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Grain orientation dependence of lattice strains and intergranular damage rates in polycrystals under cyclic loading
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Scripta Materialia 68 (2013) 265–268 www.elsevier.com/locate/scriptamat
Grain orientation dependence of lattice strains and intergranular damage rates in polycrystals under cyclic loading L.L. Zheng,a Y.F. Gao,a,b,⇑ Y.D. Wang,c A.D. Stoica,d K. And and X.L. Wange a
Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USA Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA c School of Materials Science and Engineering, Beijing Institute of Technology, Beijing 100081, China d Chemical and Engineering Materials Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA e Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China b
Received 16 October 2012; accepted 23 October 2012 Available online 26 October 2012
Neutron diffraction experiments show that lattice strains in polycrystals under cyclic loading critically depend on the crystallographic orientations of diffracted grains, which can be explained by our crystal plasticity simulations and a micromechanical analysis based on slip anisotropy and the Taylor model. Experiments also show that the residual lattice strains gradually vanish with increasing number of fully reversed loading cycles. The corresponding decay rate correlates quantitatively with the grain-orientation-dependent total cumulative slip strain and qualitatively with grain boundary damage processes. Ó 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Lattice strain; Cyclic loading; Neutron diffraction; Taylor analysis
In spite of significant advances in studies of fracture and fatigue failures, previous works typically rely on ex situ microstructural characterizations and fractography using optical and electron microscopes, and crack growth monitoring with replica techniques, among many others [1]. A direct relationship between macroscopic properties and microscopic mechanisms, however, remains elusive. Some of these difficulties can be addressed via in situ neutron diffraction measurement, which has become an easily accessible method due to a number of user-friendly facilities worldwide. These nondestructive and deep-penetrating measurements provide the unprecedented information on: (i) lattice strains (also called Type-II strains), which arise to accommodate the deformation incompatibility of neighboring grains in polycrystals and thus rely on intergranular interactions and grain-level deformation anisotropy; (ii) intragranular strains (also called Type-III strains) from the line profile analysis that provides the information on dislocation densities; and (iii) texture evolution from peak intensity development [2–7]. The above knowledge can clearly be
⇑ Corresponding
author at: Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USA. Tel.: +1 865 974 2350; fax: +1 865 974 4115; e-mail: [email protected]
applied to gain a fundamental understanding of failure mechanisms on the microstructural scale. For fatigue cracks, the plastic field in front of the crack tip and the plastic wake left behind essentially provide the resistance to the fatigue crack growth. Neutron diffraction techniques have been extensively utilized to understand the dependence of the surrounding plastic field on the loading history and materials properties from the measurements of lattice strain distributions along the crack plane [5,6]. In the immediate vicinity of the fatigue crack tip, there exists a “messy” process zone in which various grain-boundary and/or grain-interior mechanisms contribute to the intrinsic fatigue damage processes. It is believed that many of these damage processes, such as grain boundary decohesion or cavitation, critically depend on the heterogeneous, residual intergranular stress/strain field developed in the polycrystal. Therefore, the measurements of intergranular stress/strain fields under cyclic loading conditions will shed light on understanding the underlying fatigue damage processes. Prior experimental works along these lines, e.g. Ref. [3], are extended and investigated here by the crystal plasticity theory and Taylor-type analysis. The experimental intergranular data were obtained on a stainless steel specimen using the newly commissioned VULCAN diffractometer [8] at the Spallation Neutron Source (SNS) [9], Oak Ridge National Laboratory
1359-6462/$ - see front matter Ó 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.scriptamat.2012.10.033
266
L. L. Zheng et al. / Scripta Materialia 68 (2013) 265–268
(ORNL). The 316 stainless steel (face-centered cubic (fcc) structure) specimen was under uniaxial, fully reversed loading cycles. A set of specimens was obtained after interrupting the test at the end of various loading cycles. The measurement procedure essentially followed that in Ref. [3], except that with only 90° detectors, the specimen was rotated about the vertical axis of the sample table in order to have complete access of strains in grains of different orientations. The high data rate of VULCAN permits a more detailed data set than the earlier study [3]. To eliminate possible strain errors due to partial burial of the sampling volume, each specimen was carefully centered within the neutron beam [10,11]. As shown by the representative data in Figure 1a, when the angle h formed between the diffraction vector and the loading axis varies from 0° to 90°, the residual lattice strains at the end of the first loading cycle change sign twice. The h k l lattice strain can be regarded as the average strain of all grains with their h k l lattice planes perpendicular to the diffraction vector. They are obtained by fitting the individual peaks on the neutron diffraction patterns using the General Structure Analysis System (GSAS) software. Lattice strains can be interpreted using micromechanical models such as the crystal plasticity model and the self-consistent viscoplasticity model. In the latter, a grain is embedded in an effective homogeneous medium and the overall mechanical response of this composite is consistent with this effective medium [2]. In complex boundary value problems under variable loading history, it is more appropriate to use the crystal plasticity model [6,12–17]. The slip-based crystal plasticity theory describes the Schmid law, slip anisotropy, self- and latent-hardening and material flow. The elastic part of the deformation gradient arises from lattice stretching, rotating and rigid body motion. The plastic deformation is a consequence of shearing on a family of slip systems, which can be calculated from the following summation: p1 ¼ F_ pik F kj
NSLIP X
ðaÞ ðaÞ c_ ðaÞ si mj
ð1Þ
a¼1
Figure 1. Residual lattice strains at the end of a fully reversed loading cycle with respect to the angle h between the uniaxial loading axis and the diffraction vector for a fcc stainless steel: (a) neutron diffraction experiments, (b) crystal plasticity simulations and (c–f) finite element setup and grains that satisfy diffraction conditions. Contours of r33 along the loading axis at the end of a full loading cycle on the entire specimen in (c) and on the selected grains (i.e. their h k l planes are perpendicular to the diffraction vector direction h) in (d–f).
