Strain rate jump induced negative strain rate sensitivity (NSRS) in aluminum alloy 2024: Experiments and constitutive modeling

Materials Science & Engineering A 683 (2017) 143–152

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Strain rate jump induced negative strain rate sensitivity (NSRS) in aluminum alloy 2024: Experiments and constitutive modeling

MARK

⁎

Satyapriya Guptaa, , Armand Joseph Beaudoin Jr, a, Juliette Chevyb a Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Mechanical Engineering Building, 1206 W. Green St. Urbana, IL 61801, USA b C-TEC Constellium Technology Center, Parc Economique Centr'alp CS10027 Voreppe, 38341 Cedex, France

A R T I C L E I N F O

A BS T RAC T

Keywords: Negative strain rate sensitivity Aluminium alloy Digital image correlation Shear bands Jump test Field dislocation mechanics

Negative strain rate sensitivity (NSRS) leading to the strain localization and ultimately failure of material is one of the greatest challenges in the efficient usage of the Al alloys. NSRS in Al-Cu alloy (AA2024) is investigated with the help of uniaxial tension tests at constant strain rate and jump tests conducted with various strain rate jumps for different test temperatures. Digital image correlation (DIC) is adopted to locate the creation and movement of the heterogeneities. Strain controlled jump tests, in combination with DIC, prove to be an excellent tool to investigate NSRS in the alloy. Uniaxial tension tests performed at room temperature (RT) and elevated temperature (50 °C) with constant strain rate ranging from 1 × 10−3 to 1 × 10−5/sec demonstrated strain rate insensitive deformation behaviour with the absence of serrations and plastic instability. In contrast, strain controlled jump tests conducted at −50 °C, RT, and 50 °C revealed the presence of NSRS during the strain rate jumps and plastic instabilities following the jump. Local strain mapping achieved from DIC confirmed the propagation of the localized shear bands around the strain rate jumps which is not observed during uniaxial tension tests under constant strain rate. For better understanding of the underlying mechanism kicking off the plastic instabilities, the strain controlled jump test is simulated using the existing phenomenological mesoscopic field dislocation mechanics (PMFDM) model where a dynamics strain aging (DSA) module is embedded in crystal plasticity framework. Numerical predictions of grain size effect suggest that size effect depends on trade off between contributions of polar dislocations (GNDs) to internal stress and plastic deformation, and can lead to an “inverse size effect” when plasticity is the major contribution. While explaining the experimental observations related to jump test, simulation results provided an opportunity to access the experimental aging time which is a crucial material parameter for alloys exhibiting NSRS.

1. Introduction Aluminium alloys are typical materials used in the aerospace applications due to their high strength to weight ratio, resistance to fatigue, and improved corrosion properties. For that matter, aluminium alloy AA2024 – where copper is the primary alloying element – has proved to be prominent choice and is extensively utilized as plating of aeroplane, helicopter fuselage, tensioned wing parts, and ribs which are subject to shear in the aerospace industry. Despite the fact that AA2024 is fully mature and has gone through intensive research in terms of optimizing mechanical properties [1– 7], its dynamic behaviour, which is highly crucial for aeronautics community, has not been explored in depth. Dynamic behaviour includes dislocation activities such as dislocation-solute interaction

(pinning-breakaway) giving rise to dynamic strain aging (DSA) manifested in the form of negative strain rate sensitivity (NSRS) and serrated stress strain curve [8–12]. Macroscopic NSRS can be observed via comparing the flow stresses for two different strain rates Δσ m = Δ log ϵ˙ where m is the strain rate sensitivity (SRS) and should be negative if lower strain rates exhibit higher flow stress and greater strain hardening as compared to higher strain rates. DSA has been observed for dilute Al alloys in limited range of strain rate and temperature [13–17]. Technological importance of NSRS (caused by DSA) is the primary motivation for studying the dynamic behaviour of these alloys. For example, it leads to formation of localized shear bands and their movement during the deformation of alloy, which can play a prominent role in the formability and failure of alloy [18–21]. NSRS can also

⁎ Corresponding author. Current address: Postdoc Researcher, Bureau 204, Laboratoire d'étude des microstructures et de mécanique des matériaux (LEM3), Ile du Saulcy, F57045 Metz - cedex 01, France. E-mail addresses: [email protected] (S. Gupta), [email protected] (A.J. Beaudoin), [email protected] (J. Chevy).

http://dx.doi.org/10.1016/j.msea.2016.12.010 Received 27 September 2016; Received in revised form 30 November 2016; Accepted 1 December 2016 Available online 02 December 2016 0921-5093/ © 2016 Elsevier B.V. All rights reserved.

