An Integrated Crystal Plasticity-Phase Field Model For The Plastic Deformation Of Magnesium

The plastic deformation mechanisms of Magnesium(Mg)alloys with the hexagonal close packed(hcp)crystal structure include basal,prismatic,pyramidal slip and tension,compression,secondary twinning,et al.The interactions of these deformation modes with grain boundaries,and precipitates are substantially complex.Therefore,better understanding of the complicated deformation mechanisms and their interactions at the meso-scale is critical to improve the strength and ductility of Mg alloys.However,the traditional,experimental-based alloy development workflows are difficult to dynamically capture the deformation process,and quantitatively describe the relationship between the macro-mechanical behavior and microstructures.In this work,based on the advantages of crystal plasticity(CP)method to describe dislocation slip and phase field(PF)method to describe the evolution of complex microstructures,an integrated crystal plasticity-phase field model has been proposed for concurrent modeling of dislocation plasticity and heterogeneous twinning behavior including twin nucleation,propagation,and growth process and their interaction with grain boundaries and precipitates for hcp metals in a finite strain framework.The coupled model has been employed to in-situ and quantitatively study the Mg single crystal,bicrystal,and polycrystal plastic deformation.Furthermore,the full-field coupled CP-PF model has been employed to investigate the strengthening mechanisms resulting from the interaction between twin growth and precipitates in Mg alloys and systematically investigate the influence of key microstructural parameters such as precipitate orientation,volume fraction,size,and aspect ratio on the critical resolved shear stress(CRSS)of twin growth.The perspectives of the further development of the current material model were also discussed.The developed material model for the plastic deformation of Mg alloys is composed of the crystal plasticity model to describe dislocation slip,the stochastic model to describe twinning nucleation at grain boundaries,the phase field model to describe the twinning propagation and growth,and the parallel finite element solver.In the constitutive model,the plastic deformation gradient tensor contains the contributions from both twinning-induced shear and dislocation slip arising in the parent grain and the twinned region.A dislocation-density based crystal plasticity model is employed to describe the slip plasticity,where the flow rule related to dislocation slip is described by the classic Orowan equation;the hardening model is based on the mechanisms of dislocation multiplication and annihilation;the interaction strength between different dislocations is described by the interaction matrix;hardening due to the obstacle of twin boundaries to dislocation motion is considered by modifying the mean free path of dislocation slip.Based on the experimental results that tensile twin nucleation prefers at grain boundaries and assuming that the nucleation process is a Poisson process,a physics-based stochastic nucleation model is employed here to describe the influence of the heterogeneous distribution of grain boundary defects on the twin nucleation threshold.The propagation and growth of tensile twin is described by the phase field model based on Ginzburg-Landau theory,which contains the crystalline energy,interface energy,and elastic strain energy.A double obstacle function is used to describe the potential energy landscape when an original parent crystal is sheared into a twin crystal orientation.A symmetric second-order tensor is introduced to describe the anisotropic twin interface energy,i.e.the free energy difference of coherent twin boundaries and basal-prismatic interfaces.The relaxation of the elastic strain energy provides the driving force for the twin growth,which is the multiplication of the resolved shear stress on the twin plane and the characteristic shear strain.In terms of the numerical implementation of the proposed material model,the stress equilibrium equation and Ginzburg-Landau equation are converted to their equivalent integral forms.The finite element(FE)formulations of these two differential equations are derived by constructing the shape functions and discrete differential operators based on the types of the finite elements.The coupled model subjected to applied boundary conditions is then solved by an in-house large-scale parallel FE code which makes use of routines provided by the PETSc numerical library.By transforming the Ginzburg-Landau equation to the form of a variational inequality,only degrees of freedom at the twin interfaces need to be solved while points at the equilibrium positions in the parent grain or in the fully twinned region are fixed.The number of degrees of freedom in the phase field model for which a solution is sought is expected to be relatively small in comparison to the total number of discretization points.The simulations of the Mg single crystal plastic deformation show that the nucleated twin propagates quickly into the grain interior along both the twinning direction and lateral direction and subsequently grows in the thickness direction,where the CRSS of twin nucleation,basal slip,prismatic slip,and pyramidal slip are set to 17 MPa,15MPa,73 MPa,and 115 MPa,respectively.Twinning propagation is a non-stable plastic deformation process,which causes an abrupt stress drop from 46 MPa to 3MPa.The shear stress substantially concentrates ahead of the twin tips(37MPa)and is relaxed near the transverse twin boundaries(16MPa)which therefore facilitates twin propagation and suppresses twin growth in thickness.This stress re-distribution behavior is critical for the formation of the elliptical morphology of tensile