FEM simulation of micro-crystalline materials during ECAP based on the dislocation evolution method

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

FEM simulation of micro-crystalline materials during ECAP based on the dislocation evolution method ShaoRui Zhang∗, YingHong Peng, Dayong Li School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200030, China

a r t i c l e

i n f o

Article history: Available online xxx Keywords: Aluminum alloys Crystal plastic model ECAP Dislocation density

a b s t r a c t Based on severe plastic deformation, the equal channel angular pressing (ECAP) method has been used for producing metal materials with the ultrafine grain size and specific mechanical properties, particularly high yield strength. The grain sizes and the mechanical properties of ECAP processed materials strongly depend on the degree of plastic deformation, which is congregated by the evolution of dislocation slipping in the slipping planes. It is very important to analyze the dislocation density and strain hardening evolution in the slipping planes. In this paper, based on the crystal plastic model, the strain hardening & grain refinement of aluminum alloys were calculated with a dislocation evolution model during equal channel angular pressing. Next, the simulated strain, stress and grain size evolution were analyzed. Although the maximum value of the strain is very similar, the stress is rapidly increased when the materials pass the shear areas. Regarding the congregation of the dislocation density, the grain sizes decrease with the process continuing. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Grain refinement materials processed by severe plastic deformation (SPD) methods have become a focal point of material science because of their superior mechanical properties [1, 2, 6]. Among all of the SPD methods, ECAP technique, which was introduced by [11], has attracted much attention for the processing of various ultra-fine-grained materials, including metals, alloys and even composite materials [3, 8]. The mechanical properties and grain size evolution of ECAP processed materials strongly depend on the degree of plastic deformation. Determining how to exactly describe the grain refinement and dislocation evolution associated with the strain and stress development is very important for the design of an ECAP process. Analytical approaches to understand the shear strain process have been studied by researchers. Segal [12] firstly cal∗

Corresponding author. Tel.: +86 21 34206076; fax: +301 975 4052. E-mail address: [email protected], [email protected] (S. Zhang).

culated the shear strain distribution under a no-friction condition. A refined equation for shear strain was considered with some complicated die geometry [7]. Deformation simulations using the finite element method (FEM) had been also studied to analyze the effect of different processing conditions on material flow [4, 9]. The results indicated that the equivalent plastic strains are not uniform and that the locations of maximum equivalent plastic strain vary with increasing friction coefficients. As the crystal plastic theory, the macro plastic deformation is caused by the dislocation slipping along the slipping direction in the slipping plane. So the strain hardening evolution of the material depends on the dislocation slipping behavior in the slipping plane, and grain refinement is related to the evolution of dislocation density. It is necessary to understand the relation between the microstructure refinement and the macroscopic deformation behavior during ECAP. But the previous analyses didn’t consider the dislocation evolution model under a crystal plastic theory, and can’t really describe why the yield stress gets a sharp increase under a similar total strain with the process continuing

http://dx.doi.org/10.1016/j.chaos.2015.10.016 0960-0779/© 2015 Elsevier Ltd. All rights reserved.

Please cite this article as: S. Zhang et al., FEM simulation of micro-crystalline materials during ECAP based on the dislocation evolution method, Chaos, Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015.10.016

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In this paper, based on crystal plastic model, the strain hardening and grain refinement of aluminum alloys during equal channel angular pressing were calculated with a dislocation evolution model. The simulated stress, strain and dislocation density and the variation in dislocation distribution were analyzed. The results, which are simulated using FEM software based on the dislocation evolution model, were compared with experimental results, and the evolution of grain size agreed with the real evolution. 2. Dislocation model of strain hardening In the microscopic view, the strain hardening of metal materials is invoked by dislocation glide along slip systems. Thus, the flow stress (τ ) depends on the scale of the microstructure and the dislocation density [14]:

√

τ = α Gb ρ

(1)

where α is set to 0.4, G is the shear modulus of materials, and b is the modulus of the Burgers vector.ρ is total dislocation density, which is the sum of the statistically stored dislocation densities(ρ S ) and geometrically necessary dislocation densities (ρ G ), ρ = ρs + ρG .Then,

 τ = α Gb ρs + ρG

(2)

In addition, the macroscopic plastic deformation is reflection of microscopic dislocation slipping at the macro scale. So in the local crystal coordinates, the shear stress varies only due to the variation of the statistically stored dislocation densities ρ S (x), and the geometrically necessary dislocation densities are directly related to ρ S (x).

