Influence of crack tip constraint on void growth in ductile FCC single crystals

Effect of constraint (stress triaxiality) on void growth near a notch tip in a FCC single crystal is investigated. Finite element simulations within the modified boundary layer framework are conducted using crystal plasticity constitutive equations and neglecting elastic anisotropy. Displacement boundary conditions based on mode I, elastic, two term K-T field are applied on the outer boundary of a large circular domain. A pre-nucleated void is considered ahead of a stationary notch tip. The interaction between the notch tip and the void is studied under different constraints (T-stress levels) and crystal orientations. It is found that negative T-stress retards the mechanisms of ductile fracture. However, the extent of retardation depends on the crystal orientation. Further, it is found that there exists a particular orientation which delays the ductile fracture processes and hence can potentially improve ductility. This optimal orientation depends on the constraint level.     

Comparison of experimental results and finite element simulation of strain localization scheme under cyclic loading

The plastic behaviour of FCC materials is studied under cyclic tensile–compression loading at room temperature. The material is an oxygen-free high conductivity copper. The purpose of the work is to model the onset of plasticity, then the cycle by cycle evolution of the localized strain, at grain scale and at mesoscopic scale.A polycrystalline aggregate taking into account the material microstructure is developped to perform finite element simulations corresponding to the experiments. Finite element calculations are carried out on this mesh, using a constitutive law which takes into account the crystallographic orientation of each grain. An analysis of the localisation scheme is performed at different steps of the cyclic loading.     

Modeling and experimental verification of thermo-mechanical coupled behavior of face-centered-cubic polycrystals

An improved integration model based on crystal plasticity is presented to model the thermo-mechanical processes of face-centered-cubic (FCC) polycrystals. In this model, the thermal part of deformation gradient is introduced into the multiplicative decomposition of the total deformation gradient and the plastic deformation gradient is chosen as the basic integration variable. The effects of temperature, temperature changing rate and dissipation of plastic deformation are considered in the finite deformation computation. The obtained plastic deformation gradient includes the plastic deformation as well as the thermal effects. In applications, the mechanical behaviors of 1100 Al in warm forming and 22MnB5 boron steel in hot tensile deformation were computed using this model. In experiments, the hot tensile tests of 22MnB5 boron steel were performed in the isothermal and non-isothermal conditions. The predicted results can reflect the thermal effects in forming process and agree well with the experimental data.     

Quantitative re-examination of Taylor model for FCC polycrystals

The quantitative adequacy of the Taylor model for representing the behaviors of FCC polycrystals is discussed through comparison with crystal plasticity analysis using the homogenization-based finite method. The key element of the crystal plasticity theory is the constitutive relation for single crystals. The most classical way to apply it to polycrystals is the Taylor model. This model assumes that all crystal grains in a crystal aggregate are subjected to the same strain under macroscopically uniform deformation. This assumption provides a solution satisfying the continuity of displacement between crystal grains. The effect and evolution of the crystallographic texture can easily be taken into account. However, the assumption of uniform strain, the main idea in the Taylor model, has never been validated quantitatively. On the other hand, the homogenization-based finite element method can represent arbitrary microscopic deformations, i.e., each crystal grain may have nonuniform deformation, and can provide a material response under more realistic boundary conditions. In this paper, we first determine the appropriate size for the representative volume element (RVE) in the homogenization-based finite element method that can represent the macroscopic polycrystalline behavior of FCC. After that, the polycrystalline behaviors obtained using the Taylor model are compared with those obtained using the homogenization-based finite element method. Finally, the quantitative adequacy of the Taylor model is discussed. It is clarified that the Taylor model is qualitatively consistent with the homogenization-based finite element method and can be used as a practical model of polycrystalline FCC metals for a first-order approximation, although it is not quantitatively reasonable even for FCC metals.     

Effect of loading rate on dynamic fracture initiation toughness of brittle materials

Numerical simulation is carried out to investigate the effect of loading rate on dynamic fracture initiation toughness including the crack-tip constraint. Finite element analyses are performed for a single edge cracked plate whose crack surface is subjected to uniform pressure with various loading rate. The first three terms in the Williams’ asymptotic series solution is utilized to characterize the crack-tip stress field under dynamic loads. The coefficient of the third term in Williams’ solution, A ₃, was utilized as a crack tip constraint parameter. Numerical results demonstrate that (a) the dynamic crack tip opening stress field is well represented by the three term solution at various loading rate, (b) the loading rate can be reflected by the constraint, and (c) the constraint A ₃ decreases with increasing loading rate. To predict the dynamic fracture initiation toughness, a failure criterion based on the attainment of a critical opening stress at a critical distance ahead of the crack tip is assumed. Using this failure criterion with the constraint parameter, A ₃, fracture initiation toughness is determined and in agreement with available experimental data for Homalite-100 material at various loading rate.     

