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.     

The effect of different forms of strain energy functions in hyperelasticity‐based crystal plasticity models on texture evolution and mechanical response of face‐centered cubic crystals

Micro‐mechanical and macro‐mechanical behavior of face‐centered cubic (FCC) crystals is investigated by using different forms of strain energy functions in hyperelastic material models in crystal plasticity finite element framework. A quadratic strain energy function with anisotropic elastic constants, a polyconvex strain energy function with invariants associated with the cubic symmetry, and a strain energy function from an inter‐atomic potential are considered in hyperelastic material models to describe the elastic deformation of FCC crystals. In our numerical experiments, the trajectories of {111} poles in the pole figure and the accumulated plastic slips of FCC coppers under uniaxial tension and simple shear depend on the choice of strain energy functions when the slip resistance of the slip systems is high. The ability of strain energy functions in this study to represent elastic lattice distortions in crystals varies with the amount of elastic deformation and the shape of deformed lattice. However, numerical results show that the change of macroscopic mechanical behavior of FCC coppers is not significant for the choice of strain energy functions, compared with the change of crystallographic texture evolution. Copyright © 2014 John Wiley & Sons, Ltd.     

Correlation of deformation texture and microstructure

Abstract

The principles of texture evolution during the rolling deformation of fcc metals are presented, and effects which enable texture formation and local structure evolution to be correlated are analysed by comparing experimental rolling textures in fcc metals with the results of theoretical predictions based on homogeneous slip on discrete slip systems (Taylor type models). The assumptions of partially relaxed constraints conditions, and the influence of high deformation on deformation structure and its consequences for local texture effects and local grain arrangement, are discussed.

MST/1291     

On the numerical handling of fractional viscoelastic material models in a FE analysis

The paper presents the finite element (FE) implementation of linear and nonlinear fractional viscoelasticity models. To this end, a short introduction on fractional calculus is given. In addition to the fractional operators, this includes analytical and numerical solution schemes for selected fractional integral and differential equations. The presented rheological model is based on state of the art approaches and has been adopted to model the strain rate dependent material behavior of polymers. To this end, two approaches with constant and overstress dependent viscous properties resulting in linear and nonlinear evolution equations are discussed. The uniaxial constitutive relations are generalized to the multiaxial case and processed to be implemented in a FE code. The model behavior of both approaches is demonstrated and compared for selected uniaxial and multiaxial load cases.     

Rhéologie des murs en blocs de terre comprimée en compression uniaxiale: étude et modélisation

Static uniaxial compression tests have been conducted on samples of small walls made of compressed earth blocks. These tests permitted to determine, besides the compression strength, other parameters, namely Young’s modulus, the limit strain and Poisson’s ratio. To predict the mechanical behaviour of small earth-block walls, a non linear model based on the theory of elasticity-damage coupling is proposed. Indeed, this model has been applied for the block and the mortar. While confronting the experimental results with those of the model, the model proves to be satisfactory. The residue of approximation is under 10%. The model is intermediate between the two limit models proposed for the heterogeneous materials, namely Voigt’s and Reuss’. The determination by the model of the damage ratio gives information on the evolution of the damage within the walls according to the applied loading.     

On intergranular mechanical interactions and the theory of deformation crystallography of metals

The equilibrium of intergranular stress and strain can be realized simultaneously, whereas five independent slip systems of the Taylor principle and the criterion of minimal internal work are unnecessary. In fact, the Taylor principle applied in current theories is incorrect both in practice and theory, in which the activation mechanism of plastic deformation systems must violate the Schmid’s law and deviate from the elastic–plastic characteristics of deformed matrix. The intergranular reaction stress (RS) during deformation can be calculated according to Hooke’s law and elastic limit without additional subjective presupposition, therefore the RS theory is established intuitively. Under the combination of the RS (the intergranular elastic effect) and the external loading the slips penetrating grains are activated and produce deformation texture, but certain non-penetrating slips near grain boundaries will become active (the intergranular plastic effect) and produce some random texture when a RS reaches the yield strength of grains. The RS theory is simple, intuitive and reasonable, based on which the texture simulation can well reproduce the texture formation of various metals under different external loadings and under different crystallographic mechanisms.     

