Strain rate effect of high purity aluminum single crystals: Experiments and simulations

Accepted Manuscript Strain rate effect of high purity aluminum single crystals: experiments and simulations Akhtar S. Khan, Jian Liu, Jeong Whan Yoon, Raju Nambori PII: DOI: Reference:

S0749-6419(14)00196-X http://dx.doi.org/10.1016/j.ijplas.2014.10.002 INTPLA 1848

To appear in:

International Journal of Plasticity

Received Date: Revised Date:

9 July 2014 15 September 2014

Please cite this article as: Khan, A.S., Liu, J., Yoon, J.W., Nambori, R., Strain rate effect of high purity aluminum single crystals: experiments and simulations, International Journal of Plasticity (2014), doi: http://dx.doi.org/ 10.1016/j.ijplas.2014.10.002

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Strain rate effect of high purity aluminum single crystals: experiments and simulations Akhtar S. Khana, Jian Liua, Jeong Whan Yoonb, Raju Namboria a Department of Mechanical Engineering, University of Maryland, Baltimore County, Baltimore, MD 21250, USA b Faculty of Science, Engineering and Built Environment, Deakin University, Geelong Waurn Ponds, VIC 3216, Australia

Abstract In order to study the strain rate effect on single crystal of aluminum (99.999% purity), aluminum single crystals are fabricated and subjected to uniaxial compression loading at quasi-

static and dynamic strain rates, i.e., from 10   to 1000   . The orientation dependence is also investigated with single slip or multi slip. The stress-strain curves of pure Al single crystals along two orientations and at different strain rates are obtained after measuring initial orientation using the Laue Back-Reflection technique. Crystal Plasticity Finite Element Method (CPFEM) with three different single crystal plasticity constitutive models is used to simulate the deformations along two orientations under various strain-rates. The classical and two newly developed single crystal plasticity models are used in the investigation. The simulation results of these models are compared to experimental results in order to study their abilities to predict finite plastic deformation of single crystalline metal over a wide strain rate range.

Keywords: A. dislocations, B. constitutive behavior, B. crystal plasticity, C. finite elements

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1 Introduction Most metals used in engineering are polycrystalline, which are aggregates of single crystals (grains). In many models for polycrystal metals, mechanical behavior of a polycrystalline has been viewed as the collective behavior of grains and their interactions with each other. Therefore, a physical-based modeling of polycrystals involves understanding, quantifying, and predicting the constitutive behavior of the constituent single crystal grains. It is important to investigate the deformation responses of high purity single crystals both experimentally and numerically to gain further insight into deformation mechanisms of crystalline materials, since high purity single crystals eliminate the effect of grain boundary, grain interaction (single crystal), second phase particles and even solute atoms (high purity). This study only focus on FCC structures, so it should be noted that the following review only concerns FCC as well. The study of the physics of single crystal plasticity can be traced to the early 20th century (Taylor, 1934). It has been established that crystalline materials deform plastically by the crystallographic slips of dislocations on discrete slip systems. Therefore, the plastic deformation of a single-crystal material point is the sum of the crystalline slips in all the activated slip systems. Plastic slip of a slip system only occurs when the resolved shear stress (driving force of slip) onto a crystallographic plane reaches a critical value (critical resolved shear stress, or CRSS) in the direction of slip. Depending on the relative orientation of the crystal lattice to the loading axis, it is possible that only slip system is active (single slip mode). The slip mode in which more than one system is active is called multislip (slip mode in this study denotes the activation of single or multiple slip systems of the same slip system family of FCC, it should not be confused with the activation of different slip system families as in BCC and HCP). Single

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crystals deformed by different modes (single or multislip) usually show different mechanical behavior and microstructure evolution. The CRSS of a slip system is a function of plastic strain and usually increase due to the material hardening caused by plastic deformation. Three conventional stages of hardening have been found for FCC single crystal deformation in the early studies: stages I or easy glide, stage II characterized by a much higher, linear hardening and stages III with a decreasing hardening. Late stage IV, V and VI have been identified at large strains (Argon and Haasen, 1993, Les et al., 1996, 1997, Mecif et al., 1997). The existence and extent of each stage depend on many factors such as the initial crystal orientation, temperature, strain rate and the type and purity of metal, etc. In low to intermediate strain, while the welldefined three stages are typical of pure single crystal copper, most investigations of single crystal aluminum revealed a small or no stage I (Hosford et al., 1960; Howe et al., 1961; Bell and Green, 1967). The number of investigations in which high strain rates are attained is far smaller for single crystal than for polycrystal specimens (Lindholm and Yeakley, 1964, Ferguson et al., 1967, Edington et al., 1968). It is important to note that physically, the slipping and hardening of each slip system are the result of dislocation movement, density evolution and interaction. Crystal plasticity (CP) theory and modeling are based on the aforementioned deformation mechanism of crystallographic slip, i.e., the plastic deformation of single-crystal material point is the sum of the shear of slip systems. Taylor (1938) established a quantitative description of plastic deformation in single crystals based on crystallographic shearing. Rice (1971) and Rice and Hill (1972) developed a general finite deformation elastic-plastic framework for analyzing the deformations of rate independent