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deteriorate the surface finish of the shaped product [22–25], which is a prime concern to the industries. Although AA2024 is not known to have highly serrated flow curve at room temperature as observed in AA5xxx series, yet NSRS present in these alloys [26–28] can influence their deformation behaviour remarkably. For instance, it can promote the failure of material via strain localization before the onset of necking. Ample work focusing on strain localization and its role in failure of the alloy [18,19,21,29] can be found in the literature but surprisingly, none of them looked at AA2024 in this regard. A couple of theoretical studies suggested that the effect of NSRS depends on the material type: it can be beneficial for a brittle material and destructive for a ductile material [30,31]. [32] noticed that shear failure predominantly occurs in the range of temperature and strain rate favourable for NSRS, which suggests that shear failure is related to DSA and associated heterogeneous plastic deformation. For Al-Li 2091, [33] observed an increase in tear resistance for temperature range concurrent to DSA, which indirectly indicates the beneficial effect of NSRS. Apart from the macroscopic realization, optical investigation of DSA facilitates better understanding of the phenomenon at micro scale. Therefore, a number of optical techniques have been adopted to examine and visualize the creation and propagation of localized shear bands associated with NSRS [34–37]. Among these techniques, digital image correlation (DIC) [38–40,21,41] has evolved as an excellent and widely adopted choice owing to its versatility in terms of scale of observation, simple setup and easy post-processing. Objective. In the light of above facts, there exists a gap regarding investigation of DSA and associated NSRS in case of AA2024, which is of prime importance for further development and extended utilization of this alloy. This work aims for gaining some insight into NSRS and plastic instabilities demonstrated by the alloy with the integration of experiment and simulation. Two different tempers T4 and T351 of AA2024 are investigated using tension tests coupled with DIC optical measurements. In addition to uniaxial tension, strain controlled jump tests at different temperatures are conducted to enhance the aging effect dynamically. Limited information obtained from experiments motivated us to simulate the experimental observations related to jump test using already existing phenomenological mesoscopic field dislocation mechanics (PMFDM) model. A standard dynamics strain aging (DSA) formulation is coupled with crystal plasticity framework to capture the NSRS numerically. The present model takes a lead over the previously proposed gradient theories on the following grounds:

presence of NSRS in AA2024. The DSA coupled PMFDM model used for FE simulations is introduced in Section 3. Section 4 presents the details of FE mesh and the numerical algorithm used for calculations. Section 5 discusses the simulation results associated with jump tests including the effect of grain size and shape on DSA. Finally, important conclusions are listed in Section 6. 2. Experimental investigation: materials and method Uniaxial tension tests at constant strain rate (or constant crosshead speed) and strain controlled jump tests (with various strain rate jumps) were conducted to investigate DSA in AA2024. Tests were performed at different temperatures for two kinds of tempers T4 and T351 of AA2024. In-situ measurements of nominal axial strain and local strains were carried out with the help of strain gauge and digital image correlation (DIC) technique, respectively. Detailed description of the test procedure and experimental observations are presented in the following section. Test procedure. Flat standard tensile dog-bone shape specimens of gauge length ≈60 mm (along rolling direction) and width ≈10 mm were machined out from thin (≈1.2 and 2.5 mm thickness) rolled plates for both constant strain rate tension tests and strain controlled jump tests. Two different tempers of AA2024, i.e. T4 and T351 were investigated, T4 can be described as solution heat treated-quenched-naturally aged whereas T351 includes stretching of alloy after quenching which induces higher flow stress. Specimens were further prepared for DIC technique by grinding and spraying black speckles on white painted background. Tensile tests were carried out on a screw-driven load frame Instron 5900 R with 20 kN load cell at a constant cross-head speed in tension. A clip-on extensometer with 25 mm gauge length (Instron 2630–106) was used to evaluate the nominal axial strain which acted as a cross validation of strain mapping obtained from DIC. Uniaxial tension tests were performed at three different temperatures (−50 °C, RT, 50 °C) for three different strain rates (ranging from 1 × 10−3/sec to 1 × 10−5/sec ) to figure out the suitable combination of temperature and strain rate manifesting DSA. Two different specimen thicknesses (≈1.2 and 2.5 mm) were used for uniaxial tension tests and strain controlled jump tests were carried out for both T4 and T351 at different temperatures. Several strain rate jumps from higher to lower strain rates and vice versa were taken at different strain magnitudes. For mapping the local strain distribution at surface of specimens, the DIC technique was employed. The 2D DIC set up consists of an imaging source USB camera (DMK 31AU03), a Navitar zoom 7000 series lens to control focus, zoom, and f-stop and naturally or artificially pattern on the specimen to be examined. Digital images were captured during the tension tests and thereafter, post processing of the images was done using VIC2D commercial software in order to get the pixel by pixel strain and displacement maps at the specimen surface.