where F pij is the plastic part of the total deformation gradient F ij ¼ @xi =@X j , c_ ðaÞ is the slip strain rate on the ath slip system and s(a) and m(a) are the corresponding slip direction and slip plane normal vectors. Crystal plasticity models distinguish from one another in terms of specific forms used in the flow and hardening equations. Following the Peirce–Asaro–Needleman (PAN) model [12], the plastic shear strain rate, c_ ðaÞ , is given as a function of the resolved shear stress sðaÞ : sðaÞ n ðaÞ c_ ¼ c_ 0 ðaÞ sgnðsðaÞ Þ ð2Þ s flow
ðaÞ sflow
where is the flow strength, c_ 0 is a reference strain rate and n is the stress exponent. The hardening equaP ðaÞ tions are given by s_ flow ¼ b hab j_cðbÞ j, where hab are the hardening moduli, haa ¼ h0 sech2 jh0 c=ðss s0 Þj (no summation R t Pover a), h0 is the initial hardening modulus, c ¼ 0 a j_cðaÞ jdt is the total cumulative plastic strain and ss and s0 are the saturated and initial yield stresses, respectively. The latent hardening moduli are given by hab ¼ qhðcÞ; ða–bÞ, where q is the latent hardening coefficient. The single crystal plasticity theory has been implemented into the finite element software ABAQUS through the user defined material (UMAT) subroutine by Huang [13], which is further modified for our neutron analysis [6]. When predicting the lattice strains, one can construct a polycrystalline material by an aggregate of a number of single crystal grains, which are assigned with a set of constitutive models and orientations from the material texture information. The geometric setup in Figure 1c has 5324 cubic grains, each of which consists of eight eight-node elements. Material constants for the fcc stainless steel include c11 = 204.6 GPa, c12 = 137.7 GPa, c44 = 126.2 GPa, n = 10, c_ 0 = 0.001 s1, h0 = 205 MPa, s0 = 87 MPa, ss =140 MPa and q = 1.0. The specimen is subjected to uniaxial, fully reversed loading cycles. The diffraction condition selects a family of specific orientated grains with the crystallographic plane normal parallel to the direction of diffraction vector as shown in Figure 1d–f. A number of grains which satisfy the diffraction condition, typically 2–5% out of all the grains, will be chosen. And lattice strains are the projected elastic strains (in the diffraction vector direction) averaged over these selected grains. These selected grains are certainly different with respect to different h k l and h. In Figure 1a and b, which plots lattice strains at the end of a fully reversed loading cycle vs. h, neutron diffraction measurements and crystal plasticity finite element method give essentially the same trend. Residual lattice strains, e100 and e311 , change their signs from negative to positive and then to negative again as h increases, while e110 and e331 display the opposite trend. All these curves cross over zero strain at h 30° and 75°. The relationship between the residual lattice strain and h in Figure 1 can be interpreted by a Taylor analysis [15–17]. Although it is highly inhomogeneous on the grain scale, the strain field can be approximated by the faraway applied macroscopic strain field. The resulting stresses on the neighboring grains are thus different because of the elastic and plastic anisotropy. Asaro and
L. L. Zheng et al. / Scripta Materialia 68 (2013) 265–268
Needleman [15] have found that the Taylor assumption can nicely predict the hardening behavior and texture evolution from the crystal plasticity constitutive model. Wong and Dawson [17] have used the Taylor approximation to understand the splitting of different lattice strains on the stress vs. lattice strain plots during loading. We now apply this method to understand the lattice strain evolution in cyclic loading with varying h. Since Figure 1 shows that lattice strains e100 and e110 have the extreme variations, in Figure 2, we construct crystal plasticity finite element simulations of two grains in parallel, one of which has h1 0 0i direction and the other of which has h1 1 0i direction parallel to the diffraction vector (being h = 0°, 10° and 40°, respectively). Due to the uncertainty of the orientations in the h k l planes, we randomly choose ten cases and the lattice strains are calculated by taking their average. The macroscopic strain history obtained from Figure 1b is used as inputs for these calculations. When only two grains are subjected to the same strain fields (i.e. the Taylor assumption), the load partitioning behavior will be governed by their respective yield strain. That is, the grain with larger yield strain will yield later, so that it will experience larger elastic strain when the other grain yields. Wong and Dawson [17] have shown that the yield strain is approximately inversely proportional to the product of the Schmid factor and directional modulus. Therefore, it can be easily demonstrated that the {1 0 0} grain is a “hard” grain when h = 0°. Indeed, in Figure 2a σ 33 2 τ0
ε100 ε110
0
θ = 0ο
-2
(a) -0.002
0.000
ε hkl
0.002
σ 33 2 τ0 0
θ = 10ο
-2
(b) -0.002
0.000
ε hkl
267
