Crystal Plasticity Theory Research Articles

A thermomechanical constitutive model for investigating the fatigue behavior of Sn‐rich solder under thermal cycle loading

AbstractBased on crystal plastic theory, a novel thermomechanical constitutive model was proposed for Sn‐rich solder. Material parameters of the proposed constitutive model were determined by fitting the stress–strain curves under uniaxial tensile testings. Combined with the proposed model and the material parameters, crystal plastic finite element method was used to investigate the effects of the microstructure features of grain orientation, grain boundary, and intermetallic compound on the fatigue behaviors of Sn‐rich solder joint under the condition of thermal cycle loading. Results showed that the mean absolute percentage error between the simulation and experimental tensile curves was 3.50%, and the three microstructure features affected the fatigue behaviors of Sn‐rich solder from different aspects, which were analyzed in detail. Besides, it was concluded that the proposed constitutive model could be effectively used to investigate the fatigue behaviors of Sn‐rich solder joint from micro perspective.

Low-Cost Magnesium Alloy Sheet Component Development and Demonstration Project

<div class="section abstract"><div class="htmlview paragraph">Most of the applications of magnesium in lightweighting commercial cars and trucks are die castings rather than sheet metal, and automotive applications of magnesium sheet have typically been experimental or low-volume serial production. The overarching objective of this collaborative research project organized by the United States Automotive Materials Partnership (USAMP) was to develop new low-cost magnesium alloys, and demonstrate warm-stamping of magnesium sheet inner and outer door panels for a 2013 MY Ford Fusion at a fully accounted integrated component cost increase over conventional steel stamped components of no more than $2.50/lb. saved ($5.50/kg saved). The project demonstrated the computational design of new magnesium (Mg) alloys from atomistic levels, cast new experimental alloy ingots and explored thermomechanical rolling processes to produce thin Mg sheet of desired textures. A new commercial Mg alloy sheet material was sourced and pretreated with protective coil coatings, and its properties fully characterized. The Mg sheet was successfully warm-formed using novel lubricants into intermediate size benchmark parts and full-size automotive door inner and outer panels. The project also explored conventional welding processes for joining of Mg sheet, developed novel corrosion treatments for multi-metal assembly coatings, performed computer simulations of door panel forming using two new material cards based on crystal plasticity theory, and concluded with a door static and dynamic performance analysis. An overall cost driver and sensitivity assessment task compared the final cost penalty depending on the cost of the primary magnesium sheet.</div></div>

Study on oxidation-creep behavior of a Ni-based single crystal superalloy based on crystal plasticity theory

Long-term high temperature oxidation seriously affects the creep properties of the Ni-based single crystal superalloy. By using experimental research together with finite element simulation, the effects of high temperature oxidation on creep properties of a Ni-based single crystal superalloy (Ni-9.0Co-4.0Cr-5.7Al-7.0Ta-8.0W-2.2Re, wt.%) were studied. The specimens were pre-oxidized at 1100 °C for different periods of time, and then the pre-oxidized specimens were conducted by creep tests at 980 °C/270 MPa. By data fitting and analysis, the thickness evolution law and dynamic model of oxide layer/heat affected layer were established and the quantitative effect laws of pre-oxidation on creep life and deformation were established. The evolution and influence mechanism of oxide layer, heat-affected layer and substrate on creep failure were systematically studied by scanning electron microscope combined with energy dispersive spectrum analysis (SEM-EDS). Based on the thickening dynamic model of heat-affected layer and crystal plasticity theory, two oxidation-creep models were established. The initial damages caused by oxidation and microstructure evolution are both considered in the initial pre-oxidation damage model, which is fast and efficient. The distribution and evolution of stress, strain, resolved shearing stress and strain of the substrate and the heat-affected layer are obtained by another model, the two-phase model. The two models are compared and analyzed, which is also the uniqueness and innovation of this study. The simulations results of the two models are in good agreement with the creep tests.