Numerical simulation of plastic deformation and fracture in polysynthetically twinned (PST) crystals of TiAl

Plastic deformation and fracture in polysynthetically twinned (PST) crystals of TiAl have been simulated by using periodic unit cells representing the relaxed-constraint model recently proposed by Lebensohn et al. [Acta Mater. 46 (1998) 4701–4709] for the co-deformation of the lamellar compound of PST-TiAl. The unit cells contain both intermetallic phases, ₂-(Ti₃Al) and γ-(TiAl). Furthermore, the six orientation variants of the γ-phase are also considered. The constitutive behaviour of both phases is described by crystal plasticity, and the damage behaviour has been implemented by means of cohesive elements. The unit cells have been used as submodels for multi-scale finite element simulations of compression tests and fracture mechanics tests of notched micro-bend specimens. It is shown that the anisotropy of plastic deformation and damage in PST-TiAl can be well represented.     

Indentation creep surface morphology of nickel-based single crystal superalloys

The crystallographic orientation dependence of surface morphology of indentation creep on a nickel-based single crystal superalloy is investigated by using crystal plasticity slip theory with a three-dimensional (3D) finite element model. The numerical results show that the pile-up patterns developed around the indentation imprint exhibit four-, two-, and threefold symmetry on the surfaces of [0 0 1]-, [0 1 1]-, and [1 1 1]-oriented single crystals, respectively. The evolution of radial and hoop stresses around the crater provides important information for possible radial crack nucleation, whose critical locations depend on crystallographic orientations. These characteristics can be well explained in the viewpoint of crystallographic anisotropy. The findings may shed some light on understanding of the crystal structures and its time-dependent deformation mechanisms with the indentation method.     

Yield surface simulation for partially recrystallized aluminum polycrystals on the basis of spatially discrete data

The paper presents simulations of the yield surface evolution of plastically deformed aluminum polycrystals during recrystallization. The yield surfaces are calculated using a viscoplastic Taylor–Bishop–Hill strain rate polycrystal homogenization method. The input data for the yield surface calculations are the crystal orientations, their volume fractions, and their shear stresses. While the crystal orientations determine the kinematic portion of the yield surface the threshold shear stress of each individual orientation determines the kinetic portion of the yield surface. The input data for the homogenization calculations are generated through a spatially discrete simulation, where crystal deformation and primary static partial recrystallization are simulated by coupling a viscoplastic crystal plasticity finite element model with a cellular automaton. The crystal plasticity finite element model accounts for crystallographic slip and for crystal rotation during plastic deformation using space and time as independent variables and the crystal orientation and the accumulated slip as dependent variables. The cellular automaton uses a switching rule which is formulated as a probabilistic analogue of Turnbull's rate equation for the motion of grain boundaries. The actual decision about a switching event is made using a simple-sampling Monte Carlo step. The automaton uses space and time as independent variables and the crystal orientation and a stored energy measure as dependent variables. The kinetics produced by the switching algorithm are scaled through grain boundary mobility and driving force data. The crystallographic texture and the orientation-dependent resistance to shear are for each interpolation point extracted after each time step during recrystallization. The data serve as input for the calculation of discrete yield surfaces.     

A computational framework of configurational-force-driven brittle fracture based on incremental energy minimization

A variational formulation of quasi-static brittle fracture in elastic solids at small strains is proposed and an associated finite element implementation is presented. On the theoretical side, a consistent thermodynamic framework for brittle crack propagation is outlined. It is shown that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global Clausius–Planck inequality. Here, the canonical direction of the crack propagation associated with the classical Griffith criterion is the direction of the material configurational force which maximizes the local dissipation at the crack tip. On the numerical side, we first consider a standard finite element discretization in the two-dimensional space which yields a discrete formulation of the global dissipation in terms of configurational nodal forces. Next, consistent with the node-based setting, the discretization of the evolving crack discontinuity for two-dimensional problems is performed by the doubling of critical nodes and interface segments of the mesh. A crucial step for the success of this procedure is its embedding into a r-adaptive crack-segment re-orientation algorithm governed by configurational-force-based directional indicators. Here, successive crack propagation is performed by a staggered loading-release algorithm of energy minimization at frozen crack state followed by nodal releases at frozen deformation. We compare results obtained by the proposed formulation with other crack propagation criteria. The computational method proposed is extremely robust and shows an excellent performance for representative numerical simulations.     