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single crystals. Later Peirce, Asaro and Needleman (1982,1983) presented a rigorous constitutive theory for rate dependent material response. Following the single crystal visco-plasticity modeling framework laid out by Rice (1971), and Hill and Rice (1972), the kinematic deals with the geometrical aspect of deformation without considering the stresses. It assumes a multiplicative decomposition of the deformation rate into elastic and plastic parts. The velocity gradient tensor can be related to the scalar shear rates of all the slip systems. The kinetics (or flow rule) determines the shear strain rate of a system at a certain applied resolved shear stress and a microstructure state. In the classical phenomenological approach, one “strength” for each slip system is used as the internal state variables (ISV) to represent the state of material. Conventionally, a power law was used as the flow rule (Hutchinson, 1976). More realistic flow rules based on the physics of thermal activation were also used by Nemat-Nasser et al. (1998), Balasubramanian and Anand (2002), Hansen et al. (2013), and other researchers. The hardening law refers to the evolution equations of slip system strengths as the material hardens with plastic deformation. The framework and kinematics are the same for all crystal plasticity models that are based on the deformation mechanism of crystallographic slip. The classical phenomenological model uses strengths of slip systems as internal state variables and a power law as the flow rule. Considering that the slipping and hardening of each slip system are the results of dislocation movement, evolution and interaction, models based on the dislocation density have been proposed by many authors. The simplest type is to use a scalar (total) dislocation density for each slip system as ISV (Tabourot et al., 1997, Kocks and Mecking, 2003, Beyerlein and Tomé, 2008, Lee et al., 2010). More physical and advanced models includes those that separate the mobile and forest dislocations (Kubin and Estrin, 1990; Barlat et al., 2002; Austin and

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McDowell, 2011; Hansen et al., 2013), those that separate the edge and screw dislocations (Arsenlis and Parks, 2002, Alankar et al., 2009), those that consider the development of the dislocation densities in cell walls and grain interiors (Estrin et al., 1998; Roters et al., 2000; Tóth et al., 2002) and those consider the geometrically necessary dislocations (GND) in addition to the scalar statistically stored dislocations (SSD) (Ma and Roters, 2004, Ma et al., 2006a, 2006b, Gurtin, 2010). It is worth mentioning that as the complexity of models increases, the number of parameters that need to be identified also increases and it is usually more difficult to perform the numerical simulations. Combining crystal plasticity with finite element method, crystal plasticity finite element method (CPFEM) is popular. CPFEM is a powerful simulation tool as it combines the FEM’s ability of solving complicated boundary problem and CP’s ability of taking the microstructure and deformation mechanisms into account. The FEM serves as the boundary problem solver, while at each integration point the material constitutive model is based on crystal plasticity theory. The first CPFEM simulation was performed by Peirce, Asaro and Needleman (1982). With the increase of computation power, CPFEM simulations have been extended from simplified 2D setups with two or three slip systems for single crystal (Peirce et al., 1982) and polycrystal (Harren et al., 1988; Harren and Asaro, 1989) to 3-D setup with the all the 12 slips system for FCC (Becker et al., 1991) and complex grain arrangements. A recent comprehensive review of CPFEM is done by Roters (2010). There are small scale and large scale CPFEM regarding the scale of volume a material point represents.

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In small scale CPFEM simulations, the size of the finite element mesh is equal to or smaller than the grain size. In such cases, one integration point represents a single crystal (a part of a grain), so the single crystal plasticity model serves directly as the constitutive model. With such sub-grain resolutions, heterogeneity of deformation with a single crystal is taken into account (Zhang et al., 2009, Lee and Chen, 2010, Saito et al., 2012, Mayeur et al., 2013, Rossiter et al., 2013, Sabnis et al., 2013, Ardeljan et al., 2014, Knezevic et al., 2014) as well as the influence of grain boundaries and free surfaces (with special treatments of mesh at/crossing GB and free surface). Large scale CPFEM refers to simulations with more than one crystal assigned to one integration point and the number of grain is usually large enough to study the average behavior of the material such as the texture evolution. Usually, the deformation within each grain is assumed to be uniform. Homogenization schemes are needed to connect a material point (an integration point in the FEM) to each constituent grain (modeled by single crystal models). Commonly used homogenization schemes are iso-strain (Taylor, 1938), iso-stress (Sachs, 1928) and the self-consistent schemes (Kröner ,1958 , Molinari et al.,1987, Lebensohn and Tomé 1993, 1994, Knezevic et al., 2013a, 2013b). CPFEM, whether small scale or large scale, use single crystal plasticity models for the mechanical constitutive behavior of the material. So the single crystal plasticity constitutive model is a fundamental part of CPFEM simulation. The validation of a single crystal plasticity model can be accomplished by comparing the numerical simulation with experimental data. In the earlier days, stress-strain responses of polycrystals at isothermal (usually room temperature) and quasi-static (at one strain rate) deformation condition were used to back-fit the parameters and validate the applicability of a single crystal plasticity model.