The evolution of porosity in bulk nanocrystalline materials during plastic deformation and its effect on the mechanical behavior

Uniaxial compression tests were carried out to completely understand the evolution of porosity in porous bulk nanocrystalline materials, and a new evolution law of porosity under uniaxial compression was proposed. Based on the energy principle, we built a mechanical model to calculate the overall mechanical properties of bulk nanocrystalline materials. The comparison between predicted results and the corresponding experimental data indicates that the established model is capable of describing the plastic mechanical behaviors of porous nanocrystalline materials.     

Effect of strains on textural evolution of hydrostatically extruded niobium tubes

The influence of tube strains on tube deformation ratios and textural evolution during the deformation of pure Nb tubes subjected to hydrostatic extrusion was investigated. The tube deformation ratios increased with increasing tube strains. A very strong <111> fibre texture at a lower strain transformed to a dual strong <100> and weak <111> fibre texture at a higher strain. The texture evolution led to an increase in the Taylor factor at the drawing tensor.     

不同应力路径下砂岩能耗变化规律试验研究

为研究不同应力路径下岩石的能耗变化规律,采用MTS815岩石力学试验系统开展了砂岩的单轴压缩、常规三轴和卸荷三轴试验。结果表明:耗散能曲线变化是岩石内部损伤和破裂产生的表现,在弹塑性变形阶段,常规三轴耗散的能量占岩石吸收总能量的比例最大,卸荷三轴次之,单轴压缩最小;岩石的储能极限与围压具有明显的线性关系,单轴压缩试验中岩石的储能极限最低,卸荷三轴次之,常规三轴试验岩石的储能极限最高;岩石峰前和峰后的能量耗散速率与围压也具有良好的线性关系,峰后应力跌落阶段能量耗散速率明显较峰前能量耗散速率大数倍至数十倍,说明岩石峰前损伤速率较小,而峰后却快速损伤破裂,耗散能曲线的突然变陡表明岩石破坏发生。     

Deformation band and texture of a cast Mg-RE alloy under uniaxial hot compression

Microstructural evolution and texture of a cast Mg-9Gd-4Y-0.6Zr ingot under hot compression were studied in this paper. Post-deforming microstructures were characterized by optical microscopy, scanning electron microscopy and transmission electron microscopy, while crystallographic orientation information was obtained from X-Ray macro-texture measurement and EBSD micro-texture analysis. Dynamic recrystallization (DRX) initiated from the deformation bands (DB) forming on original grain boundaries; the DB became widen with continuously conversion of low-angle-boundary grains into high-angle-boundary grains. The tendency of strain localization increased with Z parameter. The macro-texture analysis indicates that uniaxial compression yielded out the randomized basal texture component. This texture component was found to be strengthened with increasing Z parameter. The micro-texture analysis shows that the deviation from the ideal basal texture arose from orientated growth within DBs. Moreover, the localization deformation promoted dynamic precipitation within DBs, which inhibited the development of DRX.     

Comparison of simulated and experimental strain ratio R for different grain shapes in sheet metal

Abstract

Predicting mechanical properties by means of a simple indicator is of great importance to sheet metal forming. An important parameter characterising the formability of a rolled sheet is the plastic strain ratio R which is strongly determined by the texture. The angular variation of R value in the rolling plane has been calculated from the orientation distribution function using the Bunge method. The following grain interaction models have been tested: two Taylor full constraint models ({hkl}〈111〉 and {110}〈111〉 plus {112}〈111〉), three relaxed constraints models (RC₄ , RC₃ , RC₂ ), and the Sachs–Kochendörfer model. The shapes of the grains were investigated by means of the secant method which allowed the spatial parameters of the two- dimensional structure image to be measured. The comparison of simulated and experimental data has proved that in the case of aluminium killed steels, the relaxed constraints model RC₄ is the best predictor of the plastic strain ratio. Good results were also obtained using the Sachs-Kochendörfer model.     