 τloc(x) = α Gb ρs (x)

(3)

Then the local shear stress gradient can be obtained:

dτloc dρx dτloc = dx dρx dx

(4)

As Eq. (2), the local geometrically necessary dislocation densities can be calculated as follows:

α d ρs ρG(x) = − √ 2 ρs dx

(5)

Then, for the sth slip system of metal materials, the relationship between the plastic strain gradient ∇γ and the edge or screw dislocation density takes the following form [10]:

ρCs = −∇γ s · Ls

(6)

ρSs = −∇γ s · ks

(7)

Ls

ns

Where is a unit vector in the slip direction, is the unit normal vector of the slip plane, andks = Ls × ns . Additionally,

ρtotal =



ρC2 + ρS2

(8)

The edge and screw dislocation densities cause two distinct stresses (τ C and τ S ) in the crystal interiors and the crystal boundaries, respectively:

 s 1/m  γ˙ s τ = α Gb ρC C γ˙ 0 s C

(9)

Table 1. Factors for the density model. f0

f∞

b(m)

γ˜

0.25

0.06

2.86e−10

3.2

 s 1/m  γ˙ τSs = α Gb ρSs S γ˙ 0

(10)

where γ˙ cs are the shear rates in the crystal and γ˙ Ss are the shear rates in the crystal boundaries, respectively. γ˙ 0 is the initial strain rate, and m is the strain rate sensitivity parameter. According to the continuity of the deformation in metal materials, the equation must be of the following form:

γ˙ = γ˙ c = γ˙ s

(11)

The total shear stressτ can be obtained using the weighted average applied to the two types of stresses:

τ = f τc + (1 − f )τs

(12)

where f is the volume fraction of shear stressτ c , which is caused by dislocation hardening in the crystal. According to previous reports [5, 13], the evolution of f is calculated by the following empirical function:



f = f∞ + ( f0 − f∞ ) exp

−γ f˙



(13)

where f0 is the initial value of volume fraction f, f∞ is the saturation value of f and the parameter f˙ describes the rate of variation of volume fraction with resolved shear strainγ . For aluminum, these factors are set as presented in Table 1 [13]: Then, the total shear stress in sth slip system can be rewritten as

 s 1/m   γ˙ s s τ = α Gb[ f ρC + (1 − f ) ρ S ] γ˙ 0 s

(14)

The shear rates and the dislocation density are governed by the following equation [15] :

γ˙ = ρ bv

(15)

The mean dislocation velocity v is calculated under the assumption of stress assisted and thermally activated overcoming of short range obstacles:

   Q  τe f f V v = λv0 exp − sinh KB T

KB T

(16)

where λ is the jump width, v0 the attack frequency, Q the effective activation energy for passing of obstacles, τ eff the effective shear stress (for crystal interior and crystal boundaries it is τ c and τ s, respectively), and V is the activation volume. The jump width corresponds to the mean spacing of obstacles, which consist of dislocations in cell walls and interiors. The approach of linear reciprocal superposition of the different obstacle spacing is adopted. The equation for the jump width is:

1

λ= √ ρtotal

(17)

Please cite this article as: S. Zhang et al., FEM simulation of micro-crystalline materials during ECAP based on the dislocation evolution method, Chaos, Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015.10.016

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3. Dynamic finite element model The kinetic equation for the plastic deformation of metal materials is

∂σ i j ¨ i − λu˙ i = 0 + Fi − ρ u ∂xj

(18)

Where: σ ij —Cauchy stress Fi —external force

ρ —the mass density λ—damping coefficient

Fig. 1. Schematic diagram of ECAP.

¨ i —deformation velocity and acceleration u˙ i , u

According to the divergence of the thermo and boundary conditions, the virtual work is

 V

ρ u¨ i δ u˙ i dV +

Fig. 2. The final configuration aluminum.