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The purpose of this study is to investigate the deformation responses of high purity aluminum single crystals both experimentally and numerically. Section 2.1 introduces the experimental procedures and section 2.2 presents the experimental deformation responses of aluminum single crystals along two orientations, at different strain rates. Strong orientation and strain rate effects were observed. Section 3 describes three rate-dependent single crystal plasticity models of different complexity. Model 1 is the classical phenomenological model with strengths of slip systems as ISV and a simple power-form flow rule. Model 2 is a phenomenological model with a new hardening law based on KHL model. Model 3 is a dislocation density-based model with a more accurate and physical flow rule based on thermal activation. Section 4 introduces the simulation procedures of the compressive deformation of single crystal aluminum using small scale crystal plasticity finite element method. Section 4 also shows the results of the simulations of CPFEM with three different single crystal plasticity models and the comparison of the results with experimental data. Section 5 is the conclusion of this study. It is concluded that the power-form flow rule of model 1 with one constant m was not sufficient to capture the strain rate effect over a wide range of strain rates. Model 2 can better capture the strain rate effect due to the introduction of an addition strain rate related term into the hardening moduli. Model 3 can also predict the strain rate effect well due to the more accurate and physical flow rule based on thermal activation. Model 3 can predict the multi-slip better than model 1 and 2 because all the possible interactions between slip systems are taken into account by using interaction matrices.

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2 Experimental procedures and results 2.1 Experimental procedures A high purity aluminum single crystal (99.999%) was used in this study. Cylindrical compression specimens were cut from the single crystal using wire electrical discharge machining (EDM). The samples were cut in such a way that the loading direction of the compression testing has different orientation with respect to the crystal lattice therefore different slip mode, i.e., single slip or multi-slip, can be expected.

The initial crystallographic

orientations of the samples and the orientations after deformation were measured by the BackLaue diffraction technique. Compression tests were conducted at room temperature, at a range of strain rates from 10-4 to 1000 s-1. Uniaxial quasi-static compression tests at different quasi-static strain rates were performed using a MTS 809 Axial/ Torsional Material Testing System. Specimens for quasi-static tests were 0.5 inch in diameter and 0.725 inch in length. A uniaxial strain gage, of type KFEL-2-120C1 (2mm gage length) by Kyowa Electronic Instruments was used during each experiment. Teflon sheets and Dow Corning high vacuum grease were applied to the top and bottom platens to reduce friction and to maintain uniform uniaxial stress state. The split-Hopkinson pressure bar technique (SHPB) was used to perform dynamic uniaxial compression experiments on the single crystal samples. The high strain rate specimens were 0.55 inch in diameter and 0.168 inch in length. More details of the SHPB technique were given in Khan and Liang (1999), Khan and Zhang (2000), Khan et al. (2004) and Khan et al. (2007).

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2.2 Experimental results of single crystal Al The results presented here are from single crystals of two different loading orientations: S1 and S2 as shown in Figure 1. Orientation S1 is well within the standard stereographic triangle, therefore a single slip mode is expected. Although to be more accurate, in the rate-dependent (visco-plastic) theory, there will always be small amount of shear strain rate on every slip system since usually there will always be some amount of resolved shear stress on every slip system. However, with a loading direction well within the standard triangle such as S1, the shear strain rate on slip systems other the primary one is negligible. S2 is close to the <110> direction with a small mis-orientation, so multi-slip is possible according to the rate-dependent approach. In the rate-independent approach, the loading direction needs to coincide exactly with the <110> (or other boundary lines or points of the standard stereographic triangle) direction to activate the multi-slip. In rate-dependent approach, depending on the model parameters (such as the strain rate sensitivity index m in the power-law flow rule), it is possible that slip systems other than the primary one have significant amounts of shear strain rate when the S2 is close to <110> direction. Figure 2 shows the experimental stress-strain curves of S1 under compressive deformation at different strain rates from 10-4 to 1000 s-1. Figure 3 is the experimental stress-strain curves of S2 under compressive deformation at strain rates of 10-4 and 1000 s-1. Due to the high purity, the yield stresses are small for both orientations at all strain rates; hence the strain rate effect on yield stresses is not clearly visible. Strain rate effect is more obvious at higher strain levels, especially after 2% of strain. For both orientations, the strain hardening rate is higher at higher

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strain rates. Hence, the flow stress is higher at higher strain rates after a certain amount of plastic strain. Positive strain rate effect on the flow stress is observed for both S1 and S2. Figure 4 shows the comparison of the responses of S1 and S2 in one figure. Orientation dependence is clearly visible. At similar strain rates, the hardening rate and the flow stress of S2 are much higher than those of S1. S2 at a strain rate of 10-4 s-1 is comparable to S1 at strain rate 1000 s-1. The high hardening rate of S2 is believed to be due to multi-slip.

