The Role of Intergranular Stresses in Plastic Deformation Studied Using a Diffraction and Self-Consistent Model

Diffraction methods are commonly used for the determination of the elastic lattice deformation from the displacement and broadening of the diffraction peak. The measurements are performed selectively, only for crystallites contributing to the measured diffraction peak. When several phases are present in the sample, measurements of separate diffraction peaks allow the behaviour of each phase to be investigated independently [e.g. 1-4]. Comparison of experimental data with a multi-scale model allows us to understand the physical phenomena which occur during sample deformation at the level of polycrystalline grains. In the present work the methodology combining diffraction experiment and self-consistent calculation was used to study the mechanical behaviour of groups of grains within stainless duplex steel and Al/SiC composite. Special attention has been paid to the role of second order stresses on the yield stresses of the phases, as well as on the evolution of these stresses during the deformation process. The intergranular stresses were determined from lattice strains measured “in situ” during tensile tests. The diffraction measurements were done using synchrotron (ID15B, ESRF, Grenoble, France) and neutron (EPSILON, FLNP, JINR, Dubna, Russia) radiations. Experimental methodology To study the elastoplastic behavior of two phase materials the diffraction measurements were performed “in situ” during a tensile test. Two materials, i.e., an aged duplex steel UR45N (50% of ferrite and 50% of austenite; microstructure, texture and composition are given in [3,4]) and a particle reinforced Al/SiCp composite (Al2124 matrix with 17% of SiC particles with size of 0.7 μμ subjected to the T1 treatment ) were studied using diffraction methods of two types. The Al/SiCp Residual Stresses 2016: ICRS-10 Materials Research Forum LLC Materials Research Proceedings 2 (2016) 551-556 doi: http://dx.doi.org/10.21741/9781945291173-93 552 composite was obtained by powder metallurgy processing followed by the T1 thermal treatment (air cooled from elevated temperature forming process). For the steel specimen monochromatic synchrotron radiation with an energy of about 90keV (λ = 0.14256Å) and a beam size of 100 × 100 μμ2 was applied. The measurements were carried out at the European Synchrotron Radiation Facility in Grenoble on the ID15B beamline. The diffraction pattern in the range of 2θ = 1.7 − 7.5° was collected on two-dimensional detector PIXIUM4700 in the form of concentric rings corresponding to different hkl reflections for both phases: ferrite and austenite. The necessary conversion of two-dimensional images into 2θ diffractograms was performed with FIT2D software [5]. The peak positions were determined using the MULTIFIT software [6] and after that Braggs’ law was employed to determine the interplanar spacings dhkk. The second experiment was performed for the particle reinforced Al/SiCp composite specimen using the time-of-flight (TOF) neutron diffraction method enabling simultaneous measurement of different hkl reflections for both phases in the studied composite. For data acquisition two of nine detector banks covering a 2θ-range of 82°≤2θ≤98° were employed on the EPSILON-MDS diffractometer at the Joint Institute For Nuclear Research, Dubna, Russia [7]. A geometry of this kind allowed to determine stress tensor if the lattice strains are measured for two sample orientations with respect to the experimental setup. The measurement was performed for the initial material and after deformation of the sample during a tensile test. The incident beam of 10 mm width was pointed at the sample of 4.4 mm x 4.4 mm cross-section. Evolution of deviatoric stresses in duplex steel during plastic deformation Initial state of the specimen The analysis of the initial stresses and of the evolution of the lattice parameter for both phases was performed for the duplex steel measured using synchrotron radiation. Firstly the stress tensor was determined for the initial non-deformed sample. The principal stresses were decomposed into two parts: hydrostatic (p) and deviatoric (q, r, s), according to Eq. 1. The deviatoric stresses were determined directly from the measured lattice strains, while the additional assumption p = 0 (zero values of hydrostatic stresses) was introduced in order to calculate the initial values of the lattice parameter a0 for both phases. The results are presented in Table 1. � σ11 0 0 0 σ22 0 0 0 σ33 � = � p 0 0 0 p 0 0 0 p � + � q 0 0 0 r 0 0 0 s � (1) Table 1. Initial stresses and lattice parameters determined for both phases of the studied steel assuming zero value of hydrostatic stresses. The parameters used in the self-consistent model are also presented. Austenite Ferrite Initially measured values (q) σRR[MMa] 134 ± 15 -155 ± 19 (r) σTR[MMa] 84 ± 15 -44 ± 19 (s) σNR[MMa] -218 ± 15 199 ± 18 a0 [Å] 3.6102 ± 0.0001 2.8791 ± 0.0001 Single crystal elastic. constants c11, c12, c44[GMa] 198 , 125 , 122 231, 134, 116 Slip systems 〈111〉{110} and 〈111〉{211} 〈110〉{111} Parameters of Voce law τ0 [MMa] 170 370