1. A clear distinction between excess dislocations (GNDs) and statistical dislocations (SSDs) is made via constitutive formulation. 2. Contributions of both SSDs as well as GNDs are considered in plastic strain rate which were neglected in previous works [42–45]. Apart from numerical explanation of experimental findings related to DSA, a parametric study is performed to identify the microscopic aging time associated with experiment. In order to do so, aging and hardening parameters of the model are systematically varied to capture the experimental stress profile for strain rate jump. Moreover, consideration of dislocation transport in the model formulation provided the opportunity to predict the influence of grain size and grain morphology on DSA. Investigation of grain size effect highlighted the plasticity contribution of GNDs and suggested the possibility of inverse grain size effect on DSA for the deformation dominated by mobile dislocations. This work can be broadly divided into two parts. The first part of the paper includes the details of experimental investigation and the second part deals with the simulation of strain controlled jump test with the help PMFDM coupled with a constitutive formulation for DSA. In the previous section, we have provided the motivation to study dynamic behaviour of AA2024 and its technical importance. Section 2 of the paper illustrates the experimental procedure and findings related to

3. Experimental results Global stress strain curves for constant strain rate uniaxial tension tests performed at RT with varying strain rates are plotted in Fig. 1. Strain rate comparison of flow curves for both the tempers T4 (Fig. 1(a)) and T351 (Fig. 1(b)) demonstrate negligible change in flow stress with changing applied strain rates. This deformation attribute can be referred as strain rate insensitive behaviour. Besides that, no significant plastic instabilities or serrations were observed for these tests irrespective of test temperature and alloy temper. However, the room temperature test for T351 carried out at strain rate 1 × 10−3/sec exhibited small serrations later in the test. A clear effect of different temper can be observed by comparing the flow stresses presented in Fig. 1(a) and (b) where T351 shows higher yield stress (for all the strain rates) as compared to T4 due to stretching treatment involved in T351 144

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Fig. 1. Stress strain curves with varying strain rates highlighting the strain rate insensitive flow, (a) for T4 temper; (b) for T351 temper.

Fig. 2. Serration check for tests performed with strain rate 1 × 10−3/sec at RT where primary y axis indicates flow stress of material and secondary y axis signifies serration stress which is running average subtracted from flow stress (a) temper T4; (b) temper T351.

plastic activities shown in Fig. 3(b). For example, a couple of plastic transients for T351 at 50 °C jump test were found to be revealing: here, two different strain rate jumps demonstrated different type of strain front propagation for the same test. Local strain ϵyy and strain rate ϵ˙ yy maps for the plastic instability emphasized by the first inset ( jump: 1 × 10−5/ sec to 1 × 10−3/sec ) are presented in Figs. 4 and 5, respectively. Well organised movement of a highly localized strain front and shear bands in the gauge section of the specimen length can be the suggested reason for the highlighted plastic instability in Fig. 3(b). The inclination angle of the shear band or strain front did not change throughout the propagation and is 50–55° with respect to the tensile axis. However, DIC strain maps for other plastic activity highlighted by second inset ( jump: 1 × 10−3/ sec to 1 × 10−5/sec ) shows the unorganised propagation of strain front (see Fig. 6) and no recognizable pattern for shear bands.