3 Crystal plasticity modeling 3.1.1 Kinematic The kinematical theory for the classical crystal plasticity model follows the pioneering work of Taylor (1938) and its precise mathematical theory by Hill (1966), Rice (1971), and Hill and Rice (1972). The deformation of a material body can be expressed mathematically by the deformation

gradient tensor F. It is assumed that  can be multiplicatively decomposed into a plastic deformation,  and an elastic part,  :

 =  

(1)

where  denotes the plastic shear of material from reference configuration to intermediate configuration in which the lattice orientation and spacing are unchanged.  denotes the

combination of elastic deformation and rigid body motion of the crystal lattice from intermediate to current configuration.

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Since the plastic deformation is assumed to be due to dislocation slip on slip systems, the velocity gradient tensor in the current configuration is related to the slipping rates of the slip systems by

= ∑ γ   

(2)

where  and  are the slip direction vector and the slip plane normal vector for the slip system α (Asaro, 1983).

The kinematic is the mathematical foundation of crystal plasticity that is based on the deformation mechanism of crystallographic slip, and it is the same for all three models used in this study. The differences between the three models are the flow rule and hardening law (and the addition of dislocation evolution in the dislocation-based model).

3.2 The classical phenomenological model (model 1) The power law flow rule in its most basic form applies to materials deformed via quasistatic strain rates under isothermal conditions and has the form (Hutchinson, 1976):  

γ  = γ   sign  



(3)

where γ 0 is the reference strain rate and m is the rate sensitive parameter. τ is the resolved

shear stress on slip system α and it is the driving force for slip. g  is the strength of slip system α and it is the internal state variable that represents the material state.

In the limits as m approaches zero this power law reaches that of the rate-independent material. As a matter of fact, in the early days, m was chosen to be a very small value so that the non-uniqueness of rate-independent approach was circumvented. Later, m was chosen to be equal to the macroscopic experimental strain rate sensitivity parameter to enable the model to

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predict the strain rate effect (Bronkhorst et al., 1992). However, because of the much-simplified form of power law, the strain rate effect was only applicable within a small range of strain rate. After initial yield, as the plastic deformation processes the strength of each slip system increases. This working hardening is characterized by the evolution of the strengths through the incremental relation:

g  = ∑# h# γ #

(4)

where h# are the tangent slip hardening moduli. In this context, the diagonal components h

characterize self-hardening while the off-diagonal terms, h# (α ≠ β) , characterize latent hardening.

Self-hardening moduli h is usually proposed as a phenomenological function of the

cumulative shear strain. Piece, Asaro and Needleman (1982) used a simple form for the selfhardening moduli: ,  h = h(γ) = h sech*  

+ -

,

(5)

where h and g  are material constants, and γ is the cumulative shear strain on all slip systems, i.e. γ = ∑ . |γ  | dt 0

(6)

The latent hardening moduli are given by

h# = qh (α ≠ β)

(7)

where q is a constant denoting the latent/self-hardening ratio. The value of q was usually chosen as 1 < 5 < 1.4 based on experimental results.

Various hardening models with different forms have been developed. An example of a more complex form was proposed by Bassani and Wu (1991). In the UMAT developed by Yoon (Yoon et al, 2005), the hardening law is the following: 12

8 (γ) = K(γ + γ ); h =

(8.1)

<=(-) <-

(8.2)

h# = qh

(8.3)

The hardening parameters to be determined are K, γ , n and the latent/self hardening ratio q.

3.3 A phenomenological model with a hardening law based on KHL model (model 2) Classic crystal plasticity modeling uses a single phenomenological strength for each slip system as internal state variables and a simple power law as flow rules. It does not account for the effect of grain size and temperature. For the strain rate effect, although there is a simple power form of flow rule with a strain rate sensitive index m, it will be demonstrated in section 4 that a constant m is not sufficient to capture the strain rate effect over a wide range of strain rates. To improve the classical phenomenological model, the approach adopted here is to modify the hardening law to incorporate the effect of strain rate, temperature and grain size into the self-hardening modulus. It is motivated by the successful continuum plasticity model: the KHL model (Khan and Liang, 1999, Khan and Zhang, 2000, Khan et al., 2004, Khan et al., 2007). The proposed hardening laws are as follows: f(γ) = ?A + BC∗ + B(1 − GHIJ )H γH, K (L A

h =

GH-

,

<=(-) <-

L L

 LMNO

)P

(9.1) (9.2)

h# = qh

(9.3)

In this model, the power law of equation (3) is still used as the flow rule. But the parameter m is no longer the only parameter that represents the rate sensitivity of the slip systems. Rather, additional rate dependence (and temperature dependence) is incorporated explicitly into the 13

hardening law of the slip system strengths. Similar approach has been adapted by Knezevic et al. (2013c, 2013d).