A multiscale constitutive model for metal forming of dual phase titanium alloys by incorporating inherent deformation and failure mechanisms

Ductile metals undergo a considerable amount of plastic deformation before failure. Void nucleation, growth and coalescence is the mechanism of failure in such metals. α–β titanium alloys are ductile in nature and are widely used for their unique set of properties such as specific strength, fracture toughness, corrosion resistance and resistance to fatigue failures. Voids in these alloys have been reported to nucleate on the phase boundaries between α and β phase. Based on the findings of crystal plasticity finite element method investigations of the void growth at the interface of α and β phases, a void nucleation, growth, and coalescence model has been formulated. An existing single-phase crystal plasticity theory is extended to incorporate underlying physical mechanisms of deformation and failure in dual phase titanium alloys. Effects of various factors [stress triaxiality, Lode parameter, deformation state (equivalent stress), and phase boundary inclination] on void nucleation, growth and coalescence are used to formulate a phenomenological constitutive model while their interaction with a conventional crystal plasticity theory is established. An extensive parametric assessment of the model is carried out to quantify and understand the effects of the material parameters on the overall material response. Performance of the proposed model is then assessed and verified by comparing the results of the proposed model with the RVE study results. Application of the constitutive model for utilisation in the design and optimisation of the forming process of α–β titanium alloy components is also demonstrated using experimental data.

Crystal Plasticity Modeling of Grey Cast Irons under Tension, Compression and Fatigue Loadings

The study of the micromechanical performance of materials is important in explaining their macrostructural behavior, such as fracture and fatigue. This paper is aimed, among other things, at reducing the deficiency of microstructural models of grey cast irons in the literature. For this purpose, a numerical modeling approach based on the crystal plasticity (CP) theory is used. Both synthetic models and models based on scanning electron microscope (SEM) electron backscatter diffraction (EBSD) imaging finite element are utilized. For the metal phase, a CP model for body-centered cubic (BCC) crystals is adopted. A cleavage damage model is introduced as a strain-like variable; it accounts for crack closure in a smeared manner as the load reverses, which is especially important for fatigue modeling. A temperature dependence is included in some material parameters. The graphite phase is modeled using the CP model for hexagonal close-packed (HCP) crystal and has a significant difference in tensile and compressive behavior, which determines a similar macro-level behavior for cast iron. The numerical simulation results are compared with experimental tensile and compression tests at different temperatures, as well as with fatigue experiments. The comparison revealed a good performance of the modeling approach.

Effect of loading path on grain misorientation and geometrically necessary dislocation density in polycrystalline aluminum under reciprocating shear

Solid phase processing (SPP) is a promising alloy fabrication technique to produce fine and homogeneous grain structures for high-performance alloys. However, there is very limited modeling capability to understand and predict the grain refinement during SPP. In this work, the crystal plasticity theory was used to study elastic–plastic deformation in polycrystalline aluminums under large shear deformation. Two approaches, kernel averaged misorientation (KAM) and grain reference orientation deviation (GROD), were used to assess the grain misorientations. The geometrically necessary dislocation (GND) density was computed with the plastic strain rate. The deformation simulations were carried out under two loading conditions to investigate the effect of loading paths on the evolutions of grain misorientation and GND density. The results show that the regions with high misorientation and GND density first appear near grain boundaries. These regions then extend toward interior grains. The loading path affects plastic deformation accumulation and recovery, hence dislocation evolution and misorientation. Both two- and three-dimensional simulations showed that the spatial and temporal evolutions of GROD, KAM, and GND density are closely correlated, which indicates they all can be used as criteria of grain refinement or recrystallization, the essential information for grain refinement simulations.