Crystallography-based prediction of plastic anisotropy of polycrystalline materials

The on-line prediction of metal sheet formability requires that both material characterization (texture identification) and yield loci predetermination be done in very shor time intervals. Of two applicable approaches, i.e., continuum mechanics and crystallography-based methods, only the latter are suitable for this purpose. Several models of plasticity of a polycrystalline material were reviewed, and their applicability to the prediction of plastic anisotropy of face-centered cubic (FCC) metals was evaluated. A tailored set of cold-rolled copper alloy samples was designed and manufactured, representing the wide spectrum of textures and cold work levels typical for the sheet metal industry. The texture was quantitatively described in the form of the orientation distribution functions derived by the inversion of four incomplete pole figures. The Taylor-Bishop-Hill model was applied in order to calculate the planar variation of the plastic strain ratio. The continuum mechanics of textured polycrystals approach was also used for the prediction of the plastic strain-rate ratio for the same set of deformed materials. The theoretical predictions were compared with the plastic strain ratios measured in tensile tests using strain gauges. The applicability of the models for prediction of the plastic anisotropy of FCC metals was discussed in view of the operating deformation mechanisms and other factors such as strain hardening sensitivity and grain size/shape effects.     

Experimental study on damage evolution of rock under uniform and concentrated loading conditions using digital image correlation

Damage evolution and crack propagation in sandstone specimens have been observed by digital image correlation method. To investigate deformation and failure process of rock under different loading conditions, uniaxial compression and indentation tests were performed. Through the experiment, displacement and strain fields are simultaneously obtained that can visually display the distribution, mode and evolution of deformation and cracking in rock. Experimental results show that the damage distributes diffusely in rock at early loading stage, and the measured apparent strain increasingly concentrates with loading because of the nucleation of crack; propagation of the crack leads to the eventual failure of the specimen. Damage factor is calculated on the basis of deviation of apparent strain, and localization factor is presented to describe the level of deformation localization. The combined use of two factors can well represent the damage evolution of rock under compression.     

THE INFLUENCE OF THE HARDENING STATE ON TIME DEPENDENT DAMAGE AND ITS CONSIDERATION IN A UNIFIED DAMAGE MODEL

Abstract— A model is presented for the prediction of the lifetime of metals in the high-temperature range under arbitrary variable multiaxial load. The definition of an internal variable for damage in continuum damage mechanics is adopted, which allows indirect measurement of damage via the deformation behaviour. To acquire some knowledge of damage evolution, damage is measured in two ways during uniaxial strain controlled cyclic tests: (a) a change of the modulus of elasticity and (b) a decrease of the peak stress. Surprisingly, both methods lead to results which are in good agreement. A new damage law is then developed (with reference to known models and lifetime rules) which is a modification of the creep damage law of Rabotnov that is extended by a dependence on the inelastic strain rate instead of the dependence on internal variables to take into account the hardening state. Uniaxial as well as multiaxial formulations of the new damage model (Inelastic Strain Rate Modified (ISRM) model) are presented.
The parameters of the ISRM model are determined with a view to applying them to AISI 316 L(N) austenitic steel. Some of the parameters are derived from standard creep experiments. To determine further parameters, the ISRM model is applied to uniaxial cyclic tests. Both failure behaviour and damage evolution are well described.     

Modeling of large plastic deformation behavior and anisotropy evolution in cold rolled bcc steels using the viscoplastic ?-model-based grain-interaction

In this paper, a micromechanical approach is used to predict the mechanical response and anisotropy evolution in BCC metals. Particularly, cold rolling textures and the corresponding yield surfaces are simulated using the newly developed viscoplastic intermediate ?-model. This model takes into account the grain interactions but without the Eshelby theory. In this work, we compare our results to those predicted by the upper and lower bounds (Taylor and Static) as well as those of the viscoplastic self-consistent (VPSC) model. The results are compared in terms of predicted slip activity, texture evolution and yield loci. For the simulations, we considered two cases: the restricted slip, {1 1 0}〈1 1 1〉, and the pencil glide, {1 1 0}〈1 1 1〉 + {1 1 2}〈1 1 1〉 + {1 2 3}〈1 1 1〉. In addition, we present a qualitative comparison with experimental cold rolling textures taken from the literature for several BCC metals: electrical, ferritic, Interstitial-Free (IF) and low carbon steels. Our results show that the pencil glide assumption is adequate for low carbon and IF-steels and that the restricted slip assumption is well suited for ferritic and electrical steels.     