3.4 A dislocation-density based model (model 3) A single crystal plasticity model (model 3) based on dislocation densities was developed, considering that the slipping and hardening of each slip system are the results of dislocation movement, evolution and interaction. The framework and kinematic of the classical model still holds with now a new flow rule and a hardening law that are based on dislocation densities. In addition, an evolution model of dislocation densities with deformation was added. The internal state variables are one scalar dislocation density for each slip system. The strengths of slip systems are assumed to be functions of the dislocation densities. A thermal strength g 0+ and an athermal strength g Q0+ are assigned to each slip system. Thermal strengths represent the short-range resistances that are due to short-range obstacles (less than ≈10 atomic diameters) such as lattice friction (Peierls-Nabarro stress), solute atoms and forest dislocations. Short-range resistances can be overcome at a lower applied shear stress with the help of thermal activation. At the same time, the athermal strengths are the long-range resistances that cannot be overcome with the help of thermal activation, which are due to long-range obstacles such as large incoherent precipitates and long-range interaction with other dislocations. The flow rule is an Arrhenius type equation of thermal and athermal strengths based the physics of thermal activation movement of dislocations. Based on experimental evidence, the macroscopic shear strength of metals can be related to the density of dislocations through the “robust” Taylor equation (Taylor, 1934). The hardening law of model 3 is the generalized Taylor equation (from macroscopic shear stress and the total dislocation density to shear strengths and dislocation densities of slip systems). Finally, the evolution of macroscopic dislocation density was 14

successfully modeled by the phenomenological “storage VS recovery” Kocks-Mecking framework. So the evolution of dislocation densities of each slip systems is modeled through generalization of the classical Kocks-Mecking relation. In summary, the flow rule, hardening law and dislocation evolution of model 3 are: 0 |τ | ≤ g Q0+

] ` γ  = R   ∆X γ  exp V− YL ?1 − (  Z[\ ) K ^ |τ | > g Q0+ [\

g 0+ = μbc∑ A# ρ#

g Q0+ = BC,∗ + μbc∑ B# ρ# A

ρ  = e ( 

c∑ Pf gf Y

− 2yj ∑ D# ρ# )γ l

(10.1)

(10.2)

(10.3)

(10.4)

The novelties of this model are: 1. It uses the flow rule based on thermal activation, which is more accurate and physicalbased than the conventional power form. The model also differentiates the thermal strength and athermal strength instead of using one “total” strength. 2. The athermal strength, thermal strength, dislocation accumulation rate and dislocation annihilation rate of each slip system are all related to the dislocation densities of all slip systems through different interaction matrices. All the possible interactions between slip systems are taken into account. 3. The grain size effect is incorporated into the athermal strength through a “micro” HallPetch term. Although the elements of this model were proposed and used by researchers before in different contexts, this is the first attempt to combine them together within the category of

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crystal plasticity models using scalar-dislocation-densities of slip systems as internal state variables.

4 Simulation and discussion The simulations of the compressive deformation of Al single crystals were accomplished using the commercial finite element program ABAQUS/Standard with the single crystal plasticity models serving as the material constitutive model. The three single crystal plasticity models described in the previous section need to be implemented into the user material subroutine, UMAT, for the ABAQUS. The classical phenomenological model 1 was already implemented into UMAT by many researchers. In this study, the ABAQUS UMAT subroutine developed by Yoon (Yoon et al., 2005) was used for the model 1 simulation. For the model 2, the hardening law part of the UMAT by Yoon was modified and the modified UMAT was used for simulation. For model 3, modifications were made to the UMAT developed by Huang (1991) and the modified UMAT was used for simulation. In the FEM simulation of uniaxial compression tests, the effect of friction was not considered so the interfaces between the end surfaces of the sample and the platens of the testing machine were assumed to be frictionless (well lubricated). As a result, the deformation within the sample was expected to be quite homogeneous especially at relative small strains (<25%) as in this study, so the number of elements has a small effect on the simulated stressstrain responses. In order to have computational efficiency, one element simulation was adopted. The use of one element was justified for the purpose of this study, i.e., validating the single crystal plasticity constitutive models. If the purpose was to study the compression deformation

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of single crystals in a more realistic sense, the deformation within the single crystal sample would be considered inhomogeneous. In that case, multiple meshes will be needed. A single 8-node reduced integration element (C3D8R) in ABAQUS was used to represent the uniaxial compression test sample. The compression direction was aligned with one of the element axes. Out of the two faces of the cube element that were perpendicular to the loading axis, one was assigned with a boundary condition that the displacement along the loading direction was zero. The other face was assigned with a certain amount of displacement along the loading direction with a certain rate to simulate the compression deformation at a certain strain rate. Also, stress-free boundary conditions were assigned to the faces that are parallel to the loading axis. In the following the identification of the parameters for three models will be discussed. In the classical phenomenological model 1 (equation 8), the parameters to be identified are the reference shear strain rate γ  and the strain rate sensitivity parameter m for the flow rule and

