On the mechanical behaviour of AA 7075-T6 during cyclic loading

International Journal of Fatigue 25 (2003) 267–281 www.elsevier.com/locate/ijfatigue

On the mechanical behaviour of AA 7075-T6 during cyclic loading Halit S. Turkmen, Roland E. Loge, Paul R. Dawson ∗, Matthew P. Miller Sibley School of Mechanical and Aerospace Engineering, Rhodes Hall, Cornell University, Ithaca, NY 14853, USA Received 25 January 2002; received in revised form 22 July 2002; accepted 8 October 2002

Abstract The mechanical behavior of an aluminum alloy during uniaxial cyclic loading is examined using finite element simulations of aggregates with individually resolved crystals. The aggregates consist of face centered cubic (FCC) crystals with initial orientations assigned by sampling the orientation distribution function (ODF) determined from the measured crystallographic texture. The simulations show that the (elastic) lattice strains within the crystals evolve as the number of cycles increases. This evolution is attributed to the interactions between grains driven by the local plasticity. Under constant amplitude strain cycles, the average (macroscopic) stress decays with increasing number of cycles in concert with the evolution of the lattice strains. Further, the average number of active slip systems also decreases with increasing cycles, eventually reaching zero as the material response becomes totally elastic at the grain level. During much of the cyclic history only a single slip system is activated in most grains. The simulation results are compared to experimental data for the macroscopic stress and for lattice strains in the unloaded state after 1, 30 and 1000 cycles.  2003 Elsevier Science Ltd. All rights reserved. Keywords: Lattice strains; Cyclic loading; Polycrystalline material; Neutron diffraction; Crystal plasticity

1. Introduction Many structural components are subjected to cyclic loading conditions throughout their service lives. They are designed so that the stress always remains in the elastic regime, except perhaps for an occasional overload. However, the stress may exceed the elastic limit locally in some crystals even if the macroscopic response is essentially elastic. This is a consequence of the anisotropic properties of the crystals comprising the metal. Anisotropy is observed in both the (elastic) stiffness and (plastic) strength of single crystals. The anisotropic properties result in heterogeneous responses of the stress and strain over the dimensions of a polycrystal when it is loaded. As the load increases in intensity, some crystals will yield before others. This situation is compounded by the presence of any pre-existing stress at the crystal level that can cause yielding to occur either earlier or later than expected for a stress-free material. Processing methods that invoke plastic flow, such as rolling

Corresponding author. Tel.: +1-607-255-3466; fax.: +1-607-2559410. E-mail address: [email protected] (P.R. Dawson). ∗

or forging, can leave the material with residual stresses that vary over the scale of crystals as well as over macroscopic dimensions. In the absence of a stress relief operation, the residual stresses from processing become the pre-existing stresses in service. The subsequent stress response at the crystal scale during cyclic loading is biased by the pre-existing state of stress. In this paper, we examine the uniaxial cyclic response of an aluminum alloy (AA 7075-T6 plate). The examination is based on finite element simulations of cyclically loaded polycrystals and coordinated experiments. The intent is to better understand the stress response at the crystal scale and to relate this local response to the macroscopic behavior. The results improve our understanding of the influence of grain interactions on the evolution of microstructural quantities such as the lattice strains and the slip system activity. We begin with a review of prior experimental observations of the micromechanical behavior during cyclic loading in Section 2. A summary of the equations of polycrystal elastoplasticity used in this study follows in Section 3. Section 4 presents the experimental work of the present study, which mainly includes cyclic tests and lattice strain measurements. The results of modeling of the cyclic tests are given in Section 5. A discussion on the sig-

0142-1123/03/$ - see front matter  2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0142-1123(02)00149-4

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nificance of the results is presented in Section 6, followed by conclusions in Section 7.

2. Background Variations of deformation over a polycrystal are substantial and give rise to regions that are much more highly stressed than others within the same aggregate. As has been shown by polycrystal plasticity simulations with each crystal represented by one or many finite elements, these variations arise due to the orientations of the individual crystals as well as the orientations of the crystals that surround it ([1], [2]). Other simulations have shown that the crystal stress distribution evolves significantly during small strain cyclic loading, consistent with the macroscopically-observed stress amplitude relaxation [3]. The residual stresses on the crystal level are a result of complicated couplings between orientation and misorientation distributions, single crystal elastic and plastic properties and external loading conditions. The relationship between the crystal stress distribution and fatigue life is an obvious one—more highly stressed crystals can serve as microcrack initiation sites (cf. [4], [5] and [6]). Seemingly identical fatigue specimens may have very different fatigue lives due to small differences in extremal crystal stress values. Phenomena like decreased life under nonzero cycle-averaged macroscopic stress (mean stress) conditions ([7], [8], [9] and [10]) can also be explained from the perspective of the crystal stress distribution. Under a tensile mean stress, a crystal or group of crystals may experience tension during the entire cycle—a condition that is known to promote the nucleation of microcracks. It is important to experimentally validate crystal stress prediction methodologies such as those described above. While it is currently not generally feasible to measure lattice strains point by point in a polycrystal, neutron and high energy X-ray diffraction techniques represent methods for determining lattice strain distributions within the aggregate [11]. One can then compare the predicted distributions to the experimental results (cf. [6], [12] and [13]). Over the past 20–30 years, many fatigue-related studies have been conducted on AA 7075-T6. The resulting body of empirical knowledge related to establishing relationships between loading conditions and fatigue life or fatigue crack growth rates, including environmental effects is vast (c.f. [7], [14], [15], [16], [17], [18]). One consistent observation is that fatigue cracks tend not to propagate in a stable manner in these high strength alloys. Instead, the number of cycles it takes to initiate a fatigue crack (initiation life) is a significant fraction of total life. It is understood that fatigue cracks typically initiate at grain boundaries, often near particles and that corrosion processes can play a large role [19]. A funda-