when h = 0°, the {1 0 0} grain yields later, which results in a negative residual lattice strain at the end of a fully reversed loading cycle. The two-grain simulations in Figure 2b and c show that the h = 10° case is similar to h = 0° but with the decrease of the absolute value of the lattice strains, while for h = 40°, both strains change their signs as opposed to results in Figure 2a and finally the residual lattice strains e100 and e110 become positive and negative, respectively. These trends agree nicely with the polycrystal simulation results in Figure 1b, which further confirms the validity of the Taylor assumption in analyzing polycrystal plasticity. A similar analysis to Wong and Dawson [17] for h = 10° and 40° cannot be easily done, because the inplane orientations in the chosen h k l planes are random so that a statistical average is needed. Refer to the inverse pole figure in Figure 3 and consider the {1 1 0} grains as an example. In Figure 3, all the crystallographic orientations that make an angle h with h1 1 0i direction will form circle-like contours about the 110 vertex. In the diffraction analysis, for any chosen grain (i.e. its {1 1 0} plane normal is parallel to the diffraction vector, or equivalently, makes an angle of h with the loading direction), its crystallographic orientation of this grain in the loading direction will be located on the h contour around the 1 1 0 vertex. The distribution density will clearly depend on the texture. Simulations in Figure 2 suggest that the grain-orientation dependence of the residual lattice strain can be characterized 110 contours 100 contours and ðS ijk Eijk =c11 Þave , by ðS ijk Eijk =c11 Þave where the average is conducted over the h contours about 110 and 100 vertices, respectively. Here S ijk and Eijk are Schmid factor and directional modulus when the hi j ki direction is parallel to the loading direction. Results in Figure 3b nicely predict the transition of lattice strain as a function of h. This simple model can be easily generated to consider other kinds of slip systems (thus changing Schmid factor contours in the inverse pole figure), elastic anisotropy (thus changing directional modulus contours in the inverse pole figure) and texture (thus requiring a weighted average over the h contours).
Sijk Eijk c11
0.002
(a)
σ 33 2 τ0
100 contours
(Sijk Eijk / c11) ave or
θ = 40ο
0
110 contours
(Sijk Eijk /c11) ave
0.5 0.4 0.3 0.2 0.1 0.0
-2
(c) -0.002
0.000
ε hkl
0.002
Figure 2. The applied stress vs. lattice strain curves from our Taylor model with varying angle h. See text for model construction.
(b)
<100> <110> 0 10 20 30 40 50 60 70 80 90
( θ ο)
Figure 3. (a) Contours of S ijk Eijk =c11 in the inverse pole figure with hi j ki being the uniaxial loading direction. (b) The lattice strain contours evolution is governed by the comparison between ðS ijk Eijk =c11 Þ110 ave contours and ðS ijk Eijk =c11 Þ100 , where the average is conducted over the h ave contours in (a). See text for model construction.
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L. L. Zheng et al. / Scripta Materialia 68 (2013) 265–268
The predicted residual lattice strains for all the orientations are given in the standard triangle in the inverse pole figure in Figure 4a. These polycrystal finite element simulation results agree with the measured lattice strains in Figure 1a. In addition, this plot is very similar to the inverse pole figure of residual stresses in Wang et al. [3], which is constructed from all the lattice strain data obtained by the spherical harmonic analysis from their neutron diffraction experiments [18,19]. It is hypothesized that such residual stress or residual lattice strain maps can be correlated to fatigue damage processes. It was found by Wang et al. [3] that with the increase of cyclic loading numbers, the residual lattice strains gradually vanish, and this is accompanied by the appearance of intergranular cracks. The intergranular damage may start from void nucleation or slip-incompatibility-induced micro-crack nucleation on the grain boundaries, followed by growth and coalescence of these defects into macroscopic cracks [20–22]. The growth rates of voids or micro-cracks are qualitatively governed by the total slip strain as summed from all the slip systems. The total cumulative slip strain is plotted in the inverse pole figure in Figure 4b. The ratio of the total cumulative slip strain to the macroscopic von Mises strain by the finite element simulation is denoted as the macromechanical Taylor factor by Raabe et al. [16], which differs slightly from the Taylor factor in the homogenization theory in which the total cumulative slip strain is determined by assuming that each grain is subject to the same macroscopic strain. The maximum values are near {1 1 1} and {1 1 0} grains, while a low value is found near {1 0 0} grains. Therefore, it is anticipated that the decay rate of the residual lattice strain is higher for {1 1 1} and {1 1 0} grains, while that for {1 0 0} is lower. Indeed, the same trend has been reported in Wang et al. [3], i.e. within the same number of cycles, the residual lattice strain for {1 0 0} grains decays by 30%, while those for {1 1 0} and {1 1 1} grains are 55% and 64%, respectively. In summary, a crystal plasticity simulation has been performed to elucidate the evolution of grain-orienta-
Figure 4. (a) Contours of residual lattice strains and (b) total cumulative slip strains plotted on the standard triangle in the inverse pole figure, based on crystal plasticity simulations of the fcc stainless steel. These results correspond to h = 0°.