temper. Subtraction of the running average from the original stress data confirms the generally uniform response in the constant strain rate uniaxial tension test. Examination of Fig. 2(a) and (b) reveals no serrations for T4 and very small unorganised plastic activities (1– 2 MPa of magnitude) for T351 around the strain close to fracture, respectively, for the room temperature tests performed with strain rate 1 × 10−3/sec . Fig. 3(a) shows global stress strain response of the strain control jump tests for T4 and T351 conducted at different temperatures. Several strain rate jumps can be observed between 1 × 10−3/sec and 1 × 10−5/sec at different strain magnitudes. In contrast to constant strain rate uniaxial test results, flow curves for all the jump tests (strain rate jumps took place during the test) indicated the presence of NSRS in the alloy. Alloy exhibited NSRS during the strain rate jumps, and significant plastic instabilities followed by the jump such as Lüders band formation. Strain rate jumps clearly demonstrate higher work hardening rate and higher flow stress for lower strain rate and vice versa, which is a manifestation of NSRS in the alloy (note the strain rate jumps in Fig. 3(a)). In the absence of DSA, the jump test would have resulted in positive SRS, i.e. higher flow stress for higher strain rate and vice versa [46–48]. Plastic instabilities initiated by the strain rate jumps were also examined through in-situ DIC mapping of point to point deformation. Although DIC analysis was performed throughout the test for all the experiments, results presented here corresponds to few interesting

4. Numerical model description This section presents a brief description of the model developed by [49,50]. The foundation of the model follows from the theory developed by Acharya [51–53]. In the present work, a reduced version of the model is used (refer [49] for more details) where reductions are made in FDM part. Following are the important constitutive equations of reduced PMFDM presented in the framework of infinitesimal deformation. 145

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Fig. 3. Stress strain curve for strain controlled jump tests demonstrating NSRS and significant plastic instability initiated via strain rate jumps, (a) T4 and T351 temper at various temperatures; (b) T351 at 50 °C with zoomed-in view of plastic activity.

dislocation (SSDs) or they are required for accommodation of strain gradients caused by inhomogeneous plastic deformation called geometrically necessary dislocations (GNDs). Storage of dislocations result in hardening of material. The concept of GNDs is mathematical and scale dependent (see Fig. 7), for example, at a very small length scale, the GND density will be significant and comparable to SSDs, however, at larger scale GND density will be very small as compared to SSDs and can be neglected. GNDs accounting for length scale effects take part in plasticity, hardening and internal stress whereas SSDs only contribute to plastic deformation and material hardening. Therefore, plastic p distortion rate U̇ governed by motion of both SSDs and GNDs and can be stated as

Field dislocation mechanics (FDM). Dislocation as a primary source of permanent deformation in the crystalline materials carry an elastic interaction stress field around it. The stress field of dislocation along with the stress field due to applied boundary conditions (traction or displacement) drives the dislocation motion [51–53]. The mathematical description of the above mentioned process is known as field dislocation mechanics (FDM). In the reduced version of the model, we directly start with equilibrium equation used to find the displacement field u and thereby stress in the material due to posed displacement or traction boundary conditions.

div σ = 0, div σ = div(C e: Ue ) = 0 , σ ·n = T on St , u = u on Su , (1)

p

U̇ = α × v + L p,

(3)

e

where C is a forth order elastic stiffness tensor, T and u are applied traction and displacement boundary conditions, respectively. Ue is elastic distortion in lattice which can be stated with the help of additive decomposition of total distortion grad u as

Ue = grad u − U p.

where α is known as the Nye dislocation density tensor [54] or GND density tensor used to quantify GND density and v is dislocation velocity vector. Lp is plastic velocity gradient which is the plasticity contribution from SSDs and must be modelled phenomenologically. Phenomenological crystal plasticity is used to describe the Lp. In the absence of a dislocation source, time evolution of GND density α can be given by conservation of Burgers vector, which is also known as transport equation

(2) p

Plastic distortion U is caused by the presence and motion of dislocations which can be stored for two reasons: they get accumulated by random trapping of each other known as statistically stored

Fig. 4. DIC strain ϵyy maps for plasticity event displayed by first inset ( jump: 1 × 10−5/sec to 1 × 10−3/sec T351) within Fig. 3(b), well organised propagation of strain front with increasing strain magnitude.