Continuum modeling of dislocation channels in irradiated metals based on stochastic crystal plasticity

As a common feature observed in irradiated metallic materials, the formation of dislocation channels has been extensively studied and is considered to play a key role in irradiation embrittlement. However, modeling dislocation channels with the conventional crystal plasticity theory has been a theoretical challenge due to the difficulty of capturing microstructural inhomogeneities. Here a continuum crystal plasticity framework incorporating a stochastic distribution model of critical resolved shear stress (CRSS) is developed to describe the formation of dislocation channels and further plastic flow localization in irradiated materials. We show that the stochastic model is capable of capturing the heterogeneity of microscale plastic strain, which is an inherent feature of metallic materials during plastic deformation. It acts as an important microscale perturbation to trigger the dislocation channel nucleation in irradiated metals, especially for single crystals that lack mesoscale perturbations such as intergranular incompatibility and grain anisotropy. Without predetermining the potential nucleation position of dislocation channels, the stochastic irradiation crystal plasticity framework successfully simulates the dislocation channel formation and plasticity localization in both irradiated single- and polycrystalline copper (Cu), and further indicates the irradiation defect density threshold for the dislocation channel formation. This stochastic model broadens the application of the conventional crystal plasticity framework, and might provide new insights for other studies on plasticity localization, including shear bands formation of metals, mechanical behaviors with heterogeneous deformation and so on.

Numerical analysis of subgrain formation during metal cutting and rolling based on the crystal plasticity theory

The grain refinement technology is important in improving the metallic material properties without the requirement of additional alloy elements. Previously, we developed an efficient method for producing ultrafine-grained steel strips using a combination of cutting and heat treatment. However, the effect of cutting on recrystallization was not apparent. The objective of this study is to investigate the effects of metal cutting on static recrystallization and outline its advantages in grain refinement using numerical simulations based on the crystal plasticity theory. Simulation results show that shear deformation in metal cutting activates more slip systems than plane strain compression via rolling, even when considering the same equivalent plastic strain. The geometrically necessary dislocations are assumed to accumulate in the crystal because many slip systems are activated in shear deformation and improve grain refinement via static recrystallization in the subsequent heat treatment. This result indicates that the deformation type plays an important role in the recrystallization process. Thus, cutting is more efficient than rolling for the production of ultrafine-grained steel.

Polycrystal plasticity with grain boundary evolution: a numerically efficient dislocation-based diffuse-interface model

Grain structure plays a key role in the mechanical properties of alloy materials. Engineering the grain structure requires a comprehensive understanding of the evolution of grain boundaries (GBs) when a material is subjected to various manufacturing processes. To this end, we present a computationally efficient framework to describe the co-evolution of bulk plasticity and GBs. We represent GBs as diffused geometrically necessary dislocations, whose evolution describes GB plasticity. Under this representation, the evolution of GBs and bulk plasticity is described in unison using the evolution equation for the plastic deformation gradient, an equation central to classical crystal plasticity theories. To reduce the number of degrees of freedom, we present a procedure which combines the governing equations for each slip rates into a set of governing equations for the plastic deformation gradient. Finally, we outline a method to introduce a synthetic potential to drive migration of a flat GB. Three numerical examples are presented to demonstrate the model. First, a scaling test is used to demonstrate the computational efficiency of our framework. Second, we study the evolution of a tricrystal, formed by embedding a circular grain into a bicrystal, and demonstrate qualitative agreement between the predictions of our model and those of molecular dynamics simulations by Trautt and Mishin (2014 Acta Mater. 65 19–31). Finally, we demonstrate the effect of applied loading in texture evolution by simulating the evolution of a synthetic polycrystal under applied displacements.

The role of dissipation regarding the concept of purely mechanical theories in plasticity

The concept of a purely mechanical theory is revisited within the framework of continuum thermodynamics. It is based on three assumptions: (i) the additive split of the specific free energy into a mechanical contribution not depending on the temperature and a contribution that depends on the temperature only, (ii) a homogeneous and stationary temperature field, and (iii) a negligible heat supply. The consequences of these three assumptions on the dissipation inequality in form of the Clausius–Duhem inequality are discussed. It is shown, that these three assumption yield a vanishing dissipation. This result is valid for the classical Cauchy–Boltzmann continuum as well as for extended continua. Many plasticity theories consider the first two assumptions tacitly, however the role of the heat supply is not discussed. Here, the consequences of a vanishing dissipation with respect to a local crystal plasticity theory are discussed, exemplarily. The paper does not favor a purely mechanical theory but highlights the implications of such an approach.