three parameters K, γ and n for the self hardening and the latent/self hardening ratio q. Unlike the flow rule based on thermal activation theory, which was adopted in model 3, the reference strain rate in the simple power-form flow rule doesn’t have a clear physical meaning. In Bronkhorst et al. (1992), the reference strain rate was chosen to be 0.001 /s, and many simulations after that adopted this value. As pointed out by Beyerlein and Tome (2007), when the reference shear strain rate is set equal to the experimental shear strain rate induced by the macroscopic rate of deformation, the effect of the strain rate sensitivity parameter m will be removed. In this study we chose the uniaxial stress-strain curve of orientation S1 with a macroscopic strain rate of ε = 0.001   to fit the hardening parameters, and the reference

shear strain rate was selected as γ = Mε ≈ 0.002   (M is the Schmidt factor and since S1 is 17

in single slip model, the simple relation γ = Mε exists between the macroscopic strain rate and the slip system shear strain rate). As a result, strain rate sensitivity parameter m was not in play when simulating the stress-strain curve of S1 at ε = 0.001   , then the three parameters of self hardening laws can be determined (again, since S1 is in single slip model, the effect of q also doesn’t come into play). When fitting the stress strain curve of S1 at a strain rate different than the reference one, the parameter m comes into play. Then the value of m can be determined. The latent/self hardening ratio q shows an effect when simulating a multi-slip deformation, so the experimental responses of S2 were used to find the best q value. In summary, by setting γ = 0.002   , when simulating S1 at ε = 0.001   , m and q have no effect. The self-

hardening parameters K, γ and n can be determined. Then the stress-strain curve of S1 at a different strain rate can used to determine the strain rate sensitivity m. At last, the responses of S2 (multi-slip) were used to find the best latent/self hardening ratio q value. In the phenomenological model 2 with a modified hardening law based on the KHL model, more model parameters were introduced. Since single crystal was studied, the grain size A

dependent term BC∗ is negligible. Also, all compression tests were done at the room temperature testing condition (the temperature rise at dynamic tests is also small due to low flow stresses), the temperature term has small effect in this study. As a result, the constants related to the temperature and grain size effect were obtained directly from (Farrokh and Khan, 2009) and will not be listed here. The parameters need to be identified for model 2 are γ , m, A, B, D , n and n . The

flow rule was not changed compared to model 1. However, an additional strain rate related term is introduced into the self-hardening modulus (equation 9.1) along with a grain size and a temperature dependent term. So three parameters (m, D and n ) are related to the strain rate

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sensitivity. As in the macroscopic KHL model, D is the arbitrary chosen upper bound strain

rate which was chosen to be 106 s-1. When simulating S1 at ε = 1   , the (1 − GHIJ )H is close GH-

,

to 1 so the effect of n is minimized. Meanwhile, by setting γ = Mε ≈ 2   , the effect of m is also not in play. As a result, the stress-strain curve of S1 at ε ≈ 1   can be used to determined

the self-hardening parameters A, B and n (q is not in play due to single slip mode). The value

of m and n were then adjusted until a good correlation between simulation and experiment was obtained for S1 at all other strain rates. At last, the responses of S2 (multi-slip) were used to find the best latent/self hardening ratio q value. In the physical dislocation density-based model 3, physical complexity and number of parameters are increased compared to the classical phenomenological model. However, the parameters have physical meaning. At least their range is known and extensive literature exists for parameter identification. In this study, most of the parameters for model 3 were collected from literature. Only a few parameters were adjusted to get better experimental correlation. Parameters of the flow rule were adopted from Balasubramanian and Anand (2002), which were obtained by fitting against the experimental data of 99% percent pure polycrystal aluminum by Carreker and Hibbard (1957).

Parameters for the thermal and athermal strength are two interaction matrices A# and B#

that relate thermal and athermal strength of each slip system to the dislocation density of each

slip system. There are only six independent components (p , p , p* , pr, p, ps for A# , and similar t , t , t* , tr , t , ts for B# ) in each interaction matrix, each of which represents a type

of dislocation interaction (Franciosi and zaoui, 1982). The interaction coefficients enter the interaction matrices according to the type of interaction they represent between the particular pair of slip systems. p , p are coefficients that describe the interactions of two parallel slip 19

planes (in-plane interactions) with the same and different Burgers vectors, respectively.

p* , pr, p, ps account for out-of-the-plane interactions between slip systems pairs that are colinear, leading to Hirth lock, Glissile junctions and Lomer-Cottrell sessile lock, respectively. The parameters we used are those adopted by Arsenlis and Parks (2001). The thermal and athermal strengths were usually not considered separately in the early models, so there were very few reports on the athermal strength interaction matrix. Based on the temperature-decrement tension tests on aluminum single crystals oriented for single slip by Cottrell and Stokes (1955), Balasubramanian and Anand (2002) concluded that the ratio of athermal and thermal strength is equal to 1 for pure Al. Thus, the parameters for the athermal strengths were set equal to those of the thermal strengths. Also, since for single crystal and large grain polycrystals, the contribution of grain size is negligible, the “micro” Hall-Petch term was ignored in this study. There were scarce reports on the interaction parameters for the dislocation evolution. So the dislocation evolution equation was simplified to a commonly used form (equation 11.3). In summary, for model 3, a simplified version of equation 10 was used:

0 |τ | ≤ g Q0+

] ` γ  = R   ∆X γ  exp V− YL ?1 − (  Z[\ ) K ^ |τ | > g Q0+ [\

g 0+ = g Q0+ = μbc∑ A# ρ# ρ = e ( 



c∑ gf Y

− 2yj ρ )γ l

(11.1)

(11.2)

(11.3)

Now in equation 11, only the parameters p , ρ  , K and yj were adjusted to get a better experimental and simulation correlation.