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Fig. 5. DIC strain rate ϵ˙yy maps for plasticity event displayed by first inset ( jump: 1 × 10−5/sec to 1 × 10−3/sec T351) within Fig. 3(b), propagation of shear band with increasing strain magnitude.

α˙ = −curl(α × v + L p ), α (v·n ) = F on Si ,

(4)

where Si and F are the inflow boundary and prescribed flux of dislocation on boundary, respectively. According to evolution equation, −curl(L p ) will act as source of GNDs in the absence of α. Macroscopic stress field σ drives GNDs and SSDs through average dislocation velocity v and Lp, respectively. Dislocation velocity field v associated with GND density tensor is defined as

v=v

d |d |

a

where v ≥ 0, d = b−(b · |a| ) a, 1

b ≡ X (σ ′·α ), a ≡ 3 tr(σ ) X (α ),

(5)

where v is the magnitude of velocity vector given by Eq. (6), X represents permutation tensor, σ′ signifies deviatoric part of stress tensor and tr is the trace operator. The velocity v described in above equation is taken as average of crystallographic slip velocity magnitude and can be written as

v =

1 nc

∑ |vs |, s

Fig. 7. Mathematical description of dislocations; GND or SSD interpretation of dislocation is related to the reference volume taken for the observation.

strain aging can be coupled with FDM. Crystal plasticity coupled DSA. Dislocation density based formulation is used where SSD density can be classified as mobile (ρm) and forest (ρf) dislocation density. In a classical manner, Lp is defined as summation of the shear rate of dislocation glide for all activated slip systems, with shear rate obtained from Orowan relation

(6)

where nc is the total number of activated slip systems. Above mentioned equations serve as a basis for a closed FDM theory (eliminating the non-uniqueness) except for the fact that behaviour of SSDs still need to be defined where Lp and dynamic

γ ̇ = ρm bv, L p =

∑ ρm bvs Ps, s

where τs = Ps: σ ,

(7)

Fig. 6. DIC strain maps for plasticity event displayed by second inset ( jump: 1 × 10−3/sec to 1 × 10−5/sec ) within Fig. 3(b), unorganised propagation of strain front with increasing strain magnitude.

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where symbols vs , τs, Ps denotes averaged dislocation velocity, resolved shear stress and Schmid tensor, respectively for a particular slip system s. Effective plastic strain rate Γ˙ can be illustrated as summation of plastic activity caused by motion of both GNDs and SSDs

Γ˙ = α × v F +

∑ ρm b|vs |, s

which affects the velocity of dislocation at particular slip system (see Eq (9)) and can be defined as

H=

s

3

∑i, j =1 Aij Aij . Dislocation velocity in Eq (7) evolves as a function of resolved shear stress and strength of material. Power law for evolution of dislocation velocity can be written as

p=

⎞M

∂ρf ∂t

=

⎞ ⎛ C0 C A H + ⎜C2 ρm + 3 ρf − C4 ρf ⎟ Γ˙ + 1 ρm p (Γ˙ ), ⎠ ⎝ b b tl

∂t Γ˙ ⎛ ta ⎞ tl = Ω / Γ˙a, Ω = tl Γ˙a, a = 1 − ⎜ ⎟. ∂t Γ˙a ⎝ tl ⎠

⎛ ⎛ t ⎞2/3⎞ Cs = 1 − exp ⎜⎜ −⎜ a ⎟ ⎟⎟ , ⎝ ⎝τ⎠ ⎠

(13)

(16)

5. Finite element simulation of strain rate jump tests In accordance with experiments, the strain controlled jump tests are more interesting as the plastic instabilities and the presence of NSRS were demonstrated observations otherwise absent in uniaxial test at constant strain rate. For this reason, we made an effort to describe the experimental findings associated with the jump test with the help of finite element simulation using the model described in Section 3. In addition, inclusion of dislocation transport facilitated the investigation of grain size and grain morphology effect on simulated NSRS behaviour of the alloy. Finite element model. A three dimensional polycrystalline FE mesh with hex elements was generated using Neper software package [56] to perform a virtual jump test. A simulation domain with 20 grains was discretized in 64000 C3D8 regular elements introducing a stepped boundary between the two grains where grain orientations were assigned with the help of three random Euler angles. As shown in Fig. 8, two different morphologies, i.e equiaxed and bamboo shaped

(11)

(12)

(15)

In the elastic regime where Γ˙ is nearly zero, ta will evolve linearly with time and we should expect a sharp change in ta when plasticity kicks in the deformation.