tion-dependent lattice strains in polycrystals, which compares well with the neutron diffraction experiments on a fcc stainless steel. A Taylor-type analysis (i.e. equal strain assumption for all grains) explains well the observed lattice strain vs. diffraction vector relationship. These grain-orientation-dependent residual lattice strains decay in magnitude with increasing fully reversed loading cycles. The corresponding decay rate is found to correlate well with the Taylor factor, indicating that the total cumulative slip strain dominates the intergranular damage process and thus eventually leads to the vanishing residual lattice strain. This work was supported by the US National Science Foundation CMMI 0800168 and a graduate fellowship from the Joint Institute for Neutron Sciences at the University of Tennessee (LLZ and YFG), the National Natural Science Foundation of China (NSFC) under contract No. 51231002 (YDW) and the Scientific User Facilities Division, Office of Basic Energy Sciences, US Department of Energy, at Oak Ridge National Laboratory (ADS and KA). [1] S. Suresh, Fatigue of Materials, Cambridge University Press, Cambridge, UK, 1998. [2] B. Clausen, T. Lorentzen, T. Leffers, Acta Mater. 46 (1998) 3087. [3] Y.D. Wang, H. Tian, A.D. Stoica, X.L. Wang, P.K. Liaw, J.W. Richardson, Nat. Mater. 2 (2003) 101. [4] A.A. Korsunsky, K.E. James, M.R. Daymond, Eng. Fract. Mech. 71 (2004) 805. [5] R.I. Barabash, Y.F. Gao, Y. Sun, S.Y. Lee, H. Choo, P.K. Liaw, D. Brown, G.E. Ice, Philos. Mag. Lett. 88 (2008) 553. [6] L.L. Zheng, Y.F. Gao, S.Y. Lee, R.I. Barabash, J.H. Lee, P.K. Liaw, J. Mech. Phys. Solids 59 (2011) 2307. [7] K. An, H.D. Skorpenske, A.D. Stoica, D. Ma, X.L. Wang, E. Cakmak, Metall. Mater. Trans. A 42 (2011) 95. [8] X.-L. Wang, T.M. Holden, G.Q. Rennich, A.D. Stoica, P.K. Liaw, H. Choo, C.R. Hubbard, Physica B 385–386 (2006) 673. [9] T.E. Mason, D. Abernathy, J. Ankner, A. Ekkebus, G. Granroth, M. Hagen, K. Herwig, C. Hoffmann, C. Horak, F. Klose, S. Miller, J. Neuefeind, C. Tulk, X.-L. Wang, in: I. Hoffmann et al. (Eds.), High Intensity and High Brightness Hadron Beams, American Institute of Physics, Melville, NY, 2005, pp. 21–25. [10] S. Spooner, X.-L. Wang, J. Appl. Crystallogr. 30 (1997) 449. [11] X.-L. Wang, Y.D. Wang, J.W. Richardson, J. Appl. Crystallogr. 35 (2002) 533. [12] D. Peirce, R.J. Asaro, A. Needleman, Acta Metall. 30 (1982) 1087. [13] Y. Huang, Mechanics Report 179, Division of Engineering and Applied Science, Harvard University, 1991. [14] A.F. Bower, E. Wininger, J. Mech. Phys. Solids 52 (2004) 1289. [15] R.J. Asaro, A. Needleman, Acta Mater. 33 (1985) 923. [16] D. Raabe, M. Sachtleber, Z. Zhao, F. Roters, S. Zaefferer, Acta Mater. 49 (2001) 3433. [17] S.L. Wong, P.R. Dawson, Acta Mater. 58 (2010) 1658. [18] Y.D. Wang, R.L. Peng, R.L. McGreevy, Philos. Mag. Lett. 81 (2001) 153. [19] W.D. Wang, R.L. Peng, X.-L. Wang, R.L. McGreevy, Acta Mater. 50 (2002) 1717. [20] A. Needleman, J. Appl. Mech. 54 (1987) 525. [21] V. Tvergaard, Adv. Appl. Mech. 27 (1989) 83. [22] L.C. Lim, R. Raj, Acta Metall. 32 (1984) 727.