mental understanding of the initiation process, however, is elusive—not all particles or grain boundaries initiate microcracks. Even though the 7000 series alloys are precipitation hardened and are more complicated than pure metals in many respects, microplasticity processes like the formation of persistent slip bands (cf. [20], [21] and [22]) are sensitive to crystallographic orientation and can play a large role in microcrack propagation. For instance, electron channeling pattern studies on 7000 series aluminum have shown that the plastic zone around the tip of a microcrack varies with crystal orientation and proximity to the grain boundary [16]. This emphasizes both important aspects of the role of the single crystal in fatigue crack initiation. Our current understanding indicates that under nominally uniform macroscopic stress, there is considerable spatial variation in the stress when examined at the level of the crystals. This occurs due to both elastic and plastic anisotropies—some crystals may be “stiffer” than others and some may be more favorably oriented for yielding (lower Schmid or Taylor factors) than others. Spatial heterogeneities imply that there exist regions of the polycrystal with a lower critically-resolved shear strength (such as the precipitate free zones in each crystal). In addition, prior processing episodes may leave the material in a pre-stressed condition. Basically, one can argue that developing an understanding of the micromechanical loading environment on the crystal scale is of primary importance to unraveling the mysteries of fatigue crack initiation in alloys like AA 7075-T6. 3. Finite element polycrystals Polycrystal elastoplasticity is a microstructural approach for modeling the mechanical behavior of crystalline solids. Models of this type have two basic parts: single crystal equations and grain interaction relations. The macroscopic properties are derived from the collective responses of crystals comprising an aggregate and are strongly influenced by both the single crystal behavior and the grain interaction assumptions. In particular, the intensity of the mechanical anisotropy of the polycrystal depends on the single crystal anisotropy and on the methodology for averaging this behavior over the distribution function for the lattice orientations (the crystallographic texture). In this work we employ a finite element formulation to link the single crystal responses together to form a polycrystal. The equations for a single crystal and the finite element formulation are summarized in the following paragraphs; a more detailed description may be found in [23,24]. 3.1. Single crystal response The deformation of a crystal is a combination of elastic and plastic contributions. The elastic strains are

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related to the stretching of atomic bonds while the plastic strains result from slip between atomic planes that is accomplished by movement of dislocations. To formulate this in a framework that properly accounts for large strains and rotations, the motion of a deforming crystal is idealized as a superposition of plastic straining, rotation of the material associated with the lattice reorientation, and elastic straining. Based on this description, the kinematics of crystal deformation is represented using the multiplicative decomposition of the deformation gradient, F [25,26,27] (1) F ⫽ V∗·R∗·Fp ⫽ V∗·Fˆp ∗

p

where F is the purely plastic part of F, R is the lattice rotation, and V∗ is the symmetric left elastic stretch tensor. The deformation gradient Fˆp defines the relaxed configuration Bˆ obtained by unloading elastically, but without rotation, from the current configuration B to a stressfree state. The constitutive equations are formulated in the relaxed configuration, Bˆ . It is convenient to write equations for slip in terms of rate of deformation. To this end, the expression for the deformation gradient is differentiated in time and separated into its symmetric and skew parts. The symmetric part, which includes the elastic and plastic rates of deformation, is related to the stress through constitutive equations for the elastic and plastic responses. The elastic strains depend linearly on the crystal stress according to a stiffness tensor that reflects cubic crystal symmetry: s ⫽ £ : e∗

(2)

where £ is the fourth-order elastic stiffness tensor [28], and the (small) elastic strain is given by: e∗ ⫽ V∗⫺I.

(3)

The plastic deformation occurs via dislocation motion through the crystal lattice on a limited number of slip systems. The complex motion of dislocations is idealized such that the plastic portion of the deformation rate is a linear combination of the slip on the contributing slip systems: ˆp ⫽ D

冘

ˆ a )S ⫽ g˙ a(sˆa丢m

a

冘

g˙ aPˆa

(4)

a

where the subscript S denotes the symmetric part of the ˆ a, and sˆa and m ˆ a are the Schmid orientation tensor sˆa丢m slip direction and slip plane normal in the configuration, Bˆ, respectively. For the aluminum alloy modeled here we use the 12 {111}具110典 slip systems. The shearing rate on the a-slip system, g˙ a, is related to the corresponding resolved shear stress, ta, by a power law of the form:

冉冊

g˙ a ⫽ g˙ a0

ta ga

1/m

sgn(ta)

(5)

where ga is the strength of the a-slip system, g˙ a0 is a

269

constant reference shear rate, and m is the rate sensitivity of slip, which is taken to be small (0.005) for AA 7075T6 at room temperature. The resolved shear stress ta is the plastic work rate conjugate to g˙ a and is computed as the projection of the deviatoric portion of crystal stress, s⬘ onto the a-slip system: ta ⫽ s⬘:Pˆa.