20

Figure 5 (a) and (b) are the simulations along with the experimental data using the classical phenomenological model (model 1) for the S1 orientation at different strain rates. In both 5 (a) and 5 (b), the reference shear strain rate was set to be 0.002 s-1, the hardening parameters were obtained by fitting the stress-strain curve of 0.001 s-1. The fitted parameters are listed in Table 1. In Figure 5(a), the stress-strain curve of S1 at the strain rate 0.01 s-1 was used to determine the value of m, which was 0.08 in order to get a good correlation. Using m=0.08, the predictions of the curves of S1 at 1 s-1 and 1000 s-1 are clearly deviated from the experimental results. In Figure 5(b), the stress-strain curve of S1 at the strain rate 1000 s-1 was used to determine the value of m, which turned out to be 0.04. Using m=0.04, the predictions of the curves of S1 at 0.01 s-1 and 1 s-1 are not satisfactory. It’s demonstrated that a constant m is not enough to capture the strain rate effect over a wide range of strain rates. Figure 6 is the simulation for the S2 orientation at different strain rates along with the experimental data using model 1. The reference shear strain rate and hardening parameters are those that were obtained when fitting S1 (Table 1). Both cases with m=0.04 and 0.08 were investigated. The latent/self hardening q ratio begins to show an effect since S2 is multi-slip, so the value of q was adjusted in the simulations of orientation S2. The measured value of q from latent hardening experiments was in the range of 1
21

Figure 7 shows the simulation for the S1 orientation at different strain rates along with the experimental data using phenomenological model with a modified hardening law based on KHL model (model 2). The reference shear strain rate was set to be 2 s-1 and the hardening parameters were obtained by fitting the stress-strain curve at 1 s-1. One more adjustable parameter (n ) that

is related to the strain rate sensitivity was added in addition to m. n and m were adjusted simultaneously to get a good correlation for all the strain rates of S1. The obtained parameters are listed in Table 2. From Figure 7, it is clear that with an optimization of the two adjustable parameters n and m, the strain rate effect of S1 was accurately captured.

Figure 8 is the simulation for the S2 orientation along with the experimental data using model 2. The parameters are those that were obtained when fitting S1 (Table 2) except latent/self hardening ratio q. Similarly, the numerically investigated range was 1
22

5 Conclusions High purity aluminum single crystals of two orientations were experimental investigated using compression tests at quasi-static and dynamic strain rates. Strong orientation and strain rate effects were observed. Three single crystal plasticity constitutive models were used to simulate the deformation behavior of single crystal aluminum by one element CPFEM simulation. By the simulations of single crystals subjected to single slip, it was demonstrated that the classical simple power-form flow rule (model 1), with one constant m, was not sufficient to capture the strain rate effect over a wide range of strain rates. In the phenomenological model with a new hardening law based on KHL model (model 2), an additional strain rate related term was introduced into the selfhardening modulus. In the dislocation density-based model (model 3), the power-form flow rule was replaced by a more accurate and physical flow rule based on thermal activation. Both models 2 and 3 were shown to better capture the strain rate effect. Between those two, model 2 has less material constants. Model 3 has more parameters but those parameters have more clear physical meanings and extensive literature exists for parameter identification. All the possible interactions between slip systems were taken into account by using interaction matrices in model 3, while only a latent/self hardening ratio q was used in model 1 and model 2. The simulations of single crystals subjected to multi-slip showed that the predication of model 3 correlated better with experimental results compared to model 1 and model 2. Only one set of parameters was used for model 3 over the whole investigated strain rate range even though the existence and extent of each hardening stage are supposed to change with strain rates. In the future, detailed analysis of hardening stages and possible changing model parameters with hardening stages could be investigated.

23

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29

`

Flow rule

γ  (s-1) 0.002

Latent

Self hardening

m

q

K(MPa)

n

0.04-0.08

1.0-2.0

60

0.61

u

10E-6

Table 1. The parameters for the classical phenomenological model (model 1)

Flow rule

γ  (s-1) 2

Latent

Self hardening

m

q

A(MPa)

B(MPa)

0.11

1.0-2.0

0.5

33.2

v

0.62

v

0.85

wx (s-1) 10E6

Table 2. The parameters for the phenomenological model with hardening law based on KHL model (model 2).

Flow rule

γ  (s-1)

1.732 × 10}

∆F (J)

3 × 10~

Hardening law

Dislocation evolution

p

q

p

p

p*

pr,

p

ps

0.141

1.1

0.08

0.22

0.3

0.38

0.16

0.45

ρ  (m-2)

10 × 10

K

yj

38

3.56

Table 3. The parameters for the physically dislocation-based model (model 3).

