain at the crack tip, which might dictate the subsequent crack growth through grains. Individual slip systems near the crack tip tend to have different amounts of accumulated shear deformation, which was utilised as a criterion to predict the crack growth path.     

Effect of Ambient Temperature on Stress Measurement Method Using Copper Foil

The copper electroplating stress measurement method uses the grain growth in the copper on a machine element that has been subjected to repeated loads. Because this growth is also caused by thermal energy, the effect of the ambient temperature on grain growth density and grain orientation was investigated. Cyclic torsion tests were carried out at temperatures from 293 to 353 K. The relationship among the grain growth density, maximum shear stress, number of cycles, and ambient temperature was formulated to measure the maximum shear stress occurring on the machine element. Moreover, cyclic bending–torsion tests were also performed, and the orientations of grown grains were analysed by electron backscatter diffraction. The slip directions of grown grains corresponded closely with the direction of shear stress in spite of the ambient temperatures. This means that principal stresses can be measured by using the pole figure or the inverse pole figure of grown grains at temperatures up to 353 K.     

The influence of temperature and grain size on substructure evolution in stainless steel 316L

The flow stress of polycrystals is controlled by the processes occurring in the grain interior as well as in the mantle, i.e. at the grain boundary and its immediate vicinity. The early stages of evolution of dislocation substructure in these two regions with strain in 316L stainless steel polycrystals have been studied at 293 K, 673 K and 1123 K representing the low temperature thermal, the intermediate temperature athermal and the high temperature thermal regimes respectively. Specimens with grain sizes of 4 and 12 m were employed to determine the effect of grain size.Transmission electron microscopy studies on deformed specimens show the different roles of grain boundary and grain interior in different temperature regimes. In the low temperature regime grain boundaries act as obstacles to moving dislocations and as such high density of dislocation is found in the grain boundary vicinity. In the intermediate temperature regime the dislocations which are easily spread into the grain interior rearrange to form cell walls. In the high temperature regime grain boundaries transform to the equilibrium state and do not contain any grain boundary dislocations, and the distribution of dislocations within grains is homogeneous at all strains. Significantly higher values of dislocation densities in the vicinity as well as in the grain interior were found in the finer grain size material in the whole strain region employed.     

Effects of grain size and mechanical pretreatment on strain localization in FCC polycrystals

Polycrystalline copper samples where the grains in the cross-section had crystallographic axes parallel to the load related to ‘single slip’ directions and three grain sizes, were ramp loaded and then step-tested to obtain their Cyclic Stress–Strain Curves (CSSCs). Plateaux were found for medium grain size at about 89 MPa, whereas large grained samples showed plateaux at 72 MPa, which correlated with the measured Taylor and Sachs factors, respectively. No plateaux were found when grains were smaller than 200 μm or ramp-loading stresses were below 87 MPa. Comparisons are made with nickel polycrystals and it is found that the plateaux in copper are narrower than those reported in nickel. The differences are attributed to a more homogeneous dislocation structure in nickel, due to lower elastic interactions across grain boundaries and easier cross-slip behavior as compared to copper.