(10)

∂ta Γ˙ = 1 − ta ∂t Ω

2 Γ˙ exp(−(Γ /ϵ˙0 )2 ), ϵ˙0 ϵ˙0

where ϵ˙0 is the reference strain rate and an input for the model. Constant incremental strain Ω provides a constant loading time tl for particular applied strain rate Γ˙a . With constant tl, the evolution of aging time mentioned in Eq. (12) can be written as

(9)

where v0 is reference dislocation velocity, M is inverse of strain rate sensitivity. Stresses acting against applied stress τa, τh, τ0 and τs are athermal stress, forest hardening, initial material strength, and solute strength, respectively. In the forest and solute hardening contributions, β and f are fitting parameters, μ and Cs are shear modulus of the material and solute concentration, respectively. Evolution of forest and mobile dislocation densities are coupled with aging time ta (can be defined as the residence time of dislocation at obstacles) and solute concentration Cs to make the model suitable for DSA [55].

⎞ ⎛C ∂ρm C A = ⎜ 21 − C2 ρm − 3 ρf ⎟ Γ˙ − 1 ρm p (Γ˙ ), ⎠ ⎝b ∂t b tl

(14)

In the dynamical system, A1 is a constant term accounting for DSA by an indirect coupling of ρm and ρf with solute atoms, s is the summation index over all the activated slip systems, Ω is incremental strain (Ω is kept constant for the present work) which plays an important role in evolution of aging time ta. p (Γ˙ ) represents the probability of dislocation pinning by solute atoms at some obstacle such as forest dislocation and is described as

(8)

where |A|F represents Forbenius or Euclidean norm described as

⎛ |τ | vs = v0 sgn(τs ) ⎜ τ + τ +s τ + τ ⎟ ⎝ a h 0 s⎠ with τh = βμb ρf , τs = fCs

∑ |α·ns |ρm b|vs | + ∑ s|α·ns ∥ α × v|F .

where τ is the characteristic diffusion time, coefficients C0 represents the forest interaction with GNDs, C1 and C2 are the terms for multiplication and annihilation of mobile dislocations, C3 is for immobilization caused by interaction with forest dislocations and C4 is defined for dynamic recovery. H is the GND contribution in forest hardening (ρf)

Fig. 8. Morphologies of finite element mesh with adopted displacement boundary conditions, (a) equiaxed grains; (b) bamboo grains.

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The standard Galerkin method was used to discretize the mechanical equilibrium equation in space. However, the least squares FEM was used to discritize the transport equation due to first-order character of the PDE. Linear brick elements were found to be well suited for both equilibrium and transport problem. Crystal plasticity coupled dynamic strain aging equations (dynamical system) are stiff due to simultaneous existence of slow and fast time scale. The CVODE integrator [57] was used to solve the dynamical system due to its adaptive stepping character. Calculations were performed in a parallel computing environment provided by the PETSc libraries [58] for easy handling of large problems. For these particular calculations, the SuperLU_DIST direct solver was used [59,60].

Table 1 Material parameters for DSA simulations.