30

Fig.1. The orientations of S1 and S2 in a standard stereographic triangle.

80

S1 1000 /s S1 1 /s 60

S1 0.01 /s

True Stress (MPa)

S1 0.001 /s

40

20

0 0

0.05

0.1 True Strain

0.15

0.2

Fig.2. Experimental stress-strain curves of single crystals with orientation S1 subjected to compression at different strain rates.

31

100 S2 1000 /s

True Stress (MPa)

80

S2 0.0001 /s

60

40

20

0 0

0.05

0.1 True Strain

0.15

0.2

Fig.3. Experimental stress-strain curves of single crystals with orientation S2 subjected to compression at different strain rates. 100 S2 1000 /s S2 0.0001 /s 80

S1 1000 /s

True Stress (MPa)

S1 1 /s S1 0.01 /s

60

S1 0.001 /s 40

20

0 0

0.05

0.1 True Strain

0.15

0.2

Fig.4. Experimental stress-strain curves of single crystals with both orientations of S1 and S2 for comparison.

32

120

S1 1000/s S1 1 /s S1 0.01 /s S1 0.001/s S1 1000 /s Simu m=0.08 S1 1 /s simu m=0.08 S1 0.01 /s simu m=0.08 S1 0.001 /s Simu m=0.08

100

True Stress (MPa)

80

60

40

20

0 0

0.05

0.1 True Strain

0.15

0.2

(a) 80

S1 1000/s S1 1 /s S1 0.01 /s S1 0.001/s S1 1000 /s Simu m=0.04 S1 1 /s simu m=0.04 S1 0.01 /s simu m=0.04 S1 0.001 /s Simu m=0.04

True Stress (MPa)

60

40

20

0 0

0.05

0.1 True Strain

0.15

0.2

(b) Fig.5. Simulations of stress-strain curves of single crystals with orientation S1 at different strain rates using model 1 with (a) m=0.08, (b) m=0.04. 33

200

S2 1000 /s S2 0.0001 /s S2 1000 /s simu q=2 m=0.08 S2 1000 /s simu q=1.4 m=0.08 S2 1000 /s simu q=1 m=0.08 S2 1000 /s simu q=2 m=0.04 S2 1000 /s simu q=1.4 m=0.04 S2 1000 /s simu q=1 m=0.04 S2 0.0001 /s simu q=2 m=0.08 S2 0.0001 /s simu q=1.4 m=0.08 S2 0.0001 /s simu q=1 m=0.08 S2 0.0001 /s simu q=2 m=0.04 S2 0.0001 /s simu q=1.4 m=0.04 S2 0.0001 /s simu q=1 m=0.04

True Stress (MPa)

160

120

80

40

0 0

0.05

0.1 True Strain

0.15

0.2

Fig.6. Simulations of stress-strain curves of single crystals with orientation S2 at different strain rates using model 1.

34

80 S1 1000/s S1 1 /s S1 0.01 /s S1 0.001/s S1 1000 /s Simu S1 1 /s simu S1 0.01 /s simu S1 0.001 /s Simu

True Stress (MPa)

60

40

20

0 0

0.05

0.1 True Strain

0.15

0.2

Fig.7. Simulations of stress-strain curves of single crystals with orientation S1 at different strain rates using model 2. 120 S2 1000 /s S2 0.0001 /s S2 1000 /s simu q=2 S2 1000 /s simu q=1.4 S2 1000 /s simu q=1 S2 0.0001 /s simu q=2 S2 0.0001 /s simu q=1.4 S2 0.0001 /s simu q=1

100

True Stress (MPa)

80

60

40

20

0 0

0.05

0.1 True Strain

0.15

0.2

Fig.8. Simulations of stress-strain curves of single crystals with orientation S2 at different strain rates using model 2. 35

80 S1 1000/s S1 1 /s S1 0.01 /s S1 0.001/s S1 1000 /s simu S1 1 /s simu S1 0.01 /s simu S1 0.001 /s simu

True Stress (MPa)

60

40

20

0 0

0.05

0.1 True Strain

0.15

0.2

Fig.9. Simulations of stress-strain curves of single crystals with orientation S1 at different strain rates using model 3. 120

S2 1000 /s S2 0.0001 /s

100

S2 1000 /s simu S2 0.0001 /s simu

True Stress (MPa)

80

60

40

20

0 0

0.05

0.1 True Strain

0.15

0.2

Fig.10. Simulations of stress-strain curves of single crystals with orientation S2 at different strain rates using model 3. 36

Highlights •

Deformation responses of pure aluminum single crystals were studied experimentally.

•

Strong orientation and strain rate effects were observed.

•

Deformation of single crystal aluminum was simulated by one element CPFEM.

•

Three single crystal plasticity models served as the constitutive models for CPFEM.

•

The classical and two newly developed single crystal plasticity models were used.

37