Scripta Materialia 68 (2013) 265–268 www.elsevier.com/locate/scriptamat
Grain orientation dependence of lattice strains and intergranular damage rates in polycrystals under cyclic loading L.L. Zheng,a Y.F. Gao,a,b,⇑ Y.D. Wang,c A.D. Stoica,d K. And and X.L. Wange a
Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USA Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA c School of Materials Science and Engineering, Beijing Institute of Technology, Beijing 100081, China d Chemical and Engineering Materials Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA e Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China b
Received 16 October 2012; accepted 23 October 2012 Available online 26 October 2012
Neutron diffraction experiments show that lattice strains in polycrystals under cyclic loading critically depend on the crystallographic orientations of diffracted grains, which can be explained by our crystal plasticity simulations and a micromechanical analysis based on slip anisotropy and the Taylor model. Experiments also show that the residual lattice strains gradually vanish with increasing number of fully reversed loading cycles. The corresponding decay rate correlates quantitatively with the grain-orientation-dependent total cumulative slip strain and qualitatively with grain boundary damage processes. Ó 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Lattice strain; Cyclic loading; Neutron diffraction; Taylor analysis
In spite of significant advances in studies of fracture and fatigue failures, previous works typically rely on ex situ microstructural characterizations and fractography using optical and electron microscopes, and crack growth monitoring with replica techniques, among many others [1]. A direct relationship between macroscopic properties and microscopic mechanisms, however, remains elusive. Some of these difficulties can be addressed via in situ neutron diffraction measurement, which has become an easily accessible method due to a number of user-friendly facilities worldwide. These nondestructive and deep-penetrating measurements provide the unprecedented information on: (i) lattice strains (also called Type-II strains), which arise to accommodate the deformation incompatibility of neighboring grains in polycrystals and thus rely on intergranular interactions and grain-level deformation anisotropy; (ii) intragranular strains (also called Type-III strains) from the line profile analysis that provides the information on dislocation densities; and (iii) texture evolution from peak intensity development [2–7]. The above knowledge can clearly be
⇑ Corresponding
author at: Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USA. Tel.: +1 865 974 2350; fax: +1 865 974 4115; e-mail: [email protected]
applied to gain a fundamental understanding of failure mechanisms on the microstructural scale. For fatigue cracks, the plastic field in front of the crack tip and the plastic wake left behind essentially provide the resistance to the fatigue crack growth. Neutron diffraction techniques have been extensively utilized to understand the dependence of the surrounding plastic field on the loading history and materials properties from the measurements of lattice strain distributions along the crack plane [5,6]. In the immediate vicinity of the fatigue crack tip, there exists a “messy” process zone in which various grain-boundary and/or grain-interior mechanisms contribute to the intrinsic fatigue damage processes. It is believed that many of these damage processes, such as grain boundary decohesion or cavitation, critically depend on the heterogeneous, residual intergranular stress/strain field developed in the polycrystal. Therefore, the measurements of intergranular stress/strain fields under cyclic loading conditions will shed light on understanding the underlying fatigue damage processes. Prior experimental works along these lines, e.g. Ref. [3], are extended and investigated here by the crystal plasticity theory and Taylor-type analysis. The experimental intergranular data were obtained on a stainless steel specimen using the newly commissioned VULCAN diffractometer [8] at the Spallation Neutron Source (SNS) [9], Oak Ridge National Laboratory
1359-6462/$ - see front matter Ó 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.scriptamat.2012.10.033
266
L. L. Zheng et al. / Scripta Materialia 68 (2013) 265–268
(ORNL). The 316 stainless steel (face-centered cubic (fcc) structure) specimen was under uniaxial, fully reversed loading cycles. A set of specimens was obtained after interrupting the test at the end of various loading cycles. The measurement procedure essentially followed that in Ref. [3], except that with only 90° detectors, the specimen was rotated about the vertical axis of the sample table in order to have complete access of strains in grains of different orientations. The high data rate of VULCAN permits a more detailed data set than the earlier study [3]. To eliminate possible strain errors due to partial burial of the sampling volume, each specimen was carefully centered within the neutron beam [10,11]. As shown by the representative data in Figure 1a, when the angle h formed between the diffraction vector and the loading axis varies from 0° to 90°, the residual lattice strains at the end of the first loading cycle change sign twice. The h k l lattice strain can be regarded as the average strain of all grains with their h k l lattice planes perpendicular to the diffraction vector. They are obtained by fitting the individual peaks on the neutron diffraction patterns using the General Structure Analysis System (GSAS) software. Lattice strains can be interpreted using micromechanical models such as the crystal plasticity model and the self-consistent viscoplasticity model. In the latter, a grain is embedded in an effective homogeneous medium and the overall mechanical response of this composite is consistent with this effective medium [2]. In complex boundary value problems under variable loading history, it is more appropriate to use the crystal plasticity model [6,12–17]. The slip-based crystal plasticity theory describes the Schmid law, slip anisotropy, self- and latent-hardening and material flow. The elastic part of the deformation gradient arises from lattice stretching, rotating and rigid body motion. The plastic deformation is a consequence of shearing on a family of slip systems, which can be calculated from the following summation: p1 ¼ F_ pik F kj
NSLIP X
ðaÞ ðaÞ c_ ðaÞ si mj
ð1Þ
a¼1
Figure 1. Residual lattice strains at the end of a fully reversed loading cycle with respect to the angle h between the uniaxial loading axis and the diffraction vector for a fcc stainless steel: (a) neutron diffraction experiments, (b) crystal plasticity simulations and (c–f) finite element setup and grains that satisfy diffraction conditions. Contours of r33 along the loading axis at the end of a full loading cycle on the entire specimen in (c) and on the selected grains (i.e. their h k l planes are perpendicular to the diffraction vector direction h) in (d–f).