Model parameters

Notation

Value

Burgers vector magnitude

b

Modulus of elasticity Poisson's ratio Reference dislocation velocity

E ν v0

2.7 × 10−10 m 70.2 GPa 0.3

Reference strength Athermal stress

τ0 τa f τ α C0 C1

Fitting parameters

C2

Arrest term

C3 C4 A1

Reference strain rate

ϵ˙ 0

Loading time

tl

3.5 × 10−7 s−1 2.27 MPa 133 MPa 9.25 MPa 10 s 0.35 20

2.43 × 10−4

6. Numerical results and discussion

8.18 × 10−2 1.4 3.33

Effect of DSA. Analogous to experiment, strain rate jump from 1 × 10−3/sec to 1 × 10−5/sec was simulated to explain the experimentally observed NSRS governed by DSA. For experimental conditions, results are presented for largest simulated domain approximation of size, i.e. 100 × 100 × 100 μm3 (corresponds to largest grain size). Since, we want to focus on transient response through the strain rate jump, simulations were performed for small magnitude of strain. For the sake of comparison, simulations for equiaxed morphology were carried out with and without DSA effect included in the model. Fig. 9 illustrates the significant difference in the flow stress and hardening behaviour caused by DSA. It is quite evident that DSA leads to NSRS, i.e development of greater stress for lower strain rate (1 × 10−5/sec ) as compared to that before the jump where applied strain rate is higher (1 × 10−3/sec ). In contrast, the simulation performed without DSA demonstrated positive SRS which can be observed from lower hardening rate of flow curve after the jump. The effect of DSA is more evident through the respective strain hardening rate plots shown in Fig. 9(b). Moreover, we have also observed a higher flow stress for simulations performed with DSA, indicating the occurrence of DSA before the strain rate jump. This can be explained on the basis of smaller magnitude of aging time considered for this calculation which can cause a rapid strain aging. Figs. 10 and 11 present two dimensional contour plots for stress component in z direction and α taken as a slice perpendicular to x direction at the origin. Plots signifying the variation in the stress and polar dislocation density distribution during the deformation of FE mesh, respectively. One can observe a prominent stress relaxation during the strain rate jump 1 × 10−3/sec to 1 × 10−5/sec (see Fig. 10) and significant stress increase followed by jump through higher hardening rate arising from DSA. Moreover, Fig. 11 indicates that strain rate jump also triggered the avalanche of polar dislocations (GNDs) which promotes the formation of sub-grain boundaries at later stage. Note

1 × 10−4 s−1 5 × 10−4 s−1 0.5 s

Table 2 Initial conditions for DSA simulations.

Model parameters

Notation

Value

Mobile dislocation density

ρm

5 × 1013 m2

Forest dislocation density

ρf

Initial aging time

ta

5 × 1011 m2 1s

grains were used to examine the effect of grain morphology on simulated NSRS behaviour of the FE model. The equiaxed morphology shown in Fig. 8(a) was chosen to study the influence of grain size on NSRS and to that end, variation in the grain size was achieved by altering the size of the simulation domain while keeping the number of grains constant. The strain rate jump was obtained via displacement controlled boundary conditions where bottom nodes were fixed for z translation and displacement was specified on the top nodes. The material properties and initial conditions used for the this work is listed in Tables 1, 2, respectively. As mentioned in the model description, the magnitude of incremental strain ω was kept constant, taken as 5 × 10−4 for these particular calculations. Numerical strategy. The coupled set of non-linear transient partial differential equations (PDEs) of PMFDM were solved by using appropriate finite element techniques depending on the nature of the PDE.

Fig. 9. Effect of DSA on deformation behaviour of simulated strain rate jump test, (a) global stress strain curves; (b) strain hardening rate curves.

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Fig. 10. Two dimensional contour plots demonstrating the evolving distribution of stress state in z direction, (a) just before the jump at 0.58% strain; (b) just after the jump at 0.61% strain; (c) after jump at 0.93% strain.

simulations for smallest domain 1 × 1 × 1 μm3 are performed with equiaxed and bamboo shaped grains. We could observe only a negligible difference between the deformation behaviour of the two morphologies with slightly higher hardening for bamboo shaped grains. However, we expect a notable difference in the hardening behaviour of the two with increasing strain magnitude due to the fact that forest hardening contribution of GNDs will increase with increasing plastic deformation. Access to experimental aging time through variation in Omega. As mentioned in Eq. (12), incremental strain Ω is directly related to aging time tl in our model. A parametric study involving the variation in Ω provided an opportunity to mimic the experimental jump profile (where a quick recovery of stress with respect to strain followed by strain rate jump is observed) and thus suggests the aging time relevant to experiments. Therefore, FE simulations with different Ω (while keeping the other model parameters constant) were performed so as to draw an estimate of aging time consistent with experimental observations. Fig. 13 shows stress strain curves for strain rate jump simulations conducted with different Ω ranging from 1 × 10−4 to 1 × 10−3. Variation in the nature of strain rate jump with changing Ω can be clearly observed where lower magnitude of Ω (i.e. smaller aging time) provided faster recovery of stress and higher strain hardening rate with respect to strain followed by strain rate jump. For the large value of Ω such as 1 × 10−3, the DSA effect nearly disappeared. This study provides a pathway to access the experimental aging time (an important material parameter to understand the DSA phenomenon) via continuum modelling.