Experimental mechanical characterization of plastic-bonded explosives

This article deals with the characterization of the static mechanical behavior of an energetic material. Due to its constituents (crystals and a polymeric binder), the behavior is complicated to model. A specific experimental protocol has been proposed in this article. It involves uniaxial tensile and compressive tests, compression under confinement and dynamic mechanical analysis. A constitutive law has been developed. The behavior is described using a Maxwell’s model, in which all the components are influenced by an isotropic damage. The first component takes into account an elasto-plastic behavior. The yield stress evolution is described using a parabolic criterion and an isotropic hardening law. The plastic flow rule is non-associated. A linear visco-elastic behavior is used for the other components. Numerical simulations show that experimental data are quite well reproduced. The last part of the article is devoted to a discussion highlighting the future improvements.     

Deformation twinning in zircaloy 2

Abstract

Fully recrystallised zircaloy 2 samples were subjected to different degrees of uniaxial compression. Grains of high Taylor factors showed {1012}〈1011〉 deformation twins, noticeable up to 13–16% compression. Twinning strongly affected the crystallographic texture and also brought in clear differences in stored energy and residual stress between the suspected parent and product grains/orientations of twinning. At later stages of deformation, where presence of twinning was insignificant, aforementioned heterogeneity was further supplemented by heterogeneity in microstructure – clear presence of fragmenting and non-fragmenting grains. Direct observations on twin fraction, twin deviation and twin continuity had shown an apparent peak in twinning by ～7·5% compression, an observation explainable through a simple model of twin decay by in grain misorientation development.     

Seismic fragility analysis of structures based on Bayesian linear regression demand models

Bayesian linear regression (BLR) based demand prediction models are proposed for efficient seismic fragility analysis (SFA) of structures utilizing limited numbers of nonlinear time history analyses results. In doing so, two different BLR models i.e. one based on the classical Bayesian least squares regression and another based on the sparse Bayesian learning using Relevance Vector Machine are explored. The proposed models integrate both the record-to-record variation of seismic motions and uncertainties due to structural model parameters. The magnitude of uncertainty involved in the fragility estimate is represented by providing a confidence bound of the fragility curve. The effectiveness of the proposed BLR models are compared with the commonly used cloud method and the maximum likelihood estimates methods of SFA by considering a nonlinear single-degree-of-freedom system and a five-storey reinforced concrete building frame. It is observed that both the BLR models can estimate fragility with improved accuracy compared to those common analytical SFA approaches considering direct Monte Carlo simulation based fragility results as the benchmark.     

A MICROCRACK GROWTH LAW FOR MULTIAXIAL FATIGUE

Within the past decade, critical plane approaches have gained increasing support based on correlation of experimentally observed fatigue lives and microcrack orientations under predominately low cycle fatigue (LCF) conditions for various stress states. In this paper, we further develop an engineering model for microcrack propagation consistent with critical plane concepts for correlation of both LCF and high cycle fatigue (HCF) behavior, including multiple regimes of small crack growth. The critical plane microcrack propagation approach of McDowell and Berard serves as a starting point to incorporate multiple regimes of crack nucleation, shear growth under the influence of microstructural barriers, and transition to linear crack length-dependent growth related to elastic-plastic fracture mechanics (EPFM) concepts. Microcrack iso-length data from uniaxial and torsional fatigue tests of 1045 steel and IN 718 are examined and correlated by introducing a transition crack length which governs the shift from nonlinear to linear crack length dependence of da/dN. This transition is related to the shift from strong microstructural influence to weak influence on the propagation of microcracks. Simple forms are introduced for both the transition crack length and the crack length-dependence of crack growth rate within the microcrack propagation framework (introduced previously by McDowell and Berard) and are employed to fit the 1045 steel and IN 718 microcrack iso-length data, assuming preexisting sub-grain size cracks. The nonlinear evolution of crack length with normalized cycles is then predicted over a range of stress amplitudes in uniaxial and torsional fatigue. The microcrack growth law is shown to have potential to correlate microcrack propagation behavior as well as damage accumulation for HCF-LCF loading sequences and sequences of applied stress states.