A duality‐based coupling of Cosserat crystal plasticity and phase field theories for modeling grain refinement

AbstractHigh‐rate deformation processes of metals entail intense grain refinement and special attention needs to be paid to capture the evolution of microstructure. In this article, a new formulation for coupling Cosserat crystal plasticity and phase field is developed. A common approach is to penalize kinematic incompatibility between lattice orientation and displacement‐based elastic rotation. However, this can lead to significant solution sensitivity to the penalty parameter, resulting in low accuracy and convergence rates. To address these issues, a duality‐based formulation is developed which directly imposes the rotational kinematic compatibility. A weak inf‐sup‐based skew‐symmetric stress projection is introduced to suppress instabilities present in the dual formulation. An additional least squares stabilization is introduced to suppress the spurious lattice rotation with a suitable parameter range derived analytically and validated numerically. The required high‐order continuity is attained by the reproducing kernel approximation. It is observed that equal order displacement‐rotation‐phase field approximations are stable, which allows efficient employment of the same set of shape functions for all independent variables. The proposed formulation is shown to yield superior accuracy and convergence with marginal parameter sensitivity compared to the penalty‐based approach and successfully captures the dominant rotational recrystallization mechanism including block dislocation structures and grain boundary migration.

Simulation of crystal plasticity of P92 heat resistant steel considering martensitic lath structure

Based on the theory of crystal plasticity, coupled with dislocation and lath hardening models, this paper establishes a crystal plasticity finite element model describing the high temperature creep of P92 steel. Open source software was used to generate lath models with an average size of 350nm, 650nm and 950nm to explore the effect of lath coarsening on the high-temperature creep behavior of P92 steel. The results show that the roughening of the slats increases the rate of creep deformation, resulting in a decrease in the service life of the material. Observing the slat model, it can be seen that the roughening of the slats enlarges the numerical gradient of stress and strain, and aggravates the overall plastic strain of the model. The coarsening of the slats accelerates the movement of dislocations, causing the density of movable dislocations to increase, and at the same time the shear strain amplitude of the slip system increases, thereby reducing the hardening behavior of the material.

Strain Localization Modes within Single Crystals Using Finite Deformation Strain Gradient Crystal Plasticity

The present paper aims at providing a comprehensive investigation of the abilities and limitations of strain gradient crystal plasticity (SGCP) theories in capturing different kinds of localization modes in single crystals. To this end, the small deformation Gurtin-type SGCP model recently proposed by the authors, based on non-quadratic defect energy and the uncoupled dissipation assumption, is extended to finite deformation. The extended model is then applied to simulate several single crystal localization problems with different slip system configurations. These configurations are chosen in such a way as to obtain idealized slip and kink bands as well as general localization bands, i.e., with no particular orientation with respect to the initial crystallographic directions. The obtained results show the good abilities of the applied model in regularizing various kinds of localization bands, except for idealized slip bands. Finally, the model is applied to reproduce the complex localization behavior of single crystals undergoing single slip, where competition between kink and slip bands can take place. Both higher-order energetic and dissipative effects are considered in this investigation. For both effects, mesh-independent results are obtained, proving the good capabilities of SGCP theories in regularizing complex localization behaviors. The results associated with higher-order energetic effects are in close agreement with those obtained using a micromorphic crystal plasticity approach. Higher-order dissipative effects led to different results with dominant slip banding.