where F pij is the plastic part of the total deformation gradient F ij ¼ @xi =@X j , c_ ðaÞ is the slip strain rate on the ath slip system and s(a) and m(a) are the corresponding slip direction and slip plane normal vectors. Crystal plasticity models distinguish from one another in terms of specific forms used in the flow and hardening equations. Following the Peirce–Asaro–Needleman (PAN) model [12], the plastic shear strain rate, c_ ðaÞ , is given as a function of the resolved shear stress sðaÞ : sðaÞ n ðaÞ c_ ¼ c_ 0 ðaÞ sgnðsðaÞ Þ ð2Þ s flow
ðaÞ sflow
where is the flow strength, c_ 0 is a reference strain rate and n is the stress exponent. The hardening equaP ðaÞ tions are given by s_ flow ¼ b hab j_cðbÞ j, where hab are the hardening moduli, haa ¼ h0 sech2 jh0 c=ðss s0 Þj (no summation R t Pover a), h0 is the initial hardening modulus, c ¼ 0 a j_cðaÞ jdt is the total cumulative plastic strain and ss and s0 are the saturated and initial yield stresses, respectively. The latent hardening moduli are given by hab ¼ qhðcÞ; ða–bÞ, where q is the latent hardening coefficient. The single crystal plasticity theory has been implemented into the finite element software ABAQUS through the user defined material (UMAT) subroutine by Huang [13], which is further modified for our neutron analysis [6]. When predicting the lattice strains, one can construct a polycrystalline material by an aggregate of a number of single crystal grains, which are assigned with a set of constitutive models and orientations from the material texture information. The geometric setup in Figure 1c has 5324 cubic grains, each of which consists of eight eight-node elements. Material constants for the fcc stainless steel include c11 = 204.6 GPa, c12 = 137.7 GPa, c44 = 126.2 GPa, n = 10, c_ 0 = 0.001 s1, h0 = 205 MPa, s0 = 87 MPa, ss =140 MPa and q = 1.0. The specimen is subjected to uniaxial, fully reversed loading cycles. The diffraction condition selects a family of specific orientated grains with the crystallographic plane normal parallel to the direction of diffraction vector as shown in Figure 1d–f. A number of grains which satisfy the diffraction condition, typically 2–5% out of all the grains, will be chosen. And lattice strains are the projected elastic strains (in the diffraction vector direction) averaged over these selected grains. These selected grains are certainly different with respect to different h k l and h. In Figure 1a and b, which plots lattice strains at the end of a fully reversed loading cycle vs. h, neutron diffraction measurements and crystal plasticity finite element method give essentially the same trend. Residual lattice strains, e100 and e311 , change their signs from negative to positive and then to negative again as h increases, while e110 and e331 display the opposite trend. All these curves cross over zero strain at h 30° and 75°. The relationship between the residual lattice strain and h in Figure 1 can be interpreted by a Taylor analysis [15–17]. Although it is highly inhomogeneous on the grain scale, the strain field can be approximated by the faraway applied macroscopic strain field. The resulting stresses on the neighboring grains are thus different because of the elastic and plastic anisotropy. Asaro and
L. L. Zheng et al. / Scripta Materialia 68 (2013) 265–268
Needleman [15] have found that the Taylor assumption can nicely predict the hardening behavior and texture evolution from the crystal plasticity constitutive model. Wong and Dawson [17] have used the Taylor approximation to understand the splitting of different lattice strains on the stress vs. lattice strain plots during loading. We now apply this method to understand the lattice strain evolution in cyclic loading with varying h. Since Figure 1 shows that lattice strains e100 and e110 have the extreme variations, in Figure 2, we construct crystal plasticity finite element simulations of two grains in parallel, one of which has h1 0 0i direction and the other of which has h1 1 0i direction parallel to the diffraction vector (being h = 0°, 10° and 40°, respectively). Due to the uncertainty of the orientations in the h k l planes, we randomly choose ten cases and the lattice strains are calculated by taking their average. The macroscopic strain history obtained from Figure 1b is used as inputs for these calculations. When only two grains are subjected to the same strain fields (i.e. the Taylor assumption), the load partitioning behavior will be governed by their respective yield strain. That is, the grain with larger yield strain will yield later, so that it will experience larger elastic strain when the other grain yields. Wong and Dawson [17] have shown that the yield strain is approximately inversely proportional to the product of the Schmid factor and directional modulus. Therefore, it can be easily demonstrated that the {1 0 0} grain is a “hard” grain when h = 0°. Indeed, in Figure 2a σ 33 2 τ0
ε100 ε110
0
θ = 0ο
-2
(a) -0.002
0.000
ε hkl
0.002
σ 33 2 τ0 0
θ = 10ο
-2
(b) -0.002
0.000
ε hkl
267