(6)

Equations (4), (5) and (6) are combined to obtain a relation for the viscoplastic response of a crystal: s⬘ ⫽

冋冘 a

g˙ a ˆ a ˆ aT P P ta

册

⫺1

ˆ p. D

(7)

All slip systems within a crystal are assumed to have the same strength (ga=g). This is a simplification of the actual case for crystals in which each slip system might have a strength that is distinct from that for other slip systems. Our attention in this paper is on the influence of crystal interactions on the stress response, and for this purpose, the simpler representation of a single slip system strength for each crystal suffices. The strength evolves with straining according to an equation of the form: g˙ ⫽ h0

冉

冊

gs(g˙ )⫺g a g˙ gs(g˙)⫺g0

(8)

where gs(g˙ ) = gs0(g˙ / g˙ s0)m⬘ and g˙ = ⌺a|g˙ a| is the total shearing rate over all of the slip systems. The parameters appearing in the evolution equation for the slip system strength (h0, gs0, g0, g˙ s0, m and m⬘) were chosen to match the macroscopic stress response of the aluminum, as discussed in Section 5. From the skew part of the velocity gradient we obtain an equation for the rate of lattice ˙ ∗: reorientation, R

冘

ˆ p⫺ ˙ ∗ ⫽ (W R ˆ p⫺ (W

冘

ˆ a)A)·R∗ ⫽ g˙ a(sˆa丢m

a

(9)

ˆ a)·R∗. g˙ aQ

a

The subscript A denotes the skew part of the Schmid orientation tensor. Equations (8) and (9) provide the relations necessary to evolve the variables that quantify the material state, namely the strength and orientation of the crystals. 3.2. Polycrystal response A finite element formulation links the single crystals together to form a polycrystal. Elements have a one-toone association with crystals. The influences of neighboring grains and grain shape thus enter directly since the topology of the aggregate is defined by the mesh. The formulation fulfills local compatibility exactly and imposes local equilibrium in a weak sense. The defor-

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mation rate can vary crystal-to-crystal and, in addition, the intracrystal variations in deformation rate are determined by the trial functions for the velocity. There are a number of other approaches available for linking the crystals together [29]. Ones that are commonly used include the Taylor and Sachs assumptions and self-consistent models. For our purposes, most of these models of polycrystalline systems have some inadequacies. For example, the Taylor (Taylor–Lin, to be more precise) assumption imposes equal strains on all crystals. Within the elastic regime, this can result in large stress differences, depending on the level of the elastic anisotropy, and miscalculation of the onset of yielding. In the case of cyclic loading, immediately after a reversal in stress, the response of each individual crystal is predominantly elastic so the condition is encountered frequently. The Sachs model assumes that the stress direction is identical in all crystals. This results in strains that deviate substantially from the average value and can violate compatibility strongly. In reality, when the material is subjected to deformation, the enforcing compatibility of combined elastic and plastic deformation between neighbor grains promotes local multiaxial loading environment and consequently multiple slip. While self-consistent models fulfill compatibility and stress equilibrium in an average sense, they do not account for the precise topology of the aggregate. The finite element formulation bypasses these limitations and gives a more reliable assessment of the deformation and stress throughout the aggregate. We restrict our attention to the isothermal, quasistatic, finite deformation of polycrystalline materials. A weighted residual is formed over the body in the current configuration, B, using the equilibrium equations plus appropriate surface traction t¯ and velocity u¯ boundary conditions on the surface boundaries ∂Bs and ∂Bu, respectively:

where q is a weighting function for the pressure field p, e¯ ∗ is the elastic strain corresponding to the start of the time step, K is the elastic bulk modulus, and b=det(I+e∗). We note that the deformation rate is written in terms of the velocity as D=1/2(ⵜu+(ⵜu)T). The global system of equations governing the motion of the polycrystalline solid is obtained by summing the residuals of Eqs (10) and (11) for all the elements comprising the discretized domain and setting the total residual equal to zero. This procedure yields a nonlinear algebraic system of equations for the nodal velocity and pressure variables. A parallel implementation based on Fortran90 and Message Passing Interface (MPI) was employed to expedite the simulations.

冕

The composition of the AA 7075-T6 aluminum alloy employed in this study is given in Table 1. The nominal processing schedule of the 25.4 mm thick plate includes hot rolling, stretching and a final heat treatment to produce the precipitate structure that provides the dominant strengthening mechanism. The resulting microstructure consists of grains that are thinned in the normal direction (S) and elongated in rolling direction (L). This flattened microstructure is depicted in Fig. 1 with Electron Backscatter Diffraction (EBSD) orientation maps of the L/T and S/L planes at the midthickness point, where T refers to the transverse direction. Average grain dimensions of 2000:200:20 microns for the L:T:S directions were determined from the EBSD data. The texture of the AA 7075-T6 plate was measured by using neutron diffraction and EBSD. Pole figures shown in Fig. 2 depict a rolling texture with the presence of “cube” oriented crystals (grains oriented close to {001}具100典 S–L) which may be the result of recrystalliz-

冕

s⬘·ⵜv dV⫺ p div v dV ⫽

B

冕

¯t·v dA.