A numerical study on influence of strain gradients on lattice rotation in micro-machining of a single crystal

In latest years small scale machining has been widely used in advanced engineering applications such as medical and optical devices, micro- and nano-electro-mechanical systems. In micromachining of metals, a depth of cut becomes usually smaller than an average crystal size of a polycrystalline structure; thus, the cutting process zone can be localized fully indoors of a single grain. Due to the crystallographic anisotropy, development of small scale machining models accounting for crystal plasticity are essential for a precise calculation of material removal under such circumstances. For this purpose, a 3D finite-element model of micro-cutting of a single grain was developed. A crystal-plasticity theory accounting for gradients of strain, implemented in ABAQUS/Explicit via a user-defined material subroutine VUMAT, was used in the computations. The deformation-induced lattice rotations in micro-cutting of a single crystal were analyzed extensively.

Anisotropic constitutive model coupled with damage for Sn-rich solder: Application to SnAgCuSb solder under tensile conditions

With the rapid development of microelectronics and nanoelectronics, Moore law has significantly slowed down and More than Moore based system in packaging (SiP) is expected to be more and more important, at least for next one to two decades. Mechanical behaviors of interconnect materials such as solders are critical for yield in processes and reliability in testing and operation. Based on the framework of crystal plastic theory and continuum damage mechanics, an anisotropic constitutive model coupled with damage was developed to describe the deformation behaviors of Sn-rich solder. In the proposed model, the inelastic shear rate function was presented by hyperbolic sinusoidal form and power law form. For the damage evolution law, the total shear strain was chosen as the damage function variable. The proposed model was implemented into the general finite element software ABAQUS by forward Euler integration procedure. Some simulation examples were performed to verify the proposed model by comparing the simulation results with the experiments at uniaxial tensile conditions with SnAgCuSb solder chosen as the Sn-rich solder. The tensile stress-strain curves of the simulation results agreed well with the experiments at small strain under different temperatures and strain rates. The simulated stress-rupture stages showed reasonable accuracy with the experiments under four representative tensile conditions. Different tensile stress-strain curves of single grains with orientation of (0-0-0)°, (0-45-0)°, and (0-90-0)° were obtained under the same loading conditions, with an inverse relationship between the tensile strength and elongation. This relationship was in accordance with a referable literature. All these results indicate that the proposed model can describe the deformation behaviors of SnAgCuSb solder well under the tensile conditions in consideration of the mechanical anisotropy and the damage evolution.

Study on creep properties of nickel-based superalloy blades based on microstructure characteristics

Creep experiments were performed on miniature K480 nickel-based superalloy specimens at 835 °C and 400 MPa. These specimens were cut from turbine blades processed with different casting processes and pouring temperatures. The microstructure analysis, fracture morphology analysis, and grain size analysis were carried out by scanning electron microscopy (SEM), indicating that the influence of the moulding process and pouring temperature on the microstructure of the alloy is mainly reflected in the difference in the size of crystal grains, grain boundary precipitates and casting defects, which in turn affects the high temperature creep properties of the alloy. Based on crystal plasticity theory, a polycrystalline alloy viscoplastic constitutive equation considering the moulding process and pouring temperature parameters is established, meanwhile, the geometric models of the K480 alloy were established based on the observations of its microscopic structure under SEM. The study revealed the relationship between the casting process of the blade, the microstructure of the material, and the creep property of the specimen. The established plastic constitutive model can predict blade stress rupture life under different casting processes.

Creep life prediction for a nickel-based single crystal turbine blade

The life of turbine blades often determines the life of the aero-engine, and creep rupture is one of the main fracture forms of turbine blades in high temperature environments. In order to accurately predict the life of turbine blades, creep tensile tests of smooth specimens were carried out according to typical loads of 760 °C/800 MPa, 900 °C/445 MPa, 980 °C/400 MPa, 1000 °C/283 MPa and 1050 °C/210 MPa. With the help of crystal plasticity theory, creep constitutive equations with temperature interpolation were constructed. The FSI heat transfer analysis was performed to obtain the creep life of the turbine blades,. The temperature difference on the blade had reached 300 °C. With creep constitutive model and temperature boundary, the creep analysis of the blade was carried out and the creep process was revealed. The dangerous section of the global-model was obtained, and the sub-model was used to re-analyze the dangerous area. This paper reveals the deformation of the blade after creep failure: the middle part of the blade shrinks which is also the potential damage area, the top part expands, and the trailing edge of the blade tip produces a radial displacement of 0.313 mm after creep failure. The creep life of the turbine blades is determined to be 91 h. Finally, a method to determine the creep life of turbine blades was given based on skeletal point stress method.