Microstructure and texture evolution during the drawing of tungsten wires

Tungsten wires develop during their forming process a pronounced fibre texture that causes anisotropic deformation of single grains. The aim of this work is to simulate the crystallographic texture and microstructure evolution that arises during wire drawing using two different texture models. A visco-plastic self-consistent model that allows simulations using a large number of grains is compared with a crystal plasticity finite element model that provides a more detailed insight into the wire’s microstructure. Texture predictions of both models are discussed and quantitatively compared with experimental texture measurements obtained by neutron diffraction. The developed fibre texture causes plane strain deformation of single grains, which induces grain curling. The prediction of grain curling is of importance because it allows studying the residual stresses that trigger splits, at the grain level.     

A model for short crack propagation in polycrystalline materials

A model for microstructurally short crack propagation in a grain structure of a polycrystalline material is developed. The crack propagation model is based on a crystal plasticity model and a microstructurally short crack propagation model in the spirit of the model by Navarro and de los Rios [A model for short fatigue crack propagation with an interpretation of the short-long crack transition. Fatigue Fract Eng Mater Struct 1987;10:169-86]. Numerical examples, where the combined crystal plasticity and crack propagation model is implemented in a model of a microstructure representing a duplex stainless steel, concludes the paper. Results showing how the misorientation of the crack- and slip-directions between two adjacent austenitic grains influences the crack propagation rate, as the crack propagates across their common grain boundary, are given.     

Grain size effects on the tensile properties and deformation mechanisms of a magnesium alloy, AZ31B, sheet

The grain size dependence of the tensile properties and the deformation mechanisms responsible for those properties are examined for Mg alloy, AZ31B, sheet. Specifically, the Hall–Petch effect and strain anisotropy (r-value) are characterized experimentally, and interpreted using polycrystal plasticity modeling. {1 0 . 2} extension twins, {1 0 . 1} contraction twins, and so-called “double-twins” are observed via microscopy and diffraction-based techniques, and the amount of twinning is found to increase with increasing grain size. For the sheet texture and tensile loading condition examined, {1 0 . 2} extension twinning is not expected, yet the polycrystal plasticity model predicts the observed behavior, including this ‘anomalous’ tensile twinning. The analysis shows that the Hall–Petch strength dependence, of the polycrystal as a whole, is primarily determined by the grain size dependence of the strength of the prismatic slip systems.     

Modeling the response of HCP polycrystals deforming by slip and twinning using a finite element representation of the orientation space

A continuum approach is presented for predicting the constitutive response of HCP polycrystals using a simple non-hardening constitutive model incorporating both slip and twinning. This has been achieved by considering a physically based methodology for restricting the amount of the twinning activity. A continuum approach is used in modeling the texture evolution that eliminates the need for increasing the number of discrete crystal orientations to account for new orientations created by twinning during deformation. The polycrystal is represented by an orientation distribution function using the Rodrigues parameterization. A total Lagrangian framework is used to model the evolution of microstructure. Numerical examples are used to show the application of the methodology for modeling deformation processes.     

Multiple scale meshfree methods for damage fracture and localization

The Reproducing Kernel Particle Method (RKPM), which utilizes the fundamental notions of the convolution theorem, multiresolution analysis and meshfree properties, is reviewed. The multiple-scale RKPMs are then proposed as an alternative to commonly used numerical methods such as the finite element method. The elimination of a mesh, combined with the filtering properties of window functions, makes a particle method suitable for problems with large deformations, high gradients, and localization problems. This class of methods has been applied to shear band problems, and large deformation fracture and damage problems.     