when h = 0°, the {1 0 0} grain yields later, which results in a negative residual lattice strain at the end of a fully reversed loading cycle. The two-grain simulations in Figure 2b and c show that the h = 10° case is similar to h = 0° but with the decrease of the absolute value of the lattice strains, while for h = 40°, both strains change their signs as opposed to results in Figure 2a and finally the residual lattice strains e100 and e110 become positive and negative, respectively. These trends agree nicely with the polycrystal simulation results in Figure 1b, which further confirms the validity of the Taylor assumption in analyzing polycrystal plasticity. A similar analysis to Wong and Dawson [17] for h = 10° and 40° cannot be easily done, because the inplane orientations in the chosen h k l planes are random so that a statistical average is needed. Refer to the inverse pole figure in Figure 3 and consider the {1 1 0} grains as an example. In Figure 3, all the crystallographic orientations that make an angle h with h1 1 0i direction will form circle-like contours about the 110 vertex. In the diffraction analysis, for any chosen grain (i.e. its {1 1 0} plane normal is parallel to the diffraction vector, or equivalently, makes an angle of h with the loading direction), its crystallographic orientation of this grain in the loading direction will be located on the h contour around the 1 1 0 vertex. The distribution density will clearly depend on the texture. Simulations in Figure 2 suggest that the grain-orientation dependence of the residual lattice strain can be characterized 110 contours 100 contours and ðS ijk Eijk =c11 Þave , by ðS ijk Eijk =c11 Þave where the average is conducted over the h contours about 110 and 100 vertices, respectively. Here S ijk and Eijk are Schmid factor and directional modulus when the hi j ki direction is parallel to the loading direction. Results in Figure 3b nicely predict the transition of lattice strain as a function of h. This simple model can be easily generated to consider other kinds of slip systems (thus changing Schmid factor contours in the inverse pole figure), elastic anisotropy (thus changing directional modulus contours in the inverse pole figure) and texture (thus requiring a weighted average over the h contours).
Sijk Eijk c11
0.002
(a)
σ 33 2 τ0
100 contours
(Sijk Eijk / c11) ave or
θ = 40ο
0
110 contours
(Sijk Eijk /c11) ave
0.5 0.4 0.3 0.2 0.1 0.0
-2
(c) -0.002
0.000
ε hkl
0.002
Figure 2. The applied stress vs. lattice strain curves from our Taylor model with varying angle h. See text for model construction.
(b)
<100> <110> 0 10 20 30 40 50 60 70 80 90
( θ ο)
Figure 3. (a) Contours of S ijk Eijk =c11 in the inverse pole figure with hi j ki being the uniaxial loading direction. (b) The lattice strain contours evolution is governed by the comparison between ðS ijk Eijk =c11 Þ110 ave contours and ðS ijk Eijk =c11 Þ100 , where the average is conducted over the h ave contours in (a). See text for model construction.
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The predicted residual lattice strains for all the orientations are given in the standard triangle in the inverse pole figure in Figure 4a. These polycrystal finite element simulation results agree with the measured lattice strains in Figure 1a. In addition, this plot is very similar to the inverse pole figure of residual stresses in Wang et al. [3], which is constructed from all the lattice strain data obtained by the spherical harmonic analysis from their neutron diffraction experiments [18,19]. It is hypothesized that such residual stress or residual lattice strain maps can be correlated to fatigue damage processes. It was found by Wang et al. [3] that with the increase of cyclic loading numbers, the residual lattice strains gradually vanish, and this is accompanied by the appearance of intergranular cracks. The intergranular damage may start from void nucleation or slip-incompatibility-induced micro-crack nucleation on the grain boundaries, followed by growth and coalescence of these defects into macroscopic cracks [20–22]. The growth rates of voids or micro-cracks are qualitatively governed by the total slip strain as summed from all the slip systems. The total cumulative slip strain is plotted in the inverse pole figure in Figure 4b. The ratio of the total cumulative slip strain to the macroscopic von Mises strain by the finite element simulation is denoted as the macromechanical Taylor factor by Raabe et al. [16], which differs slightly from the Taylor factor in the homogenization theory in which the total cumulative slip strain is determined by assuming that each grain is subject to the same macroscopic strain. The maximum values are near {1 1 1} and {1 1 0} grains, while a low value is found near {1 0 0} grains. Therefore, it is anticipated that the decay rate of the residual lattice strain is higher for {1 1 1} and {1 1 0} grains, while that for {1 0 0} is lower. Indeed, the same trend has been reported in Wang et al. [3], i.e. within the same number of cycles, the residual lattice strain for {1 0 0} grains decays by 30%, while those for {1 1 0} and {1 1 1} grains are 55% and 64%, respectively. In summary, a crystal plasticity simulation has been performed to elucidate the evolution of grain-orienta-
Figure 4. (a) Contours of residual lattice strains and (b) total cumulative slip strains plotted on the standard triangle in the inverse pole figure, based on crystal plasticity simulations of the fcc stainless steel. These results correspond to h = 0°.