(10)

∂B

B

Here, n is a unit normal to the surface; body forces are neglected; v is a weighting function for the velocity field u; and ⵜ is the gradient operator. The constitutive behavior is introduced separately for the deviatoric and volumetric parts. First, the elastic strain rate e˙ ∗ is written as a forward Euler difference over the time interval (⌬t). Using this representation, the deviatoric stress is written in terms of the deviatoric deformation rate by combining Eq. (2) through (6). The result is substituted in Eq. (10). The volumetric response is constructed in a weak form over the configuration B:

冕冉 B

p⫹

冊

K⌬t K trD ⫹ tre¯ ∗ q dV ⫽ 0 b b

(11)

4. Experiments and material In the experimental component of this work, we sought to examine the evolution of lattice strains within a polycrystalline aggregate during cyclic loading. In addition, we sought to quantify relevant microstructural components for use in the simulations. Mechanical tests and microstructural characterization experiments were conducted including neutron diffraction probes on AA 7075-T6 plate material. The goals of the characterization experiments were to quantify grain morphology, crystallographic texture and lattice strain distributions. The mechanical tests consisted of prestraining then cycling a uniaxial test specimen to a preset number of cycles. Lattice strains were then measured in the cycled specimens using neutron diffraction. In this section of the paper, we describe the material and present the details of the experimental methods. 4.1. Material

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271

Table 1 Composition of AA 7075-T6 plate by wt% Element

Zn

Mg

Cu

Fe

Si

Mn

Ti

Al

wt.%

5.63

2.54

1.63

0.29

0.08

0.03

0.02

bal.

Fig. 1. EBSD orientation maps depicting the AA 7075-T6 microstructure at the midplane on the L/T plane (top) and S/T plane (bottom).

Fig. 3. Measured and computed axial stress–strain data for the AA 7075-T6 specimens.

Fig. 2. Pole figures determined from the crystal orientation distribution of the AA 7075-T6 plate. The location of the scattering vectors used in the neutron diffraction studies are shown and correspond to the directions given in Table 3.

The tests were performed at room temperature in laboratory air on a servo-hydraulic test machine operating in true strain control employing an axial extensometer with a 12.75 mm gauge length. A true strain rate of 10⫺3 s⫺1 was employed and kept constant during the test. The L and T specimens were cycled at a constant strain amplitude of 0.004 following a tensile prestrain of 0.03. A schematic of the strain and stress vs time histories is shown in Fig. 4. We anticipated that there would be an uncertainty with respect to the residual stress in the asreceived condition. To compensate for this, an initial prestrain was imposed under monotonic loading before the start of cyclic loading. The intent is similar to a technique used in rolled sheet to reduce residual stresses by

ation during the production of plate material [30]. To obtain data to evaluate the slip system parameters, tensile tests were conducted on specimens taken from midplane of the plate with their axes aligned with either the rolling direction (L) or the transverse direction (T) of the plate. The data are shown in Fig. 3. 4.2. Uniaxial cyclic tests Uniaxial tests were conducted on cylindrically shaped specimens with a gauge length of 19 mm. Flat sections were machined into the gauge sections of each specimen for a companion study on fatigue crack initiation. The effective diameter of the cross-sections was nominally 5.46 mm. All specimens were taken from the midplane of the 25.4 mm thick plate with their axes in either the L or T direction.

Fig. 4. A schematic of the strain–time and stress–time histories for both the AA 7075-T6 experiments and simulations.

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Table 2 Number of cycles to failure, Nf, as a function of strain amplitude and specimen orientation. All specimens were prestrained to 0.03 ⌬⑀/2

L

T

0.0025 0.003 0.0035 0.004

12,902 8,652 – 2,402

– 6,302 4,402 3,402

4.3. Lattice strains

Fig. 5. Measured and computed stress–time results for an L specimen.

stretching the sheet beyond the elastic limit. Following the monotonic loading, the specimens were subjected to a cyclic loading history. For a number of specimens, the loading was interrupted after a set number of cycles (namely, cycles 1, 30, and 1000) and the specimens were saved for subsequent neutron diffraction tests. Points 2 and 4 in Fig. 4 represent the unloading points after one and N cycles, respectively. The experimental and simulated stress–time histories for the first 30 cycles are shown in Figs 5 and 6. As can be seen, the initial positive average stress (mean stress) decreases with cycles in a decreasing rate. After the initial prestrain, the subsequent material response was nearly elastic due to the small strain amplitude that was imposed. The simulated stress–time histories will be discussed in detail in Section 5. As part of the companion study, several specimens at various strain amplitudes following the prestrain were cycled to failure. The results are shown in Table 2. The strain amplitudes were high enough to yield a relatively brief fatigue life. As expected the number of cycles to failure increased when the strain amplitude was decreased.

Fig. 6. Measured and computed stress–time results for a T specimen.

The residual lattice strains in specimens having been subjected to 1, 30 and 1000 strain cycles were measured using neutron diffraction at Chalk River Laboratories, Chalk River, Ontario, Canada. Figure 7 depicts the experimental setup and basic principles of the neutron diffraction method, which has been employed for many years as a means of determining lattice strains [11]. The reader is referred to [12], [6] and [13] for the details of the neutron diffraction experiments. Basically, incoming neutrons diffract off crystallographic planes oriented such that Bragg’s law is satisfied. The scattering vector, q, bisects the incoming and diffracted neutrons as depicted in Fig. 7. This is the direction in which the normal strain for a given family of crystallographic planes is determined from changes in the lattice spacing. The orientations of the scattering vectors employed in this work (shown in Fig. 2) were chosen because their orientations coincide with high intensities on the pole figures. This increases the relative number of sampled crystal planes that contribute to the strain measurement. The scattering vectors indicating the directions of the

Fig. 7. Schematic of a neutron diffraction experimental setup for the measurement of lattice strains. The change in the lattice spacing, d, is used to determine the normal strain in the direction of the scattering vector, q.

H.S. Turkmen et al. / International Journal of Fatigue 25 (2003) 267–281

Fig. 8. The scattering vectors relative to the L specimen. The orientations of the vectors are further emphasized by showing their intersections with an imaginary sphere.