that α quantifying the GNDs is taken as Forbenius norm of 9 3

components of the Nye tensor given by ∑i, j =1 Aij Aij . Influence of grain size and grain morphology on NSRS. Variation in the grain size and grain morphology can significantly influence the extent to which DSA affects the hardening and flow stress of the material. However, the nature of the influence is unclear from the literature. Therefore, jump test simulations were conducted with varying grain size and two different grain morphologies mentioned in Fig. 8. It is important to mention that variation in grain size can be achieved in two ways: 1) by changing the number of grains for fixed simulation domain size; 2) by varying the size of simulation domain for a fixed number of grains. The latter option was adopted for grain size investigation because it keeps the relative orientations between the grains unchanged for different grain sizes used for simulations and provides the sole effect of grain size by eliminating the influence of crystal orientation. Simulations with four different cubic domain sizes (smaller domain size corresponds to smaller grain size), i.e 10 × 10 × 10 μm3, 100 × 100 × 100 μm3, 5 × 5 × 5 μm3, and 1 × 1 × 1 μm3 were performed for equiaxed and bamboo shaped grains. However, the result is presented only for equiaxed case due to very similar size effect obtained for two different morphologies. Fig. 12 illustrates the grain size influence on DSA controlled hardening behaviour for equiaxed morphology. Simulated stress strain curves demonstrated a complicated size effect for both equiaxed and bamboo shaped grains where the character changes over the duration of the transient. Response immediately following the jump is dominated by mobile dislocations and the contribution to plasticity by GNDs, mentioned in Eq. (8), impact the deformation: smaller grains, having higher GND density, are softer, i.e. an inverse size effect exists. In contrast, as the transient evolves, the forest hardening contribution of GNDs unfolds and provides (the more usual) smaller is stronger behaviour to material. This complicated size effect becomes more visible through strain hardening rate curve for equiaxed grain shown in Fig. 12(b). Furthermore, deformation behaviour is found to be overlapping for domain sizes greater than 100 × 100 × 100 μm3 irrespective of grain shape which suggests that there exist an upper limit of grain size and exceeding this limit might vanish the grain size dependence of DSA. The effect of morphology on NSRS was also investigated – where

7. Conclusions Conclusions for this work can be divided into experimental and numerical parts. From the experimental and numerical perspective, following are the novel findings. 1. Experiments reveal the existence of an underlying mechanism initiating the plastic instabilities during strain rate jumps in AA2024 which is otherwise absent for constant strain rate tests exhibiting strain rate insensitive behaviour. During the jump tests

Fig. 11. Two dimensional contour plots demonstrating the evolving distribution of alpha, (a) just before the jump at 0.58% strain; (b) just after the jump at 0.61% strain; (c) after jump at 0.93% strain.

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Fig. 12. Influence of grain size on strain rate jump test for equiaxed morphology, (a) global stress strain curves; (b) strain hardening rate curves.

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Fig. 13. Global stress strain curves for simulated strain rate jump tests illustrating the effect of aging time on strain rate jump profile.

2.

3.

4.

5.

temper T351 demonstrated more plastic instability events as compared to temper T4 of AA2024. Strain rate jump tests suggest the presence of NSRS and plastic instabilities in the alloy, however, they are initiated dynamically, for instance, by employing a sudden change in deformation rate. In-situ DIC results confirmed the creation and movement of well organised, as well as poorly organised, Lüders bands in the same test at different strain magnitude. Simulations results captured the experimental observation showing NSRS during strain rate jumps and provided validation for the DSA coupled PMFDM model, even for complicated loading conditions such as strain rate jumps. Most importantly, the parametric study conducted to achieve the experimental stress profile of the strain rate jump provides an access to experimental aging time. The combination of experimental procedure and modelling expressed in the present work offers a means of gaining detailed insight into the kinetics of plasticity in the presence of solute. Prediction of grain size effect revealed the importance of plasticity contribution of GNDs which can lead to inverse size effect for deformation dominated by mobile dislocations. However, at later stage of deformation forest contribution of GNDs kicks in and causes conventional size effect, i.e. smaller is stronger.

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