A Crystal Plasticity Simulation on Strain-Induced Martensitic Transformation in Crystalline TRIP Steel by Coupling with Cellular Automata

In transformation-induced plasticity (TRIP) steel, the strain-induced martensitic transformation (SIMT) has a close relationship with the shear band formation. At a small length scale such as that of a crystal, the explicit analysis of the shear band structure with the formed microstructure is quite important for an adequate understanding of the SIMT. Here, a study on the microstructures formed by SIMT, related to shear band formation in both single and polycrystal TRIP steels, is presented. The constitutive equation for single crystal TRIP steel considering the transformation strain on each variant system is derived based on a rate-dependent crystal plasticity theory. To express the martensitic transformation, the cellular automata approach, including a transformation criterion acting as a local rule, is introduced. Numerical simulation is conducted with patterning processes of the martensitic phase at an infinite medium under the plane strain tension. It is found that the similar distributions of the plastic strain and the martensitic phase are dependent on the initial crystal orientation and appear as the shear band structures. In addition, the sizes of embryo and cell strongly influence the shear band formation and the martensitic volume fraction of crystal TRIP steel.

Applying crystallographic orientation analysis for exploring deformation mechanism of 6.5 wt.% Si electrical strip during twin-roll thin strip casting process

The study described here is aimed at illuminating deformation mechanism involved in Twin-roll Thin Strip Casting Process (TRSC) with respect to high-permeability 6.5 wt.% Si electrical strip (6.5 wt.% Si strip) in terms of crystal plasticity (CP) theory, and a Visco-plastic Self-consistent model (VPSC) was utilized to explore the deformation mechanism by applying crystallographic orientation (OR) analysis. Results underscore that the activation of overall potential single slip system with regard to studied 6.5 wt.% Si strip including { 110 } < 111 > , { 112 } < 111 > as well as { 123 } < 111 > can contribute to the OR conversion from Cube texture to Rotated-Cube texture, α-fiber texture as well as γ-fiber texture, while with different evolution rate during deformation. The potential multi-slip system of { 112 } < 111 > + { 123 } < 111 > with average number of activated slip system per grain ~9.1–9.4, and multi-slip system of { 110 } < 111 > + { 112 } < 111 > + { 123 } < 111 > with average number of activated slip system per grain ~10.4–10.8 are identified to be the predominant slip systems during deformation by comparing to the actual OR features obtained from TRSC-processed 6.5 wt.% Si strip. The CP-based method proposed in present study is deemed as a reference to investigate deformation behavior involved in the other semi-solid forming process.

A microstructure-based modeling approach to predict the mechanical properties of Zr alloy with hydride precipitates

In nuclear reactors, hydrides can form in fuel cladding due to hydrogen absorption in Zircaloy and cause embrittlement. This work presents a microstructure-based finite element model to predict the stress–strain response of Zircaloy containing hydrides. Microstructural details extracted from scanning electron microscopy (SEM) images were used to generate heterogeneous microstructures including the morphology and spatial distribution of hydrides. The constitutive material model for Zircaloy in this study is based on crystal plasticity theory which considers the hexagonal close-packed (HCP) atomic structure of Zircaloy material. The hydrides were modeled as brittle material along with a damage model. Hydride formation inside the Zircaloy matrix results in residual stress. This phenomenon is also captured in this model. A parametric study has been conducted to understand the effect of volume fraction, orientation, and lamellae width of the hydride phase on the mechanical properties of the overall material and the findings were validated against experimental results from the literature.