273

Fig. 9. The scattering vectors relative to the T specimen. The orientations of the vectors are further emphasized by showing their intersections with an imaginary sphere.

normal strain measurement relative to the L and T specimens are shown in Figs 8 and 9; direction cosines of these scattering vectors are given in Table 3. The lattice strains measured at the first, thirtieth and one thousandth unloading (point 2 and point 4 in Fig. 4) with their uncertainties are shown in Figs 10 and 11 for the L and T specimens, loaded as shown in Figs 5 and 6, respectively. The differences between the lattice strains at the first, thirtieth and one thousandth unloading are obviously within the uncertainties for most of the plane families. Furthermore, the measurements were taken from different specimens for different number of cycles. Therefore, one can conclude that the evolution Table 3 Direction cosines of the scattering vectors employed for the determination of lattice strains. These orientations are depicted in the pole figures in Fig. 2 and relative to the test specimens in Fig. 8 and Fig. 9 Family of plane x(=T)

y(=L)

z(=S)

1 2 3 4 5 6(L) 6(T) 7 8 9 10(L) 10(T) 11

0.940 0.423 0.0 1.0 0.799 0.469 0.395 0.866 0.0 0.985 0.766 0.530 0.0

0.0 0.906 0.0 0.0 0.0 0.723 0.846 0.5 1.0 0.0 0.527 0.848 0.906

0.342 0.0 1.0 0.0 0.602 0.506 0.358 0.0 0.0 0.174 0.369 0.0 0.423

Fig. 10. The lattice strains measured using neutron diffraction on unloaded L specimens after cycle numbers 1, 30 and 1000.

of the lattice strains measured at the unloaded state (residual strains) was minimal.

5. Simulation The AA 7075-T6 polycrystalline specimens were modeled using a finite element polycrystal as outlined in Section 3. Analyses of the cyclic tests were performed for both the L and T specimens described in Section 4. The evolution of the lattice strains, the crystal stresses, the average stress, and the shear strain rates of the individual slip systems within crystals were monitored throughout the cyclic loading histories. The results of the simulations were compared to the experiments directly in two ways: the macroscopic stress history and

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Fig. 11. The lattice strains measured using neutron diffraction on unloaded T specimens after cycle numbers 1, 30 and 1000.

the lattice strains in the unloaded state following various numbers of cycles. 5.1. Polycrystal model In the finite element polycrystal, each crystal was represented by an eight-noded brick element. The aspect ratio of the brick elements was chosen as 4:2:1 (L:T:S) to partially reflect the elongated grain shape indicated by the EBSD measurements. A total of 4096 elements were used in each simulation, arranged in a 16×16×16 rectangular prism as shown in Fig. 12. An orientation distribution function was first determined from the reduction of the measured pole figure data. A set of lattice orientations was chosen by sampling the ODF and subsequently assigned to the finite elements without regard to spatial position. Although a different spatial assignment of lattice orientations could alter the details of the observations, the trends that emerge are not sensitive to the particular assignments made (see [2,31] for other

results on modeling finite element polycrystals in this regard.) The test sequence of monotonic loading to a strain of 0.03 (prestraining) followed by cyclic loading was applied to the polycrystal. The boundary conditions were chosen to replicate the loading shown in Fig. 4. Symmetry conditions were applied to three adjacent sides of the finite element polycrystal (zero tangential traction and zero normal velocity). In addition, a normal velocity was imposed on one plane (the (ST) plane of the L specimen and the (S–L) plane of the T specimen.) Finally, the remaining two sides for each specimen were completely traction-free. The analyses were performed on a Velocity Cluster parallel computer. Using twelve processors, each simulation required approximately ten days to run to completion. To obtain the single crystal constitutive parameters, the experimental stress–strain histories over the monotonic portion of the test (up to a strain of 0.03) were fit by adjusting the model parameters as shown in Fig. 3. The values obtained for the elastic moduli and slip system parameters appearing in Eqs (2), (5) and (8) are listed in Tables 4 and 5, respectively. The influence that partial recrystallization may have on reducing the strength of the slip systems was estimated by reducing the initial strength of those crystals having a cube orientation by ten percent below other crystals in the aggregate [31]. This adjustment achieved better results for the lattice strains associated with family 4 without significantly altering the strains of the other families. As noted later in this section, there are very few crystals associated with this family so this adjustment had little impact on the simulations overall. With these parameters, the simulations show higher stresses for the T specimens than for the L specimens, whereas the experiments indicate the reverse. When grain shape is introduced using a mesh having crystals with a 4:2:1 aspect ratio, the computed stress is closer to the measured value for both the L and T specimens, but neither is in the correct relative order. There are several possible explanations, including influence of the initial stress, variability of the texture among different specimens, or attributes of the material response that are not modeled. What is important is that by 3% strain the two are comparable and the material has strain hardened sufficiently to mitigate the influence of uncertainty in the initial yielding. Reducing this initial uncertainty was the main motivation for applying the prestrain.

Table 4 Cubic elastic moduli (GPa) appearing in £ of Eq. (2).