Experimental methodology
To study the elastoplastic behavior of two phase materials the diffraction measurements were performed "in situ" during a tensile test.Two materials, i.e., an aged duplex steel UR45N (50% of ferrite and 50% of austenite; microstructure, texture and composition are given in [3,4]) and a particle reinforced Al/SiC p composite (Al2124 matrix with 17% of SiC particles with size of 0.7  subjected to the T1 treatment ) were studied using diffraction methods of two types.The Al/SiC p composite was obtained by powder metallurgy processing followed by the T1 thermal treatment (air cooled from elevated temperature forming process).For the steel specimen monochromatic synchrotron radiation with an energy of about 90keV ( = 0.14256Å) and a beam size of 100 × 100  2 was applied.The measurements were carried out at the European Synchrotron Radiation Facility in Grenoble on the ID15B beamline.The diffraction pattern in the range of 2 = 1.7 − 7.5° was collected on two-dimensional detector PIXIUM4700 in the form of concentric rings corresponding to different hkl reflections for both phases: ferrite and austenite.The necessary conversion of two-dimensional images into 2θ diffractograms was performed with FIT2D software [5].The peak positions were determined using the MULTIFIT software [6] and after that Braggs' law was employed to determine the interplanar spacings  ℎ .
The second experiment was performed for the particle reinforced Al/SiC p composite specimen using the time-of-flight (TOF) neutron diffraction method enabling simultaneous measurement of different hkl reflections for both phases in the studied composite.For data acquisition two of nine detector banks covering a 2θ-range of 82°≤2θ≤98° were employed on the EPSILON-MDS diffractometer at the Joint Institute For Nuclear Research, Dubna, Russia [7].A geometry of this kind allowed to determine stress tensor if the lattice strains are measured for two sample orientations with respect to the experimental setup.The measurement was performed for the initial material and after deformation of the sample during a tensile test.The incident beam of 10 mm width was pointed at the sample of 4.4 mm x 4.4 mm cross-section.

Evolution of deviatoric stresses in duplex steel during plastic deformation Initial state of the specimen
The analysis of the initial stresses and of the evolution of the lattice parameter for both phases was performed for the duplex steel measured using synchrotron radiation.Firstly the stress tensor was determined for the initial non-deformed sample.The principal stresses were decomposed into two parts: hydrostatic () and deviatoric (, , ), according to Eq. 1.The deviatoric stresses were determined directly from the measured lattice strains, while the additional assumption  = 0 (zero values of hydrostatic stresses) was introduced in order to calculate the initial values of the lattice parameter  0 for both phases.The results are presented in Table 1.As illustrated in Fig. 1 and presented in Table 1, the initial deviatoric stresses acting in ND direction on austenite grains are compressive, while the stresses acting on the ferrite grains are tensile.Meanwhile the initial stresses in direction RD are tensile for austenite grains and compressive for the ferrite ones.The absolute values in both cases (for ND and RD directions) are approximately equal but they have opposite signs for each phase, therefore the specimen containing the same fractions of ferrite and austenite is in a state of equilibrium.The initial stresses were generated during cooling of the material, after aging process (details of the thermal treatment are given in [3,4]).The deviatoric type of stress state would be explained probably due to various interaction of the grains in different directions caused by the ellipsoidal shape of inclusions (see Fig. 1) and the difference in thermal expansion coefficients of austenite and ferrite.The initial deviatoric stresses (shown in Table 1) and the previously measured crystallographic textures [3,4] were used as the input data for elasto-plastic self-consistent model [8].The calculation were performed for the single crystal elastic constants and slip systems given in Table 1.Additionally, the ellipsoidal shape of inclusions corresponding to microstructure presented in [3,4]:   ⁄ = 5,   ⁄ = 10 (the a and c axes are defined in Fig. 1, while b axis is parallel to TD direction) was assumed for both phases.The aspect ratios of inclusions were determined from the EBSD orientation maps shown in [3,4].

Lattice parameter evolution
Due to the small range of deformation (about 17 %) and the almost linear character of hardening in the plastic range of deformation (Fig. 2) only two parameters of Voce law (Eq.2) were adjusted for each phase of the studied steel.The model results were fitted simultaneously to the macroscopic stress-strain curve (Fig. 2), as well as to the dependence of lattice parameters (〈〉 ℎ ) measured vs. applied stress (Σ 11 ) for different hkl reflections (Fig. 3).The values of  0 and  0 obtained by trial and error optimization are shown in Table 1 for both phases of the studied steel.The information about the value of the lattice parameter was obtained for the experimental data directly using Eq. 3 and for the model data according to Eq. 4.
〈〉 ℎ = 〈〉 ℎ √ℎ 2 +  2 +  2 (3) where  0 is the initial lattice parameter calculated from the initial sample (Table 1), 〈〉 ℎ is the lattice strain measured using diffraction and 〈〉 ℎ  is the model predicted lattice strain.Figs. 2 and 3 show a very good agreement of model and experimental data both for the macroscopic and microscopic/grain states.We can distinguish three different stages, the elasticity of both phases (before Γ point), the elasticity of ferrite and the plasticity of austenite (between Γ and Ω points) and the plasticity of both phases (after Ω point).