tion-dependent lattice strains in polycrystals, which compares well with the neutron diffraction experiments on a fcc stainless steel. A Taylor-type analysis (i.e. equal strain assumption for all grains) explains well the observed lattice strain vs. diffraction vector relationship. These grain-orientation-dependent residual lattice strains decay in magnitude with increasing fully reversed loading cycles. The corresponding decay rate is found to correlate well with the Taylor factor, indicating that the total cumulative slip strain dominates the intergranular damage process and thus eventually leads to the vanishing residual lattice strain. This work was supported by the US National Science Foundation CMMI 0800168 and a graduate fellowship from the Joint Institute for Neutron Sciences at the University of Tennessee (LLZ and YFG), the National Natural Science Foundation of China (NSFC) under contract No. 51231002 (YDW) and the Scientific User Facilities Division, Office of Basic Energy Sciences, US Department of Energy, at Oak Ridge National Laboratory (ADS and KA). [1] S. Suresh, Fatigue of Materials, Cambridge University Press, Cambridge, UK, 1998. [2] B. Clausen, T. Lorentzen, T. Leffers, Acta Mater. 46 (1998) 3087. [3] Y.D. Wang, H. Tian, A.D. Stoica, X.L. Wang, P.K. Liaw, J.W. Richardson, Nat. Mater. 2 (2003) 101. [4] A.A. Korsunsky, K.E. James, M.R. Daymond, Eng. Fract. Mech. 71 (2004) 805. [5] R.I. Barabash, Y.F. Gao, Y. Sun, S.Y. Lee, H. Choo, P.K. Liaw, D. Brown, G.E. Ice, Philos. Mag. Lett. 88 (2008) 553. [6] L.L. Zheng, Y.F. Gao, S.Y. Lee, R.I. Barabash, J.H. Lee, P.K. Liaw, J. Mech. Phys. Solids 59 (2011) 2307. [7] K. An, H.D. Skorpenske, A.D. Stoica, D. Ma, X.L. Wang, E. Cakmak, Metall. Mater. Trans. A 42 (2011) 95. [8] X.-L. Wang, T.M. Holden, G.Q. Rennich, A.D. Stoica, P.K. Liaw, H. Choo, C.R. Hubbard, Physica B 385–386 (2006) 673. [9] T.E. Mason, D. Abernathy, J. Ankner, A. Ekkebus, G. Granroth, M. Hagen, K. Herwig, C. Hoffmann, C. Horak, F. Klose, S. Miller, J. Neuefeind, C. Tulk, X.-L. Wang, in: I. Hoffmann et al. (Eds.), High Intensity and High Brightness Hadron Beams, American Institute of Physics, Melville, NY, 2005, pp. 21–25. [10] S. Spooner, X.-L. Wang, J. Appl. Crystallogr. 30 (1997) 449. [11] X.-L. Wang, Y.D. Wang, J.W. Richardson, J. Appl. Crystallogr. 35 (2002) 533. [12] D. Peirce, R.J. Asaro, A. Needleman, Acta Metall. 30 (1982) 1087. [13] Y. Huang, Mechanics Report 179, Division of Engineering and Applied Science, Harvard University, 1991. [14] A.F. Bower, E. Wininger, J. Mech. Phys. Solids 52 (2004) 1289. [15] R.J. Asaro, A. Needleman, Acta Mater. 33 (1985) 923. [16] D. Raabe, M. Sachtleber, Z. Zhao, F. Roters, S. Zaefferer, Acta Mater. 49 (2001) 3433. [17] S.L. Wong, P.R. Dawson, Acta Mater. 58 (2010) 1658. [18] Y.D. Wang, R.L. Peng, R.L. McGreevy, Philos. Mag. Lett. 81 (2001) 153. [19] W.D. Wang, R.L. Peng, X.-L. Wang, R.L. McGreevy, Acta Mater. 50 (2002) 1717. [20] A. Needleman, J. Appl. Mech. 54 (1987) 525. [21] V. Tvergaard, Adv. Appl. Mech. 27 (1989) 83. [22] L.C. Lim, R. Raj, Acta Metall. 32 (1984) 727.