Fig. 12. The finite element mesh employed in the simulations. Quarter symmetry was employed.

c1

c2

c3

75.84

41.45

63.30

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275

Table 5 Slip system parameters appearing in Eqs (5) and (8) gs0 (MPa)

g0 (MPa)

h0 (MPa)

m

m⬘

g˙ 0 (s⫺1)

495

213

350

0.005

0.0

1.0

g˙ s0 (s⫺1) 5.0×1010

5.2. Analyses of the cyclic tests With all of the single crystal parameters fixed from the fits to monotonic loading, cyclic loading with a constant strain amplitude of 0.004 was applied to the finite element polycrystals following the initial monotonic extension in the L or T directions. The comparisons of calculated stress–time histories to the corresponding experimental data are shown in Figs 5 and 6 for both the L and T specimens during the first 30 cycles. The decay in the stress amplitude over the duration of the loading that is observed in the experimental records is predicted well for both the L and T specimens. For the L case the computed mean stress drops more rapidly than observed in the experiments, but the stress amplitudes remain comparable. In addition to the results shown here in which texture and slip system strength both evolve, simulations were also performed without hardening, without texture evolution, and with neither hardening nor texture evolution to understand the effect these factors have on the decay of the peak stress. These variations had negligible influence on the trends because neither the texture nor the hardness change appreciably over the course of the experiment. This is noteworthy as it points to the need to identify other potential mechanisms for the decay in the stress. Here, we consider the role of grain interactions in this regard as our simulations clearly show their influence on the evolution of the peak stresses. Quantitative changes in lattice strains and slip system activity indicated by the simulations will be examined in detail.

Fig. 13. Computed lattice strains in the loaded and unloaded states in the L specimen for cycle 1 and cycle 1000.

specimen in Fig. 13 and for the T specimen in Fig. 14. One can readily see that the magnitudes of the average strain under load diminish between the 1st and 1000th cycles. The residual strains generally are much lower than the lattice strains under load, principally due to the low level of single crystal elastic anisotropy. The response of an individual crystal to the cyclic loads depends not only on the crystal’s orientation but also on those of adjacent crystals. Crystals within the same family share a common direction, but sit in different crystallographic neighborhoods. Having different neighbors increases the variability of lattice strains within the same family. These variations were investigated by calculating the standard deviation of the lattice strains for each fam-

5.3. Micromechanical behavior Lattice strains were extracted from the simulations at a number of points in the loading history. Applying a “light-up” algorithm [12], the crystals in the finite element polycrystal that satisfied the same diffraction conditions used in the experiments were identified and their strains averaged. Although the resolution angle was between 0.2° and 2.0° in neutron diffraction measurements, a value of 5° was necessary for the simulations to obtain sufficient grains for each family to guarantee meaningful statistics. The simulations do not distinguish between the lower order and higher order lattice planes ({hkl}=n{hkl}). Therefore, in all of the descriptions below the lowest order value will be used. The simulated average lattice strains at the 1st and 1000th loaded and unloaded points are shown for the L

Fig. 14. Computed lattice strains in the loaded and unloaded states in the T specimen for cycle 1 and cycle 1000.

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ily. The standard deviations appear as the vertical bars with their corresponding average lattice strains in Figs 13 and 14. The values of the standard deviations do not change significantly when the material is unloaded, indicating that the residual stresses in some crystals can be high even though the macroscopic stress is zero. For the L specimen, the average lattice strain of family 4 is the highest at the loaded state because the direction of the corresponding scattering vector is aligned with the loading axis in this case. For families 3, 8 and 11, which have scattering vectors that are perpendicular to the loading axis, the average lattice strains are negative. The ratio of the magnitude of the average lattice strains of families 3, 8 and 11 to the magnitude of the average lattice strain of the family 4 is approximately Poisson’s ratio, as expected. Unloading is essentially an elastic phenomenon and the lattice strains are reduced by amounts proportional to the directional elastic modulus of the associated crystallographic direction. Turning to the T specimen, the average lattice strain of family 3 is the largest under load as its scattering vector is most closely aligned with the loading axis of the specimen. In contrast, the scattering vectors associated with families 2, 4, 7, 8 and 10 are perpendicular to the loading axis and the average lattice strains for these families are compressive. Again, the ratio of the magnitude of the average lattice strains of these families to the magnitude of the average lattice strain of the family 3 is approximately Poisson’s ratio. For easier comparison, the measured and computed lattice strains only in the unloaded state after the first cycle are shown on a reduced scale in Figs 15 and 16, respectively. The lattice strains are shown together with the 95% confidence intervals of the simulated average lattice strains and experimental uncertainties for each family. The confidence intervals were computed assuming that the distributions are Gaussian. A more detailed discussion of this assumption is given in [31].

Fig. 15. Measured and computed lattice strains in the unloaded state for the L specimen after the first cycle.

Fig. 16. Measured and computed lattice strains in the unloaded state for the T specimen after the first cycle.

The average lattice strains of seven families (families 2, 6, 7, 8, 9, 10 and 11) correlate well with the experimental results in the first unloaded state. One of the reasons is the presence of initial residual stresses, indicated by the nonuniform {hkl} lattice spacings for different scattering vectors. Indeed, our simulations implicitly assume the absence of residual strains in the initial state. According to our simulations, however, the distribution of lattice strains following the 0.03 tensile prestrain is very insensitive to the distribution of lattice strains in the initial state (which were introduced in our finite element mesh by imposing a plane strain compression of 0.04, mimicking the hot rolling operation). Therefore, a more meaningful comparison between experimental and numerical results is done by changing the initial value of (measured) lattice spacing by the assumed equilibrium one. This correction is especially useful for {111} and {200} families, where the largest variations in the initial lattice spacings are found. The chosen value for the uniform lattice spacing was: (a) approximately the average value (over both L and T measured reflections), for {111}, {110} and {311} families, and (b) the family 4 value for {200} families, which appeared uniform over L and T specimens, and is supposed to be close to the equilibrium value because the corresponding grains have recrystallized. As can be seen in Figs 15 and 16, this correction restores better trends, in particular for families 1, 3 and 5. Family 4 should be disregarded because of the link to recrystallization which induced, among other things, a reduction in the slip system strength. The simulation actually showed improvement for the family 4 results when a 10% reduction was introduced, without altering much the results of the other families, as was said earlier. Shown in Figs 17 and 18 are the crystal stresses and their standard deviations as computed in the simulations for cycles 1 and 1000. For crystal sets with scattering vectors near the specimen axial direction (families 1, 4

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Fig. 17. Computed crystal stresses and their standard variations in the loaded and unloaded states in the L specimen for cycle 1 and cycle 1000.