The lattice parameter for initial and deformed sample
The dependence of the lattice parameter measured for different hkl direction vs. orientation factor 4 for the initial (a and c) and the deformed (a and c) samples (where the deformed/residual state is after unloading of the applied stress).Analysing the obtained results we can see a good agreement between experimental and model results.
We can conclude that important phase deviatoric stresses exist in the initial sample (observed large difference between the lattice parameter in RD and ND directions), and that the initial stresses do not depend significantly on grain orientations (linear dependence of 〈〉 ℎ vs. 3).On the other hand, for the majority of grain orientations, the residual stresses decreased in the deformed and unloaded sample, i.e. for many hkl reflections in each phase the 〈〉 ℎ values are similar in RD and ND directions, and they approach the value of the initial  0 parameter.It means that in general the deviatoric stresses decrease and an insignificant evolution of hydrostatic stresses was observed.However, the residual stresses after plastic deformation depend strongly on grain orientations and the 〈〉 ℎ vs. 3 dependence is no longer linear (significant deviations were observed for the 200 reflection in both phases).

Thermal stresses in Al/SiC p composite during plastic deformation
In the previous section the evolution of deviatoric stress was studied, but insignificant changes of hydrostatic stress were measured and predicted using a self-consistent model.The second experiment described in this section will show a significant evolution of hydrostatic type of stresses.The experiment was performed using TOF diffraction and the measurement was performed for the Al/SiC p composite subjected to the T1 treatment.It is well known that a significant difference in the thermal coefficient of aluminium and silicon carbide (Table 2) causes the appearance of initial thermal stresses between those phases in the composite.In stress analysis the method allowing determination of stress tensor as well as c/a parameter was applied [9], assuming single crystal elastic constants for the 6H polytype of SiC [10] (Rietveld analysis of X-ray diffraction results shown about 80% of this polytype in the SiC powder).3 contains information about the thermal hydrostatic stresses in SiC (denoted by  in Eq. 1) and about the average response of the matrix Al for both stages: for the initial and the deformed specimen (tensile test).In the case of the Al matrix we have calculated the hydrostatic component of the average stress tensor seen by the diffraction experiment due to averaging over many grains, although local stresses in the matrix are not of purely hydrostatic nature.The main problem in evaluating the absolute  values is that the measurement performed on the reference stress free powder specimen is not reliable.That is why we show also a relative value Δ which is the difference between the value for the deformed specimen and for its initial state.We can conclude that an important increase of hydrostatic stress occurred in the SiC reinforcement corresponding to the relaxation of initially large compressive thermal stress.Also, the decrease of the hydrostatic part of average stress in the Al matrix has been observed as a consequence of stress relaxation of the stress around SiC inclusions.
As seen in Table 3 the values of Δ are not correctly predicted by the self-consistent model in which the hydrostatic stresses do not influence plastic behavior and do not change during plastic deformation.Therefore to explain the relaxation of the hydrostatic stresses occurring in the Al/SiC p composite the heterogeneity of the stresses in the matrix around reinforcement particles must be taken into account.In Table 3 the deviatoric stresses are presented as well.In this case we can observe the evolution which is qualitatively (but not quantitatively) predicted by the self-consistent model.

Fig. 1 .
Fig. 1.The initial deviatoric stresses acting on the austenite and ferrite grains in the initial sample.

Fig. 2 .
Fig. 2. The dependence of macrostress   vs. macrostrain   for experimental data and model predictions.

Fig. 3 .
Fig. 3.The evolution of lattice parameter during the in-situ tensile test of duplex steel for different hkl reflections for a-c) ferrite d-f) austenite.

Fig. 4 .
Fig. 4. The measured lattice parameter as a function of the orientation factor 3 for a) ferrite in the initial state, b) ferrite in the residual state, c) austenite in the initial state and d) austenite in the residual state.

Table 1
. Initial stresses and lattice parameters determined for both phases of the studied steel assuming zero value of hydrostatic stresses.The parameters used in the self-consistent model are also presented.

Table 2 .
Material data of Al/SiC p composite.

Table 3 .
Stress values of phases in Al/SiC composite in initial state and after deformation.Experimental (exp) and model predicted values (mod) are shown.