A new implementation of the spectral crystal plasticity framework in implicit finite elements

We present a new implementation of a computationally efficient crystal plasticity model in an implicit finite element (FE) framework. In recent publications, we have reported a standalone version of a crystal plasticity model based on fast Fourier transforms (FFTs) and termed it the spectral crystal plasticity (SCP) model. In this approach, iterative solvers for obtaining the mechanical response of a single crystal of any crystallographic orientation subjected to any deformation mode are replaced by a database of FFTs that allows fast retrieval of the solution. The standalone version of the code facilitates simulations of relatively simple monotonic deformation processes under homogeneous boundary conditions. In this paper, we present a new model that enables simulations of complex, non-monotonic deformation process with heterogeneous boundary conditions. For this purpose, we derive a fully analytical Jacobian enabling an efficient coupling of SCP with implicit finite elements. In our implementation, an FE integration point can represent a single crystal or a polycrystalline material point whose meso-scale mechanical response is obtained by the mean-field Taylor-type homogenization scheme. The finite element spectral crystal plasticity (FE-SCP) implementation has been validated for several monotonic loading conditions and successfully applied to rolling and equi-channel angular extrusion deformation processes. Predictions of the FE-SCP simulations compare favorably with experimental measurements. Details of the FE-SCP implementation and predicted results are presented and discussed in this paper.     

Understanding local deformation in metallic polycrystals using high energy X-rays and finite elements

A methodology for understanding the stress and elastoplastic deformation responses within a loaded polycrystal is presented along with illustrative examples. High energy synchrotron X-rays are used to penetrate bulk metallic samples and produce diffracted intensity from each deforming crystal – revealing the evolving internal structure. A virtual representation of the microstructure is constructed using the finite element method (FEM) to simulate the evolution of the elastoplastic deformations, stress fields, and lattice orientations within the deforming crystals as the polycrystal is loaded. Simulations are compared directly to experimental diffraction data. In the case of powder experiments, lattice strain pole figures (SPFs) measured experimentally are compared to SPFs calculated by projecting X-rays through the finite element mesh. During in situ loading experiments, the stress states are found to differ from one crystal to the next and to vary from the stress being applied at the macroscale. A SPF/FEM-based methodology for quantifying residual stress fields within processed polycrystalline components is described. SPFs were measured at many points within a shrink-fit sample. Finite element discretizations of both the sample and orientation space of each diffraction volume were used to formulate an optimization for the distribution of the stress tensor within the sample. A different experiment, one in which the X-ray beam and the crystals are closer to the same size, is used to investigate the aggregate crystal by crystal. The Debye–Scherrer rings reduce to a set of spots associated with each crystal within the diffraction volume. This method is demonstrated by tracking deformation of four grains within a deforming BCC titanium aggregate loaded in situ within the elastic regime to determine the single crystal elastic moduli. Plastic deformation can also be investigated by monitoring the size and shape of individual diffraction spots. Each spot contains geometrically exact information regarding the internal structure of the crystal. Instead of reconstructing the crystal structure by inverting the diffraction data, virtual diffraction experiments are performed on the finite element mesh and the resulting simulated diffraction patterns are compared directly to the experimental results. Once the experimental/simulation methodology is validated, the approximation of the subgrain distribution of stress and lattice orientation from the finite element model can be used to construct theories for failure phenomena such as microcrack initiation. As opposed to other methods of discretizing a polycrystalline aggregate, the finite element framework enables a seamless transition to analyses associated with mechanical design.     

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.     

Analysis of a new specimen for mixed mode fracture tests on brittle materials

Numerical and experimental studies were performed on a new fracture test configuration called the diagonally loaded square plate (DLSP) specimen. The mode I and mode II stress intensity factors were computed for different crack lengths and crack orientation angles using finite element analysis. The numerical results show that the DLSP specimen is able to provide pure mode I, pure mode II and any mixed mode loading conditions in between. Fracture experiments were also conducted on Plexiglas using the DLSP specimen. It is shown that the results obtained from the fracture tests are consistent very well with mixed mode fracture theories.     

Effect of thermal ageing on creep and oxidation behaviour of Type 316H stainless steel

The UK has unique experience in operating high temperature civil nuclear power systems, known as advanced gas cooled reactors (AGRs). One of the primary challenges for extending the lifetime of the AGR power stations is to understand the interaction that occurs between the AGR CO₂ environment and creep-fatigue cracking behaviour. This is one of the life limiting degradation mechanisms for steel components within the reactor pressure vessel. This paper addresses the effect of thermal aging on material internal state that controls both the creep deformation and oxidation behaviour of Type 316H stainless steels when they are exposed at a simulated AGR environment. Experimental results from creep tests are discussed with respect to a multi-scale self-consistent model, while experimental results from oxidation tests are considered with respect to the application of measured short term data to predict the long term oxidation behaviour. Finally, the interaction between oxidation and creep and its impact on high temperature structural integrity of AGR nuclear systems are discussed.