 V

γ u˙ i δ u˙ i dV =

 V

pi δ u˙ i dV +

 V

σ i j δεi j dV (19)

Fig. 3. Principle total strain max distributions.

Please cite this article as: S. Zhang et al., FEM simulation of micro-crystalline materials during ECAP based on the dislocation evolution method, Chaos, Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015.10.016

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Fig. 4. The distribution of equivalent stress.

By discretizing, a new equation can be obtained:

¨ + cu˙ = Fout − Fint mu

(20)

where m —uniform mass matrix c —damping matrix Fout —the node outer force vector Fint —the node inter force vector

γ s = [θ γ˙ s (t + t ) + (1 − θ )γ˙ s (t )] t (0 ≤ θ ≤ 1) (23)

Then, the velocity can be calculated using a time interval algorithm with central difference

u˙ n+1/2 =

2m − c tn−1 u˙ n−1/2 2m − α c tn−1 +

(1 + α) tn−1 (Fout n − Fintn ) 2m − α c tn−1

By using the Taylor series expansion, Eq. (23) can be rewritten as



Ns j γ s = (γ˙ ts + Qs : ε) t

( j)

Qs = (21)

where α = tn / tn−1 The strain rate can be calculated by the node velocity

εˆ˙ = Bˆ u˙

To improve the computational stability, the tangential coefficient method is introduced into the finite element model. In a time step t, the shear strain increment γ (s) can be determined by linear interpolation for the shear strain rate at time t and time t+ t:



 θ t γ˙ ts s R mτ s

(25)

Rs = Ce : Ps + Ws · σ − σ · Ws

 (22)

(24)

Ns j = δs j +

θ t γ˙ ts mτ s



Rs : P j

τs

(26)

+ sgn(τ j )Hs j

(27)

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Fig. 5. Typical grains and dislocation densities observed after severe deformation.

where Hst is the hardening modulus, and

1 Ps = (Ls ns + ns Ls ) 2

(28)

1 W = (Ls ns − ns Ls ) 2 s

(29)

The scheme for the finite element method is performed as follows. (1) At the beginning of every step, suppose there is no plastic deformation, no active slip system. Then, setγ˙ s = 0, and read the dislocation valuesρ s S,n , ρ sC,n andρ n in the nth step. (2) Calculate the trial stress:

σˆ = D εˆ L

e

(30)

σ t+ t = σ t + σˆ · t (3) Calculate the shear strain

(5) Compare the value of the trial stress and critical shear stress (a) for elastic deformation

σ t+ t <

s γ s = [θ γ˙ n+1 + (1 − θ )γ˙ ns ] t (0 ≤ θ ≤ 1)

σ t+ t ≥



fn Pcs τcs + (1 − fn )Pss τss

s s γn+1 = γns + γ˙ n+1

(34)

(35)

(8) Calculate two distinct stresses (τ C and τ S ) by solving the Eqs. (24–27). (9) Then, the grain size can be calculated by total dislocation in the cell walls and interiors, and :

K

ρtotal

(32)

(33)

(7) For plastic deformation, amend the shear strain.

d= √

(4) Obtain the critical shear stress τn+1 = fn τC,n+1 + (1 − fn )τS,n+1 using Eqs. (9) and (10)

fn Pcs τcs + (1 − fn )Pss τss

(6) (b) for plastic deformation

(31) increment γ s



(36)

where K is the materials constant. Therefore, with the dislocation congregating, the grain refinement can be formulated during metal forming.

Please cite this article as: S. Zhang et al., FEM simulation of micro-crystalline materials during ECAP based on the dislocation evolution method, Chaos, Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015.10.016

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Fig. 6. The distribution of the grain size obtained by the finite element of ECAP.