Fig. 18. Computed crystal stresses and their standard variations in the loaded and unloaded states in the T specimen for cycle 1 and cycle 1000.

and 9 of the L specimen), the changes in average stress from the loaded to unloaded configurations are largest as this is the active loading direction. For these crystal sets the average values under load are close to the nominal stress shown in Fig. 5. The variations in crystal stresses as given by the standard deviation are about 15% of the change in the average stress in the active direction. The variations are similar for all crystal sets regardless of scattering vector direction and of the loading state (loaded or unloaded). This is a result of the multiaxial state of stress in polycrystals that by nature of the crystal anisotropy is very heterogeneous at this scale. The simulations indicate that the lattice strains under load diminish in magnitude with increasing numbers of loading cycles as seen earlier in Figs 13 and 14 for the average lattice strains obtained at the first and one thousandth loading (points 1 and 3 of Fig. 4, respectively). The relative decreases are much larger than the corresponding changes at the unloaded state (between points

277

Fig. 19. The decrease in computed lattice strains under load during cyclic loading for the L specimen.

2 and 4). The change in magnitude of the average lattice strain, ⌬⑀lattice, was obtained by calculating the difference between the average lattice strain at the first loading and the average lattice strain after a designated number of cycles. The result is shown for each family in Fig. 19 for the L specimens (similar trends were observed for the T specimen). The decreases are most pronounced in the early cycles, which correlates with the more rapid decay in the peak stress in the early cycles (Figs 5 and 6). Both the macroscopic stress and the lattice strains continue to diminish in magnitude until the material behavior becomes entirely elastic. The slip system activity of each crystal is a measure of the rates of plastic deformation within individual crystals. The simulations indicate a trend toward fewer active slip systems as the numbers of cycles increase. The average number of active slip systems na, was recorded for each family throughout the loading history (a slip system is considered to be active if it carries at least 1% of the total deformation rate at full load). These values are shown in Fig. 20 for the L specimens (again, the trends

Fig. 20. The number of active slip systems averaged over the crystals comprising each crystallographic family after various loading cycles for the L specimen.

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are comparable for the T specimen). The number of active slip systems is seen to decrease toward an asymptotic value of zero, with a significant proportion of this decrease occurring in the first 90 cycles. The decrease in the number of active slip systems implies a decrease of the plastic shear strain rate and an offsetting increase of the elastic portion of the deformation.

6. Discussion 6.1. Macro and micro strain predictions Consistent with other experimental data, we observed a progressive relaxation of the average or mean macroscopic stress during cycling of the prestrained AA 7075T6 specimens. The evolution of the predicted crystal stresses during such small plastic deformation conditions is largely due to slight realignments of the straining vector relative to the single crystal yield surface as it migrates towards a yield surface vertex ([3,32]), rather than a change in mechanical state resulting from hardening or texture evolution. This realignment is sufficient to cause the downward trends in the maximum stress values in Figs 5 and 6. While not evident from these figures, we observed that the decreasing trend continued throughout the lives of specimens cycled to final fracture. This evolution in material response occurred even though the macroscopic loading was virtually elastic. As can be seen in the figures, the finite element polycrystal model captured this trend in both the L and T specimens by showing a decrease in the crystal stresses at the maximum point in the cycle and little evolution in the minimum stress, which remains close to zero. This behavior can be seen either in terms of lattice strains (in Figs 13 and 14) or crystal stresses (in Figs 17 and 18). The cycle-induced decrease seen in the crystal responses in the axial directions at maximum load (family 4 in the L specimen and family 3 in the T specimen) directly reflect the drop in peak stress seen in the macroscopic data of Figs 5 and 6. Measuring the lattice strains at peak load during cyclic deformation is an extremely challenging experiment; hence, we have no experimental results to compare with our predictions. However, the accurate predictions of both the decrease in the macroscopic mean stress (Figs 5 and 6) and the unloaded lattice strains (Figs 13 and 14) provide self-consistent evidence that the finite element polycrystal model is accurately tracking the mean values of the lattice strain and crystal stress distributions. We also see that the values of the 95% confidence intervals calculated from the numerical results and the error in the mean lattice strains determined from the experimental data points, with reference to the Gaussian distribution, are of the same order. In the case of the simulations, the spread in lattice strains is largely due to variations of the crystallographic neigh-