twins.Profuse basal <a> dislocations are activated near the twin boundaries during the twin growth stage to accommodate the local strain compatibility.The simulations of the Mg bi-crystal plastic deformation indicate that in the case of an initial twin with a high nominal Schmid factor of 0.5,with increasing the grain boundary misorientation angle,the nominal Schmid factor of the twin in the other grain decreases while the ability of the basal slip to accommodate the initial twinning induced shear increases,and therefore causes the decrease of the propensity of twin transmission across the grain boundary.No transmission is expected for misorientation angles exceeding approximately 50°.Twin transmission can also occur in the case even with a relatively small nominal Schmid factor but a high twin-twin geometric compatibility factor(>0.75).Strain compatibility,quantified by the geometric compatibility factor,is a better indicator to predict whether a twin is transmitted than the nominal Schmid factor.The simulations of the Mg polycrystal plastic deformation show that the shear stress distributes heterogeneously within the twins and the parent grains.The stress substantially concentrates near the intersections of the twins and the grain boundaries.The distance between twin bands within one grain plays an important role in the twinning behavior,i.e.the overlap of the stress relaxation zone near the transverse twin boundary could suppress twin propagation.The orientations of the neighboring grains have a profound influence both on the development of twin chains and the evolution of back-stress in the twinned grain.Profuse basal <a> dislocations are observed to accommodate the twinning induced heterogeneous strain even though most of the grains have relatively low nominal Schmid factors for basal slip(0.24).No non-basal slip are observed during the twin growth stage.The simulations of the interaction of precipitates with the tensile twin,where the CRSS of twin growth,basal slip,prismatic slip,and pyramidal slip are set to 28 MPa,15MPa,73 MPa,and 115 MPa,respectively,show that local plastic relaxation by basal,prismatic,and pyramidal slip occurs as a precipitate enters a twin.Even once the precipitate is fully embedded in the twin and no longer in contact with the twin boundary,increase in CRSS(6MPa)is required for continued growth due to the plastic relaxation and redistribution of stresses.The model predicts that a high volume fraction of small,high aspect ratio,shear-resistant plate-shaped precipitates are desirable for both maximum strengthening,and decreasing the difference of the CRSS between tensile twinning and prismatic slip,which could improve the yield strength and decrease the yield asymmetry.The simulations of the interaction of the tensile twin with plate-shaped precipitates with different habit planes show that the decrease of the angle between the habit plane and the twinning plane causes the increase of the area of twin boundary that has to overcome the precipitate,and therefore results in the higher CRSS for twin growth.For the plate precipitates with a volume fraction of 4% and an aspect ratio of 10,precipitates with their habit plane parallel to the twinning plane are predicted to provide the maximum strengthening effect(61MPa)and trigger extensive basal and non-basal slip;those perpendicular the minimum effect(22MPa).Plates with basal or prismatic habit planes provide a similar(median)level of strengthening.There is approximately a doubling of strengthening between the orientations that give the maximum and minimum effects.The simulations of the interaction of the tensile twin with sphere precipitates with different volume fractions show that increasing precipitate volume fraction at a constant precipitate size is predicted to produce a proportional increase in the CRSS increment for twin growth.The CRSS for twin growth with 8% precipitates is 105 MPa,more than 3 times higher than that without precipitates(29MPa).Increasing volume fraction is predicted not only to increase the unrelaxed back-stress but also increase the dislocation activity required to sustain twin growth.The simulations of the interaction of the tensile twin with sphere precipitates with different sizes show that comparing the same volume fraction,smaller and more finely spaced precipitates give higher strengthening.The twin growth CRSS increases from 56 MPa to 71 MPa,i.e.it is 54% higher when the precipitate size decreases from 20 to 6 elements.For a fixed volume fraction,the effect of precipitate size falls into two regimes.Below a critical size,Orowan-like behavior is predicted where a reduction in size(and spacing)produces a strong increase in CRSS increment.Above this size,the CRSS increment is size independent(Eshelby-like behavior).The simulations of the interaction of the tensile twin with plate-shaped precipitates with different aspect ratios show that increasing the aspect ratio is predicted to proportionally increase the CRSS increment for twin growth.The twin growth CRSS increases from 56 MPa to 83 MPa when increasing the plate aspect ratio from 1 to 16 at the same precipitate size and volume fraction.The strong effect of aspect ratio on plastic relaxation behavior and the level of unrelaxed backstress is important in explaining this behavior.Although the proposed material model has gained the success in studying the relationship between the macro-mechanical behavior and microstructures in Mg alloys,it still can be improved in the following perspectives: incorporating multiple tensile twin variants,compression twinning and secondary twinning modes;considering the influence of the different dislocation activities and the underlying physics of grain boundaries on the twinning nucleation threshold;explicitly describing the pile-up of dislocations near twin boundaries,et al.