4. Results and conclusion The aluminum samples were used to conduct ECAP experiments. The samples were cut with dimensions of 10 mm ×10 mm × 80 mm. Fig. 1 shows the die dimensions for ECAP experiments. The angle between the entrance and the exit channels was 90°, and the die corner angle was 90°. Fig. 2 shows the final configuration of the aluminum samples. Based on the above theory, the ECAP method for aluminum was simulated by an FEM, and 2500 threedimensional elements were adopted. The following material properties for aluminum are assumed: elastic modulus E = 69 GPa, α = 0.3, m = 0.1. In this ECAP simulation, the end values of the equivalent stresses, ρ c and ρ s , were used as the initial values in the subsequent process. The distributions of the principle total strain max after each process obtained by FEM, which is based on dislocation density model, are shown in Fig. 3. The deformation mainly occurs in the intersecting region of the two channels. This deformation induces a rapidly increasing value of dislocation density along the shear area, and then the dislocation density stops concentrating. As a result, the strain value remains stable during the following process. The maximal values of the principle total strain max are limited by the physical dimensions of the die and the corner angle, which are very similar for each process step. However, the distribution of the strain becomes more homogeneous as the number of process steps increases.

As the deformation of ECAP is focused on the shear region, the equivalent stress values reach a maximum along the shear area where the two channels meet, followed by a decrease (Fig. 4). This behavior results in a rapid increase in the dislocation density and the critical shear stress, which induce higher strain hardening. For the next pass, the initial values of yield stress, ρ c and ρ s , both increase. Then, the equivalent values of stress increase during the following process, although the stain values remain similar (Fig. 3). The ECAP method for grain-refinement of the aluminum samples was successfully extruded up to six passes. The microstructures of the samples were examined by transmission electron microscopy (TEM). Fig. 5 provides evidence of a lower dislocation density inside the larger grains during the second pass. With the number of passes increasing, the dislocation density of the grain boundary increases. The deformation caused by ECAP is characterized by separation of the grains by heavy dislocation walls. For each dislocation wall, the wall inclination angle is with respect to the extrusion axis. Following the deformation process, the dislocation densities of the cell blocks rapidly increase in the shear area, then the grain refine in size to become close to the cell size. Fig. 6 shows the calculated grain sizes during ECAP processes. Using the value of the grain size determined by the finite element analysis, it is possible to compare the grain size of the deforming region among various processing conditions. The calculated grain sizes decrease with subsequent ECAP passes. Comparing Fig. 6(a)–(d), the grain size was

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Fig. 7. The grain size of pure aluminum by ECAP.

found to rapidly decrease with the accumulation of dislocations in the intersecting region of the two channels. Fig. 7 shows optical and the SEM micrographs of the initial Al billet and the billet processed by ECAP route BC for

6 passes. Fig. 7(a)–(e) reveal that ECAP results in different grain sizes in the work pieces. The average grain sizes in these conditions are measured as ∼ 50, 13, 5, and 0.35 μm. Fig. 7(b) shows that the grains are apparently elongated in the

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longitudinal lateral section. During the different passes, the samples are placed in the cross-sectional direction of the channel. Each of the grains exhibits an elongated deformation in a direction different from the crossed shear strain direction. After six passes, the microstructure becomes homogeneous, with average grain size of < 200 nm (Fig. 7(f)). It is apparent that ECAP produces ultrafine grains that are nearly equiaxed. However, in some areas, some slightly elongated grains exist. Comparing the FEM results with the experimental results, the finite element analyses with dislocation model were found to be able to precisely represent the local deforming behavior because the calculated grain sizes represent the averaged values of the grain sizes within the deforming region, and the calculated grain sizes are agreement with the tested grain sizes. 5. Conclusion Equal channel angle processes for aluminum were simulated by FEM based on the dislocation evolution method. The grain size and distribution of the principle total strain maximum and the equivalent stress were investigated. The simulation and test results revealed the following: 1. When the metal materials pass the intersecting areas, the total strain was observed to rapidly increase. And then the strain maintained a stable value until the end of the process. 2. The equivalent stress achieved a maximal value along the shear area. This behavior resulted in a rapid increase in the dislocation densities of metal materials. 3. For the concentration of dislocation densities, maximal stress increased with the process continually repeats, where the maximal values of total strain almost was similar for each process step. 4. The calculated grain sizes decreased with the ECAP process, and the evolution of grain size agreed with the real evolution. The FEM with dislocation evolution can be used to simulate the evolution of strain, stress and grain size during ECAP process.

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Please cite this article as: S. Zhang et al., FEM simulation of micro-crystalline materials during ECAP based on the dislocation evolution method, Chaos, Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015.10.016