borhoods—and resulting mechanical loading conditions—experienced by crystals satisfying the Bragg condition for the various q vectors. 6.2. Slip system activity Cyclic strain hardening processes result in a continuous increase in the critically resolved shear strength during constant strain amplitude cycling. At some point, the resolved shear stress on many of the slip systems remains below the strength value and the total number of active systems decreases. This reduction in the average number of slip systems has been linked to microcrack initiation processes through the development of persistent slip bands (PSBs) [33]. Prediction of slip system activity reduction employing polycrystal plasticity formulations has typically involved invoking latent hardening formulations (cf. [34] and [35]). In the present work, grain interactions and the relationship between the straining vector and the yield surface play a more important role on the reduction of the number of active slip systems than the material hardening character. Examining Table 7, we see that the analyses performed with or without hardening predict similar reductions of average number of active slip systems. Grain interactions are taken into account more realistically by the finite element polycrystal because the influence of neighboring grains on the response of each individual crystal enters it directly. It is also possible—via the explicit representation of individual grains—to study spatial variations in slip system activity. This is done in Table 7 where the average number of active slip systems per crystal in the entire aggregate, on the free surfaces and on symmetry surfaces (interior) are tabulated. Consistent with the idea that fatigue cracks initiate on surfaces because slip system activity is lower there compared to the interior (cf. [36]), we see that the surface grains in both the L and T specimens begin the deformation with relatively fewer number of active slip systems. While the data for the remaining cycles are inconclusive with respect to this particular fatigue crack initiation hypothesis, the finite element modeling framework provides the vehicle for such studies. 6.3. Shared crystals The number of crystals lit-up for each family and the number of crystals it shares with other families is shown in Table 6 for the case of the L direction specimen (similar numbers were obtained for the T direction specimen). These values are presented to show that only a subset of the crystals contribute to average lattice strain for each family. In this table, the diagonal entries are the numbers of crystals in each of the families and the off diagonal entries give the numbers of crystals shared by two families. For instance, 311 crystals contribute to

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Table 7 Average numbers of active slip systems for the two specimens. The two numbers are for the L and T oriented specimens, respectively. A slip system was considered active if its shearing rate was 1% of the macroscopic deformation rate at full load Cycle

Aggregate

Free surface

Symmetry surface

No hardening

1 3 16 31 91 1000

2.894/2.814 1.467/1.484 0.714/0.894 0.384/0.419 0.0579/0.083 0.0154/0.021

2.508/2.755 1.326/1.431 0.578/0.863 0.308/0.393 0.062/0.080 0.0222/0.018

2.786/2.870 1.161/1.463 0.515/0.895 0.313/0.434 0.126/0.071 0.087/0.003

2.900/2.809 1.466/1.538 0.925/0.897 0.666/0.660 0.202/0.270 –/–

Table 6 Matrix of the number of shared crystals (elements) for each lattice family in the L specimen calculations of lattice strain Family

1

2

3

4

5

6

7

8

9

10

11

1 2 3 4 5 6 7 8 9 10 11

311 26 54 0 0 43 105 56 0 49 59

26 321 0 0 0 0 47 0 11 13 5

54 0 206 0 60 28 119 110 61 0 39

0 0 0 76 0 0 0 2 0 0 0

0 0 60 0 155 0 79 54 96 0 46

43 0 28 0 0 127 41 58 0 15 57

105 47 119 0 79 41 629 130 133 0 131

56 0 110 2 54 58 130 293 64 18 58

0 11 61 0 96 0 133 64 412 55 76

49 13 0 0 0 15 0 18 55 307 80

59 5 39 0 46 57 131 58 76 80 317

the lattice strain computation for family 1. Generally, the number of crystals common to any two families is relatively small, indicating that the average strains are largely independent of each other. As to be expected, the number of crystals in each average is strongly linked to the crystallographic texture and crystal symmetry. Family 7 has almost twice as many crystals as any other family. As can be seen from Fig. 2, family 7 (and its associated q) comes from a high density region on the {110} pole figure and the multiplicity of {110} planes within a cubic crystal is relatively high. Off diagonal terms in Table 6 indicates the number of crystals participating in both calculations. For instance, of the 311 crystals included in the family 1 measurement, 54 of those also contributed to family 3. As can be seen in Figs 8 and 9 the directions associated with vectors 1 and 3 are nearly orthogonal, but 54 crystals contained {111} planes oriented such that the Bragg condition was satisfied by either vector 1 or vector 3. These trends carry over to the neutron diffraction experiments as well. That is, one should expect that the proportions seen in the diagonal terms of Table 6 will also be present in the physical experiment. Family 7 should have many more crystal planes contributing to its lattice strain distribution than family 4 will.

7. Conclusions A number of conclusions can be reached from the simulations and experiments reported here. These are summarized in the following list. 앫 There is plastic straining in many crystals at macroscopic stress levels lower than the nominal yield strength as a result of grain interactions. Over the course of many cycles at constant strain amplitude, the small amount of plastic strain is sufficient to allow realignments of the crystal stresses in a manner that reduces the average stress at the peak load. 앫 The slip system activity associated with this local plasticity evolves with cyclic loading. Under constant strain amplitude loading, the average number of active slip systems decreases nearly to zero with cyclic loading as stresses redistribute among crystals in an aggregate. Changes continue to occur as long as there are active slip systems. 앫 There is a slightly lower level of slip system activity in the surface grains in comparison to the interior grains. This is consistent with the eventual formation of persistent slip bands observed in other work that is attributed to a condition of single slip within crystals.

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앫 Macroscopic stress relaxation observed during the cyclic test of AA 7075-T6 is predicted well using the finite element method. The computed stresses relax during the cyclic loading regardless of whether or not slip system hardening or texture evolution occurs (see first conclusion).

[14]

Acknowledgements

[15]

Support for this work has been provided by the Air Force Office of Sponsored Research under Grant #F49620-98-1-0401. Neutron diffraction experiments were performed on a National Research Council (Canada) neutron diffractometer located at the NRU reactor of AECL (Atomic Energy of Canada Limited). The authors thank Dr Ronald Rogge for his assistance in performing these experiments. Computing resources were provided by the Cornell Theory Center. EBSD measurements were performed at the Cornell Center for Materials Research.

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